It will not have escaped the notice of most readers that we are passing through a period of crisis and general unease. The amount of rigour being applied in public affairs seems to have fallen to an all-time low. Lying seems to have become an accepted mode in parts of the media, politics and even the affairs of state. Probably the most shocking aspect of this tendency is that a lot of people who call themselves ‘scholars’ and ‘intellectuals’ are going round saying that there is no such thing as truth. They are spreading the notion that we are now living in a ‘Post Truth’ Age. 

This is not something likely to implode or pass by quickly. Rather it is part of a tsunami of sick attitudes which appear to have come to stay. Do we need to say it: these attitudes lack any respect for the values embodied in mathematics? 

They are part of a trend has been growing bolder and bolder since the 1970s. Smartphones, tablets and computers —gadgets which operate thanks to their precise mathematically devised technology— are paradoxically amplifying the cynical, sleazy and irrational side of human nature. We can’t go on shrugging this off for ever: it poses dangers. It is like a disease which has spread so much that it has already turned into a pandemic. And let’s not beat about the bush —it poses a serious threat to the continuance of mathematics as a great subject, vocation or career.  

Can anything be done about this dangerous situation? Well, those of us who care about the standards and values of maths, are on the spot. Unless we start fighting this apparently unstoppable slide into chaos and unreason, it will consume us all. To think that “nothing can be done” is to acquiesce to defeat.

Everyone who holds mathematics dear should turn the full power of their analytical thinking onto this existential struggle. If we don’t, we are heading for oblivion.

A full-hearted commitment to defending the values which underly mathematics must be our norm. Anything less is a cowardly capitulation.

So what can be done?

Well, first, we must take heed of Morris Kline’s book (1980) Mathematics, the uncertain science. As a result of sloppy thinking, a great deal of fog has been allowed into mathematics. Much of it arises from David Hilbert’s impassioned statement in 1900 that <<nothing would induce him to give up the paradise that Cantor has opened to us>>. If he said this today a host of critics would pounce on him and point out that he was saying that <<you can believe what you want to believe in mathematics>>.

This is the ‘Woozy Belief Problem’.

Yes. David Hilbert, the most brilliant mathematician of his generation, was unconsciously accepting that you can believe what you want to believe in modern maths.  

We simply can’t accept this attitude, which, by implication, trashes the idea that truth is absolute.  Truth is non-negotiable. This truth about truth is especially important in mathematics, because the whole subject has emerged historically as a result of attempts to prove (=establish the truth of) precise mathematic conjectures. This is where maths comes from: this is the heart of the subject.

So is Cantor’s theory of the transfinite still a “paradise” today? Hardly. It has thoroughly alienated our companion communities, computer science and physics. 

In any case its legitimacy is hanging by a thread, because consensus opinion has conceded that transfinite sets are mostly made up of indefinable mathematical objects.  (There are only, at most, a countable totality of mathematic objects which rest on bona fide definitions.  So the vast mass of the elements of any transfinite set —if there are transfinite sets— must be indefinable objects.)

But do indefinable mathematic objects really exist?   It is a question similar to asking if non-existent objects exist. 

We can’t believe what we want to believe in logic either. This is another aspect of the Woozy Belief Problem. We will never see an indefinable mathematical object on a whiteboard (or blackboard) during a maths lecture. How could it be legitimately introduced? Bona fide mathematical objects are introduced via definitions. 

Anyone who accepts these phony indefinable objects is inadvertently bringing a lot of fog into the house of mathematics.  Mathematics’ ultimate credibility with the public rests on our reputation that we only work with well-defined concepts. 

So the ingredients (indefinable real numbers) needed to bake the transfinite cake simply don’t exist.

Cantor made a great discovery when he realised that some infinite totalities are countable while others are uncountable. But he misinterpreted the uncountable, because he assumed that all mathematic objects are timeless. In the 1880s probably no-one gave the matter any thought.  So the general opinion was that there was a set of ‘all the real numbers’. 

Fifty years later that Kurt Godel proved that mathematics is incompleteable. This is something on the other side of the Woozy Belief Problem: something we need to believe, which is too often being sidelined.  A proof of this kind is not optional.  It means, inter alia, that the totality of well-defined real numbers is incompleteable. Of course it is. We can’t stop creative young mathematicians dreaming-up new definitions. There will never be a moment when it is possible to list categories of real numbers and say “This is all there are”.  

This means that the totality of bona fide real numbers will never settle.  In the year 2100 it won’t be possible to say “At last we know the categories of all the real numbers!”. Our descendents won’t be able to make this claim in the year 3000 either…

Godel’s incompleteability theorem should have been a culture shock of the first magnitude when it appeared in 1930. But none of the opinion leaders in maths saw the wood for the trees. The “wood” was that if mathematics is incompleteable, it is certainly not timeless. So an assumption which had been in place for more than two millennia was wrong. Perhaps it was a sign that opinion about everything at the time (especially in Germany) was so fragile that they could not take-in a major conceptual change like this. Whatever the explanation,  it was an early sign that leading figures were no longer trying to make sense or understand their subject. Reason had been abandoned. 

A totality which never settles cannot be a set. It is an ever-changing thing like the shivering mountain in the Peak District. Such a totality of real numbers may be called a SET (a Systematically Elastic Totality).

Who was the first person to perceive this truth that there is never going to be a final ordering of the real numbers?  His name was Georg Cantor! 

Oh dear. Henri Poincare warned everyone that the transfinite was a dangerous illusion in 1905.  Few took any notice. They thought that it meant we all had to become finitists, another barbarism.  Horror of finitism was used as a bogey to scare critics off.

This is hard, because it implies that a considerable body of supposedly valid 20^th century mathematics, e.g. measure theory, transfinite induction… is based on an invalid base. It will all have to be re-thought and re-constructed. We need a reform movement in mathematics to re-establish rigour. Without rigour, maths is not worth the paper it is written on.

Author’s email: per4group@gmail.com 

ANOTHER INFLUX OF FOG

Another moment when fog was officially welcomed into the house of mathematics was the official adoption of Zermelo-Fraenkel (ZF) set theory in the 1920s. 

How did this extraordinary capitulation  —using corporate willpower to push-through a result— come about? 

Well, Hilbert’s “paradise” seemed to be threatened, because set theory itself contained a necessary contradiction. (Russell’s Paradox, which was really a very serious contradiction. More than twenty years had passed and there had been not the slightest glimpse of how a solution —i.e. a lucid, clear, exemplary explanation— might be found.) If set theory had to be abandoned, Hilbert’s much admired paradise would have to go.

So ZF theory contained an axiom which outlawed set self-membership —in blatant disregard of obvious truths such as the fact that “the set of all sets mentioned in this blog” undoubtedly satisfies its own membership criterion.

It was a brazen example of using willpower to give the appearance of a solution when no solution had been found.

The long-term effect of the acceptance of ZF theory was quite insidious, because it soon became common practice to treat it as the norm. This meant that no one was going to bother to try to figure out what had gone wrong —in some unknown extremely fundamental way— in Russell’s Paradox. (Quite a lot of fruitless arcane mathematical logic was pitched at the problem, but no satisfactory solution was ever found.)

Decades passed. ZF theory, which was originally only, at best, a temporary device, graduated gradually into becoming an “official truth”. (Though later versions of restrictive set theory, e.g. Quine’s, were slightly less blatantly arbitrary.)

The present author, Christopher Ormell, first published a proper solution to the paradoxes of self-reference in 1959. It was noticed by Karl Popper, but otherwise comprehensively ignored. Later he gave a much fuller treatment in a monograph Some Varieties of Superparadox (1993).  This was re-published on-line by an Austrian group in 2003.  The solution is quite simple: it is that there is a second category of contradiction, contradiction-over-time or dynamic contradiction.  Such an account can only be accommodated into set theory, though, if we are prepared to re-think some hoary Platonic assumptions about mathematic truth, which have been handed down over a period of more than 2000 years and are now hopelessly out-of-date.

Mathematics used to be the Heartland of Truth. Whilst the leading mathematicians believed this, the educated public followed suit. It is a shaming fact, unfortunately, that some mathematic gurus of 1900 and the 1920s quietly compromised their position. They didn’t make a big thing of it, but they started believing what they wanted to believe. Once the Heartland of Truth had been abandoned, though, it was only going to be a matter of time before Joe Soap followed suit.