Questions involving philosophy of mathematics

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9
votes
2answers
597 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
464 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
458 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 ...
40
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
211 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
953 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
681 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
271 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 ...
23
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
660 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
191 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.
13
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
364 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 ...
13
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 ...
29
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 ...
282
votes
35answers
30k 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 ...