That the advanced democratic world is stuck in a quagmire of muddled, unresolved thinking… hardly needs to be said.    We are in a parlous state, which is potentially dangerous, and which could —all too easily— lead to a second “Dark Age”.We might even be heading for extinction.

So what are the most dangerous, unrecognised-toxic propositions which are still being treated as unproblematic axioms today?

What are the root causes of a situation in which virtually nothing is being done to address the heart of today’s most dangerous, perilous crises?

I’m afraid today’s core problems are mostly in maths, and they have been brought-about by seriously misinterpreting maths.   This is not good news, because, of all the subjects on the curriculum, maths is probably the most unchanging and least self-critical.  In the 1900s the gurus of maths found they were unable to    e x p l a I n    Russell’s Contradiction about the set of all sets which are not members of themselves.

Even after an impasse of more than twenty years —into the shell-shocked 1920s— they were still unable to throw any “light of reason” onto this utterly baffling contradiction.  So they brazenly, disgracefully, decided to ban the contradiction by raw willpower… It took the form of promoting Zermelo-Fraenkel axioms for set theory.  This became what Wittgenstein and Ramsey called the ‘Party Line’. They were not fazed by the obvious fact that these Zermelo-Fraenkel axioms did notdescribe set theory: what they described was a mathematical “zet” theory —quite similar to set theory, but one which did not allow its objects, “zets”, to be members of themselves.

If a zet could not be a member of itself, Russell’s Contradiction couldn’t occur —for zets. There was one snag:  it could still happens for sets.

In other words, they missed the $64 question which was:  however does the concept of the “set of all sets which are not members of themselves” manage to end-up being embroiled in a devastating necessary contradiction?

They were saying in the most visible way that they were out of their depth, and were dodging the question.

They were making a tacit statement to the effect that:

<<We are the supreme authority in maths, and our considered opinion is that this Contradiction is insoluble. We know Zermelo-Fraenkel axioms don’t explain how or why the contradiction arrives, but at least this no-go area can be closed down and forgotten if we all agree to adopt the new axioms>>.

Their statement expressed a total collapse in the authority of maths.

Maths, I’m afraid, can’t make do with this kind of second-best fudgery.  Either the truths it presents are 100% true —and self-evidently absolutely clear— or it still has a lot of work to do… getting there.  It has now —very slowly— leaked-out that the ploy was a fudge, and so the gurus of the 1920s effectively demolished their own previously unquestioned authority.

A PRIZE OFFER

To mark the fifth-year anniversary of the author’s so far unchallenged “bare hands” analysis of Fermat’s “Last Theorem” exposition, the author decided to offer a reward of £3,000 to the first person who could find a self-evidently exposition-derailing mistake in the on-line argument.  He or she would have to recruit two independent colleagues to check and confirm his or her claim —they will each receive £1000.

It is now July 1^st and  no such counter-exposition has been received.

The exposition in question is on blogs 1 and 2 of this website.

[However a new version of the offer will now take-over. The bounty is to be increased to £5,000. It is now being offered to the first person who can find a seriously disabling fault in the putative proof.  He or she will also have to recruit two checkers, who will verify the counter-thinking, and will receive £1000 each if such a disproof is found.  The new deadline is Christmas Day 2025.]

This is not rocket science… or even really difficult cutting-edge maths:  it is really elementary… and in terms of skills it barely needs more than double maths A-level in the to follow the reasoning.  Its validity or otherwise urgently needs to be established.

The author now intends provisionally to adopt the default assumption that his bare hands reasoning on the Fermat Problem <<Is most likely to be correct>>.  This is not a project to undermine Wiles’ brilliant advanced modern proof of Fermat: on the contrary.  Its object is to remove the intense embarrassment associated with more than 350 years of the mathematic community’s seeming abject inability to reconstruct what Fermat must have had in his hands in the 17^th century. To enter for the extended prize: send your name, putative disproving reasoning, and confirming colleague names,  to:  per4group@gmail.com .

PS The P E R Group’s new BULLETIN can be found on-line at:  PERtinence.co.uk/pertinence-news-1/

chrisormell@aol.com