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Foundations of mathematics are the
The term "foundations of mathematics" was not coined before the end of the 19th century, although foundations were first established by the ancient
These foundations were tactily assumed to be definitive until the introduction of
.During the 19th century, progress was made towards elaborating precise definitions of the basic concepts of infinitesimal calculus, notably the
The resolution of this crisis involved the rise of a new mathematical discipline called
.It results from this that the basic mathematical concepts, such as
Ancient Greece
Most civilisations developed some mathematics, mainly for practical purposes, such as counting (merchants),
The Pythagorean school of mathematics originally insisted that the only numbers are natural numbers and ratios of natural numbers. The discovery (around 5th century BC) that the ratio of the diagonal of a square to its side is not the ratio of two natural numbers was a shock to them which they only reluctantly accepted. A testimony of this is the modern terminology of irrational number for referring to a number that is not the quotient of two integers, since "irrational" means originally "not reasonable" or "not accessible with reason".
The fact that length ratios are not represented by rational numbers was resolved by Eudoxus of Cnidus (408–355 BC), a student of Plato, who reduced the comparison of two irrational ratios to comparisons of integer multiples of the magnitudes involved. His method anticipated that of Dedekind cuts in the modern definition of real numbers by Richard Dedekind (1831–1916);^{[2]} see Eudoxus of Cnidus § Eudoxus' proportions.
In the
Aristotle's logic reached its high point with
Before infinitesimal calculus
During Middle Ages, Euclid's Elements stood as a perfectly solid foundation for mathematics, and philosophy of mathematics concentrated on the ontological status of mathematical concepts; the question was whether they exist independently of perception (realism) or within the mind only (conceptualism); or even whether they are simply names of collection of individual objects (nominalism).
In Elements, the only numbers that are considered are
Nevertheless, this did not challenge the classical foundations of mathematics since all properties of numbers that were used can be deduced from their geometrical definition.
In 1637,
Infinitesimal calculus
This needed the introduction of new concepts such as continuous functions, derivatives and limits. For dealing with these concepts in a logical way, they were defined in terms of infinitesimals that are hypothetical numbers that are infinitely close to zero. The strong implications of infinitesimal calculus on foundations of mathematics is illustrated by a pamphlet of the Protestant philosopher George Berkeley (1685–1753), who wrote "[Infinitesimals] are neither finite quantities, nor quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities?".^{[3]}
Also, a lack of rigor has been frequently invoked, because infinitesimals and the associated concepts were not formally defined (
Despite its lack of firm logical foundations, infinitesimal calculus was quickly adopted by mathematicians, and validated by its numerous applications; in particular the fact that the planet trajectories can be deduced from the
19th century
In the 19th century, mathematics developed quickly in many directions. Several of the problems that were considered led to questions on the foundations of mathematics. Frequently, the proposed solutions led to further questions that were often simultaneously of philosophical and mathematical nature. All these questions led, at the end of the 19th century and the beginning of the 20th century, to debates which have been called the foundational crisis of mathematics. The following subsections describe the main such foundational problems revealed during the 19th century.
Real analysis
The modern
Karl Weierstrass (1815–1897) formalized and popularized the (ε, δ)definition of limits, and discovered some pathological functions that seemed paradoxical at this time, such as continuous, nowheredifferentiable functions. Indeed, such functions contradict previous conceptions of a function as a rule for computation or a smooth graph.
At this point, the program of arithmetization of analysis (reduction of mathematical analysis to arithmetic and algebraic operations) advocated by Weierstrass was essentially completed, except for two points.
Firstly, a formal definition of real numbers were still lacking. Indeed, beginning with
Several problems were left open by these definitions, which contributed to the
The third problem is more subtle: and is related to the foundations of logic: classical logic is a
NonEuclidean geometries
Before the 19th century, there were many failed attempts to derive the parallel postulate from other axioms of geometry. In an attempt to prove that its negation leads to a contradiction, Johann Heinrich Lambert (1728–1777) started to build hyperbolic geometry and introduced the hyperbolic functions and computed the area of a hyperbolic triangle (where the sum of angles is less than 180°).
Continuing the construction of this new geometry, several mathematicians proved independently that if it is
Later in the 19th century, the German mathematician
These proofs of unprovability of the parallel postulate lead to several philosophical problems, the main one being that before this discovery, the parallel postulate and all its consequences were considered as true. So, the nonEuclidean geometries challenged the concept of
Synthetic vs. analytic geometry
Since the introduction of
Mathematicians did not worry much about the contradiction between these two approaches before the midnineteenth century, where there was "an acrimonious controversy between the proponents of synthetic and analytic methods in projective geometry, the two sides accusing each other of mixing projective and metric concepts".^{[6]} Indeed, there is no concept of distance in a projective space, and the crossratio, which is a number, is a basic concept of synthetic projective geometry.
Apparently, the problem of the equivalence between analytic and synthetic approach was completely solved only with
Natural numbers
The work of
Giuseppe Peano provided in 1888 a complete axiomatisation based on the ordinal property of the natural numbers. The last Peano's axiom is the only one that induces logical difficulties, as it begin with either "if S is a set then" or "if is a
This was not well understood at that times, but the fact that infinity occurred in the definition of the natural numbers was a problem for many mathematicians of this time. For example, Henri Poincaré stated that axioms can only be demonstrated in their finite application, and concluded that it is "the power of the mind" which allows conceiving of the indefinite repetition of the same act.^{[7]} This applies in particular to the use of the last Peano axiom for showing that the successor function generates all natural numbers. Also, Leopold Kronecker said "God made the integers, all else is the work of man".^{[a]} This may be interpreted as "the integers cannot be mathematically defined".
Infinite sets
Before the second half of the 19th century,
Sets, and more specially infinite sets were not considered as a mathematical concept; in particular, there was no fixed term for them. A dramatic change arose with the work of Georg Cantor who was the first mathematician to systematically study infinite sets. In particular, he introduced cardinal numbers that measure the size of infinite sets, and ordinal numbers that, roughly speaking, allow one to continue to count after having reach infinity. One of his major results is the discovery that there are strictly more real numbers than natural numbers (the cardinal of the continuum of the real numbers is greater than that of the natural numbers).
These results were rejected by many mathematicians and philosophers, and led to debates that are a part of the foundational crisis of mathematics.
The crisis was amplified with the
With the introduction of the
Mathematical logic
In 1847,
Independently, in the 1870's,
Frege pointed out three desired properties of a logical theory:^{[}citation needed]consistency (impossibility of proving contradictory statements), completeness (any statement is either provable or refutable; that is, its negation is provable), and decidability (there is a decision procedure to test every statement).
By near the turn of the century,
Foundational crisis
The foundational crisis of mathematics arose at the end of the 19th century and the beginning of the 20th century with the discovery of several paradoxes or counterintuitive results.
The first one was the proof that the
Several schools of philosophy of mathematics were challenged with these problems in the 20th century, and are described below.
These problems were also studied by mathematicians, and this led to establish
This led to unexpected results, such as
Zermelo–Fraenkel set theory with the axiom of choice (ZFC) is a logical theory established by Ernst Zermelo and Abraham Fraenkel. It became the standard foundation of modern mathematics, and, unless the contrary is explicitly specified, it is used in all modern mathematical texts, generally implicitly.
Simultaneously, the
In summary, the foundational crisis is essentially resolved, and this opens new philosophical problems. In particular, it cannot be proved that the new foundation (ZFC) is not selfcontradictory. It is a general consensus that, if this would happen, the problem could be solved by a mild modification of ZFC.
Philosophical views
When the foundational crisis arose, there was much debate among mathematicians and logicians about what should be done for restoring confidence in mathematics. This involved philosophical questions about
For the problem of foundations, there was two main options for trying to avoid paradoxes. The first one led to
Formalism
It has been claimed^{[}by whom?] that formalists, such as David Hilbert (1862–1943), hold that mathematics is only a language and a series of games. Hilbert insisted that formalism, called "formula game" by him, is a fundamental part of mathematics, but that mathematics must not be reduced to formalism. Indeed, he used the words "formula game" in his 1927 response to L. E. J. Brouwer's criticisms:
And to what extent has the formula game thus made possible been successful? This formula game enables us to express the entire thoughtcontent of the science of mathematics in a uniform manner and develop it in such a way that, at the same time, the interconnections between the individual propositions and facts become clear ... The formula game that Brouwer so deprecates has, besides its mathematical value, an important general philosophical significance. For this formula game is carried out according to certain definite rules, in which the technique of our thinking is expressed. These rules form a closed system that can be discovered and definitively stated.^{[10]}
Thus Hilbert is insisting that mathematics is not an arbitrary game with arbitrary rules; rather it must agree with how our thinking, and then our speaking and writing, proceeds.^{[10]}
We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules. Rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise.^{[11]}
The foundational philosophy of formalism, as exemplified by
Merely the use of formalism alone does not explain several issues: why we should use the axioms we do and not some others, why we should employ the logical rules we do and not some others, why "true" mathematical statements (e.g., the laws of arithmetic) appear to be true, and so on. Hermann Weyl posed these very questions to Hilbert:
What "truth" or objectivity can be ascribed to this theoretic construction of the world, which presses far beyond the given, is a profound philosophical problem. It is closely connected with the further question: what impels us to take as a basis precisely the particular axiom system developed by Hilbert? Consistency is indeed a necessary but not a sufficient condition. For the time being we probably cannot answer this question ...^{[12]}
In some cases these questions may be sufficiently answered through the study of formal theories, in disciplines such as
Intuitionism
Intuitionists, such as L. E. J. Brouwer (1882–1966), hold that mathematics is a creation of the human mind. Numbers, like fairy tale characters, are merely mental entities, which would not exist if there were never any human minds to think about them.
The foundational philosophy of
Some modern
Logicism
Logicism is a school of thought, and research programme, in the philosophy of mathematics, based on the thesis that mathematics is an extension of logic or that some or all mathematics may be derived in a suitable formal system whose axioms and rules of inference are 'logical' in nature. Bertrand Russell and Alfred North Whitehead championed this theory initiated by Gottlob Frege and influenced by Richard Dedekind.
Settheoretic Platonism
Many researchers in
Several set theorists followed this approach and actively searched for axioms that may be considered as true for heuristic reasons and that would decide the continuum hypothesis. Many large cardinal axioms were studied, but the hypothesis always remained independent from them and it is now considered unlikely that CH can be resolved by a new large cardinal axiom. Other types of axioms were considered, but none of them has reached consensus on the continuum hypothesis yet. Recent work by Hamkins proposes a more flexible alternative: a settheoretic multiverse allowing free passage between settheoretic universes that satisfy the continuum hypothesis and other universes that do not.
Indispensability argument for realism
This
... quantification over mathematical entities is indispensable for science ... therefore we should accept such quantification; but this commits us to accepting the existence of the mathematical entities in question.
However, Putnam was not a Platonist.
Roughandready realism
Few mathematicians are typically concerned on a daily, working basis over logicism, formalism or any other philosophical position. Instead, their primary concern is that the mathematical enterprise as a whole always remains productive. Typically, they see this as ensured by remaining openminded, practical and busy; as potentially threatened by becoming overlyideological, fanatically reductionistic or lazy.
Such a view has also been expressed by some wellknown physicists.
For example, the Physics Nobel Prize laureate Richard Feynman said
People say to me, "Are you looking for the ultimate laws of physics?" No, I'm not ... If it turns out there is a simple ultimate law which explains everything, so be it – that would be very nice to discover. If it turns out it's like an onion with millions of layers ... then that's the way it is. But either way there's Nature and she's going to come out the way She is. So therefore when we go to investigate we shouldn't predecide what it is we're looking for only to find out more about it.^{[13]}
And Steven Weinberg:^{[14]}
The insights of philosophers have occasionally benefited physicists, but generally in a negative fashion – by protecting them from the preconceptions of other philosophers. ... without some guidance from our preconceptions one could do nothing at all. It is just that philosophical principles have not generally provided us with the right preconceptions.
Weinberg believed that any undecidability in mathematics, such as the continuum hypothesis, could be potentially resolved despite the incompleteness theorem, by finding suitable further axioms to add to set theory.
Philosophical consequences of Gödel's completeness theorem
Gödel's completeness theorem establishes an equivalence in firstorder logic between the formal provability of a formula and its truth in all possible models. Precisely, for any consistent firstorder theory it gives an "explicit construction" of a model described by the theory; this model will be countable if the language of the theory is countable. However this "explicit construction" is not algorithmic. It is based on an iterative process of completion of the theory, where each step of the iteration consists in adding a formula to the axioms if it keeps the theory consistent; but this consistency question is only semidecidable (an algorithm is available to find any contradiction but if there is none this consistency fact can remain unprovable).
More paradoxes
The following lists some notable results in metamathematics. Zermelo–Fraenkel set theory is the most widely studied axiomatization of set theory. It is abbreviated ZFC when it includes the axiom of choice and ZF when the axiom of choice is excluded.
 1920: downward Löwenheim–Skolem theorem, leading to Skolem's paradoxdiscussed in 1922, namely the existence of countable models of ZF, making infinite cardinalities a relative property.
 1922: Proof by Abraham Fraenkel that the axiom of choice cannot be proved from the axioms of Zermelo set theory with urelements.
 1931: Publication of Gödel's incompleteness theorems, showing that essential aspects of Hilbert's program could not be attained. It showed how to construct, for any sufficiently powerful and consistent recursively axiomatizable system – such as necessary to axiomatize the elementary theory of arithmetic on the (infinite) set of natural numbers – a statement that formally expresses its own unprovability, which he then proved equivalent to the claim of consistency of the theory; so that (assuming the consistency as true), the system is not powerful enough for proving its own consistency, let alone that a simpler system could do the job. It thus became clear that the notion of mathematical truth cannot be completely determined and reduced to a purely formal system as envisaged in Hilbert's program. This dealt a final blow to the heart of Hilbert's program, the hope that consistency could be established by finitistic means (it was never made clear exactly what axioms were the "finitistic" ones, but whatever axiomatic system was being referred to, it was a 'weaker' system than the system whose consistency it was supposed to prove).
 1936: Alfred Tarski proved his truth undefinability theorem.
 1936: Alan Turing proved that a general algorithm to solve the halting problem for all possible programinput pairs cannot exist.
 1938: Gödel proved the consistency of the axiom of choice and of the generalized continuum hypothesis.
 1936–1937: completeness theorem).
 1955: Pyotr Novikov showed that there exists a finitely presented group G such that the word problem for G is undecidable.
 1963: Paul Cohen showed that the Continuum Hypothesis is unprovable from ZFC. Cohen's proof developed the method of forcing, which is now an important tool for establishing independenceresults in set theory.
 1964: Inspired by the fundamental randomness in physics, Gregory Chaitin starts publishing results on algorithmic information theory (measuring incompleteness and randomness in mathematics).^{[15]}
 1966: Paul Cohen showed that the axiom of choice is unprovable in ZF even without urelements.
 1970: Hilbert's tenth problem is proven unsolvable: there is no recursive solution to decide whether a Diophantine equation (multivariable polynomial equation) has a solution in integers.
 1971: Suslin's problem is proven to be independent from ZFC.
Toward resolution of the crisis
Starting in 1935, the Bourbaki group of French mathematicians started publishing a series of books to formalize many areas of mathematics on the new foundation of set theory.
The intuitionistic school did not attract many adherents, and it was not until
One may consider that Hilbert's program has been partially completed, so that the crisis is essentially resolved, satisfying ourselves with lower requirements than Hilbert's original ambitions. His ambitions were expressed in a time when nothing was clear: it was not clear whether mathematics could have a rigorous foundation at all.
There are many possible variants of set theory, which differ in consistency strength, where stronger versions (postulating higher types of infinities) contain formal proofs of the consistency of weaker versions, but none contains a formal proof of its own consistency. Thus the only thing we do not have is a formal proof of consistency of whatever version of set theory we may prefer, such as ZF.
In practice, most mathematicians either do not work from axiomatic systems, or if they do, do not doubt the consistency of
The development of category theory in the middle of the 20th century showed the usefulness of set theories guaranteeing the existence of larger classes than does ZFC, such as Von Neumann–Bernays–Gödel set theory or Tarski–Grothendieck set theory, albeit that in very many cases the use of large cardinal axioms or Grothendieck universes is formally eliminable.
One goal of the
See also
 Aristotelian realist philosophy of mathematics
 Mathematical logic
 Brouwer–Hilbert controversy
 Church–Turing thesis
 Controversy over Cantor's theory
 Epistemology
 Euclid's Elements
 Hilbert's problems
 Implementation of mathematics in set theory
 Liar paradox
 New Foundations
 Philosophy of mathematics
 Principia Mathematica
 Quasiempiricism in mathematics
 Mathematical thought of Charles Peirce
Notes
 ^ The English translation is from Gray. In a footnote, Gray attributes the German quote to: "Weber 1891–1892, 19, quoting from a lecture of Kronecker's of 1886."^{[8]}^{[9]}
 ^ Joachim Lambek (2007), "Foundations of mathematics", Encyc. Britannica
 ISBN 0486600890.
 ^ The Analyst, A Discourse Addressed to an Infidel Mathematician
 ISBN 9780883855690 pp. 5–13. Also available at: http://www.maa.org/pubs/Calc_articles/ma002.pdf
 ^ O'Connor, John J.; Robertson, Edmund F. (October 2005), "The real numbers: Stevin to Hilbert", MacTutor History of Mathematics Archive, University of St Andrews
 ISBN 3764350482
 ^ Poincaré, Henri (1905) [1902]. "On the nature of mathematical reasoning". La Science et l'hypothèse [Science and Hypothesis]. Translated by Greenstreet, William John. VI.
 ISBN 9781400829040. Archivedfrom the original on 29 March 2017 – via Google Books.
 ^ Weber, Heinrich L. (1891–1892). "Kronecker". Jahresbericht der Deutschen MathematikerVereinigung [Annual report of the German Mathematicians Association]. pp. 2:5–23. (The quote is on p. 19). Archived from the original on 9 August 2018; "access to Jahresbericht der Deutschen MathematikerVereinigung". Archived from the original on 20 August 2017.
 ^ ^{a} ^{b} Hilbert 1927 The Foundations of Mathematics in van Heijenoort 1967:475
 ^ p. 14 in Hilbert, D. (1919–20), Natur und Mathematisches Erkennen: Vorlesungen, gehalten 1919–1920 in Göttingen. Nach der Ausarbeitung von Paul Bernays (Edited and with an English introduction by David E. Rowe), Basel, Birkhauser (1992).
 ^ Weyl 1927 Comments on Hilbert's second lecture on the foundations of mathematics in van Heijenoort 1967:484. Although Weyl the intuitionist believed that "Hilbert's view" would ultimately prevail, this would come with a significant loss to philosophy: "I see in this a decisive defeat of the philosophical attitude of pure phenomenology, which thus proves to be insufficient for the understanding of creative science even in the area of cognition that is most primal and most readily open to evidence – mathematics" (ibid).
 ^ Richard Feynman, The Pleasure of Finding Things Out p. 23
 ^ Steven Weinberg, chapter Against Philosophy wrote, in Dreams of a final theory
 PMID 16502614, archived from the original(PDF) on 20160304, retrieved 20160222
References
 Avigad, Jeremy (2003) Number theory and elementary arithmetic, Philosophia Mathematica Vol. 11, pp. 257–284
 ISBN 048669609X(pbk.) cf §9.5 Philosophies of Mathematics pp. 266–271. Eves lists the three with short descriptions prefaced by a brief introduction.
 Goodman, N.D. (1979), "Mathematics as an Objective Science", in Tymoczko (ed., 1986).
 Hart, W.D.(ed., 1996), The Philosophy of Mathematics, Oxford University Press, Oxford, UK.
 Hersh, R. (1979), "Some Proposals for Reviving the Philosophy of Mathematics", in (Tymoczko 1986).
 Hilbert, D. (1922), "Neubegründung der Mathematik. Erste Mitteilung", Hamburger Mathematische Seminarabhandlungen 1, 157–177. Translated, "The New Grounding of Mathematics. First Report", in (Mancosu 1998).
 Katz, Robert (1964), Axiomatic Analysis, D. C. Heath and Company.
 ISBN 0720421039.
 In Chapter III A Critique of Mathematic Reasoning, §11. The paradoxes, Kleene discusses Formalismin depth. Throughout the rest of the book he treats, and compares, both Formalist (classical) and Intuitionist logics with an emphasis on the former. Extraordinary writing by an extraordinary mathematician.
 Mancosu, P. (ed., 1998), From Hilbert to Brouwer. The Debate on the Foundations of Mathematics in the 1920s, Oxford University Press, Oxford, UK.
 Putnam, Hilary (1967), "Mathematics Without Foundations", Journal of Philosophy 64/1, 5–22. Reprinted, pp. 168–184 in W.D. Hart (ed., 1996).
 —, "What is Mathematical Truth?", in Tymoczko (ed., 1986).
 Sudac, Olivier (Apr 2001). "The prime number theorem is PRAprovable". Theoretical Computer Science. 257 (1–2): 185–239. .
 Troelstra, A. S. (no date but later than 1990), "A History of Constructivism in the 20th Century", A detailed survey for specialists: §1 Introduction, §2 Finitism & §2.2 Actualism, §3 Predicativism and SemiIntuitionism, §4 Brouwerian Intuitionism, §5 Intuitionistic Logic and Arithmetic, §6 Intuitionistic Analysis and Stronger Theories, §7 Constructive Recursive Mathematics, §8 Bishop's Constructivism, §9 Concluding Remarks. Approximately 80 references.
 Tymoczko, T. (1986), "Challenging Foundations", in Tymoczko (ed., 1986).
 —,(ed., 1986), New Directions in the Philosophy of Mathematics, 1986. Revised edition, 1998.
 van Dalen D. (2008), "Brouwer, Luitzen Egbertus Jan (1881–1966)", in Biografisch Woordenboek van Nederland. URL:http://www.inghist.nl/Onderzoek/Projecten/BWN/lemmata/bwn2/brouwerle [20080313]
 Weyl, H. (1921), "Über die neue Grundlagenkrise der Mathematik", Mathematische Zeitschrift 10, 39–79. Translated, "On the New Foundational Crisis of Mathematics", in (Mancosu 1998).
 Wilder, Raymond L. (1952), Introduction to the Foundations of Mathematics, John Wiley and Sons, New York, NY.
External links
 Media related to Foundations of mathematics at Wikimedia Commons
 "Philosophy of mathematics". Internet Encyclopedia of Philosophy.
 Logic and Mathematics
 Foundations of Mathematics: past, present, and future, May 31, 2000, 8 pages.
 A Century of Controversy over the Foundations of Mathematics by Gregory Chaitin.