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Results tagged with mathematical-philosophy
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user 23141
Philosophical aspects of logic and set theory; truth status of mathematical axioms; Philosophy of Mathematics; philosophical aspects of mathematics in general; relation of mathematics to philosophy; etc. Consider also posting at http://philosophy.stackexchange.com/, where philosophy-of-mathematics is one of the most popular tags.
5
votes
In what sense does the sentence $\operatorname{con}(\mathsf{PA})$ "say" that $\mathsf{PA}$ i...
Let's go back to the continuum hypothesis. Some, probably most, people think that it really is objectively true or false, despite having different truth values in different models, because they believ …
- 42.8k
16
votes
Why not adopt the constructibility axiom $V=L$?
Keith Devlin wrote a book called The Axiom of Constructibility --- A Guide for the Mathematician which espouses just this point of view.
My impression is that most working set theorists regard $V = L …
10
votes
Accepted
Explaining the consistency of PRA and ZF from predicative foundations
From the point of view of what you call predicative set theory --- I would say "predicativism given the natural numbers" --- I don't think there are any known arguments for the consistency of ZF, and …
- 42.8k
4
votes
Accepted
"Mathematics is the science of the infinite"
I think there is a fairly straightforward answer to this question coming from reverse mathematics. According to Wikipedia, "finitistic reductionism" is represented by the system WKL${}_0$, and reverse …
- 42.8k
17
votes
Accepted
Quantum functional analysis
Okay, I'll take this one. First let me say that the English term is "completely bounded" (or "complete isometry", etc.).
About the term "quantum". The general principle is that analyzing some aspect …
- 42.8k
31
votes
Contemporary philosophy of mathematics
I've written fairly extensively on predicativism and on the paradoxes of set theory and logic. My central claims are (1) it makes better sense, both mathematically and philosophically, to regard all u …
10
votes
Reasoning Using Countable Subsets of Real Numbers
See my paper Analysis in $J_2$, where I explain how core mathematics, particularly abstract analysis, can be developed within a concrete countable set $J_2$ (the second set in Jensen's constructible h …
- 42.8k
43
votes
Axiom of choice, Banach-Tarski and reality
It's notable that most of the "bread and butter" mathematical consequences of the axiom of choice are actually consequences of countable choice. (Every infinite set contains a countable subset, a coun …
- 42.8k
5
votes
In what sense is the "descending chain principle" for ordinals less than $\epsilon_0$ 'infin...
Tait (Finitism, Journal of Philosophy 78 (1981), 524-546) has argued that finitistic reasoning coincides with PRA, on the grounds that this is as far as you get with "finitist types". What makes a typ …
- 42.8k
11
votes
Interpretation of the Second Incompleteness Theorem
Yudkowsky and Herreshoff have a (messy but) great paper which relates the second incompleteness theorem to issues in theoretical artificial intelligence. (This paper of mine might be a more accessible …
- 42.8k
16
votes
Is platonism regarding arithmetic consistent with the multiverse view in set theory?
I'm still uncertain of its appropriateness here, but since Joel asked, here is a quote from my book that discusses this issue:
The two kinds of independence … in geometry
and number theory offer …
- 42.8k
5
votes
Formal/rigorous treatment of (im)predicativity/predicativism
I have a brief survey on predicativism here. But it may be more of the kind of "verbal" explanation that you've been unsatisfied with.
Maybe proof theoretic ordinals could provide the kind of rigorou …
- 42.8k