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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.
13
votes
Essential reads in the philosophy of mathematics and set theory
Last fall I taught a course in the Philosophy of Set Theory at NYU and you can find the reading list available on my web page. This course was more narrowly focused on the question of realism and plur …
- 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 …
- 4,713
19
votes
Accepted
What governs our "perception?" about the platonic realm of sets?
The recent developments on the consistency of NF bring welcome closure to the longstanding open question about whether NF was consistent. And this is naturally a very important matter for those who fi …
- 236k
8
votes
Accepted
Concept of bedrock and mantle in the multiverse view in the philosophy of mathematics
The confusion seems to arise from your statement "the mantle of $V$ which is the smallest ground for $V$." But this not quite right.
In a bottomless model of ZFC, the mantle is not a ground. It is the …
- 236k
11
votes
Silver's approach to the inconsistency of $\mathrm{ZFC}$
When I was a graduate student at Berkeley in the early 90s, I had heard that Silver's approach to refuting ZFC involved the idea that somehow we make a mistake in our thinking about ZFC by conflating …
- 236k
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 …
- 4,713
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)\ \ …
- 4,713
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 …
- 4,713
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 …
- 4,713
23
votes
Accepted
Are the categories of sets, abelian groups, and commutative rings unique?
Introduction to pluralism
A version of this question lies at the heart of the ongoing dispute on pluralism in the philosophy of mathematics. Is there at bottom just one mathematical reality? Does ever …
- 236k
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 …
- 4,713
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
12
votes
Abstract thought vs calculation
Arguments by mathematical induction seem to provide an entire class of examples of the phenomenon, where computation is replaced by a higher level of reasoning.
With induction, one uses a comparativel …
- 4,713
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 …
- 35.5k
21
votes
Siegel zeros and other "illusory worlds": building theories around hypotheses believed to be...
I believe that there are many instances of this phenomenon in set theory, where an elaborate theory is developed over a period of years by many people, even though the theory is not viewed ultimately …
- 236k