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 Title: Lecture: Woman and Society (Die Frauenfrage)
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 Title: Jesus and Christ
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 mathematical precision when Christ must have lived in the man Jesus,
 mechanical forces through mathematics, so is it possible to understand
 Title: Lecture: The Moral as the Source of WorldCreative Power
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 machine in terms of mathematics and the principles of mechanics, or
 reality a picture built up purely on the basis of mathematical
 principles, mathematicalmechanical principles. The rudiments of it
 that of today, for it was based, not upon mathematical principles but
 another in accordance with mathematical laws. To be sure, in every
 Title: Lecture: The Path to Freedom and Love and their Significance in World Events
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 mathematics. Mathematical thinking streams up entirely from our
 inner being, but it has a mirrorexistence only. Through mathematics
 determined; but the mathematical thoughts in themselves are only
 They say: mathematical concepts are a priori. — A priori,
 But why are mathematical concepts a priori? Because they stream
 had a dim inkling that we bring mathematics with us from our existence
 before birth, and therefore they called the findings of mathematics
 Title: Lecture: Search for the New Isis, the Divine Sophia: The Quest for the IsisSophia
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 built one of these pyramids with such mathematical precision that the
 luciferic world picture, at the mathematicalmechanical world picture of
 the grave of Isis, a mathematical abstraction. Then comes the search for
 Title: Lecture: The Two Christmas Annunciations
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 on, but it is merely mathematical and mechanical. It only speaks of
 mathematically, whereas the ancient wisdom of, the stars perceived in
 mathematics as we know them could not then exist. The whole of life
 our geometry and mathematics. Euclid was the first to give geometry to
 also in the case of such purely mathematical considerations, the
 sensed when he felt mathematics to be like great poetry —
 poured into mathematical mould, that the pupil of the ancient
 thus brought to a mathematical understanding of the universe, he
 The mathematics of the universe, which have become so thoroughly
 introduction of mathematics into their soulcondition, when this was
 found especially mature and ready? Kant speaks of mathematics as being
 mathematics is something that rises up within us, that rises to
 that when he does mathematics, experiences of the time before
 true insight into these matter the mathematical capacity is in itself
 mathematics is.
 mathematical forms and reckoning to them, we are spreading out again
 symbolize in mathematics. With my innermost forces I moved from star
 Magi? It has become our mathematics, with its knowledge of the
 cramped into our mathematical, mechanical grasp of the heavens, so
 that we apply nothing but the laws of mathematics and mechanics to
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 Title: Lecture: The Threshold In Nature and In Man
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 overborne by purely mathematical pictures of the world, purely
 Title: Lecture: The Alphabet
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 music today — the mathematical physicist reckons the pitch of a note,
 from the mathematician. Man is led away from immediate experience into
 In itself it is really interesting — if one has a mathematical bent
 Title: Mathematics and Occultism
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 Mathematics and Occultism
 of Mathematics, from participating in the teachings of the Master.
 to mathematics within the domain of human knowledge. Plato intended to
 Now Plato looked upon mathematical science as a means of training for
 mathematical images hover over the borderline between the material
 case of all mathematical pictures. They relate to the
 mathematically, I do indeed think about something my senses can
 property of mathematical perception is this: that a single
 senseperception what is spiritual. From the mathematical figure I can
 steps in mathematical knowledge for this purpose, and understand
 clearly what it is that we really gain from a mathematical figure.
 “Learn to emancipate thyself from the senses by mathematics,
 world can be based on mathematical ideas, but only that the first
 supersensible in mathematical thought. When a man reaches the stage of
 senseperception in the same way as he is able to think mathematically
 understand. He must learn through the elementary mathematical
 of senseperception as the mathematician thinks of the circle and its
 supersensible mathematical perception; no experience will ever
 Gnostics and all Occultists conceive mathematical science as an
 of mathematics to natural science. Kant and many others like him, for
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 Title: Lecture: Elemental Beings and Human Destinies
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 diffusion of light with mathematical ideas, they will say: You see,
 Title: Lecture: Richard Wagner and Mysticism
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 thought to be akin to mathematics but because it was known that the
 mathematical concepts. Men must have patience to find their bearings
 Title: Lecture: Gnostic Doctrines and Supersensible Influences in Europe
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 mathematics and its kindred sciences, all of which are the legacy of
 Title: Lecture VI: Ancient Myths
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 space events take place which are calculated purely mathematically
 merely calculate, as the modern mathematical astronomer does, the
 this, even I though it is not proved mathematically, which would here
 Title: Lecture VII: Ancient Myths
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 calculate by mathematics or mechanics. But what was formerly
 often quoted it — ‘Mathematics is in truth a great poem’.
 But in order to see how mathematics, by which one also calculates the
 poetry of mathematics. For mathematics is phantasy. Mathematics is
 with the immediate external reality. Mathematics in fact is Maya
 but learns to know mathematics in its substance, learns to know
 mathematical calculations in Astronomy. But the reality is certainly
 mathematically, is there; it is only passed completely through a
 Title: Lecture: The Souls Progress through Repeated Earth Lives
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 the following: All that we conceive as mathematics, all that comes
 mathematically, we develop really out of our inner being; it is the
 into some kind of unintelligible expression. Mathematics is said to
 then one knows whence this curious mathematical knowledge springs.
 The astral body has gone through the mathematics of the whole
 more in the purity of mathematicalgeometrical lines.
 Title: Lecture: Goethe and the Evolution of Consciousness
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 partiality for mathematics, for geometry, for mechanics.
 Title: Lecture: Awakening to Community  I
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 objective mathematics, objective chemistry, that it is working toward
 Title: Between Death and Rebirth: Lecture Five
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 heavenly bodies purely in terms of their mathematical, spatial and
 Title: Between Death and Rebirth: Lecture Eight
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 presented as something only brought about by mathematicalphysical
 Title: Christ and the Spiritual World: Lecture Six
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 mathematical ideas and proofs, for example, are created by the soul
 come to grasp what he set out in his mathematicalmechanical laws of
 Title: Mission of FolkSouls (1929): Lecture 3
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 In order to arrive at geometrical, mathematical knowledge it is not
 only attainable by mathematical construction, then you have some
 mathematical concepts only; it is man who is so imperfect that he is
 only able by means of abstractions such as the truths of mathematics,
 Title: Mission of Spiritual Science and of Its Building at Dornach
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 mathematics, because we are dealing with external space. The lifegerm,
 Title: Lecture: The Inexpressible Name, Spirits of Space and Time.
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 furthest of all along the path of mathematicalgeometrical knowledge.
 mathematicalgeometrical sphere, the less transparent things become.
 Title: Lecture: The Meaning of Easter: St. Paul and the Christ Impulse
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 most that which is given him in mathematics and mechanics, and now in
 grasp that which is beyond the earth with mathematical and mechanical
 Title: Agriculture Course: Lecture 6
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 Stars.” Modern astronomy serves as a mere mathematical orientation,
 Title: Paths to Knowledge of Higher Worlds
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 mathematics above all as an excellent example for the scientific
 mathematical culture.
 What particular result can the human soul gain through mathematics?
 mathematics can be inwardly surveyed, is inwardly transparent, and
 that mathematics contains, as it were, nothing to which we submit
 The spiritual science of Anthroposophy is naturally not mathematics.
 penetrates into mathematical thought. It is not mathematics in itself
 mathematical thinking. If such a “mathematizing” culture
 Title: Errors in Spiritual Investigation
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 existed a kind of belief, a certain mathematicalgeometrical belief
 mathematicalanalytical, constructive thinking for that square
 thing to exist, but it is impossible to show with mathematical
 mathematical thinking does not suffice to prove something that is
 mathematicians proved that it is impossible to solve the problem in
 the circle is considered not to know mathematics in this realm.
 one can prove the squaring of a circle mathematically.
 Title: The Supersensible Being of Man and the Evolution of Mankind
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 presentiment of the spiritual world if clear, mathematical thinking
 Title: Lecture: The Peoples of the Earth in the Light of Anthroposophy
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 in mathematical and mechanical formulae. The Westerner must realise
 cold, barren mathematics is the only science which remains to him for
 by mathematical calculation, by equally dry spectroanalysis or
 Title: Lecture: The Position of Anthroposophy among the Sciences
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 of mathematics among the sciences has already been mentioned. Kant's
 mathematics, is widely known. Now I have not to deal here with
 mathematics itself, with its value for the other sciences and in
 “mathematicising” — if I may use this word; that
 is, when actively engaged in mathematical thinking. His attitude of
 characterise it by speaking, first, of that branch of mathematics
 what he does when “mathematicising”, he would be able to
 transfer, to some extent, the mathematical attitude of soul to other
 mathematics, although it remains unnoticed. (I have given a
 mathematicians nor philosophers have paid attention to this peculiar
 mathematical or unmathematical. What can live in us in like
 (in which mathematics is our sure guide), and will open up a new
 of things as mathematical perception is with mathematical
 mathematics in the right frame of mind, we come to see precisely in
 the mathematician's attitude when “mathematicising”
 For mathematics is simply the first stage of supersensible
 perception. The mathematical structures we “perceive” in
 who knows the intrinsic nature of “mathematicising” knows
 for our eternal human nature, mathematical thinking has all the
 learn from a study of “mathematicising” what clairvoyance is
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 Title: Lecture: Fundamentals of the Science of Initiation
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 we have it now, merely containing mathematical laws of the
 Title: Lecture: Cosmogony, Freedom, Altruism
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 earth's environment in terms of mathematics, but they have
 perceive a mathematical order, there a spiritual
 really believe in this purely mathematical world, in the
 purely mathematical Cosmos. His particular civilisation leads
 Title: Lecture: The Shaping of the Human Form out of Cosmic and Earthly Forces
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 the truths of mathematics. These are certainly evident though
 Title: Lecture: Hygiene  a Social Problem
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 physics, mathematics, chemistry, physiology, biology, anthropology.
 Title: Lecture: On the Dimensions of Space
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 within the mathematical domain, the thoughts which we arrive at in
 Title: Lecture: Thinking and Willing as Two Poles of the Human SoulLife
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 that we cannot understand it by means of purely mathematically
 Title: Evolution/Aspect: Lecture 1: Introductory Lecture
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 distinguish an Eastern and a Western system of mathematics; yet some
 Title: Fall/Darkness: Lecture 13: The Fallen Spirits' Influence in the World
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 Goethe's Faust, mathematics, mechanics and technology at
 again in the late 70s. In between he spoke only on mathematics,
 teaching the exact sciences of mathematics, physics and
 that someone who had used physics, mechanics, mathematics and
 Title: Lecture: The Spirit in the Realm of Plants
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 thoroughly mathematical thinker, a deeply thoughtful man, who was
 Title: On the Mysteries of Ancient and Modern Times: Lecture 1
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 Mathematics. The latter were strictly preserved and guarded;
 kind of symbolic treatment of the mathematical, and indeed,
 mathematical and intellectual knowledge. These things they
 aspect of mathematical and mechanical knowledge, not letting
 Title: On the Mysteries of Ancient and Modern Times: Lecture 2
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 mathematics, it is no doubt impossible to stir your heart and
 an abstract mathematics. Yet this is what humanity needs to
 Title: On the Mysteries of Ancient and Modern Times: Lecture 3
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 no more, as yet, than a mere mathematical, mechanical and
 today; you will find mathematical and mechanical
 Title: Lecture: The World Development in the Light of Anthroposophy
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 powerful thought impression (to mathematicians I might say: this
 mathematics, so that it is out of the question to speak of
 Title: Lecture: The Supersensible in the Human Being and in the Universe
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 try to solve problems in mathematics. This fact has given rise to
 which natural science contains mathematics. For this reason,
 mathematically. We speak of the exact sciences.
 exercises — as the tackling of a mathematical problem
 apply to mathematics and natural science, so that thought is
 Title: Lecture: Foundations of Anthroposophy
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 with mathematical precision. Since the whole process can be
 Title: Lecture: Knowledge Pervaded with the Experience of Love
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 exact as mathematical or other scientific ideas. We do not think
 way in which mathematical or geometrical figures confront him in
 mathematics; but the human being must bring along something, for
 Title: Lecture: Hermes
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 mathematical concepts of Natural Science may be so deeply absorbed
 mathematical laws, in the plants with their lifefilled forms which
 mathematical sciences, geometry (which Pythagoras afterwards learnt
 was brought down by Hermes into mathematics and geometry and he
 for the method was really a mathematical convenience. After three
 mathematics, geometry, the science of landsurveying and
 Title: Lecture: Galileo, Giordano Bruno, and Goethe
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 mankind to conceive mathematically and mechanically such an
 Science to Mathematics, which has continued on his lines from
 to any single field; to the field of Mathematics. We see at
 about Mathematics, he will have or to toil at Mathematics for
 years. Then, as you know, man's conception of Mathematics
 cast in a mathematical mould and because his method of
 thinking was so rigidly mathematical. In illustration of this
 mathematical mind, the question of time, which was completely
 humbly and mathematically.
 Title: Lecture: On the Occasion of Goethe's Birthday
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 in our day, — I mean mathematics, although even here we
 carried on in certain branches of mathematics during the
 already been worked out with all possible mathematical
 Title: Mission/FolkSouls (1970): 3. The inner Life of the Folk Spirits. Formation of the Races.
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 geometrical and mathematical knowledge sensedata are superfluous;
 the logic of mathematics you will have some idea of how the
 the Archangels are not Beings who are limited to mathematical
 mathematics. These truths are the normal experiences both of man and
 Title: Lecture: Wonders of the World: Lecture 3
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 mathematicalmechanical laws. Such a nature — a nature which is
 calculated according to purely external mathematicalmechanical
 could not explain according to the strict laws of mathematics and
 as the facts which follow the physicalmathematicalchemical laws, but
 Title: Lecture: About Horses That Can Count and Calculate
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 I discovered that mathematical
 thinking, the whole mathematical conception, is something far more objective
 than one generally thinks. The whole mathematical thinking is something
 The foundation of mathematical thinking can be explained by the fact
 that all mathematical concepts are contained in the structure of the
 mathematical ability is principally dependent on three channels situated
 cord. If a human being arrives at mathematical conclusions, we are able
 we generally imagine. For mathematical conclusions are formed, as it
 were, by themselves, and particularly in the sphere of mathematics the
 of mathematics that we really feel impelled to develop it as it were
 connected with the Earth through a kind of mathematical automatism.
 in check the whole mathematical automatism. Now it is strange to see
 need to calculate the various mathematical problems which he gives to
 the horse, but merely touches upon something mathematical. In doing
 so, he becomes incorporated within the mathematical automaton of the
 become aware so strongly of the way in which the mathematical automaton
 the earth, in the case of the mathematical automaton, there will be a
 very large numbers together. A famous professor of mathematics once
 he was being connected in a very strange way with the mathematical automaton.
 the horse the programme; this will be enough, for the whole of mathematics
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 Title: Tension Between East and West: Lecture 1: Natural Science
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 mathematics to natural processes. Why is this? It is
 mathematical knowledge. And Kant's saying is often quoted and
 mathematics. This is exaggerated if we are thinking of ordinary
 mathematics. And yet, when we apply these to lifeless natural
 with mathematics what otherwise stands outside us. Why? Because
 mathematics is experienced inside us with immediate certainty:
 certainty, the truth. Things mathematical are seen and
 of mathematics, we can penetrate into the facts and
 that of mathematics with which to penetrate into the higher
 we have mathematics, geometry; for the organic, the
 them: not with the ordinary mathematics of numbers and figures,
 penetrating with mathematics of this kind into worlds that we
 quantitative mathematics, in the broadest sense, that can
 clairvoyance is, in reality, to the mathematical outlook.
 Title: Tension Between East and West: Lecture 2: Psychology
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 as are operations in mathematics or geometry.
 Although attained by as exact a method as are mathematical
 Title: Tension Between East and West: Lecture 3: East and West in History
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 mathematical problems. It is just when we turn our attention to
 call a qualitatively mathematical way, what cannot be
 comprehended with ordinary mathematics and geometry: living
 with a living structure in the way that mathematical thought
 outlook, yet one as exact as ever mathematics or mechanics can
 creation, and unless from a qualitative mathematics we
 Title: Tension Between East and West: Lecture 4: Spiritual Geography
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 think of mathematical truths to realize that all kinds of
 Title: Tension Between East and West: Lecture 6: Individual and Society
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 mathematically formulated laws; to comprehend something so
 Title: The Social Question and Theosophy
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 to think at all, and just as no one can become a mathematician
 Title: Goethe's Secret Revelation: Lecture II
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 Because regarding such truths such as mathematical truths, most
 everyone can in mathematics. But there are methods of purifying
 the thoughts in us think, as we let mathematical thought do. Thus,
 civilization, in history, in mathematics and all other sciences can
 Title: Purpose of the Goetheanum and Anthroposophy
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 mathematician works out his problems; so that it is all carried
 mathematics that one can have less than nothing.
 gifts of a mathematician. It is very difficult for most people
 the lifeless through mathematics, we come to an understanding
 when we proceed in a purely mathematical way, and develop a
 mathematician. The forming of mathematical concepts is
 mathematics in order to apply it to the lifeless things
 content when we know it mathematically —
 mathematics is elementary Anthroposophy — and we
 speak of exact mathematics, in contrast with the mystical,
 Title: Meditation and Concentration
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 our head. Instead of mathematics, logic metaphysics, we
 Title: Karma of Vocation: Lecture III
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 a mathematical concept, it would not have been possible without
 Title: Karma of Vocation: Lecture VIII
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 instruments, was also familiar with geometry and mathematics,
 humanities with a monk, learned mathematics from his father
 mathematics and mechanics, he continued to excel in these
 country, teaching mathematics and also practicing medicine, of
 he lectured on mathematics for sixty scudi a year and also
 lectured with great distinction on mathematics and related
 Title: Mysteries of the Sun: Lecture III
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 help of the simplest mathematical concept and when I say
 Title: Reincarnation and Immortality: Lecture III: The Supersensible Being of Man
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 a mathematician that he should draw a triangle whose angles do
 Title: Reincarnation and Immortality: Lecture IV: Nature of Anthroposophy
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 mathematical kind, one is exclusively concerned with a simple
 mathematician making his calculations, one gradually becomes
 Title: Necessity for Spiritual Knowledge: Lecture 2
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 Earth came to be built up, a mathematicalgeometrical
 Space, in terms of mathematics, of geometry and of mechanics.
 essential feature is a mathematicalmechanical picture of the
 naturally arise as to whether this mathematicalmechanical
 mathematical phantasy. Phantasy it is, none the leer
 for being mathematical. The essential feature in the attitude
 and to embrace what is beyond the Earth by mathematical and
 are merely mathematical and nothing else. The concepts which
 conceived of in terms of mathematics.
 superterrestrial concept of space, this mathematical and
 experienced. For this mathematicalmechanical concept of
 do today. The mathematicalmechanical conception yields no
 world that is merely mathematical and mechanical. Just as the
 know, not by means of external mathematical considerations,
 mathematicalmechanical space conceived of by Copernicus,
 purely mathematical and mechanical Copernican world
 I have often said that this purely mathematicalmechanical
 of this, a kind of convenient mathematicalmechanical world
 not mathematically and dynamically as it is built up through
 mathematical space is only a schematic image. There is a
 Title: Eternal Soul of Man in the Light of Anthroposophy
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 much prudence as the mathematician takes with his arithmetic
 for the mathematical sciences; so it is known exactly and
 as the mathematician customarily carries out his work. Only the
 works of the mathematician are in objective forms, while here
 fullest deliberation, like the course of mathematical
 mathematician says — into the negative, into debts that
 Title: Supersensible Knowledge: Lecture XII: Richard Wagner and Mysticism
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 called it “mathesis,” mathematics, not because
 mysticism is mathematics, but because genuine mystics have
 Title: Supersensible Knowledge: Lecture XIII: The Bible and Wisdom
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 mathematician without consulting any authority. Someone needs
 Title: Necessity and Freedom: Lecture II: The Legend of the Prague Clock
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 mathematically on the basis of past ones. Ideally, scientists
 Title: Necessity and Freedom: Lecture V: The "I" is Found on the Physical Plane in Acts of Will
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 more a matter of mathematics and geometry. Just as today we can
 Bernhard Bolzano, 1781–1848, mathematician and logician
 Title: The Building at Dornach (Bn/GA 289): Lecture I: The Goetheanum
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 has here been made to depart from the usual purely mathematicalgeomatricalmechanical
 and intellectual, from everything merely mechanical and mathematical,
 the merely mathematicalgeometricaldynamic column formation has been
 Title: Problem of Faust: Lecture I: The Problem of Faust
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 sphere of knowledge, unless it be Mathematics, that there
 Title: Anthroposophy as a Demand of the Times
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 scientific result, but as the mathematical or geometric results
 sense just as a mathematician would for a mathematical problem.
 Title: Haeckel, "The Riddle of the Universe," Theosophy
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 know is true, and all higher mathematics are only imaginary
 Title: Cosmic Forces in Man: Lecture I: Cosmic Forces in Man
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 terms of mathematics and mechanics. We look at the stars through
 attached to them because we fall back upon mathematical formulae when
 worldmachinery which can be expressed in terms of mathematics.
 the Earth can be explained in terms of mathematics and mechanics. In
 Title: Esoteric Cosmology: Lecture XVIII: The Apocalypse
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 might develop by dint of precise, minute and mathematical observation
 Title: Third Lecture: The Gospel of St. John
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 teaching them higher mathematics, I would be teaching them
 Title: At the Gates: Lecture VII: Workings of the Law of Karma in Human Life
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 Bernoulli was a great mathematician, and eight other gifted mathematicians
 Title: At the Gates: Lecture XI: The PostAtlantean CultureEpochs
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 associated with logic and mathematical calculation began to develop,
 Title: At the Gates: Lecture XIII: Oriental and Christian Training
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 mathematical concepts, for example. No perfect triangle exists in the
 Title: Festivals/Easter: Lecture I: Easter: The Festival of Warning
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 most that which is given him in mathematics and mechanics, and now in
 mathematical and mechanical knowledge alone. In a world in which he
 Title: Lecture: Foundations of Esotericism: Lecture XXI
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 thoughts, that is in mathematics. When people calculate, their
 they have no love for mathematics. Here one cannot vote in
 parliamentary fashion. Mathematical truth is recognised by man through
 emotion, as objectively, as in mathematics. In Europe one can only
 matters was expressed so objectively, with such pure mathematical
 mathematician. Through this he wished to create emotionally free
 thoughts and introduce the spirit of mathematics into higher regions.
 Devachanic plane except when he is thinking mathematically. Certain
 Title: Lecture: The Human Spirit and the Animal Spirit
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 criminologist Moritz Benedict, celebrated for his mathematical
 Title: Lecture I: Human Questions and Cosmic Answers
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 which at present holds good only for mathematics; we shall be able to
 What we do today only when we are thinking in terms of mathematics, we
 the human soul. The study of mathematics does not, as a rule, give
 mathematics. For Novalis, the science of mathematics was a great and
 devout mood of soul when studying mathematics!
 Title: Lecture III: Human Questions and Cosmic Answers
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 formed part of mathematical science in the course of its evolution on
 mathematics all the numbers were contained in the unit. This
 Title: Lecture IV: Human Questions and Cosmic Answers
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 example a particularly good mathematician, develops between death and
 Title: First Lecture (First Scientific LectureCourse)
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 the phenomena of Nature to mathematical thinking as Goethe had.
 Goethe himself not having been much of a mathematician, this is
 relation of natural phenomena to those mathematical formulations
 Increasingly in modern time, the mathematical way of studying the
 phenomena of Nature — I do not say directly, the mathematical
 the study of natural phenomena in terms of mathematical formulae
 confuse what can still be seen in purely mathematical ways, and what
 he knew what is the only possible relation of Mathematics to Natural
 Science, though Mathematics be ever so idolized even for this domain
 Title: Second Lecture (First Scientific LectureCourse)
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 thought. We derive mathematical formulae concerning all that can be
 Title: Eighth Lecture (First Scientific LectureCourse)
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 enough to entertain a few mathematical notions about wavemovements
 Title: Tenth Lecture (First Scientific LectureCourse)
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 mathematics, in our geometry, in our ideas of movement. These
 real mathematical perception into that realm of phenomena which is
 mathematical certainty that air could not be liquefied. Yet air was
 Title: Macrocosm/Microcosm: Lecture 8: Mirrorimages of the Macrocosm in Man. Rosicrucian Symbols.
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 entrusts himself to a tutor in mathematics. If he did not assume that
 case of those given by a tutor in mathematics, only now our whole soul
 solution of a mathematical problem. It is the essence of the new
 Title: Macrocosm/Microcosm: Lecture 11: Man and Planetary Evolution
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 quoting experiments, adducing mathematical proofs and the like, so
 Title: Man/Being/Spirit/Soul: Lecture III: The Science of the Spirit and Modern Questions
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 we can comprehend as exactly as mathematical or geometrical
 mathematical ideas as fantastic, for what I have
 mathematics, astronomy, physiology, biology, in history,
 Title: Origins of Natural Science: Lecture I
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 mathematical conceptions accessible to men of that time. He was an
 exceptionally good mathematician. His soul, therefore, was filled on
 that come to a man who immerses himself in mathematics. Thus he was
 mathematician. He has the disciplined thinking that derives from the
 pursuit of mathematics. But he shies away, as it were, from applying
 his mathematical skills to the docta ignorantia. He approaches
 the docta ignorantia with all kinds of mathematical symbols
 mathematics. He says to himself: Mathematics is the last remnant left
 when I apprehend mathematics with my mind. — At the same time,
 mathematical skills in the field of the docta ignorantia except
 soul, Cusanus was almost as much of a mathematician as was Descartes
 later on, but he dared not try to grasp with mathematics what
 realm has withdrawn. Mathematics, the most reliable of the sciences,
 1440, Nicholas Cusanus applied mathematical thinking and mathematical
 with the admission that even with mathematics one can no longer
 removed from human perception that even mathematics can approach it
 1440. “Conceive of mathematics as so powerful and reliable that
 it can force the sense world into mathematical formulas that are
 mathematical treatment.
 Title: Origins of Natural Science: Lecture II
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 theology nor the precise perceptions of mathematics can lead any
 ideas and concepts. These may have the certainty of mathematics, but
 faintly, in symbols taken from mathematics, did Cusanus want to
 1543. one century later, Copernicus, with his mathematical mind, took
 fully grasp, not even symbolically, with mathematics. Today, we see
 how in fact the application of this mathematical mind to nature
 conceptual systems, namely, the mathematical, is regarded as the best
 from outside with mathematics although mathematics cannot penetrate
 into the inner essence. For this reason, Cusanus applied mathematics
 broadest sense possible. The third is mathematical. Considering the
 Title: Origins of Natural Science: Lecture III
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 depends on the way in which mathematical thinking was gradually
 perception in regard to mathematical thinking itself. We are much
 position today in regard to mathematics and to the relationship of
 mathematics to reality. We assume that this is the way it has to be
 felt differently concerning mathematics. We need only recall what
 system with mathematically oriented thinking of the kind to which we
 began to apply this mathematical thought to the whole physical and
 postulates, if not in mathematical formulae — for actual
 from earlier ones, is based on the mathematical pattern. By and by it
 appeared selfevident to the men of that time that in mathematics
 mathematical or geometrical system, they would thereby achieve
 through the special way in which man relates to mathematics and
 mathematics relates to reality. Mathematics had gradually become what
 The mathematics existing in the age of Descartes
 as apply today. Take a modern mathematician, for example, who teaches
 mathematician starts from the concepts of Euclidean geometry, the
 mathematics that has been thoughtout but not experienced. As an
 Try to sense the tremendous difference between this mathematical
 feeling bound to human experience, and the bare, bleak mathematical
 understand the origin of the later mathematical way of thinking that
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 Title: Origins of Natural Science: Lecture IV
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 the formrelated or geometrical elements of mathematics, the
 mathematics — is originally experienced by man in direct
 The kind of mathematics prevalent today, by means of which man wants
 mathematics as something that he experienced within himself together
 no surprise therefore to discover this mathematical order in the
 some other mathematical formula on natural phenomena — even if
 such abstract mathematical concepts can be identified with
 The most that can be said of such an imposition of mathematics on
 natural phenomena is that what has first been mathematically thought
 nature — one experienced as mathematics through one's own
 corporeal nature the mathematical images of what one had experienced
 were images of the divine spiritual. The instant that mathematics is
 Only when man severed himself from mathematics, only then was it
 possible also to sever mathematics from the earth and to found an
 mathematically so much as lyrically. There is something musical in
 order to do this, mathematics would have had to be also mysticism,
 ancient mystics, and not because he experienced the mathematical laws
 Copernicus tried to grasp astronomy with abstract mathematical ideas.
 On the other hand, Newton shows mathematics completely on its own.
 Here I do not mean single mathematical deductions, but mathematical
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 Title: Origins of Natural Science: Lecture V
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 The isolation of man's ideas (especially his mathematical
 this view because, as I have said, a man experienced mathematics by
 If man is to apply mathematics, for example, to the world, then it is
 mathematical. He impresses the whole mathematical system upon the
 follows a mathematical method but it is not mathematics in the modern
 sense. He expounds his philosophy in a mathematical style, joining
 idea to idea as a mathematician would. He still retains something of
 the former qualitative experience of quantitative mathematical
 man's inner life, we can say that his style is mathematical.
 a mathematical style to psychology, let alone ethics.
 Title: Origins of Natural Science: Lecture VI
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 These images are the mathematical geometrical, and arithmetical
 longer saw through this, mathematical forms and numbers became
 purely mathematicalmechanistic form of motion for what is supposed
 Therefore they did not find either mathematics or the primary
 primary qualities, everything mathematicalmechanical, into
 itself to the body if it is to comprehend mathematics. Nor was it
 indispensable. A healthy and open mind can feel that mathematics
 connections. Failing to grasp how the mathematical with its
 longer produce anything other than mathematical, mechanistic,
 Title: Origins of Natural Science: Lecture VII
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 absent in the mathematicalmechanistic sphere. This sphere has been
 divorced from man, as if man were absent from the mathematical
 divorced from man. This is most evident in the case of mathematics.
 mathematics, a way of thinking which from then on was combined with
 physics was divorced from man, along with mathematics. The progress
 did not make a radical turn toward the mathematical attitude that
 By permeating all external phenomena with abstract mathematics, this
 Title: Origins of Natural Science: Lecture VIII
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 mathematicalmechanical element in human movement, the experiencing
 the age of Galileo and Copernicus, together with the mathematics that
 emerged, as did mathematics and physics, and became external
 even iatromathematics.
 Title: Warmth Course: Lecture III
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 space you speak in mathematical terms of length, breadth, and
 anything if I remain in our threedimensional space. The mathematical
 cannot be expressed as a solid. In purely mathematical space, it is
 formula. There is such a possibility in purely mathematical formulae.
 Title: Warmth Course: Lecture V
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 the realm of the mathematical concepts. In so far as they are purely
 mathematical, there is no trace of the tonal or the visible. Now we
 sort of thing. I mean all that is obtained from pure mathematics. Such
 greater than the part, etc. The basis of our mathematical concepts
 willconcept, that in reality you carry into the pure mathematical
 concept. That is the essential distinction between mathematical
 inner determination of mathematical concepts. This distinction arises
 from the fact that mathematical concepts are so rigidly bound up with
 subsists in the sphere of the will is brought into mathematical
 priori, to speak of them as we do in mathematics. When you go over to
 faculties, whence comes the remoteness of the mathematical and
 the mathematical or kinematical concepts.
 Title: Warmth Course: Lecture VII
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 really physical terms and not merely in mathematical terms, which
 see that in this way we can suffuse mathematics with vitality? The
 Title: Warmth Course: Lecture IX
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 you will then, if you treat these things mathematically, have to
 Title: Warmth Course: Lecture X
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 Title: Warmth Course: Lecture XI
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 heat is related to the application of mathematics and how it affects
 Title: Warmth Course: Lecture XII
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 mathematical quantities merely as such if we at the same time expect
 calculation has to involve the mathematical relations expressed in
 turned, then none of the mathematical expressions so far given would
 handled mathematically and have, so to speak, more than one meaning.
 Some mathematicians even question whether there is any justification
 It is necessary for us to recognize how the purely mathematical leads
 interesting parallelism between thinking in mathematics and thinking
 one. For the purely mathematical difficulty arises of itself, and
 mathematical thinking, similar to the one du BoisReymond delivered on
 different. Within mathematics, unless the matter slips us because it
 is too complicated, in this realm of the purely mathematical it must
 cannot apply mathematics if we wish to enter reality unless we keep in
 mathematics of physics, and aside from the cases where negative
 quantities arise from purely mathematical relations, physicists do not
 and negative signs, but only because of purely mathematical
 Title: Warmth Course: Lecture XIV
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 line. The mathematician reads but the medical man does not listen. And
 when the medical man reads the thoughts of the mathematician are busy
 things further. You will be able to develop them. Our mathematical
 Title: Philosophy/Cosmology/Religion: Lecture I: The Three Steps of Anthroposophy
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 mathematical problem. The point is that at first this organization is
 ‘the spiritual eye’, we then say: as the mathematician has
 become only a superstructure of Natural Science gained by Mathematics,
 Title: Philosophy/Cosmology/Religion: Lecture II: Exercises of Thought, Feeling and Volition
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 consciously exact as the mathematical and natural scientific method of
 Title: Philosophy/Cosmology/Religion: Lecture III: Methods of Imaginative, Inspired and Intuitive Knowledge or Cognition
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 thinks out a mathematical problem can be fairly certain that he is
 Title: Lecture: Rosicrucian Esotericism: Lecture I: Rosicrucian Esotericism
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 era when men's thinking would be guided by mathematical principles.
 Title: The Story of the Green Serpent and the Beautiful Lily: Lecture I
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 (dry and mathematical as he was in his other writings) — when
 Title: Lecture 3: Spiritual Beings in the Heavenly Bodies and in the Kingdoms of Nature
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 mathematics, numbers and calculation. The fact that three times three
 inner being. In dry, prosaic mathematics begins what we may call
 begins with dry mathematics, because most people take dry mathematics
 Indeed education through mere mathematics is very good. For instance,
 individual, but mathematics and moral obligations do not depend on
 than what the mathematician does in his dry science. Nowhere is there
 Title: Lecture: Younger Generation: Lecture V
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 nineteenth century, it was wellnigh only mathematics. That was the
 Title: Lecture: Younger Generation: Lecture VIII
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 is not limitless at all. For the mechanisticmathematical way of
 Title: Lecture: Mission of Michael: Lecture IV: The Culture of the Mysteries and the Michael Impulse.
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 such as mathematics and geometry, which are employed in the endeavor
 mathematical truths. If we have once grasped them we know they are
 These mathematical truths are different from the truths we arrive at
 The reason is to be found in the fact that mathematical truths
 then perceives them inwardly. We possess mathematical truths through
 the fact that we ourselves behave mathematically in the world. We
 receive the inner perception of mathematics. Mathematics arises below
 we are aware of the mathematical and geometrical concepts. The middle
 mathematics — this is the purely Ahrimanic element, which arises
 we know that mathematics is correct; but this sovereign correctness of
 mathematics we owe to Ahriman who sits in us. The most uncertain
 significance of this matter. In reality, the whole mathematical
 through logic and the Ahrimanic wisdom through mathematics, geometry,
 Title: Lecture: Lecture I: Occult Signs and Symbols
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 built up mathematically out of simple elements. Much that at first
 I am sure will appear to mathematical astronomers as sheer madness.
 Title: Lecture: Lecture III: Occult Signs and Symbols
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 Title: The East in the Light of the West: Lecture II
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 mathematics or geography today by means of old documents, written in
 Title: The East in the Light of the West: Lecture IV
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 present time can be studied at its source; even mathematics may also
 Title: Lecture: The Proclamations to the Magi and the Shepherds
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 mathematical science, speaking only of conditions and relationships in
 mathematics. What plays with a higher significance into man's inner
 possible. Nor could mathematics or geometry be grasped in the way they
 kind of knowledge resembling our geometrical or mathematical sciences.
 in mathematics which, in utter contrast to the vast majority today,
 mathematical forms that the pupil of the ancient Mysteries was led.
 And when this mathematical understanding of the universe had developed
 described as the three Magi from the East. The mathematics of the
 through the mathematics imparted to them?
 The philosopher Kant says of the truths of mathematical science that
 mathematics comes from within ourselves, rises into consciousness from
 engrossed in mathematical thought. For one who can truly apprehend
 these things, mathematical cognition is in itself a proof that before
 origin of mathematics.
 see in them the expressions of mathematical, arithmetical progression,
 mathematics. In that other existence my own inner forces led me from
 has become our mathematical astronomy. The Magi possessed
 mathematicalmechanistic conception of the heavens, to the phenomena
 of which we apply only mathematical laws. What wells up from within us
 in our mathematical astronomy is the modern metamorphosis of the
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 Title: Life Between ... IX: Life After Death
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 Bernoulli family in whom there were eight mathematicians! Surely that
 Title: Life Between ... XV: Intercourse With the Dead
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 Today we find there a mathematical formula. Originally there was a
 Title: Lecture II: Human and Cosmic Thought
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 real science as there is mathematics. This means that from being a
 mathematical terms, we shall not be able to explain the simplest
 perception, but if we go outside a mathematical explanation, we shall
 materialism will become a mathematical thinker, and will recognize as
 valid only whatever can be treated mathematically. From this results
 mathematical formulae. This may be called Mathematism.
 world is ordered mathematically. If mathematical ideas are found to
 Title: Lecture IV: Human and Cosmic Thought
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 would bring to expression a mathematical worldstructure permeated by
 — and if he would inquire into the nature of mathematical
 Title: Lecture II: Human and Cosmic Thought
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 real science as there is mathematics. This means that from being a
 mathematical terms, we shall not be able to explain the simplest
 perception, but if we go outside a mathematical explanation, we shall
 materialism will become a mathematical thinker, and will recognize as
 valid only whatever can be treated mathematically. From this results
 mathematical formulae. This may be called Mathematism.
 world is ordered mathematically. If mathematical ideas are found to
 Title: Lecture IV: Human and Cosmic Thought
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 would bring to expression a mathematical worldstructure permeated by
 — and if he would inquire into the nature of mathematical
 Title: Lecture: Secrets/Threshold: Lecture IV
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 mathematics. At one time the university Math departments were
 the realm of mathematics. What actually is he demanding? If we know
 Title: Inner Impulses: Lecture IV
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 mathematical knowledge, with which one can do but little harm, or
 Title: Inner Impulses: Lecture VII
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 externally, mathematically or physically as Copernicus, Kepler,
 Title: Mystery Centres: Lecture XII: The Mysteries of the Samothracian Kabiri
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 something mathematical. He begins to calculate the size of the stars
 since man learnt only to know of the heavens mathematically, which
 Title: Mystery Centres: Lecture XIV: Human SoulStrivings During the Middle Ages the Rosicrucian Mysteries
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 mathematical grandeur and mathematical movements, which can be
 Title: Fundamentals of Anthroposophical Medicine: Lecture I
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 the way that a mathematician uses figures to illustrate a
 mathematical problem, I would say the following: imagine that a
 Title: Lecture: Lecture IV: Physiology and Therapeutics
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 mathematics the point has been reached of hatching such hypotheses or
 Title: Anthroposophical Approach to Medicine: Lecture I
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 a mathematician uses figures to illustrate a mathematical problem, I
 Title: Social Future: Lecture VI: National and International Life in the Threefold Social Organism
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 subjective, and cannot be proved. Mathematical truths are also
 individuals can confirm the truths of mathematics. Anyone acquainted
 Title: World Economy: Lecture IV
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 Mathematics we only have to take what is given as number; but when we
 inwardly understand it. A mathematician may well object: You have
 our mathematical formulas in this way. True enough; for as a rule we
 only build up a mathematical formula as a result of contemplating
 Title: World Economy: Lecture VIII
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 speak mathematically — treat S and D as variables and P, the
 Title: World Economy: Lecture IX
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 mathematical formulas, but, like the ones of which I spoke before,
 Title: World Economy: Lecture X
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 well as any that can be clothed in mathematical formula. Of course
 Title: The Earth As Being with Life, Soul, and Spirit: Lecture 2
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 Mathematics is certainly spirit, indeed pure spirit; but employed
 world, as there is mathematics. But from the other point of view,
 Title: Lecture: Michaelmas VII: The Creation of A Michael Festival Out Of The Spirit (Extract)
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 time mathematically, by our earthly means alone, whereas for the real
 Title: Lecture: The Mysteries of Ephesus The Aristotelian Categories
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 to be calculated mechanically, mathematically, namely when they become
 Title: Arts and Their Mission: Lecture VIII
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 its own particular medium. In speaking about mathematics, mechanics,
 Title: Kingdom of Childhood: Lecture 5
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 in spiritual life isdone from the body. Mathematics are done by the
 Title: Spiritual Science and Medicine  Lecture VI
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 There are indeed points of view better than the mathematics of
 Title: Spiritual Science and Medicine  Lecture XV
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 are mathematically inclined, i.e., are used to live much in
 mathematical shapes and to think in mathematical forms. For this way
 other hand, individuals with a certain aversion to mathematics,
 difference between mathematical and nonmathematical minds. And
 Title: Lecture: Curative Education: Lecture 10
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 Vreede [the original leader of the MathematicalAstronomical Section
 Title: Art of Healing: Lecture I
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 as any method leading maybe to the attainment of mathematics or
 mathematics or geometry, only in Anthroposophy we are not
 Title: Spiritual Science and the Art of Healing: Lecture I
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 maybe to the attainment of mathematics or geometry. In fact we
 same way as we do with regard to mathematics or geometry, only
 Title: Principle/Economy: Lecture IV: Results of Spiritual Scientific Investigations of the Evolution of Humanity: I
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 Title: Lecture III: Man in the Past, the Present and the Future
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 mathematicians of all times applied to his own life. When he had
 Title: From Jesus to Christ: Lecture I: Jesuit and Rosicrucian Training
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 dispute over mathematics, because we have raised mathematics entirely
 Title: Education: Lecture II: Principles of Greek Education
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 cases until about the age of twenty — so mathematics and
 Title: Education: Lecture IX: Arithmetic, Geometry, History
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 geometry, indeed all mathematics, occupy a unique position in education.
 physics and mathematics written by Dr. von Baravalle (a teacher at the
 chiefly with higher mathematics. If we penetrate to its underlying
 essence, it is a splendid guide for teaching mathematics in a way
 teaching mathematics and physics from earliest childhood up to the
 Title: Human Values in Education: Lecture V
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 name or another: Diploma in Geometry or Mathematics or whatever
 mathematics for instance, who boasted of the fact that when
 Title: Human Values in Education: Lecture VIII
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 and mathematical. In the Goetheanum at Dornach an attempt was
 Title: Human Values in Education: Lecture X
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 of mathematics (albeit, as soon as one enters into the
 spiritual, mathematical coolness engenders enthusiasm, for
 if one is quite cool in the mathematical sense, when one has to
 Title: Practical Course/Teachers: Lecture XII: How to Connect School with Practical Life
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 Title: Study of Man: Lecture III
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 concrete life — purely out of his geometrical, mathematical
 Title: Study of Man: Lecture XIV
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 perhaps not with mathematical accuracy, but in a form filled with
 Title: Spiritual Ground: Lecture III: Spiritual Disciplines of Yesterday and Today
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 To be sure the multiplication table is not the whole of mathematics,
 and integral calculus is built up on elementary mathematics, — it
 Title: Spiritual Ground: Lecture V: How Knowledge Can Be Nurture
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 intellectual element can be given the child too soon. Mathematics as
 operations of mathematics are not foreign to human faculty in the way
 arithmetic and mathematics in the right way. And what this is can
 with mathematics in the right way during these past years we should
 Title: Lecture: Inner Nature of Music: Lecture II
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 a mathematical gift was inherited in a similar way through several
 generations, and eight of the family members were mathematicians of
 Title: Lecture: Inner Nature of Music: Lecture III
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 family within a period of 250 years and that the mathematical talent was
 handed down through the generations just as mathematical talent was
 the mathematical sense, which is tied to these three semicircular
 musical ear. The mathematical family shows a special development of
 Title: Behind the Scenes: Lecture 1
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 nor mathematical. That is one of the things I wanted to say.
 Title: Anthroposophic Movement: Lecture Four: Spiritual Truths and the Physical World
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 corresponding fashion any truth from the realms where mathematics and
 Title: Anthroposophic Movement: Lecture Six: The Emergence of the Anthroposophic Movement
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 Title: Anthroposophic Movement: Lecture Eight: Responsibility to Anthroposophy
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 has become purely a business in which physical mathematics is simply
 Title: True/False Paths: Lecture Three: Form and Substantiality of the Mineral Kingdom in Relation to the Levels of Consciousness in Man
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 activities find consummation in the mathematicalspatial forms of the
 Title: True/False Paths: Lecture Eight: Potential Aberrations in Spiritual Investigation
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 mathematically. Life, however, does not proceed mechanically or
 mathematically, but dynamically. We must not say, therefore, that the
 purely mathematically and geometrically. We must say: the medium is
 Title: True/False Paths: Lecture Nine: Abnormal Paths into the Spiritual World and their Transformation
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 is that it is written like a mathematical textbook, that I sought to
 mathematical detachment the whole path of development I have been
 along and say: I understand nothing about mathematics so I cannot say
 whether the statements of mathematicians are true or false. The
 a great mathematician or not. — That is approximately the
 Title: True/False Paths: Lecture Three: Form and Substantiality of the Mineral Kingdom in Relation to the Levels of Consciousness in Man
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 activities find consummation in the mathematicalspatial forms of the
 Title: True/False Paths: Lecture Eight: Potential Aberrations in Spiritual Investigation
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 mathematically. Life, however, does not proceed mechanically or
 mathematically, but dynamically. We must not say, therefore, that the
 purely mathematically and geometrically. We must say: the medium is
 Title: True/False Paths: Lecture Nine: Abnormal Paths into the Spiritual World and their Transformation
Matching lines:
 is that it is written like a mathematical textbook, that I sought to
 mathematical detachment the whole path of development I have been
 along and say: I understand nothing about mathematics so I cannot say
 whether the statements of mathematicians are true or false. The
 a great mathematician or not. — That is approximately the
 Title: Art/Mystery Wisdom: Lecture Eight
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 merely from the geometricalmathematical point of view and
 Title: Metaporphoses/Soul One: Lecture 1: The Mission of Spiritual Science
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 justified for that period, for the time had passed when certain mathematical
 Title: Metaporphoses/Soul One: Lecture 3: The Mission of Truth
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 mathematics that humanity in general has reached the goal of truth, for here
 as far with them as they have with mathematics. The truths of mathematics are
 Title: Metaporphoses/Soul Two: Lecture 6: Positive and Negative Man
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 geometry and mathematics cannot be accepted on the authority of another
 Title: Art of Lecturing: Lecture I
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 whether we can teach mathematics or geography well, the very,
 mathematicians, or good geographers. We can be outstanding
 Title: The Development of Thought from the 4th to the 19th Century  1
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 It is the same in mathematics, when objective truths are proclaimed,
 never departing from the principles of mathematical calculation,
 Title: Anthroposophy Science: Lecture I
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 nature, experimentation, and the use of mathematics to establish
 entire field of natural science through mathematics, and with
 these mathematical thoughts, arrive at mathematical results.
 mathematics.
 mathematics must be introduced. Through this, one feels
 an overview of a series of facts with the use of mathematical
 treating knowledge mathematically, has been characteristic of
 certain and safe with the mathematical handling of the
 accompanying the use of mathematics will lead us to
 mathematical foundation.
 of certainty derived from the mathematical treatment of
 only so much real knowledge as there is mathematics? How
 modern psychology, by the use of mathematics, has developed
 introduce mathematics into actual knowledge.
 significance of this mathematical certainty in the context of
 opposed to the more mathematical type of consciousness that
 actually know things mathematically when one applies
 mathematics to the facts of the outer senseaccessible world?
 And what distinguishes this mathematical approach from other
 facts of the world when we do not use our mathematical
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 Title: Anthroposophy Science: Lecture II
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 nature, experimentation, and the use of mathematics to establish
 knowledge of the world around us to mathematical knowledge,
 science that is free of mathematics to one that includes it.
 methodical way the transition from a mathematical penetration
 that has most to do with cognition, at least mathematical
 abstractions of mathematicalgeometrical space.
 leading us to an exact understanding of the mathematical
 Title: Anthroposophy Science: Lecture III
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 nature, experimentation, and the use of mathematics to establish
 explanation of the nature of mathematical thinking.
 and on the other hand the absorption in mathematical thought.
 this, in mathematical thought, in the activity of pure
 mathematical thinking, things are different. The difference
 “high” mathematical thinking — if I may
 soul, is then found in today's mathematicallyoriented
 our mathematics can be characterized as purely pictorial. One
 can say: what we experience mathematically has as such no
 natural surroundings. In this regard, mathematical thinking
 mathematical thinking only makes images of, is actually real
 and full of content; but mathematical thinking itself is
 recognize the pictorial character of mathematical thought, we
 can realize that we experience these mathematical pictures
 experience as the mathematical thinking itself.
 mathematical thinking we are right inside what actually takes
 mathematical activity, permits us to have a clear
 why we really know that when we work in mathematics we are in
 field of pure mathematics.
 is the fact that in the process of mathematical thinking, one
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 Title: Anthroposophy Science: Lecture IV
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 nature, experimentation, and the use of mathematics to establish
 to be built up the way mathematical imagery is built up, the
 way mathematical constructs are developed, analyzed, and so
 results of our inner mathematical activity to the outer
 and algebraic mathematics to approach plant growth and plant
 formation, we do not find that this mathematical form of
 activities as it is in mathematical activity — happens
 it we can proceed. We experience clarity as in mathematics
 created it, acquaints us inwardly, "mathematically," with
 being. Just as mathematical mental images match and explain
 possibility, through a higher kind of qualitative mathematics
 mathematical type of experience. Easily pictured mental
 power of imagination (in a manner modeled on mathematical
 love — that world which if approached only mathematically
 mathematical attitude of soul, and are not fools, we will see
 Title: Anthroposophy Science: Lecture V
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 nature, experimentation, and the use of mathematics to establish
 through a mathematical approach. Further, I pointed out that
 processes are understood through mathematics. Nor will one
 to us, as a mathematical approach makes physical phenomena
 mathematical understanding of the entire natural realm. I
 mathematical construction into a realm where things become
 transformation of welldefined mathematically constructible
 figures into something that cannot be grasped mathematically
 life and nerve life that evade direct, purely mathematical
 just as consciously as a mathematical figure, but going
 beyond the mathematical. I mean, of course, imagination.
 first step along the way from purely mathematical
 study of mathematics can give. Of course, it would be a very
 good thing indeed if physicians had more mathematical
 No matter how good their mathematical conceptions, doctors
 transforming mathematical knowledge and advancing to
 physicalmineral structure reveals itself to the mathematical
 Title: Anthroposophy Science: Lecture VI
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 nature, experimentation, and the use of mathematics to establish
 view as a whole, just as easily as the mathematical images:
 “mathematical” attitude of soul, we can make sure
 Title: Anthroposophy Science: Lecture VII
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 nature, experimentation, and the use of mathematics to establish
 of consciousness as is mathematical thinking. This intuition
 mathematical reference point — a reference point for
 is inwardly grasped in mathematical thinking and what is
 mathematical thought and what we find given in outer
 “gulfs.” Mathematical lines and forms which we
 perceptions must be penetrated by mathematical thinking. In
 our perceptions to what has been worked out mathematically or
 mathematics. When we seek for the realm in which mathematics
 mathematical thinking, of the results of mathematical
 of fanciful or dilettante activity. For even in mathematics,
 receive a training in geometry and mathematics. In a similar
 goal it must have inner mathematical clarity. I hope you have
 Title: Anthroposophy Science: Lecture VIII
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 nature, experimentation, and the use of mathematics to establish
 Title: Boundaries of Natural Science: Lecture I
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 everything that one finds in one's environment with mathematical formulae.
 One wants to translate data gathered from nature into clear mathematical
 formulae, into the transparent language of mathematics.
 to being able to give a mathematicalmechanical explanation of natural
 its position and momenta in mathematical formulae. They believed they
 movements wholly intelligible in terms of mathematics. Hence there arose
 between the various heavenly bodies in mathematical formulae, so too
 of lucid mathematics. This was a striving that climaxed in the last
 against this striving for a crystalclear, mathematical view of the
 mathematics, in all explanations, proceeding all the way up to the explanation
 world view within the lucid realm of mathematics, while on the other
 in thought of mathematical clarity or, indeed, even of constructing
 mathematical formulae and calculate the movements of matter in terms of
 concepts, we see that it is an external, mathematicalmechanical lucidity.
 apply a mathematicalmechanical explanation, apply the theory of evolution,
 sobriety, as Herbart has done, to apply mathematical computation to
 to gain a foothold, because the mathematical formulae simply cannot
 of nature. Consciousness rejects this lucidity. This mathematical clarity
 Title: Boundaries of Natural Science: Lecture II
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 which I commenced yesterday, the modern striving for a mathematicalmechanical
 man into a cog in a social order modeled on mathematics or mechanics
 mathematics, of the number, weight, measure, or temporal relation that
 within our systematic science of nature as a whole do mathematics, do
 to us differently from that which we could represent as subject to mathematicalmechanical
 of touch, and that which confronts us in the mechanicalmathematical
 pose the question: how is it that we arrive at any mathematicalmechanical
 judgment? How do we arrive at a science of mathematics, at a science
 of mechanics? How is it, then, that this mathematics, this mechanics,
 is a difference between the mathematicalmechanical qualities of external
 Title: Boundaries of Natural Science: Lecture III
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 to our use as an immediate necessity: mathematics and that part of mechanics
 comprehended by mathematics and analytical mechanics, we have before
 the notions of mathematics and analytical mechanics, this process within
 manner and the more inward mode of apprehension we find in mathematics
 between that which is still actually mathematical — as we still
 conceive mathematics today — and that which leads over into
 this phenomenon of mathematics as such. We comprehend mathematical truths.
 We proceed from mathematical phenomena to certain axioms. We weave the
 fabric of mathematics out of these axioms and then stand before an
 we must see in this fabric of mathematics something that arises through
 experience. We thus must ask: where does mathematics originate? Nowadays
 how is this inner activity of the soul that we need in mathematics,
 in the wonderful architecture of mathematics — how is this inner
 is it, then, that we acquire when we engage in mathematics? We want
 is that which manifests itself as the ability to perform mathematics
 as the ability to perform mathematics and something like mathematics
 to the change of teeth, the soul faculties enabling one to perform mathematics
 same way we must be entirely clear that the capacity to perform mathematics,
 the nature of mathematics — mathematics taken, of course, in the
 seventh year there works an inner mathematics, an inner mathematics
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 Title: Boundaries of Natural Science: Lecture IV
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 pellucid is that of mathematics and analytical mechanics.
 in the soul as mathematics, as analytical mechanics, actually rests
 the same spirit from which we now draw only mathematics and analytical
 that underlies complex phenomena is, inwardly, the same as the mathematician's
 search for the axiom underlying complex mathematical constructs. Goethe,
 therefore, who himself admitted that he had no conventional mathematical
 training, nevertheless sensed the essence of mathematics so clearly
 rigorous enough to satisfy a mathematician. It is just this that the
 reveals the same inner necessity as mathematics and analytical mechanics.
 Goethe did have this — of the true inner structure of mathematics.
 discipline, as, for example, mathematics or analytical mechanics.
 of mathematical thinking. If one does this, one enters via a natural
 to consciousness in the same way that mathematics or the faculties and
 structure of mathematics, I flew in the face of every kind of philosophy
 can be experienced when one performs mathematics in the right way, when
 this performance of mathematics itself becomes an experience that can
 his experiences of an inner mathematics in the Vedanta. This path is
 within an element of primal mathematical thought and was able to enter
 Title: Boundaries of Natural Science: Lecture V
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 of contemporary science, even of the mathematical sciences. He must
 one has formulated as a natural law, or perhaps mathematically.
 “mathematicizes” within us during the first seven years up
 occurs unconsciously), when one enters into this “living mathematics,”
 within them with the extended mathematical representations. Tomorrow
 Title: Boundaries of Natural Science: Lecture VI
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 can only in mathematical, geometrical, analyticmechanical thinking. When
 as mathematics and analytical mechanics. On the other hand, spiritual
 Title: Boundaries of Natural Science: Lecture VIII
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 philosophers who maintain this have never really studied mathematics
 of mathematical thinking. The pursuit of philosophy is actually impossible
 without a grasp of at least the spirit of mathematical thinking. We
 have seen what Goethe's attitude was toward this spirit of mathematical
 in mathematics. Many thus would deny the existence of the very faculty
 Title: Colour: Part Two: Thought and Will as Light and Darkness
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 lifeless mathematics. It is true Hegel used more pregnant concepts
 than the mathematical ones to understand the world; but what attracted
 Title: Colour: Part Two: The Connection of the Natural with the MoralPsychical. Living in Light and Weight.
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 line, this line could also, if we altered the mathematical equation,
 it has also only one mathematical equation; it is a simple line, only
 mathematicians, if they would, could certainly do today, one would
 possibility does not exist; things especially like Mathematics, etc.,
 Title: Colour: Part Two: Dimension, Number and Weight
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 In physical mathematics it is a matter of indifference to divisions
 Title: Colour: Part Three: The Creative World of Colour
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 those which approach, one might say, the most abstract Mathematics,
 transcends the abstractmathematical, or anything that is similar to
 the abstractmathematical. And so it is not surprising if our time has
 be the greatest mathematician or the greatest metaphysician, but he
 Title: Colour: Part Three: Colours as Revelations of the Psychic in the World
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 Mathematics or Mechanics or Physics from artistic sensibility, but
 Title: The Building at Dornach: Lecture II
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 there is something inorganic, purely mathematical in the
 Title: Ways/Architecture: Lecture III: The New Conception of Architecture
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 above all when we pass from a purely mathematical conception
 mathematician in the substrata of his consciousness and by
 Title: Ways/Architecture: Lecture IV: True Aesthetic Laws of Form
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 in the depths just as it contains the secrets of mathematics,
 everything, even about mathematical principles. When one man
 applies a mathematical principle and gets a different result
 Title: Ways/Architecture: Lecture V: The Creative World of Colour
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 based on utterly abstract mathematical principles — and
 reality which transcends all abstract mathematics or systems
 great mathematician, or a great metaphysician, but he does
 Title: Art of Lecturing: Lecture I
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 whether we can teach mathematics or geography well, the very,
 mathematicians, or good geographers. We can be outstanding
 Title: Christmas Conference: Lecture 8: Continuation of the Foundation Meeting, 27 December, 10 a.m.
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 to encompass mathematics and astronomy. And the last, for the
 given with almost mathematical precision. But in order to
 Title: Christmas Conference: Lecture 9: Continuation of the Foundation Meeting, 28 December, 10 a.m.
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 realm of mathematics and astronomy it comes from her. You,
 instruction — by means of mathematics. Everything which
 Title: Christmas Conference: Lecture 11: Meeting of the Vorstand of the General Anthroposophical Society
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 currencies, down to what in mathematics is termed the
 point, as they say in mathematics, will have a definite part
 Title: Awakening to Community: Lecture II
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 The truth of a mathematical statement can be discerned in a flash,
 rationalistic science of today, particularly mathematics, which is
 Title: Awakening to Community: Lecture IV
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 objective mathematics, objective chemistry, that it is working toward
 Title: Awakening to Community: Lecture VII
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 complete rebirth of chemistry, physics, mathematics, and so on,
 have continued there into 1923, I gave the mathematical physicists an
 finding a mathematical formula to express the difference between
 Title: Hegel, Schopenhauer, Thought, Will
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 natural laws and we can practically only use mathematics to
 Title: Excursus/Mark: I: A Retrospect
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 path leading to mathematics, to botany, or any other branch of
 Title: Excursus/Mark: III: Excursus: Lecture III
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 the case of mathematics, for instance, or any other science, that
 Title: Spiritual Teachings of Soul/World: Course I: Lecture II: The Origin of the Soul
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 corresponds to intricate mathematical calculations prove it. But also
 Title: Spiritual Teachings of Soul/World: Course II: Lecture I: The Epistemological Basis of Theosophy I
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 are sure judgments. These are the mathematical judgments. Is the mathematical
 This is the peculiar of mathematical proofs. For everybody it is clear
 How is mathematical knowledge possible? — Kant now calls those
 Wolff (1679–1754), German philosopher and mathematician
 Title: Spiritual Teachings of Soul/World: Course II: Lecture III: The Epistemological Basis of Theosophy III
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 absurdum as the mathematician calls it.
 So we have mattered what the mathematician calls a reductio ad absurdum, like
 Wilhelm Leibniz (1646–1716), German mathematician and philosopher
 Title: Spiritual Teachings of Soul/World: Course III  Lecture I: Theosophical Teachings of the Soul. Part I: Body and Soul
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 anybody to his philosophers’ school who had not learnt mathematics first,
 nature of the mathematical, know something about the nature of this peculiar
 mental activity which the human being exercises if he does mathematics. However,
 this is clear also to Aristotle that it does not depend on doing mathematics
 rather than on understanding: the human being is able to do mathematics.
 mathematical theorem could never be gained by mere observation. In nature nowhere
 but in mathematics we investigate these, and we apply the world which we have
 sentence. Those who had the biggest conquests in astronomy by their mathematical
 He knew which farreaching scope mathematical thinking has up to the most distant
 The mathematical thinking does not suffice. It is the first step where we completely
 soul in the same way as mathematics gets out the truth of the starry heaven
 everybody who wants to enter into my school must have gone through a mathematical
 course first. — Not mathematics is necessary, but a knowledge which has
 the independence of the mathematical thinking. If one sees that the human being
 mathematical truth is. There are still savage people. Indeed, they become extinct,
 described it as the spirit of mathematics as nous which results easily like
 Title: Spiritual Teachings of Soul/World: Course III  Lecture III: Theosophical Teachings of the Soul. Part III: Soul and Mind
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 penetrated into the spirit of mathematics before they tried to take up his teachings
 disciple should have understood the spirit of mathematics. We heard in the first
 lecture what this spirit of mathematics offers. It offers truths in the most
 this way. Mathematics only gives the most trivial supersensible truth, but it
 that they learn in mathematics how one gets to the supersensible truth. What
 Why does the mathematical truth
 in the spirit of mathematics. After they had got used to looking up to truth
 questions at first also uninterested, without desire and harm, as the mathematical
 pupil can look in the eye of spirit as well as somebody who delves in a mathematical
 field sees in the eye of pure objective mathematics: without passion, without
 most trivial supersensible truth. Mathematics is widely avoided just because
 to him as the mathematical truth speaks indifferently to him, but in such a
 way as mathematical truth also speaks in another sense. Mathematical truth namely
 is true and eternal in certain way. What appears to us in the language of mathematics,
 state where he faces spiritual truth really as the human being faces the mathematical
 Title: Spiritual Teachings of Soul/World: Course IV  Lecture I: Theosophy and Spiritism
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 become a competent mechanic must study mathematics. Who wants to be at home
 Title: Spiritual Teachings of Soul/World: Course IV  Lecture II: Theosophy and Somnambulism
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 human being. However, think only once about the fact that the mathematician
 Title: Spiritual Teachings of Soul/World: Course IV  Lecture III: The History of Spiritism
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 everything by the appearance to the senses and by mathematical calculations,
 Wallace, the mathematician Morgan. Also
 researchers like Zöllner, Weber, Fechner and also several famous mathematicians
 De Morgan (1806–1871), British mathematician and logician
 mathematician and physicist: On Spiritistic Manifestations from the
 Title: Spiritual Teachings of Soul/World: Course IV  Lecture IV: The History of Hypnotism and Somnambulism
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 mathematician) in his book Deliciae physicomathematicae (1636).
 Schwenter, however, had taken it from another book Recreationes mathematicae
 (1624), written by the French (medical writer) mathematician and philosopher
 S. J., German mathematician and physicist
 Title: Spiritual Teachings of Soul/World: Course V  Lecture I: What Does the Modern Human Being Find in Theosophy?
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 Bruno (1548–1600), Italian Dominican monk, philosopher, mathematician,
 Title: Spiritual Teachings of Soul/World: Course V  Lecture II: What Do Our Scholars Know about Theosophy?
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 you can see through the telescope and what you can achieve with mathematics.
 has understood the truth of a mathematical theorem does not care whether a million
 Title: Spiritual Teachings of Soul/World: Course V  Lecture IV: Is Theosophy Buddhist Propaganda?
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 anybody who wants to teach mathematics today does not teach what he himself
 mathematics or physics. What is announced as a principle of wisdom becomes immediate
 mathematics or natural sciences, but it is a lively life. What the science of
 Title: Esoteric Christianity: Jeshu ben Pandira  Lecture 1
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 study a book of mathematics, for instance. This will help us to fall
 the chemist, the mathematician, proceeds without any accompanying
 Title: Esoteric Christianity: The Christ Impulse as Living Reality  Lecture 1
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 of the ether.’ The answer was: Purely mathematical constructions,
 Title: Novalis: On his Hymns to the Night
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 knowledge of law and mathematics was necessary. He became a
 mathematicmaterialistic sciences, which lay at the foundation
 mathematician. What is most important is that Novalis as a
 spiritual being allowed mathematics to penetrate into his inner
 When mathematics showed him how it is suitable for the
 mathematical imagination became a great poem which filled him
 when he dived into numbers and sizes. For him mathematics
 power and crystallize out there. Mathematics became for his
 people, who only know mathematics from outside, it remains
 Title: Temple Legend: Lecture 7: The Essence and Task of Freemasonry from the Point of View of Spiritual Science  1
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 knowledge of our presentday mathematics. We behold the wonderful
 Title: Temple Legend: Lecture 11: Concerning the Lost Temple and How It Is To Be Restored  1
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 mathematics. Then he will have to learn how it is to be made,
 Title: Temple Legend: Lecture 20: The Royal Art in a New Form
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 proclaims mathematical truth through their official spokesmen? If I
 want to learn mathematics, I must occupy myself with it, and gain an
 Title: Fruits/Anthroposophy: Lecture 5: From Sense Perception to Spirit Imaging
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 just as accurately defined as the solution of a particular mathematical
 is there in the soul in the same way as mathematical, geometrical forms
 with a sense of our own reality being maintained as a mathematical soul
 Title: Fruits/Anthroposophy: Lecture 6: From Imaginative Knowledge to Inspirational Knowledge
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 is done with the same inner clarity as that experienced in forming mathematical
 mind proceeding with the clarity of mathematical thought. The steps
 spirit and with the lucidity we know in mathematics, not with a dreamy,
 working with geometry and mathematics. The work which has to be done
 Title: Where/How/Spirit: Lecture I: Where and How Does One Find the Spirit?
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 mathematics, for example, gives us something spiritual, for
 example, in mathematics. Nobody can understand by experience
 Title: Where/How/Spirit: Lecture III: Goethe's Secret Revelation  Esoteric
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 such truths, as the mathematical ones are, most human beings
 the mathematical thoughts think in us. If we have purified the
 in history, in mathematics and all other sciences — it
 Title: Where/How/Spirit: Lecture IV: Bible and Wisdom I
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 in its first beginnings by Euclid, the great mathematician.
 mathematics regardless how it has developed in the course of
 Title: Preparing for a New Birth
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 of numbers. The other was a mathematics student. Now, we know
 that at certain moments in the study of mathematics, you do
 pleased with himself, while the mathematics student was not,
 student woke up and saw the mathematics student get out of bed
 when they both got up, the mathematics student said, “We didn't
 Title: Chance/Necessity/Providence: Lecture 1: Probability and Chance
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 been what is called in mathematics the calculus of probability. It's
 dice cup and throw the dice. The mathematician makes his calculation
 in mathematical formulas the degree of probability of any particular
 you see that it is possible to express in mathematical formulas what
 mathematical terms.
 the mathematical genius of Archimedes is not up to grasping so vast
 Title: Spiritual Science/Treasure for Life: Lecture IV: On Death
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 once the mathematicians were towards the problem that one
 one sent solutions to mathematical circles how one could
 the views of those mathematicians changed. For somebody still
 Title: Child's Changing Consciousness: Lecture I
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 French astronomer and mathematician. Immanuel Kant (1724–1895)
 there three hundred years from now. Though mathematically
 Title: Child's Changing Consciousness: Lecture III
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 mathematician horses that, for awhile, caused a sensation in
 horse was at simple mathematical calculations. The whole thing
 contemporary mathematical sciences, one might ask this lecturer
 Title: Child's Changing Consciousness: Lecture IV
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 pervades every subject, even mathematics. This fundamental
 Title: Child's Changing Consciousness: Lecture V
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 essentially the same as the principles of mathematics and
 around in space, or at least something that is mathematically
 Title: Child's Changing Consciousness: Lecture VIII:
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 mathematics and physics at the Waldorf school in Stuttgart.]
 effects. Mathematics is built on purely causal sequences, so it
 Title: Spiritual/Physical: Lecture II:
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 century; for this writer tries to prove mathematically that the earth
 Title: How Can Mankind Find Christ Again?: Lecture 5: The Change in the Human Soul Constitution
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 was alive in ancient times, but today it is dead, it is simply mathematical
 through mathematics. In saying this, no particular objection is intended
 to external mathematics; it can even shed light on many things —
 Title: How Can Mankind Find Christ Again?: Lecture 8: Experiences of the Old Year and Outlook over the New Year (part 2)
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 people love mathematics. Certain rules are established and the rest
 is mathematical reckoning. There is no need to follow the details with
 is on mathematics or botany or politics or socialism or anarchy; that
 Title: Community Life: Address 2: The GoeschSprengel Situation  Part 2
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 days ago on various mathematical and geometrical ideas.
 Title: Human History: Lecture I: The Relation of the Human Being to the Supersensible Worlds
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 ether, namely purely mathematical equations. These are
 mathematical signs on the paper something is included that is
 Title: Human History: Lecture VIII: The Origin of the Human Being
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 Snell (Karl S., 1806–1886, mathematician and natural
 Title: Human History: Lecture XII: Copernicus and His Time in the Light of Spiritual Science
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 that summarised by logic and mathematics what spread out in
 Title: Spirit and Matter: Lecture III: Immortality, the Forces of Destiny, and the Course of Life
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 Title: Spirit and Matter: Lecture VI: Life, Death, and Immortality in the Universe
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 of mathematics stop. As absurd as it sounds, but it is true. I
 Title: Answers to Big Questions: Lecture I: The Nature of Spiritual Science and Its Significance for the Present
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 Indeed, the mathematical field, the field of geometry seems to
 anybody can allege that the statements of mathematics, of
 who have soared to invent geometries, mathematics purely
 mathematically, by strict mathematical investigations that do
 That means there were mathematically thinking people who felt
 mathematics and geometry and could invent a geometry that
 also several such geometries. Mathematically trained people
 for them! This is very indicative that even in mathematical
 Title: Answers to Big Questions: Lecture VII: How Does One Attain Knowledge of the Spiritual World?
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 anyhow? Transfer this example on mathematics that somebody may
 say of it: the incomprehensibility of mathematics mostly
 Mathematics, geometry teaches the human being the knowledge of
 what is dry outer science, dry outer philosophy, mathematics,
 Title: Answers to Big Questions: Lecture XV: What Has Astronomy to Say about the Origin of the World?
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 mathematically what arises from the atoms and their movements,
 difficult, if one must not have such a lot of mathematical
 Title: Riddles of the World: Lecture I: Haeckel, the Riddles of the World and Theosophy
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 true, and the higher mathematics is only a fantastic stuff.
 Title: Riddles of the World: Lecture III: Basic Concepts of Theosophy. Soul and Spirit of the Human Being
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 trivial I can say that mathematics is something that is
 Title: Riddles of the World: Lecture V: The Question of Women's Rights
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 as mathematical judgements for the mathematicians. What the
 Title: Riddles of the World: Lecture VI: The Basic Concepts of Theosophy. Human Races
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 consciousness, our reason, our mathematical, logical
 Semitic language is the combinatorial, the mathematical, and
 Title: Riddles of the World: Lecture VII: The Core of Wisdom in the Religions
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 Pythagoras (~570~495 B.C., philosopher, and mathematician)
 Title: Riddles of the World: Lecture VIII: Fraternity and the Struggle for Existence
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 mathematics to get a right result — for 3 times 3 is
 Title: Riddles of the World: Lecture IX: Inner Development
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 incomprehensible as higher mathematics is for the usual farmer
 Title: Riddles of the World: Lecture XVI: German Theosophists at the Beginning of the Nineteenth Century
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 mathematics like a religion. It is something sacred to him.
 something like music with the abstract lines of mathematics and
 Novalis speaking about mathematics. However, you can only take
 space. Mathematics is the access to the infinite truth.
 Title: SpiritualScientific Consideration: Lecture 1: Prelude to the Threefold Commonwealth
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 elementary mathematics and technical knowledge you can
 Title: Knowledge of Soul and Spirit: Lecture I: The Mission of Occult Science in Our Time
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 nothing but an example, a suggestion to us: the mathematical
 comparison completely suitable. The mathematical truth is
 inside. Thus, it is with all geometrical and all mathematical
 have convinced yourselves of any mathematical theorem, then two
 mathematical truths, as true it is that — if the conditions are
 mathematics, but because this occult science has the character
 of mathematical truth. However, it is long ago that one looked
 mathematically that the three angles of the triangle amount to
 occult science just as mathematics is an occult
 Title: Knowledge of Soul and Spirit: Lecture IV: Initiation
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 today. I can produce evidence. Mathematics, geometry, can
 This is thinking free from sensuousness. Mathematics is not
 mathematics free from sensuousness and create as one can do it
 between the dry abstractions that one puts in mathematics or in
 Title: Knowledge of Soul and Spirit: Lecture VI: The SoCalled Dangers of Initiation
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 certainty, like mathematics, in itself, and an immense
 Title: Knowledge of Soul and Spirit: Lecture XII: Sun, Moon and Stars
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 wholly mathematical forces, but effects based on wholly
 Title: Knowledge of Soul and Spirit: Lecture XIV: The Hell
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 musicians and a mathematician from a family of mathematicians,
 (famous Swiss family of mathematicians and scientists)
 become mathematicians repeatedly, it is
 line, in that what is the ground for the mathematical head, he
 Title: Knowledge of Soul and Spirit: Lecture XV: The Heaven
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 and mathematician) and others have their successors who can
 then: this law contains nothing else than the old mathematical
 Title: The Social Question as a Question of Consciousness: Lecture 3
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 say, of chemistry, physics, mathematics, is of course true and these
 the point is not that the truths of chemistry and mathematics are true
 Title: The Social Question as a Question of Consciousness: Lecture 7
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 a thinking entirely dominated by mathematical technics. We have become
 Title: The Cyclic Movement of Sleeping and Waking
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 Title: The Apocalypse: Lecture VI
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 doesn't mean our rather abstract geometry or mathematics, but
 Title: Mystery Trinity: Part 2, Lecture I
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 Title: Freedom/Immortality/Social: Lecture VIII: How Natural Sciences Justify the Supersensible Knowledge
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 hours that I did the very best progress in mathematics or
 geometry that I had, however, no talent to keep mathematical
 the others did their calculations with the mathematical
 develop these mathematical formulae ad hoc from the basic
 Title: Freedom/Immortality/Social: Lecture IX: How Does One Justify the Anthroposophical Psychology?
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 for him as pupil to keep mathematical formulae in mind that he
 Title: Redemption of Thinking: Lecture II:
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 that the pure thinking proceeds with mathematical security from
 Title: Redemption of Thinking: Lecture III:
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 mathematically, but develop in proving at the same time and
 certainty, as it exists in mathematics? He gets to such
 Title: Reincarnation and Karma: Lecture I
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 good arithmeticians, good mathematicians, often reappear with no gift
 for mathematics at all. If we wish to discover what gifts we may
 may happen that a man had a special talent for mathematics in a
 with a talent for mathematics and this will represent a continuation
 of the previous incarnation. Abel, the mathematician who died young,
 mathematical talent.
 But when a mathematician
 mathematics. I knew a man who had so little gift for mathematics that
 good mathematician.
 exceptionally good mathematician in one incarnation, the mastery he
 mathematical talent has passed into the eyes and no longer exists as
 a gift for mathematics.
 Hermite said: “He has left mathematicians something to keep
 to be appointed Professor of Mathematics in the University of
 Title: Reincarnation and Karma: Lecture V
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 chemistry or mathematics. How an individual assimilates Anthroposophy
 Title: Theosophy/Rosicrucian: Lecture I: The New Form of Wisdom
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 teacher of mathematics to his students. Can it be said that the
 student of mathematics depends upon his teacher simply out of belief
 in authority? No! And can it be said that the student of mathematics
 source of all spiritual truths. If the teacher knows the mathematical
 little accept them on authority as in mathematics he need accept on
 mathematics, in philosophy, or in practical life. The difference in
 functioning of the semicircular organs in the ear. Mathematical
 a succession of fine mathematicians. An individual may possess great
 Title: Theosophy/Rosicrucian: Lecture VII: The Technique of Karma
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 Mathematical talent also needs something quite specific. A particular
 mathematicians. Thinking, logic, is the same in the mathematician as
 space. Special development of these canals determines mathematical
 talent — herein lies the gift for mathematics. This is a physical
 remembered that eight firstclass mathematicians were born in the
 Title: Theosophy/Rosicrucian: Lecture XIV: The Nature of Initiation
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 Study in the Rosicrucian sense is the ability to immerse oneself in a content of thought not taken from physical reality but from the higher worlds. This is called the life in pure thought. Modern philosophers for the most part deny this; they say that every thinking must have a certain vestige remaining from sense perception. This, however, is not the case, for no one, for example, can see a true circle; a circle must be seen in the mind; on the blackboard it is only a collection of tiny particles of chalk. One can only attain to a real circle if one leaves aside all examples, all actual things. Thus thinking in Mathematics is a supersensible activity. But one must also learn to think supersensibly in other fields.
 Title: Anthroposophic Movement (1938): Lecture IV: Blavatsky's Orientation: Spiritual, but AntiChristian
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 regions, where mathematics are no longer valid and the senses
 Title: Anthroposophic Movement (1938): Lecture VI: The Two First Periods of the Anthroposophic Movement
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 apparatus and see whether he is a mathematician or not.’
 Title: Anthroposophic Movement (1938): Lecture VIII: Conclusions: The Anthroposophical Society and its Future Conduct.
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 mathematical aspect only; which amounts to nothing more, than
 that men carry the mathematics of earth up into the stars of
 Title: Turning Points: Lecture 2: Hermes
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 abstract notions, particularly the mathematical concepts of
 the people the elements of the science of mathematics, while he
 script, our Mathematics, Geometry, Surveying, our general conduct
 Title: Agriculture Course (1938): Lecture VI
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 only serves as a means of mathematical orientation, and cannot
 Title: Cosmogony/Freedom/Altruism: Lecture I: Social Impulses for the Healing of Modern Civilization
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 earth's environment in terms of mathematics; but they have not
 realise that where he could only perceive a mathematical order,
 does not really believe in this purely mathematical world, in
 the purely mathematical cosmos. His particular civilisation
 Title: Toward Imagination: Lecture 6: The Feeling For Truth
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 professor of physics and mathematics at the Vienna Agricultural College,
 Title: Social Life: Lecture I
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 Mathematics, — can be furnished by the intellect, but not
 Title: Karmic Relationships, VIII: Lecture I
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 the arguments of the mathematicians, physicists and chemists; on the
 Title: Karmic Relationships, VIII: Lecture V
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 a mathematical truth and point from certain phenomena in the present
 Title: Karmic Relationships, Volume I: Lecture II
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 Title: Karmic Relationships, Volume I: Lecture VII
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 mathematics, including branches of knowledge such as political
 principle to mathematical concepts of many kinds.
 on a variety of subjects: national economy, philosophy, mathematics.
 (Great Men of Letters). There you will find his strictly mathematical
 he was really able to think only mathematically. In philosophy, in
 political economy, in mathematics itself, he thinks mathematically,
 with mathematical precision and clarity. In natural science, too, he
 thinks with clarity but, again, in terms of mathematics. He is not a
 Title: Karmic Relationships, Volume I: Lecture VIII
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 physics and mathematics, especially by the treatise Neue
 Title: Karmic Relationships, Volume I: Lecture X
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 Title: Karmic Relationships, Volume I: Lecture XII
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 onesided. A geometrician knows the whole of mathematics; when he
 equations for it. He knows the mathematical, calculating side of it
 properly speaking no mathematician at all. He understood, for
 generally realised) that man does not get geometry or mathematics out
 Title: Karmic Relationships, Volume II: Lecture I
Matching lines:
 of the period. Translate it into the mathematicaldialectical mode of
 Title: Karmic Relationships, Volume IV: Lecture II
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 were to produce mathematical results and, instead of testing their
 laboratory, to see whether or not he was a real mathematician.
 Title: Karmic Relationships, Volume IV: Lecture VI
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 other hand consider the stars only in mathematical calculations, but
 Title: Karmic Relationships, Volume IV: Lecture IX
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 veritable antipathy to the modern mathematical form of science. In the
 sciences or mathematics. But the very curious thing is this, so curious
 Title: Karmic Relationships, V: Lecture III
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 — mathematics, philosophy, architecture, commerce, industry,
 Title: Karmic Relationships, V: Lecture IV
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 word he was no mathematician; he was a geometrician and nothing else.
 deeply versed in mathematics. He lived at a time when all descriptive
 Title: Karmic Relationships, VII: Lecture Five
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 physical world than the mathematical axiom: the whole is
 Title: Foundations of Anthroposophy: Lecture I: Foundations of Anthroposophy
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 whole process with mathematical precision. Since the whole
 Title: Foundations of Anthroposophy: Lecture III: World Development in the Light of Anthroposophy
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 mathematicians I might say: this thoughtintegral in comparison
 study of mathematics, so that it is out of the question to
 Title: Supersensible Knowledge: Lecture I: Supersensible Knowledge: Anthroposophy as a Demand of the Age
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 in the sense in which the external scientist, the mathematician for
 mathematician surveys his operations when he undertakes to solve a
 as conscientious in regard to his knowledge as a mathematician must
 be at every step he takes. For just as the mathematician must see
 Title: Spiritual Development: Lecture I: The Inner Experience of the Activity of Thinking
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 exercises if one has had some practice in mathematics,
 field of mathematics if, for instance, one makes a mental
 goes on. Here the mathematicians have
 Title: Lecture: Theosophic/Esoteric Cosmology: Spiritual Cosmology
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 that to the telescopes, the mathematical and photographic methods that
 that someone tells you a mathematical truth. If you haven't studied
 mathematics you don't have the necessary knowledge to verify this
 truth. No one will deny that to judge a mathematical truth the
 nothing more than every mathematician also demands.
 Title: Theosophy and Rosicrucianism: Lecture VI: Man's Descent into an Earthly Incarnation
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 produced eight mathematicians in a short time. How can we
 matters stand with the brothers Bernoulli? Mathematical
 mathematical logic does not differ from any other), but the
 mathematical gift is based upon the specially exact
 longer orientate yourself in space. The mathematical talent,
 developed mathematical gifts will incarnate in a family in
 Title: Theosophy and Rosicrucianism: Lecture XIII: The Rosicrucian Training
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 mathematics must be grasped independently of all sensory
 with the circle. Through mathematics we learn to think of the
 Title: Gospel of John: Lecture IX: The Prophetical Documents and the Origin of Christianity
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 just as anyone wishing to learn mathematics at present does
 mathematics have first been communicated. What, for example,
 Title: Gospel of John: Lecture X: The Effect of the Christ Impulse Within Mankind
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 mathematical and geometrical truths. These cannot be put to
 mathematics. Men agree upon these mathematical truths at
 come into consideration in simple mathematical truths, many
 Title: Occult History: Lecture 1
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 daughter of a great mathematician, Theon. We see how there flashes up
 vision of the great mathematical, lightwoven texture of the universe.
 individuality to have a mathematician as father in order that something
 Title: Occult History: Lecture 3
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 vividly in the great woman mathematician Hypatia, in whom all the mathematical
 Title: Occult History: Lecture 4
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 constructed on the clearest mathematical principles are here derived
 today once again to recognise the spiritual, mathematical law prevailing
 laws, are to be found again microcosmically in the mathematical numbers
 Title: Occult History: Lecture 5
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 Nature, especially where this knowledge is clothed in mathematical forms.
 Title: Lecture I: Man in the Light of Occultism
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 symbols. Those of you who know mathematics will readily
 occultism, — any more than there are different mathematics. It is
 we can imagine there might be different standpoints in mathematics.
 Title: World of the Senses and the World of the Spirit: Lecture I
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 have so ordered things that mathematical laws appear to us as they
 Title: Destinies of Individuals and Nations: Lecture 8: Three Decisions on the Path to Imaginative Perception
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 transfer ordinary arithmetic, physical mathematics, to
 Title: Destinies of Individuals and Nations: Lecture 10: Problems on Spiritual Path  National Characteristics in Europe Moulded by Folk Spirits
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 Title: Things Past and Present: Lecture V: Comenius and the Temple of PanSophia
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 external is geometry and mathematics. That person who today
 allows geometry and mathematics to work upon him in a true
 mathematics, a different kind of knowledge from the other
 something within geometry and mathematics which one can
 mathematics goes back to a much more ancient knowledge than
 Title: Lecture: Riddle of Humanity: Lecture One: Greetings to the Builders Working on the Goetheanum. Otto Weininger, a Decadent Genius. Distorted Pictures of Imaginative Knowledge.
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 observed, with mathematical rigour, from this point of view.
 body, in the way one would demonstrate a mathematical truth.
 Title: Life Gifts: Lecture II: The Relativity of Knowledge, and Spiritual Cosmology
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 Certainly mathematics do belong to the realm of the spirit, pure
 world as there is mathematics.” But one might also say, from
 Title: Life Gifts: Lecture III: Thoughts about the Life Between Death and Rebirth
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 Science as we might perhaps take up mathematics, but we shall
 Title: Symptom 2 Reality: Lecture VII: Incidental Reflections on the Occasion of the New Edition of 'Goethes Weltanschauung'
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 mathematics, physics, chemistry and especially for projective
 Title: Fundamental Social: Lecture 1: The Transforming of Instinctive into Conscious Impulses
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 ratio. Malthus expresses it somewhat mathematically. He says:
 Title: Fundamental Social: Lecture 5: Understand OneAnother
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 necessity — a kind of mathematical necessity. So long
 Title: Mysteries of Light: Lecture II: The Development of Architecture
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 quality: hence the Gothic forms also produce an abstractmathematical
 Title: Karmic Relationships, VI: Lecture V
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 understood), Geography, Mathematics, Poetry, Chemistry, Medicine,
 Title: Karmic Relationships, VI: Lecture VII
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 heavens only in terms of mathematical and mechanical laws. But this
 nothing but lines must remain — mathematical lines!’
 Title: Man: Hieroglyph: Lecture One
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 mathematicalmechanical calculation, to which in recent times have
 physical, and even in the physical domain, mechanicalmathematical.
 true to say that this mechanicalmathematical method of observation
 mathematicalmechanical construction of the Universe reckon? It
 in the mechanicalmathematical way in which Galileo or Copernicus
 In mathematics it is a matter of indifference whether some object is a
 appears as a mathematical picture only, in which the universal origin
 Title: Man: Hieroglyph: Lecture Three
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 mathematical astronomy. That would necessitate beginning over again
 Title: Man: Hieroglyph: Lecture Five
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 There simply does not exist such a thing as mathematics calls
 mechanical or mathematical means, but only through a knowledge of the
 Title: Responsibility of Man: Lecture II
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 human being as modern science — from mathematics to
 Title: Materialism/Anthroposophy: Lecture V
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 mathematics, and mechanics and observes only with the
 Title: Materialism/Anthroposophy: Lecture X
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 all that developed as mathematics, as modern natural science
 Title: Materialism/Anthroposophy: Lecture XVI
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 attempts to grasp the matter outwardly through mathematical
 again but not as something new. The mathematical and
 Title: Therapeutic Insights: Lecture III
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 Title: Cosmosophy 1: Lecture V
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 mathematical, geometrical way, we find that we can think out
 Title: Cosmosophy 1: Lecture IX
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 shadows, making some mathematical calculations that proceed
 Title: Cosmosophy 1: Lecture X
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 person may be a great mathematician, may have myriad
 Title: Man's Being: Lecture III: Our Experiences at Night, Life after Death
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 miraculous unfolding of mathematicalmechanical, physical
 Title: Evolution of Consciousness: Lecture III: InitiationKnowledge  New and Old
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 doubt that mathematical truths are among the most difficult things. They
 spiritual world we find that our mathematics and geometry are no
 Title: Evolution of Consciousness: Lecture IV: Dream Life
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 a mathematical problem, for instance, such as we may strain every
 in relation to space and time we can grasp through mathematics. When
 nature and from the laws of mathematics — from the latter laws
 threedimensional space. It has its own mathematics, following
 Title: Evolution of Consciousness: Lecture VIII: During Sleep and after Death
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 this for mathematicians — all equations for the ether must be
 Title: Esoteric Studies: Easter: Lecture IV: Decline of the Mystery System and the Rise of Freedom, IAO is Man, Aristotle's Categories
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 no longer remain merely something to be mathematically and
 Title: Easter Festival: Lecture IV: Decline of the Mystery System and the Rise of Freedom, IAO is Man, Aristotle's Categories
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 longer seen merely as objects of mathematical and mechanical
 Title: Festival of Easter: Lecture 4: Decline of the Mystery System and the Rise of Freedom, IAO is Man, Aristotle's Categories
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 mathematical calculations but when they are letters in a
 Title: Occult Movement: Lecture Three: Materialism of the 19th Century
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 abnormal way. He was a distinguished mathematician,
 conversant with the whole range of higher mathematics at that
 “differential” in mathematics is the atomic, the
 Title: Occult Movement: Lecture Six: The Dangers of Aberation Along the Path into the Spiritual World
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 the man whom you see with us, as doctor, mathematician and
 mathematician and therefore obstinate, a clear mind and
 Title: Golden Blade, 1962: Lecture 2: Paths to the Spirit in East and West
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 have never really studied mathematics, or gone into the
 without the remotest understanding of mathematical thinking.
 least the spirit of mathematical thinking. Goethe's
 himself to any special training in mathematics. Many would deny
 Title: Lecture: Evolution of Human Freedom and Personal Consciousness
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 Immortality must be faced as a mathematical problem —
 Title: Lucifer and Ahriman: Lecture I: The Incarnation of Lucifer in Asia in the Third Millenium B.C.
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 the spread of the belief that the mechanistic, mathematical
 in the cosmos, not merely the mathematical, mechanistic laws
 only the mathematical aspect of astronomy. Therefore he
 Title: Lucifer and Ahriman: Lecture II: The Advance Preparation of Ahriman for His Future Incarnation
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 of friends bought themselves books on mathematics. A kind of
 way to library shelves and the mathematical knowledge was not
 Title: Influences of Lucifer/Ahriman: Lecture One: The Incarnation of Lucifer in Asia in the Third Millenium B.C.
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 belief that the mechanistic, mathematical conceptions inaugurated by
 discerned in the cosmos, not merely the mathematical, mechanistic
 mathematical aspect of astronomy. Therefore he tempts many people to
 Title: Influences of Lucifer/Ahriman: Lecture Two: The Advance Preparation of Ahriman for His Future Incarnation
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 themselves books on mathematics. A kind of “sporting
 shelves and the mathematical knowledge was not much in evidence!
 Title: Background/Mark: Lecture One: On the Investigation and Communication of Spiritual Truths
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 let us say, that of mathematics or botany or some other
 Title: Background/Mark: Lecture Five: The Two Main Streams of PostAtlantean Civilisation
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 for granted that in mathematics and science some grounding is
 Title: Esoteric Development: Lecture I: Inner Development
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 preconditions are truly necessary. Just as higher mathematics would
 Title: Esoteric Development: Lecture II: The Psychological Foundations of Anthroposophy
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 Mathematical forms may
 curve form into the other in accordance with mathematical
 law in so far as this can be expressed in mathematical concepts and
 mathematical forms is acquired within consciousness and is then
 mathematical concept when, on the one hand, this consciousness
 on the other, it visualizes this mathematical concept within
 itself in pure abstract mathematical thinking. But this signifies
 nothing else than that the ego, with its mathematical representation,
 is not outside the transcendental mathematical lawconformity
 transcendental. If, as regards mathematical thinking, one has
 Title: Esoteric Development: Lecture III: Supersensible Knowledge: Anthroposophy as a Demand of the Age
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 external scientist, the mathematician, for example, is
 thinking just as a mathematician surveys his operations when he
 as a mathematician must be with every step he takes. For just as the
 mathematician must see clearly into everything that he writes on the
 Title: Esoteric Development: Lecture IV: The Attainment of Spiritual Knowledge
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 working at mathematics I evolve thoughts purely out of myself.
 Mathematics in the real sense is a web which I spin out of my
 pure as a mathematical formula.
 Title: Esoteric Development: Lecture VII: The Great Initiates
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 highest thoughts of mathematics and geometry, that have an eternal
 Title: Esoteric Development: Lecture X: The Three Decisions on the Path of Imaginative Cognition
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 same time. Nor can ordinary calculation, physical mathematics, be
 Title: Mystery of Death: Lecture IV: The Intimate Element of the Central European Culture and the Central European Striving
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 mathematical formulae what Kepler, the German astronomer, had
 Title: Old/New Methods: Lecture Seven: Methods of Initiation, Old and New  2
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 or, over and above that, in mathematics. Here we really do restrict
 Title: Old/New Methods: Lecture Fourteen: The 5th PostAtlantean Period, the French Revolution, Schiller, Goethe, the Freedom Problem, or Berlin University Course Report  2
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 clever as requiring someone who solves mathematical problems to be
 mathematical problems. To such a person it is said: If you want to
 find out whether these mathematical problems have been solved
 Title: Social Forms: Lecture I: The Waldorf School, Spiritual Science, Outer World, Inner World
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 prouder still, think of them as mathematical fiction.
 small pellets, sources of energy, or mathematical fiction,
 Title: Social Forms: Lecture V: Forming Sound Judgment
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 mathematical certainty if one turns to the sources of true
 Title: Social Forms: Lecture IX: Hegel
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 firmness, having nothing mathematical about it, yet
 Title: Truths and Errors: Lecture II: How Does One Disprove Spiritual Science?
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 must take place according to the mathematically computable
 Title: Truths and Errors: Lecture III: How Does One Defend Spiritual Science?
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 such a way, as I have just characterised: mathematics. Already
 with the simplest mathematical operations, you can convince
 must get to the same result. Thus, it is with the mathematical
 takes place after the same principle as in mathematics that one
 different about a mathematical theorem, somebody who has
 one attains the mathematical truths in the deepest inside of
 mathematical knowledge — somebody may say —, but no
 mathematics can decide on whether beings really exist who carry
 these mathematical principles in themselves; one has to
 experience reality in other ways than with mathematical
 mathematical judgements what he experiences if he rises to a
 supersensible world. No mathematical judgement can give the
 as an object with mathematical judgements. The view of the own
 ego is essential. With the mathematical judgements, we remain
 of reality. This is the difference, the fact that mathematics
 Title: Truths and Errors: Lecture IV: Truths of Spiritual Research
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 The mathematician forms, at least theoretically, mental
 manifolds in the mathematical thinking. If anybody is familiar
 is familiar with mathematics at the same time, for that many
 Simony (18521915, mathematician, physicist) in Vienna.
 mathematicians I have to add that then the speculations of the
 better mathematics if one possibly wants to count in the things
 supersensible world, also mathematical ideas of infinity become
 real, in particular things of the border area of mathematics. I
 mathematical element should not be overestimated if it concerns
 the supersensible world. Mathematics is useful only formally;
 it cannot get to reality. However, mathematics can be
 applied to any other human being. Mathematics has this in
 Title: Truths and Errors: Lecture V: Errors of Spiritual Research  1
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 chemistry, mathematics, or any other science is something that
 Title: Truths and Errors: Lecture VI: Errors of Spiritual Research  2
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 studies to become a botanist, a zoologist or a mathematician.
 Title: Truths and Errors: Lecture VII: The Questions of Life and the Riddle of Death  1
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 medallion. Mathematics shows such symbols of regularities
 that he applies this mathematical method to a higher area not
 Title: Truths and Errors: Lecture IX: Spiritual Science and Natural Sciences  their Relationship to the Riddles of Life  1
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 investigates the mathematical laws, one comes to a certain
 Title: Truths and Errors: Lecture X: Spiritual Science and Natural Sciences  their Relationship to the Riddles of Life  2
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 real scourge of mathematical societies. Since they received a
 Title: Esoteric Christianity: The Gospel of St. John and Ancient Mysteries
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 Mathematics as elaborated by Leibnitz, Newton and others, such works
 Title: Lecture: The Four Human Group Souls (Lion, Bull, Eagle, Man)
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 Theosophical learning is not like the mathematical. First it was
 Title: The Theory of Categories / Kategorienlehre
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 construct other concepts than mathematical ones, and could
 only arrived at pure concepts in mathematics. Most men, of
 Title: Effects of Occult Development: Lecture IV
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 to mathematics, the evening hours to poetry. We are now living
 Title: Deeper Education: Lecture III: A Comprehensive Knowledge of Man as the Source of Imagination in the Teacher
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 teacher, a different kind of mathematics teacher. In
 Title: Man/Symphony: Lecture I: Man as Microcosm
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 Yes, the mathematics, the intellectual element, was the same in the
 Title: Man/Symphony: Lecture II: The Sun in Relation to the Outer Planets
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 important mathematician of our times, in response to the question:
 Title: Lecture: The Ahrimanic Deception
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 with mathematics, with the science of mechanics. What lies around the
 mechanism, in fact, only to be grasped by the aid of mathematical,
 mechanistic laws. With the help of these mathematical, mechanistic
 understood and calculated by mathematical formulae. This indeed is
 worldmathematics, mere worldschematism of a mechanical order, does not
 these illusions of a mathematical nature about the universe, we must
 instructing men in mathematics, but not in instructing them that
 mathematicalmechanistic concepts of the universe are merely illusions.
 Title: Lecture: The Concepts of Original Sin and Grace
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 inherited the principles of mathematics or other such concepts from
 Title: Jesuit and Rosicrucian Training
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 we no longer dispute over mathematics, because we
 have raised mathematics entirely into consciousness.
 Title: Prophecy  Its Nature and Meaning
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 gaze. In other words he made no special mathematical calculations but
 Title: Lecture: The Nature of Eternity
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 Title: Lecture: How Can We Gain Knowledge of the Supersensible Worlds?
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 mathematics. Consequently one must say that at the present time
 Title: Cosmic/Human Metamorphosis: Lecture 7. Errors and Truths.
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 best: Friederick Daniel Schubart; Hahn, the mathematician; Steinhofer;
 of Bengel, Ötinger, Steinhofer and the mathematician Hahn.
 Title: Brotherhood and the Fight for Survival
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 had to come, but as little as one can vote in mathematics to obtain a
 Title: Esoteric Lessons Part II: Berlin, 11711
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 the world mathematical thoughts are the most sensefree ones; but
 Title: Oswald Spengler: Lecture I: On Spengler's "Decline of the West"
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 mathematician the rapidity of our decline and comes out
 Title: Oswald Spengler: Lecture II: Oswald Spengler  I
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 thoughts. Aside from mathematical ones, thoughts always lead
 Title: Oswald Spengler: Article I: Spengler's "Perspectives of World History"
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 as transparent as those of mathematics, he has moved toward
 developed this mathematical picturethinking in the lifeless is
 Title: Education as a Social Problem: Lecture VI: The Inexpressible Name, Spirits of Space and Time, Conquering Egotism
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 knowledge of mathematics and geometry. Everything is
 mathematicalgeometrical the more untransparent do matters
 Title: Knowledge of Healing: Lecture II
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 knowledge, developed by man himself, chiefly in mathematics, knowledge,
 unconcerned attitude usually adopted towards mathematics. It is not
 Title: Egyptian Myths: Lecture 9
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 arose geometry, mathematics. With the help of this, man could rule the
 Title: Egyptian Myths: Lecture 11
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 colors in the aura, especially in those who have keen mathematical
 Title: Influence of Spiritual Beings Upon Man: Lecture IX
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 of mathematics alone, for these are the most trivial of all. No one
 can say that he finds mathematical truths through external experience;
 If mathematical truths were not so simple and obvious then the passions
 Title: Universe, Earth and Man: Lecture I: The Egyptian period, and the present time.
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 is not the absolute void of which our ordinary mathematicians and
 Title: Universe, Earth and Man: Lecture II: Ancient Wisdom and the new Apocalyptic Wisdom. Temple sleep. Isis and the Madonna. Past stages of Evolution. The bestowing of the Ego. Future Powers.
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 case of a person who occupies himself much with mathematical ideas
 state of his external health. It is not, of course, as if mathematical
 applying himself to mathematics and one who does not. Suppose two
 anything about mathematical ideas; the other is intensely interested
 nonmathematical person whereas it might be possible to heal the other
 difference between a man who likes mathematics and one who does not.
 count on their fingers, and then follow it with pure mathematical
 Title: Universe, Earth and Man: Lecture XI: The reversing of Egyptian remembrance into material forms by way of Arabism. The harmonizing of Egyptian remembrance. The Christian impulse of power in Rosicrucianism.
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 mathematics to him; it gave him the musical secrets of the world in
 the harmony of numbers. Mathematics, which is only something abstract
 Title: Jeshu ben Pandira: Lecture I
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 mathematical book, for instance. This will help us to fall asleep;
 chemist, the mathematician, runs its course without any sort of
 Title: Earthly/Cosmic Man: Lecture 1: Introductory Lecture. Winter Session, 19111912
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 and Western mathematics. But on account of intrinsic characteristics,
 Title: Earthly/Cosmic Man: Lecture 9: Formcreating Forces
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 individuality. Mathematics in itself is an illustration of this fact.
 no different from a teacher of mathematics, who quite obviously has
 his function. But mathematics are not accepted merely on the authority
 of the teacher of mathematics; every individual accepts mathematics
 an affair of humanity, like mathematics — not the affair of any
 Title: Presence of the Dead: Lecture One: Understanding the Spiritual World (Part One)
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 mathematics, not only as far as we know it today, but also everything
 read the mathematics contained there and read it consciously, we
 Title: Presence of the Dead: Lecture Seven: Robert Hamerling: Poet and Thinker
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 examination. He had good grades in Greek, Latin, and mathematics.
 Title: Building Stones: Lecture Three
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 millions of years hence may be mathematically correct, but
 Title: Challenge/Times: Lecture II: The Present from the Viewpoint of the Present
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 as mathematics. In connection with the things I have
 Title: Challenge/Times: Lecture VI: The Innate Capacities of the Nations of the World
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 mathematics, and I can, therefore, build a great railway
 of mathematics, of mechanics, of dynamics. Thus must a person
 Title: Roman Catholicism: Lecture I
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 free scientist. After having argued at length that even mathematics
 of science because one is bound by the truths of mathematics, he goes
 Title: Astronomy Course: Lecture I
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 Biology, of Mathematics, and so on. Special branches have
 quite estranged from life. They turn out mathematicians,
 they are concerned with Astronomy, in a mathematical and
 mathematics and mechanics. In observing the phenomena, men
 abstract mathematical science, or an abstract science of
 out that the mechanical and mathematical way of treating what
 fashion without understanding mathematics or mechanics, we
 mathematical and mechanical kind. No doubts of any kind exist
 ideas, i. e., to the mathematical, mechanical mode of
 precisely the mathematical, mechanical faculties for its
 phenomena in the picture formed by the mathematical,
 — namely, mathematical ideas — , and on the other
 Mathematics satisfies this requirement, at least in most
 in mathematics and mechanics men think they
 attitude towards a mathematical, mechanical
 only so much real science as there is mathematics in
 think how remote the simplest mathematical ideas are to
 nothing in the way of mathematical ideas.
 mathematical, mechanical web over them. This web of thought
 mathematics. We may talk to him perhaps of the need for a
 Maximum number of matches per file exceeded.
 Title: Astronomy Course: Lecture II
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 grasped through mathematics and mechanics, while
 dealing with them anything of the nature of mathematics or
 mathematicalmechanical is brought into some kind of relation
 development, might truly be expressed in mathematical and
 phenomena merely through the mathematics which we apply to
 Chaldeans were satisfied with a mathematical picture of the
 system, when employed purely mathematically, supplies the
 mathematics, can thus be foretold. One system is as well
 Universe is thus pictured in purely mathematical curves or in
 they had before. A mathematical description is thus obtained
 science because it is mathematical. The question arises: Why
 certainty through a purely mathematical approach? Only think
 that in considering a thing mathematically we lift the
 Mathematics is something that ascends from our inner
 being; in mathematics we lift ourselves out of external
 mathematical standpoint, man does not put reality into his
 phenomena than the purely mathematical one.
 mathematical astronomy we speak of the daily rotation of the
 Title: Astronomy Course: Lecture III
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 phenomena in their mathematical and geometrical aspect alone.
 be looked on from a purely spatial, mathematical point of
 we interpret rightly the mathematical pictures presented to
 the cosmic Mathematics indwells man. Man is himself in the
 midst of the living Mathematics. Of this we shall speak more
 Title: Astronomy Course: Lecture IV
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 development, to think of the planetary system mathematically
 in such a way that the mathematical results are no longer
 commensurable. We are at the place, where in the mathematical
 come to the incommensurable. The mathematicians among you
 mathematics reminds me of a scene in which I was once a
 sciences, Mathematics, to the celestial phenomena, and in the
 follow mathematically, actually does when it slides away into
 must take leave of what we are able to master mathematically.
 approximately by mathematical or at any rate geometrical
 Mathematics in Astronomy we come at one point into chaos. And
 no longer follow mathematically. In Embryology on the other
 the standpoint of Mathematics. It appeared that at one point
 the mathematician comes up to a limit — a limit he
 encounters too in purely formal Mathematics. Now there is
 application of mathematics to reality. How do we proceed? We
 develop Mathematics as a formal science and it appears to us
 see that Mathematics is only applicable to outer reality on
 to continue Mathematics beyond certain limits. First, certain
 from the mathematical process itself. They are in fact
 inductive laws, developed within Mathematics. They
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 Title: Astronomy Course: Lecture V
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 can no longer declare the mathematical method to be
 mathematical analysis; we can only admit that analysis and
 mathematical formulae. We do not contain the whole
 least similar to mathematics even though it be not pure
 mathematics. We therefore see, from the very history of
 mechanical, mathematical method of research. It is the
 mathematics and mechanics.
 the totality of the starry Heavens in the mathematical and
 Title: Astronomy Course: Lecture VI
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 mathematical domain. That we do not dispute the justification
 of a mathematical approach is implicit in the whole spirit of
 beyond which it is impossible to go with mathematical
 Title: Astronomy Course: Lecture VII
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 do, in quantitative treatment? Mathematics is the obvious
 than merely mathematical and phoronomical ideas. Approach the
 world with the premises of Phoronomy and Mathematics, and we
 mathematically ordered. We shall find other contents in the
 mathematical and phoronomical. Even in experimental research
 of our Euclidean Mathematics would be at most a kind of
 another, drawing logical and mathematical conclusions without
 mathematical calculations and mechanics. It will be lifted
 indeed a kind of qualitative Mathematics. How, then, shall we
 by no means easy to express in mathematical form.) In a
 modern Mathematics, but a form of space derived from the
 Title: Astronomy Course: Lecture VIII
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 mathematical and geometrical laws — they arrived at an
 the reason that the mathematical method itself is
 Title: Astronomy Course: Lecture IX
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 have to use the element of figure and form, the mathematical,
 permeated with mathematical thoughts. What Astronomy
 mathematics and to show you how, in a different domain from
 Mathematics itself this element of the incomprehensible, the
 found in any mathematical textbook relating to the subject.)
 where the mathematics itself confronts us with what I called
 mathematical equation, this is not two curves, but one; it is
 cannot remain in space. So you see, Mathematics itself
 of space. Even in Mathematics therefore we are confronted
 that most people who concern themselves with mathematics take
 on to what should accompany the mathematical formula in true
 treatment of Mathematics as subject matter for education any
 mathematicians present, Herr Blümel, Herr Baravalle, if
 understand it from the mathematical aspect, then, to anyone
 Mathematics, where — as in Synthetic Geometry —
 dear friends, when you apply Mathematics in one way or
 mathematically. You apply to the outer form what has been
 between a man thinking mathematically and the empirical
 in addition to the simple relation of mathematical man to
 Title: Astronomy Course: Lecture X
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 function of b, a = φ(b). Mathematically, it is of course
 mathematically speaking, in all its points. At every point
 first. In this way you can create, even in mathematics
 mathematics capable of dealing with realities; and it is
 Title: Astronomy Course: Lecture XI
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 mathematical Astronomy.
 kind of qualitative mathematics, learned from the human form
 constants is open on one side. The curve is mathematically
 mathematical standpoint or with the help of mathematical
 still farther. Ordinary mathematics, reckoning in the main on
 mathematics and geometry as such, by thinking of the
 possible, mathematically to penetrate what is organic. You
 mathematical understanding of organic Nature. This
 Title: Astronomy Course: Lecture XII
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 mathematical line, that tends to vanish as we prolong it.
 fresh extension of mathematics. You are led to recognise a
 distinction — a purely mathematical one — when
 continued grow shorter. It is a fully valid mathematical
 a genuine, mathematical idea.
 Title: Astronomy Course: Lecture XIII
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 been studying, he propounds peculiar mathematical formulae.
 mathematical lines. In mathematical lines he represents to
 mathematical pictures coincides in the same way with what is
 the Ptolemaic school, the mathematical imagination does not
 the mathematical thoughtpictures of the Ptolemaic school,
 mathematical lines he would comprise the logic of the
 The model he makes, the mathematical picture living in his
 Ptolemy the Mathematical picture is quite detached from what
 return to the former method, simply uniting by mathematical
 planet. These mathematical lines correspond to what was there
 mathematically, to empirical reality in general. You may
 of a priori mathematical reasoning and them applied it to the
 Heavens by mathematical lines, building it all into a system
 namely in Galileo and others, whose mathematical thinking is
 mathematical results, was able to construct his cosmic
 course of time. Then Ptolemy conceived a mathematical
 times, all they retained of this was the mathematical aspect,
 mathematical configuration there was at least some remnant of
 Title: Astronomy Course: Lecture XIV
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 mathematical or kindred figures what has in fact been
 worse to come, for the mathematician at least. In effect, we
 mathematician's bugbear — the “problem of three
 Moon. Though perhaps mathematically you cannot quite lay hold
 Title: Astronomy Course: Lecture XV
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 cosmic space with our accustomed mathematical method and way
 not be easy to take hold of mathematically. Yet without
 finding some mathematical way of access we shall never
 in mathematics generally. It cannot be objected that the
 outer senseperceptible reality you never find mathematical
 forward to begin with as a purely formal, mathematical
 Title: Astronomy Course: Lecture XVI
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 mathematics, — shall we make essential progress. Of
 mathematicians, to find the transition to a qualitative
 mathematics. Indeed this possibility is there in a high
 find your way into a qualitative form of mathematics.
 Title: Astronomy Course: Lecture XVII
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 We cannot not now bring all the mathematical paraphernalia to
 mathematics in general and the way we are applying this
 qualitative mathematic (which we have first had to develop)
 Title: Astronomy Course: Lecture XVIII
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 mathematical ideas, why should you not think of a certain
 conceive a truly mathematical relation — or at least, a
 relation analogous to mathematical — between the
 Astral. Thus you would find a mathematical way of approach to
 them, No matter what mathematical curve I may devise, once it
 only seem to have been working with mathematical curves and
 presented in seemingly mathematical and geometrical guise.
 analytical treatment of mathematics to the projective
 treatment — to a projective form of mathematics and
 Title: Young Doctors Course: Appendix: Evening Gathering with Young Medical People
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 mathematical or qualitative way is that one may suddenly ask
 Title: Young Doctors Course: Bridge Lecture 2: The Moral as the Source of WorldCreative Power
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 exactitude we envisage a machine in terms of mathematics and
 a picture built up purely on the basis of mathematical
 principles, mathematicalmechanical principles. The rudiments
 mathematical principles but upon ancient clairvoyant
 one another in accordance with mathematical laws. To be sure,
 Title: Young Doctors Course: Bridge Lecture 3: The Path to Freedom and Love and their Significance in World Happenings
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 thinking in its purest form — in mathematics.
 Mathematical thinking streams up entirely from our inner
 mathematics the makeup of external objects can, it is true,
 be analyzed and determined, but the mathematical thoughts in
 They say: mathematical concepts are a priori. — A
 independent of experience’. But why are mathematical
 inkling that we bring mathematics with us from our existence
 mathematics ‘a priori’. But the term ‘a
 Title: Origin and Destination of Humanity: Lecture III: Reincarnation and Karma
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 ancestors (Isaac N., 1642–1727, English physicist, mathematician
 Title: Origin and Destination of Humanity: Lecture IV: Theosophy and Darwin
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 mathematical and other laws, how should he be able to find the laws
 beaver, carries out. They are so artistic, of such a mathematical regularity
 Title: Origin and Destination of Humanity: Lecture XI: Origin and Goal of the Human Being
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 knows this thought only in one single field, in the field of mathematics,
 inside, intuitively. Nobody could obtain a mathematical theorem only
 the mathematical in space, you get to the great mystics who inform us
 about soul and spirit. Hence, the mystic also calls mysticism “mathematics”
 – mathesis , not because mysticism is mathematics, but because
 it is built up corresponding to the sample of mathematics.
 to the higher spheres like the thought in mathematics, then you experience
 Title: Origin and Destination of Humanity: Lecture XII: Goethe's Secret Revelation I
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 one does not feel any passion with the truths of mathematics, no quarrel
 mathematical knowledge. In Goethe's estate I found a notebook
 mathematical studies even during later years, even up to the highest
 Title: Origin and Destination of Humanity: Lecture XVI: The Great Initiates
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 of mathematics and geometry to which the human being dedicates himself
 Title: Origin and Destination of Humanity: Lecture XVIII: The Future of the Human Being
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 G., 1777–1855, mathematician) and Weber (Wilhelm Eduard W., 1804–1891,
 mathematical one, it is too constructivemathematical, and is not
 Title: Origin and Destination of Humanity: Lecture XX: The Divinity Faculty and Theosophy
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 physics, chemistry, botany, zoology, mathematics etcetera. Not before
 Title: Origin and Destination of Humanity: Lecture XXI: The Faculty of Law and Theosophy
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 service of a reform movement. Who has a mathematically trained thinking
 of mathematics and mechanics and of that what enables us from the start
 of thinking in all fields as it exists between mathematics, the differential
 one must have studied mathematics before one constructs a bridge. Theosophy
 The natural sciences still go back to mathematics, of course, go back
 must arise from a basic education which is as certain as the mathematical
 for this field: the Romans did not achieve a single mathematical theorem!
 A totally unmathematical and inexact way of thinking formed the basis
 as we are able to lead back the rough matters to mathematical concepts.
 one can be a lawyer, as well as one is a mathematician who has solved
 that mathematics has real significance only, since the thinking of the
 the social ethics if it is as clear as it is with mathematics which
 some concepts to a clear shape similar to mathematics. Everybody has
 as mathematics proceeds with the bridge building. Nevertheless, life
 Mathematics also does not
 to the legal life, that is positioned to life like mathematics to nature,
 However, the knowledge exists that there is a kind of mathematics, a
 Mathematics is nothing else
 than an internal experience. You can nowhere learn externally what mathematics
 is. There is no mathematical theorem which would not have resulted from
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 Title: Origin and Destination of Humanity: Lecture XXII: The Medical Faculty and Theosophy
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 Title: Origin and Destination of Humanity: Lecture XXIII: The Arts Faculty and Theosophy
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 in it. Why does one study botany, mathematics, history, for example?
 professor of theology and of mathematics in Vienna. Today, a professor
 would faint about that. However, we know that mathematical thinking
 in the way that he exercises some mathematics, learns to think quite
 what can he know about mathematics? Only mathematical concepts, having
 no inkling of the fact that mathematics introduces in the great principles
 of arguing. In the Middle Ages, mathematics was regarded as the basis
 is that the mechanics is the paradise of the mathematical sciences.
 He was an artist and mathematician at the same time. The physical education
 world the paradise of mathematics! Where he built bridges, thoughts
 of mathematics from bridge building. The gods want free beings, they
 should develop from the mathematical attitude into a spiritual direction;
 Title: Apocalypse of John: Introductory Lecture
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 related to the religious documents as mathematical
 instruction is related to the books on mathematics which have
 history versed in mathematics, namely, the geometry of
 scholastic form the mathematical and geometrical facts that
 been taught mathematics and geometry will value and
 Title: An Occult Physiology: Lecture 7: The Conscious Life of Man
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 that is either purely mathematical, or one which is equally objective
 Title: Lecture: Wonders of the World: Lecture 3
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 mathematicalmechanical laws. Such a nature — a nature which is
 calculated according to purely external mathematicalmechanical
 could not explain according to the strict laws of mathematics and
 as the facts which follow the physicalmathematicalchemical laws, but
 Title: Esoteric Lesson: Berlin, 1903 or 1904
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 way that our mathematical theory is related to ancient math books. One
 can understand mathematics out of one's own intellectual forces and
 mathematics as one tries to develop one's intellectual forces. The
 Title: Esoteric Lesson: Berlin, December 1904
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 way that our mathematical theory is related to ancient math books. One
 can understand mathematics out of one's own intellectual forces and
 mathematics as one tries to develop one's intellectual forces. The
 Title: Social Basis For Primary and Secondary Education: Lecture I
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 will to adopt the mathematical method of our spiritual science. This is
 experience by mathematical method appears to this author therefore to be
 mathematical clarity, perhaps indeed mathematical dryness, is what is
 Title: Social Basis For Primary and Secondary Education: Lecture III
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 convenience of the college of teachers, let us say mathematics,
 mathematical concepts, for example, and concepts of physics. Here we must
 but mathematics from morning to evening; I mean it in the sense of what I
 Title: Deeper Secrets: Lecture II
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 mathematical logic were both introduced through Abraham. Even until
 and measure, by mathematical logic — this human corporality was
 through mathematical logic and reasoning. But contact with the Egyptian
 was the faculty of mathematical logic — that is to say, they were
 reality in the form of a mathematical worldpicture — received
 through Moses, into the Hebrew people with their faculties of mathematical
 Title: Thomas Aquinas: Lecture II: The Essence of Thomism
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 that pure thought so runs with mathematical certainty from idea
 Title: Thomas Aquinas: Lecture III: Thomism in the Present Day
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 mathematics, but in the proof develop ourselves at the same
 gets in mathematics? And he gets such certainty in a manner
 Title: Search for the New Isis: Lecture I: A Christmas Lecture
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 it has become cold and gray mathematics and geometry. Today we see
 mathematics. This is the last remnant of that which in the
 mere mathematical and geometrical pictures, if we can raise
 mathematics again through inner experience to that glory which the
 Title: Search for the New Isis: Lecture II: The Quest for IsisSophia
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 with such mathematical precision, that its shadow disappeared in the
 mathematicalmechanical worldpicture of modern astronomy and of
 mathematical lines, then we see the grave of the world's spiritual
 into the cosmic spaces, which have become a mathematical abstraction,
 Title: Search for the New Isis: Lecture III: The Magi and the Shepherds: The New Isis
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 mechanics, of mathematics.
 into our modern mathematics and mechanics.
 mathematics and mechanics. The faculties of outer and inner
 the brain and becomes our mathematical, mechanistic world (green
 inwards and becomes the dry, geometricalmathematical, mechanistic
 mathematical, mechanistic world. It is only outstanding persons like
 deep imagination of this inner, mathematical world. This world of
 seems to us to be dry, mathematical, mechanistic.
 What today has faded into the mathematicalmechanistic
 faculty that is now dry mathematics may intensify pictorially to
 and their offspring are the mathematical faculties of today. The
 mathematical faculties of today will become those faculties which
 into mathematics, into geometry, into the science of mechanics. But
 from mathematics, phoronomy and geometry, this means the finding of
 Title: Search for the New Isis: Lecture IV
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 mathematics being fulfilled in the cosmos and to which every star was
 mathematics, abstract mechanics or phoronomy — the
 mathematicalmechanistic knowledge that arises from within us.
 mathematical view of the world we have the abstract legacies of
 spiritfilled vision of the heavens has become abstract mathematics
 Kepler, as if they were intelligible as a mere object of mathematics
 once again to unfold real visions.. The mathematical and mechanical
 true being. We have evolved mathematical conceptions and external
 mathematically because in the West — although the East has
 mathematical.
 mathematics’ which is none other than ancient astronomy or, if
 And what has remained of this? Mathematical lines and forces spread
 a network of mathematical and mechanical forces cropped up in the
 mathematical. Only so has the consciousness of freedom come to
 developed, and the mathematics and mechanics into which the old
 consciousness that otherwise occurs only in mathematics with its
 Title: Man as a Being: Lecture 1
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 Now let us consider mathematical truths, the truths of geometry. A
 mathematics out of his head, or from somewhere or other (ideas on the
 subject are not very precise). But it is not so. Mathematics derives
 you will get to know the sphere from which mathematics comes. It is
 depths that mathematical thought comes, depths to which we no longer
 mathematics lives at a deeper level than our ordinary soullife. And
 thus we see that mathematics is really rooted in that part of us which
 Title: Man as a Being: Lecture 2
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 the scientific spirit that tries to apply mathematics to everything.
 Mathematics, as I explained to you yesterday, comes from the senses of
 Title: Man as a Being: Lecture 3
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 example, he says, “Mathematical concepts are a
 evolution that most men, civilised men at any rate, bring mathematical
 process of awakening mathematical concepts and that of imparting such
 Title: Driving Force: Lecture VII
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 in mathematics because method alone has been followed, is
 Title: Cosmic Christianity: Lecture I
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 the arguments of the mathematicians, physicists and chemists; on the
 Title: Cosmic Christianity: Lecture V
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 a mathematical truth and point from certain phenomena in the present
 Title: Lecture: Reading the Pictures of the Apocalypse: Part 1: Lecture Three
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 Friday” just as little as a mathematician describes Pythagorian
 Title: Lecture: Reading the Pictures of the Apocalypse: Appendix: Cosmogony
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 develop itself on the field of the sense world through minute, mathematical
 Title: History of the Middle Ages: Lecture I: Celts, Teutons, and Slavs
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 Mathematics and Physics, so also we see everywhere in actual History
 Title: History of the Middle Ages: Lecture IV: Arabic Influence in Europe
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 we find all such sciences — medicine, chemistry, mathematical
 Title: History of the Middle Ages: Lecture VI: Culture of the Middle Ages
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 — such as Mathematics, Natural Science, etc. — were not
 Title: Impulse for Renewal: Lecture I: Anthroposophy and Natural Science
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 back to the application of mathematical methods because even
 where pure mathematics aren't applied, but thoughts steered
 developed, here the mathematical way of thinking forms a
 of mathematics. The Kantian saying claims that there is only as
 much certainty in a science as there is mathematics contained
 within it. Now obviously mathematics can be introduced into
 mathematical developments of concepts. However, what has been
 according to the pattern of mathematical conceptions. When
 mathematical function. For instance, if I say: X equals Y,
 I have in mathematics, when I created a concept of function. It
 mathematics has been brought into the sphere of science.’
 as a nonmathematician; he even called himself as such.
 However, if one places Goethe there as a nonmathematician,
 Goethe couldn't achieve much with mathematical details, that he
 was not particularly talented in his time to solve mathematical
 detailed mathematical examples, if it was more algebraic. That
 paradoxical as it might sound, more of a mathematician's brain
 than some mathematicians; because he had fine insight into
 mathematized nature, in the nature of building mathematical
 mathematician, when building concepts, scrutinizes everything
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 Title: Impulse for Renewal: Lecture II: The Human and the Animal Organisation
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 lectures I've just given, I spoke about the mathematical
 Mathematic and mechanistic laws
 have not been discovered by outer experience. Mathematic and
 we create mathematics and we believe we have a purely apriori
 science. However, mathematics is no pure apriori science. We
 balance is what we translate into mathematical geometric
 within itself than a person does mathematically. We find the
 mathematical simply as a result of us being placed in the
 in our earliest childhood reflects later in us as mathematics;
 visible as mathematics and mechanics. We often take mathematics
 Title: Impulse for Renewal: Lecture III: Anthroposophy and Philosophy
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 on an understanding of a mathematical nature. Time was short to
 Title: Anthroposophy/Civilization
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 mathematicians, and they have calculated that Uranus has a
 Title: Renewal of Culture
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 the universe and admiringly looks upon his mathematical conception of the
 such a way as to supply a mathematical physical picture of the universe.
 mathematicalmechanical element now appears to us in the cosmos, although in
 mathematicalmechanical cosmos, only the great machinery, as it were, when
 through mathematics and natural science — there is spirituality.
 Title: Philosophy, Cosmology and Religion: Lecture I: The Three Steps of Anthroposophy
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 exactly; as exact, for instance, as a mathematical problem
 appears to a mathematician, or the content of his experiment
 research. Thus, as the mathematician or natural scientist
 way that mathematics or natural science is exact. But I should
 and insight as is required of a mathematician confronted by a
 mathematical problem. This will then lead to a kind of
 Title: Philosophy, Cosmology and Religion: Lecture II: Soul Exercises in Thinking, Feeling, and Willing
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 the scientific thinking we utilize in mathematics or natural
 of mind as is only developed in mathematics or the exactness of
 Title: Philosophy, Cosmology and Religion: Lecture III: The Imaginative, Inspirative, and Intuitive Method of Cognition
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 one would a mathematical problem, in order that neither
 meditation. When we concentrate on a mathematical problem we
 mathematics, the meditative exercises and the exact
 Title: The Social Question: Lecture V: The Social Will as the Basis Towards a New, Scientific Procedure
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 Certainly mathematics doesn't have a state characteristic, but
 Title: The Social Question: Lecture VI: What Significance does Work have for the Modern Proletarian?
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 such a superstructure. Certainly chemistry or mathematics can't
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