Mathematical proof

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

In mathematics, aproofis 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.

Quotes

• The great masters of modernanalysisareLagrange,Laplace,andGauss,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 andsyntheticalas possible.
  • W. W. Rouse Ball,History of Mathematics,(London, 1901), p. 463.

• 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 theVon Neumannproof, if you actually come to grips with it, falls apart in your hands! There is nothing to it. It's not just flawed, it'ssilly.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 butfoolish!
  • John Stewart Bell,Omni,May 1988, p. 88.

• NowGödel'sproof,Russell'soriginal 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 ofself-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 thetheory 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 anaxiom 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 attendedFrege'scourse,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-foundedsystemof 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 inFrege's Lectures on Logic: Carnap's Student Notes, 1910-1914(2004) ed., Tr. Erich H. Reck,Steve Awodey

• Pythagorasdid 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... hisnumerologicaldeductions....the Pythagorean theorem was known toThales....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 themetaphysicalimplications....this relation...was to Pythagoras and the Pythagoreans a basic law of nature, and... a brilliant confirmation of their number philosophy.
  • Tobias Dantzig,The Bequest of the Greeks(1955)

• Proof is the idol before whom the pure mathematician tortures himself.
  • Arthur Eddington,The Nature of the Physical World(1928)

• 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.
  • Gottlob Fregeas quoted byDagobert D. Runes,Readings in Epistemology, Theory of Knowledge and Dialectics(1962) p. 334.

• 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 myBegriffsschriftof 1879 and which I announced in myGrundlagen 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 byRalph H. Johnson,Manifest Rationality: A Pragmatic Theory of Argument(2012) p. 87.

• Gödel's incompleteness theoremsare 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 toGödel's achievement, as mathematically suspect.
  • Rebecca Goldstein,Incompleteness: The Proof and Paradox of Kurt Gödel(2005) pp. 23-24.

• The oldest definition ofAnalysisas opposed toSynthesisis that appended toEuclidXIII. 5. It was possibly framed byEudoxus.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 thePolymath Projectwas to find an elementary proof of a special case of the densityHales-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, calledSzemerédi'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 inThe Best Writing on Mathematics2010 (2011) p. 89.

• ThePythagoreansdiscovered the existence ofincommensurablelines, or ofirrationals.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 inAristotleand in a proposition interpolated in Euclid's Book X.; it is by areductio ad absurdumproving 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 tillEudoxusdiscovered the great theory of proportion (expounded in Euclid's Book V.), which is applicable to incommensurable as well as to commensurable magnitudes.
  • Thomas Little Heath,Achimedes(1920)

• When we are engaged in investigating the foundations of a science, we must set up a system ofaxiomswhich 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 inArchiv 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,Erdőscrisscrossed four continents at a frenzied pace, moving from one university or research center to the next. Hismodus operandiwas 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 oflimitsis 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 evenNewtonandLeibniz....
  Many contemporaries of Newton, among themMichel 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'Alemberthad 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 ofHeron,Nichomachus,andDiophantus,and ofArchimedesas far as his arithmetic is concerned, reads like the procedural texts of the Egyptians and Babylonians... The deductive, orderly proof ofEuclidandApollonius,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 thework 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,andRiemann,that theimpossibilityof deducing theparallel axiomfrom the others was demonstrated. This outcome was of the greatest intellectual importance....[I]t called attention... to the fact that aproofcan be given of theimpossibility of provingcertain propositions within a given system....Gödel'spaperis 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 thatEuclidwas 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 ofpure mathematiciansis toderive theorems from postulated assumptions,and that it is not their concern whether the axioms are actually true.
  • Ernest Nagel,James R. Newman,Godel's Proof(2001)

• In 1920 logic was mostly aphilosopher'sgarden.There were also a few mathematicians there, cultivating the logical roots of the mathematical tree. Today,RecursionTheory,Set Theory,Model Theoryand Proof Theory, logic's major subdisciplines, have become full-fledged branches of mathematics.
  • Anil Nerode,Richard Shore,Logic for Applications(1997)

• We could call it "proof fromnton+ 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 toinduction;this explains the name.
  • George Pólya,How to Sove It(1945)

• Those who have written the history of geometry have thus far carried the development of this science. Not much later than these isEuclid,who wrote the 'Elements,' arranged much ofEudoxus' work, completed much ofTheaetetus's and brought toirrefragableproof propositions which had been less strictly proved by his predecessors.
  • Proclus,"The Eudemian Summary," (ca. 335 BC) as quoted by James Gow,A Short History of Greek Mathematics(1884)

• 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....Heuristicarguments 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.
  • Gian-Carlo Rota,Indiscrete Thoughts(1997) p. 160, ed., F. Palombi.

• The use ofmathematical inductionin 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.Poincaréconsidered 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.
  • Bertrand Russell,Introduction to Mathematical Philosophy(1919) p. 27.

• More than any other of his predecessorsPlatoappreciated 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 byPlutarchin hisLife of Marcellus... "Plato's indignation at it and hisinvectionsagainst 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. "
  • David Eugene Smith,History of Mathematics(1925) Vol. 1.

• At the age of forty he was, for the first time, introduced to the works ofEuclid,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.Hobbestells 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 ofSir Henry Savile,Warden of Merton College, established the firstProfessorships of Geometry and Astronomyat Oxford.)
  • Alfred Edward Taylor,Thomas Hobbes(1908)

• One of the central concepts for the understanding of ancient Greek mathematics has customarily been, at least since the time ofPaul TanneryandHieronymus Georg Zeuthen,the concept of 'geometric algebra'. What it amounts to is that Greek mathematics, especially after the discovery of the 'irrational'... isalgebradressed up, primarily for the sake of rigor, in geometrical garb. The reasoning... the line of attack... the solutions...etc.all are essentiallyalgebraic... 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 possiblesui generisfeatures 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 anyhistoricallygratifying 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 ascientia 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 arecompletely 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 inClassics in the History of Greek Mathematics(2004) ed. Jean Christianidis, pp. 386-390.

• ThalesandPythagorastook their start from Babylonian mathematics but gave it a very different... specifically Greek character... in thePythagoreanschool and outside, mathematics was brought to... ever higher development and began gradually to satisfy the demands of stricter logic... through the work ofPlato's friendsTheaetetusandEudoxus,mathematics was brought to a state of perfection, beauty and exactness, which we admire in theelements of Euclid....the mathematical method of proof served as a prototype forPlato's dialectics and forAristotle's logic.
  • Bartel Leendert van der Waerden,Ontwakende wetenschap(1950) translated asScience Awakening(1954) byArnold Dresden.

• 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.
  • Bartel Leendert van der Waerden,Ontwakende wetenschap(1950) translated asScience Awakening(1954) byArnold Dresdenp.266.

• M. Poincaréfinds 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 inScienceVol. XX(Jul-Dec, 1904)

A History of Mathematics(1893)

byFlorian Cajori,source.

• The termssynthesisandanalysisare 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 inEuclid,XIII. 5, which in all probability was framed byEudoxus:"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 theElementsare [Euclid's] own discoveries. In fact, the proof of the "Theorem of Pythagoras" is the only one directly ascribed to him.

• Theanalytic methodis 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)

byMelvyn B. Nathanson,source: arXiv.org)arXiv:0905.3590[math.HO] (May 22, 2009). Also published inThe 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, thefundamental theorem of algebrastates that every polynomial of degreenwith complex coefficients has exactlyncomplex roots.D'Alembertpublished a proof in 1746, and the theorem became known as "D'Alembert's theorem," but the proof was wrong.Gausspublished 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.

• Erdőswas 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 ofGel'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... areWeil'sNumber Theory for Beginners... andArtin'sGalois 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.

See also

• History of mathematics
• Logic
• Mathematical induction
• Mathematics
• Mathematics education
• Proof

External links

• Proofs in Mathematicsby Alexander Bogomolny @cut-the-knot.org