by Christopher Ormell (July 2025)
There have been relatively few examples of conceptual mistakes in science which were overlooked and subsequently allowed to stand for years—treated as if they were fully secure, fully grounded, trustworthy theory. One was the ancient doctrine of Phlogiston, which survived as the “Official View” of combustion well into the 18th century. It’s diehard proponents were still fielding feudal-style thinking, long after Descartes had done his damnedest to debunk the visibly scholastic dogmas of the Medieval era. Another was an ideological fallacy which surfaced in the mid 20th century. It was a Marxist version of Lamarkianism, which blatently opposed Mendelian genetics, and perversively became the “Official State” view of genetics in the USSR.
After WW2 an assumption seems to have set-in, though, that such anomalous obsolete theories were likely to disappear quite quickly, because faulty logic is inevitably going to stand out like a sore thumb… They would consequently soon end up banished in the free-thinking circles of modern science. But this was naïve. Thomas Kuhn unleashed his more realistic account of scientific practice in the 1960s—pointing out that theory was often based on “paradigms” —partially understood, blindly trusted emphases. This raised reservations about the “fearless rational thinking” which was supposed to be the order of the day. Kuhn’s wide-ranging critique of “Normal Science” was really saying that modern science was not nearly as “rational” as it was supposed to be. So this dismissive point of view eventually became part of a vast, demoralising sinkhole of existential doubt which ushered-in the beginnings of despair in the post-modern era (which may be dated from the 1970s onwards). It is still hovering over us today. It was a position fiercely attacked at the time by Imre Lakatos. Sadly, though, Lakatos died in 1974, leaving a void where healthy positive dialogue ought to be.
In spite of the Post Modern Pandemonium and the despair about science which lost its special elan in the late 20th century, it is gradually becoming clear that a long serious, fantasy-driven tendency has been, and is still being, valorised in academic math. This essay is offered as a way of seeing this astonishing aberration in perspective.
It began originally with “New Math”, which was originally conceived in the 1830s. It can be described as a “doubly-pure” version of the subject. It started with bizarre new abstract concepts, plucked out of the air, like “imaginary numbers.” (They were concepts wholly unrelated to math’s practical applications, and they didn’t arise from sustained experience in empirical science either.) Envy among the nationalistic Continental Elites may have may have played a part in creating this extraordinary initiative. The Continental Elites had been seriously miffed by Newton’s spectacular breakthroughs in calculus and mechanics: they desperately wanted some alternative math bragging rights of their own.
However, the new initiative was not an easy development to “sell” in 1830, because, for example, so-called “imaginary numbers” didn’t sound sensible to lay ears. (Insisting that they “really existed” as mathematic objects, sounded quite daft.) Actually “imaginary numbers” were (and are) no more damned by this perception than negative numbers, because saying that “there are -12 passengers on a bus” doesn’t sound very sensible either. The truth is that parts of a well-defined axiomatic math system may offer perfectly good analogies for some patch of physical reality, and other parts may not. (Imaginary numbers make very good sense when used in analogies for alternating current, but remarking that “there are 6i passengers on a bus” is obviously absurd.)
Doubly-pure math grew steadily and continued to flourish throughout the 19th century.
It seems that the intellectual exceptionality and adrenaline, associated with this daring doubly-pure math movement, had the effect of blinding the vision of its practitioners. Because, when George Boole discovered the logic of set theory in the mid 19th century, these “new math exponents” seemed suddenly to forget the elephant in the room—namely that, among the vast totality of sets, the majority were inevitably going to be applied sets. The only sets which were bona fide “mathematical objects” were sets whose elements were themselves mathematic objects, e.g. the “set of all prime numbers,” the set of solutions to a cubic equation … Probably the New Math gurus, whose enthusiasm was fired by doubly-pure thinking had—in effect—quite forgotten that applications of math existed (sic) alongside “pure” math! We know the gurus had a very low opinion of applied math. They took the view that the only “real” kind of math was the heady discipline they themselves espoused, arising from their doubly-pure ideas. In effect they were treating the word ‘real’ as meaning “what is worth studying,” not as a word associated with the raw shocks and gut pains of ordinary experience.
This oversight was asking for trouble.
The most damning implication was that sets couldn’t be the basic “bricks” of math, because they would have to be mathematic objects, and you need to already know what a mathematic object is before you can even talk about collections of such things.
But such sobering thoughts were never considered. Instead a flawed movement to try to make ordinary sets the basic concept in math was launched. Its remit was to embark on the immense project of re-formulating the whole body of math in these terms. This was seen as the major, promising, exciting, way ahead. It soon became a Holy Grail—one which was taken up by Dedekind, Cantor, Frege, and , of course, famously, Bertrand Russell. Russell’s bland assumption, which he seems to have never questioned—that all sets are mathematical objects—is actually a howler of the most howling kind. Sets are things like a stack of beakers, a pack of cards (or dogs), a collection of pictures. They simply can’t be “mathematical objects,” because they are defined by words with ordinary imprecise sensory associations. Math, by comparison, has always prided itself as being a wholly abstract subject based on rigorous definitions: using abstract nouns, abstract verbs, etc. Treating all sets as mathematic objects, is a brazen breach of this basic truth.
Consider the case of Fido the dog, who has been given a birthday present containing a set of bone-shaped biscuits. At the end of the day Fido has eaten all the biscuits, so he has eaten, in effect, a “mathematical object” —according to these set-muddled gurus!
The tram system (a set of trams) in Croydon (London) can convey passengers to Elmer’s End, but it doesn’t make sense to say that you can take this trip using a mathematical object!
Dr Hoenhauer is flying from Nashville to New York with a portable mini chess set in his pocket. According to orthodox math opinion, when they land at JFK this mathematical object (a set) will have travelled 886 miles by airplane! How orthodox math gurus can square the absurdity of these conclusions … while upholding the important principle that mathematical objects have no place in either time or space—beggars belief.
The absurdity of these examples says it all: it is an existential gaffe, it blows the whole notion of basing maths on sets sky-high.
This is really an embarrassing expose of the astonishing loss of sense underlying Russell’s main work. He used this misconception as the basis for his (and Alfred Whitehead’s) Principia … and his entire supposedly canonical notion of the nature of math and mathematical logic. Russell, along with his fellow set-intoxicated friends seems to have completely forgotten that that math has applications! They fell into this serious mistake because they had evidently been paralysed by the supposedly privileged, ineffable glory of their doubly-pure math.
It would be hard to think of a worse lapse in the history of math. Whyever didn’t Russell’s more skeptical friends tell him he was walking onto thin ice? However could the critical corporate professional judgment of his peers in the 1900s fall so low?
Actually, earlier, in the 1880s a more technical blindness about sets had surfaced. Georg Cantor propounded his wholly unexpected, exotic, notion that “transfinite” (“super-infinite” ) sets existed … they were sets with allegedly unimaginably greater populations than a “mere” ordinary infinity. Before Cantor appeared, it had been the unquestioned consensus of the math establishment that infinity was not a number. It was a way-of-talking which drew a line where the knowability came to an end.
So Cantor’s dramatic “discovery” was initially treated as nonsense. Kroneker, the leading German mathematician at the time, poured much scorn on the idea. But shock gradually turned into worship, when the younger generation decided it was a brilliant, once-in-a-lifetime, extraordinary, mind-blowing revelation… (Only already hopelessly intoxicated younger gurus could fall for this fantasy.) It was the most spectacular spin-off imaginable from the prevailing set mania—and one with apparently boundless, mazy implications. It was amplifying the mystique around math a thousandfold. It was a bit like discovering that “God had an unsuspected elder brother”!
It should be said at once that Cantor deserves considerable credit for his discovery that the totality of real numbers is not completeable. There will never be a point at which we can say that “we have conceived all possible types of real numbers”. This conclusion stems from Cantor’s famous “Diagonal Argument.” (If you listed all the recognised real numbers less than 1, you would have an endless stack of numbers like the decimal part of the squareroot of 2, the squareroot of 3, the squareoot of 5… etc. But this list (array of digits) has a diagonal, i.e. the digits which are m decimal places along the mth real number’s decimal point.)
It looks like this:
.4 1 4 2 1 3 5 0…
.7 3 2 0 5 0 8 0…
.2 3 6 0 6 7 9 8…
.6 4 5 7 5 1 3 1…
…And so on (the diagonal digits are shown in bold).
Now think of a wholly new real number where each digit is chosen from this bold diagonal with 1 added. (9+1 is treated as 0 if the digit is 9.) In the section shown above this is 0.5478 … This newly constructed real number will of course differ from every number in the stack!
In which case, the original stack cannot be composed of “all the real numbers!”
So there is this new unexpected diagonal real number to be taken into contention as well.
Its presence says we can never finalise (close) the concept of “the set of all real numbers.” We can never legitimately claim that we have conceptualised them “all.” This means that there can be no such thing as “the set of all real numbers.” The word ‘all’ implies completion: but there is no completion. Sets were conceived (originally by George Boole) as “the totality of all the xs,” where an “x” was a named kind of object. But there is no such thing as “the totality of all the real numbers”… because Cantor’s diagonal argument plainly shows us that whenever we assume that we have conceived “all the known real numbers,” our assumption will be dashed … when we focus onto this unique new real number, the modified version of the diagonal of all the previously known examples.
What Cantor had discovered was that the real numbers were incompletable, a word which only entered the vocabulary of math around 1930 with Godel’s famous proof. In the 1880s a much more simplistic, rigid notion was in place—all sets were supposed to be guaranteed, Godlike, timeless objects.
Cantor didn’t even consider the possibility that a totality might never settle.
In the end he went with the flawed view of his day that the “set of all real numbers” had closure (timelessness). He blithely reasoned that this implied that the “set of all the real numbers” must have significantly more elements than ordinary infinity. But, it can’t, because there is no such thing as “the set of all real numbers.” This is a non-starter.
What this reasoning shows is that there are ordinary infinite sets, like the set of odd numbers, and there are also SETs where “SET” stands for “systematically elastic totality.” Cantor was right to treat such SETs as uncountable … but they are uncountable because they are incompletable, not because their elements are more numerous than ordinary infinite sets. This supposed system of exotic higher numerosities is a fantasy. Numbers can only register the presence of definable objects, and the universe of discourse of definable mathematic objects is clearly countable. (We are human beings and we can only recognise definitions stateable with a finite number of different elements. We would have to “live for ever” to take-in objects defined using an infinite set of symbol tokens. There are about 60 basic symbols used in math, and the totality of their finite subsets is countable. A set is said to be “countable” when each element can be uniquely paired with a natural number. Such a set has an “ordinary infinity” of elements. )
There simply aren’t enough well-defined mathematic objects to populate the super-infinities Cantor was imagining.
So Cantor’s transfinite theory was dreamland.
But, let’s remember, the whole enterprise of doubly-pure math had already been a partial expression of dreaminess. A massive illusion had already been incubated. As time passed this dreaminess came to be the subject’s norm.
Cantor himself should have been elated, being on cloud nine, because, credulous people far removed from math were celebrating this extension of the math mystique. He was the one who had introduced an apparently miraculous new kind of math.
But he did not find himself on cloud nine. Instead he suffered several miserable mental breakdowns. His fellow gurus should have asked themselves Why? Why is Georg so mentally worried? But they didn’t. They were too mesmerised by this wonderland of fabulous “infinities beyond infinity.” It was, in their view, a wonderland anyway, but also one apparently legitimised by math, something with awesome authority in those bygone days.
So this disaster may now be seen as the consummation of a gradually growing unrealistic tendency which had begun long before Cantor was born.
George Boole’s discovery of set theory in the mid 19th century was the turning point. It came out of the blue, because Boole was a highly intelligent, largely self-taught, Irish polymath, and, as such, not a payed-up member of the myopic, intoxicated “modern math elite.” His discovery, though, was soon seen by the elite as sensational … because it offered a fresh—hugely confirming—version of what looked like doubly-pure algebra. The three old doubly-pure concepts introduced in 1830 were, by now, somewhat old-hat. Boole’s set theory seemed to be an extremely welcome, wholly new, fresh, understandable, operationally different, doubly-pure concept!
Actually it wasn’t doubly-pure at all. The objects it named were mostly very realistic, down-to-earth examples.
So it seems in retrospect that the myopic elite of so-called “modern math” had begun from the beginning to look for slightly “dreamy” concepts. As a result they became more and more estranged from the ordinary consensus thinking of their lay contemporaries. By the end of the 19th century these intoxicated myopics were ready for a showdown. The elite considered itself to be streets ahead of (=intellectually vastly superior to) the lay authorities … not to mention their unconvinced, conservative “old math” colleagues. These new young dreamers wallowed in their fearless recognition of more and more bizarre hyper-abstractions. (Their searching-energy burgeoned, no doubt as a response to the awe and admiration they were blindly receiving.)
Then the inevitable happened around 1900. The myopic elite broke ranks with the lay consensus and their unconvinced conservative oldies: and insisted that a form of mathsxit be adopted. It was a conscious withdrawl from, and rejection of, math based-on science and technology. They would henceforward wash their hands of being “the back room boys of physics,” and instead they would usher-in their superior, privileged, magical style of math. Math, they thought, was now “Coming-of-Age.” They had been lionised to the nth degree (by lay supporters who marvelled at the metaphysics of super-infinities), and they were determined to live up to the role. They were absolutely sure that they deserved—and would continue to enjoy—the adulation of the thinking classes. It was a form of naïve triumphalism veering towards anti-social defiance, even mania. Its cheerleaders were David Hilbert and Bertrand Russell. They were both, by temperament, louche, bohemian and aggressively radical. (If they were alive today, the Me-Too movement would be quite determined to put them behind bars.)
David Hilbert declared theatrically in Vienna in 1900 that nothing would induce him to give up the paradise which Cantor has opened to us! He was brazenly, absent-mindedly—ridiculously—implying that sentiment can trump logic in math.
Henri Poincare, the most admired mathematician of his day, declared that set theory (‘mengenlehrer’) and Cantor’s transfinite sets were a “disease from which mankind will eventually recover.” I don’t suppose that in his most pessimistic dreams he realised that, more than a century later, this would still be part of the math elite’s nominal Party Line.
Poincare’s scepticism was brushed aside.
So a breakaway myopic movement began … one which declared that higher math was no longer to be regarded as a mere “dogsbody of physics.” The gurus were set on declaring that “Modern Math” could “stand on its own feet.” It’s proper role, they averred, was as a quasi-religious intellectual, aesthetic, quest. Math, formerly described as the “Queen of the Sciences” wasn’t now going to be treated as a “science” at all—rather it was to be seen as being quite unique, a wholly new artform!
These mathsxiters were, no doubt, subliminally reacting to two trends which were in full swing at the end of the 19th century. The first was that physics was in a ditch. There appeared to be no possible way to make any kind of sense of the Michelson-Morely experiment. (The myopic math gurus had no intention whatever of trying to dig physics out of their ditch.) The second was a resurgence of the ‘Romantic Movement,’ and a strongly associated mood in philosophy… to the effect that at the heart of human reality was aesthetics. Most of Russell’s Bloomsbury friends were bowled-over by the sensibility on show in G. E. Moore’s iconic book Principia Ethica (1903).
In an essay in the Review in November 2023, I listed this rejection of science-based math as an UDI, one of eight howling blunders made by the leaders of the math establishment in the 20th century.
One would expect an attack targeting eight serious blunders would provoke a furious counter-argument
But there has been no attempt to reject these reflections … no counter-thesis whatever from the defenders of the status quo. Only silence—which, incidentally, has been their common reaction to all the howling mistakes. For a very long time the leadership of higher math have not missed an opportunity to miss an urgently needed rebuffing opportunity. They are probably short of ideas, disoriented by their own propaganda, and sickened by the almost complete loss of public support.
So here we have the extraordinary situation that the proponents of “modern math” foolishly chose to go off the rails in the late 19th century and have subsequently let this untenable ploy stay unchallenged. They have been in full-blown Denial about the direction in which rigorous logic points, ever since. They have now been sticking to their dreamy, illogical notions for more than a century. This, incidentally, puts them in the same bracket as the die-hards of phlogiston. But unlike the later proponents of phlogiston, they are still stubbornly treating their original notions as dreamy.
How could such an absurdity occur?
The myopic gurus of math seem to behave as if they think they have infallibility! (An unusually strong version of infallibility, actually.) What’s more, they have managed to convey their belief in this “infallibility” to many powerful slightly mathsphobic lay opinion-leaders.
If so, they are enjoying a self-serving illusion of infallibility thousands of times stronger than that of the Catholic Church, which has only succumbed twice in modern times to the temptation of claiming specific infallibility. (The concept of Papal infallibility was only established in the incumbency of Pope Pius IX.)
Thomas Kuhn alerted us in the 1960s to the unexpected presence of irrationality in science. It seems to have stemmed broadly from slap-dash mental habits rooted in crude, early rote-based teaching.
A degree of in-house intellectual arrogance seems also to have played a part, arising from the widespread belief among the gurus that “modern math” was far, far superior to ordinary common sense.
Unfortunately there was no obvious “authority” or “reference group” in place to apply checks-to, or qualms-about, the gurus’ overweening confidence. Part of the problem was that the gurus of math had disgracefully fudged the record in the 1920s about Russell’s Contradiction—they made it appear that they had “solved” the problem, when they hadn’t. This sleight-of-hand was not widely broadcast, but rumours about it have leaked out, and they could only induce contempt among perceptive observers. At best it revealed an unedifying hypocrisy, at worst a deliberate attempt to fake the true state-of-affairs … and to misinform the masses.
The later arrival of digital computers didn’t help.
There was a deep animosity from the beginning of the computer era between the established math gurus, who regarded themselves as privileged truth-seekers in a morally compromised world … and the new computerists, who didn’t even try to deny that they were in it for the money.
The Barons of Silicon Valley have consistently, punitively rubbished the absurdities of their rival, hyper-abstract, math gurus. The Barons have been circulating the preposterous mantra that Computers have nothing to do with math since the 1960s. They have also ruthlessly airbrushed-out the immense contribution that brilliant modelling math has since made—over more than six recent decades—to their overall success. Astronauts were brought back from the moon using only a few gallons of fuel. The helical structure of DNA was plucked out of a mass of foggy, confusing X-ray imagery.
Naturally this public put-down provokes gut opposition from the gurus of higher math, but they have been seriously disengenuous about the fact that they themselves weren’t interested in the feats pulled-off by their “applicable colleagues” in the de facto Golden Age of Modelling which has followed since 1960. It was the gurus themselves who tacitly gave away their colleagues’ ownership of these dramatic modelling feats, most of which were illuminating important commongood projects. The math gurus could have easily used their academic standing to contest the propaganda put out by the Barons, but they didn’t.
Another, more recent, quite unexpected, factor which has played a major part in pushing the gurus of higher math into believing in their own infallibility, has been the attitude of a class of leaders in the higher echelons of society. They mostly consist of able, intelligent, normally confident, individuals who tend to feel somewhat sheepish and defensive about their own embarrassing “failure” to see any significant meaning in math during their early years. Now ensconced in administrative, political and traditional media circles, they tend still to be blindly accepting the “official story” handed down decades ago by the gurus of math (namely, their infallibility). The net result is that many of them are unaware that there are unconsciously propping-up a now shell-shocked, burnt-out, despairing body of disenchanted, once proud math gurus. When lively authors submit articles for general publication questioning the math status quo … timid gate-keepers still tend to turn the articles down. Those in the political and media establishments are evidently unaware just how much the public credibility of formerly triumphant math stance has faded-away and disintegrated.
So what can be done about today’s shaky (systematically IT-blackened) state of affairs in higher math, where some professional mathematicians in academia are beavering away? They still think that their lot is—grudgingly—to do math modelling, instead of their dreamy original vocation— “doubly-pure mathematics.” Where, in spite of a mountain of evidence (and reasoning) to the contrary, they still think that what Henri Poincare called “mengenlehre” makes sense, and that the indefinable elements (those needed for transfinite totalities to exist) must be treated as “real and genuine” even though these indefinables are utterly shadowy. They have never been, and never will be, seen.
Well, everything hinges on how soon the over-hyped AI bubble bursts … the moment when it becomes generally evident that AI effectively sabotages any inner motivation students might harbour to “think hard and immersively” for themselves.
At this moment a lot of people will painfully realise that they have to start thinking differently. Anti-maths (AM) is the new exciting path leading towards a human-friendly future, in which thoroughly satisfying explanations of transient reality can be expected, recognised and celebrated. An alternative interpretation of what “AI” really means is that it signals the arrival of Attacks on human Intelligence. It is, after all, essentially a vast data-grab of the entire fruit of our illustrious ancestors’ curiosity and creativity. It has been grabbed unceremoniously by Big Tech … to flaunt a mind-numbing, demoralising show of endless “designed-to-impress” IT-enabled information.
We are at a fork in the road. Either we go down a feverish, lavishly financed, glossy, AI motorway, leading to more and more anonymous presentations of (generally dull, sometimes ludicrous, sometimes dangerously unreliable) info. Or we go down a new, unexpected, still barely mapped trail—one likely to lead towards far more understandable, far more humanly satisfying, experience (understanding). of ourselves and the world our parents pitchforked us into? Do we really want “success” to be hailed as the moment when the entire human race has been reduced to intellectual despair—reduced to a dumb demoralised acceptance that humanity’s capacity to understand anything is pretty feeble?
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Christopher Ormell is an older philosopher who was supervised at Oxford long ago by Gilbert Ryle and John Austin, and who set himself the target of applying linguistic analysis to find the real meaning of math. Websites: philosophyforrenewingreason.com, and mathsforrenewingreason.com.
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Follow NER on Twitter @NERIconoclast
