Why mathematics works: The mind-reality connection
Reading | Metaphysics
Brian Fang, BSc | 2025-10-24
Brian Fang discusses the many instances in which mathematics developed without empirical motivation turned out to precisely describe the physical patterns of nature. Why would primates evolved to hunt and gather develop the cognitive ability to unveil the underlying mathematical structure of the cosmos? He argues that the most plausible explanation is that nature is itself the expression of mind-like structures also directly present in the human intellect. Mathematical introspection is thus an exploration of the underlying mental landscapes of the cosmos as a whole.
The most incomprehensible thing about the universe is that it is comprehensible.
Albert Einstein
The core mystery
Here’s a puzzle that should keep anyone awake at night: A physicist develops equations for particles she’ll never see, describing conditions from microseconds after the Big Bang. Her brain—evolved to track prey and navigate tribal politics—somehow grasps quantum mechanics and spacetime curvature. This isn’t just remarkable; it’s deeply strange.
This strangeness has a name: the unreasonable effectiveness of mathematics. We create abstract mathematical structures with no experimental motivation, then discover decades later that nature runs on exactly those structures. This happens so regularly that we’ve stopped noticing how bizarre it is.
The question is: what does this tell us about reality itself?
The pattern that demands explanation
Consider the timeline. In 1854, Riemann explored a radically general geometry where the rules for measuring distance could vary from point to point. Pure imagination—no experiment suggested space worked this way. Sixty years later, Einstein realized Riemann’s mathematics perfectly described gravity as curved spacetime. He didn’t invent new geometry; he recognized a structure that was waiting.
This pattern repeats with uncanny regularity:
- Group theory (abstract algebra) → backbone of particle physics
- Hilbert spaces (functional analysis) → natural language of quantum states
- Topology (rubber-sheet geometry) → describes phases of matter
- Category theory (mathematics of mathematics) → appears in quantum foundations
The key point: mathematical structures typically emerge decades before physicists need them. We’re not curve-fitting data; we’re discovering a prepared architecture.
It is important to point out that the mathematical formulation of the physicist’s often crude experience leads in an uncanny number of cases to an amazingly accurate description of a large class of phenomena.
Eugene Wigner
Why “just a byproduct” fails
The common evolutionary explanation for human mathematical ability is that it’s a spandrel—an accidental byproduct of general intelligence honed for survival. But this view collapses under scrutiny.
First, there’s the cost problem: the human brain consumes 20% of our energy while making up just 2% of body mass. Evolution paid a steep price for this organ—difficult childbirth, long childhoods, and massive caloric demands. Natural selection is not wasteful; it’s implausible that it would incidentally produce neural architecture capable of understanding infinite-dimensional Hilbert spaces or the 196,883-dimensional symmetries of the Monster group.
Second, there’s the interference problem: rather than extending our evolved cognition, mathematics often contradicts it. Our spatial intuitions resist non-Euclidean geometry; our temporal sense fails to grasp infinity. The very heuristics that help us survive—approximate reasoning, bias, social instinct—tend to obstruct mathematical thought. If math were an extension of survival reasoning, we would expect cognitive harmony, not deep internal friction.
Perception, by contrast, is fast, heuristic, and contextual—prone to illusions and bound to the present moment. Mathematical reasoning is none of these: it is abstract, slow, universal, and structurally alien to the messy adaptiveness of the senses. The two are not merely different; they reflect fundamentally divergent modes of knowing.
Evolution is a tinkerer, not an engineer. It works with what is already there and takes the path of least resistance. It is not always the most efficient solution, but it is the dumbest solution that works.
François Jacob (Nobel Prize-winning biologist)
The anthropic dodge and its limits
A subtler response invokes anthropic selection: intelligent observers can only arise in universes governed by consistent mathematical laws, so it’s no surprise that we find ourselves in a world where mathematics works.
But this addresses only the background conditions for observation, not the specific, puzzling nature of mathematics’ success in physics. The anthropic principle might explain why we exist in an orderly universe, but it does not explain:
- Why we often discover the relevant mathematical structures before experimental data demands them.
- Why mathematics developed with no application in mind—like number theory or group theory—turns out to be essential for technologies and physical theories.
- Why independent branches of mathematics converge on the same structures later revealed in physics.
- Why each domain of physics tends to have a uniquely appropriate mathematical framework, rather than many equivalent alternatives.
Here’s the deeper issue the anthropic account misses: We should expect mathematics to be full of discarded structures that once seemed promising but proved irrelevant to the physical world. We should find a landscape dominated by abandoned theories—consistent, expressive, even elegant, but ultimately rejected by nature.
And while such structures do exist—think of p-adic numbers, large parts of pure logic, or games like chess—they only emphasize how unusual the successful cases are. The striking fact isn’t that some math is useful. It’s that the math most often developed without physical motivation keeps proving physically indispensable.
Riemann’s abstract geometry, developed decades before general relativity, now guides satellites and GPS. Euler’s identity—uniting complex exponentials, π, and imaginary numbers—underlies quantum interference and wave functions. These aren’t cherry-picked coincidences; they’re part of a repeated pattern where mathematics, pursued for its own internal structure, turns out to match the architecture of physical reality.
The mystery isn’t that mathematics works; it’s that the universe keeps speaking in mathematical structures we discovered before we knew there was anything to say.
Mathematics appears as a storehouse of abstract forms—the mathematical structures; and it so happens—without our knowing why—that certain aspects of empirical reality fit themselves into these forms, as if through a kind of pre-adaptation.
The Bourbaki group
How discovery actually feels
Mathematicians consistently describe their work as recognition, not invention. Poincaré’s famous account is typical: “At the moment when I put my foot on the step of the omnibus, the idea came to me… that the transformations I had used to define the Fuchsian functions were identical with those of non-Euclidean geometry.”
Ramanujan claimed formulas arrived in dreams from his goddess. Gauss grasped the arithmetic series formula in a childhood flash. These moments share a phenomenology: insight arrives complete and unchosen, as if uncovering something pre-existing.
Independent rediscovery strengthens this impression. When Newton and Leibniz developed calculus simultaneously, when Bolyai, Gauss, and Lobachevsky independently discovered non-Euclidean geometry, we witnessed discovery of objective structures, not parallel invention.
The idealist resolution
If mathematics consistently proves to be reality’s deep grammar, the most economical explanation is idealist: reality is fundamentally mental or structural, rather than material.
This isn’t mysticism—it’s taking physics seriously. Modern science already traffics in pure structure, relation, and information. Quantum mechanics dissolves particles into probability amplitudes. General relativity replaces solid matter with curved geometry. What we call ‘physical’ increasingly means ‘mathematical.’
Mathematical Platonism holds that mathematical objects exist independently in some abstract realm. But this creates an epistemological puzzle: how do physical brains access causally inert abstract objects?
Analytic idealism offers a cleaner solution: on this view, mathematical structures exist as stable patterns within a universal mind-like reality. Physical matter consists of how these patterns appear from localized perspectives (i.e. our individual minds). Mathematical discovery becomes a form of introspective access to transpersonal mental structure.
This resolves the central mystery: evolved brains can grasp mathematical truth because both brains and mathematical truth emerge from the same underlying mental reality. We’re not imposing order on alien matter—we’re decoding patterns in which we participate.
The connection is straightforward:
- Platonism says: mathematical objects exist independently and somehow guide physical reality.
- Idealism explains: they exist as patterns in the fundamental mind-like structure that is reality.
- Result: no mysterious causal gap between abstract math and concrete physics.
I regard consciousness as fundamental. I regard matter as a derivative of consciousness.
Max Planck
…there appears to be some profound reality about these mathematical concepts, going quite beyond the mental deliberations of any particular mathematician. It is as though human thought is, instead, being guided towards some external truth—a truth which has a reality of its own.
Roger Penrose
But perhaps some mind, rather than producing eternal truths, always has eternal truths in mind. Such a mind would have to be eternal itself if the truths it possesses are eternal.
St. Augustine
The things which are temporal arise by their participation in the things which are eternal. The two sets are mediated by a thing which combines the actuality of what is temporal with the timelessness of what is potential.
Alfred North Whitehead
The beauty connection
This view also explains why aesthetic criteria work in mathematics and physics. Beauty isn’t cultural decoration—it’s a detection mechanism for deep structural harmony. When physicists chase ‘mathematical elegance,’ they’re following their nose toward fundamental symmetries.
Maxwell’s equations, Einstein’s field equations, Yang-Mills theories—these emerged through aesthetic polishing, not data fitting. The fact that beautiful mathematics keeps making successful predictions suggests we’re tracking real structural features, not imposing arbitrary preferences.
A physical law must possess mathematical beauty.
Paul Dirac
What this means
None of this implies every mathematical speculation will prove physically relevant, or that physics is finished. It means the effectiveness of mathematics is telling us something metaphysical about reality’s nature.
The modest conclusion: mathematical structures exist independently and are somehow built into reality’s foundation. The bolder conclusion: reality is the unfolding of mind-like logical structure, accessed through our participation in it.
Either way, evolution didn’t create mathematical ability from nothing. It tapped into pre-existing logical order. The fact that primates can prove theorems about perfect objects and decode cosmic structure isn’t an accident—it’s a clue about existence itself.
The bottom line
When abstract equations dreamed up for pure intellectual joy turn out to be nature’s source code, we’re receiving a message about reality’s deepest nature. The universe gave rise to beings who can understand it through mathematics. Perhaps it’s time we took that hint seriously.
The most reasonable explanation is that mind and mathematics aren’t accidental features of a fundamentally material cosmos. Instead, they’re windows into what the cosmos actually is: a vast, evolving pattern of logical structure that we call physical reality when viewed from the outside, and consciousness when experienced from within.
The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve.
Eugene Wigner
Bibliography
- Balaguer, Mark. “Mathematical Platonism.” Stanford Encyclopedia of Philosophy, 2021. https://plato.stanford.edu/entries/platonism-mathematics/
- Colyvan, Mark. The Indispensability of Mathematics. Oxford: Oxford University Press, 2001.
- Dirac, Paul A.M. “The Evolution of the Physicist’s Picture of Nature.” Scientific American, vol. 208, no. 5 (May 1963): 45-53.
- Einstein, Albert. “Physics and Reality.” Franklin Institute Journal, March 1936.
- Jacob, François. “Evolution and Tinkering.” Science, vol. 196, no. 4295 (10 June 1977): 1161-1166.
- Kastrup, Bernardo. Why Materialism Is Baloney: How True Skeptics Know There Is No Death and Fathom Answers to Life, the Universe, and Everything. Winchester, UK: Iff Books, 2014
- Penrose, Roger. The Road to Reality: A Complete Guide to the Laws of the Universe. London: Jonathan Cape, 2004.
- Planck, Max. Interview. The Observer, 25 January 1931, p. 17.
- Poincaré, Henri. “Mathematical Creation.” In Science and Method. London: Thomas Nelson and Sons, 1908.
- Steiner, Mark. The Applicability of Mathematics as a Philosophical Problem. Cambridge, MA: Harvard University Press, 1998.
- Wigner, Eugene P. “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.” Communications in Pure and Applied Mathematics, vol. 13, no. 1 (February 1960): 1-14.
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