Human Mathematics

Epistemic status: High confidence from pattern recognition across many conversations. Both math-phobia (“I’m just not a math person”) and math-smugness (“I’m naturally gifted”) are illusions—–opposite reactions to dumb teaching procedures.

During my university years I had this piece of conversation over and over, in many different flavours:

I never got to properly investigate why and if I had the chance to ask, the answer converged almost always to either “I was just incapable to do math” or “I had a terrible teacher”. It seems to be socially acceptable to not know/like math, in a rather interesting way. Imagine the same conversation:

—Or—

Wouldn’t it feel a bit odd? Why is it unremarkable—almost expected—to declare defeat in mathematics?

The normalized illusion of “math is too hard for me” comes from the fact that most of us meet mathematics devoid of its narrative¹. We are handed procedures² without the intuitions they compress, syntax without semantics, facts without reasons. When the story that gives the rules meaning is invisible, confusion feels like a personal failing rather than a missing layer of explanation.

The irony becomes even more convoluted. Because the few for whom the dull rules do make sense—whether through raw intuition, clearer teaching, or sheer persistence—are quickly labelled “naturally smart”. The label comforts everyone: if understanding math is an innate gift, then struggling is inevitable and quitting is blameless. After all, if you ain’t smart there is nothing you can do about it. Plus, everybody else seems to struggle with it, and—as the Italian saying goes “if everyone’s in it, it’s almost a party”.³

So if you do get math, then you’re automatically “special”. But, you see, there is nothing mystical about grasping an idea once it is built from first principles⁴.

So, the main idea of this post is to expose how mathematics only comes alive when we start from first principles—when we seek the underlying ideas and reach for patterns they realize. Re-inserting that narrative does more than make the subject engaging; it restores agency to the learner and reveals how every layer of abstraction emerges from the one beneath it.

First Principles

Mathematics⁵ is fundamentally an exploratory process. We discover true statements from reasonable assumptions; these statements exist because they capture an internalized idea. It is worth underscoring that these assumptions are devised by—and for—humans, not gods. Anything can be replaced and remodelled provided it preserves the same expressive power. The ideas behind any specific implementation are what really matters; the implementation itself is merely an artefact of the fatty lump of tissue we carry in our skulls; meaning and value arise from the expressive reach of the underlying concepts.

Hence, logical coherence alone is not sufficient. A mathematics that does not also tell us why something is true—what necessity binds premise to conclusion—risks becoming a sterile bookkeeping of symbols. The formal grammar that composes it must expose the deeper structure, the web of concepts that it expresses, showing how each statement compels.

Below is a short “guideline” of the style I’m striving for:

• Start blind: every definition, property or inference step must be designed from first principles.
• Objects/mathematical entities embody ideas: each entity instantiates a distilled intuition. For example, a Group embodies the idea of symmetry, a Vector Space captures the idea of directed magnitude⁶.
• Relationships first: treat entities as black boxes; prescribe only their interactions. The emergent structure is the collection of recurring patterns you see in the web of connections.
• Results shape the entities: In practice, theorems just compress the information that we gathered from the web of relationships.
• Layered abstractions: relationships themselves can be promoted to objects, after which we study interactions among those newly minted entities, recursively. This ascent through meta-levels—objects, relations, relations-of-relations⁷—amplifies expressive power.

Taken together, these principles compose the living architecture of mathematics. Each layer feeds into the next, yet the instantiation of their content is not sacred; at any rate, we are free to refactor the foundations provided the edifice retains (or improves) its conceptual integrity. Such flexibility embodies epistemic⁸ humility: we acknowledge our blindness yet keep reaching for the light.

The Layers

Mathematics is often taught in strings of discrete steps—definition → lemma → theorem—marching from whatever is considered “common ground” to the highest peaks of abstraction. This local scaffolding is pedagogical, not ontological. The subject itself behaves more like geological strata laid down by successive intellectual eruptions. Each layer solidifies, then becomes the bedrock on which the next eruption deposits fresh ash.

However the lived practice is subtler. Layers overlap, bleed, and sometimes reorder themselves retroactively when a new idea reframes everything beneath it.

Yesterday’s theorems ossify into today’s definitions; yesterday’s definitions dissolve into tomorrow’s tacit background. To move between layers you do not crack them open with a pick-axe—you basically have to wait within one until the immersion teaches your cortex a new grammar, so it allows you to express ideas in the next layer. Tom Garrity calls this slow osmosis mathematical maturity.⁹

Acquiring a fresh layer of abstraction rewires the brain rather than simply adding facts. The process parallels any learning experience—picking up a new musical instrument, mastering a new language, etc. Once the neural circuitry for that skill or concept crystallises, the world itself appears richer: faint harmonic overtones previously indistinct fit cleanly into a bigger structure; mathematical structures that once tangled together resolve into a crisp landscape.

After the neurons myelinate, you cannot “un-see” the new patterns, therefore it becomes obvious.

Textbooks slice the mathematical wilderness into neat regions—algebra (groups → rings → …), analysis (calculus I → II → …), topology (point-set → algebraic → …)—to avoid cognitive overload. The partition is a navigational aid, not reality: the map is not the territory. But it spares beginners the burden of walking every path themselves, outsourcing, locally, some of the meta-cognition¹⁰.

Eventually, when all the connected components merge up, the boundaries blur and the layers dissolve into a single fractal-like tangle of ideas. A glance at Wikipedia’s Glossary of Areas of Mathematics confirms the interbreeding: there is almost no field that sits alone without another adjectivized before it (e.g., algebraic geometry, geometric algebra, analytic number theory, etc).

Trying to operate in a stratum whose sub-structure you have not yet internalised is the most common pothole on the mathematical road. Tackling spectral sequences without first digesting basic homological algebra is the canonical example: the machinery looks arbitrary, the “obvious” steps are opaque, and frustration masquerades as stupidity. The failure is architectural, not cognitive; the scaffolding was yanked a floor too soon.¹¹

The cure is not grit but gradient: retreat, patch the missing layer, then climb again. Once the neurons have myelinated the requisite circuits, yesterday’s nightmare step becomes today’s one-liner—and, inevitably, tomorrow’s subconscious assumption.

Conclusions

The title of this post was inspired by Tom Leinster¹²