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Sep
30
GIANLUIGI OLIVERI. A Realist Philosophy of Mathematics. Texts in Philosophy; 6
Filed Under Philosophia Mathematica | Leave a Comment
Source:GIANLUIGI OLIVERI. A Realist Philosophy of Mathematics. Texts in Philosophy; 6
Sep
29
CHARLES PARSONS. Mathematical Thought and Its Objects
Filed Under Philosophia Mathematica | Leave a Comment
Source:CHARLES PARSONS. Mathematical Thought and Its Objects
Sep
29
Identity, Indiscernibility, and ante rem Structuralism: The Tale of i and -i
Filed Under Philosophia Mathematica | Leave a Comment
Some authors have claimed that ante rem structuralism has problems with structures that have indiscernible places. In response, I argue that there is no requirement that mathematical objects be individuated in a non-trivial way. Metaphysical principles and intuitions to the contrary do not stand up to ordinary mathematical practice, which presupposes an identity relation that, in a sense, cannot be defined. In complex analysis, the two square roots of –1 are indiscernible: anything true of one of them is true of the other. I suggest that ‘i’ functions like a parameter in natural deduction systems.
Source:Identity, Indiscernibility, and ante rem Structuralism: The Tale of i and -i
Sep
29
Wittgenstein on Circularity in the Frege-Russell Definition of Cardinal Number
Filed Under Philosophia Mathematica | Leave a Comment
Several scholars have argued that Wittgenstein held the view that the notion of number is presupposed by the notion of one-one correlation, and that therefore Hume’s principle is not a sound basis for a definition of number. I offer a new interpretation of the relevant fragments from Wittgenstein’s Nachlass, showing that if different uses of ‘presupposition’ are understood in terms of de re and de dicto knowledge, Wittgenstein’s argument against the Frege-Russell definition of number turns out to be valid on its own terms, even though it depends on two epistemological principles the logicist may find too ‘constructivist’.
Source:Wittgenstein on Circularity in the Frege-Russell Definition of Cardinal Number
Sep
29
The Epistemological Status of Computer-Assisted Proofs
Filed Under Philosophia Mathematica | Leave a Comment
Several high-profile mathematical problems have been solved in recent decades by computer-assisted proofs. Some philosophers have argued that such proofs are a posteriori on the grounds that some such proofs are unsurveyable; that our warrant for accepting these proofs involves empirical claims about the reliability of computers; that there might be errors in the computer or program executing the proof; and that appeal to computer introduces into a proof an experimental element. I argue that none of these arguments withstands scrutiny, and so there is no reason to believe that computer-assisted proofs are not a priori.
Source:The Epistemological Status of Computer-Assisted Proofs
