It is a scandal of the greatest magnitude that the maths establishment have flatly ignored the insights of Charles Peirce and Ludwig Wittgenstein… into the nature of meaning, and particularly into the meaning of maths.  It is also an insult to their colleagues in Academia, because the gurus of maths have paid no attention —worth speaking about— to these revolutionary insights. The gurus of maths have simply continued to assume that they are “above” such “minor” issues. They still seem to think that their judgments are infallible, indeed timelessly infallible… a ludicrous conceit, because they demolished the priceless culture of education in the 1960s with their wholly unrealistic, unwise “New Maths for Schools”, and ruined the mathematic education of a whole generation of students.  Since that disastrous fiasco they have left maths education withering in the wind, thus showing no responsibility at all for their own discipline.

 But… the opinion leaders among today’s maths gurus know that they have a secret weapon:  one which may save their bacon for years to come.  It is that the brightest of their lay school contemporaries —and especially those who have since risen to posts of influence in the media—  have belatedly realised that they ought to have taken more notice of maths when they were at school. They have, in other words, come to realise that maths is actually an area of important knowledge, which should not be lightly rubbished. This is a source of acute embarrassment, and it tends to lead them to back the old-fashioned opinions of the wayward gurus in spite of the gurus’ loss of ordinary public support.

These influential voices are not aware that the Official Story in Higher Maths already came off the rails around 1900, and entered a fantasyland which involved championing “indefinables”. These dominant gurus will simply not admit that the exotic notion of transfinite sets is self-contradictory, or indeed that the vast majority of sets are not bona fide mathematic objects.  Their predecessors at the turn of the 20^thcentury thought that their standing with the educated subset of society was sky-high and cast-iron: so cast-iron in fact that they could discontinue their quest for truth, and could luxuriate in the possibilities of formal fantasies.  But when the Peircean applicability of maths began to flower —after the arrival of computers— it became evident that it was this feet-on-the-ground Peircean applicability which was impressing the consensus of the educated, not the counter-intuitive, exotic notions of their  intoxicated, creative virtuosi.

There are two brazen Denials which they will not admit: (1) There can’t be sets composed of more than a countable infinity of elements, because every maths object requires a definition, if it is to be bona fide. And there is only, at most, a countable total of possible definitions. (2) most sets are not mathematic objects, because they are collections of artefacts, people, animals, plants, etc. and these everyday words are not remotely “well defined” by mathematic standards.

It follows that their membership criteria are not mathematically bone fide. The only sets which are genuine mathematic objects are those sets which are wholly composed of bona fide mathematic objects.

The result: higher maths became “an uncertain, aesthetic activity” around 1900… and a majority of its practitioners have still not conceded that this was a serious blunder. Maths stopped, in a word, being the heartland of absolute truth.  Absolute truth used to occur in maths, because its results can normally be checked twice, three-times, four-times, five-times etc. But this heavy re-checking becomes impossible once it is decreed from above that some of the items are indefinable.

For nearly 13 decades rigour in maths has been shaky, but no one in-the-house-of-maths seems to be willing to admit this monumentally embarrassing lapse.

NOTE: You can send your comments on the reasoning promoted on this website by sending your thoughts to: per4group@gmail.com
CHRISTOPHER ORMELL around March 1^st 2026. chrisormell@aol.com