Manan Pal Singh

What Does It Mean to Discover Mathematics?

Recently I came across a discussion in a private discord channel -- how good AI is at Math? Can they prove theorems that haven't been proven yet? A person said

Its value in math has been to bruteforce insane permutations of stupid ideas that humans don't have the ability to try. Whether or not that's intelligence, I don't know.

AI can rapidly explore many candidate approaches: algebraic maanipulations, substituions, lemma combinations, computational cases, at a scale, a human cannot. But calling it "bruteforce" misses several things

Still, the skepticism is justified. Generating thousands of plausible steps is not the same as understanding why a theorem is true. In serious mathematics, the hard part is often choosing the right definitions, inventing a helpful abstraction and recognizing which direction is meaningful, not merely searching permutations.

AI can be an extremely fast mathematical explorer, but exploration alone is not discovery. Discovery alone is not a verified proof.

But if AI can explore different techniques and try to combine them to come up with something meaningful, is that merely a bruteforce or is it a mathematical discovery?

That question quicly turns into another question:

Is human maehtematical creativity fundamentally different?

If a system combines known techniques in a genuinely new way and that combination yields a new theorem, proof or method, that is a mathematical discovery indeed. Discovery doesn't have to come from a mystical flash of insight, in fact, it never has.

Human mathematicians also search. They try examples, change representations, apply known theorems, combine techniques, reject dead ends and follow patterns that appear promising. Expert intuition makes this search much more efficient, but it remains, at least partly, a search through possible ideas.

This suggests that the difference between human and machine mathematical work may not be as simple as "understanding versus brute force". It may instead involve several dimensions

To understand the issue, it helps to examine what human mathematical discovery has historically looked like.

My tentative answer is yes - a machine can discover mathematics without possessing conscious understanding. But what it discovers, how independently it discovers it, and whether it can recognize the significance of its own discovery are separate questions. A system may produce a novel and correct theorem while still depending on humans to formulate the problem, judge its importance, explain the proof, and place it within mathematics.

Is successful combination already discovery?

Suppose a system takes two known techniques that have never previously been used together and combines them to prove a theorem that nobody had proved before.

That would ordinarily count as a mathematical discovery.

Discovery does not require creating every ingredient from nothing. Scientific and mathematical discoveries are almost always constructed from inherited concepts, notation, previous results, examples and unanswered questions. If a new combination produces a previously unknown result, then the result can be genuinely novel even though its ingredients are old.

This gives us at least three distinct questions:

Was the result new?

If the theorem or proof was previously unknown, it possesses historical novelty.

Was it correct?

A plausible-looking argument is not enough. A proof must survive careful scrutiny, formal checking where possible and evaluation by experts.

Was the method significant?

A proof may be technically correct but uninteresting. Another proof may reveal a new structure, unify several areas or create a method that solves many further problems.

These are separate from the philosophical question of whether the discoverer consciously understood the result.

A computer search can discover a new mathematical object without possessing human-like awareness. A chess engine can produce a brilliant move without experiencing insight in the way a human player does. Its competence is still real.

How can something combine ideas without understanding them?

The answer depends on what "understanding" means.

There are at least three levels.

Current AI systems can display substantial amounts for the first two. The third, however, is unknown and may not be present at all.

A system therefore does not need conscious awareness to combine ideas successfully. It can learn patterns such as

That can produce a valid proof / solution even if the system lacks the richer human experience of understanding, knowing why the idea is natural, what it means conceptually, or how it changes the broader picture.

The difference may be that human intuition is embedded in consciousness, embodiment, motivation, taste and a broader understanding of mathematical meaning. But at the operational level, both humans and AI may use internalized patterns to direct search.

Mathematical taste operates before proof search begins. Choosing which question to ask may require more understanding than answering it. A system that solves a carefully formulated conjecture demonstrates one kind of competence; a system that notices an overlooked phenomenon, formulates the right conjecture, and explains why it matters demonstrates another. Any claim about autonomous mathematical discovery should therefore account for how much of the problem formulation came from the machine and how much came from its human environment.

Were historical mathematicians also searching?

Yes!

A mathematician might spend months testing an approach that a computer could reject in seconds. However, this does not mean human discovery was merely random brute force. Expert mathematicians possess mathematical taste. They can often identify

This reduces the search space dramatically.

The strongest human contribution is often not just finding a path through an existing search space. It is changing the search space.

A mathematician may realize that

This is representation-level creativity. The central question for AI is therefore not only whether it can search quickly. It is whether it can reliably invent new conceptual spaces in which the search becomes easier.

One sign of understanding is explanatory compression. A brute force calculation may establish a thousand separate cases, and a good theorem explains all thousand through one structure. The deepest discoveries reduce the number of independent facts we must remember. They show that apparently different phenomena follow from the same underlying reason. An AI system that merely accumulates results is less impressive than one that discovers such compression.

Does mathematical discovery ever come from absolute scratch?

Probably not.

Even the most revolutionary mathematical ideas depend on prior concepts, structures etc. A thought that had absolutely no relationship to any prior concept would be impossible to formulate or understand.

This does not make creativity trivial. It means creativity is better described as transformation rather than creation from nothing.

A major mathematical advance often follows this pattern

This is why calling discovery "combination of past concepts" can be misleading. The word suggests attaching existing tools together mechanically. But an important discovery may change the relationships among all the tools.

It is less like combining two puzzle pieces and more like realizing that the pieces belong to a three-dimensional object rather than a flat picture.

To see why this distinction matters, it helps to study one of history's greatest mathematical discoveries. Calculus is often remembered as the work of Newton and Leibniz, yet its origins stretch across thousands of years of accumulated ideas. If we misunderstand how calculus was discovered, we risk asking the wrong questions about whether AI can discover mathematics.

Newton and the Long Ancestry of Calculus

Newton is often described as one of the inventors of calculus. That is correct, but it can create the impression that calculus appeared suddenly in his mind.

It did not.

Calculus emerged from several ancient and medieval streams

Newton and Leibniz unified these streams into general systems.

Object discrimination and one-to-one correspondence

Before mathematics, there had to be a cognitive distinction between

Imagine an early human group distributing food. Nobody needs the word "four" to notice that four people require four portions. They can pair one portion with one person.

This is one-to-one correspondence, one of the deepest foundations of counting.

Two collections have the same size if their elements can be paired without leftovers. This is a primitive but deep mathematical structure. Even modern set theory defines equality of cardinality through bijections, which are generalized one-to-one pairings.

At this stage, there was no such thing as numbers. Only concrete judgments such as

The key achievement was treating quantity as something detectable even before it had a name.

Stable counting sequences

The next step was attaching a repeatable sequence of words, marks or gestures to objects

one, tow, three, four...

A successful counting procedure required several principles

Suppose five sheep pass through. A person counting them may make one notch for each - |||||.

The marks store quantity externally. Mathematical information no longer disappears when the objects leave sight.

External notation is essential to every later form of mathematics. Long calculations, proofs and symbolic systems depend on storing intermediate structure outside the unaided mind.

The abstraction of number

At some point, five sheep, five stones and five days become instances of the same abstraction - they had "five-ness" to them. This asbtraction is more profound than it initially appeared. Because now, a collection's number becomes independent of

Five widely separated stars, five tightly packed seeds have the same cardinality. That is a major conceptual leap. It allows statements such as "5 and 3 is 8" without specifying 5 of what and 3 of what.

Mathematics begins to separate structure from content. This same separation later allows a variable such as x to stand for length, time, mass, money or an abstract element of a system.

Arithmetic operations - Addition and Subtraction

Repeated practical situations generate general operations.

For example

Once operations become abstract, people can reason about relationships rather than solve each practical case separately.

This becomes important for geometry. A rectangle of length 5 and width 4 has area

Integration will eventually become a generalization of addition, accumulating many small contributions whose sizes vary.

Fractions and measurement

Whole numbers cannot express every practical magnitude. Half a load, three quarters of a field require fractions.

Measurement differs from counting. Counting asks how many separate objects exist. Measurement asks how many copies of a chosen unit fit into a magnitude.

Fractions allow quantities to lie between whole numbers.

This is an early movement from discrete mathematics toward continuous magnitude.

Practical geometry

Agriculture, architecture, taxation and construction create problems involving

Rectangles are easy to measure because they can be tiled by unit squares. Curved boundaries are more difficult. A circle or irregular field cannot be tiled exactly by a finite number of ordinary square units.

This introduces one of the central questions that eventually produces integration

How can a curved area be measured using simpler shapes?

At first, practical approximations were sufficient. But mathematics eventually asked for exact methods and proofs.

Astronomy and continuously changing quantities

Atsronomy forced mathematicians to reason about quantities that vary continuously

This was different from merely counting livestock. Astronomical quantities vary with time, and predictions required interpolations between observations.

Ancient babylonian astronomy included sophisticated numerical procedures. Evidence also exists of babylonian reasoning relating a planet's displacement to the area under a time-velocity type graph, though this did not become a general calculus.

Suppose an object moves with varying speed. If the speed were constant, distance would be

But when speed changes, no single value of v is sufficient.

The future calculus question becomes

How does a continuously changing speed accumulate into total distance?

Astronomy supplied both motivation and numerical sophistication for later calculus.

Greek deductive proof

Greek mathematics transformed many mathematical practices into deductive systesm. A result was no longer accepted only because it worked repeatedly. It was derived from definitions, assumptions and earlier propositions.

This difference is substantial!

A claim could no longer rest only on repeated observation. It could be established through definitions, assumptions and a chain of necessary reasoning.

There is a difference between observing that many triangles appear to have an angle sum of 180° and proving that every Euclidean triangle must have that angle sum.

This created a standard that would later challenge early calculus. Infinitesimal calculations often produced correct answers, but mathematicians still had to explain why the procedures were valid.

The history of calculus is partly the history of closing the gap between successful computation and rigorous justification.

Irrational magnitudes

The diagonal of a unit square has length . There is no fraction p/q that has square exactly equal to 2.

For Greek mathematics, this created a problem because the ratios of whole numbers were not sufficient to describe every geometric magnitude. They responded by developing a theory of proportions for magniutes, without reducing everything to arithmetic numbers.

This allowed continuous geometry to progress before arithmetic had fully absorbed irrational numbers.

Zeno and infinite subdivision

Zeno’s paradoxes forced attention onto infinite subdivision.

To move from one point to another, an object must first travel halfway. It must then travel half the remaining distance, and so on.

The total distance can be written as

This route contains infinitely many conceptual subdivisions. How can they be completed in finite time?

Modern mathematics handles this using convergent series. Ancient mathematicians did not yet possess that theory. Zeno’s paradoxes made clear that infinity, continuity and motion could not be handled casually.

Antiphon approximates curved figures with polygons

A circle cannot be tiled exactly by a finite number of ordinary triangles or rectangles. One response is to inscribe a polygon.

Begin with a square inside a circle. Then double the number of sides:

The polygon occupy more and more of the circle. The unfilled curved slivers become smaller. Antiphon is associated with early polygonal approach.

The essential idea is not merely to draw a polygon. It is to construct a sequence of approximations whose error becomes progressively smaller. This is one of the central metal structures of limit reasoning.

Eudoxus and the method of exhaustion

Exodus developed a rigorous method for establishing exact geometric results through approximation.

The structure is roughly

This does not speak explicitly of a numerical limit. But it proves that an error cannot remain positive because repeated refinement can eventually make the approximating discrepency smaller than any given magnitude.

Exodus used such methods to prove results including

Archimedes explicitly credited these results and built extensively on the method.

A major idea appears here

An exact result can be established through arbitrarily accurate approximation.

Euclidean geometry

Euclid organized geometry, ratios and earlier mathematical results into an axiomatic deductive system.

Later mathematicians inherited

For centuries, mathematical questions about tangents and areas were expressed geometrically rather than algebraically.

A parabola was not initially the graph of . It was a conic section defined through geometric relationships.

Conic sections create sophisticated curved objects

Greek mathematicians studied the ellipse, parabola and hyperbola as geometric curves.

These curves provided important test cases for questions that later became calculus

The parabola was especially significant because it was curved, yet structurally regular.

Archimedes combines discovery heuristics with rigorous exhaustion

Archimedes brought ancient methods of area and volume calculation to an extraordinary level. In the quadrature of the parabola, he showed that the area of a parabolic segment equals of the area of a certain inscribed triangle. His construction generates areas of the form

The geometric series sums to

Archimedes didn't merely declare the infinite sum. He used exhaustion style proof to establish the exact result. Even more revealing, Archimedes distinguished between

He sometimes imagined figures as composed of infinitely thin slices balanced on a lever. His preface explicitly says he first found the parabolic result mechanically, and then demonstrated it geometrically.

This looked remarkably modern

That pattern remains central in modern mathematics and AI-assisted research.

Place-value notation and zero

Indian place-value notation transformed computation.

The numeral 507 represents

The position of a digit determines its value, and zero preserves an empty position. This greatly simplifies arithmetic, approximation and algebraic calculation. The development of notation is not merely cosmetic. Better notation changes what humans can think and calculate.

Algebra becomes a general problem solving language

In ancient and medieval aglebraic traditions, equations gradually became objects that could be transformed according to rules. Example

can be solved by completing the square

The important conceptual change is that an unknown quantity can be manipulated before it's value is known.

Trigonometry turns geometry into computable relationships

Astronomy required accurate relationships among angles, chords, sines and cosines. A sine table approximates a continiously varying function at select inputs

To consruct such tables, mathematicians needed

This work created a setting in which mathematicians repatedly asked how a function changes when it's argument changes slightly.

For a small "h"

Such relationships are close to derivative thinking, even when no general derivative concept is present.

Medieval motion theory separates uniform and varying speed

14th Century Oxford scholars studied velocity and acceleration in abstract mathematical terms.

The mean speed theorem states that an object underoing unfiorm acceleration, travels the same distance as an object moving for same duration of time, at the average of initial and final velocities.

This is equivalent to area of a trapezoid beneath a velocity-time graph.

The Kerala school develops infinite trigonometric series

From roughly 14th century onward, members of the Kerala school developed remarkable infitie series expansions. They also developed correction terms that improved convergence when computing π. A series turned a difficult transcendental function into arithmetic

That is an enormous conceptual tool because powers are easier to differentiate and integrate, than an arbitrary gemotrical curve.

However, the Kerala school did not produce the same general unification later achieved by Newton and Leibniz, a systematic differential-integral calculus with widely transferable notation and rules.

Renaissance recovery of Greek texts restores old area methods

European scholars recovered, translated and studied Greek mathematical works, including Archimedian geometry. This mattered because quadrature of curves and volume of solids again became active research programs rather than isolated ancient accomplishments.

But exhaustion proofs were cumbersome. A separate geometric argument might be needed for each new shape. The need became

Can we replace individually crafted proofs with a reusable method?

Kepler uses infinitesimal-style slicing for volumes

Johannes Kepler investigated volumes of barrels and other solids by imagining them divided into many thin slices.

A barrel is not a cylinder, so it's volume cannot come from simple single formula. But one may think of it as assembled from thin circular disks

The important shift is from exhaustion's contradiction-based proof to a more direct constructive imagination.

Cavalieri formalizes the method of indivisbles

Bonaventura Cavalieri treated a plane area as if it were composed of indefinitely many parallel line segments and a solid as if composed of planar slices.

If two solids have equal cross-sectional areas at every corresponding height, then they have equal volume

Cavalieri's method made many calculations faster and became an important factor in development of integral calculus.

Symbolic algebra becomes compact enough for general rules

During 16th century and 17th century, algebraic notiation became increasingly symbolic.

Compare The cube of the unknown together with two times the unknown equals 4 with

The symbolic form exposes structure. It can be transformed, generalised and re-used. Expressions like these made it possible to develop rules that apply to entire families of objects.

Descartes and Fermat create analytic geometry

Rene Descartes and Pierre de Fermat connected alegbraic equations to geometric loci.

A curve could now be represented by an equation such as

This was revolutionary, because now a geometric curve could be investigated by algebraic manipulation.

A point on the curve is not merely a location in a diagram. It's a pair satisfying an equation.

The tangent problem becomes algebraic

For a circle, a tangent can be defined geometrically as a line touching at one point. For more complicated curves, "touching" is inadequate because a tangent may cross the curve.

The deeper idea is that the tangent captures the curve's local direction. Given two nearby points

the secant slope is

As becomes smaller, the secant approaches the tangent. The proble was that at , the expression became 0 / 0. Calculus had to develop a meaningful way to obtain limiting ratio without simply performing an illegal operation.

Fermat's adequality handles tangents, maxima and minima

Fermat developed a method, often called adequality.

Consider finding maximum of

Compare with

Set this "adequal" to and cancel common terms

Discard the remaining small

Fermat used this technique for maxima, minima and tangents, and his methods became a major contribution to differential calculus.

Roberval and Torricelli treat tangents through motion

Another approach imagined curves as trajectories produced by combined motions. For example, a point on a curve might result from

The tangent direction is then the direction of the combined instantaneous motion. The connects tangents to mechanics rather than static geometry. The idea strongly anticipates Newton, for whome, changing quantities would be generated by continious motion.

Wallis generalizes powers and infinite interpolation

John Wallis extended algebaric and infinitary techniques. He worked with generalized powers and quadrature formulas, and treated sequences of cases in ways that encouraged interpolation to broader formulas.

For positive integer "n", one can ask for the area under for x from 0 to 1. The modern result is

Finding patterns across encourages the belief that one rule governs an entire family.

Wallis also influenced Newton's work on infinite series and generalized binomial expansions.

Barrow recognizes differentiation and integration as inverse processes

Isaac Barrow, Newton's teacher, developed a geometrical relationship between tangent problems and quadrature problems.

Suppose is the area accumulated under a curve from a fixed starting point to x. Increase x by a small amount "h". The new thin strip has approximate area . Therefore,

Dividng by "h"

In modern notation,

This is the conceptual heart of the fundamental theorem of calculus.

Newton conceives variables as fluents

Newton described changing quantities as fluents - maginutes generated by continuous motion. If "x" and "y" change with time, their instantaneous rates are fluxions - and

Suppose,

Let "x" increase momentarily by where "o" represents a tiny interval of time. Then

Since

Divide by

As the moment vanishes, the last term disappears

Newton's framework connected calculus naturally to velocity, acceleration and physical motion.

Newton connects quadrature and fluxions

Newton understood that finding an accumulated area and finding a rate of change were inverse problems.

If , then it's fluxional relationship gives the equivalent of

Reversing the rule gives

That means area calculations could often be solved by finding an expression, whose fluxion produced the curve.

This unification is a strong reason Newton deserves credit as an inventor of calculus rather than merely another precursor.

Leibniz develops differentials

Leibniz appraoched the subject through infinitesimal differences. If "x" changed by "dx", then "y" changes by "dy". The ratio represents the local relationship between those changes.

Leibniz invents the integral notation

Leibniz chose to represent sum of infinitely many small contributions. Thus suggests adding strips with height "y" and infinitesimal width "dx". This notation was not purely cosmetic, it helped mathematicians reason alegbraicly.

You may ask what was the point of going through all of this?

The history of calculus shows that mathematical discovery does not require creation from nothing. Newton and Leibniz inherited numbers, geometry, algebra, infinite series, tangent methods, quadrature, theories of motion and centuries of approximation techniques. Their achievement was not that they possessed ingredients nobody had ever seen. It was that they reorganized those ingredients into a general framework and revealed a relationship that had previously remained fragmented - accumulation and instantaneous change were inverse aspects of the same structure.

This suggests that combination can indeed be discovery—but not every combination is equally profound.

A machine that searches through known techniques and finds a previously unknown proof has discovered something. If the proof is correct, the achievement does not disappear merely because the system lacked a subjective flash of insight. Mathematics is not made true by the emotional experience of its discoverer.

But discovery has degrees. A machine that proves a supplied conjecture has done less than one that formulates the conjecture. A machine that formulates a conjecture has done less than one that invents the definition or representation that makes an entire family of conjectures visible. The deepest mathematical advances do not merely find answers inside an existing space of possibilities. They change the space in which future questions are asked.

That may be the real boundary to watch.

Can a machine notice that a problem has been expressed in the wrong language? Can it invent an invariant because the existing quantities obscure the structure? Can it recognize that two distant fields are describing the same object? Can it replace hundreds of isolated results with one explanatory principle—and then understand which new questions become possible because of it?

These are not merely engineering questions. They are the same abilities that distinguish incremental mathematical progress from conceptual revolutions.

Perhaps machine mathematics will not arrive as a sudden artificial Newton producing a complete theory in solitude. It may emerge gradually, first as accelerated search, then as reliable proof, then as conjecture generation, then as the invention of new representations and finally as the construction of theories that humans had not thought to seek.

At that point, asking whether the machine "really understood" may remain philosophically important. But it will no longer decide whether a discovery occurred.