February 8, 2021. Some philosophical reflections on the nature of scientific explanation, structure, emergence, and the unreasonable effectiveness of mathematics.

Introduction

Explanations must come to an end somewhere.

Reductionism is the idea that you explain stuff with smaller stuff, and keep going until you stop. In many ways, this describes the explanatory program of 20th century physics, which, starting from the 19th century puzzles of statistical mechanics, conjured up atoms, subatomic particles, the zoo of the Standard Model, and even tinier hypothetical entities like strings and spin foams. Most physicists spend their time in a lab, on a computer, or in front of a blackboard, trying to reduce complex things to simple things they understand. So like Platonism in mathematics, reductionism in physics simply makes a philosophy out of everyday practice. We break stuff down, so things reduce; we play abstractly with mathematical objects, so they exist abstractly.

But also like Platonism, reductionism is a convenient fiction, or rather, a caricature in which some things are emphasised at the cost of others. And given the reverence which which philosophers hold the considered ontological verdicts of science, it’s worth asking: what does science really tell us about the universe? What sorts of objects are necessary for explanation? Does explanation go only upwards, or can it go downwards or sideways? Should we eliminate the things we explained? And what has explanation to do with existence anyway? This post is an attempt to unconfuse myself about some of these questions.

The existence of shoes

… our common sense conception of psychological phenomena constitutes a radically false theory, a theory so fundamentally defective that both the principles and ontology of that theory will eventually be displaced, rather than smoothly reduced, by completed neuroscience.

Physical objects can be described at different levels. A shoe is constructed from flat sheets of material, curved, cut, marked, and stuck together in clever ways; materials curve and stick by virtue of their constituent chemicals, usually long, jointed molecular chains called polymers; polymers, in turn, are built like lego from a smorgasboard of elements; and each elemental atom is a dense nuclear core, surrounded by electrons whirring around in elaborate orbitals.

From the properties of the neutrons, protons and electrons, it seems we can work our way upwards, and infer everything else. The laws of quantum mechanics and electromagnetism determine the orbital structure of the atom. The valence shell of the atom determines how it can combine with other atoms to form chemicals. Finally, the structural motifs and functional groups of the polymers gives it the properties the industrial chemist, the designer, and the cobbler exploit to make a shoe. Thus, some philosophers conclude, only electrons, protons, and neutrons exist. The rest can be eliminated as unnecessary ontological baggage. This view is called eliminative reductionism. It is a hardcore philosophy which does not believe in shoes [¹].

There is a gentler, less silly form of reductionism which grants the existence of shoes, but insists that they are (in the phrase of Jack Smart) nothing “over and above” the constituent subatomic particles. The shoe “just is” electrons and protons and neutrons, in some order; this is what we mean by a shoe. There are others way to characterise the reduction, and a whole literature devoted to the attendant subtleties, but most fall under the heading of analytic micro-quibbles. Instead, we will make a much simpler observation: order matters.

Clearly, if we took those subatomic particles, and arranged them in a different way, we would get different elements, different chemicals, and a duck or a planetesimal instead of a shoe. Arrangement is important. It is patently absurd to try and explain the bulk properties of the shoe—the fact that it fits around a human foot, for instance—without appeal to arrangement, since a different order yields objects which do not fit around a foot. Since order has explanatory significance, it should presumably be tarred with the same ontic brush we apply to things like electrons.

Of course, one may object that explanation does not equal existence. I can handily account for the continual disappearance of my socks by the hypothesis of sock imps. But this is a bad explanation! It’s not consistent with other reliably known facts about the world. Sock imps don’t make the ontic cut, not because there is no link between explanation and what we deem to exist, but because that link should only be made for robust explanations, and the poor little sock imps collapse at the first empirical hurdle. That different arrangements of things have different properties is robust, almost to the point of truism, and there seems to be no principled reason to ban order from our ontology.

Emergence vs structure

More is different.

It’s worth noting the parallel to emergence. In his famous article “More is Different”, Philip W. Anderson argued for the idea of domain-specific laws and dynamical principles which did not follow the strict, one-way explanatory hierarchy of reduction, particularly in his field of condensed matter physics. And indeed, condensed matter makes a science of order itself, studying how properties of macroscopic wholes (such as phases of matter) “emerge” from the arrangement of microscopic parts. Anderson thought of emergence as patterns that appear when you “zoom out” from the constituents, but which are still made from the constituents; we are just describing those constituents at a different level.

But this seems to suffer from the same problem as a reductionist account of shoes. The “emergent properties” are not properties of the constituents at all! The symmetries, order parameters, and collective excitations studied by condensed matter physicists belong only to the arrangements. In fact, systems made from totally different materials can exhibit the same emergent behaviour [²]! They are something new, something “over and above” the spins of the lattice, or the carbon atoms of a hexagonal monolayer, since different arrangements of those same parts would have different properties. We can turn Anderson’s snappy slogan around: different is more. If arranging things differently gives them new and different properties, it is a sign of structure, and structure is something over and above the component parts themselves.

What is a particle?

It is raining instructions out there; it’s raining programs; it’s raining tree-growing, fluff-spreading, algorithms. That is not a metaphor, it is the plain truth. It couldn’t be any plainer if it were raining floppy discs.

We don’t need emergence to argue for structure; we can use the elementary components themselves. When philosophers talk about reductionism, they tend to imagine subatomic particles as small, indivisible blobs, without internal organisation or further ontological bells and whistles. An electron might have properties like mass or charge, and obey the curious dictates of quantum mechanics, but all this is packaged irreducibly and not worth further discussion. But if we try and unpack all these “simple” properties, we will find that, like the magic bag of Mary Poppins, a particle is much deeper than it first appears! The Large Hadron Collider does not produce evidence for tiny, structureless blobs. Rather, it confirms at a rate of petabytes per second that the universe is made of mathematics.

The state-of-the-art definition of a particle is a bit of a mouthful: an irreducible representation of the Lorentz group. In plain English, being a representation means that particles are objects which have or “transform with” symmetries, in the same way a circle looks the same however you rotate it. That it is irreducible means that it cannot be split into smaller parts which have the same symmetry, which is the mathematical avatar of being “indivisible”. Finally, the symmetry itself, the Lorentz group, is the same group describing the shape of empty space according to special relativity. So, in summary, a particle transforms with the symmetries of empty space, and cannot be split into parts with this symmetry. Lurking implicitly in the background is the whole framework of quantum mechanics, and in particular, that particles are states in a Hilbert space. In plain English, we can add and subtract states of a particle, and compare them to each other.

Thus, every particle is like a mathematical diamond: indivisible, multifacted, and structured up to the hilt. When philosophers of science eagerly assent to believe whatever the particle physicists tell them, they may not realise what they signed up for! Spacetime, quantum mechanics, and symmetries, the Lorentz group and Hilbert spaces; these are all welded indissolubly to form the most robust and fundamental objects in the universe. Even with something as “simple” as an electron, order is inescapable.

Unreasonable effectiveness and natural patterns

It is difficult to avoid the impression that a miracle confronts us here, quite comparable… to the two miracles of the existence of laws of nature and of the human mind’s capacity to divine them.

It may feel like we have jumped from physical to mathematical objects in one fell, tendentious swoop. Do we need Hilbert space, or might another mathematical concept suffice? And does Hilbert space really exist, or is it merely a useful human invention? If the latter, why so useful? This is intentionally designed to rhyme with our earlier statement that order is a robustly explanatory feature of the world, and distinct from the things that are ordered. Mathematics really just is the study of order, or patterns, according to their own peculiar and abstract logic. Physics (and to a lesser extent the other sciences) study natural patterns, the way these structures or forms of order are realised in the natural world. That applies not just to emergent behaviour like phases of matter, but even the crystalline makeup of an elementary particle.

I have tried to motivate this perspective from the nature of physical explanation, but perhaps it can teach us about mathematical explanation and its relation to the physical world. A common criticism of Platonism is that, if mathematical objects exist in some non-physical realm, the ability to do mathematics must involve extrasensory perception. Clearly, since we are physical beings, this ability is grounded in physical experience, and now we have a simple explanation: patterns are naturally realised everywhere, from cardinal numbers in counting cows to topology in tying a knot to representation theory in colliding protons. We don’t need magical access to the World of Forms to see these things; they are all around us.

Similarly, the unreasonable effectiveness of mathematics for describing the world, first noted by Eugene Wigner, seems no more miraculous that the utility of integers for counting loaves of bread rather than proving results about number theory. We get the patterns from the world, clean them up, rebrand a little, and start connecting them together. The meta-patterns that emerge are remarkable, but the appearance of “unreasonable effectiveness” is the result of a largely successful PR campaign to divorce mathematical structures from their physical origins. As Einstein quipped, “Since the mathematicians have invaded the theory of relativity, I do not understand it myself anymore.” The abstraction of pseudo-Riemannian geometry follows from the more concrete act of bouncing light off mirrors.

More and more, we are seeing this converse of unreasonable effectiveness, where deep mathematical ideas are inspired by physics. The living embodiment of this trend is Ed Witten, a string theorist whose contributions to mathematics have been so profound and wide-ranging that he earned a Fields Medal (the Nobel prize in mathematics), the only physicist to have ever done so! Once again, there is no mystery here; it is just the usual state of affairs, but without the Platonist guff to distract us. The patterns are out there and always have been.

What is a pattern?

Everything comes to be from both subject and form.

All this raises the question: what is a pattern? The first and most famous philosophical treatment of these issues is the hylomorphism of Aristotle, who argued that objects are a compound of both form (the structure, order, or patterns I have discussed here) and matter (energy or “raw potentia”). I won’t discuss Aristotle’s ideas in greater detail. Suffice to say they have deeply informed this post, and the interested reader should check out James Franklin’s modern take. Instead, I will approach the question by picking on two smaller problems, taking Newton’s laws as a concrete example.

Newton formulated his laws of motion (such as $F = ma$) in terms of forces and acceleration. Does the empirical robustness of these laws mean that this is the only way to formulate them? Not at all! There are two other distinct but equivalent versions of classical mechanics: Lagrangian and Hamiltonian. They explain the same things, make the same predictions, and thus seem to describe the same natural patterns. This suggests to me that although patterns are discovered, formalisms are invented. A pattern is the equivalence class of descriptions.

Students of physics will be aware that, although Hamiltonian and Lagrangian mechanics are equivalent to Newton’s laws in the mechanical context, they have taken on a life of their own. The Lagrangian approach involves the mathematics of optimising functions, while the Hamiltonian approach in its most abstract form becomes the mathematical field of symplectic geometry. Both Lagrangian and Hamiltonian mechanics can be upgraded (with some inspired retrospective guesswork) to frameworks for quantum mechanics, which Newton’s laws simpliciter cannot. There is much more going on than a simple isomorphism of description! A more nuanced view is that humans invent formalisms which can agree on a domain of interest, a restricted equivalence class of explanation if you will. But the formalisms will tend to grow beyond the selvage lines of the original use case. Formalisms are only perspectives on patterns.

This hints at certain structural “metalaws”. Patterns are big and rhizomatic; human-invented mathematical frameworks are a single mathematical glance, if you like, and can only take in part of the pattern. Even if formalisms agree on some domain, they will suggest different corridors of growth. A rectangle may be both an equiangular quadrilateral, or a parallelogram with diagonals of equal length, but the notions involved and corresponding generalisations are distinct. This also helps explain the phenomenon of deep connections between apparently unrelated mathematical objects, sometimes only revealed by a clever change of perspective. It could be that there is a paucity of structure, so that by dumb luck (and the pigeonhole principle), we often unknowingly describe the same thing in a different guise. But to my mind, it is more likely that patterns tend to sprawl and overlap in complex ways. They are less like a few items of furniture in a crumbling garret—paucity of structure—and more like the interwined flora of a tropical jungle.

The second issue is how accurate our descriptions must be. We know that Newton’s laws are not exactly correct, and break down in regimes far-removed from those of everyday experience, such as the very small (where quantum mechanics applies) or the very fast (where special relativity applies). Does this mean we should stop believing in forces, or Lagrangians, or Hamiltonians? This is like the old Platonist quibble that there is no such thing as a perfect circle in the real world, so we must be reasoning about circles in some other realm. In both cases, the pattern is only approximately realised in nature, with bumps and fuzzy edges. But approximation is itself subject to structural laws, exhibiting patterns treated by mathematics (in, e.g., topology) and physics (effective field theory). Perhaps an even better example is statistics, which is literally all about extracting structure from noisy realisations. So structural approximations are clearly robust, lawlike and explanatory, even if they are subtle. Incidentally, this suggests another metalaw: patterns can stand in patterned relations to other patterns.

This ties back to our original question about the nature of physical explanation. Reductionism instructs us to boil things down to their smallest elements. The Aristotelian view is that, really, we should be searching for form and structure at whatever level they happen to occur. This is not only the nature of emergence, but physics more broadly. How else can we connect the study of the large-scale structure of spacetime, quarks, bowling balls, planetesimals, or storm clouds? Physicists almost never boil things down to their smallest elements! Rather, it seems much more accurate to say that they look for patterns “in the wild”. (In contrast, mathematicians study patterns “in captivity”, which gives them that air of artifice and pedigree.)

One upshot is that, for better or worse, physicists often wade into other disciplines armed with the lassoo of an Emergent Pattern to corral the apparent complexity. See for instance scaling laws, self-organised criticality, small-world networks, and thermodynamic explanations for life itself. They’re not always right (and they’re not always respectful), but they are just doing their thang.

Conclusion

I’ve argued that the nature of physical explanation is richer and less boringly hierarchical than the reductionist would have us believe. In order to explain the properties of shoes or particles, it seems not only parsimonious but necessary to commit to the existence of patterns in addition to the things which make those patterns up. This not only jives with (and ontologically grounds) the notion of emergence, but also provides a handle on the metaphysics and epistemology of mathematical explanation. Put simply, mathematicians study patterns; physicists study natural patterns.

Clearly, I’ve left many questions unanswered. Must patterns be instantiated in the physical world, and if not, where do such patterns live? What is the “mereology” that allows them to combine, or to recursively describe their relationships? And finally, what grounds the truth about patterns, in physics, mathematics, or elsewhere? Most of these I defer to Aristotle, though I hope to write more in future. In the mean time, discussion and debate are welcome!

Acknowledgments and references

I’d like to thank Leon Di Stefano for introducing me to Aristotelian structuralism and many enriching conversations over the years. His ideas inspired and informed this post. I’ve also been heavily influenced by James Franklin’s book, An Aristotelian realist view of mathematics. Aristotle himself writes with characteristic brevity on form and matter in Physics (i). Finally, I fitfully consulted the SEP entries on reductionism and mathematical structuralism.

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