In Jose Ferreiros Jeremy Gray (ed.), The architecture of modern mathematics (2006)

Jeremy Avigad
Carnegie Mellon University
Philosophical concerns rarely force their way into the average mathematician’s workday. But, in extreme circumstances, fundamental questions can arise as to the legitimacy of a certain manner of proceeding, say, as to whether a particular object should be granted ontological status, or whether a certain conclusion is epistemologically warranted. There are then two distinct views as to the role that philosophy should play in such a situation. On the first view, the mathematician is called upon to turn to the counsel of philosophers, in much the same way as a nation considering an action of dubious international legality is called upon to turn to the United Nations for guidance. After due consideration of appropriate regulations and guidelines (and, possibly, debate between representatives of different philosophical factions), the philosophers render a decision, by which the dutiful mathematician abides. Quine was famously critical of such dreams of a ‘first philosophy.’ At the opposite extreme, our hypothetical mathematician answers only to the subject’s internal concerns, blithely or brashly indifferent to philosophical approval. What is at stake to our mathematician friend is whether the questionable practice provides a proper mathematical solution to the problem at hand, or an appropriate mathematical understanding; or, in pragmatic terms, whether it will make it past a journal referee. In short, mathematics is as mathematics does, and the philosopher’s task is simply to make sense of the subject as it evolves and certify practices that are already in place. In his textbook on the philosophy of mathematics (Shapiro, 2000), Stewart Shapiro characterizes this attitude as ‘philosophy last, if at all.’.
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References found in this work BETA

Philosophy of Mathematics: Structure and Ontology.Stewart Shapiro - 2000 - Philosophical Quarterly 50 (198):120-123.
The Nineteenth-Century Revolution in Mathematical Ontology.Jeremy Gray - 1992 - In Donald Gillies (ed.), Revolutions in Mathematics. Oxford University Press. pp. 226--248.

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Idealization in Cassirer's Philosophy of Mathematics.Thomas Mormann - 2008 - Philosophia Mathematica 16 (2):151 - 181.
Space of Valuations.Thierry Coquand - 2009 - Annals of Pure and Applied Logic 157 (2-3):97-109.

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