Pen exercise: My Feelings on Grothendieck

Constraint: write about what you feel when you read excerpts about The God of Abstraction.


I’ve been a physics student for seven years now, and if you include my youthful incursions into the subject, even longer. In my journey I have seen a lot of clues, a lot of suggestions from a variety of sources: the motion and push and pull of things I can see, the way our experiments (or what I would like to think of as ‘penetration tests’ of reality) resolve themselves in particular ways, even the solutions we get from the idealised world in our problem sets. There’s a unity behind all those things, a vantage point from which everything becomes simple, or if not as simple as possible then a vantage point from which all have their own natural places, at least.

Whereas The God of Abstraction has bequeathed us a unifying worldview for the discrete and the continuous (and indeed a proof-by-example that abstraction, or rather the right abstraction, can be used to resolve confusion), the situation in physics is not as fortunate. We live in only one of many possible realities, whereas mathematics is a vast ocean that extends infinitely in all directions and suggests an incomprehensible depth we can only sense in vague ways via computational complexity considerations.

Where then is the conception of reality that is simple, true, calculable, and inevitable? We express our models in austere equations yet they stumble from even minute deviations from the idealised case. We attempt to explain reality by introducing small deviations from our ideal models, and in a particular way where we can ignore higher-order effects? Two days ago I was confronted with a humongous equation describing in full analytic glory the dynamics of a (2+2)-body system, by which one can take to mean two Earth-Moon systems orbiting each other. This equation filled many pages and I daresay no mortal can ever hope to understand it if only on its own terms. And this is okay? So long as it reduces to nearby ideal cases?

Models of equivalent explanatory power are interchangeable, hence we are free to choose whichever suits our human preconceptions the best. If we cannot take in the most accurate map of reality in our minds (and we cannot, for the only truly accurate model is reality itself), then perhaps we can get away with a simpler model with simple boundaries and simple regions. And there are only very few things which are simple to us, conceptions which we can take for granted a priori by virtue of being a species forged from the dry African savannah. Barry Mazur gives us an incomplete list, with a particular emphasis on our mathematical cognitions (really they aren’t as specialised and as particular to mathematics as they seem):

First, our physical intuition. Our understanding of causes and effects, of push and pull, of movements and their causes as we see them in the world. It is what enables us to take objects or phenomena out of reality and into our heads and run them as if in a computer simulation.

Second, our computational intuition. Processes, algorithms, counting, our counterargument to the Halting problem. It is what we use to anticipate the completion of processes and induct from a laughably few examples (really, an impossibly huge number of examples obscured only by our unawareness of our own sensory data).

Third, our geometric intuition. To build from basic notions a specific thing, like a bird building its nest from humble twigs. The fuel that powers our engineering marvels and the metaphorical fuel that will (eventually) bring us to the stars.

Lastly, our algebraic intuition. Our capacity for abstraction, to sense the essence of things, or in the spirit of one of its greatest possessors, to see the vantage point from which things in consideration appear easy and most natural. This is our capacity to compress our models of reality and draw simple boundaries along its joins.

Four intuitions from a single cluster in mindspace. The task to win, to survive until the last black hole evaporates and perhaps a little longer.

Physics brands itself as the fundamental vantage point from which everything can be understood (in principle). Really, there is no ‘physics’, no ‘chemistry’, no ‘biology’, no ‘philosophy’. There is only reality, the vast swaths of which we are yet bereft of an explanation, let alone of control. To our children we shall seem blind and deaf. To our children’s children, a mere higher-order effect they can handwave away, impossibly small and weak, like motes of dust in a sandstorm. But this future is not guaranteed. It is but one of many, one where we do not fail in our inexorable march towards winning this race towards the natural chaos where all things eventually go (and perhaps a bit further).

So unity. We have so many toy models, submodels, barely compatible pockets of coherence in our best-possible-map-of-reality we call our ‘physics’. Rare are those who simplify and compress: the last great practitioner of this lost art was Maxwell, and even then he failed to utilise the simplest possible formalism available in his time. Now there is no one. A tradition in dire need of resurrection. A true resurrection, not the goings-about by textbook authors who seem more concerned with imitation and deference to Nobel Prize-winning authorities than with a genuine presentation (really, the charitable interpretation is that it’s difficult to write down our best-possible-model at all rather than a conscious attempt at copying).

But how will this resurrection be made possible? A clue is found in both the youth of the God of Epistemic Rationality and the God of Abstraction. The former arrived at and extended Quinean naturalism by himself, the latter reinvented Lebesgue integration by himself. It is therefore this brave and arduous act of independent cartography, of seeing what is out there and making the map yourself which produces the quality of thinking needed to even begin at the task of simplification and generalisation of our best-possible-map.

This then is our crossroads. Of the people who have the physiological capacity think independently for extended periods of time, there are very few who get the corresponding (and it seems, necessary) period of isolation in which they would be able to practice this capacity in full. The opportunity for a Lebesgue-integration moment is not available to all, especially at a time when it is most needed (i.e., during one’s formative years).

If one has missed this fertile period of one’s intellectual development, can one still compensate by undergoing such an activity at a later date? Perhaps, but it should be noted that the fertile ground available to a would-be cartographer recedes in proportion to one’s schooling, for it might happen that one exhausts cursory familiarity with all fields accessible to a beginner, thus forever tainting the independence of one’s work.

Still, I believe reality is sufficiently large that we are not yet, in these modern times, at the point where such an exhaustion of fields can take place. There will always be fertile ground for an aspiring rationalist-empiricist, and if that ground recedes unreachable for everyone it shall be our duty as a society to enforce a temporary ignorance in special institutions in order to allow such an important activity to those who shall need it.

Coddle our cartographers. They are our only hope for sanity in this vast and lonely universe.


Definitions

The God of Abstraction
Alexander Grothendieck
The God of Epistemic Rationality
Eliezer Yudkowsky, since E. T. Jaynes is dead
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