Editing David Hilbert (section)

===Formalism===
In an account that had become standard by the mid-century, Hilbert's problem set was also a kind of manifesto that opened the way for the development of the [[formalism (mathematics)|formalist]] school, one of three major schools of mathematics of the 20th century. According to the formalist, mathematics is manipulation of symbols according to agreed upon formal rules. It is therefore an autonomous activity of thought.

====Program====
{{Main|Hilbert's program}}
In 1920, Hilbert proposed a research project in [[metamathematics]] that became known as Hilbert's program. He wanted mathematics to be formulated on a solid and complete logical foundation. He believed that in principle this could be done by showing that:

# all of mathematics follows from a correctly chosen finite system of [[axiom]]s; and
# that some such axiom system is provably consistent through some means such as the [[epsilon calculus]].

He seems to have had both technical and philosophical reasons for formulating this proposal. It affirmed his dislike of what had become known as the [[ignorabimus]], still an active issue in his time in German thought, and traced back in that formulation to [[Emil du Bois-Reymond]].<ref>{{Cite book |last=Finkelstein |first=Gabriel |title=Emil du Bois-Reymond: Neuroscience, Self, and Society in Nineteenth-Century Germany |date=2013 |publisher=The MIT Press |isbn=978-0262019507 |location=Cambridge; London |pages=265–289 |language=English}}</ref>

This program is still recognizable in the most popular [[philosophy of mathematics]], where it is usually called ''formalism''. For example, the [[Bourbaki group]] adopted a watered-down and selective version of it as adequate to the requirements of their twin projects of (a) writing encyclopedic foundational works, and (b) supporting the [[axiomatic method]] as a research tool. This approach has been successful and influential in relation with Hilbert's work in algebra and functional analysis, but has failed to engage in the same way with his interests in physics and logic.

Hilbert wrote in 1919:

{{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>Hilbert, D. (1919–20), Natur und Mathematisches Erkennen: Vorlesungen, gehalten 1919–1920 in G\"ottingen. Nach der Ausarbeitung von Paul Bernays (Edited and with an English introduction by David E. Rowe), Basel, Birkh\"auser (1992).</ref>}}

Hilbert published his views on the foundations of mathematics in the 2-volume work, [[Grundlagen der Mathematik]].

====Gödel's work====
Hilbert and the mathematicians who worked with him in his enterprise were committed to the project. His attempt to support axiomatized mathematics with definitive principles, which could banish theoretical uncertainties, ended in failure.

[[Kurt Gödel|Gödel]] demonstrated that any consistent formal system that is sufficiently powerful to express basic arithmetic cannot prove its own completeness using only its own axioms and rules of inference. In 1931, his [[Gödel's incompleteness theorem|incompleteness theorem]] showed that Hilbert's grand plan was impossible as stated. The second point cannot in any reasonable way be combined with the first point, as long as the axiom system is genuinely [[finitary]].

Nevertheless, the subsequent achievements of proof theory at the very least ''clarified'' consistency as it relates to theories of central concern to mathematicians. Hilbert's work had started logic on this course of clarification; the need to understand Gödel's work then led to the development of [[recursion theory]] and then [[mathematical logic]] as an autonomous discipline in the 1930s. The basis for later [[theoretical computer science]], in the work of [[Alonzo Church]] and [[Alan Turing]], also grew directly out of this "debate".<ref>{{Cite journal |last=Reichenberger |first=Andrea |date=31 January 2019 |title=From Solvability to Formal Decidability: Revisiting Hilbert's "Non-Ignorabimus" |url=https://scholarship.claremont.edu/jhm/vol9/iss1/5 |journal=Journal of Humanistic Mathematics |volume=9 |issue=1 |pages=49–80 |doi=10.5642/jhummath.201901.05 |s2cid=127398451 |issn=2159-8118|doi-access=free }}</ref>