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

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462
votes
36answers
59k 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 ...
255
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24answers
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 ...
182
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4answers
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 ...
180
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25answers
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 ...
143
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10answers
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 ...
114
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16answers
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"?
93
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22answers
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 ...
88
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10answers
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?
81
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7answers
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 "...
80
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10answers
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 ...
77
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9answers
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 ...
76
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3answers
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 ...
72
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11answers
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, ...
61
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13answers
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? ...
60
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12answers
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 ...
59
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18answers
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.
58
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15answers
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 ...
57
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10answers
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,"...
53
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17answers
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/...
50
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8answers
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. ...
47
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5answers
2k views

What are examples of unexpected algebraic numbers of high degree occured in some math problems?

Recently I asked a question about a possible transcendence of the number $\Gamma\left(\frac{1}{5}\right)\Gamma\left(\frac{4}{15}\right)/\left(\Gamma\left(\frac{1}{3}\right)\Gamma\left(\frac{2}{15}\...
45
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6answers
6k views

Does advanced math “power” more rudimentary math?

I came across this quote by Eric Weinstein that "when things got supposedly more advanced, they actually got simpler because mathematicians started revealing what was powering all the things that you ...
42
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9answers
7k views

Is complex analysis more “real” than real analysis?

In physics, in the past, complex numbers were used only to remember or simplify formulas and computations. But after the birth of quantum physics, they found that a thing as real as "matter" itself ...
41
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8answers
6k views

Does the existence of a mathematical object imply that it is possible to construct the object?

In mathematics the existence of a mathematical object is often proved by contradiction without showing how to construct the object. Does the existence of the object imply that it is at least possible ...
41
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3answers
3k views

Avoiding proof by induction

Proofs that proceed by induction are almost always unsatisfying to me. They do not seem to deepen understanding, I would describe something that is true by induction as being "true by a technicality". ...
40
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11answers
5k views

Why not both true and false?

Why can't some mathematical statement (or whatever is the correct term) be both true and false? For example we can prove (e.g. by induction) that $1+2+3+\cdots+n=\frac{n(n+1)}{2}$ for all positive ...
40
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5answers
4k views

What's the best way to measure mathematical ability?

Very soft question I admit, but it's something that's been bothering me for a while. I've been thinking that being self taught has the problem of accreditation. You can't evaluate a mathematician ...
38
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12answers
11k views

How big is infinity?

This might be more philosophy than math, but it’s been bothering me for a while. Question: If there’s an infinite amount of real numbers between $ 0 $ and $ 1 $, shouldn’t there be twice the amount ...
38
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4answers
3k 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 ...
38
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13answers
15k 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?
38
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1answer
4k 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. ...
37
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17answers
4k 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 ...
37
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15answers
6k views

The problem of instant velocity

The concept of velocity is by definition the movement divided by the interval of time between initial position and final position. If $f(t)$ is the position of a particle at time $t$; the velocity in ...
37
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8answers
5k views

Should a mathematical proof be 'convincing'?

I just read a description of what is a mathematical proof in my mathematical logic textbook, and I'm a bit puzzled by it. It goes like this: A mathematical proof is a finite sequence of mathematical ...
37
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2answers
2k views

Difference between provability and truth of Goodstein's theorem

I have been thinking about the difference between provability and truth and think this example can illustrate what I have been wondering about: We know that Goodstein's theorem (G) is unprovable in ...
36
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6answers
8k views

How to create new mathematics? [closed]

How do scientists and mathematicians create new mathematics for describing concepts? What is new mathematics? Is it necessarily in format of previous mathematics? Can one person make (invent or ...
35
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1answer
1k views

What is the role of mathematical intuition and common sense in questions of irrationality or transcendence of values of special functions?

I got the number $$\frac{\Gamma\left(\frac{1}{5}\right)\Gamma\left(\frac{4}{15}\right)}{\Gamma\left(\frac{1}{3}\right)\Gamma\left(\frac{2}{15}\right)}=0.824326275998351470388591998726842...$$ in the ...
34
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5answers
2k views

Is $\mathbb{N}$ impossible to pin down?

I don't know if this is appropriate for math.stackexchange, or whether philosophy.stackexchange would have been a better bet, but I'll post it here because the content is somewhat technical. In ZFC, ...
33
votes
8answers
4k 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 numbers,...
32
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2answers
3k views

Examples of theorems that are true conceptually, but false computationally

I am currently reading Theorem Proving in Lean, a document dealing with how to use Lean (an open source theorem prover and programming language). My question stems from chapter 11. The particular ...
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 ...
31
votes
4answers
1k views

What is(are) the reason(s) for defining things in the following way?

In this answer it is written that, In modern mathematics, there's a tendency to define things in terms of what they do rather than in terms of what they are. My questions are, What is(are) ...
30
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8answers
4k views

Why do we call complex numbers “numbers” but we don’t consider 2-vectors numbers?

We refer to complex numbers as numbers. However we refer to vectors as arrays of numbers. There doesn’t seem to be anything that makes one more numeric than the other. Is this just a quirk of history ...
30
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8answers
5k 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. ...
29
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5answers
2k views

Can mathematics get from other sciences what it got from physics?

Throughout history, physics has been an unparalleled source of '' inspiration'' for discovering/inventing mathematical ideas, which is due to its ability to describe the physical world. But can this ...
29
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2answers
2k 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 ...
29
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2answers
1k 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 ...
28
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7answers
5k views

Some confusion about what a function “really is”.

Despite my username, my background is mostly in functional analysis where (at least to my understanding), a function $f$ is considered as a mathematical object in its own right distinctly different ...
27
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9answers
3k views

Is formal truth in mathematical logic a generalization of everyday, intuitive truth?

I'm trying to wrap my head around the relationship between truth in formal logic, as the value a formal expression can take on, as opposed to commonplace notions of truth. Personal background: When I ...
27
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5answers
2k views

What does it mean for a set to exist?

Is there a precise meaning of the word 'exist', what does it mean for a set to exist? And what does it mean for a set to 'not exist' ? And what is a set, what is the precise definition of a set?