March ?, 2020. Stuff
Mathematics is the study of patterns. Physics is the study of patterns in nature. It is no surprise, then, that the two subjects are deeply intertwined. But we can think about feature of this relationship in terms of what I will (only half-seriously) call “metapatterns”. These are empirical observations about patterns which explain how maths and physics work and are related.
Order and chaos
Humans have long believed in an orderly cosmos, with natural laws governing cause and effect. In a sense, even mythology is a search for principles underlying regularity. Even if the laws are fanciful, the premise is in fact an empirical one: there is regularity in the world above and around us. Things need not have turned out that way. We could live in a disorderly universe, where “laws” changed from moment to moment, and patterns vanished the moment they were observed. But in our universe, even chaos is a highly structured affair. This leads to our first metapattern:
Metapattern 1. Nature is patterned.
It seems, at first, like every branch of observational science is separate and all things are different. Nature may be patterned, but perhaps there are too many patterns to usefully catalogue or transform into a science. But one of the key lessons of maths and physics is that many patterns are secretly the same. Newton saw the same laws governing the motion of the planets and the fall of an apple. By labelling points on the plane with $x$ and $y$, Descartes united algebra and geometry. This suggests a second metapattern:
Metapattern 2. Patterns recur.
Shallow and deep patterns
If we could read patterns directly off their manifestations, recurrence would be a trivial matter. But physics is an experimental science; the laws of gravitation are not written in the stars or the arc of a tennis ball. We reason backwards from observed regularities or shallow patterns to the simplest mathematical laws which generate them. But these guesses can be wrong. There is always a deep pattern, the true mechanism, which we can never observe directly and can never confirm we have found. At best, we can match our patterns against the patterns generated by nature and show there is (as far as we know, and for the moment only) no difference. That might change.
At first sight, the situation in mathematics is very different. Mathematicians define the patterns they study into being; there is is no experimental uncertainty when we specify the angles of a triangle. Put differently, definitions hardcode regularities into mathematical objects. Instead of guessing laws of nature, mathematicians prove theorems. These are logically consequences of the hardcoded regularities, and cannot be disproved by subsequent definitions. Maths, it seems, is not subject to the empirical vagaries of physics.
But this is misleading. Why do mathematicians bother to prove theorems in the first place? If a triangle is completely specified by the angles, why don’t we know everything about it once the angles are given? The answer is that hardcoded regularities — the assumptions and properties true by definition — are shallow patterns not dissimilar from empirical observation. They are in no sense generative.
In contrast, theorems provide classifications, equivalences, transformations, computational methods, bounds, and so on, which are not part of the shallow data we put in by hand. They are novel outputs, and give us power over the objects of mathematics in the same way that natural law gives us power over physical objects. And like natural laws, theorems are partial: they do not and cannot yield complete knowledge. The deep patterns are the structures themselves, with all the regularities manifest by virtue of that structure, and they remain unseen. We arrive at another metapattern:
Metapattern 3. Deep patterns are unobservable.
The different faces of patterns
What do I really mean by “deep patterns” and “structures themselves” in mathematics? This seems a bit obscure.
Emergence and approximation
The regularities that take so much work to uncover in physics appear to be hardcoded into mathematical objects, and no subsequent theorems can refute them.
But why do mathematicians spend so much time proving theorems? A theorem is a mathematical result that follows the “hardcoded regularities”. For instance, if a triangle is hardcoded to be right-angled, with short sides of length $a$ and $b$, the long side has length $c$ obeying\[c^2 = a^2 + b^2.\]
This is not obvious simply from the definition of the triangle as right-angled. In fact, the hardcoded regularities — the assumptions we built into the mathematical object itself — are more like experimental observations than natural laws.
Unlike physics, we do not “explain” the regularities. But the work of mathematics is to reason from the simple regularities to theorems: properties we did not expect. I think that theorems hint at deep patterns in mathematics, in just the same way that natural laws are, or hint at, deep patterns in physics. It is these deep patterns which are of real interest in both domains, since they require work and insight to reach, and tend to be of greatest leverage.
Metapattern 3. Patterns are deep.
Comment re deep patterns not being knowable?
Deep patterns help explain the mathematical phenomenon of duality. Often, two completely different looking objects in mathematics turn out to be equivalent. “Equivalent” means that we can turn one into the other, and vice versa, using some sort of structure-preserving transformation or isomorphism (Greek for “same shape”).
To understand what this means, we need to say more about what a pattern is. Regularities are often only revealed by careful observation, and the patterns we invoke to explain them may be refuted by a single observation. Nature obeys deep patterns, which we can never directly observe, but only attempt to reconstruct While physicists have to wade through a morass of observation to arrive at mathematical truth, mathematicians can coin it for free.