We know that the formerly superordinate power of rationality crashed, sadly, in June 1901 when Bertrand Russell discovered his deadly contradiction.  It was not an ordinary,  freestanding, optional contradiction like that which appears when someone remarks idly that <<the weather is fine, even though it is dark and teeming with rain!>>. No one is forced to make such a self-stultifying claim. No “remedy” is needed, because any sensible person will simply avoid making such an obviously barbed assertion.

Russell’s deadly contradiction was much more serious than this, because it presented a fork, one of the prongs of which had to be true. EITHER the set of all non-self-qualified sets was a member of itself, or it was NOT.

The disaster Russell uncovered was that a contradiction followed from both these hypotheses.

So Russell’s ‘Deadly Contradiction’ was really a pair of contradictions, one of which had to be the necessary case.  Both hypotheses were shown to be simultaneously necessarily true, and necessarily false. Logic was supposed to be the pursuit of clarity of reasoning. Here it was destroying clarity of meaning.

It is much to Russell’s credit that he realised just how dangerous this was, and he spent three years trying to find a solution, that is, a cogent explanation of what had gone wrong.

But he could not find the slightest hint of an “explanation”. It was like looking for a shoe in an empty shoebox.  It was not that a “solution” had been somehow lost in a vast, mazy space too large to search.  The “solution” —if one existed— had been lost in a space too small to search, i.e. a nonexistent space.

It was less to Russell’s credit that he gave up at this point, and crassly proposed an unplausible red herring,  which tried to impose an arbitrary stratification onto sets in general. He also tried to give this proposal “authority” by throwing all his energy into writing Principia Mathematical an immense, grandiose, un-self-evident symbolic structure, probably the greatest attempt in history at trying to blind intelligent people with science.

Later, in the 1920s, the European mathematic hierarchy adopted Zermelo-Fraenkel axioms for set theory, a thinly disguised way of legislating the problem out of existence. It was a way to make it appear that the problem had been solved, when it hadn’t. 

Finding the twin Contradictions was bad news: not being able to find the slightest hint of an explanation was worse: pretending that a solution existed, when it didn’t, was sickening.  It was an act of betrayal.  It said —to anyone wide-awake to the issues involved, that the leadership of Western logic and mathematics had given up trying to understand their own subject. They had effectively abandoned rationality.

They were admitting —in a tendentious way— that the immense intellectual authority they had previously enjoyed in the eyes of the educated public had departed.

It has taken a hundred years for the full implications of this abject  failure, of the mathematic hierarchy to understand their own subject, to sink in.

Why?  Because making millions of ordinary intelligent young people conscious of their own mathematic inferiority has been virtually guaranteed by the methods used to teach maths in schools.  School mathematics has been pitched at ordinary schoolchildren in a way which is fully suited to the mindset of the mathematic hierarchy, and hopelessly unsuited to the mindset of all-but a tiny minority of children.

The subject is taught as being all about “manipulating uninterepreted numbers and uninterpreted symbols”. This is what professional mathematicians do, but ordinary people don’t need a preparation for doing this. They need a firm grip on the meaning of maths and what mathematics does for the human race.

A raw symbolic mathematics, as typically taught in classrooms, does not make sense in the ordinary sense of ‘sense’. It treats maths as a language in which the adjectives have been turned into nouns… which, for those who do not warm to maths, look empty and uninterpreted.

This is why we need a serious campaign to insist that maths is taught in schools via the realistic, narrative contexts in which it makes perfectly good sense, of the ordinary kind. We have been sleepwalking… effectively treating maths —which is at the heart of the modern world— as if it were meaningless.  We need to wake up and teach the compulsory mathematics curriculum in a way which makes sense. Those children, who fully identify with maths and intend to pursue a mathematic career, can be taught uninterpreted maths. The mass of ordinary children, though. require constant interpretations.

CHRISTOPHER ORMELL 1^st December 2022
[If you would like an online copy of the P E R Group’s Narrative Maths Manifesto, send an email to per4group@gmail.com asking for this.]