by Christopher Ormell (July 2026)
We urgently need a new worldview. We are trying to understand an immensely strange, baffling, dangerous world while unconsciously allowing old, tired conventions to tie our hands behind our backs. Many remain in thrall to wholly literal interpretations of ancient religions, and we are still awed into silence by the prestige of “modern math,” which quietly went off the rails around 1900 when its protagonists started assuming that ordinary sets were mathematic objects. They had long since stopped paying attention to mathematical applications, and in doing so forgot that ordinary everyday sets are applications. The whole point of set theory is to provide precise definitions that can serve as membership criteria. But ordinary things do not have precise definitions. Only sets whose elements are already mathematic objects can themselves qualify as mathematic objects. This lapse of attention to the precision required for membership criteria cancels set-based mathematics.
A strange syndrome has emerged. Few laymen, it seems, are willing to say “Boo!” to the mathematicians because, as mature adults, they realise they should perhaps have paid more attention to mathematics at school. Yet today a range of ominous existential crises seriously threatens the wellbeing of the human race. When all the warning lights are flashing at once, it is hardly the time to play safe by clinging to old conventions.
Perhaps the deepest source of the chill that seems to have settled over Western civilisation is the thought that the physical universe may ultimately prove alien—even hostile—to us. This thought sent shivers down Blaise Pascal’s spine nearly four hundred years ago, and it may help explain why the truth embedded in Christianity is now under sustained, if often unfocused, attack. We used to call the New Testament, optimistically, “the bringer of Good News.” More objectively, Christianity can be understood as a challenging social contract: if we agree to uphold high ethical standards, life becomes happier, more pleasant, and more liveable for everyone. The means of achieving this is person-to-person conversion—people power.
Today, however, Christianity, which emerged two thousand years ago, is increasingly overshadowed by cosmological certainties that point to the profoundly alien nature of deep space—a comparison unimaginable in the centuries before Christ. The implicit suggestion is that a universe so alien cannot be reconciled with the Good News, which rests upon personal moral discipline. The implication is that the “real,” physical universe may itself be incompatible with the social contract Christianity proposes.
A Cambridge theologian friend, the late Michael Langford, once confessed to me that the concept of “God” was really only a way of talking about the mysterious, unknown origin of everything. It is rather like infinity in mathematics, which common sense tells us is simply a way of talking about symbolic mathematics’ open-endedness. This, I believe, is where the young firebrands of “modern mathematics” around 1900 went wrong. They were determined—against the accepted understanding of the time—to treat infinity as though it were a number.
This brazen idea proved irresistible. The young firebrands soon became the new mathematical establishment. A new century was dawning, bringing hopes of radical change. They naturally rejected the assumptions that had produced what they regarded as Victorian suffocation. They embraced the superficial notion that “pure mathematics” was essentially an aesthetic art form and could not resist the mystique that the exotic concept of the transfinite seemed to offer. We now know that bona fide mathematic objects must have strict definitions. Mathematics’ credibility with the intelligent public depends upon it. We also know that the totality of all possible definitions of mathematic objects—all possible permutations—amounts only to an ordinary infinity. There are simply not enough possible definitions to accommodate the transfinite as mathematic truth.
Around 1900, many theologians welcomed these mathematical firebrands—Russell, Hilbert, and others—believing they had finally mastered the transfinite. Today their successors unexpectedly find themselves on the wrong side of the issue.
Another, previously much-discussed, source of existential doubt was the emergence of a crude version of Darwinism, which appeared to convey the message that evolution is “red in tooth and claw.” The Nazis and Fascists of the twentieth century eagerly embraced the simplistic conclusion that human nature had been shaped over the ages by an evolutionary struggle in which only the most brutal survived.
The evidence points in a different direction. Species do not usually attack their own kind. There are exceptions—for example, male lions killing the cubs of rival males—but they are rare. Most tellingly, Goebbels never dared tell the German people about the Holocaust. In the end, the Nazi and Fascist regimes were utterly defeated by the Allies. Might did not prove to be right. It led instead to the collapse of their evil ideology, the devastation of Europe (and two great cities in Japan), and the deaths and suffering of millions. It is sickening that some bombastic national leaders appear to have forgotten that lesson today.
Alas, since the fall of the Berlin Wall, neo-liberal, dog-eat-dog economics have gradually fostered a more materialistic society than existed, at least in part, during the 1930s. At that time, “idealism” was still regarded as a respectable word. Today it is often treated as a sign of derangement. This is an aberration. The Anglophone tradition was built upon far-sighted acts such as the abolition of the slave trade. We went to war in support of plucky Belgium in 1914 and plucky Poland in 1939. Churchill’s long-sightedness was far from a consensus view in Britain in 1940. When he vowed to fight on, he defied many of his own colleagues. Had he succumbed to short-sighted thinking, we might still be living under the thrall of the Third Reich.
This worldwide drift towards materialism has coloured almost every aspect of public life. Economics increasingly dominates politics across the globe. The unspoken implication is that the sheer power and ferocity of deep space hang over us like a dark cloud. A universe so alien by nature, we are encouraged to believe, is unlikely to reflect the sensitive, hard-won sources of human happiness. Yet healthy human relationships require effort: care, empathy, self-restraint, and long-sightedness. Compared with these, the rewards of materialism are dull and second-rate.
The actual physical effect of deep space upon us may be very slight. Cosmologists examine minute patches of colour and proceed to draw vast conclusions from these tiny fragments of evidence. They have largely forgotten how extensively cosmology rests upon far-reaching extrapolation. If light bent as much in deep space as it does on Earth, photons arriving from even the nearest star would have travelled in complete circles on their journey. Cosmology is therefore an intrinsically fragile semi-science. Many of its claims deserve to be treated with caution. When presented as the final word, they become classic examples of the tail wagging the dog.
Returning to the central question, we must ask whether the sources of order and structure in the universe are actively hostile towards us. The only primary sources of order we know are physical and biological processes, as studied by science, and—following Schopenhauer—human willpower guided by imagination. Modern science is essentially the search for secure, reliable mathematic models of carefully observed, finite fields of phenomena. It is an intentionally modest, piecemeal enterprise. After the Renaissance, this approach largely replaced Aristotle’s brilliant, intuitive—though by then obsolete—synoptic vision. The mathematics employed in this piecemeal approach treats its first principles and axioms as given. They are assumed to exist a priori. But what, ultimately, is the status of those “given” principles?
They remain unresolved. The central lesson of mathematical science is that it offers only a partial, patchwork account of the universe. Karl Popper described most scientific conclusions as “not yet falsified opinions.” Science undoubtedly provides explanation, but it can never amount to a complete synthesis. Most theorists assume that this is the best we can do.
But the serious existential crises we are currently facing require more. They urgently require an overarching conception that makes sense of the plasmas, black holes, novae, dark matter, dark energy, and the rest of deep space. Only by developing a convincing understanding of why such a vast and strange physical universe exists can we hope to diminish the chill wind now blowing through our civilisation.
Kant, in effect, brilliantly anticipated the solution. He pointed out that the way we perceive the physical universe is inevitably shaped by the still unknown mechanisms that make sentience and consciousness possible. A person wearing rose-tinted spectacles sees the world as rosy. Likewise, the unknown mechanisms that produce consciousness are our inbuilt “rose-tinted spectacles.” They are the principles and axioms that make consciousness possible—a remarkable achievement, contrary to the self-delusions of those hoping to get rich quickly by hyping AI. Artificial intelligence is an extraordinary software achievement, but it cannot predict a future that does not yet exist, whereas we know we possess the freedom to change the future. A significant part of the future is inevitably undetermined and remains open.
Unfortunately, Kant’s reputation suffered after his death because he failed to recognise the possibility of non-Euclidean geometry and could offer no account of how consciousness itself arises.
A caveat is needed here, just as it was with the over-extrapolation of cosmology. We cannot expect a clear, rational account of the universe until we have solved the problem of consciousness—especially the extent of this unobvious “spectacle effect.”
Here, anti-math enters the stage. It is still in its infancy, at what might be called its “Stonehenge stage,” but the underlying concepts are already visible. An immense, misty vista of structured, transient reality is beginning to emerge. There will be countless opportunities to discover new anti-math reifications that no one has yet imagined. It is impossible to foresee where all this may eventually lead, but one thought experiment is particularly revealing. A wise Druid would have found it just as impossible to conceive of de Moivre’s theorem, Hilbert space, or elliptic functions as he or she walked through the darkly wooded Wiltshire Downs three thousand years ago. Yet three millennia of intelligent, rational inquiry gradually produced precisely those ideas. It is therefore reasonable to suppose that consciousness itself may only be possible as part of a larger reality in which far-flung, uninhabitable regions of deep space serve as necessary counterweights to the freedom of choice made possible here on Earth.
The defining feature of human consciousness is that we can choose and create. This is a form of reality quite different from the predictable, physical-mathematic world we have been discussing.
Both math and anti-math arise from the same human faculty. The difference is that mathematics can, at best, create timeless sets of abstract objects. Anti-math, by contrast, can eventually generate structures possessing liveliness, imagination, diversity, and creativity—though always with transience to follow.
Most theorists have regarded this as impossible. They failed to recognise that consciousness and creativity may be priceless privileges, achievable only because a balancing counterweight exists elsewhere.
Put differently, human willpower may eventually create the blueprint for human existence through anti-math-driven modelling. The very existence of this capacity then underwrites the willpower needed to realise it. Here lies a remarkable circular logic. Suddenly the universe begins, at least in outline, to make sense.
This idea is far removed from the sociological fantasies of the 1960s. It also dissolves much of the chill produced by modern cosmology. What has seemed so threatening may simply be the distant consequence—indeed, the necessary counterpart—of our precious consciousness.
It is ironic that the diehard advocates of higher mathematics have long insisted that the mathematic objects generated by their preferred definitions are as “genuinely real” as material objects. This is, frankly, an in-house metaphysics that has outlived its usefulness. It feels convincing because mathematicians spend their professional lives exploring the properties of those objects. Each day they return to precisely the same abstract landscape they left the day before, creating an illusion of permanence. A more balanced view is that these are bona fide mathematic objects, brought into being through the collective reifying activity of generations of mathematicians. Few seem aware that the objects they regard as having descended from Mount Olympus were actually created, by reification, through the careful work of their predecessors.
When mathematicians eventually recognise that anti-math is a comparable symbolic language capable of generating a different kind of reality—a landscape of symbolic artefacts created by human will—they may also recognise its unique potential. Mathematical reifications cannot achieve this because they are static, inert, and timeless: the very opposite of living organisms. Deterministic electronic neural networks occupy an intermediate position. They can approximate intelligence, but their predictability means they cannot produce genuine human intelligence rooted in consciousness and free will. If genuine intelligence requires an entire universe to sustain it, it seems improbable that a few million computer chips, however ingeniously connected inside vast data centres, could produce the same result.
By contrast, anti-math reifications can already give us transient objects that are perfectly reliable—but only for a time. As such, they may prove capable of sustaining, and explaining, the vast realm of living biological reality. From this, ultimately, emerges the extraordinary phenomenon of human consciousness. Human consciousness, which includes self-consciousness, is unique. Animals plainly possess consciousness, but they appear to possess little, if any, self-consciousness. Some primates may exhibit the beginnings of it.
How can we be confident that the physical cosmos is itself an immensely distant by-product of this circular logic? Because human beings can conceive of—and even materialise—new realities, including fields of endlessly jumping random tallies. We can then reify selected patterns that emerge within them, following the example of three thousand years of mathematical development. Mathematics began by reifying simple tallies before progressively elaborating ever more sophisticated abstract structures. Prime numbers, for example, were recognised, named, and studied because they possessed distinctive structural properties.
Human beings, however, can ask unlikely questions. The question here is whether we can develop a new abstract modelling language with a built-in capacity for representing liveliness and creativity.
I believe we can. Anti-math appears capable, in principle, of doing precisely this, although its models are inherently transient.
Today we can reify selected random tally patterns by applying rigorous definitions—that is, consistent human willpower—thereby producing anti-math waves. The result is a vast potential landscape of flexible, ongoing, transient objects that remain perfectly reliable while they exist, yet ultimately disappear. Grass, for example, has been faithfully reproducing itself for millions of years. It is therefore reasonable to suppose that this process, if systematically applied across science (“Mark 2”), may eventually enable us to construct accurate, active models of the human brain. In time, this research programme could produce credible models of human mental performance—that is, genuinely human-like consciousness.
At first sight, this may seem optimistic. Yet the proposed waves of jumping random tallies are, in effect, the waves of wave mechanics. Orthodox science (“Mark 1”) already tells us that the brain can, in principle, be represented through wave mechanics. Achieving such a model will undoubtedly require the efforts of generations of future pioneers, just as modern mathematics rests upon the accumulated work of countless thinkers over the past three thousand years.
This does not mean we shall easily predict the consciousness of other people. That remains an intimate art in its own right. It does mean, however, that we may eventually understand the kinds of processes that make consciousness possible.
Yes, this is a difficult idea to accept. But so was the discovery that the Earth is a sphere. So was the realisation that the Earth revolves around the Sun, rather than the reverse, which generations regarded as obvious common sense. So was the discovery that vaccines can use disease to prevent disease. So was the fact that ships made of iron can float. One of the great strengths of mathematics is that it teaches its practitioners how often unfamiliar logical processes lead to profoundly counter-intuitive conclusions. Catastrophe theory was once illustrated by suggesting that a butterfly flapping its wings in London might, through a chain of events, contribute to a thunderstorm in Madrid. It is just one example of how apparently insignificant logical details can produce unexpected consequences.
Over the past seventy years we have learnt that the performance of the human brain gives rise to the human mind. No one has encountered a human mind apart from a biological brain. It therefore seems implausible that an infinite, God-like mind could exist as a vast network of bloodless neurons and synapses somehow operating in the vacuum of deep space. Artificial intelligence creates the illusion of such a possibility by processing immense quantities of existing human knowledge. But it can only reorganise what already exists. Where the record is incomplete, it invents one—it hallucinates. That limitation gives AI an enormous capacity for error, although it also has genuine uses, such as checking difficult mathematical proofs.
My late friend’s description of God as “a way of talking” raises an obvious question: why have millions of intelligent people throughout history been convinced that God is a supreme reality? Perhaps there is something in human experience that supports that conviction. There is, after all, a powerful common ethical subconscious that creates the sense that something greater than the individual is present. Carl Jung pioneered this line of thought. It may be understood as a largely unconscious network of shared moral intuitions and emotional perceptions. That is a genuine form of experienced reality. What it appears to lack is any power to create or sustain the physical status quo. This, I believe, is where the discovery of anti-math changes the picture.
If anti-math proves equal to this task, it represents a genuine turning point. It finally gives us an abstract modelling language suited to representing biological reality. Mathematics is too rigid and timeless to model life itself, let alone consciousness, with its diversity, vitality, and creativity.
Anti-math may at first seem difficult to accept, but its underlying principles are as simple as elementary arithmetic. If that proves to be true, then our understanding of consciousness—and of humanity’s place in the universe—will have changed fundamentally.
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Christopher Ormell is an older philosopher of mathematics who shook hands with Alan Turing in 1952, and solved Russell’s Contradiction in 1958. The solution was published in Mind, and was noticed with approval by Karl Popper, but otherwise ignored. Later he found a mirror image of Descartes’ classic Cogito argument. He launched it in a six-article series in the journal Cogito (1992-4, also ignored. It was a proof that absolute unpredictability is logically possible, but is not identifiable—definable—as a mathematical state.) He had earlier discovered superparadoxes, which generate vast numbers of contradictions (a monograph was eventually put online about this in 2003). He found the first formula for the nth prime number without using trigonometric functions. (Math Gazette 1966) He discovered explicit formulas for calculating [x] and |x|. He later spent 29 years searching for an elementary solution to Fermat’s last theorem. This putative reasoning has now been on-line for six years. (A prize has been 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 quite unexpectedly brings methods, similar to those of math, to bear on the logical implications of transient forms. These stable forms are imposed—by willpower—onto absolutely random sequences of four tally types. The result: Anti-Math enables the brilliant, civilised, moral thinking of Kant to come back to centre-stage. We are beings who manage to secure our own existence by unconsciously imposing life-affirming, reifying, definitions onto a wholly independent, inexplicable, neutral-random dynamic substratum. Websites: philosophyforrenewingreason.com, and mathsforrenewingreason.com.
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