# 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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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 ...
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 ...
4answers
1k views

### What is the correct reading of $\bot$?

I have some doubts about the "natural" interpretation of $\bot$ in Natural Deduction and sequent calculus. In Prawitz (1965) $\bot$ (falsehood or absurdity) is called a sentential constant [page 14] ...
6answers
9k views

### Why is mathematical induction a valid proof technique? [duplicate]

Context: I'm studying for my discrete mathematics exam and I keep running into this question that I've failed to solve. The question is as follows. Problem: The main form for normal induction over ...
4answers
2k views

### How do we know what natural numbers are?

Do I get this right? Gödel's incompleteness theorem applies to first order logic as it applies to second order and any higher order logic. So there is essentially no way pinning down the natural ...
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 ...
7answers
10k views

### How to interpret material conditional and explain it to freshmen?

After studying mathematics for some time, I am still confused. The material conditional “$\rightarrow$” is a logical connective in classical logic. In mathematical texts one often encounters the ...
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 ...
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 ...
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". ...
1answer
1k views

### Why is CH true if it cannot be proved?

Continuum hypothesis (CH) states that there can be no set whose cardinality is strictly between that of integers and real numbers. Godel, 1940 and Paul Cohen,1963 showed that CH can neither be proved ...
9answers
11k 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 ...
9answers
2k views

### what is the definition of Mathematics ? [closed]

we all study mathematics , and all of us learn mathematical methods to solve problems , we learn how to prove , how to think mathematically but the question is, what is mathematics ? how can we ...
3answers
495 views

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"?
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?
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 ...
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 ...