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Results tagged with mathematical-philosophy
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user 1946
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.
18
votes
Accepted
Are all mathematical theorems necessarily true?
Apart from the storm of comments, let me just try to answer the question.
There are several ways in a which a mathematical theorem
can be contingent.
First, the independence phenomenon in set theo …
- 236k
20
votes
Supervenience in mathematics
To my way of thinking, the most natural example of
supervenience in mathematics---and the most similar to how
this term is used in the philosophy of mind, where one uses
it to describe the relation of …
- 236k
5
votes
What is against having distinct membership relations on sets in the Platonic realm?
It turns out that one cannot have two fundamentally different parallel set membership relations $\in$ and $\in^*$, if they both satisfy ZF with respect to the common language, for in this case they mu …
- 236k
8
votes
Mathematical analysis of Lewisian concepts, esp. natural properties
I have engaged with the Lewis-style set-theoretic mereology in a few papers, undertaken jointly with Makoto Kikuchi. My interest in this topic was inspired originally by a MathOverflow question, Why h …
- 236k
6
votes
Proof by `universal receiver'
It is very common in set theory to prove that a particular model or structure is well-founded by mapping it into a fixed well-founded structure. The point is that if $j:\langle M,{\in^M}\rangle\to \la …
3
votes
Accepted
Formal definition of 'useful' ?
It seems to me that you have two questions here.
First, you inquire about a formal account of "usefulness". I believe that this is already provided by the formal mathematical accounts of utility in …
- 236k
25
votes
The Importance of ZF
There is a very active ongoing debate within set theory about whether mathematics needs new axioms, and philosophers of mathematics are weighing in on all sides. Relevant considerations include many v …
- 236k
5
votes
Unprovable sentence about integers
Let $A$ be any set of natural numbers that is computably enumerable, but not decidable. There are numerous natural instances of such sets, such as the set of all finite presentations of the trivial gr …
- 236k
14
votes
Are proper classes objects?
Let me offer another answer in counterpoint to Andreas's answer, by pointing out a number of cases in set theory where it seems that a second-order treatment of classes, as in Goedel-Bernays set theor …
- 236k
5
votes
Accepted
Are ITTM's necessary to compute Turing's "computable numbers" and what does that mean for or...
No, classical computability theory as you point is quite capable of dealing with infinitary computable enumerations and computability-in-the-limit from its earliest stages. I believe that Turing is to …
- 236k
13
votes
Accepted
Ontological status of some "sets" in ZFC
Although you've been given a hard time in the comments, I think that this is actually a serious question in the philosophy of mathematics, whose answer depends on one's philosophical position concerni …
- 236k
48
votes
Contemporary philosophy of mathematics
Let me mention a few current issues on which I have been involved in the philosophy of
mathematics. Of course there are also many other issues on which people are working.
Debate on pluralism. First, …
2
votes
Quantification over uncountable sets
There are several things one can say.
The theory of ZFC without powerset is often denoted by $\newcommand\ZFCm{\text{ZFC}^-}\ZFCm$. One has to be a little careful with what it means, since collection …
- 236k
36
votes
Accepted
Meta$^{n{-}th}$ mathematics
My opinion is that there is no crisp distinction between
mathematics, metamathematics and meta-metamathematics, and the
subjects thoroughly blend one into another in such a way that
prevents any coher …
- 236k
14
votes
Physics and Church–Turing Thesis
There is a body of literature on the topic of
supertasks, which are computational tasks involving
infinitely many steps. A large part of this work involves a
purely mathematical analysis and developme …
- 236k