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Results tagged with mathematical-philosophy
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user 49
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.
16
votes
Taking a theorem as a definition and proving the original definition as a theorem
A Grothendieck topos can be defined either as the category of sheaves on a site, or as a category satisfying Giraud's axioms, or as an elementary topos that is bounded over $\rm Set$. I believe any o …
42
votes
Accepted
Is pure mathematics useful outside of mathematics itself?
This is not really an answer to the question as asked, but I believe it's important and relevant to your problem, and too long for a comment.
I will not here express any opinion about the validity or …
16
votes
Variable-centric logical foundation of calculus
Here is another approach, which I believe I first learned from Toby Bartels. Suppose $X$ is an arbitrary differentiable manifold (think of the state space of some physical system), and define a varia …
- 66.8k
8
votes
How are material set theory and structural set theory related from the point of view of cate...
I think this question has not been satisfactorily studied, but we can say something. Firstly, note that since a model of SEAR (or other structural set theory) is really a category, such models form n …
- 66.8k
11
votes
Why would the category of sets be intuitionistic?
There are many answers that could be given, but I think the standard "algorithmically oriented" answer is the BHK/Curry-Howard interpretation, according to which a "subset $A$ of $X$" is specified by …
- 66.8k
23
votes
How should a "working mathematician" think about sets? (ZFC, category theory, urelements)
There are already some excellent answers explaining in what senses ZFC can still be a foundation for most mathematics. But it also seems appropriate to mention some ways in which ZFC is insufficient …
- 66.8k
6
votes
Extensionality in HoTT versus extensionality in internal language of a category
In this context, an equality is said to be "extensional" if it depends on the behavior of the objects in question, and "intensional" if it depends only on their definition. I'm not entirely sure of t …
- 66.8k
21
votes
Are proper classes objects?
(Insert standard and obvious disclaimers about opinion vs. fact.)
Of course proper classes are mathematical objects. The fact that we can say things like "the proper class M is a model of set theory …
- 66.8k
6
votes
Accepted
Is finitism an extreme form of constructivism?
this is not a rejection of infinitism methods, but just an attempt to find a better foundation. While, constructivism is really a rejection of certain arguments.
I think that's arguable. There d …
- 66.8k
13
votes
Accepted
evil properties, higher category theory and well-chosen tensor products
There are a lot of questions here, but I'll try to answer them all.
Should every mathematical theory take place in a ∞-category? Or is 'real' mathematics basically evil?
I would say that all mat …
- 66.8k
28
votes
Accepted
Category of categories as a foundation of mathematics
My personal opinion is that one should consider the 2-category of categories, rather than the 1-category of categories. I think the axioms one wants for such an "ET2CC" will be something like:
Firs …
- 66.8k