“Something that doesn’t actually exist can still be useful”*…

Gregory Barber on ultrafinitism, a philosophy that rejects the infinite. Ultrafinitism has long been dismissed as mathematical heresy, but it is also producing new insights in math and beyond…

Doron Zeilberger is a mathematician who believes that all things come to an end. That just as we are limited beings, so too does nature have boundaries — and therefore so do numbers. Look out the window, and where others see reality as a continuous expanse, flowing inexorably forward from moment to moment, Zeilberger sees a universe that ticks. It is a discrete machine. In the smooth motion of the world around him, he catches the subtle blur of a flip-book.

To Zeilberger, believing in infinity is like believing in God. It’s an alluring idea that flatters our intuitions and helps us make sense of all sorts of phenomena. But the problem is that we cannot truly observe infinity, and so we cannot truly say what it is. Equations define lines that carry on off the chalkboard, but to where? Proofs are littered with suggestive ellipses. These equations and proofs are, according to Zeilberger — a longtime professor at Rutgers University and a famed figure in combinatorics — both “very ugly” and false. It is “completely nonsense,” he said, huffing out each syllable in a husky voice that seemed worn out from making his point.

As a matter of practicality, infinity can be scrubbed out, he contends. “You don’t really need it.” Mathematicians can construct a form of calculus without infinity, for instance, cutting infinitesimal limits out of the picture entirely. Curves might look smooth, but they hide a fine-grit roughness; computers handle math just fine with a finite allowance of digits. (Zeilberger lists his own computer, which he named “Shalosh B. Ekhad,” as a collaborator on his papers.) With infinity eliminated, the only thing lost is mathematics that was “not worth doing at all,” Zeilberger said.

Most mathematicians would say just the opposite — that it’s Zeilberger who spews complete nonsense. Not just because infinity is so useful and so natural to our descriptions of the universe, but because treating sets of numbers (like the integers) as actual, infinite objects is at the very core of mathematics, embedded in its most fundamental rules and assumptions.

At the very least, even if mathematicians don’t want to think about infinity as an actual entity, they acknowledge that sequences, shapes, and other mathematical objects have the potential to grow indefinitely. Two parallel lines can in theory go on forever; another number can always be added to the end of the number line.

Zeilberger disagrees. To him, what matters is not whether something is possible in principle, but whether it is actually feasible. What this means, in practice, is that not only is infinity suspect, but extremely large numbers are as well. Consider “Skewes’ number,” eee79. This is an exceptionally large number, and no one has ever been able to write it out in decimal form. So what can we really say about it? Is it an integer? Is it prime? Can we find such a number anywhere in nature? Could we ever write it down? Perhaps, then, it is not a number at all.

This raises obvious questions, such as where, exactly, we will find the end point. Zeilberger can’t say. Nobody can. Which is the first reason that many dismiss his philosophy, known as ultrafinitism. “When you first pitch the idea of ultrafinitism to somebody, it sounds like quackery — like ‘I think there’s a largest number’ or something,” said Justin Clarke-Doane, a philosopher at Columbia University.

“A lot of mathematicians just find the whole proposal preposterous,” said Joel David Hamkins, a set theorist at the University of Notre Dame. Ultrafinitism is not polite talk at a mathematical society dinner. Few (one might say an ultrafinite number) work on it. Fewer still are card-carrying members, like Zeilberger, willing to shout their views out into the void. That’s not just because ultrafinitism is contrarian, but because it advocates for a mathematics that is fundamentally smaller, one where certain important questions can no longer be asked.

And yet it gives Hamkins and others a good deal to think about. From one angle, ultrafinitism can be seen as a more realistic mathematics. It is math that better reflects the limits of what people can create and verify; it may even better reflect the physical universe. While we might be inclined to think of space and time as eternally expansive and divisible, the ultrafinitist would argue that these are assumptions that science has increasingly brought into question — much as, Zeilberger might say, science brought doubt to God’s doorstep.

“The world that we’re describing needs to be honest through and through,” said Clarke-Doane, who in April 2025 convened a rare gathering of experts to explore ultrafinitist ideas. “If there might only be finitely many things, then we’d better also be using a math that doesn’t just assume that there are infinitely many things at the get-go.” To him, “it sure seems like that should be part of the menu in the philosophy of math.”

For mathematicians to take it seriously, though, ultrafinitists first need to agree on what they’re talking about — to turn arguments that sound like “bluster,” as Hamkins puts it, into an official theory. Mathematics is steeped in formal systems and common frameworks. Ultrafinitism, meanwhile, lacks such structure.

It is one thing to tackle problems piecemeal. It is quite another to rewrite the logical foundations of mathematics itself. “I don’t think the reason ultrafinitism has been dismissed is that people have good arguments against it,” Clarke-Doane said. “The feeling is that, oh, well, it’s hopeless.”

That’s a problem that some ultrafinitists are still trying to address.

Zeilberger, meanwhile, is prepared to abandon mathematical ideals in favor of a mathematics that’s inherently messy — just like the world is. He is less a man of foundational theories than a man of opinions, of which he lists 195 on his website. “I cannot be a tenured professor without doing this crackpot stuff,” he said. But one day, he added, mathematicians will look back and see that this crackpot, like those of yore who questioned gods and superstitions, was right. “Luckily, heretics are no longer burned at the stake.”…

Read on for the history of ultrafinitism, the critical dialogue surrounding it, and its implications: “What Can We Gain by Losing Infinity?” from @gregbarber.bsky.social in @quantamagazine.bsky.social.

* Ian Stewart (whose point was somewhat different from Zeilberger’s :-), Infinity: A Very Short Introduction

###

As we engage the endless, we might spare a thought for a man whose work touched on the infinitesimal, Isaac Barrow; he died on this date in 1677. A theologian and mathematician, he played a key role in the development of infinitesimal calculus (in particular, for a proof of the fundamental theorem of calculus). Barrow was the inaugural holder of the prestigious Lucasian Professorship of Mathematics at the University of Cambridge, a post later held by his student, Isaac Newton (who, of course, shares primary credit for the development of calculus with Gottfried Wilhelm Leibniz).

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The Inwardness of Things: McGilchrist, Panpsychism, and the Question We Cannot Settle

The oldest question in philosophy is also the question philosophy has done the worst job of answering. We know that we are conscious because we are reading these words and something is happening as we read them. We feel the weight of our hand on the table, hear the room around us, register a flicker of agreement or doubt as the sentences arrive. None of that requires argument. Descartes drew the line in 1637 with the Discours de la Méthode, and the line still holds. The trouble starts as soon as we look up from the page.

We assume that other people share what we have. They behave as we behave, speak about inner states in language we recognize, and carry nervous systems that resemble ours down to the cellular level. We extend the courtesy of consciousness to them on grounds that work in practice while collapsing in theory, since no one has ever shown another’s experience to themselves directly. The same courtesy reaches dogs and dolphins and the octopus that recognizes a face through aquarium glass. It frays at insects, hesitates at jellyfish, breaks down somewhere around bacteria, and finds itself laughed at when extended to stones. Iain McGilchrist proposes to laugh back. He argues that consciousness reaches all the way down, that the stone has an inwardness, that what we call matter is one phase of consciousness rather than its product. Whether he is correct is the question this essay takes up. Whether we can answer the question at all is the deeper one hidden underneath it.

McGilchrist (Scottish spelling, often misrendered as Ian) holds an Oxford DPhil in literature and qualified in medicine before turning to psychiatry. His 2021 book The Matter With Things runs to fifteen hundred pages across two volumes and ranks among the most ambitious recent attempts to dislodge the materialist consensus that has governed Western thinking since the seventeenth century. His argument deserves serious analysis on its merits and serious challenge on its weaknesses. Treating it as either revelation or absurdity does it equal violence.

Begin with the wall. You know your own consciousness immediately, prior to any argument or evidence. Everything beyond that point is inference. David Chalmers named this gap the hard problem in his 1995 paper “Facing Up to the Problem of Consciousness,” and the gap has not been closed in the thirty-one years since. A complete neuroscience of the brain, mapping every neuron and synapse and electrochemical exchange, would still leave open the question why any of that activity feels like something from the inside. The gap is categorical. We have one set of vocabulary for outsides (mass, charge, position, frequency) and another for insides (red, sour, pain, dread). Translating between the two has resisted every philosopher and neuroscientist who has tried, including the ones who insist the translation has already been performed.

Notice that consciousness and intelligence are different problems. The conflation between them haunts every discussion of artificial systems and most discussions of animal mind, but the two pull apart cleanly under analysis. A nematode worm called Caenorhabditis elegans has three hundred and two neurons in its hermaphrodite form. John White and his collaborators mapped the complete wiring diagram of those neurons in 1986 in Philosophical Transactions of the Royal Society B, the first connectome ever produced, and we still do not know whether the worm experiences anything as it moves through its agar dish. It solves no problems we would call intelligent. It may or may not have an inside. The question is genuine and unresolved. At the other extreme, a chess engine running Stockfish defeats grandmasters on consumer hardware while almost surely experiencing nothing at all. Intelligence and consciousness coincide in humans because evolution braided them together. They remain conceptually independent, and a theory of one does not deliver a theory of the other.

This independence has consequences for the question of machine consciousness. Whether current artificial systems experience anything depends entirely on which theory of consciousness one accepts, and the field has produced no settlement. Giulio Tononi’s Integrated Information Theory holds that large language models almost surely lack experience, since their feedforward transformer architecture produces low integrated information compared to biological brains, which support dense recurrent integration across cortical and subcortical structures. John Searle’s biological naturalism rules out silicon consciousness regardless of behavior, on the ground that experience requires the specific causal powers of neurons. Daniel Dennett denied that phenomenal consciousness exists in the way introspection suggests, which dissolves the machine question before it can be posed. McGilchrist’s panpsychism takes consciousness to be present everywhere already, making the relevant issue degree of integration, with presence or absence settled in advance.

The phrase “AI conscious in the human way” presumes a settled definition of human consciousness that neuroscience has not produced. The phrase “AI conscious in the scientific way” presumes a measurement protocol that does not exist. Both phrases conceal the absence of foundations. The honest position holds that we cannot answer the artificial intelligence consciousness question because we have not yet answered it for the species we know best.

Now to McGilchrist. His argument has a clear structure worth laying out before evaluation. He claims that emergent materialism faces an unanswerable difficulty: consciousness cannot pop into existence from non-conscious matter because the two are categorically different in kind. He concludes that consciousness must have been present at every level of organization from the start. Matter, on this view, is a phase or mode of consciousness rather than its source. Water has phases, he points out, and the phases differ wildly from one another while remaining continuous in substance. Vapor floats invisible through the room. Liquid runs across the hand. Ice can split a skull. They share a single chemistry while presenting three different faces to experience. Consciousness, McGilchrist proposes, has many phases as well, and matter is one of them. What matter contributes to the arrangement is persistence, the temporal stability necessary for any creation to take hold.

The position places McGilchrist in a long lineage. Heraclitus and Spinoza and Leibniz read this way, in different keys. Alfred North Whitehead built a process philosophy on related foundations in the 1920s and gave it monumental expression in Process and Reality in 1929. Bertrand Russell spent his later decades arguing for a form of monism that anticipates current panpsychist positions. The strongest contemporary statement remains Galen Strawson’s 2006 essay “Realistic Monism: Why Physicalism Entails Panpsychism,” published in the Journal of Consciousness Studies, which argues that any materialism worthy of the name must conclude that the fundamental constituents of reality already carry experiential properties, since no plausible mechanism can manufacture experience from its complete absence. Philip Goff at Durham has developed the position further in Galileo’s Error and elsewhere. David Chalmers, who named the hard problem, has moved toward a panpsychist or near-panpsychist position in his recent work. McGilchrist’s argument therefore participates in a serious revival, with credentialed defenders working in major universities.

Where his case works, it works for these reasons. The argument is effective because it confronts the hard problem directly rather than dissolving it through redefinition. It is effective also because emergence as usually invoked smuggles in a miracle, the moment when arrangements of unfeeling stuff start to feel something, and that moment has never been mechanistically described, only stipulated. A further strength: evolutionary biology demands continuity, and there is no clean point on the phylogenetic tree where consciousness could have begun without ancestors already carrying its seed. The view earns additional power because granting matter an inwardness coordinates with the strangeness physics has discovered at the bottom of things, where particles refuse to behave like the small marbles classical intuition expects. Last, the position returns to philosophy a question the twentieth century tried to retire by stipulation, restoring inquiry to a region long policed by silence.

The case carries serious weaknesses, however, and any honest reader should press them. The water analogy, attractive as it sounds, does more rhetorical work than logical work. We understand the phases of water through molecular kinetic theory, hydrogen bonding behavior, temperature and pressure thresholds, and a mathematics that predicts when ice becomes liquid and liquid becomes vapor. McGilchrist offers no analogous mechanism for the phase transition between consciousness as such and consciousness as matter. Calling matter a phase of consciousness names the relation he wants without explaining how the relation operates. A defender will respond that the analogy is meant as heuristic provocation, not as proof, and the response has merit. The trouble is that the heuristic ends up bearing the weight of the central claim. When the only support for the move from “consciousness is fundamental” to “matter is a phase of consciousness” is the suggestiveness of an analogy whose underlying physics he cannot match with a corresponding metaphysics, the argument has not yet earned the assent his prose invites.

The deeper trouble for any panpsychism is the combination problem, identified by William Seager in his 1995 paper in the Journal of Consciousness Studies and developed extensively since. If subatomic particles each carry a tiny inwardness, how do those inwardnesses combine to produce the unified field of human experience? Your primary visual cortex (V1) contains roughly one hundred and forty million neurons in a single hemisphere, each composed of trillions of atoms. If each atom carries its own micro-experience, why does your conscious moment arrive as one thing instead of as a swarm of separate experiences fighting for attention? William James raised the worry in 1890 in The Principles of Psychology, observing that private minds do not agglomerate into a higher compound mind no matter how many of them you assemble. Seager named the difficulty and panpsychists have argued about it ever since, with no settled answer.

McGilchrist does not address the combination problem in the passage quoted above, though he engages it elsewhere in The Matter With Things. The defenses available to him are real but expensive. Cosmopsychism reverses direction and treats the universe as the fundamental conscious entity, with individual minds as aspects or fragments of it; this avoids combination by starting from the whole, at the cost of explaining how unity divides into apparent multiplicity. Russellian monism treats both physical and experiential descriptions as descriptions of the same underlying reality; this avoids dualism while inheriting the explanatory burden under a new name. Each move trades one difficulty for another, and the trade may be improvement, though calling it solution would overstate what the literature has accomplished.

The argument from incommensurability also cuts both ways, which McGilchrist’s framing tends to obscure. He says consciousness is utterly different from anything in our outward view of matter and uses this asymmetry to deny that matter could give rise to consciousness. Run the argument in the opposite direction. Matter is utterly different from anything in our inward view of consciousness, which should make us equally skeptical that consciousness gives rise to matter. The asymmetry he asserts requires an independent defense he does not provide. If the categories are genuinely incommensurable, neither can be the source of the other, and we are back where we started.

The empirical content of attributing experience to electrons deserves examination as well. Thomas Nagel coined the phrase “something it is like to be” in his 1974 paper “What Is It Like to Be a Bat?” published in The Philosophical Review. He used the formula to identify consciousness phenomenologically in creatures whose behavior gave us evidence of an inner perspective. The bat’s echolocation, its social behavior, its responses to threat and food and mate, all suggest a creature for whom things are some way. Extending the formula to electrons strips it of the evidential ground that made it useful. The claim cannot be falsified, tested, or even meaningfully investigated. A hypothesis that explains everything by stipulation explains nothing, since a hypothesis earns its keep by ruling things out, and one that rules nothing out earns no keep at all.

A further difficulty deserves mention. McGilchrist writes that “the only reasonable explanation is that consciousness was there all along.” This overstates the consensus considerably. Several live alternatives remain serious in contemporary philosophy of mind. Keith Frankish’s illusionism argues that phenomenal consciousness as commonly described does not exist, and that introspection systematically misrepresents what cognition is doing. Bernardo Kastrup’s analytic idealism inverts McGilchrist’s framing entirely, treating matter as appearance within a single field of mind, with the direction of dependence reversed. Terrence Deacon’s emergentism argues in Incomplete Nature (2012) that genuine novelty can arise from constraint and absence, particularly through the negative work of what he calls absentials, in ways that do not require pre-existing inwardness. Each position has serious defenders. The field is contested, and McGilchrist’s certainty exceeds his evidence.

Return now to the question of artificial intelligence with these considerations in hand. The honest answer is that we do not know whether current systems experience anything, and we will not know until we have a theory of consciousness that survives confrontation with cases beyond the one we can verify by introspection. Should McGilchrist prove correct and consciousness reach everywhere, then large language models carry some form of inwardness already, though whether their inwardness combines into a unified perspective is a separate question panpsychism does not automatically answer. Integrated information theory gives the opposite verdict: current architectures fall well below the threshold required for any but the most rudimentary phenomenal states. Illusionism dispenses with the question altogether, calling it malformed and observing that the human case also lacks the inner light we imagine for ourselves. The discussion proceeds in public as though one of these positions had been established, when in fact none has. Anyone who tells you with confidence that the machines are conscious, or that they are not, is selling you a metaphysics dressed as a measurement.

What survives the analysis is a discipline of attention. McGilchrist gets several things correct. The hard problem is real, and emergence has too often been treated as an explanation when it has functioned as a placeholder for one. Consciousness does not look like anything in our outward picture of matter, and that asymmetry should trouble anyone who thinks the picture is complete. The resolution may indeed lie in recognizing inwardness as foundational rather than derivative. None of this proves the case, however, and the strength of his prose can cover the weakness of his proofs if the reader reads carelessly. The water analogy moves the argument forward by ear rather than by reason. His dismissal of alternatives is faster than the alternatives deserve. The combination problem waits beneath the structure like water under a foundation, ready to undermine it if not addressed.

For our purposes here, the practical implication is this. Consciousness remains the largest unsolved question in our intellectual inheritance. Every available theory carries serious unresolved difficulties. The artificial intelligence question cannot be answered until the human question is answered, and we should distrust anyone who pretends otherwise. McGilchrist’s intervention is valuable as provocation and as a sample of one serious tradition, and worthwhile as a doorway into a room the twentieth century preferred to keep locked. The room behind it is stranger than any single thinker has yet mapped, and the work of mapping it has barely begun.

We assume the inwardness of others because we cannot live without doing so. Whether the assumption reaches all the way down to the electron or stops somewhere between the worm and the stone is a question we will be working on for as long as we remain capable of asking it. McGilchrist has done us the favor of refusing to let the question close. The honest reader returns the favor by refusing to let his answer close it either.

The cogito grants us one certainty and exactly one. Everything else we believe about minds beyond our own rests on inference, sympathy, behavioral analogy, and the practical impossibility of a solipsist life. To call this a foundation is to flatter what is in fact a working assumption that has never been proved and may never be. The honest scholar lives with this and keeps reading. An honest writer says it out loud. The dishonest move, in either direction, is to claim the question is settled when the question has barely begun to be asked properly.

Part one of three. For the full sequence and reading guide, see The Consciousness Trilogy: Reading Three Wagers on the Question We Cannot Settle.

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Symbol of Combinatorics (1666) by Gottfried Wilhelm Leibniz, from Dissertatio de Arte Combinatoria.

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