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Results tagged with mathematical-philosophy
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user 1176
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.
8
votes
What is an oracle, really?
The notion of "inner working" of an oracle is meaningless, because oracles are not machines. They are (infinite) input strings.
In fact, it would be better to call oracles just input, as that would cr …
- 48.8k
91
votes
Accepted
Is rigour just a ritual that most mathematicians wish to get rid of if they could?
I was not going to write anything, as I am a latecomer to this masterful troll question and not many are likely going to scroll all the way down, but Paul Taylor's call for Proof mining and Realizabil …
24
votes
Are the Foundations of Mathematical Logic Shaky?
No, they are not shaky.
I answered a similar question before, but let me try again.
You make a number of assumptions about logic. You are assuming that it is a goal of logic to bootstrap itself and …
- 48.8k
7
votes
Sets = structured sets without structure
I find your approach somewhat puzzling. You seem to be saying that it useful to look at the property of "being a structure" rather than at the structure itself. What interesting mathematics do you int …
- 48.8k
12
votes
Set-theoretical foundations of Mathematics with only bounded quantifiers
Most mathematics can be done in logical systems which are far weaker than Zermelo-Fraenkel set theory. For example, something like structural set theory will suffice for a great deal of ordinary mathe …
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12
votes
Identity types: What makes Intuitionistic Type Theory *intuitionistic*?
As far as I can tell Martin-Löf's analysis of identity and his formulation of the identity types is the intuitionistic explanation of identity. In terms of BHK it would be an algorithmic version of Le …
- 48.8k
15
votes
Accepted
synthetic differential geometry and other alternative theories
Perhaps I can make the implications of what Harry said a bit more explicit. A well-adapted model of SDG embeds smooths manifolds fully and faithfully. This in particualar means that the SDG model and …
- 48.8k
13
votes
What types are to mathematical proofs as types à la Martin-Löf are to constructive proofs, a...
A good starting point to learn about type theories for classical logic is the $\lambda\mu$-calculus introduced in 1992 by Parigot in λμ-Calculus: An algorithmic interpretation of classical natural ded …
- 48.8k
10
votes
Accepted
How to tell a paradox from a "paradox"?
Many paradoxes are first expressed in a semi-formal way, for example "the least number not describable by fewer than eleven words". They are warning signs that lead us to further analysis and can be r …
- 48.8k
24
votes
Does the "propositions-as-types" paradigm match mathematical practice?
There are many aspects to the question "does a logical formalism reflect mathematical practice?" I will focus just on a very simple but important detail that every mathematician is familiar with.
In …
- 48.8k
9
votes
Categorification of logic
Honestly, I think your motivation is a bit misdirected, but apart from the answers already given, you should look at the general topic of categorical logic. Within that, there are category-theoretic t …
- 48.8k
10
votes
Extensionality in HoTT versus extensionality in internal language of a category
The extension is the observable behavior, where "observation" means roughly "application of an eliminator". For a predicate $p$ we may observe whether it is true for a given argument, so by collecting …
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13
votes
Are simplicial sets the intended model of HoTT?
My impression is as follows.
While I cannot speak for Voevodsky, he certainly gives the impression that simplicial sets is his favorite, if not the intended model. For example, he would suggest axiom …
- 48.8k
17
votes
Variable-centric logical foundation of calculus
I would like to argue in the opposite direction. The 17th century notation that is still in use today is a syntactic hodgepodge which does great disservice to students and their teachers alike, despit …
- 48.8k
24
votes
Accepted
The Lucas argument vs the theorem-provers -- who wins and why?
Yes, computers can infer that the Gödel sentence is true. This is performed in a meta-theory which is stronger than the object theory, as it has to be.
For example, Russell O'Connor formalized Gödel' …
- 48.8k