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 ...

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History of Logic Development

Where can I find a book which explains the development of modern logic, e.g. Tarski, Frege, Peano, up untill Wittgenstein, Russel?
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0answers
393 views

a priori grounds of mathematics [closed]

Hi, from the title of the question you may guess I'm new to the site, and I am. Even though I've read the FAQ and I know that this isn't the place for such open questions (I believe it is open, but ...
8
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3answers
602 views

Variable-centric logical foundation of calculus

Since calculus originated long before our modern function concept, much of our language of calculus still focuses on variables and their interrelationships rather than explicitly on functions. For ...
23
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4answers
2k views

In what ways did Leibniz's philosophy foresee modern mathematics?

Leibniz was a noted polymath who was deeply interested in philosophy as well as mathematics, among other things. From my mathematical readings I have the impression that Leibniz's stature as a ...
7
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0answers
974 views

Is there a finite-dimensional vector space whose dimension cannot be found? [closed]

Is there a finite-dimensional vector space whose dimension cannot be found? Assume, we have somehow constructed a vector space whose dimension is finite, but yet unknown. Is there always an algorithm ...
7
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3answers
1k views

Unprovable sentence about integers

Is there any natural* statement S about the natural integers such that if PA contains no contradictions then neither PA+S nor PA+not S contains a contradiction? If unknown, where can I read about the ...
8
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0answers
2k views

Group Theory, Game Theory, a bit of Philosophy and a post in Tao's blog

I've decided to write this post after reading the incredibly beautiful and highly recomended post by Terence Tao http://terrytao.wordpress.com/2007/06/25/ultrafilters-nonstandard-analysis-and-epsilon-...
6
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5answers
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A meta-mathematical question related to Hilbert tenth problem

I am reading Bjorn Poonen's very nice survey on Hilbert's Tenth problem (http://www-math.mit.edu/~poonen/papers/uniform.pdf), and while I believe I understand the mathematics well, I have widespread ...
5
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2answers
1k views

Is beauty at the high school level even possible? [closed]

This question is a follow up to 74841, and follows from a suggestion by Gian-Carlo Rota that beauty as judged by the educated public differs from that experienced by mathematicians (he gives Euclidean ...
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5answers
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What is the “reason” for modularity results?

The question is a little wishy-washy, but I take my cues from other popular questions that relate to the philosophy behind the mathematics as Why do Groups and Abelian Groups feel so different? . I ...
2
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1answer
838 views

Notion of Truth and Axioms

Hello, The proofs in logic often use the notion of truth. Can we ignore the notion of truth, if we add axioms to the Peano's axioms ? Is it possible to prove Gödel's first incompleteness theorem ...
41
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5answers
3k views

How to resolve a disagreement about a mathematical proof?

I am having a problem which should not exist. I am reading what I believe to be an important paper by a person - let me call him/her $A$ - whom I believe to be a serious and talented mathematician. A ...
18
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4answers
2k views

Are proper classes objects?

Many of us presume that mathematics studies objects. In agreement with this, set theorists often say that they study the well founded hereditarily extensional objects generated ex nihilo by the "...
6
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1answer
925 views

How are mathematical objects defined from an ultrafinitist perspective?

I remember attending a lecture given by an ultrafinitist who denied that curves are a set of points, he would only say that any particular point may or not be on the curve. Similarly for algebraic or ...
12
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1answer
902 views

Why should I believe the Singular Cardinal Hypothesis?

The Singular Cardinal Hypothesis (SCH) is the statement that $\kappa^{cf(\kappa)} = \kappa^+ \cdot 2^{cf(\kappa)}$ for every singular cardinal $\kappa$ (or various equivalent statements). It is ...
-1
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1answer
973 views

Is it possible to construct a finite mathematical universe? [duplicate]

Possible Duplicate: Is there any formal foundation to ultrafinitism? Very recently I have come across the skeptic opinions of a school of mathematicians(ultrafinitist) over a physically ...
12
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5answers
3k views

What should philosophers know about math? [closed]

I know, I know... this is not a technical question. Nevertheless, I believe this is the right place to ask such question. I am sure many of you read about philosophy, including philosophy of ...
13
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3answers
1k views

Are there examples of nonconstructive metaproofs?

This came up in a question on the xkcd forums. Is it possible to have a nonconstructive metaproof, i.e. a proof that there exists a proof in some formal system which does not construct said proof? Are ...
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10answers
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Is PA consistent? do we know it?

1) (By Goedel's) One can not prove, in PA, a formula that can be interpreted to express the consistency of PA. (Hopefully I said it right. Specialists correct me, please). 2) There are proofs (...
15
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1answer
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Martin's “Philosophical Issues about the Hierarchy of Sets”

Some months ago (October 2010), in the context of the Workshop on Set Theory and the Philosophy of Mathematics, Professor Donald A. Martin gave a talk entitled "Philosophical issues about the ...
7
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6answers
990 views

Seemingly emergent structures in mathematics

I rather suspect that this must have come up here on MO already, but my handful of searches didn't turn up the thread, so... I'm curious about examples of mathematical structure that seems to arise "...
21
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6answers
2k views

Philosophical Question related to Largest Known Primes

The other day while discussing math, and primes specifically, the following question came to mind, and I figured I'd ask it here to see what people's opinions on it might be. Main Question: ...
55
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13answers
11k views

Logic in mathematics and philosophy

What are the relations between logic as an area of (modern) philosophy and mathematical logic. The world "modern" refers to 20th century and later, and I am curious mainly about the second half of ...
1
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4answers
881 views

Condition of possibility = Co-Implication

Sorry, but I do not know another place to post this question. Condition of possibility is an important philosophical concept. Naively, this concept could be formally defined this way: $q$ is a ...
45
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7answers
5k views

Why hasn't mereology succeeded as an alternative to set theory?

I have recently run into this wikipedia article on mereology. I was surprised I had never heard of it before and indeed it seems to be seldom mentioned in the mathematical literature. Unlike set ...
3
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1answer
200 views

Class Separation, Oracles, Relativization

It is known, there exists oracles A, B s.t.: $P^A = NP^A; P^B \neq NP^B$, showing that any proof of P vs NP must be non-relativizing. Questions: (1) Can we actually use Oracles to separate ...
39
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9answers
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Examples in mirror symmetry that can be understood.

It seems to me, that a typical science often has simple and important examples whose formulation can be understood (or at least there are some outcomes that can be understood). So if we consider ...
8
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3answers
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Kunen's use of Countable Transitive Models

Hi, I have a doubt concerning Kunen's exposition of forcing in his classical book (arguably $the$ book on forcing). When dealing with Countable Transitive Models to set up the forcing machinery, ...
12
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5answers
1k views

Intended interpretations of set theories

In his Set Theory. An Introduction to Indepencence Proofs, Kunen develops $ZFC$ from a platonistic point of view because he believes that this is pedagogically easier. When he talks about the intended ...
23
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10answers
4k views

Physics and Church–Turing Thesis

Is there constructed some set of physical laws from which we can logically obtain that any function that can be implemented in some device is Turing computable? EDIT I believe that if we restrict ...
13
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5answers
2k views

Consistency strength needed for applied mathematics

Given that we can never proof the consistency of a theory for the foundations of mathematics in a weaker system, one could seriously doubt whether any of the commonly used foundational frameworks (ZFC ...
8
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5answers
2k views

Why is it OK to rely on the Fundamental Theorem of Arithmetic when using Gödel numbering?

Gödel's original proof of the First Incompleteness theorem relies on Gödel numbering. Now, the use of Gödel numbering relies on the fact that the Fundamental Theorem of Arithmetic is true and thus the ...
2
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0answers
454 views

Single logic foundation vs. multi-logic foundation

Dear all, I have always wondered why I have never read anything about this topic. My question is, are there are any books or articles covering this subject? With this topic I mean the philosophical ...
4
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2answers
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where can you find Grothendieck's “Recoltes et Semailles”?

Where can you find Grothendieck's "Recoltes et Semailles"? Is it available anywhere?
10
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4answers
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Is finitism an extreme form of constructivism?

I hope this question is not too soft for MO. The Wikipedia says about finitism that it is an extreme form of constructivism. See http://en.wikipedia.org/wiki/Finitism. I doubt that this is correct. ...
11
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6answers
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Reasons for the importance of planarity and colorability?

Could it have been foreseen that - exemplarily - planarity and colorability would turn out to be such important concepts in graph theory (there's almost no textbook on graphs without two chapters ...
11
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2answers
555 views

Inconsistency and workaday independence.

Set-theoretic topologists, for example, encounter many propositions that turn out independent from set theory. Sometimes these results require novel forcing arguments, but often they simply rely on ...
22
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7answers
2k views

Supervenience in mathematics

I'm not quite sure if this is the right place to ask, and if this is the right way to ask, but I dare. In philosophy (of mind, e.g.) the concept of supervenience is used: "Supervenience [is] used ...
28
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3answers
3k views

The influence of string theory on mathematics for philosophers.

I've agreed, perhaps unwisely, to give a talk to Philosophers about string theory. I'd like to give the philosophers an overview of the status and influence of string theory in physics, which I feel ...
9
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5answers
2k views

Proper classes and their consequences

I have two main questions: What is a proper class? I've read that it's collection of objects that's "too big" to be a set, but in what sense is such a collection "too big"? Since I'd like this post ...
29
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7answers
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Can a problem be simultaneously polynomial time and undecidable?

The Robertson-Seymour theorem on graph minors leads to some interesting conundrums. The theorem states that any minor-closed class of graphs can be described by a finite number of excluded minors. As ...
19
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2answers
2k views

Should there be a true model of set theory?

As I understand it, there is a program in set theory to produce an ultimate, canonical model of set theory which, among other things, positively answers the Continuum Hypothesis and various questions ...
14
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1answer
741 views

How is Fredkin and Toffoli's Conservative Logic related to Linear Logic?

In the answers to this question, Timothy Gowers asks: I've been interested in this question for some time. I haven't put any serious thought into it, so all I can offer is a further question ...
56
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9answers
7k views

Is there any formal foundation to ultrafinitism?

Ultrafinitism is (I believe) a philosophy of mathematics that is not only constructive, but does not admit the existence of arbitrarily large natural numbers. According to wikipedia, it has been ...
54
votes
2answers
16k views

Is the analysis as taught in universities in fact the analysis of definable numbers?

Ten years ago when I studied in the university I had no idea about definable numbers, but I came to this concept myself. My thoughts were as follows: All numbers are divided into two classes: those ...
37
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7answers
7k views

Arguments against large cardinals

I started to learn about large cardinals a while ago, and I read that the existence, and even the consistency of the existence of an inaccessible cardinal, i.e. a limit cardinal which is additionally ...
14
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3answers
2k views

What was Weierstrass's counterexample to the Dirichlet Principle?

Shortly after his work on the foundations of geometry David Hilbert turned his attention to finding a suitable statement of the Dirichlet principle, from which to prove the Riemann mapping theorem and ...
15
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2answers
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Question arising from Voevodsky's talk on inconsistency

This question arises from the talk by Voevodsky mentioned in this recent MO question. On one of his slides, Voevodsky says that a general formula even with one free variable describes a subset of ...
76
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16answers
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What if Current Foundations of Mathematics are Inconsistent? [closed]

The title of the question is also the title of a talk by Vladimir Voevodsky, available here. Had this kind of opinion been expressed before? EDIT. Thanks to all answerers, commentators, voters, ...
17
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2answers
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Universe view vs. Multiverse view of Set Theory

Here I refer to Hamkins' slides: http://lumiere.ens.fr/~dbonnay/files/talks/hamkins.pdf particularly, to the "Universe view simulated inside Multiverse", p. 22. My question is: is it very unsound ...