Foundationalist programmes in mathematics face active challenges in contemporary philosophy
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Scholarly literature on the philosophy of mathematics documents that traditional foundational schools and programs face active contestation and criticism from alternative modern approaches.
Abstract
The Euclidean ideal of mathematics as well as all the foundational schools in the philosophy of mathematics have been contested by the new approach, called the “maverick” trend in the philosophy of mathematics. Several points made by its main representatives are mentioned – from the revisability of actual proofs to the stress on real mathematical practice as opposed to its idealized reconstruction. Main features of real proofs are then mentioned; for example, whether they are convincing, understandable, and/or explanatory. Therefore, the new approach questions Hilbert’s Thesis, according to which a correct mathematical proof is in principle reducible to a formal proof, based on explicit axioms and logic.
d Kitcher (1988) saw in foundationalist approaches their
primary target. To emphasize the radical nature of their view, the
latter presented themselves as part of the “maverick
tradition” in philosophy of mathematics.
In addition to creating an ideological rift within the philosophy of
mathematics, this narrative also ignored the fact that philosophers
within the foundationalist tradition itself had begun to criticize
foundationalist approaches (Putnam 1967; Quine 1951) and to pursue new
questions related to contemporary mathematics (Maddy 1996).
The ecumenical stance does not oppose foundational efforts. It
considers them to be valuable contributions to the philosophy of
mathematics while at the same time recognizing that there are other
important questions to be addressed. This stance is well-exemplified
in the volume “The Philosophy of Mathematical Practice”
(Mancosu 2008b). On the contrary, the anti-foundationalist stance
directly opposes foundationalist efforts. A contribution adopting this
stance is Corfield 2003. Even within the analytic framework, one can
perceive anti-foundationalist features in the work of Maddy (1997), at
least to the extent that her naturalism rejects a philosophy-first
approach and she embraces a methodology that sits uncomfortably with
the traditional foundationalist programs.
While the ecumenical stance sees the developments in the philosophy of
mathematics as an extension of the topics and the perspectives which
fall under the purview of philosophy of mathematics, the
anti-foundationalist stance sees it as an irreconcilable split between
different approaches and traditions. The latter is made explicit in
the “brief and biased history” of philosophy of
mathematics by Aspray and Kitcher (1988), who identify two traditions
that are centered around two general research programs: (1) the
mainstream , which started with Frege and is centered around
questions regarding the foundations of mathematics; and
Everything we examined (2)
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