The story of mathematical proof - with John Stillwell
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- Опубліковано 13 чер 2024
- 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.
Buy John's book here: geni.us/mathsproof
Watch the Q&A here: • Q&A: The story of math...
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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If you liked this video, check out our mathematics playlist here: ua-cam.com/play/PLbnrZHfNEDZyDfeVsNBMNDUu-o5j9_QMb.html
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.
Okay
Just a whisper....see if the mics or sound can be improved. The sound has to be easy on the ears especially if the video is a long video 👂👀🙉
I don't like to complain, but I wish you could get the audio back to the quality of your older videos. This was brutal on my ears.
Did you guys upload the Q&A video? If so, would you please provide the link? (As yet, there isn't one in the description.)
Also, you might want to 'pin' your own comments, so they'll always be at the top.
@@nHans Here's the Q&A - ua-cam.com/video/aPyWM5cQkeQ/v-deo.html - now added to the description too
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.
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.
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.
... your 'unknown unknowns' as it were.
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!
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.
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.
Mathematics is not a discipline of Science but rather a field in the Arts.
Stillwells math history book is incredible and I love it. ❤
Thank you for uploading this.
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!!
I have recently been looking for someone to explain the history or story of proofs! What a coincidence!
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
Thanks for this illuminating presentation. It helps me understand Godel’s incompleteness theorem, I think. Must check out the Q&A!
Great teacher.
Errors occur in proofs in practice, likely in rough proportion to how often bugs occur in software. But usually nobody checks closely enough.
This is the most exciting video I have come across in a few. Yes. I will try to buy the book.
Excellent!
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.
Thanks
Thank you
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.
Yeah & it's such a deep level of intuition that people often attribute it to non-human or paranormal forces. The mathematician Ramanujan for example said that his Goddess sent him his theorem proofs in dreams.
Best you tube channel in the world
Well I think I followed that - and the 44 minute lecture took me about 88 minutes to 'sort of' comprehend :-)
Too much echo / reverberation on the sound. Needs some long curtains or drapes to attenuate it.
Or just a headset
I also like thinking of the incompleteness theorem that Stillwell's referencing as the "no lone genius" result about axiom systems: none of them on their own can produce all facts about mathematics.
guys - u need audio soft to reduce all that noice - or put it into autotune that will change a sound and keep words)
The Greeks also had an issue with zero if I remember correctly. Apparently they had the idea that "nothing" cannot exist, which is something that the Indians didn't agree with. Indians saw Nothingness or the Void as a vessel. It's the future, like an egg. That's why Vedic & Euclidean mathematics developed so differently. Today for example we know from Quantum physics that heat is energy. E=mc². Energy turns into matter, but entropy means that the physical manifestation is always smaller or lesser (in perception) than the source energy pool because they have different densities. 2D vs 3D vs 4D etc.
I had never missed a geometry problem in school and never had to take finals. The last problem I solved was Steiner-Lehman Theorem. But today, I realize that something is completely wrong, for which I invent The Obviousness Theory of Proof based on the 16 Methods of Reason, which says that Obviousness is different for different way of reason.
The first "proof" at 36' would also "prove" that the rationals aren't countable...
Can you elaborate?
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.
A beautiful and illuminating presentation. I should direct my students here. (Added: I do think the names Davis, Robinson, and Putnam should have been added to Matiyasevich.)
Bookmark 5:50
VSauce Michael in 40 years?
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Two high school girls have proven the Pythagorean theorem by a new method evolving series that is not based on itself.
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?
Energy cannot be created or destroyed. If we really realised the impact of this statement, it would shake everything up. Cannot be created or destroyed INFINITE, it was here, is here, and will always be here. W glibbly call it energy, but what does that mean if we take away that name. It is in everything, absolutely everything, it is sustaining everything, it moves everything, it is in us, it is in our breath. This is why there is no such thing as '1', it is an illusion. The mind created '1' for convenience, because that is how the mind works, it breaks things down into segments, it never sees the whole, because it is in fact 'Finite' The real 'PROOF' is in our Breath, because without it we are no more. When we are on our last breath, all the theory's and equations in the world will not be able to help.
What are you babbling about ?
Cant watch it because the sound is not that great. Please record it again with better sounds quality. :)
1.34 million subscribers and they can't give us subscribers audio that doesn't assault the ears
ive heard way worse audio quality during some of the best lectures in science.
Petty given the content.
@Mushie_Man I think it is terible too. My phone has a much better microphone. A headset is pretty cheap and with all that noise and the camera quality at the start it is embarassing. That looked and sounds like a video call 15 years ago.
Don’t complain til you listen to an iPad. This thing 4 times the size of a phone, has 1/4 the volume of a phone. Maybe they did this to sell their AirPods.
Free knowledge is great. The budget might not be CNN, but the content by far does it up for the audio.
I was so immersed, i didn't even notice.
Since 2020 I've found it difficult to watch this channel, Covid is over, go back to the way it used to be😢
They have. About half are like this and the other half are in person.
Poor audio for such an interesting subject. It' s distraction to learners. Please revise this video.
The Greeks did not develop algebra? And all of Diophantus work what was it? Statistics? 😆
cya 😪
Sound quality sucks..🙂
So what? The content is what is important and I understood everything he said.
@@Safetytrousers are you saying quality is not important?
@@savage22bolt32 Having the best sound quality is not as important as what is being said. Better sound quality would be nice, but it is not essential.
What I don’t understand is.. X = 3…… Wrong….. It says x2 which means X=2 the math will prove it. 25 + 5x + 5x =35 that means X2 = 2 + 2 = 4. So 35 + 4 = 39.. what that other craps means is over my head 😂
The original equation to be solved is x*x + 10*x = 39. If you plug in there x = 2, you get 2*2 + 2*10 = 24. So x is not 2. Plug in x = 3 and you get the right answer.
Why do you write "It says x2 which means X=2"?
Yep your still wrong my numbers are right
come on guys, sound... quality sucks bro
So what? The content is what is important and I understood everything he said.
@@Safetytrousers Yeah, but if the audio annoys or irritates someone enough then someone else may not want to listen to the content regardless of its quality.
@@SEAWORRIER By that token any number of things may annoy someone into not wanting to watch something.
@@Safetytrousers True. But some things may not be fixable as technical matter (e.g. talk content that someone would object to, prejudicial sentiments from a potential viewer, talk content being too complex for a general audience, etc. ). Other things are a technical matter and can be made at least better or more tolerable, like recording quality.
The content is important, but not to the exclusion of everything else, otherwise no one would ever bother improving the presentation of said content. Good production serves the delivery of good content, like the substance of this talk.
@@SEAWORRIER The irony of someone who has a 3 minute video of him unwrapping a playstation complaining about quality.
Nasty sound quality... can't it be de-noised? Cool talk tho
Impossible to listen to man, come on :(
Acoustics are terrible
Audio please... as per the rest of the comments. RI are you incompetent?