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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.
5
votes
Why should I believe Martin's Maximum++?
Although it's not really a scholarly publication, the Quanta Magazine article, To Settle Infinity Dispute, a New Law of Logic, gives a good introduction to the topic. That article mentions a conferenc …
- 82.7k
49
votes
Is PA consistent? do we know it?
EDIT: I have written a paper that greatly expands on my answer here, and that in particular contains sketches of Gentzen's proof and Friedman's proof, as well as a discussion of formalism.
I have alr …
- 2,031
10
votes
In what sense does the sentence $\operatorname{con}(\mathsf{PA})$ "say" that $\mathsf{PA}$ i...
There is really nothing peculiar about Con(PA) in this regard. Let's take a simpler statement, such as
$$(\exists x \exists y \exists z : xxx + yyy - zzz = 114) \vee (\exists x \exists y \exists z : …
- 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 …
- 82.7k
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 …
- 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
6
votes
Is there a metamathematical $V$?
Kameryn Williams has already given a very good answer, but perhaps it is worth saying explicitly that there is not 100% consensus on the exact meaning of completed infinity (or actual infinity).
Havin …
- 82.7k
16
votes
Comparative analysis of history of mathematics
I'm not aware of anything exactly like what you have in mind, but here are a few things which might be close. They all take aim at the widespread belief that the intellectual development of mathemati …
- 82.7k
11
votes
The name for an assumption made for the sake of contradiction
Especially in the philosophy of religion, the term reductio premise is sometimes used. A Google Scholar search for "reductio premise" (in quotation marks) turns up a few dozen references; one of the …
- 82.7k
58
votes
Accepted
Why is integer factoring hard while determining whether an integer is prime easy?
What I think you're asking for are examples of search problems that seem to be hard, while a corresponding decision problem is solvable in polynomial time (but not totally trivial). It is true that s …
- 82.7k
4
votes
Accepted
Formalisation of intuitive concepts in the language leading to mathematical progress
An example that is closely related to computation is proof. Mathematicians have been proving theorems ever since Euclid (and presumably even earlier). But it was not until the 20th century that the …
- 82.7k
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 …
- 82.7k
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
14
votes
Swimming against the tide in the past century: remarkable achievements that arose in contras...
There are various examples of certain areas of mathematics being regarded as sterile and disconnected from the rest of mathematics, but nevertheless being pursued doggedly by some researchers, and eve …
- 82.7k
13
votes
Accepted
A meta-mathematical question related to Hilbert tenth problem
There's a general "trick" for handling all issues of this sort. Take any mathematical theorem that a platonist regards as meaningful. Formalize it as a formal theorem T in ZFC. The formalist will n …
- 2,821