# Questions tagged [philosophy]

Questions involving philosophy of mathematics. Please consider if Philosophy Stack Exchange is a better site to post your question.

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### 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 ...
20k views

### Is mathematics one big tautology?

Is mathematics one big tautology? Let me put the question in clearer terms: Mathematics is a deductive system: it works by starting with arbitrary axioms, and deriving therefrom "new" properties ...
23k views

### How do I convince someone that $1+1=2$ may not necessarily be true?

Me and my friend were arguing over this "fact" that we all know and hold dear. However, I do know that $1+1=2$ is an axiom. That is why I beg to differ. Neither of us have the required mathematical ...
15k views

### Can a coin with an unknown bias be treated as fair?

This morning, I wanted to flip a coin to make a decision but only had an SD card: Given that I don't know the bias of this SD card, would flipping it be considered a "fair toss"? I thought if I'm ...
8k views

### How far can one get in analysis without leaving $\mathbb{Q}$?

Suppose you're trying to teach analysis to a stubborn algebraist who refuses to acknowledge the existence of any characteristic $0$ field other than $\mathbb{Q}$. How ugly are things going to get for ...
16k views

### Is 10 closer to infinity than 1?

This may be considered a philosophy but is the number "10" closer to infinity than the number "1"?
16k views

### Is math built on assumptions?

I just came across this statement when I was lecturing a student on math and strictly speaking I used: Assuming that the value of $x$ equals <something>, ... One of my students just rose ...
18k 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?
14k 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 "...
13k views

### Is mathematics just a bunch of nested empty sets?

When in high school I used to see mathematical objects as ideal objects whose existence is independent of us. But when I learned set theory, I discovered that all mathematical objects I was studying ...
14k views

### Does mathematics require axioms?

I just read this whole article: http://web.maths.unsw.edu.au/~norman/papers/SetTheory.pdf which is also discussed over here: Infinite sets don't exist!? However, the paragraph which I found most ...
6k 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 ...
12k views

### Infinite sets don't exist!?

Has anyone read this article? This accomplished mathematician gives his opinion on why he doesn't think infinite sets exist, and claims that axioms are nonsense. I don't disagree with his arguments, ...
8k views

### Why do we not have to prove definitions?

I am a beginning level math student and I read recently (in a book written by a Ph. D in Mathematical Education) that mathematical definitions do not get "proven." As in they can't be proven. Why not? ...
8k 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 ...
9k 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.
9k views

### Is there such a thing as proof by example (not counter example)

Is there such a logical thing as proof by example? I know many times when I am working with algebraic manipulations, I do quick tests to see if I remembered the formula right. This works and is ...
6k views

### Is it an abuse of language to say “*the* integers,” “*the* rational numbers,” or “*the* real numbers,” etc.?

I'm finding that the more math I learn, the more concepts I thought were well-defined seem to be intuitive and naive. Here I'm asking about whether it's an abuse of language to refer to "the integers,"...
5k views

### Is a proof still valid if only the writer understands it? [closed]

Say that there is some conjecture that someone has just proved. Let's assume that this proof is correct--that it is based on deductive reasoning and reaches the desired conclusion. However, if he/...
5k views

### What does it mean to solve an equation?

This question might be more philosophical than mathematical. In school we are taught how to solve equations such as $x^2 - 1 = 0$ or $\sin(x) - 1= 0$. Solutions to these equations are quite simple. ...
2k views