Mathematics is, broadly speaking, a highly disciplined, rational, rigorous, carefully-led human activity.  In spite of this general reputation, however, there were four moments in history when the leadership of the subject lapsed into uncharacteristic, palpable, mind-boggling, inexcusable irrationality. This blog asks the question <<How could such huge lapses of judgment happen?>>. How could the discipline, reason and rigour we are so proud to represent in mathematics go so glaringly and hopelessly astray?

When, in the 6^th century BCE, the brotherhood of Pythagoras discovered that the squareroot of 2 cannot be a fraction, they decided that <<God had made a mistake when he created the universe!>>. It was., they thought, a profoundly shocking piece of inside information which must be kept from the intelligent public at all costs. The brothers were told that they must not leak this secret to the hoi polloi under any circumstances.

It was, by modern standards, a preposterous line.

How could such silliness pass unchallenged? If the universe was indeed created by a God (=an infinite supermind)  He would certainly not have overlooked this supposed “mistake”.

If anyone put out such a secretive denial today they, would be regarded as unhinged… treated with derision, and steered towards psychiatric therapy.  There was only one saving grace: that this lapse of sensible judgment was confined to the brothers.

In the mid 19^th century George Boole published his Laws of Thought and thus became the founding father of Set Theory. It came out of the blue. It was an event which must have brought great cheer to the modern mathematics movement of the day.  Here was another totally new kind of algebra! The mathematical traditionalists, who were still complaining about “imaginary numbers” and “funny geometry”, would have to eat their hats.  It showed plainly that Galois’s new abstract algebra was not any kind of freak, rather a first exciting glimpse of whole new provinces of algebraic abstraction!  So Boole’s new algebra was wholeheartedly accepted, and later perceptive figures like Gottlob Frege were thinking about using it to create a subject-wide road map based on sets.

But there was an extremely simple question which, it seems, no one was minded to ask: <<Was this new algebra pure mathematics or applied mathematics?>>. Why, why, why was this question not brought into the public domain?  Of course grammar was against the “applied mathematics” answer, because there is a natural tendency to think that one can only “apply” something to the real world which already exists.  (This is a fallacy however, because there is no doubt that an entirely new kind of paint can be “applied” to walls and woodwork.)

The other reason why the answer could be presumed to be a foregone conclusion, was that very few higher mathematicians were taking an interest in applications anyway. Yes, they would try to solve difficult differential equations brought to them by the physicists, but they didn’t any longer involve themselves in the direct study of physics… or any other area of applications for that matter.  There were very few mathematical polymaths around in the mould of Descartes, Newton, Leibniz when Boole published his bombshell.

So the question, it seemed, didn’t need to be asked:  Boolean algebra, as it was initially called, was obviously pure mathematics!

Except that it wasn’t. All Boole’s examples of sets were collections of real things.  We are talking about sets of plates and saucers, tiles, knives and forks, animals, stars, planets, atoms, minutes, days, years. Sets of these real things could not possibly be “mathematical objects”!  A set of cows could be milked, but we don’t expect to be able to milk a mathematical object.  Sets of cups can be glazed, but we don’t expect to be able to glaze a mathematical object. (Bitcoins, it should be said, are electronic manifestations of mathematical objects.) 

Boolean algebra was, unquestionably, applied mathematics.  Later Bertrand Russell set himself the daunting task of re-casting the whole edifice of pure mathematics in terms of sets, which, however he had not looked at sufficiently carefully. According to Russell’s thesis in Principia Mathematica Part I ‘the number 2’ means ‘the set of all possible pairs in the universe’.  These pairs included pairs of gloves in Selfridges, sculling pairs who have won bronze medals in the last four Olympics, and pairs of shoes left outside Bhuddist temples in Thailand. If so, ‘the number 2’ cannot be a mathematical object, because mathematical objects have never made any permanent reference to things in the real world.  (When we apply mathematics to the real world we use mathematical objects to make-up temporary analogies in thought experiments about real circumstances.)

Does it matter?  Yes!  The only sets which can count as bona fide mathematical objects, are collections of elements which are already mathematical objects. (For example, the set of roots of an equation, or the set of prime numbers less than a thousand which end in the digits 3,7.) This has the serious consequence that sets cannot be the building blocks of mathematics.  This is because you need to know what counts as a ‘mathematical object’ before you can have any mathematical set. Some mathematicians, e.g. Conway, have tried to build a theory which treats all mathematical objects as sets constructed from the null set.  This gets round the difficulty, because the null set, lacking elements, does not rely on any prior definition of what counts as a ‘mathematical object’. However the theory lacks the kind of robust believability needed to be the basis of a subject of such potential importance as mathematics.  It smacks, rather, of sophistry, because it consciously tries to build an immense edifice out of a peculiarly airy notion. (It can be argued that there is no such thing as ‘the standard null set’ anyway: rather every universe of discourse has its own null set.) 

<<Is set theory pure mathematics or applied mathematics?>>  is probably the greatest unasked question in the history of mathematics. If it had been asked in the 1860s it would have saved Frege and Russell from spending large amounts of their valuable time on a doomed notion of the meaning of numbers.

In the 1920s the continental mathematical consensus backed  Zermelo-Fraenkel (ZF) axioms for set theory. Their axioms studiously forbad a set from being a member of itself. It was a desperate attempt to put Russell’s Paradox to bed. But it was also a denial of the obvious, namely that a set like ‘the set of all sets mentioned in this blog’ exists. Of course it exists… indeed it so obviously exists that it is difficult to understand what the statement <<There is no such thing as the set of all sets mentioned in this blog>> could possibly mean.

The whole beauty of set theory rests on the fact that a collection of items is precisely defined by “what satisfies the membership criterion”. ZF theorists were trying to impose an arbitrary, ad hoc exception onto this; “except when that thing is the set itself”.  Unfortunately at the time almost everyone had Russell Paradox fatigue, and they went along with it. Thank God that such blind scholastics were not around when the Arab mathematicians invented algebra. An equation like x = 4x – 9 would have been roundly condemned. <<How can this possibly make sense, if you have to know what 4x-9 is before you can know what x is?>>.

At the time Russell’s Paradox had already been an unsolved problem for more than twenty years.  It was an enormity, because it consisted of a proof that what was necessarily true was also necessarily false. If so, the necessity of logic disappears and with it the raison d’etre of mathematics. During those twenty years thousands of logicians and mathematicians had tried to grapple with the problem, a challenge of the greatest difficulty. ZF theory was an attempt to ban the difficulty by refusing to recognise self-membership.  But it is a misuse of the discipline on which mathematics rests, to try to use it to deny what all those thousands of logicians and mathematicians had taken as self-evident.  Rubbishing self-reference tacitly accepts that the necessity of logic can be treated with contempt.

The fourth moment of poor judgment shown by the leadership of maths was the New Maths for Schools revolution of the 1960s. That has beaten all previous standards for foolishness. It has, in effect, landed the human race in a state where genuine education is impossible, rationality is commonly disregarded and the leaders of mighty states (e.g. Trump) act as if there is no such thing as truth.

We need a new basis for mathematics and mathematical science: one which builds-in reason and responsibility for the future. 

PS The problem of Russell’s Paradox was finally solved by the discovery of dynamic contradiction in 1993.   

CHRISTOPHER ORMELL September 1^st 2021.