Raw Notes on Hardy’s Apology

August 25, 2018

Preface. Reading is a form of communication. It follows that one who loves reading loves socializing. Just that this particular mode of communication goes one-way only and not real-time. One ridiculously good aspect deriving from the one-way property is that it doesn’t matter how great the author was, the obscure reader entitles himself to an advantageous post, where he freely points his finger or makes satirical comments without consequences. The great body of my notes are of this worthless type. Haha! Usually I strive to write my book report coherently, focusing on the most important subject that got me thinking. But this booklet, albeit small, contains many a profound idea that branches out and is hard to merge. So this time, I go all casual and post the discrete set of raw notes \footnote{edited and expanded} in the following. There will be additional comments on one plot line regarding my conversion though.


Before Section 10 it was mostly me enduring personal attacks, if I remember correctly. There are remarks here and there that I could relate to. But all that ambition/immortality talk is alienating.

10 A painting may embody an ‘idea’, but the idea is usually commonplace and unimportant.

Never noticed this. It rings true. Just look at all the nudes, landscape, still life, mythological/historical paintings. The painterly effect that wraps the idea is usually more important than the idea itself in our appreciation of the artwork. In contrast, a piece of mathematical work has no paintbrush or words to dress itself up, only the idea in and is itself.

Beauty is the first test: there is no permanent place in the world for ugly mathematics.

This line is vehemently opposed by the twitter sphere – where I got the idea to have a look at this booklet in the first place. But one doesn’t have to take it as an offense as if Hardy meant that “ugly” mathematics do not contribute or have no place. Just not permanent place. In the long run, inevitably people will find more elegant proofs for the same thing. So those get to go into textbooks, and the “ugly” is replaced and hence loses its temporary place.

14 I do not know what is the highest degree of accuracy ever useful to an engineer—we shall be very generous if we say ten significant figures.

Thanks for your generosity. MATLAB default eps is 2.2204e-16. Not far.

As regards Pythagoras’s, it is obvious that irrationals are uninteresting to an engineer, since he is concerned only with approximations, and all approximations are rational.

Well, to this I have no objection. Control engineers love rational functions! It’s not to say there aren’t infinite-dimensionals in the control theory literature. Actually, they extensively study delay. But if an average engineer can get away with a rational approximation, why not.

At this point, I had yet to be exposed to his shocking belief. So I asked, isn’t mathematics a model of, hence an approximation to the physical reality? Which one is the ultimate? I would say the physical. In the physical world there is no infinity, an apparent contradiction with the existence of infinity in mathematics. So an engineer approximates true mathematical solutions, but is it further from the physical reality than the true mathematical solution is?

15 isolated curiosities… These are odd facts… nothing in them which appeals to a mathematician.

The two examples taken from Mathematical Recreations are to illustrate superficial theorems. They look a lot like that Hardy-Ramanujan number. I wonder if he would get upset at being attributed a trivial theorem to. But we may comfort him in saying, this attribution is more to the memory of you guys’ peculiar deathbed conversation than to the actual mathematical merit of the discovery itself. Even without resorting to this, he deemed his numerous early papers insignificant anyway. So he had enough unimportant results to not notice the additional one.

16 “It is the large generalization, limited by a happy particularity, which is the fruitful conception.”

A quote from Whitehead. Typically poetic, full of ambiguity, love, and passion. I could see his thoughts dancing. Delightful!


Hardy may rest assured that the discussion on generality and depth qualities has been helpful, no matter how imprecise the definitions are. Just like previously when it was about beauty, he admitted nobody could quite define beauty. But a man knows it when sees it.

18 There is a very high degree of unexpectedness, combined with inevitability and economy. The arguments take so odd and surprising a form… one line of attack is enough…

Oh, so all you want is drama! – I was dying to see what he thinks constitutes an elegant proof. He really consented to act as one of those men whom he considers second-rate. But I should notice that outside his first-rate contiguous four-hour morning routine, second-rate labor is the only option.

‘Enumeration of cases’, indeed, is one of the duller forms of mathematical argument… (and of cases which do not, at bottom, differ at all profoundly)

[offensive language to a certain field]

22 I believe that mathematical reality lies outside us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our ‘creations’, are simply notes of our observations.

Ohhh, that’s whyyy, many a difference in our views! I could still picture vividly how my eyebrows were raised to the moon. Like, if only you had told me earlier! It was quite shocking for someone who somehow chose the physical over the mathematical as “the ultimate” and never seriously contemplated the other, to have the other option flying into the face. Obviously I was not converted to the other camp so quickly. But thinking in a different light, even if I view mathematics as human constructs wholly dependent on the mind, as contrary to Hardy’s position, aren’t we ourselves also part of the physical reality? Mathematics as a manifestation of how we think and reason, is a surest and most unambiguous measurement of this particular kind of physical reality. Besides, the way he puts it is so charming that it starts to appeal to me – the first sign of conversion by the artful insinuation.

24 A chair or a star is not in the least like what it seems to be; the more we think of it, the fuzzier its outlines become in the haze of sensation which surrounds it; but ‘2’ or ‘317’ has nothing to do with sensation, and its properties stand out the more clearly the more closely we scrutinize it.

This “realistic” view of pure mathematics really starts to make sense. The clarity and constancy of our own reasoning stand out, whereas the hazy feel when we get in closer quarter to our physical surrounding is already stated in the uncertainty principle.

27 Its ‘tremendous effects’ have been, not on men generally, but on men like Whitehead himself.

You have overlooked Whitehead’s subtlety and thus have mistaken him. He was trying to insinuate that if average men had believed him, they would go into pure mathematics. If they were successful, mankind in general would be happier. It could be my imagined subtlety though.

28 As Bertrand Russell says, ‘one at least of our nobler impulses can best escape from the dreary exile of the actual world’.

Russell calls our existence in the actual world an exile… poignantly beautiful. If the utility of pure maths is to increase the general happiness of mankind, it can only do so by increasing the individual’s internal happiness, because the mathematical truth can only be internalized on one’s own. It’s the only constant, the invariance, the least common denominator of mankind, which no culture, propaganda, or religion can corrupt.

Mathematics is not a contemplative but a creative subject; no one can draw much consolation from it when he has lost the power or the desire to create … and it would be silly to bother about him.

The sad part. Some kind of creativity seems to last as the bearer lives. But I guess I could relate if it is the specific kind of creativity that takes no external form but the idea itself. Being another form of procreation – in the realm of mind rather than body – it can not escape the fate of losing reproductivity, as the mind is supported by the body.

Note Even if we grant that ‘Archimedes will be remembered when Aeschylus is forgotten’, is not mathematical fame a little too ‘anonymous’ to be wholly satisfying? We could form a fairly coherent picture of the personality of Aeschylus (still more, of course, of Shakespeare or Tolstoi) from their works alone, while Archimedes and Eudoxus would remain mere names.

Indeed an interesting point. And it brings the narrative back to ambition/immortality again. So let me expound on my alienation to it a bit more. The notion of leaving a name behind, or leaving one’s creation behind, has never extorted from me anything but a blank stare. How is that an attractive idea? An apparent analogy again exists in physical procreation: people often cite a sense of self-propagation as a reason of their desire to have children. But my lack of that desire is easily explained by the insensible but quite possible apprehension of the disgusting human expansion. It may sound like free will, but might be just a realization of nature’s policy on a certain individual. After all, the self-regulation of reproductive activities in accordance with environmental conditions is widely observed in other species. This line of reasoning, however, does not explain my lack of imagination about the ambition of leaving something creative in the mind realm. Does the world have trouble with too many abstract ideas or works of art? I can’t see that. Yet somehow, the opposite seems to me more like a merit, that is to leave as little trace of my former living as possible, and to anchor the meaning wholly on the present moment and for myself. If it turns out some of my works become immortal – whoa that will be great – but that is never the prime cause.