Books like A Mathematician's Apology

A Serious Business of TasteThe dominant theme of A Mathematician’s Apology, established from the first page, is one of aesthetics. Aesthetics, the study of what is inherently important and valuable, is for Hardy the fundamental power of mathematics, not an incidental result of correct thought. Aesthetics, while not unique to mathematics, is arguably more single-mindedly applied in mathematics than in any other human activity, including art of all kinds.Hardy, like many poets and artists as well as other mathematicians, is hesitant about his exposition of the aesthetics of mathematics. It seems to him vaguely disloyal and a possible waste of time. Mathematicians, after all, do mathematics, they don’t write about doing mathematics. And he has a point: either one gets the aesthetics of mathematics or one doesn’t. Mathematicians could care less about their reputations outside of mathematics. Hardy is therefore wary of going over the line from an ‘apologia’, that is, an explanation, to a defense, lest his should offend his colleagues. Most creative artists, I imagine, feel similarly.Hardy is acutely aware that an aesthetic, that is, a specific criterion or set of criteria, is that which is valuable for itself, and for no other reason. The aesthetic aims at nothing practical, nothing beyond itself. It is an argument from first principles that cannot be gainsaid by any other argument. It proves itself by its own assertion and by its own internal logic. Paradoxically, this is what makes an aesthetic so powerful: it doesn’t care what is thought of it outside the rules of its own creation. It is the scale of its own value. Its attraction is precisely its special kind of absoluteness.Aesthetics therefore is a dangerous business. It holds itself apart from criticism of any kind from those outside the circle which embraces it. It is an elitist activity. Its justification is merely the complete indifference about whether others subscribe to its views or not. There is no compulsion for others to ‘belong’ nor even to recognise its existence. It does not even claim any right to exist, for that would imply a purpose beyond itself.When a mathematician asserts that 1+1=2, there is no meaning beyond that assertion other than the expression of the aesthetic of mathematics. The assertion may have implications. Indeed this assertion has vast implications for the practice of mathematics. But that is the extent of its mathematical significance. It is a start in expressing the relevant aesthetic, but it is not intended to make the world better, or more intelligent, or more interested in mathematics.Does that make all aesthetics equal? Hardy says that the highest ambition “is to leave behind something of permanent value.” Value cannot be permanent unless it is intrinsic, of value in and of itself. But such intrinsic value is not arbitrary. The aesthetics adhered to by the professions - law, medicine, science - for example are distinctly different and incommensurate; but they are not arbitrary. Rather they are arrived at through social processes and accepted for what they are - the way we do things.Nonetheless a distinction can be made between fundamental criteria of action (I hesitate to call them aesthetics at this point) which are hidden, implicit, and unexpressed, from those that are made explicit, revealed, and given for consideration to others without the threat of compulsion. It is only the latter that are aesthetics. The former we can categorise as mere prejudice or, at best, unconsidered preferences. An aesthetic then must be articulated and expressed to be considered as such. When it is, it develops an uncanny power. The greater the precision of the articulation, the greater its attractiveness. I can’t account for this except as an empirical observation of the way in which people respond to aesthetic propositions. ‘Justice’, for example, as an abstract criterion is far less captivating than ‘the just treatment of relationships among those with same sex preferences as a matter of law’. This latter aesthetic has become accepted throughout Western Europe and North America as it has become expressed. An aesthetic is likely to attract those with a talent to employ it creatively. If for no otter reason than the social comfort of being among ‘like-minded’ people, that is others who appreciate not just the same things but the ability itself to appreciate those things. Hardy identifies curiosity, professional pride and ambition for reputation within the profession as general aesthetic aspects, applicable to many others than mathematicians. The principle aesthetic criteria of mathematics that Hardy identifies, however, is that of ‘pattern’, more specifically patterns of ideas. It has always struck me that it is precisely this aesthetic that is presented in Herman Hesse’s novel The Glass Bead Game, published almost contemporaneously with A Mathematician’s Apology. The game in question is never described except to the extent that it involves the identification of patterns across otherwise discrete fields of human knowledge - mathematics, history, politics, painting, poetry, physics, etc. A mathematical aesthetic expanded universally in other words. My discussion with other readers suggests that indeed you either get this aesthetic, and find the book a treasure, or you don’t.Pattern-seeing rather than pattern-making is the essential mathematical skill. The difference is crucial, and what makes mathematics an empirical science. Numbers are there to be explored and interrogated. Mathematicians don’t invent, they discover, patterns that numbers have always had. These patterns are as real, perhaps even more real, than the patterns proposed by, say, physicists. The latter involve themselves with ‘strings’, and ‘quarks’, and ‘dark energy’, for example. But these are mere hypotheses in comparison with the factual solidity of the number 2, or the logical necessity of more than one cardinal order of infinity.There are ugly and beautiful patterns in mathematics. One might suppose that this distinction is also arbitrary. But as Hardy explains, it is not arbitrary at all, nor is it vague even if its details are obvious only within the profession. Hardy refers to this as the ‘seriousness’ of a theorem or a proof. A serious proof, like Euclid’s proof of the infinity of prime numbers is short, un-showy, and (surprisingly) surprising. It has a seductive elegance that does not so much force as it does invite acceptance.A component of seriousness is ‘significance’. This Hardy further divides into ‘generality’ and ‘depth’. Without these characteristics, theorems, however ingenious, remain curiosities of interest only to puzzlers and hobbyists. On the other hand, too much generality and a theorem becomes abstractly insipid. Depth is even more subtle and has to do with the virtuosity involved in solving a problem that has just the right degree of generality and difficulty - again using innovative or unexpected ...“line of attack” to get to a solution. Such a proof “should resemble a simple and clear-cut constellation, not a scattered cluster in the Milky Way.”Ultimately metaphors like this are inevitable, not because the criteria are vague but because they have been so internalized by practitioners that they are almost pre- (or perhaps post-) linguistic. They ‘know it when they see it’ not because they are inexperienced and unsophisticated dilettantes, but they are able to ‘walk the talk’ so well that it is unnoticed by consciousness. The aesthetic becomes truly a ‘law written on the heart’ for better or worse.Hardy makes several mentions of how the aesthetic becomes obsolete and how it might be modified. Cambridge, for example, “crippled its mathematics for a century” by insisting on examinations that were exclusively about technique at the expense of creativity. Only by having the personal courage to move against this established norm did the situation improve. The toleration of such ‘rebellion’ is clearly, therefore, a necessary characteristic of the aesthetic-society, as it were.A Mathematician’s Apology is a highly personal statement, as its title implies. But the fact that its contents are only usually revealed over late night brandies during international academic conferences, doesn’t make it idiosyncratic or merely anecdotal. Without the mathematical talent sufficient to participate in the community that shares the aesthetic, it is perhaps impossible to appreciate the power of the aesthetic Hardy outlines. My summary of Hardy’s outline is undoubtedly inadequate. But it is, I think, sufficient to establish the almost miraculous way in which a professional discipline can create and sustain criteria of value that are not only independent of economic or commercial imperatives, but markedly antithetical to them. One might say that mathematics is serious business indeed.Appendix: Aesthetics and Fake NewsAn aesthetic has no intentional meaning beyond itself but it does have an incidental effect in the sense that it eliminates any consideration of truth. Since an aesthetic is its own truth, it cannot be compared or verified by reference to any other truth. This might appear as an aesthetic defect until it is realized that an aesthetic has a great epistemological consequence. It eliminates what has come to be known as fake news.Fake news isn’t fake because it is intentionally wrong or not (although it may be). It is fake because it is irrelevant in a given aesthetic. Fake ‘fake news’ are purported facts which are presented for some reason other than their mere presentation. It exists when there is an ulterior motive that remains unexpressed, a purpose - political, economic, or otherwise pragmatic - which is beyond the simple factual assertion.It takes some practice to know whether one is dealing with an aesthetic or some other instrumental or intermediate criterion of value. The phrase ‘this is important because...’ is a giveaway that whatever is being discussed is not an aesthetic. Even what follows the ‘because’ - it is right; it is expedient; it is effective; it is sensible - may not be an aesthetic. Often we hide our aesthetic under layers of rationalization so that we may not be aware ourselves what our aesthetic is.When a politician claims that a news story is fake, it is because he has some underlying interest he wants to promote, some hidden aesthetic, possibly Power, possibly wealth, possibly reputation. But never ‘truth’, this being the underlying aesthetic whose revelation might be damaging to itself.When a physicist or a social scientist makes a claim about reality, he or she is also making claim that is fake. They may use the term ‘truth’ to defend such a claim, saying that it ‘fits the facts’ better than alternatives, or that ‘it will cost us less to do X than to do Y’ but these are statements that confirm the existence of some other criterion that constitutes the reason that their assertion should be accepted. Only when we reach the terminal response-point ‘just because’ have we encountered what can be called the fundamental aesthetic that can’t be defended, only accepted or rejected, have these scientists approached the directness of the aesthetic of mathematics.