Review: Beyond Knowledge?

Beyond Reality

Crucially, a whole series of mathematical philosophers, including Russell, Godel and Turing, proved that Mathematics isn't the totally consistent and comprehensive discipline it appears to be. So it is difficult to comprehend how someone like Mochizuki can spend many years and 500 pages upon a wholly new branch of Maths with such weaknesses having long been established.

How could such a work possibly be checked?

And of course, you also have the fact that Mathematics is an idealised discipline, which, though it has deep resonances with reality, actually occupies an entirely different realm - a universe of pure form. It is all about pattern, as if pattern in all its forms is an integrated subject!

It certainly isn't helped by the extension of space to more than three dimensions, which certainly don't exist in reality, but can be very useful in dealing with patterns with more than three variables, by enabling the alternative "graphical view and means" to be fruitfully employed (though by proxy via Algebra).

Also, in addition, Mathematics has been regularly extended, so that it has moved beyond its initial area. The inclusion of Operators as a kind of number is the most dramatic example of far. And it can be confusing. For, using well-established methods originally developed just for numbers, but now also on an operator like "2" - the doubling operator, makes for ambiguity between numbers and operators. And the whole realm of so called Complex Numbers is actually really concerned with operators, based upon the ubiquitous "i" - the operator "turn anticlockwise through 90 degrees" - which when applied twice becomes "i²", also becomes the Inversion Operator "-1" - hence the perennial teaser "i = sort (-1)". Clearly Mathematics is no longer just about number but about the manipulation of form - including various extensions well beyond its original realm.

Now this view demands that formal relations cannot be primary. They certainly cannot be essential drivers of reality as many scientists seem to believe.

It has to be the other way around. The physical substances and their properties (matter) are primary, and the forms are common consequences.

The position of modern Sub Atomic physicists, with their Copenhagen Interpretation of Quantum Theory, must therefore be nonsense - for they always make form primary.

Nevertheless, the very manipulatability of Mathematics makes it a pre-eminent and revealing tool of students of many parts of Reality, and clearly the most powerful handmaiden of the sciences - it just cannot be the essence.

Now, early in this account in New Scientist (3036), a proof in Mathematics is said to be a series of logical steps, leading from an established starting point to an undeniable conclusion. And it must be absolute! The Mathematical proof is supposed to demonstrate this without any doubts whatsoever. The problems involved at the present time are getting impossible, as typified by Mochizuki's work. Confirmation of a 500 page proof is so daunting nobody dare take it on. And, such colossal tasks are becoming more common in the field, as Mathematics stretches ever further away from its origins in Number.

The rest of this article seems to be about attempts to garner the possibilities of using computers to "lighten the load"...