A Mathematician's Apology (1940) [pdf](math.ualberta.ca)
math.ualberta.ca
A Mathematician's Apology (1940) [pdf]
https://www.math.ualberta.ca/mss/misc/A%20Mathematician%27s%20Apology.pdf
35 comments
Some context for his ageism: He was 63 when the Apology was written, and feeling very much past his prime and no longer able to do genuinely creative work in mathematics. The ageism is directed mostly at himself.
I really liked this book in my teens. I am not sure I would enjoy it now, a couple decades later. The ageism is part of it. His examples of mathematicians doing their best work when they were young are a little silly: who knows what great work Riemann and Galois and Ramanujan would have done had they lived longer...
> He's incredibly ageist and cynical about man's abilities to do great things. Definitely a curmudgeon, but mixed in with some good quotes and advice.
You are just a curmudgeonist!
You are just a curmudgeonist!
I like Hardy, but I dislike his ageism here too.
Erdős is a good counterexample to his argument. Even Euler. Many famous mathematicians have produced good work late in their lives.
Erdős is a good counterexample to his argument. Even Euler. Many famous mathematicians have produced good work late in their lives.
Erdos isn't a counter-example to anything. I mean I agree with you. Old people can do math. Look at Yitang Zheng. But Erdos is the exception to every rule ever.
Doesn't that make him the counter example to everything :)
Yitang Zhang, not Zheng.
Oh darn it. I originally spelled it Yiteng Zheng..I caught one typo. thanks
G. D. Birkhoff proved the Ergodic Theorem when he was nearly 50. It is one of the high points of twentieth century analysis.
I didn't it read it as curmudgeon.
I think people these days are just not emotionally honest, people have to be 'positive' all the time.
I think people these days are just not emotionally honest, people have to be 'positive' all the time.
This was Aaron Swartz's inspiration for, “A Non-Programmer's Apology”, where he battles with and ultimately justifies his own decision to favour teaching and campaigning over programming, despite being a talented programmer: http://www.aaronsw.com/weblog/nonapology
“I am saved, I think, because it appears that Hardy’s logic to some extent parallels mine. Why is it important for the man who “can bat unusually well” to become “a professional cricketer”? It is, presumably, because those who can bat unusually well are in short supply and so the few who are gifted with that talent should do us all the favor of making use of it. If those whose “judgment of the markets is quick and sound” become cricketers, while the good batters become stockbrokers, we will end up with mediocre cricketers and mediocre stockbrokers. Better for all of us if the reverse is the case.
But this, of course, is awfully similar to the logic I myself employed. It is important for me to spend my life explaining what I’d learned because people who had learned it are in short supply — much shorter supply, in fact (or so it appears), than people who can bat well.”
“I am saved, I think, because it appears that Hardy’s logic to some extent parallels mine. Why is it important for the man who “can bat unusually well” to become “a professional cricketer”? It is, presumably, because those who can bat unusually well are in short supply and so the few who are gifted with that talent should do us all the favor of making use of it. If those whose “judgment of the markets is quick and sound” become cricketers, while the good batters become stockbrokers, we will end up with mediocre cricketers and mediocre stockbrokers. Better for all of us if the reverse is the case.
But this, of course, is awfully similar to the logic I myself employed. It is important for me to spend my life explaining what I’d learned because people who had learned it are in short supply — much shorter supply, in fact (or so it appears), than people who can bat well.”
Despite saying,
"I have never done anything 'useful'. No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world."
He contributed a lot of practical findings in math. The Hardy-Weinberg principle[0] comes to mind...
[0]https://en.wikipedia.org/wiki/Hardy%E2%80%93Weinberg_princip...
"I have never done anything 'useful'. No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world."
He contributed a lot of practical findings in math. The Hardy-Weinberg principle[0] comes to mind...
[0]https://en.wikipedia.org/wiki/Hardy%E2%80%93Weinberg_princip...
It's almost frightening to me that the Hardy of the Hardy-Weinberg equilibrium could be writing this. It's part of a healthy understanding of genetics at all!
I very much doubt that Hardy regarded that as a "discovery". From a pure-mathematical point of view it's completely trivial.
(I do not say that to diminish its importance, which is an entirely separate matter. And something can be mathematically trivial but still an important discovery -- the cleverness may e.g. reside in noticing that it's a thing that might be true at all. Be all that as it may, I can't imagine Hardy, given his general dismissive attitude to applications of mathematics, seeing it as a discovery rather than a triviality that happened to be useful to biologists.)
(I do not say that to diminish its importance, which is an entirely separate matter. And something can be mathematically trivial but still an important discovery -- the cleverness may e.g. reside in noticing that it's a thing that might be true at all. Be all that as it may, I can't imagine Hardy, given his general dismissive attitude to applications of mathematics, seeing it as a discovery rather than a triviality that happened to be useful to biologists.)
Yes, that is true. He really did not care about it. The paper he wrote is Mendelian Proportions in a Mixed Population (http://science.sciencemag.org/content/28/706/49.full.pdf+htm...). It is brief, barely a page in length, and he introduces the HW equation with the preamble "A little mathematics of the multiplication table type is enough to that in the next generation the numbers will be [equation]".
As an added bonus, he cites Karl Pearson
As an added bonus, he cites Karl Pearson
Math is so huge......I've taken several math classes beyond calculus. Most people I know struggled with algebra, and are confounded and confused by my math skill. My math skill is so far advanced from zero that the average person cannot even conceive what such things mean.....even famous science writers don't understand eigenvalues, but I do.
Despite all that, I consider myself horrible at math. I am aware that I know only a small portion, and even that portion, poorly.
So maybe Hardy spent more time focusing on what he didn't know, and measured his achievements based on that rather than on what he did know.
Despite all that, I consider myself horrible at math. I am aware that I know only a small portion, and even that portion, poorly.
So maybe Hardy spent more time focusing on what he didn't know, and measured his achievements based on that rather than on what he did know.
The greatness of Hardy is that some unknown person from across the world (Ramanujan) wrote him barely legible letters, and he read them, believed in the writer, and helped expose him to the world.
I was once in a position where lots of people wrote me letters seeking to tell me about their talents. All the letters went unread to my dustbin. True and sad.
I was once in a position where lots of people wrote me letters seeking to tell me about their talents. All the letters went unread to my dustbin. True and sad.
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Anyone who enjoyed reading that (especially the bit about mathematical reality) absolutely must read Anathem by Neal Stephenson. The proof is rather enumerative, which I will leave as an exercise ;)
As for discussion. Is not the quote "We live either by rule of thumb or on other people’s professional knowledge." the most antithetical to hacker culture you've ever heard?!?
As for discussion. Is not the quote "We live either by rule of thumb or on other people’s professional knowledge." the most antithetical to hacker culture you've ever heard?!?
I did not care much for Hardys opinion about how innocent pure math was. Since it was later proved wrong by nukes. But the novella was a real glimpse into the austere life of a curmudgeon and some of the oxford culture. In addition to the culture of the elite as well, which unsurprisingly respect science and math much more than proles.
Well, it's not math that kills, it's people... I mean, I don't think it is fair to blame scientists, engineers, workers, etc. - let alone the tools that they use - for the way the product of their work is used.
Also, to be fair, the elementary analysis book which Hardy wrote under the title A Course of Pure Mathematics by today's standards would hardly qualify as a book on 'pure math' anyway.
Also, to be fair, the elementary analysis book which Hardy wrote under the title A Course of Pure Mathematics by today's standards would hardly qualify as a book on 'pure math' anyway.
>I mean, I don't think it is fair to blame scientists, engineers, workers, etc. - let alone the tools that they use
I think it should be fair. It's no secret what these precision guided bombs are being used for, so why should the nerds that continue to work on them be free of culpability?
I think it should be fair. It's no secret what these precision guided bombs are being used for, so why should the nerds that continue to work on them be free of culpability?
The invention of the nuclear bomb has led to the most peaceful period in human history. So to whatever extent the scientists and engineers who had some part in creating those bombs are culpable, they should be considered heroes.
No, it didn't. Democide in the 20th century, the death toll of which far outstripped the death toll from conventional warfare, reached its peak after the use of nuclear weapons against civilians in Japan. The notion that the post-nuclear era has been "peaceful" either absolutely or relatively is completely, ridiculously false.
Nuclear power is a fact of nature. It simply exists. Learning this doesn't cost you any innocence; it is nature itself that has conspired to make such things possible, not humans.
I'd like to hear more on this idea that nukes prove pure math not innocent. Surely it's applied maths (and physics, and engibeerung) that created nukes?
Sure, but what you think is pure maths today might become applied maths tomorrow. So nothing we consider today to be pure is necessarily so.
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Is the apology for posting something that should have been HTML as a PDF?
Here it is in HTML if you prefer: http://www.math.ualberta.ca/mss/misc/A%20Mathematician's%20A...
Why Hypertext Markup Language rather than Portable Document Format? Because you like ads?
PDF is meant for paper. HTML is meant for screens.
Learn LaTeX and then come back here. Or better yet, learn LaTeX, write some documents, wait 10 years, then come back after revisiting those documents.
I think I first started learning LaTeX in 2007 or 2008. I began writing HTML around 1995. IMO, anything resembling a "document" that you expect to be useful longer than a year or two should be written in LaTeX, or at least something based directly on TeX. Alternatively, just use plain text. Anything else just doesn't have a comparable shelf-life.
Whether it was intended to render to screen or print is totally irrelevant. If anything, stay away from screen-oriented formats because there hasn't been a standard "screen" format like, ever, with the possible exception of TTY geometries.
And output format isn't even the half of the relevant qualities to worry about when it comes to shelf-life. TeX is basically written in Pascal. TeX is a standard as well as its own future-proof implementation, permitting pixel-perfect reproduction across decades. I expect direct ports of TeX to Web Assembly not long after the standard sees adoption.
Of course, TeX is not the same thing as PDF; not even remotely. But TeX is oriented toward the world of hard copies, and I hardly see that as a fault. But even if so, it's de minimis in the grand scheme of things.
I think I first started learning LaTeX in 2007 or 2008. I began writing HTML around 1995. IMO, anything resembling a "document" that you expect to be useful longer than a year or two should be written in LaTeX, or at least something based directly on TeX. Alternatively, just use plain text. Anything else just doesn't have a comparable shelf-life.
Whether it was intended to render to screen or print is totally irrelevant. If anything, stay away from screen-oriented formats because there hasn't been a standard "screen" format like, ever, with the possible exception of TTY geometries.
And output format isn't even the half of the relevant qualities to worry about when it comes to shelf-life. TeX is basically written in Pascal. TeX is a standard as well as its own future-proof implementation, permitting pixel-perfect reproduction across decades. I expect direct ports of TeX to Web Assembly not long after the standard sees adoption.
Of course, TeX is not the same thing as PDF; not even remotely. But TeX is oriented toward the world of hard copies, and I hardly see that as a fault. But even if so, it's de minimis in the grand scheme of things.
Actually, 99% of written materials might be perfectly readable (on a computer screen) in plain text format. For them even HTML is an overkill. A good reader app will let you choose the RX ("reader's experience"), but even a simple text viewer/editor would do the job most of the time.
I particularly thought this was cute:
"It is one of the first duties of a professor, for example, in any subject, to exaggerate a little both the importance of his subject and his own importance in it. A man who is always asking ‘Is what I do worth while?’ and ‘Am I the right person to do it?’ will always be ineffective himself and a discouragement to others. He must shut his eyes a little and think a little more of his subject and himself than they deserve."