I recall a university lecturer telling us that mathematicians were still working to try and *rigourously* verify the validity of the techniques involved in the finite element method, long after engineers had started actually using it on real world problems. Very different mindset, in that respect!
I only know about perfect numbers because my wife once told me our wedding anniversary was perfect, which I incorrectly took as a compliment. We shan't live long enough for the next one...
That's interesting about the finite element method - and of course on some level if it works, it works! I think an interesting case at the moment is AI algorithms - a lot of them "just work" without any formal performance guarantees, but personally I'd definitely feel happier if we had a better understanding of what the limits and the edge cases of that are.
I’ve always liked maths for its demand that you don’t just solve for the particular, you solve for the general case (my father did a maths degree and he liked encouraging to think in that way).
I’m interested by two elements of “professional maths”:
- how the little or big breakthroughs come about (it feels like they’re indescribably intellectual leaps, not slogging away like someone learning times tables) and
- how what seem like abstract concepts are applied to the real world - logs being obviously the one we are most familiar with lately, but one wonders how often knot theory or similar gets applied in factories. I’d find that interesting to read about.
In terms of breakthroughs, I think yes, it's leaps to some extent. Though a lot of the time for me it's probably more routine than that, because leaps require a degree of inspiration. An analogy might be that there's a locked door to get through, and a key ring with some number of keys on which represent the "standard approaches". And of course maybe none of them work, but sometimes you can jiggle them around in the lock to make one work, or by seeing which ones nearly work then you can get a clue as to what might actually work.
And that's an interesting one. I'm not sure about knot theory specifically. The classic one is that number theory (not specifically the perfect numbers, but things not a million miles away) is the basis for a big chunk of the cryptography that's used these days, on your web browser for example. On some level the basis for this is that multiplying two 100 digit numbers together to get a 200 digit answer is doable (by computer at least!), but splitting a 200 digit number into two 100 digit factors is incredibly hard. https://en.wikipedia.org/wiki/RSA_(cryptosystem) But there's various other bits of pure maths underpinning various other cryptosystems as well.
Roger Penrose has some wonderful anecdotes of mathematicians talking to mathematicans about maths, often apparently incomprehensibly, and of maths emerging as it were from background and prior thought. Increasingly I think we do not know science or intellectual endeavour unless we know more of the history: logic, philosophy, demonstration (e.g. counting?) and coherence ('proof'?). Interesting indeed.
Yes, I think sometimes there's a common understanding between people in the same field, so someone can say "can you try the approach of [mathematician X]", and the other person will know what they mean - whereas to an outsider it can be meaningless
I love your dedication to things I would have a clue how to go about starting. You’ve reminded me of “Wondrous Numbers” that I read about probably about 40 years ago. So simple, and yet never proved. So far, all numbers have turned out to be wondrous, but there still might be be one that isn’t:
An interesting thing is that at least one person argues that it might be false https://twitter.com/AlexKontorovich/status/1172715174786228224 - he has a fun analogy with Conway's game of life, so the question becomes almost if you can construct something like a "glider gun" there that fires off patterns that don't eventually hit 1. But such an example might be *ridiculously* large!
Not specifically statistical advice, I'd say - the stuff I personally do is more at the methodology end than the applied end of things, though I do work with engineers and other more applied people sometimes .. To be honest I'm always a bit mindful of the Fisher quote https://www.brainyquote.com/quotes/ronald_fisher_394439 !
Interesting, thanks.
I recall a university lecturer telling us that mathematicians were still working to try and *rigourously* verify the validity of the techniques involved in the finite element method, long after engineers had started actually using it on real world problems. Very different mindset, in that respect!
I only know about perfect numbers because my wife once told me our wedding anniversary was perfect, which I incorrectly took as a compliment. We shan't live long enough for the next one...
That's interesting about the finite element method - and of course on some level if it works, it works! I think an interesting case at the moment is AI algorithms - a lot of them "just work" without any formal performance guarantees, but personally I'd definitely feel happier if we had a better understanding of what the limits and the edge cases of that are.
I’ve always liked maths for its demand that you don’t just solve for the particular, you solve for the general case (my father did a maths degree and he liked encouraging to think in that way).
I’m interested by two elements of “professional maths”:
- how the little or big breakthroughs come about (it feels like they’re indescribably intellectual leaps, not slogging away like someone learning times tables) and
- how what seem like abstract concepts are applied to the real world - logs being obviously the one we are most familiar with lately, but one wonders how often knot theory or similar gets applied in factories. I’d find that interesting to read about.
In terms of breakthroughs, I think yes, it's leaps to some extent. Though a lot of the time for me it's probably more routine than that, because leaps require a degree of inspiration. An analogy might be that there's a locked door to get through, and a key ring with some number of keys on which represent the "standard approaches". And of course maybe none of them work, but sometimes you can jiggle them around in the lock to make one work, or by seeing which ones nearly work then you can get a clue as to what might actually work.
And that's an interesting one. I'm not sure about knot theory specifically. The classic one is that number theory (not specifically the perfect numbers, but things not a million miles away) is the basis for a big chunk of the cryptography that's used these days, on your web browser for example. On some level the basis for this is that multiplying two 100 digit numbers together to get a 200 digit answer is doable (by computer at least!), but splitting a 200 digit number into two 100 digit factors is incredibly hard. https://en.wikipedia.org/wiki/RSA_(cryptosystem) But there's various other bits of pure maths underpinning various other cryptosystems as well.
Roger Penrose has some wonderful anecdotes of mathematicians talking to mathematicans about maths, often apparently incomprehensibly, and of maths emerging as it were from background and prior thought. Increasingly I think we do not know science or intellectual endeavour unless we know more of the history: logic, philosophy, demonstration (e.g. counting?) and coherence ('proof'?). Interesting indeed.
Yes, I think sometimes there's a common understanding between people in the same field, so someone can say "can you try the approach of [mathematician X]", and the other person will know what they mean - whereas to an outsider it can be meaningless
ok, definitely a Paul Simon fan :)
Yes, definitely (among other artists of course!)
I love your dedication to things I would have a clue how to go about starting. You’ve reminded me of “Wondrous Numbers” that I read about probably about 40 years ago. So simple, and yet never proved. So far, all numbers have turned out to be wondrous, but there still might be be one that isn’t:
https://en.wikipedia.org/wiki/Collatz_conjecture
Yes, that's another tough one for sure!
An interesting thing is that at least one person argues that it might be false https://twitter.com/AlexKontorovich/status/1172715174786228224 - he has a fun analogy with Conway's game of life, so the question becomes almost if you can construct something like a "glider gun" there that fires off patterns that don't eventually hit 1. But such an example might be *ridiculously* large!
And do you also solve or offer statistical advice services to other scientific disciplines?
Not specifically statistical advice, I'd say - the stuff I personally do is more at the methodology end than the applied end of things, though I do work with engineers and other more applied people sometimes .. To be honest I'm always a bit mindful of the Fisher quote https://www.brainyquote.com/quotes/ronald_fisher_394439 !