"There is another remark which suggests itself here and which physicists may find paradoxical, though the paradox will probably seem a good deal less than it did eighteen years ago. I will express it in much the same words which I used in 1922 in an address to Section A of the British Association. [...] I began by saying that there is probably less difference between the positions of a mathematician and of a physicist than is generally supposed, and that the most important seems to me to be this, that the mathematician is in much more direct contact with reality. This may seem a paradox [...] but a very little reflection is enough to show that the physicist's reality, whatever it may be, has few or none of the attributes which common sense ascribes instinctively to reality. A chair may be a collection of whirling electrons, or an idea in the mind of God; each of these accounts of it may have its merits, but neither conforms at all closely to the suggestions of common sense. [...] A mathematician, on the other hand, is working with his own mathematical reality. Of this reality, as I explained in section 22, I take a 'realistic' and not an 'idealistic' view. At any rate (and this was my main point) this realistic view is much more plausible of mathematical than of physical reality, because mathematical objects are so much more what they seem. A chair or 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. It may be that modern physics fits best into some framework of idealistic philosophy -- I do not believe it, but there are eminent physicists who say so. Pure mathematics, on the other hand, seems to me a rock on which all idealism founders: 317 is a prime, not because we think so, or because our minds are shaped in one way rather than another, but because it is so, because mathematical reality is built that way."

That's not quite the argument you describe -- his point is more that mathematical objects as understood by the mathematician are more like mathematical objects as we encounter them casually, than physical objects as understood by the physicist are like physical objects as we encounter them casually -- the physicist will insist that the chair you're sitting on is really a sort of configuration of fluctuations in quantum fields, but if you count up to 23 then the mathematician will agree that what you just did really does reflect the underlying nature of the number 23.

(If you build all mathematics on top of set theory, then you will most likely treat the number 23 as some much more complicated thing. But you'll see that as an "implementation detail" that could be done in lots of different ways, rather than saying that really, deep down 23 is this complicated thing with lots of weird internal structure.)

For the number 23, I wonder there must at least 2 schools :

( … there are many …

One can think that there is really a number 23! It was discovered and somehow we human has accessed to it. I am not sure.

Or one can think about it. This is the mapping of empty set to 0, set of empty set to 1, set of empty set of empty set to 2 … …)

one can think of 23-ish item is a set with all 23 elements whose combination of any 2 elements does not reduce. You need a thousand page to prove 1 + 1 = 2, with the reason that the first 1 is not the same as the second 1 to avoid this operation collapse back to 1. Our counting always assume different object, but to be rigorous there is nothing in the first 1 is explicitly said in that equation is not the same as the second one, as pointed out by a previous Hn refer to latest article .

…

Or my beloved : there is no 23. Only 0 and an operation +1 exist. You can say 23 as the result of a marker after 23 +1 operation on 0. It is 0 +1 -> 1 … 1 +1 -> 0 +1 +1 -> 2, Qed. If you have 23 stones/… with you, you do a counting by doing a mapping to this 0 obj +1 ops in your head-compute somehow.

…

Thanks for finding the quote. Memory can be sketchy sometimes!

And the engineering takes their crude tools striking inanimate matter while the mathematician and the physics argue over the nature of reality.

In 1922 this claim will have seemed increasingly resonant as the contemporary quantum revolution upended the classical view of the world.

Mentioned in a footnote in that book is the following, which I have always rather liked: "A science is said to be useful if its development tends to accentuate the existing inequalities in the distribution of wealth, or more directly promotes the destruction of human life."

(He mentions it in a footnote mostly to clarify that he doesn't really quite believe it -- it was a "conscious rhetorical flourish" in something he wrote in 1915, and in the main body of the text he gives a less cynical account of what it means for something to be useful.)

Maybe that cynicism was one of the reasons he appeared to be against applications, in as much as they would be a form of monetisation or exploitation of his work. edit: Just read some more about him and it seems this may be well documented already heh.

You might be thinking of: "One rather curious conclusion emerges, that pure mathematics is on the whole distinctly more useful than applied." I wonder what he would think if he could see the ways in which number theory, once often regarded as the purest of the branches of math, is now used in things like cryptography.

"Apology" is definitely worth reading. Some of his opinions can seem rather elitist:

"Statesmen, despise publicits, painters despise art-critics, and physiologists, physicists, or mathematicians usually have similar feelings; there is no scorn more profound, or on the whole more justifiable, than that of the men who make for the men who explain. Exposition, criticism, appreciation is work for second-rate minds."

At the same time, he is very honest about himself - in fact, he seems to have been suffering from depression over what he perceived as a decline in his ability to do math at the level he was accustomed to:

"If then I find myself writing, not mathematics but 'about' mathematics, it is a confession of weakness, for which I may rightly be scorned or pitied by younger and more vigorous mathematicians. I write about mathematics because, like any other mathematician who has passed sixty, I have no longer the freshness of mind, the energy, or the patience to carry on effectively with my proper job."

Or:

"A mathematician may still be competent enough at sixty, but it is useless to expect him to have original ideas."

Or more sadly, but with some serenity:

"It is plain now that my life, for what it is worth, is finished, and that nothing I can do can perceptibly increase or diminish its value. It is very diffciult to be dispassionate, but I count it a 'success'; I have had more reward and not less than was due to a man of my particular grade of ability."

"If I had been offered a life neither better nor worse when I was twenty, I would have accepted without hesitation."

The personal reflections bookend a central portion where he illustrates with several examples (e.g. Euclid's proof of the infinitude of the primes) his feelings about the "importance" of math, its "usefulness", and the distinction between pure and applied math.

It's interesting to compare "Apology" to "Littlewood's Miscellany" (I recommend the Cambridge University Press version, which contains the essay "The Mathematician's Art of Work" - ISBN 0-521-33702-X). There is more math than in "Apology" and many anecdotes. J. E. Littlewood was Hardy's long-time collaborator.

> I wonder what he would think if he could see the ways in which number theory, once often regarded as the purest of the branches of math, is now used in things like cryptography.

I would say number theory proves his statement, although perhaps not his point.

Applied math is useful for the applications that are known at the time of its creation, and it is likely that it will remain with that level of applicability in the future, although if the real world applications that it is used for fall out of favor we might find that the applied math decreases in importance, given its importance is in its applicability, and the applicability of things has an importance contingent on the importance of the thing that they are applied to.

This is of course not 100% sure, as there can also arise new applications of things in the future.

Pure math on the other hand, being not tethered to any particular application on the time of its creation, may find all sorts of applications in the future, pure math has as such infinite potential applicability waiting to be discovered and thus infinite potential usefulness, whereas applied math has limited known applicability and thus limited known usefulness.

I've not read Littlewood's book. Thanks for the suggestion!