# Tagged Questions

Questions involving philosophy of mathematics

20answers
7k 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 ...
5answers
337 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?
3answers
107 views

### There is concept of finite sets that can have only one “interpretation”?

In our mind we have a naive idea of what a set is, and in nature we can only observe something that behave like a finite set, ZFC (or set theories in general) tries to catch these properties in ...
2answers
99 views

### The consistency of PA is falsifiable. Can the same be said of its soundness?

The statement that 'PA is consistent' is absolutely falsifiable (a term I just made up), in the sense that if it is false, then we can demonstrate its falseness independently of any metatheory (just ...
3answers
220 views

### What would be the immediate implications of a formula for prime numbers?

What would be the immediate implications for Math (or sciences as a general) if someone developed a formula capable of generating every prime number progressively and perfectly, also able to prove (or ...
1answer
450 views

### Philosophy of a Mathematician.

Introduction I don't study Mathematics at university, and probably I don't have any chances to have a little understanding of what mathematics in all its aspects. But I love to find structures and ...
1answer
99 views

### Binary vs. Ternary Goldbach Conjecture

Is there an "understandable" explanation of why the ternary Goldbach conjecture is tractable with current methods, while the binary Goldbach conjecture seems to be out of scope with current ...
2answers
180 views

### “Optical Illusion” in 4D

(Apologies in advance for the wordiness - not very mathematical, I know!) I open my book of Escher optical illusions and look at the 2D page. "Aha!" my brain says - "That image doesn't make sense ...
0answers
54 views

### Why geometry is so important part of math for physics? [closed]

I mean geometry in broad (with differential geometry, topology, etc.).
13answers
3k 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 ...
6answers
1k 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 ...
3answers
658 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 ...
0answers
58 views

### Axiom of Choice-esque argument to show that a proof of a statement exists without actually giving a proof

What if the set of all well-formed statements in ZFC formed a kind of pseudo-category where a morphism f between objects A, B represented a formal proof that A implied B? What if that category could ...
4answers
376 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, ...
1answer
335 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 ...
2answers
141 views

### Truth of Fundamental Theorem of Arithmetic beyond some large number

Let $n$ be a ridiculously large number, e.g., $$\displaystyle23^{23^{23^{23^{23^{23^{23^{23^{23^{23^{23^{23^{23}}}}}}}}}}}}+5$$ which cannot be explicitly written down provided the size of the ...
3answers
134 views

### Measure of how much information is lost in an implication

In an implication like $p \implies q$, is there some measure of how much information is lost in the implication? For example, consider the following implications, where $x \in \{0,1,\ldots,9\}$: ...
1answer
175 views

### What does it mean that a number exists? [closed]

The word exists is a very culturally loaded word. Yet mathematicians use it all the time. My answer proposing that the statement that there "exist a number Q such that Q*Q=2", only means that we can ...
33answers
17k 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 ...
6answers
864 views

### What is this physicist saying?

I do not want to poison this forum with politics. But I want to understand, precisely, what is meant by the bolded statement. It is made by a physicist who used to work at Harvard regarding the ...
2answers
52 views

1answer
152 views

### Books on the philosophy of mathematics and logic

Here is a list of some books on the philosophy of Mathematics and logic founded in an article about this matter. I would like to buy one or maybe two of these or any other suggested books. I would be ...
0answers
67 views

4answers
2k 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 ...
2answers
163 views

### why does soundness seem to be less important than consistency for the structuralist?

If I am not wrong, many mathematicians (I believe this is not only restricted to structuralists) agree that an inconsistent formal system does not have any model. By model I mean some kind of set ...