The Categorical Imperative: Pi, PhDs, and Thinking Like a Scientist‑Magus

# Pi, PhDs, and the Categorical Imperative — a tiny manifesto for tired grads

You graduated into a world that prizes tidy truths and inboxes under 100 characters. Meanwhile the internet insists reality might be a circle trying to turn itself into a square, that you could be a spontaneously conjured Boltzmann brain, and that Francis Bacon possibly moonlighted as a wizard. If you’re a Gen‑X/Millennial grad, here’s a short, slightly snarky primer on juggling these weird ideas and writing a PhD proposal that won’t get you politely escorted to the door.

## The circle and the square: metaphors before metaphysics

Think topologically: a coffee mug and a donut are homeomorphic; a circle and a square are too. That’s the point — at a certain level of abstraction the shapes tell us about qualitative sameness. But when poets read pi as “the stubborn remainder between infinity and containment,” they’re doing cognitive topology with metaphorical boundary conditions.

Math disciplines give different lenses:

– Topology: cares about deformation. Circle vs square = same shape under continuous warp. Useful metaphor for structural similarity.
– Differential geometry / analysis: cares about curvature, measure, and local structure — where the circle’s symmetry matters.
– Number theory / analysis: waves in on pi, irrational and transcendental, the precise stubbornness that refuses to reconcile discrete grid and continuous swirl.
– Category theory: stops asking what objects are and asks what maps between them do. It’s the philosopher’s Swiss Army knife for translating metaphors into formal structure.

These perspectives show how a single metaphor can be rigorous in one discipline and poetic in another. Don’t confuse evocative imagery for a substitute theorem — but do use it to point at problems worth formalizing.

## Science and philosophy: an institutional divorce, not a metaphysical one

The split between lab coats and armchairs was administrative, not epistemic. Methods changed and specializations multiplied, but science still leans on unprovable assumptions about observation, repeatability, and what counts as explanation. That’s where logic and philosophy quietly sneak back in: model theory tells us what structures satisfy our axioms; proof theory tells us how confidence accrues; Bayesian probability gives a formal language for belief updates.

In practice every experimentalist is an amateur philosopher of methods — they just call it a protocol.

## Boltzmann brains and the topology of credence

The Boltzmann brain is a beautiful, awful thought experiment because it surfaces a mismatch between cosmological measures and epistemic expectation. From a logic/probability angle:

– Bayesianism asks for priors and a likelihood model. If your cosmology wildly favors freak observers, the posterior says you’re probably one — which is cognitive suicide.
– Measure theory and algorithmic randomness put a fine point on “typicality.” If typical observers are weird anomalies at late times, either the cosmology is wrong or our reference class is.
– Modal and epistemic logics let us formalize self‑location uncertainty. They don’t make the problem go away; they make it precise.

The practical verdict: if a theory makes you the statistical equivalent of a hallucination, fix the theory, not your sanity. That’s sound methodological hygiene.

## Occult roots, respectable lineage

Alchemy and Hermeticism weren’t just lampooned superstition; they were proto‑programs for manipulating nature, readable in the same breath as early experimental notes. Newton’s notebooks included both fluxions and recipes for transmutation. Over centuries, the desire to “make nature talk” evolved into reproducible method, instrumentation, and standards of evidence.

So when you flirt with the language of animism or symbolic architecture, remember: the same hunger produced modern optics, spectroscopy, and grant applications. Keep the romance, earn the reproducibility.

## How different forms of logic help you not be a reckless mystic

If you want to play at the seam of esoterica and rigor, here are useful logical tools:

– Model theory: to see which narratives are structurally possible given your axioms.
– Proof theory and type theory: to check that your arguments don’t secretly rely on hidden assumptions.
– Modal logic: to distinguish necessity, possibility, and epistemic status (am I necessarily a Boltzmann brain? Probably not).
– Paraconsistent logic: if you must entertain contradictory intuitions without exploding into triviality, this is your safety helmet.
– Algorithmic information theory: to penalize weird, ad hoc explanations that compress poorly.

These aren’t magic wands; they’re safety protocols.

## Writing a PhD proposal without invoking thaumaturgy

You can be charmingly iconoclastic and still fundable. Practical checklist:

– One clear question; not “rewrite metaphysics,” but a crisp problem you can finish in three years.
– Situate your work across literatures — philosophy, history of science, math, logic — and say why that cross‑pollination matters.
– Concrete methods: archival work, formal modeling, case studies. Spell out data, archives, and supervisors.
– Feasibility: timelines, ethics, and a plan B if the grant agency gets bitchy.
– Anticipate objections: show you know the slippery spots and how you’ll anchor them.

Short paragraphs, direct prose, and a sentence about your contribution on page one will get you further than clever metaphors.

## Final takeaway — wonder with a permit

The posture you want is curious but skeptical. Use geometric metaphors as heuristics, not ontological claims. Treat logical tools as instruments, not talismans. Honor the Boltzmann brain as a sanity check on cosmology, not an excuse for epistemic nihilism. And if your PhD project flirts with Renaissance magic, balance scintillating conjecture with archival rigor and clear methods.

I like to end on a slightly impertinent note (because what’s philosophy without a little impudence): wonder is valuable, but so is the ability to explain to a committee why your wonder is answerable. So here’s my parting question for you, the curious reader:

If you could formalize one poetic intuition (pi as the remainder between infinity and containment; the world as a dance of circles and squares) into a mathematical or logical object, what would it be — and how would you test whether the formalization actually captures any of the charm that made you care in the first place?