We are still in a Covid-19 pandemic, which is a health crisis of very serious proportions. We are also still stuck in Ulam’s Dilemma, a mathematic credibility crisis of very serious proportions.  There is not much we can do to ameliorate the Covid Crisis, but there is a great deal we can do to ameliorate the Ulam Crisis.

The Artform Turn of 1900 is the root source of the trouble.  It was a conscious decision to break away from the painful headaches of late 19^th century mathematical physics, and to try to turn higher mathematics into a delightful, luminous, abstract intellectual artform.

Recent developments have thrown new light onto both the occasion for this UDI (“Unilateral Declaration of Independence”) and its aesthetic criteria for what counts as ‘mathematical art’.

The first step is to recognise that mathematical physics was at its wit’s end in 1900. The experimental physicists had discovered two things which were visibly shaking the foundations of Classical Physics. The first was the Michelson-Morley experiment, which showed that light, a wavelike phenomenon, could not be a wave in any medium known at the time to experimental physics. It also destroyed the concept of simultaneity, as Einstein showed five years later in his first Relativity Theory paper.  This meant that time was relative to the observer… a contradiction in the very foundations of Classical Physics.  This posed conceptual problems which were so extreme they were simply off the normal scale of difficulty.   Something very different would be needed to square this circle.

The second was a consequence of the Stefan-Boltzmann Law of Black Body Radiation. The mathematics only worked if the laws of physics operated in a jumpy, non-continuous fashion. This, too, was a death knell for Classical Physics, because continuity had been naturally taken for granted for two hundred years. It was as serious a predicament as that faced by the Pythagorean Brotherhood in the 6^th century BCE. The Pythagoreans were forced to the conclusion that God had made a mistake when he created the universe.  That God would have made the laws of physics jumpy was unthinkable to earlier scholars. Now the unthinkable (Quantum Theory) had to be thought. Something very different would be needed to square this circle.

Physicists had, of course, always turned to their mathematical colleagues for help on deep theoretical questions. The problem was that these new theoretical questions were not just “deep”, they were bottomless pits of disorientation. In the circumstances no one can blame the pure mathematicians of 1900 from throwing in the towel. Their temperament, training  and education had not prepared them to think outside the box of Classical Physics.  They had not the slightest inkling where to start.

The physicists were unhappy, and additionally unhappy because the mathematicians weren’t giving them much help.  The mathematicians were unhappy because they had not been able to help.  

UDI was the obvious thing to do. But it would not be a very glorious kind of UDI to be seen to be running away from the deep conceptual problems of physics. So this UDI was spun as a “coming of age” and an “at long last appropriately autonomous status for the most senior academic discipline”.  Hilbert had, fortuitously, at the ICM in 1900, established the principle in higher mathematics that you can believe the basic assumptions you want to believe.  So here was an UDI which the higher mathematic community wanted to believe, and consequently soon began to believe. 

The principal problem about switching to aesthetic criteria for higher maths was the problem posed by superficiality. Aesthetics are all about the appearances of things, not the reality. There was always the danger that meretricious, portentious, or even wacky concepts might slip through the net, under the guise of ‘aesthetics’. This probably happened, but it would need dedicated scholarship to establish examples of this contention, because it would involve wading through hundreds of dubious, abandoned, long forgotten, obscure analyses and definitions. 

So Ulam’s Dilemma takes the form of a quagmire composed of millions of papers arising from questionable ingenuity and creativity: an early taste of “Anything Goes” within a tight, sophisticated logical context.

So what is the problem?  The problem is that a body of problemique has been allowed (actually encouraged) to form, which is now so vast and so inpenetrable that it must test the sanity of any conscientious critical investigator to the limit.  It amounts to an un-understandable monstrosity of quasi-arbitrary aesthetic reasoning.  It has had the same kind of effect as yeast, which first produces alcohol in small, pleasing quantities, but as time goes on the alcohol becomes more and more toxic, and in the end it kills the yeast. Ulam’s dilemma is having a similar effect.  It has become a reductio ad absurdum of the quest for superb aesthetic mathematics. Higher mathematics has become a vast hornets’ nest of variant abstractions. It is so large and so multifarious that it far outstrips the patience of those most dedicated and sympathetic to its ill-defined aims.

The key to what went wrong lies in these ‘ill-defined aims’.  We now know that to conceptualise the purpose of higher mathematics —as that of seeking to explore aesthetic reasoning— is much too vague for its own good.

This Artform Turn was precipitated by the  mega crisis in physics and the discovery of Cantor’s amazing, exotic transfinite sets.  Closer examination of Cantor’s supposed ‘paradise’ would have shown it as being already a nightmare, because although Aleph 1 is vastly, indescribably, larger than countable infinity, it is unbelievably tiny and infinitesimal compared with Aleph 2.

Aleph 2, though vastly greater than Aleph 1, is vastly tinier than Aleph 3… and so on.  We are left with a panorama stretching to infinity of transfinite cardinalities which keep summersaulting between indescribably vast extremes.       

Must higher mathematics end in this excruciating defeat?

No!  But it needs a radical re-think about the taken-for-granted “excelsior” of moving towards ever higher and higher abstraction.

We need to wake up, and realise that the only way forward in pure mathematics lies in researching relatively simple arguments which, when found, can be greatly enjoyed by the entire maths-literate community. 

CHRISTOPHER ORMELL August 1^st 2021.