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Mathematical proof

rigorous demonstration that a mathematical statement follows from its premises

In mathematics, a proof is an inferential argument for a mathematical statement. In the argument, other previously established statements, such as theorems, can be used. In principle, a proof can be traced back to self-evident or assumed statements, known as axioms, along with accepted rules of inference.

Papyrus Oxyrhynchus 29, one of the oldest surviving fragments of Euclid's Elements, used for millennia to teach proof-writing techniques.

Quotes

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  • The great masters of modern analysis are Lagrange, Laplace, and Gauss, who were contemporaries. It is interesting to note the marked contrast in their styles. Lagrange is perfect both in form and matter, he is careful to explain his procedure, and though his arguments are general they are easy to follow. Laplace on the other hand explains nothing, is indifferent to style, and, if satisfied that his results are correct, is content to leave them either with no proof or with a faulty one. Gauss is as exact and elegant as Lagrange, but even more difficult to follow than Laplace, for he removes every trace of the analysis by which he reached his results, and studies to give a proof which while rigorous shall be as concise and synthetical as possible.
  • The physicists didn't want to be bothered with the idea that maybe quantum theory is only provisional. A horn of plenty had been spilled before them, and every physicist could find something to apply quantum mechanics to. They were pleased to think that this great mathematician had shown it was so. Yet the Von Neumann proof, if you actually come to grips with it, falls apart in your hands! There is nothing to it. It's not just flawed, it's silly. If you look at the assumptions made, it does not hold up for a moment. It's the work of a mathematician, and he makes assumptions that have a mathematical symmetry to them. When you translate them into terms of physical disposition, they're nonsense. You may quote me on that: The proof of Von Neumann is not merely false but foolish!
  • Now Gödel's proof, Russell's original paradox, all these things, all stem from one common root which is inherent in all symbolic languages, including the language we use. ...the problem which dogs all formal systems, the problem of self-reference; that is, the language can be used to refer to sentences in the language. Indeed, between 1900 and 1910 Russell tried to forbid this, to say you cannot do mathematics if you can do that, and so he invented the theory of types. Of course, no sooner had he invented it than it turned up you could not do mathematics at all if you obeyed the theory of types. So then he had to put in an axiom of reducibility, which allows a certain amount of self-reference. And by this time everyone was pretty bored.
    • Jacob Bronowski, The Origins of Knowledge and Imagination (1978) pp. 81-82.
  • In the summer of 1914 I attended Frege's course, Logik in der Mathematik. Here he examined critically some of the customary conceptions and formulations in mathematics. He deplored the fact that mathematicians did not even seem to aim at the construction of a unified, well-founded system of mathematics, and therefore showed a lack of interest in foundations. He pointed out a certain looseness in the customary formulation of axioms, definitions, and proofs, even in the works of the more prominent mathematicians. ...Unfortunately, his admonitions go unheeded even today.
    • Rudolf Carnap, "Intellectual Autobiography" (1963) pp.4-6, as quoted in Frege's Lectures on Logic: Carnap's Student Notes, 1910-1914 (2004) ed., Tr. Erich H. Reck, Steve Awodey
  • Pythagoras did not possess a proof of the theorem which bears his name... he was temperamentally uninterested in proofs of this nature, as may be gleaned from... his numerological deductions. ...the Pythagorean theorem was known to Thales. ...the hypotenuse theorem is a direct consequence of the principle of similitude, and... Thales was fully conversant with the theory of similar triangles. On the other hand, there is no doubt that Pythagoras fully appreciated the metaphysical implications. ...this relation ...was to Pythagoras and the Pythagoreans a basic law of nature, and... a brilliant confirmation of their number philosophy.
  • Proof is the idol before whom the pure mathematician tortures himself.
  • Another roof, another proof.
    • Paul Erdős, A Tribute to Paul Erdős (1990) ed. Alan Baker, Béla Bollobás, A. Hajnal, Preface, p. ix. His motto, as he roamed about the world, a guest of other mathematicians.
  • Paul Erdős, although an atheist, spoke of an imaginary book, in which God has written down all the most beautiful mathematical proofs. When Erdős wanted to express particular appreciation of a proof, he would exclaim "This one's from the Book!". This viewpoint expresses... that mathematics, as the... foundation on which the laws of our universe are built, is a natural candidate for what has been personified as God...
    • John Francis, Philosophy of Mathematics (2008), p. 51.
  • It really is worth the trouble to invent a new symbol if we can thus remove not a few logical difficulties and ensure the rigour of the proofs. But many mathematicians seem to have so little feeling for logical purity and accuracy that they will use a word to mean three or four different things, sooner than make the frightful decision to invent a new word.
  • The ideal of strictly scientific method in mathematics which I have tried to realise here, and which perhaps might be named after Euclid I should like to describe in the following way... The novelty of this book does not lie in the content of the theorems but in the development of the proofs and the foundations on which they are based... With this book I accomplish an object which I had in view in my Begriffsschrift of 1879 and which I announced in my Grundlagen der Arithmetik. I am here trying to prove the opinion on the concept of number that I expressed in the book last mentioned.
    • Gottlob Frege, Grundgesetze der Arithmetik, Vol. 1 (1893) pp. 137-140, as quoted by Ralph H. Johnson, Manifest Rationality: A Pragmatic Theory of Argument (2012) p. 87.
  • Gödel's incompleteness theorems are expressed using no numbers or other symbolic formalisms. Though the nitty-gritty details of the proof are formidably technical, the proof's overall strategy, delightfully, is not. ...They belong to a branch of mathematics known as formal logic or mathematical logic, a field which was viewed, prior to Gödel's achievement, as mathematically suspect.
    • Rebecca Goldstein, Incompleteness: The Proof and Paradox of Kurt Gödel (2005) pp. 23-24.
  • The oldest definition of Analysis as opposed to Synthesis is that appended to Euclid XIII. 5. It was possibly framed by Eudoxus. It states that "Analysis is the obtaining of the thing sought by assuming it and so reasoning up to an admitted truth: synthesis is the obtaining of the thing sought by reasoning up to the inference and proof of it." In other words, the synthetic proof proceeds by shewing that certain admitted truths involve the proposed new truth: the analytic proof proceeds by shewing that the proposed new truth involves certain admitted truths.
    • James Gow, A Short History of Greek Mathematics (1884) p. 177.
  • The specific aim of the Polymath Project was to find an elementary proof of a special case of the density Hales-Jewett theorem (DHJ)... The theorem was known to be true, but for mathematicians, proofs are more than guarantees of truth: they are valued for their explanatory power, and a new proof of a theorem can provide crucial insights. There were two reasons to want a new proof... First... [it] had just one—a long and complicated proof that relied on heavy mathematical machinery. An elementary proof—one that starts from first principles instead of relying on advanced techniques—would require many new ideas. Second, DHJ implies another famous theorem, called Szemerédi's theorem, novel proofs of which led to several breakthroughs over the past decade, so there was reason to expect that the same would happen...
    • Timothy Gowers, Michael Nielsen, "Massively Collaborative Mathematics", Nature (Oct. 15, 2009) Vol. 461, pp. 879-881. Also published in The Best Writing on Mathematics 2010 (2011) p. 89.
  • The Pythagoreans discovered the existence of incommensurable lines, or of irrationals. This was, doubtless, first discovered with reference to the diagonal of a square which is incommensurable with the side, being in the ratio to it of √2 to 1. The Pythagorean proof of this particular case survives in Aristotle and in a proposition interpolated in Euclid's Book X.; it is by a reductio ad absurdum proving that, if the diagonal is commensurable with the side, the same number must be both odd and even. This discovery of the incommensurable... showed that the theory of proportion invented by Pythagoras was not of universal application and therefore that propositions proved by means of it were not really established. ...The fatal flaw thus revealed in the body of geometry was not removed till Eudoxus discovered the great theory of proportion (expounded in Euclid's Book V.), which is applicable to incommensurable as well as to commensurable magnitudes.
  • When we are engaged in investigating the foundations of a science, we must set up a system of axioms which contains an exact and complete description of the relations subsisting between the elementary ideas of that science. The axioms so set up are at the same time the definitions of those elementary ideas; and no statement within the realm of the science... is held to be correct unless it can be derived from axioms by means of a finite number of logical steps. Upon closer consideration the question arises: Whether, in any way, certain statements of single axioms depend upon one another, and whether the axioms may not therefore contain certain parts in common, which must be isolated if one wishes to arrive at a system of axioms that shall be altogether independent of one another.
    But above all, I wish to designate the following as the most important among numerous questions which can be asked with respect to the axioms: To prove that they are not contradictory, that is, that a finite number of logical steps based upon them can never lead to contradictory results.
    • David Hilbert, 'Mathematische Probleme', Gŏttinger Nachrichten (1900) pp. 253-297; and in Archiv der Mathematik und Physik, 3rd series, Vol. 1 (1901) pp. 44-63 & pp. 213-237; Tr. Mary Frances Winston Newson, 'Mathematical Problems', Bulletin of the American Mathematical Society (1902) Vol. 8 (Oct. 1901-July 1902), pp. 437-479.
  • In a never-ending search for good mathematical problems and fresh mathematical talent, Erdős crisscrossed four continents at a frenzied pace, moving from one university or research center to the next. His modus operandi was to show up on the doorstep of a fellow mathematician, declare, "My brain is open," work with his host for a day or two, until he was bored or his host was run down, and then move on to another home. ...He did mathematics in more than 25 different countries, completing important proofs in remote places and sometimes publishing them in equally obscure journals.
    • Paul Hoffman, The Man Who Loved Only Numbers: The Story of Paul Erdős and the Search for Mathematical Truth (1998) p. 6.
  • The difficulty in presenting a rigorous as well as clear statement of the theory of limits is inherent in the subject. ...If the reader has found some difficulty in grasping it he may be less discouraged when he is told that it eluded even Newton and Leibniz. ...
    Many contemporaries of Newton, among them Michel Rolle... taught that the calculus was a collection of ingenious fallacies. Colin Maclaurin... decided that he could found calculus properly... The book was undoubtedly profound but also unintelligible. One hundred years after the time of Newton and Leibniz, Joseph Louis Lagrange... still believed that the calculus was unsound and gave correct results only because errors were offsetting each other. He, too, formulated his own foundation... but it was incorrect. ...D'Alembert had to advise students of the calculus... faith would eventually come to them. This is not bad advice... but it is no substitute for rigor and proof. ...
    About a century and a half after the creation of calculus... Augustin Louis Cauchy... finally gave a definitive formulation of the limit concept that removed doubts as to the soundness of the subject.
    • Morris Kline, Mathematics and the Physical World (1959) Ch. 22: The Differential Calculus pp.382-383.
  • Another feature of Alexandrian algebra is the absence of any explicit deductive structure. The various types of numbers... were not defined. Nor was there any axiomatic basis on which a deductive structure could be erected. The work of Heron, Nichomachus, and Diophantus, and of Archimedes as far as his arithmetic is concerned, reads like the procedural texts of the Egyptians and Babylonians... The deductive, orderly proof of Euclid and Apollonius, and of Archimedes' geometry is gone. The problems are inductive in spirit, in that they show methods for concrete problems that presumably apply to general classes whose extent is not specified. In view of the fact that as a consequence of the work of the classical Greeks, mathematical results were supposed to be derived deductively from an explicit axiomatic basis, the emergence of an independent arithmetic and algebra with no logical structure of its own raised what became one of the great problems of the history of mathematics. This approach to arithmetic and algebra is the clearest indication of the Egyptian and Babylonian influences... Though the Alexandrian Greek algebraists did not seem to be concerned about this deficiency... it did trouble deeply the European mathematicians.
    • Morris Kline, Mathematical Thought from Ancient to Modern Times (1972) p.144
  • It was not until the nineteenth century, chiefly through... Gauss, Bolyai, Lobachevsky, and Riemann, that the impossibility of deducing the parallel axiom from the others was demonstrated. This outcome was of the greatest intellectual importance. ...[I]t called attention... to the fact that a proof can be given of the impossibility of proving certain propositions within a given system. ...Gödel's paper is a proof of the impossibilty of formally demonstrating certain important propositions in number theory. ...[T]he resolution of that parallel axiom question forced the realization that Euclid was not the last word on the subject of geometry, since new systems of geometry can be constructed... incompatible with those adopted by Euclid. ...[I]t gradually became clear that the proper business of pure mathematicians is to derive theorems from postulated assumptions, and that it is not their concern whether the axioms are actually true.
  • We could call it "proof from n to n + 1" or still simpler "passage to the next integer." Unfortunately, the accepted technical term is "mathematical induction." This name results from a random circumstance. ...Now, in many cases... the assertion is found experimentally, and so the proof appears as a mathematical complement to induction; this explains the name.
  • Those who have written the history of geometry have thus far carried the development of this science. Not much later than these is Euclid, who wrote the 'Elements,' arranged much of Eudoxus' work, completed much of Theaetetus's and brought to irrefragable proof propositions which had been less strictly proved by his predecessors.
  • Proofs are for the mathematician what experimental procedures are for the experimental scientist: in studying them one learns of new ideas, new concepts, new strategies—devices which can be assimilated for one's own research and be further developed.
    • Yehuda Rav, "Why do we prove theorems?" Philosophia Mathematica (Feb. 1999) Vol. 7, Issue 1, pp.5-41.
  • A proof of a mathematical theorem is a sequence of steps which leads to the desired conclusion. The rules to be followed... were made explicit when logic was formalized early in the this century... These rules can be used to disprove a putative proof by spotting logical errors; they cannot, however, be used to find the missing proof of a... conjecture. ...Heuristic arguments are a common occurrence in the practice of mathematics. However... The role of heuristic arguments has not been acknowledged in the philosophy of mathematics despite the crucial role they play in mathematical discovery. ...Our purpose is to bring out some of the features of mathematical thinking which are concealed beneath the apparent mechanics of proof.
  • The use of mathematical induction in demonstrations was, in the past, something of a mystery. There seemed no reasonable doubt that it was a valid method of proof, but no one quite knew why it was valid. Some believed it to be really a case of induction, in the sense in which that word is used in logic. Poincaré considered it to be a principle of the utmost importance, by means of which an infinite number of syllogisms could be condensed into one argument. We now know that all such views are mistaken, and that mathematical induction is a definition, not a principle.
  • More than any other of his predecessors Plato appreciated the scientific possibilities of geometry... By his teaching he laid the foundation of the science, insisting upon accurate definitions, and logical proof. His opposition to the materialists, who saw in geometry only what was immediately useful to the artisan and the mechanic, is made clear by Plutarch in his Life of Marcellus... "Plato's indignation at it and his invections against it as the mere corruption and annihilation of the one good geometry, which was thus shamefully turning its back upon the unembodied objects of pure intelligence."
  • At the age of forty he was, for the first time, introduced to the works of Euclid, and at once 'fell in love with geometry,' being attracted, he says, more by the rigorous manner of proof employed than by the matter of the science. (Mathematics, we must remember, were then only beginning to be seriously studied in England. Hobbes tells us that in his undergraduate days geometry was still looked upon generally as a form of the 'Black Art,' and it was not until 1619 that the will of Sir Henry Savile, Warden of Merton College, established the first Professorships of Geometry and Astronomy at Oxford.)
  • One of the central concepts for the understanding of ancient Greek mathematics has customarily been, at least since the time of Paul Tannery and Hieronymus Georg Zeuthen, the concept of 'geometric algebra'. What it amounts to is that Greek mathematics, especially after the discovery of the 'irrational'... is algebra dressed up, primarily for the sake of rigor, in geometrical garb. The reasoning... the line of attack... the solutions... etc. all are essentially algebraic... attired in geometrical accouterments. We... look for the algebraic 'subtext'... of any geometrical proof... always to transcribe... any proposition in[to] the symbolic language of modern algebra... [making] the logical structure of the proof clear and convincing, without thereby losing anything, not only in generality but also in any possible sui generis features of the ancient way of doing things. ...[i.e., that] there is nothing unique and (ontologically) idiosyncratic concerning the way... ancient Greek mathematicians went about their proofs, which might be lost...
    I cannot find any historically gratifying basis for this generally accepted view... those who have been writing the history of mathematics... have typically been mathematicians... largely unable to relinquish and discard their laboriously acquired mathematical competence when dealing with periods in history during which such competence is historically irrelevant and... anachronistic. Such... stems from the unstated assumption that mathematics is a scientia universalis, an algebra of thought containing universal ways of inference, everlasting structures, and timeless, ideal patterns of investigation which can be identified throughout the history of civilized man and which are completely independent of the form in which they happen to appear at a particular junction of time.
    • Sabetai Unguru, On the Need to Rewrite the History of Greek Mathematics, Archive for History of Exact Sciences, Vol. 15, No. 1, 30.XII (1975) pp. 67-114, as quoted, with Introduction in Classics in the History of Greek Mathematics (2004) ed. Jean Christianidis, pp. 386-390.
  • Thales and Pythagoras took their start from Babylonian mathematics but gave it a very different... specifically Greek character... in the Pythagorean school and outside, mathematics was brought to... ever higher development and began gradually to satisfy the demands of stricter logic... through the work of Plato's friends Theaetetus and Eudoxus, mathematics was brought to a state of perfection, beauty and exactness, which we admire in the elements of Euclid. ...the mathematical method of proof served as a prototype for Plato's dialectics and for Aristotle's logic.
  • An oral tradition makes it possible to indicate the line segments with the fingers; one can emphasize essentials and point out how the proof was found. All of this disappears in the written formulation... as soon as some external cause brought about an interruption in the oral tradition, and only books remained, it became very difficult to assimilate the work of the great predursors, and next to impossible to pass beyond it.
  • M. Poincaré finds the answer to these questions in the so-called 'mathematical induction' which proceeds from the particular to the more general, but at the same time does so by steps of the highest degree of certitude. In this process he sees the creative force of mathematics, which leads to real proofs and not mere sterile verifications. ...No arithmetic could be built up, rejecting the axiom of mathematical induction, as the non-Euclidean geometries have been built up, rejecting the postulate of Euclid.
    • J. W. A. Young, "Poincare's Science and Hypothesis" (Dec. 16, 1904) a book review in Science Vol. XX (Jul-Dec, 1904)

A History of Mathematics (1893)

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by Florian Cajori, source.
  • The terms synthesis and analysis are used in mathematics in a more special sense than in logic. In ancient mathematics they had a different meaning from what they now have. The oldest definition of mathematical analysis as opposed to synthesis is that given in Euclid, XIII. 5, which in all probability was framed by Eudoxus: "Analysis is the obtaining of the thing sought by assuming it and so reasoning up to an admitted truth; synthesis is the obtaining of the thing sought by reasoning up to the inference and proof of it."
  • Comparatively few of the propositions and proofs in the Elements are [Euclid's] own discoveries. In fact, the proof of the "Theorem of Pythagoras" is the only one directly ascribed to him.
  • The analytic method is not conclusive, unless all operations involved in it are known to be reversible. To remove all doubt, the Greeks, as a rule added to the analytic process a synthetic one, consisting of a reversion of all operations occurring in the analysis. Thus the aim of analysis was to aid in the discovery of synthetic proofs or solutions.

"Desperately Seeking Mathematical Proof" (2009)

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by Melvyn B. Nathanson, source: arXiv.org) arXiv:0905.3590 [math.HO] (May 22, 2009). Also published in The Best Writing on Mathematics: 2010 (2011) pp. 13-17.
  • The history of mathematics is full of philosophically and ethically troubling reports about bad proofs of theorems. For example, the fundamental theorem of algebra states that every polynomial of degree n with complex coefficients has exactly n complex roots. D'Alembert published a proof in 1746, and the theorem became known as "D'Alembert's theorem," but the proof was wrong. Gauss published his first proof... in 1799, but this, too, had gaps. Gauss's subsequent proofs, in 1816 and 1849, were okay. It seems to have been difficult to determine if a proof... was correct. Why?
  • Proofs have gaps and are... inherently incomplete and sometimes wrong. ...There is another reason ...Humans err. ...and others do not necessarily notice our mistakes. ...This suggests an important reason why "more elementary" proofs are better... The more elementary... the easier it is to check, and the more reliable its verification.
  • Erdős was a genius at finding brilliantly simple proofs of deep results, but, until recently, very much of his work was ignored...
  • Social pressure often hides mistakes in proofs. In a seminar lecture... interrupt the speaker... to ask for more explanation... [often] the response will be that it is "obvious" or "clear" or "follows easily..." Occasionally... a look... conveys the message that the questioner is an idiot. That's why most mathematicians sit quietly... understanding very little... and applauding politely... One of the joys of Gel'fand's seminar... he would constantly interrupt... to ask questions and give elementary examples... [T]he audience would actually learn some mathematics.
  • There are... masterpieces of... exposition... Two examples... are Weil's Number Theory for Beginners... and Artin's Galois Theory. Mathematics can be done scrupulously.
  • Perhaps we should discard the myth that mathematics is a rigorously deductive enterprise... hand-waving is intrinsic. We try to minimize it and we can sometimes escape it, but not always, if we want to discover new theorems.

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