The time has come to blow away some cobwebs from the increasingly dusty body of supposedly established mathematical truth. There are a few fully accepted false results, which are there, embarrassingly, because in recent times mathematics has too often been handled in a slightly uncritical way. Consolidation has been neglected in a rush towards aesthetic novelty.  Stanley Ulam noticed in 1976 that far too much diversification had been allowed to happen in research.  The crisis this has brought about is peculiarly damaging to the subject’s good name.  It has produced a situation in which no leading figure can ever hope in future to have a clear, integrated picture of the vast sprawling mess which higher maths has become.  In this way the guardians of the subject have allowed a situation to develop which makes oversight, and therefore confident leadership, impossible.  The immediate effect of such unlimited diversification is to stultify anyone who tries to get a bird’s eye view of what has been achieved.  But it also raises the question why the leadership of higher pure maths has allowed this kind of stultification to happen.  Any professional discipline whose research effort gradually increases the overall impenetrability of the subject, is asking for trouble. Every now and then a phase of digestion, review and consolidation is needed.  

The five most serious glitches were listed in Blog no. 6.  Of these the chief mistake —the one which tends to dominate attention and controversy— must be, of course, the notion of transfinite sets.

The notion of a transfinite set is still a nominally accepted, integral part of the portfolio which gives ‘modern mathematics’ its sense of identity.

This is, by any standard, a serious aberration. It seems as if the professional discipline, mathematics, is now permanently locked into its earlier recognition of an extremely questionable notion.

QUESTION 1  How can transfinite sets exist when the universe of discourse of the entire subject is countable?

[Actually all universes of discourse are countable, because our attention span is finite and we can only recognise finite strings of symbols.]

QUESTION 2  Today most commentators seem to have followed Russell in recognising that there are not enough well-defined mathematical objects to form a single transfinite set. There are about sixty characteristic symbols used to define mathematical objects. They can be treated as a number system N₆₀ (base 60).  Every definable mathematical object automatically belongs to this number system, which includes the normal alphabet. (Treat the defining sentence of any mathematical object as a natural number in N₆₀.)   So the totality of all possible, imaginable mathematical objects is countable.  So Russell was right when he conceded that, if transfinite sets exist, they must be immense collections formed mostly of indefinable mathematical objects. The problem now becomes: How do you define an indefinable mathematical object?  How do you even show off a single example of an indefinable object in a convincing manner?

[No one has ever seen a single bona fide example of an indefinable mathematical object or ever will. We are being asked to treat them as slightly metaphysical, but invisible possibilities. However mathematics is a secure, mundane, feet-on-the-ground, discipline, quite contrary to metaphysics. Believing in these “indefinables” is like believing in ghosts.] 

QUESTION 3  Mathematics’ sterling public reputation for rigour rests on its strict observance of the principle that mathematical objects must be well-defined. So how can one square this totally non-negotiable basis, with the notion that there are shadowy indefinable, invisible mathematical objects?

[Mathematics must surely preserve its reputation for being credible in the ordinary public sense of ‘credible’.]

QUESTION 4  Can mathematics afford to go out on a limb and publicly acknowledge that it operates with shadowy items it can’t define?

[No. This puts it in the position of being a self-imagined Ivory Tower, a source of endless potential public derision.]   

QUESTION 5  It is clear that Cantor’s diagonal argument establishes that the real numbers are uncountable. But why did Cantor uncritically assume that the only possible explanation for this uncountability was that there were immense transfinite hordes of them?

[The obvious explanation for this uncountability is that the totality of real numbers keeps changing (growing), often in unexpected, previously undreamt-of, ways.  But since the definition of any newly dreamt-up real number is inevitably expressible as a natural number in N₆₀ the totality of all possible real numbers is only a subset of a countable whole. So Cantor’s approach won’t do.]

QUESTION 6  What grounds did Cantor have for assuming that the totality of real numbers has the minimal stability needed to count as being a ‘set’?

[It is clear that he was a naïve Platonist who accepted the simplification that mathematics is an entirely timeless discourse. But we now know, thanks to Godel’s theorem, that the totality of valid mathematical truths is incompletable, i.e. not timelessly fixed. So there is at least one thing which is not timeless in mathematics.]

QUESTION 7  How can the totality of real numbers be a set, when it never settles? (Cantor’s diagonal argument precisely draws  our attention to this necessary open-endedness.)

[A set in pure mathematics is a mathematical object, and as such it has to be timeless.  The corollary of this is that any ever-growing totality, like the totality of known mathematical truths, cannot count as being a closed ‘set’. The potentially ever-growing totality of new kinds of real numbers is in a similar case.]

QUESTION 8  How can there be different degrees of infinity when the word ‘infinity’ means ‘not finite’?  (The notion of degrees of notness is an absurdity.)

[Cantor thought he could avoid this dilemma by adopting Dedekind’s new definition of infinity, which said that an infinite set is one which is capable of being mapped 1-to-1 onto some of its own subsets. But he forgot that the full definition of such a mapping involved the concept of ‘ad inf’.)

QUESTION 9  How can the open-ended sequence of transfinite cardinalities fail to have a final super-ordinate cardinal item, if ‘potential infinity’ necessarily —as Cantor argued— implies ‘actual infinity’?

[We know there can be no final transfinite cardinal number because immediately provokes contradiction (Cantor’s contradiction). It is incidentally a curiously unhelful side-effect of the transfinite notion that it abolishes the concept of ‘the largest magnitude to which we can refer in mathematics’.]

These unanswerable questions show beyond any shadow of doubt that the notion of the transfinite is a chimera.  But to say that there has been a great reluctance to come to terms with this painful realisation, would be an under-statement. When the present author put these arguments into the public domain  in 2006 (in the journal Philosophy) in 2008 (in the Mathematical Gazette and The Ethical Record) and in 2011 (in The Higher). This provoked a backlash in the form of two sketchy attempts at rebuttal-by-wouffle which clearly revealed that these would-be critics had failed to make any serious effort to see the central point of the new anti-transfinite case. They had never considered for a moment that the Official Story might be wrong.

Somehow the circle of higher mathematicians will have to find an acceptable corporate process for pruning, weeding and clearing nonsense from their overgrown garden.  

CHRISTOPHER ORMELL June 1^st 2021.