The most extreme example of this <<shying away from the big picture>> has occurred in higher mathematics. Higher mathematics has suffered eight major public setbacks (acute embarrassments) since 1900. But the broad response of the higher maths community can be summed-up in twelve words <<Ignore these setbacks and just carry on as if nothing has happened>>. The gurus of higher maths are exceptionally talented people who enjoy solving very hard, abstruse formal problems. One might have expected them to arise and try to rethink their subject in the light of these disastrous setbacks. But there has been absolutely no sign of such a response. These gurus have simply opted to continue the performance activity they enjoy so much, and to consciously stop looking at the ominous gathering clouds. This is not a natural response. The natural response would be for them to try to defend their chosen, once highly regarded, vocation against its looming extinction.
So why would this depressive, doomed, short-sighted, syndrome emerge?
There is only one convincing explanation, viz. that they have fallen prey to a serious loss of intellectual confidence. They have lost their self-belief. It is a lost conviction that they —or anyone else— has the mental capacity to clarify the incoherence which the setbacks reveal.
It seems that they have, in effect, been completely spooked by three contradictions which stopped higher maths in its tracks around 1900.
The first one was that mathematics couldn’t count as a ‘science’ because it was not about real (empirical) objects, but about formal, well-defined, mathematic objects… i.e. humanly conceived conventions. This seemed to imply that mathematics could not be a genuine inquiry into truth… directly contradicting the fact that the most powerful, paradigmatic truths were to be found in maths.
The second was Cantor’s apparent discovery of transfinite sets which had far, far greater numbers of elements than ordinary infinity. But some mathematicians of the highest calibre —Kroneker, Poincare, Borel— pointed out that the universe of discourse of mathematics is limited by the fact that there is a finite set of symbols, and definitions of mathematic objects can only generate an ordinary infinity of mathematical objects. This means that Cantor’s transfinite sets can only be populated by a hosr of shadowy, indefinable, fairylike objects. (Objects we will never know.)
The third was Bertrand Russell’s discovery in 1900 that the set of all sets which are not members of themselves is necessarily a member of itself and also not a member of itself. This is logical reasoning destroying its own credibility.
There have been other embarrassing setbacks in mathematics since 1901, but these three are probably the ones which have caused most dismay.
The irony is that these contradictions can be convincingly explained by taking a commonsense, de-mystified view of mathematics. The first was solved comprehensively by Charles Peirce who pronounced that <<Mathematics is the science of hypothesis>>. The second can be solved by giving up the imagined “paradise” claimed by Hilbert in 1900, and by abandoning Cantorian theory. ‘Infinite’ means ‘not finite’. It was always absurd to think that there could be degrees of notness. The third can be comprehensively explained by recognising that there are two types of contradiction, the static (parallel) and dynamic (serial) kinds —as shown by the present author in his monograph Some Varieties of Superparadox (1993), which can be read on the internet.
So, the clouds effectively parted thirty years ago, and wake-up time has been around for long enough.
CHRISTOPHER ORMELL 1st October 2023