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6
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1answer
174 views

Hidden structures

There is a lot of talk about "hidden structures" in the realm of mathematics: hidden structures in the ZFC system, hidden structures in the natural number system, and so on. Saunders Mac Lane ...
6
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2answers
523 views

Are there any non-self-referential statements that cannot be assigned a truth value?

Statements like A) A is false. or B1) B2 is true. B2) B1 is false. cannot be assigned a truth-value due to their ...
4
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4answers
632 views

From continuity to differentiability and analyticity- what's next?

Continuity is an intuitive concept. I will not dwell on the precise definitions of continuity and the rest here. Note that differentiability is a more restrictive condition than continuity, while ...
3
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0answers
297 views

Does the concept of predicativity need to be formalized to go beyond Feferman-Schutte ordinal?

Feferman-Schütte ordinal is sometimes said to be: ....first impredicative ordinal, though this is controversial, partly because there is no generally accepted precise definition of "predicative". ...
2
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1answer
176 views

Reference request for Intuitionism

I need to write an essay on Intuitionism for my Philosophy of Science class, and I'm looking for books which cover the following topics: Brouwer's Intuitionism, from both a philosophical and ...
9
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2answers
171 views

Literature on general paradox?

I suppose this one teeters on the edge of un-mathematical, but here it goes... I've been on something of a logic binge lately and have (surprise, surprise!) especially been interested in the results ...
10
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5answers
1k views

The status of high school geometry

Okay, so we've all seen Euclidean geometry in primary and high school. Back then, I really thought of points as indivisible entities in space and lines as 'breadthless lengths'. As far as I could tell,...
10
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9answers
11k views

Does a negative number really exist?

Second Update: I see that some answers that reference my image are more closely answering my question. Here is a second image to clarify my point. Take this image representing a checkerboard like ...
10
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2answers
2k views

Definition of “non-constructive proof”

I was wondering if it is possible to define exactly what a non-constructive (nc) proof is. I have often seen the concept associated with the use of principles such as the axiom of choice or the law of ...
12
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3answers
561 views

Is there any difference between a math invention and a math discovery? [closed]

From wikipekia: The calculus controversy was an argument between 17th-century mathematicians Isaac Newton and Gottfried Leibniz (begun or fomented in part by their disciples and associates – ...
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4answers
491 views

Does $3+2=5$ have a non-physical interpretation? [closed]

Normally we consider simple arithmetic to be related to the world of objects. So the sum $3+2=5$ means $3$ three apples and $2$ apples gives $5$ apples. But is there an alternative interpretation ...
1
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3answers
629 views

What is the difference between a parametric equation and a mathematic law?

First of all sorry for my English, I'm not used to communicate with this language. I want to ask something about a thing that I discovered while studying physics (AKA applied mathematics). There is ...
2
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1answer
342 views

Truth and undecidability

I believe this is more of a philosophical question. Given a consistent theory T and a statement S independent of T. Can S be true or false in T? (I don't see any contradiction with that) I read that ...
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5answers
1k views

Inherently discrete concepts

Are there any concepts which are naturally defined only for the integers and so far has resisted any attempts at extension to other fields such as rationals or reals? Does not meet criteria: ...
32
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3answers
2k views

Rejecting infinity

I've heard about mathematicians who defend a strictly finite conception of mathematics, with no room for infinity. I wonder, how is it possible for these people to do this? Are there any concepts that ...
7
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1answer
324 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
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3answers
2k 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 ...
10
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3answers
1k views

Difference between undecidable statements in set-theory and number theory?

Do all statements about the integers have a definite truth value? For instance: Goodstein's theorem is clearly true, otherwise we could find a finite counterexample thus it would be possible to ...
9
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1answer
271 views

Exotic Manifolds from the inside

As we know, an exotic $\mathbb{R}^4$ is a manifold which is homeomorphic, but not diffeomorphic to the standard $(\mathbb{R}^4,id)$, and there are even very explicit descriptions of them (Kirby ...
5
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2answers
314 views

What is characteristic (function, polynomial, etc)?

My question is - what's the nature of characteristic functions, equations and so on? Am I right in understanding that this is just the general term for naming "ways" to find some invariants of some ...
2
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1answer
267 views

Multiple quantifier translation

Having some difficulty translating into English from Symbolic logic (the mixture of the quantifiers are confusing to me): ∀x(¬∃yBackOf(y, x) → Large(x)) Any suggestions would be appreciated. Thanks!
7
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2answers
310 views

Ideas about Proofs

If there are two different proofs for one theorem, at some level are the two proofs the same, or can they be fundamentally different? In other words, if you have two proofs of a theorem, can one show ...
10
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2answers
1k views

How many different proofs can a theorem have?

I notice some problems has many different proofs, do all theorems have multiple proofs, is there some theorems which has only 1 way to prove it? $n$ ways? infinite?
9
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4answers
777 views

Consequences of solving the Halting problem

What impact would a device (ie super-computer or relativistic computer or other method) that solves the halting problem have on math? Would there be any mathematical problems left to solve? What ...
4
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2answers
687 views

Law of Excluded Middle in Logic Proof

I'm having some difficulty doing a proof for the following: $$\neg A \vee \neg(\neg B \wedge (\neg A \vee B))$$ It is said that you could use the law of excluded middles. Any help or guidance would ...
4
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0answers
251 views

Finitistic objections to the current mathematical model

I recently read this pdf: Warning Signs of a Possible Collapse of Contemporary Mathematics, and I'm having some trouble understanding the issues it raises. The author says that the consistency of ...
13
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5answers
3k views

Common misconceptions about math

YARFMO (Yet another reposting from Mathoverflow) ;-) The more you know about math the more you find conceptions previously thought correct to be false: 1.) math is not as exact as many believe - in ...
1
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1answer
207 views

Making meaning of mathematical “bridges”

I apologize for posting such an untechnical question, but with responses it could surely be posed in a better form. I'm a math noob, but I've seen (as we all have) a few examples of "connections" ...
1
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1answer
174 views

Alternative, consistent frameworks of mathematics with isomorphic mappings to physical phenomenon

A friend of mine who is quite an aggressive Nominalist told me the other day: "Mathematics and numbers are arbitrary; they can accurately predict physical systems in real life only because they are ...
11
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4answers
2k views

What are the most important questions or areas of study in the philosophy of mathematics?

This question is intended to complement What mathematical questions or areas have philosophical implications outside of mathematics?