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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.
241
votes
Accepted
Is the analysis as taught in universities in fact the analysis of definable numbers?
The concept of definable real number, although seemingly
easy to reason with at first, is actually laden with subtle
metamathematical dangers to which both your question and
the Wikipedia article to w …
- 236k
148
votes
Accepted
Nontrivial theorems with trivial proofs
Bertrand Russell proved that the general set-formation principle known as the Comprehension Principle, which asserts that for any property $\varphi$ one may form the set $\lbrace\ x \mid \varphi(x)\ \ …
136
votes
Has philosophy ever clarified mathematics?
I find the case of Alan Turing's development of the concept of computatibility to be an example. Before Turing, the logicians had no clear concept of what it means to say that a function is computable …
72
votes
Nontrivial theorems with trivial proofs
Cantor proved that the set of real numbers is uncountable---it cannot be put in bijective correspondence with the natural numbers---but the proof is a simple diagonalization: if the real numbers could …
72
votes
Can a problem be simultaneously polynomial time and undecidable?
Consider the following simplified example of the same phenomenon, which many students find clarifying.
Let $f(n)=1$, if there are $n$ consecutive $7$s in the decimal expansion of $\pi$, and otherwise …
- 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, …
43
votes
Accepted
Why hasn't mereology succeeded as an alternative to set theory?
I have long found this question interesting, and in some recent joint work with Makoto Kikuchi, now available, we consider various aspects of the question of whether a set-theoretic version of mereolo …
- 236k
39
votes
Do set-theorists use informal set theory as their meta-theory when talking about models of ZFC?
The main fact is that a very weak meta-theory typically suffices, for theorems about models of set theory. Indeed, for almost all of the meta-mathematical results in set theory with which I am familia …
- 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
30
votes
Should there be a true model of set theory?
Two weeks ago a conference was held on precisely the topic
of your question, the Workshop on Set Theory and the
Philosophy of
Mathematics
at the University of Pennsylvania in Philadelphia. The
confere …
- 236k
28
votes
Ultrainfinitism, or a step beyond the transfinite
My view is that the large cardinal hierarchy already has all the
principal features of the project you are proposing.
Each of the large cardinal concepts can be regarded as
corresponding to a certain …
- 236k
26
votes
Siegel zeros and other "illusory worlds": building theories around hypotheses believed to be...
I have heard that Jack Silver's discovery of zero sharp ($0^\#$) was part of his attempt to show measurable cardinals inconsistent. Instead of finding the long-sought-after contradiction, however, he …
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
24
votes
Logic in mathematics and philosophy
There is a general pattern of inquiry in mathematics and the sciences by which an investigation begins in philosophy, using philosophical ideas that may be initially quite vague, but which become incr …
- 236k
24
votes
Is platonism regarding arithmetic consistent with the multiverse view in set theory?
The view you are suggesting is something close to what is held by
Solomon Feferman, who holds that the objects and truths of
arithmetic have a definite nature that is not shared when one
moves up to h …
- 236k