1
vote
2answers
425 views

What are trivial objects, in general?

Trivial objects show up in most every branch of mathematics, and we all know lots of examples: the trivial group, ring, vector space, module over a ring, graph, knot, homomorphism from one object to ...
2
votes
3answers
543 views

Assessing effectiveness of (epsilon, delta) definitions [closed]

There is much discussion both in the education community and the mathematics community concerning the challenge of (epsilon, delta) type definitions in calculus and the student reception of them. The ...
14
votes
10answers
1k views

An example of a proof that is explanatory but not beautiful? (or vice versa)

This question has a philosophical bent, but hopefully it will evoke straightforward, mathematical answers that would be appropriate for this list (like my earlier question about beautiful proofs ...
43
votes
7answers
2k views

How closed-form conjectures are made?

Recently I posted a conjecture at Math.SE: $$\int_0^\infty\ln\frac{J_\mu(x)^2+Y_\mu(x)^2}{J_\nu(x)^2+Y_\nu(x)^2}\mathrm dx\stackrel{?}{=}\frac{\pi}{2}(\mu^2-\nu^2),$$ where $J_\mu(x)$ and $Y_\mu(x)$ ...
8
votes
6answers
1k views

Intuitionistic logic as quantization of classical logic?

A classically trained mathematician is more likely to be familiar (at least anecdotally) with an area of mathematical physics such as deformation quantization than with Intuitionistic logic. It is ...
16
votes
2answers
649 views

Age of Stochasticity?

One user on MSE made an interesting question, which was unanswered so I suggested him to post it here but he refused for personal reasons and said I could ask it here. The question is this: Today ...
9
votes
3answers
784 views

Is there an observer dependent mathematics? [closed]

Is there any field of mathematics that deals with the role of the observer? E.g., some formulation in which a set is changed, in some unspecified way, when it is observed? Or maybe some philosophy of ...
13
votes
1answer
1k views

Euler's mathematics in terms of modern theories?

Some aspects of Euler's work were formalized in terms of modern infinitesimal theories by Laugwitz, McKinzie, Tuckey, and others. Referring to the latter, G. Ferraro claims that "one can see in ...
3
votes
0answers
229 views

A Question Regarding Boolean-valued Models

What were the intuitions motivating the creation (or discovery, if you will) of Boolean-valued models? I have searched for the Scott-Solovay paper on the subject, but to no avail. There also seems to ...
3
votes
0answers
709 views

Which mathematical questions are objectively true or false? [closed]

Which parts of mathematics are objectively true or false like finite arithmetic and which are only true, false or undecidable relative to a particular axiom system such as Euclidean geometry? ...
19
votes
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 ...
3
votes
2answers
888 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 ...
36
votes
5answers
2k 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 ...
17
votes
4answers
1k 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 ...
1
vote
4answers
617 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 ...
3
votes
1answer
178 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 ...
27
votes
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 ...
14
votes
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 ...
28
votes
7answers
5k 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 ...
12
votes
1answer
626 views

evil properties, higher category theory and well-chosen tensor products

Let's start with the following random example: If $F$ is a presheaf, then for every chain of open subsets $U \subseteq V \subseteq W$, the morphisms $F(W) \to F(V) \to F(U)$ and $F(W) \to F(U)$ ...
49
votes
16answers
7k views

Can a mathematical definition be wrong?

This question originates from a bit of history. In the first paper on quantum Turing machines, the authors left a key uniformity condition out of their definition. Three mathematicians subsequently ...
18
votes
9answers
4k views

Why are proofs so valuable, although we do not know that our axiom system is consistent? [closed]

As a person who has been spending significant time to learn mathematics, I have to admit that I sometimes find the fact uncovered by Godel very upsetting: we never can know that our axiom system is ...
28
votes
5answers
3k views

Why do categorical foundationalists want to escape set theory?

This is a question that I have seen asked passively in comments relating to the separation of category theory from set theory, but I haven't seen it addressed in full. I know that it's possible to ...
2
votes
3answers
1k views

The problem of infinity [closed]

Background and motivation The following is copied from my blog since someone thought it was the clearest statement I had made regarding a problem I recently posed. On their advice, it is a community ...
5
votes
5answers
3k views

Models of ZFC Set Theory - Getting Started

For just any first-order theory: What are the sets I am supposed/allowed to think of when thinking of models as sets (of something + additional structure)? Provided: I can think of models of any ...
4
votes
1answer
841 views

Co-Objects are better [closed]

This is a rather vague question, but perhaps we can talk about it. There are two types of mathematical objects (which don't exclude each other): A) There is a good description of morphisms defined ...
15
votes
10answers
1k views

Can you prove equivalence without being able to calculate it?

In mathematics we often seek to classify objects up to an equivalence relation, where two objects A and B are said to be equivalent if there exists a map $f:A\rightarrow B$ satisfying certain ...
1
vote
4answers
462 views

Does the Golden Ratio Apply to Timing as Well? [closed]

I've seen the golden section applied to art, but does it apply to sound/timing as well? Just curious.
3
votes
4answers
4k views

Badiou and Mathematics [closed]

Does anyone have an opinion on Alain Badiou's use of set theory? Is there anything interesting mathematically there? Also could anyone shed any light on the comment in the Wikipedia article link text ...
3
votes
2answers
458 views

Broken Symmetry

I have a tangled web of ideas about natural transformations, vector spaces, equivalence classes, local coordinates, etc. in my head that I'm trying to unravel. So here are some of the questions I ...
17
votes
8answers
1k views

The Importance of ZF

It seems as though many consider ZF to be the foundational set of axioms for all of mathematics (or at least, a crucial part of the foundations); when a theorem is found to be independent of ZF, it's ...
7
votes
2answers
2k views

Math History Question about the exponential function

While tutoring a student recently, I have come across the situation of explain logarithms by first introducing functions of the form $$f(x)= a^x$$ where $a \ge 0,x\in \mathbb{R}$. My student then ...
9
votes
3answers
952 views

Is formal proof (formalized mathematics) interesting to practicing mathematicians? To educators? [closed]

Formalizing mathematical proofs so that they can be checked for correctness and manipulated by computer is a recurrent proposal, most notably stated in the QED manifesto (1994). The December 2008 ...
6
votes
3answers
2k views

How platonistic is your attitude towards mathematics? [closed]

A discussion in the n-category cafe about Manin's 'emotional Platonism' made me wonder how such a perception of mathematics is distributed among mathematicians and how that influences attitudes ...
-3
votes
2answers
446 views

what is logic without a proof system [closed]

Is there a proof for no proof ?