To continue to comment on the apparent complacency of the maths hierarchy’s failure to reconstruct the elementary proof which Fermat claimed he had achieved four centuries ago…  is like a red rag to a bull.  But it is essential. It can only be acutely embarrassing to accept that no one has —apparently— tried to find this elusive elementary beast.  The honour of maths is at stake.  How can we possibly be satisfied with this state-of-affairs?

Kurt Godel proved in 1930 that maths is incompleteable.  Anyone can reflect —if they wish— on the possibilities and consequences of new “named triangular points”. They can see immediately that the exploration of new special points is possible.  For example, there is the centroid of the triangle XYZ… where X is the centroid of a given triangle ABC,  Y is the incentre of the same given triangle, and Z is its ortho-centre.

Amateur Japanese mathematicians became mesmerised by such far-fetched possibilities in the 18^th and 19^th centuries. It was called “Temple Geometry”. Worshippers at a Temple could provoke such artificial inquiries to their heart’s content, leaving the new “Problem” on a Board outside the door of the Temple. It was a challenge thrown down to anyone in the congregation, and “Solutions” were eagerly expected the following week.

Let’s face the facts: these Japanese amateurs went on long inquiries which took them a long way beyond the classic body of theorems published by Euclid in Ancient Times (BCE).   This Temple Geometry was not widely emulated by European geometers. There was an unspoken feeling in the UK and the Continent that such inquiries were artificial, and pursuing them amounted to little more than a kind of busywork.

The Japanese themselves eventually came to this conclusion, too.

So a kind of busywork had been identified. But did it go far enough?  We know that some introspective giants of mathematics such as Pascal, Russell, and Simone Weil were agonisingly aware that what they were doing was risking being on an ego-trip.  (Were they really illuminating important hidden truths, or just doing busywork… Was there a risk that they were flattering their egos?)  Higher maths during all stages of its development has looked magnificent and in-effect as important ju-ju to the ordinary person. The lay public has no standard to differentiate between busywork in maths research and serious progress. This means that there is always a danger that what looks like “serious progress” to a mathematician may be only turn out to be classifiable as “busywork” when the smoke has cleared and the ordinary intelligent fan has had time to consider its meaning.

And this, one would think, would lead to the conclusion that reconstructing Fermat’s elementary proof of his Theorem would be a much-visited blue-chip.  It must be THE prime example of thoroughly “real”, “important” maths.  So whatever does the probable fact that Fermat’s elementary proof was valid, say about four centuries of subsequent failure —on the part of the maths hierarchy— to reconstruct it?  It is not a good sign of the hierarchy’s overall appetite for targetting genuine progress in maths.

Fermat was, after all, an amateur mathematician.  He was obviously thoroughly convinced that he had put-together a valid explanation: he would hardly lie in the margin of a book only he would be likely to look at.

To comment on the reasoning, email:  per4group@gmail.com CHRISTOPHER ORMELL around July 1^st 2026.