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Results tagged with mathematical-philosophy
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user 3106
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.
4
votes
Plausibility argument for a measurable cardinal
The standard discussion of the justification of measurable cardinals is Penelope Maddy's article Believing the axioms. Regarding the "unreachability from below" argument, which she calls inexhaustibi …
- 82.7k
11
votes
What does T+non-Cons(T) mean?
While I don't disagree with the substance of what Qiaochu Yuan and Andrés Caicedo have said, I'm not happy with the terms "gibberish" or "useless."
It's important to bear in mind that when we say "con …
- 82.7k
13
votes
The sets in mathematical logic
The short answer is that there is no way to be absolutely certain that mathematics is free from contradiction.
To start with an extreme case, we all take for granted a certain amount of stability in …
- 82.7k
12
votes
Causality, if any, in mathematics itself
The short answer, as you surely suspected, is that there is no rigorous notion of the type you're asking for, that would allow us to say that (in a particular case) that $X$ definitely causes $Y$, or …
- 82.7k
10
votes
Gödel's ontological proof & Benzmüller's work
Gödel’s proof has the nice feature that one can cleanly separate the logical core of the argument (which is uncontroversial—but see the next paragraph) from its alleged application to theology (which …
- 82.7k
13
votes
The meaning and purpose of "canonical''
The term "canonical basis" is used in representation theory. One of the fundamental examples is the Kazhdan–Lusztig basis of the Hecke algebra of a Coxeter group. Why is the term "canonical" used? …
- 82.7k
34
votes
Metamathematics of buts
In a paper entitled "Contrastive Logic" (Logic Journal of the IGPL 3 (1995), 725–744), Nissim Francez introduced something he called bilogics, which are logics intepreted over a pair of structures ins …
- 82.7k
2
votes
How much should the average mathematician know about foundations?
François G. Dorais's answer is excellent, but there are two things I would like to add in connection with the original question of what every mathematician should know about foundations.
The first ha …
32
votes
Are there any fields of academic mathematics whose epistemic status as math is controversial...
There are several possible dimensions to the question, "Is it math?"
Does it belong in the mathematics department? I think you mostly want to exclude this dimension, because of your comment about pur …
14
votes
Excellent mathematical explanations
For your purposes, it may be better to exhibit pairs of proofs of the same result, one of which is considered "more explanatory" than the other.
The first example that comes to my mind is the altern …
11
votes
Uninteresting questions with interesting answers
Gerry Myerson's integral and Joe Silverman's geometry problem fall into the category of problems that seem uninteresting at first because they can be answered by a straightforward calculation that is …
14
votes
Is rigour just a ritual that most mathematicians wish to get rid of if they could?
Another MO question about rigor got me thinking about this old question again. One valuable feature of rigor, which I don't think has been said explicitly in the other answers, is that rigor allows m …
13
votes
Accepted
Has there been any mathematical study of causality?
I am converting my comments into an answer.
Setting aside the alleged parallel between causation and inference for a moment, there has indeed been some mathematical investigation of cause and effect. …
- 82.7k
4
votes
Clarification of Gödel's second incompleteness theorem
Reading between the lines, I think your question may be answered in David Auerbach's article "How to Say Things With Formalisms." Regarding whether these exotic "consistency" statements may be interp …
- 82.7k
11
votes
Conceptual structuralism and continuum hypothesis
Koellner is probably referring to the paper
Daniel Isaacson, The reality of mathematics and the case of set theory, in: Zsolt Novak and Andras Simonyi (eds), Truth, Reference, and Realism, Central Eu …
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