by Christopher Ormell (January 2026)
Math was the much admired backbone of Western Civilisation for twenty-five centuries. But since the arrival of computers, and the emergence of obscure internal lapses inside math, it has been widely seen as broken-backed. Fog has been allowed into the house of math, a place which should be the heartland of clear thinking and logic. Morris Kline pronounced this with his last book (1980) entitled Mathematics, the Uncertain Science.
However, in spite of this common perception that they have lost the argument—and as a result—their influence, its gurus seem to be managing to carry on more or less as if nothing untoward has happened. Their teachers and their leadership figures remain in place: their contacts in the media and education still mostly bow to their word, and there is still no sign in the math journals that the subject is in deep trouble. It presents itself as a kind of timeless ju-ju, which can still validly claim to enjoy the unique infallibility it used to proclaim with massive (i.e. overbearing) confidence. When I submitted an essay in this Review listing eight major blunders—a fully referenced, systematically down-played, devastating, embarrassing account (NER Nov 2023)—there was no subsequent attempt by the gurus to rebut my argument: there was no visible response at all. If an ordinary business, had made so many gross mistakes, they would have long since collapsed. But the academic higher math class manages to sail blithely on, seemingly oblivious to the mass of hidden objections and contradictions which are implicit in its baggage.
Students are still being taught that sets of common items (like a flock of birds) are mathematic objects, that totalities “unimaginably larger than infinity” exist, and that dimensions can obtain in halves! Such absurdities are, indeed, still being shamelessly applauded behind closed doors—i.e. in-house—as signs of the “triumphant magic of higher math”!
This triumphalism is no longer viable.
The gurus of math seem to think that their subject is timeless, and hence that it will easily survive the flak. But math is an activity which requires human exponents. Its symbols are timeless, but without exponents, they are as dead as a door nail.
The gurus are protected by the fact that almost no ordinary lay person is going to say “Boo!” to these High Priests of Math: probably because the lay leadership of civic society has a guilt-complex vis-à-vis the math. These lay leaders acquired it at school. They are now older and wiser, and as a result, acutely aware that they ought to have taken more interest in math at school. Meanwhile the gurus of math are maintaining their brittle bravado, and their in-house groupspeak has effectively replaced rigour in math. Relationships in math are extremely hierarchical … because its big beasts can normally easily ridicule the limited ideas of their lesser colleagues.
And it is likely to stay like this, unless a determined lay campaign can be raised to recognise that rigour (truth) matters wherever questions arise: that the admission of fog into the house of math should never have been allowed to happen.
In effect, the hierarchical relationships of math entail that the High Priests of the subject never encounter much opposition.
This is the heart of the problem. There is no adversarial counterforce. The High Priests of Maths are mostly highly intelligent, highly perceptive, creative individuals, but they are human, and they are (unbelievably) in denial about today’s humble wisdom, which is that we—all of us—need to face up-to adversarial challenges in order to maintain our credibility. A regime which categorically believes in its own 100% divine supremacy (i.e. infallibility) is effectively doomed. Some slips are inevitably going to pass, unconsciously, unnoticed, overlooked. This is what has happened to the High Priests of Math. Their first lapse of good sense arose from Cantor’s exotic transfinite totalities. Before Cantor, it was the accepted wisdom that infinity is not a number, and that totalities greater than infinity are a contradiction in terms. (The original role of the word ‘infinity’ was arguably to create a “sense of completion” where manifold open-endedness showed.)
All the great figures (Newton, Descartes, Leibniz, Euler, Gauss) agreed about this. But this venerable applecart was over-turned by radical young turks in the 1900s. They desperately wanted to abandon what they regarded as “tired 19th century notions.” They desperately wanted “to turn a new leaf”—one which would show-off the glorious modernity of their subject! Led by the young Bertrand Russell and the young David Hilbert, they let themselves believe than transfinite totalities were possible—even if the cost would be that indefinable mathematic objects (sic) would have to be accepted.
Hilbert even thought that his emotional attachment to the transfinite could validly trump its lack of logic!
This was a serious discontinuity. If indefinable mathematics objects are permitted within math, the entire consistency of the subject is thrown into dispute. Its central credibility rests on its definitions. It is this insistence on genuine definitions which gives math its bite. A subject which is willing to accept indefinable mathematic objects—when bona fide math objects are created by definitions—is self-evidently succumbing to sloppiness.
Today, a century later, we are in a situation like that of Martin Luther when he nailed his theses to the door of the church in Wittenberg. A once confident, brave, honourable culture has reduced itself—like the Catholic Church in 1517—to the pathetic all-purpose retort “We are right because WE SAY SO!”
Behind the fiasco of the transfinite, there was an even more disastrous illusion. It was that sets were the fundamental objects of mathematics. But a set like “The set of pairs of gloves in a shop window in Bromley” can’t possibly be a mathematic object. Four of the words involved here (‘gloves’, ‘shop’, ‘window’ ‘Bromley’) are obviously not defined to the precision needed in math. It beggars belief that nearly all the eminent mathematicians of 1900 swallowed this absurdity. They were determined to believe that sets were foundational. How could such a howling mistake be overlooked?
The young, self-named, “modern mathematicians” of the day were obviously thrilled-to-bits that set algebra, as enunciated by George Boole fifty years earlier, was neat, universal, and—crucially—different from, that of classical algebra. But sets themselves could only be awarded the primatur (that they were genuine “mathematic objects”) if their elements (members) were already mathematic objects. For example, {i, e, p, g} is clearly a mathematic object, whereas {some gloves, a shop, a window, Bromley} clearly isn’t.
These “modern mathematicians” of the 1900s were, we know, intoxicated with the notion that math was “unique”, “supreme” and “pure”. They had no time at all for “applied math”. They didn’t even try to hide their disgust at the very idea of “applications.” So much so, that these bemused, utterly unworldly, “modern higher mathematicians” failed to notice something which was under their noses … that most of the sets they were valorising were actually items of “applied math.”
They were also passionately dedicated to the idea that that sets “must be” the fundamental building blocks of math. They agreed that the Holy Grail of future work on modern mathematics “must be” to transform it into a unified system based on ordinary sets.
They were overlooking a hidden contradiction. A set cannot be the fundamental building block of math, because it is necessary to know “what a “mathematic object is,” before one can even gather together the items (elements) of a single bona fide mathematic set. This means that Russell and Whitehead’s Principia Mathematica was doomed from the beginning. It was predicated on the notion that math could be reduced, in principle, to a vast edifice of ordinary sets. It never seemed to cross their minds that the kind of “sets” they were championing could never be mathematic objects.
So we have a case here of a once-much-publicly-revered subject having fallen into fantasyland, and its champions into Denial. There has been a significant, but muted, public, general reaction to this aberration—namely that the word ‘mathematics’ has almost dropped out of common usage. Historians of all shades and colours are now wholly ignoring the immense difference which Newton’s mathematics made to Western Civilisation. It introduced the createability of new machines and systems. The Newtonian math created confidence in machines of all kinds. These historians have mostly convinced themselves that the Industrial Revolution was provoked by the Slave Trade, a disgusting side-effect of a new clockwork atheistic worldview which was dominating thought waves in the 18th century.
When is this extraordinary mass illusion going to collapse?
Much will depend on the fate of AI. If the full extent of the preposterous hype applied by the propagandists of Silicon Valley to AI becomes public knowledge, there is likely to be a financial crash of unprecedented proportions. Left to its own devices, AI has the capability to destroy higher education, and indeed to wipe-out the high-competence mindsets necessary to maintain a decent world order. Andy Thomas has shown (NER December 2025) that once the majority of text on the internet is AI-induced, the efficacy of AI will go haywire. But this is unlikely to happen, because this looming existential danger automatically provokes its own opposition.
The future is much more likely to be influenced by Anti-Math, which is the unimaginable, previously wholly unguessed, wholly overlooked, discipline which is dedicated to teasing-out the implications of transient systems. It introduces a “Living Logos”, because all the objects which are necessary for (responsible for) our sentient being, are objects with finite lives. Once this realisation has widely sunk-in, ordinary math will be able to settle-in as a more modest, civilised, positive, relevant agency in the public domain. (The tendency to overdo it will disappear.) Ordinary math offers a very good way of exploring the implications of projects which only involve mechanical, quantifiable factors. This is the best possible news for the gurus of math: their discipline will survive after all! It will be needed as a modelling logos—applicable widely, but not to everything, only inorganic and outline organic reality. It doesn’t and can’t, mimic the essence of living organisms.
So math has not been helped by its apparently blue-chip monopoly (as “the only possible abstract logos”). It has been left stranded, like a beached whale, when faced with living organisms. The best human minds have been dazzled for four centuries by the elegant qualities of pure math, and they have fallen over backwards (i.e. gone into Denial) defending its presumed monopoly. Now the full picture has been revealed: contrary to more than 2,000 years of dosey thinking, there are two quite distinctive 100% abstract logos, not one. We tried to get-by while only appreciating half of the space which could be opened-up by abstract exploration. So now a hopeful, wholly unexpected, bright future beckons.
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Christopher Ormell is an older philosopher of mathematics who solved Russell’s Contradiction in 1959. The solution was published in Mind, and was noticed by Karl Popper, but otherwise ignored. Later he found a mirror image of Descartes’ classic Cogito argument which he launched in a six-article series in the journal Cogito (1992-4 also ignored). He discovered superparadoxes, which generate vast numbers of contradictions (put online in 2003). He earlier found the first formula for the nth prime number without using trigonometric functions. (Maths Gazette 1967). He discovered explicit formulas for calculating [x] and |x|, and later spent 29 years searching for an elementary solution to Fermat’s last theorem. This putative reasoning has now been on-line for more than five years. (A prize was offered for its refutation, but so far no flaw has been found.) His main work, though, has been discovering a wholly unsuspected, spectacular, polar-opposite to math … Anti-Math, which unexpectedly brings methods, similar to those of math, to bear on the logical implications of transient forms. These forms are imposed—by willpower—securely onto random sequences. The result: Anti-Math enables the brilliant, civilised thinking of Kant to re-occupy centre-stage. We are beings who manage to secure our own existence by unconsciously applying life-affirming definitions to a wholly independent, inexplainable, neutral-random substratum. Websites: philosophyforrenewingreason.com, and mathsforrenewingreason.com.
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