Does formalizing reasoning require new mathematics?
Before we can answer that question, we need to understand what it would mean. And that opens onto something deeper than reasoning, deeper than mathematics: something about the strange relationship between structure, representation, and reality itself.
We use mathematics to understand reality. And it works—unreasonably well, as Wigner once put it. But where does the mathematics come from? Not historically (we know the names and dates). I mean: what happens when the mathematics we have can't capture the domain we're trying to understand? When the structures we're investigating don't fit into the formal machinery available?
The usual story treats mathematics as a fixed toolkit. You reach in, grab the appropriate tool, apply it. Algebra for this, calculus for that, statistics for the other thing, as if the tools were always already there, waiting.
The history of mathematics, and of mathematically-driven science, is punctuated by moments when the tools had to be invented because the existing ones couldn't represent the phenomenon at all. Those moments are where I want to dwell.
Motion, and the birth of calculus
Before calculus, the concept of instantaneous velocity was, strictly speaking, incoherent. Velocity is distance divided by time. At an instant, no time elapses and no distance is covered. You get 0/0, which is undefined. You could calculate average velocity over an interval, but as the interval shrinks toward zero, you approach nonsense.
Zeno's paradoxes exploited exactly this gap. The flying arrow, at any given instant, occupies a space exactly equal to itself. It isn't moving at that instant, because motion requires change over time, and an instant has no duration. So when does it move? The mathematics of antiquity couldn't answer. More precisely: it couldn't even formulate the question in a way that admitted an answer.
What Newton and Leibniz introduced was the concept of a limit: what a ratio approaches as both numerator and denominator approach zero. This has a precise definition (for any ε > 0, there exists a δ > 0 such that…). The limit concept made instantaneous rate of change well-defined, and from there came derivatives, integrals, differential equations.
The issue with Newton's mechanics before calculus wasn't clumsiness or lack of rigor. The laws couldn't have been stated at all, because the formalism that would make them well-formed sentences didn't exist. F = ma says that force equals mass times the second derivative of position with respect to time. That's a differential equation. Absent the formalism, it isn't even a proposition. The formalism is prior to the thought, not subsequent to it.
Space, and what happened to Euclid
For two thousand years, Euclidean geometry wasn't a theory of space; it was the theory. Seemingly the only coherent way to think about extension and form.
Euclid's fifth postulate (the parallel postulate) states, roughly, that through a point not on a given line, exactly one parallel line can be drawn. Unlike his other axioms, this one felt less self-evident. Mathematicians spent centuries trying to prove it from the other four, convinced it must follow.
The attempts failed. Worse: when mathematicians like Saccheri tried proof by contradiction (assume the parallel postulate is false, derive absurdity), they got strange results but no contradictions. Spaces where parallel lines diverge. Spaces where they converge. Internally consistent, just different.
In the 19th century, Lobachevsky, Bolyai, and Gauss made the breakthrough explicit: these alternative geometries are not contradictory. They're coherent formal systems, as rigorous as Euclid's. Riemann generalized further, developing the geometry of curved spaces of arbitrary dimension.
What this revealed deserves more wonder than it usually receives: geometry is not a priori knowledge of physical space. It's a formal system that may or may not describe reality. Which geometry applies is an empirical question.
And reality answered. Einstein's general relativity describes gravity as a curvature of spacetime, not as a force. The mathematics required is Riemannian geometry, the very framework built by exploring what happens when Euclid's postulate fails. Space itself turned out to be non-Euclidean.
We couldn't discover this until we could conceive it. We couldn't conceive it until the mathematics existed.
Quantum phenomena, and the need for a new kind of object
The deepest case is quantum mechanics. It eventually required a new kind of mathematical object, not just new equations.
Classical physics describes the state of a system as a point in "phase space": a list of positions and momenta for every particle. Evolution over time traces a trajectory through this space. Observables (energy, momentum, and so on) are functions that assign numbers to points. If there's probability, it's because we don't know the exact state; the underlying reality is determinate.
Quantum systems resist all of this. A quantum state is a vector in a Hilbert space (a complete vector space with an inner product, often infinite-dimensional). Observables are operators that act on these vectors. And crucially, operators can fail to commute: the order in which you apply them matters. Position and momentum satisfy ΔxΔp ≥ ℏ/2. The uncertainty principle falls directly out of their non-commutation.
Measurement projects the state onto an eigenspace of the observable, with probabilities given by the Born rule. Superposition, entanglement, interference: none of this can be represented in classical phase space, because classical structures lack the compositional properties these phenomena require. The Hilbert space formalism appears to be the only known structure adequate to what quantum systems do.
Number itself
Even the concept of number has been repeatedly reconstructed when existing versions couldn't represent what was needed.
Negative numbers: the equation x + 5 = 3 has no solution in positive integers. Merchants dealing with debt, physicists dealing with direction, needed numbers that could represent "less than nothing" relative to a reference point. The extension to negative integers required restructuring arithmetic to be symmetric around zero, a reconception of what number means.
Complex numbers: the equation x² = -1 has no solution in real numbers. For centuries, √-1 was dismissed as meaningless. But Cardano, Euler, and Gauss showed that defining a number i such that i² = -1 yields a consistent, enormously powerful system. Complex numbers turned out to be essential for describing oscillation, rotation, wave phenomena, and quantum mechanics, where the state space is a complex Hilbert space.
Real numbers: the rationals seem dense (between any two, there's another), but they have "holes." √2 is not a ratio of integers, so it doesn't exist in ℚ. Dedekind cuts and Cauchy sequences filled these holes, constructing the complete ordered field ℝ necessary for rigorous analysis, for calculus to actually work.
Each extension involved building new structure because the existing structure couldn't do what was needed. The numbers weren't sitting there waiting to be found. They were constructed under constraint, to solve problems the prior system couldn't even pose.
Formalism is constitutive
You cannot think what you cannot represent.
Before the formalism, you don't have ineffable insights waiting for expression. You have, at best, vague gestures. The representational structure determines the boundary of what's thinkable.
Calculus gave Newton the concept of instantaneous velocity, a concept that requires the mathematical structure to be thinkable at all. What Riemann constructed were the conditions under which a new kind of space could be conceived. (It turned out, somewhat remarkably, that we inhabit one.) Hilbert spaces appear to be the only known structure adequate to quantum phenomena, because superposition, entanglement, and non-commuting observables have no home in classical mathematical frameworks.
The choice of mathematical representation shapes what truths are available to us. This is the claim. Not that notation is convenient. Not that formalism is a useful aid. That representation is the medium in which precise thought becomes possible at all, and that without the representation, the thought is not muffled or incomplete—it does not exist.
Building with what
Mathematics is the tool we use to make things precise. It's how we move from vague intuition to rigorous structure. It's the medium of exact description.
New mathematics has to be built. Someone has to construct it.
With what?
Not with the mathematics that doesn't exist yet (that's what we're building). Not with pure logic (we're not deriving from axioms; we're choosing axioms, forging definitions, deciding what structures to study). Not with empirical observation alone (we're building formal machinery, not just collecting data).
There is something strange here. We extend the tools of precision from within the medium of precision, but before the extension, the medium doesn't include what we're about to add. We are reaching beyond our current representational capacity to construct new representational capacity.
Riemann sat with the problem for years, followed failed paths, and constructed something—manifolds, curvature tensors—that had no prior formal existence. His intuition guided the construction, but the intuition wasn't itself mathematical; the math didn't exist yet. It was something pre-formal that nonetheless tracked structure precisely enough to build what was needed. And then something cohered that couldn't have been mechanically derived from what came before.
The capacity, and its features
The capacity to extend mathematics is the capacity the rest of mathematics depends on, and we understand it least.
A working definition: the capacity to construct, from within an existing formal system, new formal structure that the existing system cannot generate but that, once built, is rigorous by its own standards and adequate to phenomena the prior system could not represent.
It's not "creativity" in the generic sense; not all creativity produces rigorous formal systems. It's not "intelligence" in the IQ sense; the skills don't obviously correlate. It is something like a sensitivity to structure that operates before the structure has been formalized, yet reliably produces valid formalizations.
A few features can be named without overclaiming.
It is constrained without being determined. The result has to cohere internally, has to compose with other mathematics, has to capture the phenomenon. Within those constraints, choices have to be made, and the choices are made by something that isn't itself algorithmic. There is more than one path to the right answer, and not every path arrives.
It is sensitive to fit. Riemann didn't try arbitrary structures. He tried structures that felt like they were tracking something. The feeling of tracking something is doing work that nothing in the formalism can explain, and that the resulting formalism cannot retrospectively justify.
It operates pre-formally and produces formally valid results. Whatever the capacity is, it can reach beyond the current formalism to construct new formalism that is then, by the standards of the new system, fully rigorous. The pre-formal intuition is not itself rigorous, but it is reliable enough that the work it guides ends in rigor.
It is recognized only after the fact. We cannot test for it in advance. There is no benchmark, no examination, no procedure that identifies people who can extend mathematics before they have extended it. We know it exists because we can see, retrospectively, that someone did it.
That last feature has consequences. The capacity to construct new formalism cannot be evaluated by any test administered from within the existing formalism, because the test would have to specify in advance what counts as success, and the whole point is that the new formalism includes things the old formalism couldn't specify. We can grade calculations. We cannot grade the construction of calculus.
The tools of precision cannot fully account for their own extension. This is a structural feature of the situation, not a mystical claim.
The strangeness of it
Step back from the features and look at what is being described.
Human cognition has many capacities we have learned to name and study. Language, with its recursive grammar and infinite expressive range. Tool use, which extends the body's affordances. Theory of mind, which lets us model other minds. Episodic memory, abstract reasoning, mathematical computation, aesthetic judgment. Each has been studied, measured, given a vocabulary, located in cognitive architecture.
The capacity at issue here has none of that. It has no agreed measure, no diagnostic, no developmental trajectory we know how to track. The people who exercise it cannot reliably describe what they did. Newton said little about how he came to calculus. Riemann's intuitions remained, by his own admission, ahead of his ability to articulate them. Ramanujan attributed his formulas to a goddess. This isn't false modesty. The process is opaque to the people doing it, in a way few other cognitive processes are.
The capacity is also rare in a distribution we don't understand. Not one person in history but a thin line through history: a handful of names, separated by centuries, working in different traditions, who each did something the rest of the species could not. The capacity is real (we have the proofs), unevenly distributed (most who try cannot do it), and resistant to the predictive measures we have for other cognitive abilities.
It is hard to comprehend even as a fellow human. We can read the papers, follow the derivations, verify each step. The construction itself, considered as a cognitive event, remains outside what we can model. We made the tools of precision with something that has stayed imprecise.
This is a strange position for the species to be in. The capacity that produced every formal system we have is the capacity we have the least formal grip on.
What it has made possible
Pre-formal traditions could be remarkably sophisticated. Alchemy was reproducible and disciplined. Steam engines worked, often beautifully, before thermodynamics existed to explain them. Pre-modern medicine, navigation, metallurgy, agriculture—all accumulated real knowledge through careful empirical work, transmitted across generations, refined by trial. To dismiss what people did without formal articulation is to misunderstand what they did.
What formal articulation adds is different in kind. It makes the space of possibilities visible. It separates what's hard from what's impossible. It lets knowledge compound across generations and traditions in a way trial-and-error knowledge cannot, because the formalism itself is the carrier. The next generation doesn't have to retrace the path; they inherit the structure and can extend it.
The semiconductor in the device you are reading this on is downstream of a long chain of people, across centuries and traditions, who each extended the formalism a little further—calculus, complex analysis, electromagnetism, quantum mechanics, solid-state physics, information theory. The vaccine is downstream of a different chain. So is the climate model. Each is a precipitate of formal articulation, performed serially, by people who mostly never met each other.
The shift is one-directional. Once a domain has been formally articulated, it cannot be un-articulated. The thoughts are now available. A creature with our other capacities—language, tool use, culture, art, abstract reasoning—but without formal articulation would not have anything like modernity. The relationship to reality that distinguishes the last few centuries from the previous several millennia is built on the slow, uneven accumulation of formal structure, and the accumulation is what this capacity makes possible.
What is strange—what brings the question back—is that the capacity responsible for all of this is the one we understand least. We live inside a civilization built by a process we cannot describe, performed by a small number of people we cannot reliably identify in advance, exercising a capacity for which we have no test. The cumulative product is visible everywhere. The cognitive operation that produces it remains, by the standards of everything else it has built, unformalized.
Discovered, invented, or articulated
This puts pressure on an old question: is mathematics discovered or invented?
The discovery view says mathematical structures exist independently of us. The integers, the continuum, Hilbert spaces: they're out there, and mathematicians are explorers mapping pre-existing terrain. This explains why mathematics is objective (we don't get to vote on whether a proof works), why different cultures converge on the same structures, and why math is so effective in physics.
The invention view says mathematics is a human construction. We create formal systems, define rules, explore their consequences. This explains the creative element in mathematical work, the existence of multiple incompatible systems, and the way new mathematics gets built for specific purposes.
Consider what actually happens when new mathematics is constructed. Riemann didn't discover Riemannian geometry the way you discover a continent, stumbling upon something fully there all along. He also didn't invent it the way you invent a game, making arbitrary choices constrained only by internal consistency.
He was constrained. The geometry had to cohere. It had to capture something about the structure of possible spaces. It had to compose with other mathematics. Not anything would work.
And yet the geometry didn't exist until he built it. There was no fact about Riemannian manifolds before there was a concept of Riemannian manifolds. The structure became determinate through the process of articulation.
The dichotomy seems false. Mathematics is the articulation of structure that becomes determinate only through being articulated. The structure is real—we are constrained by it, we do not get to vote. But it does not sit fully formed in some Platonic realm waiting to be uncovered. It comes into focus through the work of formalization. The figure emerges through being drawn, but the figure isn't arbitrary; only some drawings work.
This is what happens in any domain where articulation is constitutive. A poem isn't discovered (it didn't exist before being written) but isn't invented arbitrarily (not any words will do; the poet is constrained by meaning, form, sound). The poem becomes real through being written, and what's written has to work. Mathematical construction is like that, but for structure as such.
Why structure at all
One level deeper: why is mathematics so effective at describing reality?
The standard framing puts two things on the table, mathematics and reality, and asks why they fit. But the framing may be wrong. There are three things, not two.
There is reality. There is mathematics. And there is the human cognitive apparatus that sits between them—the apparatus that perceives, carves the world into parts, notices regularities, builds formalism. Mathematics is not fitting a wholly mind-independent reality directly. Mathematics is fitting the intelligible arrangement of phenomena: the version of reality that has already passed through cognition, that has been parsed into objects and properties and relations before any equation is written.
If this is right, the fit isn't between two independent things. It's between two products of the same apparatus. The phenomena that get presented to us as phenomena have already been organized by perception and intuition. The mathematics built to capture those phenomena is built by the same cognition that organized them in the first place. The fit isn't a miracle; it's a near-tautology, downstream of cognitive structure on both sides.
This is not the claim that reality is mind-dependent, or that physics is psychology. The claim is more careful: mind-independent reality exists, but intelligible structure—the structure mathematics formalizes—is the joint product of what's there and how we're built to apprehend it. The reason math fits intelligible reality so well is that intelligible reality is partly what cognition makes it, and math is what cognition formalizes about itself.
This relocates the puzzle rather than dissolving it. A residual mystery remains: why does mathematics built within a cognitive niche shaped for medium-sized objects in low-dimensional space then go on to describe quarks and black holes and Hilbert space, things our cognition cannot picture at all? The fit between cognition and what cognition can directly apprehend is one thing. The fit between the formalism cognition builds and what cognition can never see is another. The second is harder to explain, and it is where the unreasonableness Wigner pointed at actually lives.
We don't have a settled answer. What can be said is that the question changes shape once we stop treating mathematics and reality as two independent terms whose alignment is mysterious, and start asking why the cognitive apparatus between them sometimes builds formalisms that reach beyond their cognitive origins.
The thoughts we cannot have
If formalism determines what's thinkable, then inadequate formalism doesn't merely slow us down. It stops us. And it stops us in a way we can't perceive, because the thoughts we can't have don't announce themselves as missing. We experience them only as absence, as the place where inquiry trails off without knowing why.
The scientists before Newton weren't stupid. They couldn't think correctly about motion because the mathematics didn't exist. The physicists before Riemann weren't incurious. They couldn't conceive of curved space because the structure hadn't been articulated.
What are we currently unable to think because we lack the formalism?
We can't know directly; that's what it means to lack it. But we can notice symptoms: domains where existing tools feel forced, where formalizations distort rather than illuminate, where the conceptual structure we intuit doesn't match the formal structure we're imposing.
Those symptoms are invitations. They mark places where new mathematics might be needed, and where building it might crack open what currently seems intractable.
Reasoning
Reasoning might be such a place.
We have logics. We have probability theory. We have computational models. And yet something doesn't quite fit: the way reasoning can evaluate its own standards, transform its own frameworks, operate on its own structure.
Existing formalisms tend to treat that capacity as external, as "meta," as something other than reasoning itself. Add a meta-level to reason about the base level, and you've relocated the problem: the meta-level operates within its own fixed rules, untouchable from within. The regress doesn't terminate in genuine reflexivity. It terminates in another frozen layer.
This is the symptom. Not proof that new formalism is needed, but the kind of friction that, in the history of mathematics, has sometimes preceded the construction of new tools. Calculus was preceded by Zeno's paradoxes and centuries of inadequate accounts of motion. Riemannian geometry was preceded by two thousand years of failed attempts to prove the parallel postulate from the others. Hilbert spaces were preceded by decades of classical physics breaking on quantum phenomena.
Whether reasoning is in that kind of position is not something I can settle here. The symptoms are real. The history is suggestive. The capacity we'd be trying to formalize is the same capacity, ultimately, that does the formalizing—which is the place where the work would be hardest and the payoff largest.
The question is open. That's the honest answer. And open questions of this shape are where the next chapter, if there is one, gets written.
