Questions involving philosophy of mathematics

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6
votes
1answer
251 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
631 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
votes
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
votes
1answer
374 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
394 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
160 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 ...
17
votes
8answers
866 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. ...
46
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 ...
14
votes
8answers
1k 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
203 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 ...
5
votes
2answers
758 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
195 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
votes
2answers
600 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
275 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 ...
22
votes
6answers
2k views

What are natural numbers?

What are the natural numbers? Is it a valid question at all? My understanding is that a set satisfying Peano axioms is called "the natural numbers" and from that one builds integers, rational ...
8
votes
4answers
466 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 ...
0
votes
1answer
145 views

Need help performing a tree method to test for satisfiability

For those who commented on my previous questions, sorry for the lack of information and explanation. Clearly I did not do a good job of explaining myself so I deleted the question and hope this one ...
25
votes
11answers
5k views

Good books on Philosophy of Mathematics

Where can I learn more about the implications, meta discussions, history and the foundations of mathematics? Is Russell's Introduction to Mathematical Philosophy a good start?
4
votes
2answers
464 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 ...
41
votes
17answers
4k views

What's the goal of mathematics?

Are we just trying to prove every theorem or find theories which lead to a lot of creativity or what? I've already read G. H. Hardy Apology but I didn't get an answer from it.
4
votes
0answers
213 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 ...
0
votes
2answers
354 views

Equality of abstract structures

Philosophical questions concerning the difference between equality, isomorphism, equality upto (unique) isomorphism, undistinguishability, and the like are not very popular among practicing ...
5
votes
10answers
4k 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 ...
11
votes
9answers
962 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 ...
6
votes
5answers
696 views

Philosophy (Logic)

I was reading my daily reddit and came accross this link to a new double major at Oxford, Computer Science and Philosophy. http://www.comlab.ox.ac.uk/admissions/ugrad/Computer_Science_and_Philosophy ...
6
votes
2answers
272 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 ...
33
votes
17answers
3k views

Non-Scientific questions solved by mathematics

I have a general question about the applications of mathematics. What are some applications of mathematics that are not scientific, perhaps maybe literary or philosophical, or political. I am ...
26
votes
1answer
2k views

$e^{e^{e^{79}}}$ and ultrafinitism

I was reading the following article on Ultrafinitism, and it mentions that one of the reasons ultrafinitists believe that N is not infinite is because the floor of $e^{e^{e^{79}}}$ is not computable. ...
10
votes
5answers
2k 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 ...
3
votes
3answers
277 views

categorization of logic

(1). I was wondering about what are the relation and differences between formal and informal logic? What topics does each of them have? For example, topics such as Meaning and Definition, Syllogistic ...
9
votes
5answers
665 views

Time in Mathematics

I claim that it is commonly believed that Mathematical objects can be seen as genuinely static, with no "Platonic" time in which they do genuinely evolve. Nevertheless time has its place in ...
1
vote
1answer
192 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" ...
0
votes
1answer
160 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 ...
7
votes
4answers
1k 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?
15
votes
7answers
1k views

What mathematical questions or areas have philosophical implications outside of mathematics?

Please list both the problem/area and justify why it is important philosophically. This question doesn't cover questions that are only important within the philosophy of mathematics itself.
14
votes
7answers
2k views

Why do statements which appear elementary have complicated proofs?

The motivation for this question is : Rationals of the form $\frac{p}{q}$ where $p,q$ are primes in $[a,b]$ and some other problems in Mathematics which looks as if they are elementary but their ...
7
votes
4answers
368 views

Are the computable reals finitary?

In the comment thread of an answer, I said: The computable numbers are based on the intuitionistic continuum, and are not finitary. To which T.. replied: Computable numbers are not based on ...
14
votes
5answers
2k views

Why does Benford's Law (or Zipf's Law) hold?

Both Benford's Law (if you take a list of values, the distribution of the most significant digit is rougly proportional to the logarithm of the digit) and Zipf's Law (given a corpus of natural ...
47
votes
4answers
3k views

What is “ultrafinitism” and why do people believe it?

I know there's something called "ultrafinitism" which is a very radical form of constructivism that I've heard said means people don't believe that really large integers actually exist. Could someone ...
30
votes
6answers
2k views

If all sets were finite, how could the real numbers be defined?

An extreme form of constructivism is called finitisim. In this form, unlike the standard axiom system, infinite sets are not allowed. There are important mathematicians, such as Kronecker, who ...
284
votes
35answers
31k views

Do complex numbers really exist?

Complex numbers involve the square root of negative one, and most non-mathematicians find it hard to accept that such a number is meaningful. In contrast, they feel that real numbers have an obvious ...