But where is the fourth?

Continuous maths, discrete maths.

Maybe another good fourfold, orthogonal to the augmented Bourbaki-related fourfold above, would be (at least within the classical, non-intuitionistic framework):
Nonstandard analysis. Standard analysis. Discrete infinite maths. Finitistic (discrete finite) maths. Google AI has said in Feb. 2026 that nonstandard analysis is not widely used in math research. Maybe it's just that standard (i.e., non-infinitesimal, non-transfinitary) analysis, using ideal limits, is built into tradition, at least partly in response to criticisms by George Berkeley, one of the few non-mathematicians whose ideas on mathematics have been respected by mathematicians. But I just have no strong opinion on the question. ■

Engineer-esque classifications.

I think I hit upon the idea of many to many, one to many, etc. as a way to classify areas of math, but I found that others had seen it before me. I lost much info when I stupidly crashed my hard drive in 2005.
Case 1: Somebody on the Internet had laid the idea out, dealing also with cases that didn't fit neatly into place. Later I found that that person had gotten into some sort of personal trouble. Well, maybe AI will help me find more on such classification, it's been years since I tried.
Case 2: On the Internet, I happened upon a mathematical subjects classification by an engineer for engineers. I understood some of it and it fit my classification idea to that extent. I thought, yes, an engineer might well appreciate systematic classing according to many to many, etc. I've been unable to find it in Internet searches since then. ■

The Four Causes meet special relativity. Hilarity does not ensue.

Notice E−pc at the end of the Special Relativity row above. E−pc, or equivalently E⁄c−p, stands to net momentum p as SLOWNESS stands to SPEED. I noticed decades ago that E−pc would be more useful to consider for a system at high speed than for one at rest. It doesn't take Einstein to see this. In a rest frame, the quantity's magnitude is simply equivalent to the rest mass (rest energy). This is likewise as linear energy is more useful to consider for a system at low speeds than for one at lightspeed, where it is equivalent to magnitude of momentum. It's almost a mirror-image kind of thing. It's right there in the arithmetic. Anyway, now (2026-2-17 or so) I find it said by Google AI that E−pc is a formula for light-cone momentum and is useful to consider at high speeds. Call it "slowmentum"?—no, that already is defined for another physical quantity. In doodles I used to label it lusk and symbolize it k. I'm not a physicist or a mathematician, but I was fiddling around with these things in search of Aristotle's Four Causes or the like. I think I found them and it's not something lame.

Assume relativistic units (i.e., lightspeed c = 1).
p = net momentum.
e = linear energy.
E = total energy.
m = rest mass (often symbolized by "m₀" as I do earlier in this post).
k = gross minus net momentum, corresponding to internal spatial structure but not invariant in magnitude like rest mass across all rest frames soever varied.

Equations:

Remember, set lightspeed c equal to 1 (to temporarily streamline it out of the equations).
{Relativistic slowness in a direction}
= 1−{velocity v as a fraction of lightspeed}.
k = E(1−v) = E−p.
E² = p² + m².
E = p+k = m+e.
p−m = e−k.
p−e = m−k = p+m−E = E−e−k = √(2ke).
Example:

Aristotle's Four Causes' principles favor special relativity over Newtonian mechanics.

The above distinction between bearer and borne works only in Einsteinian physics, not in Newtonian physics, where the bottom two corners are flatly conflated (always m=k and bearer = borne, so there's no occasion at all to extract and label k and borne) since Newtonian physics recognizes no signal-speed limit such as lightspeed universal to events and invariant across all rest frames, and so recognizes no non-arbitrary quantity of slowness and does not distinguish among rest energy, total energy, and total energy minus (magnitude of) net momentum (E−pc).

I used not to know that Ancient Latin patiens served to translate Ancient Greek páschonta (undergoer, sufferer, e.g., "which is cauterized"); páschonta had connotations like those of passion rather than of patience which are nearly the opposite. Instead, I assimilated agent:patient to make:let and must:can. So I had to make a fourfold out of the four causes' traditional three causal principles (agent, patient, act). Now I have FOUR causal principles (I suspect I'm the first to see the fourth) for the FOUR causes. I think that's an improvment.