4
votes
4answers
378 views

The standard role of intuitive numbers in the foundations of mathematics

In my career I've been formed mostly in the formal side of mathematics, that is, standard set theory and every classical branch of mathematics that uses set theory. However, I am not quite sure about ...
1
vote
2answers
96 views

Abstract Objects in Logic

I am confused on the concept of extensionality versus intensionality. When we say 2<3 is True, we say that 2<3 can be demonstrated by a mathematical proof. So, according to mathematical logic, ...
3
votes
1answer
169 views

First and Second Fundamental Form Intuition

I was just wondering what various quantities relating to the first and second fundamental forms of a regular surface mean intuitively. First of all, another explanation as to what the first and second ...
4
votes
1answer
85 views

What is the intuition behind $\Delta_1^0$ sets and $\Delta_1^1$ sets?

In the context of first-order arithmetic, if $\phi$ is a formula with only bounded quantifiers, then if you put existential quantifiers in front it becomes a $\Sigma_1^0$ formula according to the ...
4
votes
2answers
221 views

why does soundness seem to be less important than consistency for the structuralist?

If I am not wrong, many mathematicians (I believe this is not only restricted to structuralists) agree that an inconsistent formal system does not have any model. By model I mean some kind of set ...
6
votes
1answer
283 views

Age of Stochasticity?

Today I came across D. Mumford's 1999 article The Dawning of the Age of Stochasticity, which is quite remarkable even after more than a decade. The title already indicates the theme, but I copy the ...
4
votes
2answers
269 views

What is the role of Topology in mathematics?

What is the role of Topology in Mathematics? Is it like Logic that you need in every areas of math?
5
votes
1answer
247 views

Generalisation of dualities, what concept do dualities represent?

Duality is a concept that pops up in different areas of mathematics as well as other science, but besides being a "woo isn't that nice?", is there anything more to duality (than loosely stated some ...
6
votes
3answers
843 views

What is a physical “dimension” - in the sense of “dimensional” analysis?

Mathematically speaking, what does it mean to say that a physical quantity is some numerical value with a “dimension” associated with it? When we say that the velocity of light is some constant, c ...
45
votes
12answers
5k views

I need mathematical proof that the distance from zero to 1 is the equal to the distance from 1 to 2 [closed]

I didn't know how to phrase the question properly so I am going to explain how this came about. I know Math is a very rigorous subject and there are proofs for everything we know and use. In fact, I ...
11
votes
9answers
937 views

Problems that are largely believed to be true, but are unresolved

Are there unsolved problems in math that are large believed to be true, but for reasons other then statistical justification? It seems that Goldbach should be true, but this is based on heuristic ...