The story of mathematical proof – with John Stillwell

Discover the surprising history of proof, a mathematically vital concept.

In this talk John covers the areas of number theory, non-Euclidean geometry, topology, and logic, and peer into the deep chasm between natural number arithmetic and the real numbers.

This video was filmed by the Ri on 24 January 2023.

John Stillwell was born in Melbourne, Australia, and taught at Monash University from 1970 until 2001, before moving to USF in 2002.

He was an invited speaker at the International Congress of Mathematicians in 1994, and his mathematical writing has been honoured with the Chauvenet Prize of the Mathematical Association of America in 2005 and the book award of the Association of Jesuit Colleges and Universities in 2009.

Among his best-known books are 'Mathematics and Its History' and 'Yearning for the Impossible'. His interests are history of mathematics in the 19th and 20th centuries, number theory, geometry, algebra, topology, foundations of mathematics.

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[TheRoyalInstitution]

Edited to say - we hear you (no pun intended) and acknowledge your complaints about the problems we've been having with our sound. We do now have a full AV team in place, but we're still working through the backlog of videos from when this was an issue. Despite our fancy name, we're an independent charity and don't receive any government funding, so we're often working with a tiny team and a shoestring budget to bring you these incredible lecturers. We promise that we are working very hard to fix the sound issues and you'll hear the difference soon.

[Streetsy]

For someone who never got access to this level of math education, I am really enjoying this kind of video presentation. Thank you to everyone involved.

[SEAWORRIER]

Please give your guest speakers a guide on how to record better audio or provide them with the resources to do so. The content of this talk is good but the audio quality is like nails on a blackboard.

[phitsf]

After having studied calculus for the sake of applying it to problems i.e. with very little attention paid to much pure maths involving proofs; This video has answered lots of questions I didn't know I had.

[dwdei]

Thank you! For years I've never rid myself of never having been able to understand why dx was sometimes 1 and sometimes 0. I thought it was me inventing a confusion out of nothing and accepting that I simply didn't have a maths brain.
My sticking plaster solution was that sometimes the dx is a "grammar" thing, reminding me that the equation is in reference to a changing x, and sometimes it is a measurement inside the equation which tends to zero and might as well count as zero.
A fascinating talk, and what a terrific subject!

[Maplecook]

I would have been able to understand high school math, if my teacher had used visuals like this. For a kid, the pictures are very helpful, even if they seem superfluous to seasoned math people.

[andycordy]

Algebra was a nonsense to me at school. I liked geometry because I was a visually oriented person although I didn't know it then. Nobody showed me that x squared was actually a square I could draw and understand.😕 I shut down and the idea that mathematics had a connectedness was never apparent to me, I made do without it.

[nHans]

I know that this lecture was about mathematical proofs and not practical applications of mathematics. Still, I gotta defend Boole—his eponymous logic (or algebra) is the foundation for digital electronics, digital control systems, computer science and all modern digital computers.

[jarrodanderson]

Stillwells math history book is incredible and I love it. ❤

[HebaruSan]

Errors occur in proofs in practice, likely in rough proportion to how often bugs occur in software. But usually nobody checks closely enough.

[YunocTV]

What a great video! I really enjoyed it. Very well presented by John Stillwell.

[JianYZhong]

Thanks for this illuminating presentation. It helps me understand Godel’s incompleteness theorem, I think. Must check out the Q&A!

[johnfitzgerald]

Yes, these are really great, first time and for review. I'd seen the first proof of Pythagorean's Theorem but not the second. It is great to have a tie in between geometry and algebra. The video is packed with extensions to the handful of examples I've picked up. Thank you

[PlayNowWorkLater]

Very interesting and well presented. The significance of the square root of two in Pythagorean theorem really popped for me. Thank you for allowing my mind to expand just a little bit more today. Good stuff!!

[JavierBonillaC]

This is the most exciting video I have come across in a few. Yes. I will try to buy the book.

[tmann]

I have recently been looking for someone to explain the history or story of proofs! What a coincidence!

[josephe]

Too much echo / reverberation on the sound. Needs some long curtains or drapes to attenuate it.

[fraiopatll]

Computation in mathematics, normally understood, comes at the feet of mathematical results or theorems. But where do the very ideas expressed by the theorems come from? That's where and how math is created. Simply said, deep mathematical ideas are the product of human creativity. Theorems, as such, confirm the truth of the deep mathematical ideas so created, or discovered, as some would say. The creative process leading to mathematical insights are often haphazard, messy, subconscious, and even fortuitous and unexpected. The initial formulation of the deep mathematical discoveries are usually inchoate, incomplete and even possibly wrong. After several attempts to prove the initial formulations fail, new formulations are devised and renewed attempts to prove them are exercised, until eventually a final formulation is proved (in the proper sense of the word), at which point in time, the final formulation attains the status of theorem, which means a proven proposition (or provable, as some would say).

It seems to me that at every level of mathematical instruction from elementary school all the way to college levels, there is very little effort put into teaching the students to exercise their creativity so as to produce mathematical results and insights. Theorems and their proofs are absolutely important and indispensable, and every student of mathematics must acquire a high degree of proficiency in proofs. But, to make progress in mathematics, students of mathematics need to be taught to become creative in discovering or producing mathematical results.

[phitsf]

I'm sure the content is amazing but the audio quality from webcam software/zoom/whatever is just the worst. Lockdowns inspired people to make lots of great content, remotely, but unfortunately the vast majority of it is of the quality I just mentioned. I'm unable to count the number of excellent presentations I've seen, not to mention all of the ones I haven't. It's impractical and simply unreasonable to have everyone re-record everything with better hardware and setups, I have hopes someone will make the effort to scrape all the poor quality content and run it all through some AI to clean it up. Unfortunately by the time AI is being used so broadly I will have taken a staunch stance against AI because the better it will get, the more it will scare me.

Thank you, Royal Institution, for all the content; Good and bad. I think I will start working to transcribe libraries onto stone tablets so generations 20, 000 years from now might have some useful material to help rebuild/build-new. There's no point re-inventing the wheel or the standard model of the universe is there?

[marcvanleeuwen]

Indeed (36:39) I am worried that you measure theoretic proof of the uncountability of the real numbers does not hold water. While the source of contradiction (assuming an enumeration of the real numbers) is never stated clearly, I suppose it is based on the assumption that if the orignal length of the line was originally strictly greater than 1 (like infinitely long, but one could do with a bit less), then after removing (infinitely many) segments whose summed-up lengths never exceed 1, some segment of positive length must remain (in fact many, and the sum of the lengths of the remaining segments makes up for the difference of original and removed lengths). However that is not true, since one could remove all the _rational_ points, which are countable, and no segment would remain (there would remain an uncountable number of isolated irrational numbers). So a proof must somehow use that the real numbers are unlike the rational numbers, which this proof does not do.