# Tag Info

Accepted

### When writing proofs, is logical notation a crutch?

It's perfectly normal. In fact, I think that's how a mathematitian's mind grows. First, you are naive and "intuistic", and you do a lot of "well, of course this is so!" like statements that are not ...

### Should a mathematical proof be 'convincing'?

The primary reason to write down a proof is in order to communicate with other mathematicians. If mathematicians acquainted with the relevant literature cannot understand your argument (i.e: do not ...

### Each person has at most 3 enemies in a group. Show that we can separate them into two groups where a person will have at most one enemy in the group.

Split the house however you like. Let $E_i$ be the number of enemies person $i$ has in their group, and let $E = \sum E_i.$ For any person having more than $1$ enemy in their group, i.e. at least $2$, ...

### What is the theorem that has the most proofs?

Well, there was a book with 367 proofs of the Pythagorean Theorem published. I'm sure there's more.

### Each person has at most 3 enemies in a group. Show that we can separate them into two groups where a person will have at most one enemy in the group.

The question is ambiguous in two ways: we are not told whether being an enemy is a symmetric or asymmetric relation, nor whether the parliament has a finite or infinite number of members. I will show ...
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### What is "magic" about the combination of addition and multiplication in formal arithmetic?

Looking back at this question I'm quite dissatisfied with my original answer: it wasn't wrong, but it missed some important content. I believe this new version is much better. The end of your post ...

### Is an axiom a proof?

One can say that a proof of a statement $X$ is a finite sequence of statements where a) each statement in the sequence is either an axiom or follows from some previous staements via one of a handful ...
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### What is the reason we usually don't use formal proofs in mathematics?

tl;dr There is little reason to make paper-and-pencil formal proofs other than learning about formal logic. Machine-checked proofs take less effort to produce and are more valuable than paper-and-...

### What is the theorem that has the most proofs?

Proofs of Euler's polyhedral formula in The Geometry Junkyard: Proof 1: Interdigitating Trees Proof 2: Induction on Faces Proof 3: Induction on Vertices Proof 4: Induction on Edges Proof 5: ...

### Each person has at most 3 enemies in a group. Show that we can separate them into two groups where a person will have at most one enemy in the group.

I will give a counter-example for the case where the enemy relationship is not assumed to be symmetric (but is assumed to be irreflexive). Imagine four kingdoms called North, South, East, and West. ...

### Should a mathematical proof be 'convincing'?

Everything about a mathematical proof is psychological. Including the valid and convincing part. Who gets to decide if it is valid and if it is convincing? The primary audience for a proof is the ...

### What is the theorem that has the most proofs?

There are also a lot of ways to prove that there are infinitely many primes. See for example: Euclid's theorem on the infinitude of primes: A hisorical survey of its proofs on arXiv, by Romeo ...

### Should a mathematical proof be 'convincing'?

Mathematical proofs often, if not always, have some gap, and consequently, we have the 'convincing' part of the definition. Note that the author indicates gaps in his/her proof as he/she talks about ...
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### Trying to understand the difference between metatheory and theory and circularity

Most of the time in mathematics we don't study logic; we study other things like real numbers, groups or differential equations. When we're studying these things we don't often stop to worry about ...
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### Proving $\vdash \neg \neg P \to P$ (double negation elimination) in first order logic, preferrably without deduction theorem

First, here is proof that shows $\neg \neg P \vdash P$: \begin{array}{lll} 1&\neg \neg P & Premise\\ 2&\neg \neg P \to (\neg \neg \neg \neg P \to \neg \neg P) & Axiom \ 1\\ 3&\neg ...

### What is the theorem that has the most proofs?

There are actually a surprisingly large number of ways to prove the Fundamental Theorem of Algebra, ranging from real analysis to complex analysis to Galois theory to Riemannian Geometry. At least one ...
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### Higher inductive type: what for?

Good question. To start with I should be clear that I think HITs are probably much more interesting to a homotopy theorist than to a computer scientist. However, it seems that they are not ...
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### In Homotopy Type Theory, how do the continuous notions of spaces and paths, match the discrete notions of constructible terms and proofs?

I would say there is no mistake in your thinking. Rather, the mistake happened many decades ago when algebraic topologists gradually came to use the word "space" for a discrete object that ...

### A simple proof of Descartes's rule of sign

This is an old question that I think deserves an answer that is at the level of a high-schooler. There are however a couple of details that although intuitively obvious, require some basic results ...

### Direct Proof for sum of $n$ integers equation?

Do what Gauss did, reverse the sequence and add it with the original. Now each term is 2n + 4, and there are n such terms. The sum is n(2n+4). Since this results from two sequences, divide by 2 and ...

### natural deduction: introduction of universal quantifier and elimination of existential quantifier explained

Example Let $\Gamma$ the set of first-order Peano axioms: no variables free. 1) $\Gamma \vdash \exists x (x = 0)$ --- easily provable 2) $\Gamma, x=0 \vdash x=0$ --- obvious 3) $\Gamma \vdash x=0$ ...
The easiest (and in my opinion cleanest) way to do this is to augment the context. In the sequent calculus you presented, you have the left-hand of the sequent being a set $Γ$ of formulae. Instead of ...