Editing Foundations of mathematics (section)

==== Formalism ====
{{Main|Formalism (mathematics)}}

It has been claimed{{by whom|date=June 2024}} that formalists, such as [[David Hilbert]] (1862&ndash;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:

{{blockquote|And to what extent has the formula game thus made possible been successful? This formula game enables us to express the entire thought-content 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.<ref name="ReferenceA">Hilbert 1927 ''The Foundations of Mathematics'' in van Heijenoort 1967:475</ref>}}

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.<ref name="ReferenceA"/>

{{blockquote|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.<ref>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).</ref>}}

The foundational philosophy of formalism, as exemplified by [[David Hilbert]], is a response to the paradoxes of [[set theory]], and is based on [[formal logic]]. Virtually all mathematical [[theorem]]s today can be formulated as theorems of set theory. The truth of a mathematical statement, in this view, is represented by the fact that the statement can be derived from the [[Zermelo–Fraenkel set theory|axioms of set theory]] using the rules of formal logic.

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 [[Peano axioms|laws of arithmetic]]) appear to be true, and so on. [[Hermann Weyl]] posed these very questions to Hilbert:

{{blockquote|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 ...<ref>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{{snd}} mathematics" (ibid).</ref>}}

In some cases these questions may be sufficiently answered through the study of formal theories, in disciplines such as [[reverse mathematics]] and [[computational complexity theory]]. As noted by Weyl, [[formal logical system]]s also run the risk of [[consistency proof|inconsistency]]; in [[Peano axioms|Peano arithmetic]], this arguably has already been settled with several proofs of [[consistency proof|consistency]], but there is debate over whether or not they are sufficiently [[finitism|finitary]] to be meaningful. [[Gödel's incompleteness theorem|Gödel's second incompleteness theorem]] establishes that logical systems of arithmetic can never contain a valid proof of their own [[consistency proof|consistency]]. What Hilbert wanted to do was prove a logical system ''S'' was consistent, based on principles ''P'' that only made up a small part of ''S''. But Gödel proved that the principles ''P'' could not even prove ''P'' to be consistent, let alone ''S''.