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55
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10answers
13k views

How is a system of axioms different from a system of beliefs?

Other ways to put it: Is there any faith required in the adoption of a system of axioms? How is a given system of axioms accepted or rejected if not based on blind faith?
16
votes
2answers
783 views

Is First Order Logic (FOL) the only fundamental logic?

I'm far from being an expert in the field of mathematical logic, but I've been reading about the academic work invested in the foundations of mathematics, both in a historical and objetive sense; and ...
10
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4answers
505 views

What are the reasons for not supporting constructive mathematics

It is obvious that in constructive mathematics, you cannot use the law of excluded middle. What else would be the reasons for not adopting constructive stance in mathematics?
4
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2answers
337 views

Did Structuralism influence the formulation of Category Theory?

Having only the a very cursory knowledge of Structuralism ( it's a movement generally held to have originated in linguistics, then moving on to philosophy & literature), there does appear to be ...
6
votes
6answers
2k views

Michael Spivak in “Calculus” asserts that $\sqrt2$ cannot be proven to exist, and that such a proof is impossible. What does he mean by “exist”?

Michael Spivak in "Calculus" asserts that $\sqrt2$ cannot be proven to exist, and that such a proof is impossible. What does he mean by "exist"? How are you to prove that any number "exists"? Why ...
8
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6answers
2k views

Why accept the axiom of infinity?

According to my readings, Russell showed that a principle Frege used to reduce Peano arithmetic to logic lead to a contradiction. So, Russell tried to reduce mathematics to logic a different way but ...
5
votes
1answer
751 views

How do mathematical objects relate to the real world? (a little philosophy)

I am just going to give an example of what I mean using Skolem's Paradox. I don't want to get into Skolem;s Paradox itself or its "resolution." Skolem's showed that in first-order formulations of ...
26
votes
2answers
779 views

A few questions about intuitionistic mathematics

I have to write a paper on Intuitionism for my Philosophy of Science class and I'm struggling with a few concepts I have encountered in my self-study. The (intuitive) characterization of valid ...
5
votes
4answers
366 views

Are there any situations where you can only memorize rather than understand?

I realize that you should understand theorems, equations etc. rather than just memorizing them, but are there any circumstances where memorizing in necessary? (I have always considered math a logical ...
5
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4answers
3k views

Is McGee's counterexample to Modus Ponens accepted by the mathematical community?

In the mid 1980's Vann McGee proposed a counterexample to Modus Ponens: (a) If a Republicans will win the election, then if Reagan will not win, Anderson will win. (b) A Republican will win the ...
1
vote
1answer
273 views

Equivalence of sequences and subsets of natural numbers

For me, facts like the independence of the continuum hypotheses from ZFC cast a doubt on the "law of the excluded middle". (In this context, the doubt is that there might be no "final set theory" such ...
25
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3answers
1k views

What did Gauss think about infinity?

I have someone who is begging for a conversation with me about infinity. He thinks that Cantor got it wrong, and suggested to me that Gauss did not really believe in infinity, and would not have ...
3
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1answer
409 views

Probability and Axiom of Choice

I'm not a logician, so I apologize if what follows translates to nonsense. I would like to try to define a different theory of random choice. I hesitate to call it probability theory because I do not ...
5
votes
3answers
353 views

How can any statements be proven undecidable?

As I understand it, undecidability means that there exists no proofs or contradictions of a statement. So if you've proved $X$ is undecidable then there are no contradictions to $X$, so $X$ always ...
13
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6answers
2k views

What philosophical consequence of Goedel's incompleteness theorems?

I want to write a philosophical essay centered about Goedel's incompleteness theorem. However I cannot find any real philosophical consequences that I can write more than half a page about. I read the ...
54
votes
6answers
6k views

In what sense are math axioms true?

Say I am explaining to a kid, $A +B$ is the same as $B+A$ for natural numbers. The kid asks: why? Well, it's an axiom. It's called commutativity (which is not even true for most groups). How do I ...
2
votes
1answer
324 views

Russell Paradox and set theories

The Russell paradox arise in the Cantor set theory, but it can be avoided in the $ZF$ and in $NGB$ axiomatic set theory. Are there other axiomatic set theories in which this paradox can be avoided? ...
6
votes
1answer
151 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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1answer
374 views

Structuralist slogans

I am afraid to make a bad impression by misusing this forum but I am looking for as-many-as-possible mathematically inspired formulations and references to one (sometimes vague) idea. The idea is ...
6
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2answers
406 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 ...
3
votes
0answers
228 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
votes
1answer
99 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 ...
3
votes
5answers
334 views

Essays on the real line?

Are there any essays on real numbers (in general?). Specifically I want to learn more about: The history of (the system of) numbers; their philosophical significance through history; any good ...
7
votes
2answers
149 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 ...
5
votes
4answers
478 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 ...
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 ...
8
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4answers
967 views

Is the set of all mathematical truths countable or uncountable?

Is the set of all theorems countable or uncountable? Maybe its a stupid question. I just wanted to know. I am led to think that since, we use a finite set of symbols and English letters, the set of ...
5
votes
0answers
577 views

Mac Lane and Eilenberg's motivations for category theory

I'm looking to understand the conceptual process that brought Eilenberg and Mac Lane in developing the basic concepts of category theory. I quote Mac Lane's book "Category theory for working ...
10
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2answers
1k 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
414 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 – ...
1
vote
4answers
473 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
vote
3answers
352 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 ...
6
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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: ...
3
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4answers
427 views

Evidence of Absence = Absence of Evidence?

Any clever-cloggs out there who can explain the formula below in more simple English please? - Do you agree with the formula?
2
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1answer
252 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 ...
28
votes
3answers
1k 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 ...
6
votes
1answer
264 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 ...
10
votes
3answers
700 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 ...
6
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3answers
1k 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 ...
12
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1answer
394 views

Formalizing metamathematics

I am reading historical/philosophical stuff on the concept of "metamathematics" and am by now quite confused. Several questions emerged, but they are probably somehow confused and interrelated, I ...
8
votes
1answer
407 views

Ultrafinitism and the denial of existence of $\lfloor e^{e^{e^{79}}} \rfloor$

I was reading about Ultrafinitism and the denial of existence of $\lfloor e^{e^{e^{79}}} \rfloor$ by ultrafinitists. I am wondering if they were to deny the existence of $\lfloor e^{e^{e^{79}}} ...
6
votes
1answer
167 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 ...
18
votes
7answers
1k views

Definition of definition

I was wondering if there is a good way to "define" what definition means exactly in mathematics. Since the answers may be subjective or philosophical, I want to ask only for references on this topic. ...
51
votes
12answers
6k 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 ...
14
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8answers
2k views

Reference request: is mathematics discovered or created?

I have to write a short monograph as an assignment for a course on the philosophy of science. Being a math student, of course I want to opt for something math-related. After some initial ideas which ...
5
votes
2answers
217 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 ...
4
votes
2answers
892 views

Can one rigorously define “meaningful” versus “arbitrary” in math?

Often we regard certain mathematical expressions, or elements thereof, as arbitrary, in the sense that they have no apparent reason or cause, whereas more beautiful or natural seeming expressions feel ...
2
votes
1answer
199 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! ...
9
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2answers
645 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?
8
votes
1answer
277 views

When can we say that a theorem has been proven?

I'm taking a Data Structures and Algorithms course for a CS program. The introductory material was all mathematics, mostly a series of formulas that we are to remember. I can work through the formulas ...