Editing Philosophy of mathematics (section)

===Formalism===
{{Main|Formalism (philosophy of mathematics)}}
Formalism holds that mathematical statements may be thought of as statements about the consequences of certain string manipulation rules. For example, in the "game" of [[Euclidean geometry]] (which is seen as consisting of some strings called "axioms", and some "rules of inference" to generate new strings from given ones), one can prove that the [[Pythagorean theorem]] holds (that is, one can generate the string corresponding to the Pythagorean theorem). According to formalism, mathematical truths are not about numbers and sets and triangles and the like—in fact, they are not "about" anything at all.

Another version of formalism is known as [[deductivism]].<ref>{{cite book |author1=Alexander Paseau |author2=Fabian Pregel |title=Deductivism in the Philosophy of Mathematics |date=2023 |publisher=Metaphysics Research Lab, Stanford University |url=https://plato.stanford.edu/entries/deductivism-mathematics/}}</ref> In deductivism, the Pythagorean theorem is not an absolute truth, but a relative one, if it follows deductively from the appropriate axioms. The same is held to be true for all other mathematical statements.

Formalism need not mean that mathematics is nothing more than a meaningless symbolic game. It is usually hoped that there exists some interpretation in which the rules of the game hold. (Compare this position to [[structuralism (philosophy of mathematics)|structuralism]].) But it does allow the working mathematician to continue in his or her work and leave such problems to the philosopher or scientist. Many formalists would say that in practice, the axiom systems to be studied will be suggested by the demands of science or other areas of mathematics.

[[File:Hilbert.jpg|thumb|[[David Hilbert]]]]
A major early proponent of formalism was [[David Hilbert]], whose [[Hilbert's program|program]] was intended to be a [[Gödel's completeness theorem|complete]] and [[consistency proof|consistent]] axiomatization of all of mathematics.<ref>{{Citation|last=Zach|first=Richard|title=Hilbert's Program|date=2019|url=https://plato.stanford.edu/archives/sum2019/entries/hilbert-program/|encyclopedia=The Stanford Encyclopedia of Philosophy|editor-last=Zalta|editor-first=Edward N.|edition=Summer 2019|publisher=Metaphysics Research Lab, Stanford University|access-date=2019-05-25|archive-date=2022-02-08|archive-url=https://web.archive.org/web/20220208161851/https://plato.stanford.edu/archives/sum2019/entries/hilbert-program/|url-status=live}}</ref> Hilbert aimed to show the consistency of mathematical systems from the assumption that the "finitary arithmetic" (a subsystem of the usual [[arithmetic]] of the positive [[integers]], chosen to be philosophically uncontroversial) was consistent. Hilbert's goals of creating a system of mathematics that is both complete and consistent were seriously undermined by the second of [[Gödel's incompleteness theorem]]s, which states that sufficiently expressive consistent axiom systems can never prove their own consistency. Since any such axiom system would contain the finitary arithmetic as a subsystem, Gödel's theorem implied that it would be impossible to prove the system's consistency relative to that (since it would then prove its own consistency, which Gödel had shown was impossible). Thus, in order to show that any [[axiomatic system]] of mathematics is in fact consistent, one needs to first assume the consistency of a system of mathematics that is in a sense stronger than the system to be proven consistent.

Hilbert was initially a deductivist, but, as may be clear from above, he considered certain metamathematical methods to yield intrinsically meaningful results and was a realist with respect to the finitary arithmetic. Later, he held the opinion that there was no other meaningful mathematics whatsoever, regardless of interpretation.

Other formalists, such as [[Rudolf Carnap]], [[Alfred Tarski]], and [[Haskell Curry]], considered mathematics to be the investigation of [[formal system|formal axiom systems]]. [[Mathematical logic]]ians study formal systems but are just as often realists as they are formalists.

Formalists are relatively tolerant and inviting to new approaches to logic, non-standard number systems, new set theories, etc. The more games we study, the better. However, in all three of these examples, motivation is drawn from existing mathematical or philosophical concerns. The "games" are usually not arbitrary.

The main critique of formalism is that the actual mathematical ideas that occupy mathematicians are far removed from the string manipulation games mentioned above. Formalism is thus silent on the question of which axiom systems ought to be studied, as none is more meaningful than another from a formalistic point of view.

Recently, some{{Who|date=July 2012}} formalist mathematicians have proposed that all of our ''formal'' mathematical knowledge should be systematically encoded in [[machine-readable medium|computer-readable]] formats, so as to facilitate [[proof checking|automated proof checking]] of mathematical proofs and the use of [[proof assistant|interactive theorem proving]] in the development of mathematical theories and computer software. Because of their close connection with [[computer science]], this idea is also advocated by mathematical intuitionists and constructivists in the "computability" tradition—{{Crossreference|see [[QED project]] for a general overview}}.