| It is the merest truism, evident at once to unsophisticated observation, that mathematics is a human invention. Percy W. Bridgman; (1882-1961); The Logic of Modern Physics; 1927/1951; p60 Is geometry derived from experience? Careful discussion will give the answer- no! We therefore conclude that the principles of geometry are only conventions … Whence are the first principles of geometry derived? Are they imposed on us by logic? Lobatschevsky, by inventing non-Euclidean geometries, has shown that this is not the case. Henri Poincaré; (1854-1912); Science & Hypothesis; 1905/1952; pxxv … number is entirely the creature of the mind. George Berkeley; (1685-1753); Principles of Human Knowledge; 1710; s12  … mathematics … although it can be applied to an exterior world, neither in its origin nor in its methods depends on an exterior world.
L. E. J. Brouwer; (1881-1966); Brouwer’s Cambridge Lectures on Intuitionism; Dirk van Dalen, ed.; 1951/1981; p92  Mathematics is a human artefact, a human conception, in which there is no truth … there is no absolute unity; no absolute space and no absolute time, there is no mathematics.
Gerrit Mannoury; (1867-1956); Quoted in Mystic, Geometer, and Intuitionist: The Life of L. E. J. Brouwer; Dirk van Dalen; 1999; p121 |
Non-Euclidean geometry is proof that mathematics … is man’s own handiwork, subject only to the limitations imposed by the laws of thought. … [its] creation signalized the realization that mathematics in no sense depends upon our environment. E. Kasner; (1878-1955); & J. Newman; (1907-1966); Mathematics and the Imagination; 1940; p359,361 Mathematics is the language of physical science and certainly no more marvelous language was ever created by the mind of man. R. B. Lindsay; (1900-1985); On the Relation of Mathematics & Physics; 1963; p151 The scientist’s world
is perfectly mathematical,
but the sense world is not. Gordon Clark; (1902-1985); A Christian View of Men & Things; 1952/1981; p210 
Modern astronomers might agree with Kepler that the heavens declare the glory of God and the firmament showeth His handiwork; however, they now recognize that the mathematical interpretations of the works of God are their own creations … Morris Kline; (1908-1992); Mathematics and the Search for Knowledge; 1985; p85 All mathematics begins with a set of axioms. Any set of axioms is as valid as any other as long as it avoids contradictory assumptions. Billy E. Goetz; (1904-1986); The Usefulness of the Impossible; 1963; p189 … Science, and especially Mathematics, the ideal form of science, are creations of Intellect in its quest for Harmony. Cassius J. Keyser; (1862-1947); The Human Worth of Rigorous Thinking; 1913/1925; p23 … the number 2 … is a metaphysical entity about which we can never feel sure that it exists… Bertrand Russell; (1872-1970); Introduction to Mathematical Philosophy; 1919/1993; p18 … the mathematician … derives from the axioms only what he puts into them, since all conclusions that follow are logically implicit in the axioms. Morris Kline; (1908-1992); Mathematics: Method and Art; 1963; p165 
… numbers are free creations of the human mind … Richard Dedekind; (1831-1916); Essays on the Theory of Numbers; 1901/1963; p31 Nature does not count nor do integers occur in nature. Man made them all, integers and all the rest … Percy Williams Bridgman; (1882-1961); The Way Things Are; 1959; p100
… mathematics is something that has been created over time as a means of conceptualizing the natural world. We should not be surprised by its effectiveness at doing what it is designed to do. Why is it that so much of science can be explained mathematically? Because, so much of mathematics is speculative brainstorming.
The unreasonable effectiveness is an illusion. Paul Cox; What is Mathematics? Part 2  The mathematician is entirely free, within the limits of his imagination, to construct what worlds he pleases. … he is not thereby discovering the fundamental principles of the universe nor becoming acquainted with the ideas of God.
J. W. N. Sullivan; (1886-1937); Mathematics as an Art; 1963; p271 … we see that an object with identity is an abstraction corresponding exactly to nothing in nature. P. W. Bridgman; (1882-1961); The Logic of Modern Physics; 1927/1951; p35 π  | | “The most beautiful equation in mathematics.” Euler’s Identity 1 = -eiπ | | |
π No two real things are precisely equal. Billy E. Goetz; (1904-1986); The Usefulness of the Impossible; 1963; p188  … language imposes subjects and predicates on a world that does not have stable, enduring units corresponding to its terms.
F. Nietzsche; (1844-1900); Will to Power; cited in Truth in Philosophy; Barry Allen; 1993; p46 … no two objects are ever completely identical. Gottlob Frege; (1848-1925); The Foundations of Arithmetic; 1884/1999; p44 We are prepared to say that one and one are two, but not that Socrates and Plato are two … Bertrand Russell; (1872-1970); Introduction to Mathematical Philosophy; 1919/1993; p196 It is certain that all natural bodies, even those said to be of the same kind, differ from each other, that no two portions of gold are exactly alike, and that one drop of water is different from another drop of water. Nicolas Malebranche; (1638-1715); The Search After Truth; 1674/1997; p253  A typical statement of empirical arithmetic is that 2 objects plus 2 objects makes 4 objects. This statement acquires physical meaning only in terms of physical operations, and these operations must be performed in time. Now the penumbra gets into this situation through the concept of object. If the statement of arithmetic is to be an exact statement in the mathematical sense, the “object” must be a definite clear-cut thing, which preserves its identity in time with no penumbra. But this sort of thing is never experienced, and as far as we know does not correspond exactly to anything in experience.
P. W. Bridgman; (1882-1961); The Logic of Modern Physics; 1927/1951; p34 If we are to add at all, we must add unlikes, in violation of all mathematical regulations. Billy E. Goetz; (1904-1986); The Usefulness of the Impossible; 1963; p189 A simple arithmetic statement like “7+5=12” is true, not because it conforms to a set of empirical facts, but because it is a theorem of arithmetic which is deducible from certain prior theorems which in turn derive from the postulates, rules and basic concepts of that system. Joseph Gerard Brennan; The Meaning of Philosophy; 1953; p85 … we must concede that no material object is truly and simply one. St. Augustine; (354-430); De Libero Arbitrio; p45 You cannot step into the same river twice.
Heraclitus, 500 B.C. You cannot step into the same river
even once.
Cratylus, 400 B.C. Mathematics has been shorn of its truth; it is not an independent, secure, solidly grounded body of knowledge. Morris Kline; (1908-1992); Mathematics: The Loss of Certainty; 1980; p352 How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality? … In my opinion the answer to this question is briefly this:- As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality. Albert Einstein; (1879-1955); Sidelights on Relativity; Dover; 1983; p28 
The object of mathematical theories is not to reveal to us the real nature of things; that would be an unreasonable claim. Henri Poincaré; (1854-1912); Science & Hypothesis; 1902/1952; p211 
… the exact validity of mathematical laws as laws of nature is out of the question. L. E. J. Brouwer; (1881-1966); Intuitionism and Formalism; 1912 Truth to the mathematician merely means freedom from internal inconsistencies. Billy E. Goetz; (1904-1986); The Usefulness of the Impossible; 1963; p189 In fact, consistency, not truth, is the key word to mathematical thought. … The thought that the axioms underlying a mathematical system must be “obvious truths” slowly became a thing of the past. Carroll V. Newsom; (1904-1989); An Introduction to Modern Mathematical Thought; 1963; p75 In recent years consistency replaced truth as the god of mathematicians and now there is a likelihood that this god too may not exist. Morris Kline; (1908-1992); Mathematics: Method and Art; 1963; p161 … Mathematics is still the paradigm of the best knowledge available. Morris Kline; (1908-1992); Mathematics: The Loss of Certainty; 1980; p352 
Although there is at present still considerable disagreement about the ultimate foundations of mathematics, nobody can nowadays hold the opinion anymore that “arithmetical propositions” communicate any knowledge about the real world. … Their validity is that of mere tautologies; they are true because they assert nothing of any fact … Moritz Schlick; (1882-1936); Form and Content: An Introduction to Philosophical Thinking; in v2 Philosophical Papers; H. L. Mulder; ed.; 1979; p344 Thus the absence of all mention of particular things or properties in logic or pure mathematics is a necessary result of the fact that this study is, as we say, “purely formal.” Bertrand Russell; (1872-1970); Introduction to Mathematical Philosophy; 1919/1993; p198 Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.
Bertrand Russell; (1872-1970); Recent Work on the Principles of Mathematics; International Monthly; vIV; p84; 1901. In The Monist; v22
The critical mathematician has abandoned the search for truth. He no longer flatters himself that his propositions are or can be known to him or to any other human being to be true… Cassius J. Keyser; (1862-1947); The Human Worth of Rigorous Thinking; 1916; p221 In the words of the philosopher Wittgenstein, mathematics is just a grand tautology. Morris Kline; (1908-1992); Mathematics: Method and Art; 1963; p165 
The propositions of mathematics are devoid of all factual content; they convey no information whatever on any empirical subject matter. Carl G. Hempel; (1905-1997); On the Nature of Mathematical Truth; 1945 The propositions of mathematics are of exactly the same kind as the propositions of logic: they are tautologous, they say nothing at all about the objects we want to talk about. Hans Hahn; (1879-1934); Logic, Mathematics and Knowledge of Nature; 1933 … for a period of over two thousand years, mathematicians pursued truth. … Creations of the early 19th century, strange geometries & strange algebras, forced mathematicians, reluctantly and grudgingly, to realize that mathematics proper, and the mathematical laws of science were not truths. Morris Kline; (1908-1992); Mathematics: The Loss of Certainty; 1980; p3,4 
… arithmetic is a calculus which starts only from certain conventions but floats as freely as the solar system and
rests on nothing. Friedrich Waismann; (1896-1959);
Introduction to Mathematical Thinking; 1951; p121 The current predicament of mathematics is that there is not one but many mathematics … It is now apparent that the concept of a universally accepted, infallible body of reasoning– the majestic mathematics of 1800 and the pride of man– is a grand illusion. … one cannot speak of arithmetic as a body of truths that necessarily apply to physical phenomena. … Thus the sad conclusion which mathematicians were obliged to draw is that there is no truth in mathematics, that is, truth in the sense of laws about the real world. Morris Kline; (1908-1992); Mathematics: The Loss of Certainty; 1980; p6, 95  … we do not experience numbers as we experience colors and sounds, numbers are nothing in and by themselves…
Hans Hahn; (1879-1934); Empiricism, Logic and Mathematics; 1931/1980; p15 |
Gradually … mathematicians granted that the axioms and theorems of mathematics were not necessary truths about the physical world. … As far as the study of the physical world is concerned, mathematics offers nothing but theories or models. Morris Kline; (1908-1992); Mathematics: The Loss of Certainty; 1980; p97 The philosophical aims of the three schools [Intuitionism- Brouwer, Logicism- Frege, Formalism- Hilbert] have thus not been achieved, and it seems to us that we are no nearer to a complete understanding of mathematics than the founders of these schools. Andrzej Mostowski; (1913-1975); Thirty Years of Foundational Studies; 1966; p8 The fundamental concepts of mathematics are … empty space, empty time … points without extension, lines without breadth, surfaces without depth, spaces without content. All these concepts are contradictory fictions, mathematics being based upon an entirely imaginary foundation, indeed upon contradictions. Upon these foundations the psyche has constructed the entire edifice of this amazing science. Mathematicians have occasionally realized that they were dealing with contradictions, but seldom or never was this made the subject of any profound study. The frank acknowledgement of these fundamental contradictions has become absolutely essential for mathematical progress. The efforts made to conceal this fact have all worn threadbare. Hans Vaihinger; (1852-1933); The Philosophy of “As If”; 1924; p51 … all mathematics is a gigantic tussle with nonexistent impossibilities. Billy E. Goetz; (1904-1986); The Usefulness of the Impossible; 1963; p189 Geometry is not true, it is advantageous. Henri Poincaré; (1854-1912); Science & Method; 1908 … we can no more say that Einstein’s geometry is “truer” than Euclidean geometry, than we can say that the meter is a “truer” unit of length than the yard. Hans Reichenbach; (1891-1953); Philosophy of Space and Time; 1958; p35 A straight line has no width, no depth, no wiggles and no ends. There are no straight lines. We have ideas about these non-existent impossibilities; we even draw pictures of them. But they do not exist … A straight line hasn’t even a definition.  A point has no dimensions, no existence, and no definition. … Euclid lists twenty-three definitions which define more than twenty-three figments of the imagination. … He assumes all right angles are equal, although there are no right angles. … Lastly, Euclid introduces five “common notions” as axioms, that is, as self-evident truths, the very first of which is impossible, let alone true: “Things equal to the same thing are equal to each other.” No two things are precisely equal. …
The whole of geometry is consciously, willfully, deliberately antagonistic to reality.
Billy E. Goetz; (1904-1986); President of MIT 1958;
The Usefulness of the Impossible; 1963; p187ff |
… for geometry as a mathematical science, there is no problem concerning the truth of the axioms. This apparently unsolvable problem turns out to be a pseudo-problem. The axioms are not true or false, but arbitrary statements. Hans Reichenbach; (1891-1953); Philosophy of Space and Time; p5 … there is no reason to suppose that [a] triangle is a revelation of an eternally pre-existing truth– such as a thought in the mind of God. It is an arbitrary creation of the mathematician’s mind, and did not exist until the mathematician thought of it. J. W. N. Sullivan; (1886-1937); The Limitations of Science; 1933; p152 Geometry predicates nothing
about the relations of real things … Albert Einstein; (1879-1955); Sidelights on Relativity; Dover; 1983; p35 The theory contends that an innate property of the human mind, the ability of visualization, demands that we adhere to Euclidean geometry. In the same way as a certain self-evidence compels us to believe the laws of arithmetic, a visual self-evidence compels us to believe the validity of Euclidean geometry. It can be shown that this self-evidence is not based on logical grounds. Hans Reichenbach; (1891-1953); Philosophy of Space and Time; p32 It was thought till recently that geometry dealt with space and spatial points, and laymen probably thought that, even if they themselves did not succeed in grasping what a point of space was, mathematicians at least had a perfect grasp of it. But this was an enormous error: mathematicians had no better grasp of it, and they were no more capable of saying what a point really was than the nearest layman. Hans Hahn; (1879-1934); Empiricism, Logic and Mathematics; 1931/1980; p13 The ideas expressed in the preceding considerations attempted to establish Euclidean geometry as epistemologically a priori; we found that this a priori cannot be maintained and that Euclidean geometry is not an indispensable presupposition of knowledge. Hans Reichenbach; (1891-1953); Philosophy of Space and Time; p31 I shall not attempt to prove that mathematics is useful. I will admit it and so save myself the trouble that here is a great and respected discipline where all is impossible yet much is useful. The usefulness largely flows from the impossibility. Mathematical concepts have been simplified and generalized until they describe an imaginative world no part of which could possibly exist outside men’s minds. Billy E. Goetz; (1904-1986); The Usefulness of the Impossible; 1963; p189 Though the axioms of non-Euclidean geometry appeared to be contrary to ordinary human experience, they yielded theorems applicable to the physical world. Morris Kline; (1908-1992); Mathematics: Method and Art; p160 Every mathematical system contains undefined terms: for example, the words ‘point’ and ‘line’ in a geometric system. In deductive proof from explicitly stated axioms the meaning of the undefined terms is irrelevant. … pure mathematics is not … concerned with … meanings (of) undefined terms. … it is concerned with deductions that can be made from the axioms … Morris Kline; (1908-1992); Mathematics: Method and Art; p166 Mathematicians do not know what they are talking about because pure mathematics is not concerned with physical meaning. Mathematicians never know whether what they are saying is true because, as pure mathematicians, they make no effort to ascertain whether their theorems are true assertions about the physical world. Morris Kline; (1908-1992); Mathematics: Method and Art; p167 Thus, the analysis outlined on these pages exhibits the system of mathematics as a vast and ingenious conceptual structure without empirical content and yet an indispensable and powerful theoretical instrument for the scientific understanding and mastery of the world of our experience. Carl G. Hempel; (1905-1997); On the Nature of Mathematical Truth; 1945 We modify the mathematics when applications reveal misrepresentation or downright errors in the mathematics. Morris Kline; (1908-1992); Mathematics: The Loss of Certainty; p344 Freeman Dyson agrees: “we are probably not close yet to understanding the relation between the physical and the mathematical worlds.” … it is important to realize that nature and the mathematical representation of nature are not the same. The difference is not merely that mathematics is an idealization, the mathematical triangle is assuredly not a physical triangle. These … “explanations” … say rather little … in impressive language that tempers the admission that they have no answer to why mathematics is effective. Morris Kline; (1908-1992); Mathematics: The Loss of Certainty; p349  It is sometimes assumed that the effectiveness of mathematics … shows that mathematics itself exists in the structure of the physical universe. This, of course, is not a scientific argument with any empirical evidence.
George Lakoff; (1941-); Where Mathematics Comes From; 2000; p3 How then under this view can mathematics apply to the physical world and especially to physical phenomena? There are several answers. One is that mathematical axioms use undefined terms and these can be differently interpreted to suit the physical situation. Should we reject mathematics because we don’t understand its unreasonable effectiveness? … Should I refuse my dinner because I do not understand the process of digestion? … mathematics deals with the simplest concepts and phenomena of the physical world. It does not deal with man but with inanimate matter. Morris Kline; (1908-1992); Mathematics: The Loss of Certainty; p342, 350 The question of the existence of a Platonic mathematics cannot be addressed scientifically. At best, it can only be a matter of faith, much like faith in God. … The burden of scientific proof is on those who claim an external Platonic mathematics does exist … At present there is no known way to carry out such a scientific proof … as far as we can tell, there can be no such evidence, one way or the other. There is no way to tell empirically whether proofs proved by human mathematicians are objectively true, external to the existence of human beings or any other beings. George Lakoff; (1941-); Where Mathematics Comes From; 2000; p2, 342 Why then should the deductions still apply? Poincaré’s answer is that we modify the physical laws to make the mathematics fit. Morris Kline; (1908-1992); Mathematics: The Loss of Certainty; p343 Let us grant that the pursuit of mathematics is a divine madness of the human spirit. A. N. Whitehead; (1861-1947); Science and the Modern World; /1958; p22 CONCLUSION | No one has ever seen a number or a geometric shape. They are non-existent impossibilities. Number names are labels that we give to mental “objects” that in fact do not exist. The concepts that are represented by the symbols “0” and “1” on which all of mathematics is founded, do not correspond
to anything in creation. The symbol “0” as a place holder did not come into use until the middle ages. Numbers have been invented to try and organize our experiences; they do not have to exist to be useful, just as laws of science do not exist, yet they are useful for dealing with the material world. Claiming that numbers are concepts that are discovered is indefensible. Declaring that numbers exist in God’s mind, and that men are able to discover them there, is a completely unbiblical doctrine and contrary to plain biblical teaching about the nature of God. To assert that a sinful, rebellious creature is able to peer into God’s mind and extract new knowledge is deviant and unorthodox. ********************* When we calculate, we violate the mathematical regulations which we ourselves have stipulated,
i.e. we have no choice but to sin every time we do math. To ascribe to numbers a nature which belongs only to Truth and God, such as immutability or eternality, is to erect idols. Mathematics is a useful tool like mailboxes and shovels, but it can never lead to truth. ********************* Mathematics is man’s creation.
It is built upon axioms that were assumed to be “self-evident” truths.
There are no “self-evident” truths. The “creation” of numbers by man mimics God’s creation of the universe. The few propositions (axioms) on which all of mathematics is based are analogous to the few commands that God spoke to create the universe. The commands that God gave resulted in not only the creation of the universe,
but all of history itself. In the case of mathematics, these axioms can lead to an endless number of mathematical “discoveries.” |
| | Mathematics is the art of giving the same name
to different things. Henri Poincaré Poetry is the art of giving different names
to the same thing. Unknown author ******************** Out of the ground the LORD God formed every beast of the field and every bird of the air, and brought them to Adam to see what he would call them. And whatever Adam called each living creature, that was its name. So Adam gave names to all cattle, to the birds of the air, and to every beast of the field.
Genesis 2:19-20 | | |
 … the present state of affairs … is intolerable.
Just think, the definitions and deductive methods which everyone learns, teaches and uses in mathematics, the paragon of truth and certitude, lead to absurdities! If mathematical thinking is defective, where are we to find truth and certitude?
David Hilbert; (1862-1943); On the Infinite; 1925 Next >> 4- No Truth in Science ^^^ RETURN TO TOP ^^^
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