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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.

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2020 · cited by 3
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.
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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
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  1. The Philosophy of Mathematical Practice (Stanford Encyclopedia of Philosophy)referenceno side taken
  2. Anti-foundationalist Philosophy of Mathematics and Mathematical Proofspeer-reviewedno side taken
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