We Look Once, Carefully, and Then Compute: Kant Synthetic a priori

Kant's Question

Kant asked how we can know anything about the world without looking at it. He called such knowledge synthetic a priori: it says something about the world, and it owes nothing to observation. His own answer was that the knower supplies the form of experience, and nobody believes it now. The question outlived the answer, and it is answerable, because the stock of examples has grown in a direction he could not have foreseen. The knowledge now comes out of a computation.

Three Examples

Define a curve by x²+y²=1. Its circumference is 2π. Nothing was measured, no circle was drawn, no string was laid along a rim.

Take a body of any shape in slightly viscous nearly incompressible flow. Solve the Euler equations with a slip condition and the drag and the lift come out of the shape alone — no turbulence model fitted to data, no tabulated coefficient, no wind tunnel. Euler himself claimed this in 1755: that his equations contain all of such flow, water and air at subsonic speeds, with nothing further to supply. He could not act on the claim because there was no way to solve them. The solving is what was added.

Give the nuclear charges and the number of electrons. Minimise the energy of non-overlapping charge densities. Out comes the configuration of the molecule — bond lengths, angles, where the lone pair points, tetrahedral rather than square. No measured bond length went in, no force constant, no functional calibrated against molecules whose answers were already known.

What Makes Them A Priori

The objection is obvious: a computer is physical, a computation is a physical process, so how is this knowledge without observation?

The criterion is not whether some physical process occurred. It is whether the object was interrogated. A spectrometer interrogates the molecule — light goes in, the molecule answers, and the answer makes the number. A simulation consults no water. Run it in a universe containing none and the digits are the same. No circle was drawn, no wing was placed in a tunnel, there was no water in the room.

How It Is Possible

The deflationary half first, because it is true: the answer was already in the input. The definition determines 2π, the equations and the shape determine the drag, Coulomb's law and non-overlap determine the geometry. Nothing was created.

The other half is the whole point. Latent is not available. Nobody reads the H–O–H angle off Coulomb's law by inspection, and nobody ever read π off x²+y²=1. The content is implicit, and the only route from implicit to explicit is a chain of steps.

So: a priori knowledge of the world is latent information made explicit by labour. The possibility was never the mystery. What the old debate lacked was any account of the labour — which is exactly what was missing before there were machines to do it.

And the question changes shape. Kant asked whether such knowledge is possible. The answer is yes, routinely, industrially; the better question is at what rate does the implicit become explicit, and can the rate be certified? Exact for the circle. A computed bound for the drag. A mesh ladder and a step ladder for the molecular geometry. What separates the first from the other two is dissipation.

No Arbitrary Parameters

It is usual to concede at this point that all this is a priori only above an empirical foundation — that observation bought the laws and computation merely spends them. The concession is too quick.

Nothing empirical enters the circle's definition. The Euler equations are momentum balance and nothing else, and the one measured parameter they carry, the viscosity, drops out of the answer: the drag coefficient is essentially Reynolds-independent across the whole range of interest, which is precisely why the shape alone suffices. The inverse-square law is what a conserved flux in three dimensions must look like; its form is forced by geometry, not fitted to a torsion balance, and the coupling is a choice of units. RealQM's whole input is a geometric law and a list of integers.

Experience enters at one point only: in the claim that a construction applies to some given object. And that is the antecedent of a conditional — the specification, supplied rather than discovered. Nobody calls the circumference of a circle empirical because somebody had to choose the radius.

That no fitted quantity appears anywhere is moreover a requirement, and the requirement is Leibniz's. An arbitrary parameter is exactly a quantity for which a theory can give no reason, so by the principle of sufficient reason it cannot stand: either the theory determines it, and it was never arbitrary, or the theory is unfinished at that point. A fitted parameter is not a measurement incorporated into a theory. It is a hole in the theory with a number in it. Which is the charge against turbulence constants, against density functionals calibrated on a training set whose answers were known in advance, and against contracted basis sets. That such a theory then reproduces the data it was fitted to is not evidence in its favour; it is what having been fitted means.

Even the dimension of space is not an input. Below three nothing can be complicated — a network must be planar, so no circuit of any depth can be wired, and a passage through an organism severs it. Above three nothing is stable — there are no closed orbits, and the four-dimensional Coulomb potential is exactly the critical case for collapse to the centre, so there are no atoms and no chemistry. One thinkable option.

The Point

Kant asked how we can know anything about the world without looking. The answer, two and a half centuries on, is that we look once, carefully, and then compute.

The full argument is in the article, A Priori Knowledge of the World: Kant's Question and Three Computed Answers.