Philosophy of mathematics
Ideally it would be a writer with some background in mathematics as the course is intended for mathematics and philosophy students.
Here’s a list of possible questions:
— Rigor and Structure —
What is the most plausible version of structuralism about mathematics?
Are the natural numbers Von Neumann ordinals?
Does indifference in mathematical practice tell us anything about the metaphysics or semantics of mathematics?
What makes a putative proof rigorous?
The demand for rigour in mathematics has gone too far.
The demand for rigour in mathematics has not gone far enough.
Does mathematics need a foundation?
— Neo Fregeanism and Thin Objects —
“Since ‘2+2=4’ is true, and it refers to the number 4, the number 4 exists.” Discuss.
Explain how Neo-Fregeans aim to obtain arithmetic from Hume’s principle. Are they successful?
“The Julius Caesar problem is a pseudo-problem.” Discuss.
Is reference to physical bodies in better standing than reference to directions and letter types?
Explain and critically discuss one arrow in Linnebo’s Fregean triangle.
Explain Linnebo’s account how thin objects can be obtained by a criterion of identity. Is he correct?
“Predicative abstraction principles have no advantage over impredicative abstraction principles.” Discuss.
Should we adopt a dynamic approach to abstraction?
— Set theory —
Does the iterative conception of set have a role to play in justifying the axioms of set theory?
Do limitation of size ideas have a role to play in justifying the axioms of set theory?
How should the generative language of the iterative conception of set be understood?
What is the potentialist understanding of the set theoretic hierarchy and is it right?
How should the Axiom of Replacement be justified?
How should the Axiom of Infinity be justified?
Does dynamic abstraction have a role to play in the iterative conception of set?
Is the reflection principle important in the foundations of set theory?
— Defending the Axioms —
How should we assess set theoretic axioms?
Is the practice of set theory rational?
Is there any substantive difference between Maddy’s Thin Realism and her Arealism?
Does Arealism allow for any substantive notion of objectivity in mathematics?
Should extrinsic justifications (in set theory) take precedence over intrinsic justifications?
— Mathematical Practice —
Is mathematics modular?
Should mathematical proof steps be motivated?
What is mathematical beauty?
Do mathematical definitions matter?
— Indispensability —
Is the indispensability argument sound?
Can Melia weasel away the indispensability argument?
Is there any easy road to nominalism?
Can Field’s program succeed?
Should we believe in mathematical entities?
Can we have knowledge of mathematical entities?
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