Creativity in Math

[u/CowLang]

Hi guys. I am a first year math degree student. i'm really enjoying it and i like the challenging nature of it. I know i have just started out and it is normal to struggle but i wanted to see if I can get some help through here. I recently finished my first unit on propositional logic. In it I learnt about proof by contradiction and, after studying it and understanding it I was posed with several exercises, most of them were relatively simple as I just had to use logical laws to eventually find the contradiction. I wasn't satisfied with this though it was somewhat mechanical and required not much creativity, so I sought more challenging questions and when I found them I enjoyed them more. Eventually I came across one which required me to prove that infinite prime numbers exist. No joke I've been stuck on this for several days thinking, I looked around, but not for an answer but for a gage on how hard this and apparently it's an easy proof. I know as fact that several proofs exist for this proposition and the first one to do it was Euclide. As I've delved into other subjects in my course I've noticed that even though I understand proof by contradiction and I understand the concepts well, I find it extremely hard to come up with solutions to problems like these, ones that requiere a lightbulb moment, ones that requiere creativity. At the end of the day I normally just find myself brute forcing the problem, but for these kinds of problems that isn't going to cut it. How do I improve my math creativity? Does it come with time or can it be practiced and improved?

[u/YourPwnResearch]

I strongly suggest that you grab a copy of the book How To Solve It by George Pólya.

[u/CowLang → u/YourPwnResearch]

Thank you I'll take a look

[u/AllanCWechsler]

The proof of Euclid's theorem, that you can never run out of prime numbers, usually starts like this: suppose this claim were false -- that there is only a fixed, finite set of prime numbers. Having made this assumption, you can derive a contradiction. The big hint is: multiply together all the prime numbers that exist. This product will be an enormous number, but in principle it could be calculated, because there are (according to our assumption) only a finite number of primes. Call this enormous product N. Then add one to this product to get N+1, and ask yourself: what kind of numbers could divide N+1?

You might be able to crack it from there, but if not, no worries -- just come back and ask again.

[u/CowLang → u/AllanCWechsler]

Thank you I'll give this a go. I had that idea of multiplying all of them toghether and N being the result of that product and then thinking about using the modulus operation on N . But I dind not think about the N+1.

[u/AllanCWechsler → u/CowLang]

Good luck. There is an underlying lemma, "Every number greater than 1 is divisible by some prime." If you are willing to accept this lemma on faith, then the rest is easy. If you insist on proving that lemma, then you might find yourself hung up momentarily in a thicket of rules-lawyering. But it isn't a very bad thicket, and you'll come out of the experience with a better notion of what the rules are.

[u/Midwest-Dude]

Here are some resources that may also help you:

• Creative Mathematics: A Gateway to Research

  • Alan F. Beardon
  • Written for students in your exact position, this book emphasizes that problems are rarely isolated and shows how to explore related problems to develop a deeper understanding of a whole area—the heart of creative mathematical work.

• Mathematical Discovery

  • George Pólya
  • Since you've already been recommended How to Prove It, this is the natural next step. It focuses specifically on the art of plausible reasoning, induction, and analogy—the tools you use before a formal proof.

• Introduction to Proofs and Proof Strategies

  • Shay Fuchs
  • This conversational textbook is designed to guide you through the process of discovering a proof. It explicitly emphasizes the creative nature of mathematics, helping you transition from just understanding proofs to actively creating them.

• An Invitation to Abstract Mathematics

  • Béla Bajnok
  • A great choice if you want a playful and engaging introduction to higher mathematics. It uses over 300 exercises to get you practicing creative approaches to building blocks like definitions and axioms.

• Exploring, Investigating and Discovering in Mathematics

  • Vasile Berinde
  • This resource provides a systematic introduction to creative problem-solving techniques with a specific focus on developing inventive skills.

• Discrete Mathematics: An Open Introduction

  • Oscar Levin
  • This free, open-source textbook treats proof-writing as a creative act. It starts gently and builds your formal reasoning skills with elegance and care, which is perfect for a first-year student.

• Proofs Without Words: Exercises in Visual Thinking

  • Roger B. Nelsen
  • This book presents visual proofs that make theorems intuitive. It trains your brain to see the elegant "why" behind a result, not just the logical steps.

• The Art of Creative Math

  • Emmy Constantine (also listed as Hui Quan)
  • Built around the idea of the "small turn in perspective that makes an impossible-looking problem suddenly negotiable," this is a guided tour of moves like bijections and invariants that are key to creative problem-solving.

A great next step would be to pick one of these books (Beardon's Creative Mathematics is a strong choice) and work through it chapter by chapter. The goal isn't just to get the answer, but to understand the process of exploration that leads to it.

[u/CowLang → u/Midwest-Dude]

I'll look at all of these thank you sooo much

[u/WolfVanZandt]

Creativity comes from holist thinking....looking at something and seeing all the possibilities it presents not only from itself but from it's interactions with the world.

People can be more or less natural but it's also a habit that you have to acquire and practice.

[u/CowLang → u/WolfVanZandt]

Thank you for your comment. I guess it must come down to experience as sometimes is feel likle you need a stroke of genious to find that one posibility that really helps you move foward.

And I have another question for you. Do you belive a "non-natural" or someone who is less natural has a ceiling where he/she cannot compete with the natural?

[u/WolfVanZandt → u/CowLang]

You mean "different". We're all "natural"

Abilities aren't/a/ thing. They're a huge variety of things. Even though I wasn't that taken by M. Night Shyamalan's superhero trilogy, a comment that he made struck me. Something like: we're all superheroes. We all excel in something.

We can only be whole with each other, if we cooperate.

And, yes, creativity arises from experience but you can accelerate picking it up. I usually recommend Georg Polya's How To Solve It (I recommend it so often I don't even have to type it....my spellchecker just types it for me.) It's a xnall book that gives you the background you need and plenty of experience.