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I am starting to learn about logic and, consequently, about proofs. I already knew that things have to be proven in mathematics, but I did not know that a high school student was able to do it. I always thought it was extremely advanced. I know there are advanced proofs, but I always thought that it was extremely hard to prove, for example, ab=ba, but it is an axiom (a pre-established rule)!

My mathematical knowledge is ordinary. It is what you expect from a high school student. The only exception is that I am learning logic.

So I ask: is there any mathematical paper that a high school student would understand? If it helps, the high school student can google things, but not everything.

This part does not matter, so you can skip it. I want to try to read a mathematical paper because I would like to understand how rigorous proofs have to be, and because I would like to understand how to write one. I would not write one with the intention of publishing it. Write something down structured (for example, like a paper) helps me to agree with the things I just did, helps me to explain it to myself and others, helps me to not lose track of what I was doing, if a mistake was made, helps me to find it, and, if I would to see it 6 years later, I could understand it, etc. That is the reason I write, or pretend to be teaching or making a video when I am learning something: because I am trying to understand it. Another reason is that my memory is horrible, and it helps a lot.

  • I would have thought that a high school student could be expected to be able to do a proof of $1+2+3+\cdots+n=\frac12n(n+1)$, for example by induction – Henry Sep 13 '20 at 01:52
  • @Henry, I never learned that, but I am Brazilian and the education is greeeeeat: 1 + 2 = 4. –  Sep 13 '20 at 01:54
  • @Henry, actually, that is something I tried to prove today, but I don't think it's right. –  Sep 13 '20 at 01:56
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    You should try to read proofs in books or in the web rather than in papers. For example several proofs of that result are at https://proofwiki.org/wiki/Closed_Form_for_Triangular_Numbers – Henry Sep 13 '20 at 01:59
  • @Henry, thanks. –  Sep 13 '20 at 02:01
  • @Henry, I had done it defining the first number, s, and the difference between Sn and Sn+1, d. So, d = n+1 and s = 1. Then I would make an equation system (ax + bx² + c): a + b + c = 1, 2a + 4b + c = 3, and 3a + 9b + c = 6. I would solve it: c = 0, a = 1/2 and b = 1/2, so x1/2 + x²1/2, which is 1/2(x + x²). Then I would define A(x) = 1/2(x + x²) and see if A(1) = 1 and A(n+1) - A(n) = n+1. –  Sep 13 '20 at 02:09
  • @Henry, to define the difference: Qn = Q (sequence) and Sn = Q1 + Q2 + Q3 ... + Qn. So Sn+1 = Q1 + Q2 + Q3 ... + Qn + Qn+1. Then Sn+1 - Sn = Qn+1. Since Qn = n, Sn+1 - Sn = n + 1. Which means d = n + 1. –  Sep 13 '20 at 02:13
  • See here. Does this proof meet your criteria? – user926356 Sep 13 '20 at 03:03
  • Since OP is Brazilian, I'd recommend Elon's Curso de Análise Volume 1: https://loja.sbm.org.br/index.php/curso-de-analise-vol-1.html (it's cheap, R$40). – Ivo Terek Sep 13 '20 at 03:14
  • @MarceloCoto, I think it does. Thanks! –  Sep 13 '20 at 03:27
  • @Schilive, See also the following proofs of the Fundamental Theorem of Arithmetic: 1, 2, 3. – user926356 Sep 13 '20 at 03:41

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If you want to learn what constitutes as rigorous, oh boy you're in for a ride. In practice there are different levels of rigour, the ultimate level being I suppose writing a proof in Coq. Coq is a computer program that literally checks if your proof is correct or if it has a flaw. For that level of rigour, you need proof theory, for which you basically need to be a graduate student so not for you yet.

When learning rigour, you start off with very little and gradually go deeper and deeper. I would not recommend any single paper for rigour, since papers reference other papers which means to learn how they are rigorous you'd need to check every reference, then every reference's reference etc. And also for that reason, many papers tend to be not super big about rigour since otherwise they'd millions of pages long.

I suggest going for a textbook that has a reputation for its rigor. Since you're in high school, you don't have much exposure to much math other than geometry and algebra (maybe a tiny bit of calculus if you're lucky), so you don't have many options, except one. I would highly highly recommend Euclid's Elements. It's literally the most famous math book ever written (over 2000 years ago), and has the reputation as the first attempt at a rigorous theory of geometry. Well technically it's actually very flawed by modern math standards, but like I said before, start with very little, then go deeper.

  • About the rigour, I am mostly interested about the structure of a proof, so the book will probably help a lot! When I first heard about this Euclid's book, I thought about reading it, but I thought it would be all Greek to me. But I'll check it out! Thanks. By the way, the joke was not supposed to be a joke. –  Sep 13 '20 at 02:33
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You could check How to think like a mathematician, by Kevin Houston. I find it way easier to understand than most introductory books to formal maths. I am not a high school student but a 4th semester engineering student starting a math minor so i might be biased about the difficulty, but you should definitely give it a try.

CoolJedi132
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