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* MATHEMATICS IS A HUMAN INVENTION * Not only are there no two identical objects, no single unchangeable object exists in nature * THERE IS NO TRUTH IN MATHEMATICS * THERE IS No truth in Geometry * MATHEMATICS is Never true, YET it can be useful **********************************************
MATHEMATICS It is the merest truism, evident at once to unsophisticated observation, that mathematics is a human invention. Bridgman, P. W. - The Logic of Physics - 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. Poincaré, Henri - Science & Hypothesis - pxxv . . . number is entirely the creature of the mind. Berkeley, George - Principles of Human Knowledge- Sec. 12 Mathematics is the language of physical science and certainly no more marvelous language was ever created by the mind of man. Lindsay, R. B. - On the Relation of Math & Physics - p151 The scientist’s world is perfectly mathematical, but the sense world is not. Clark, Gordon - A Christian View of Men & Things - 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 - Mathematics and the Search for Knowledge - 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. Goetz, Billy E. - The Usefulness of the Impossible - p189 ... the number 2 . . . is a metaphysical entity about which we can never feel sure that it exists . . . Russell, Bertrand - Intro. to Mathematical Philosophy - p18 ... the mathematician ... derives from the axioms only what he puts into them, since all conclusions that follow are logically implicit in the axioms. Kline, Morris - Mathematics: Method and Art - p165 1. the number of Xs is > the number of Ys. Randall, J. H. Jr. - Philosophy: An Introduction - p62
Not only are there no two identical objects, no single unchangeable object exists ... we see that an object with identity is an abstraction corresponding exactly to nothing in nature. Bridgman, P. W. - The Logic of Modern Physics - p35 No two real things are precisely equal. Goetz, Billy E.- The Usefulness of the Impossible - p188 ... language imposes subjects and predicates on a world that does not have stable, enduring units corresponding to its terms. Nietzsche, F. - Will to Power - cited in Truth in Philosophy, Barry Allen, p 46. . . . no two objects are ever completely identical. Frege, Gottlob- The Foundations of Arithmetic- p44 We are prepared to say that one and one are two, but not that Socrates and Plato are two . . . Russell, Bertrand - Intro. to Mathematical Philosophy - 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. Malebranche, Nicholas - The Search After Truth - 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. Bridgman, P. W. - The Logic of Modern Physics - p34 If we are to add at all, we must add unlikes, in violation of all mathematical regulations. Goetz, Billy E. - The Usefulness of the Impossible - 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. Brennan, Joseph Gerard - The Meaning of Philosophy - p85 . . we must concede that no material object is truly and simply one. Augustine - De Libero - p45
You cannot step into the same river twice. You cannot step into the same river
THERE IS NO TRUTH Mathematics has been shorn of its truth; it is not an independent, secure, solidly grounded body of knowledge. Kline, Morris - Mathematics: The Loss of Certainty - 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; Sidelights on Relativity; p28; Dover; 1983; ISBN 0-486-24511-X The object of mathematical theories is not to reveal to us the real nature of things; that would be an unreasonable claim. Poincaré, Henri - Science & Hypothesis - p211 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. Newsom, Carroll V. - An Introduction to Modern Mathematical Thought - 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. Kline, Morris - Mathematics: Method and Art - p161 ... Mathematics is still the paradigm of the best knowledge available. Kline, Morris - Mathematics: The Loss of Certainty - p352 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.” Russell, Bertrand - Intro. to Mathematical Philosophy - p198 Truth to the mathematician merely means freedom from internal inconsistencies. Goetz, Billy E. - The Usefulness of the Impossible - p189 In the words of the philosopher Wittgenstein, mathematics is just a grand tautology. Kline, Morris - Mathematics: Method and Art -p165 The propositions of mathematics are devoid of all factual content; they convey no information whatever on any empirical subject matter. Hempel, Carl G. - On the Nature of Mathematical Truth ... 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. Kline, Morris - Mathematics: The Loss of Certainty - p3,4 ... arithmetic is a calculus which starts only from certain conventions but floats as freely as the solar system and rests on nothing. Waismann, Friedrich - Introduction to Mathematical Thinking - 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. Kline, Morris - Mathematics: The Loss of Certainty- p6 ... 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. Kline, Morris - Mathematics: The Loss of Certainty - p95 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. Kline, Morris - Mathematics: The Loss of Certainty - p97 ... all mathematics is a gigantic tussle with nonexistent impossibilities. Goetz, Billy E. - The Usefulness of the Impossible - p189
Geometry is not true, it is advantageous. Poincaré, Henri - Science & Method ... 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. Reichenbach, Hans - Philosophy of Space and Time - 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. Goetz, Billy E. - The Usefulness of the Impossible - p187 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. ... Goetz, Billy E. - The Usefulness of the Impossible - 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. Reichenbach, Hans - 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. Sullivan, J.W.N. - The Limitations of Science - p152 Geometry predicates nothing about the relations of real things ... Albert Einstein; Sidelights on Relativity; p35; Dover; 1983; ISBN 0-486-24511-X 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. Reichenbach, Hans- Philosophy of Space and Time - p32 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. Reichenbach, Hans - Philosophy of Space and Time - p31
MATHEMATICS is Never true, 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. Goetz, Billy E. - The Usefulness of the Impossible - p189 Though the axioms of non-Euclidean geometry appeared to be contrary to ordinary human experience, they yielded theorems applicable to the physical world. Kline, Morris - 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 ... Kline, Morris - 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. Kline, Morris - 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. Hempel, Carl G. - On the Nature of Mathematical Truth We modify the mathematics when applications reveal misrepresentation or downright errors in the mathematics. Kline, Morris - 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. Kline, Morris - Mathematics: The Loss of Certainty - p349 These ... “explanations” ... say rather little ... in impressive language that tempers the admission that they have no answer to why mathematics is effective. Kline, Morris - 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. Lakoff, George - Where Mathematics Comes From - p3 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. Kline, Morris - Mathematics: The Loss of Certainty - p350 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. Kline, Morris - Mathematics: The Loss of Certainty - p342 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. Lakoff, George - Where Mathematics Comes From - p2, 342 Why then should the deductions still apply? Poincaré’s answer is that we modify the physical laws to make the mathematics fit. Kline, Morris - 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. Kline, Morris - Mathematics: The Loss of Certainty - p354
Mathematics is the art of giving the same name to different things. Poincaré, Henri Poetry is the art of giving different names to the same thing.
The stunning parallel between the attempt at the mathematical representation of nature and the mystical use of numbers called Bible Numerics can be seen on Page 20.
Next >> 4- No Truth in Science
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, in Pie in the Sky, John D. Barrow, p112
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