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Pythagoras and Us

I’ve switched to posting on this personal blog site instead of on Medium, because Medium clearly looks too professional: people kept wondering how such rough and unpolished writing could ‘get published’ and lamenting the decline of standards in academic ‘publishing’. I’d hate to add to the causes for lamentation in this world, so hopefully this medium looks more appropriately bloglike and nobody will be misled.

Pythagoras on the Purpose of Life and the Meaning of Wisdom – Brain Pickings

As my first topic here, I had a thought about Max Tegmark’s Mathematical Universe Hypothesis (MUH). Tegmark is an overt Pythagorean, and Pythagoreanism, as I see it, vindicates a very strong form of rationalism: the view that a priori, non-empirical thinking can yield all the knowledge we are capable of having.

I’ve written “thinking” rather than “reasoning”, because I think the word “reasoning” is too loaded. Some people, e.g., think that you can only reason from a given premise, and so reasoning on its own can’t deliver any knowledge since it would have no premises to begin from. I want to include Cartesian intellectual intuitions in “a priori, non-empirical thinking”. I want to exclude anything that comes from the senses. “Rationalism” is thus a bad label, but history has entrenched it, and I’d rather reclaim than fight.

Tegmark’s Pythagorean hypothesis is that the whole universe – actually multiverse – is not simply describable in mathematics; rather, “our universe is mathematics”. The reason physics becomes more mathematical as it advances is that physical reality just is a mathematical object.

Our Mathematical Universe by Max Tegmark | Waterstones

The OG Pythagoreans (allegedly) held that the universe is just numbers; Tegmark holds that it is more generally a mathematical structure, which he defines as “abstract entities with relations between them“. In his book, Our Mathematical Universe, Tegmark notes that a physicist might define a particle as “an element of an irreducible representation of the symmetry group of the Lagrangian”. Such an element is, for Tegmark, a purely abstract object. And the universe consists of nothing but such abstract objects in a structure. (As an aside, Tegmark has something else in common with the OG Pythagoreans: a suspicion of irrational numbers – Tegmark is a finitist and rejects real numbers that require an infinite amount of information to specify.)

I like Pythagoreanism. Spinoza was a sort of Pythagorean, as I read him. And I’m heedful of the warning from Ross, Ladyman, and Spurrett, that metaphysics should begin from real science rather than some metaphysician’s idea of what science should say. So it’s good then that Tegmark – a real scientist – is actually saying this. In fact what he says is quite close to a position they defend: Ontic Structural Realism (OSR). But there is a big difference, and that difference matters for strict rationalism.

Steven French attempts to distinguish Tegmark’s Pythagoreanism from OSR. The latter identifies the physical world with a structure; as Ladyman et al put it: “There are no things. Structure is all there is.” But it is one thing to say that everything is structure; it is quite another to say that that structure consists of abstract objects with relations between them. The latter Tegmarkian formulation in fact seems to contradict the OSR, which denies that there are any “things”, but perhaps abstract objects are different from “things” in this sense; they are nodes in the structure with no intrinsic properties of their own. The real difference is that an abstract Tegmarkian structure exists mathematically, and Tegmark is partial to “David Hilbert’s dictum that ‘mathematical existence is simply freedom from contradiction'” (from Our Mathematical Universe).

This is a bit like Johannes Clauberg’s idea of “real being” that I mentioned once. Among other things it implies that every mathematically-conceivable universe exists just as much as “ours” does (I’ll explain the scare-quotes soon). If we think of the total mathematical description of the universe an ideal physics could give, we define one universe, but there are many, many, many other non-contradictory mathematical descriptions, and each of these, on the Pythagorean theory, defines something that exists in just the same sense. French comments: “to say this is ontologically inflationary would be an understatement”.

It’s not just ontologically inflationary. It vindicates strict rationalism. Defining mathematically possible structures is, I think, a purely a priori activity. Most scientists and philosophers would say that you then need empirical proof that some structure is actually instantiated in the universe. You need to take measurements and see if they correspond to the structure. But for a Pythagorean, measurement is irrelevant. Any possible structure is real.

This makes science easy from the armchair. Is there a Higgs boson? Well, is that a possible solution to the relevant equations? Then yes! And also no, if it’s possible to solve the equations another way. Different worlds, but all perfectly real. Is there an elephant in your living room? Well “your living room” is just its mathematical description, so if it’s mathematically possible to describe it including (the mathematical description of) an elephant, then yes there is an elephant in your living room. And also no elephant, of course. And a pig, and an aardvark, and whatever you like.

Here some people might object; in fact I think Tegmark would object. Empirical science, he would say, is for finding out truths about the mathematical structure that we are in. He makes much of a distinction between what he calls:

the outside view or bird perspective of a mathematician studying the mathematical structure and the inside view or frog perspective of an observer living in it.

We are the observers, viewing the universe from our “frog perspective”, and empirical science, Tegmark might say, is our means of working out which mathematical structure we are viewing from the inside.

But I don’t think that will do, and this is why I put “ours” in scare-quotes above. After all, what are we? On Tegmark’s MUH, we can only be mathematical structures: mathematical descriptions of our physical properties (Tegmark is a physicalist). In fact, as subjects we need only be mathematical descriptions of whatever states of our body are necessary to realise a certain conscious experience. But there are many, many, many such descriptions: of Boltzmann brains floating in deep space, of brains in vats, brains hallucinating, brains in universes like ours up to the edge of the light cone and then radically different. Which of these many brains is you?

To formulate the question, you must refer to something other than a mathematical structure. Again, mathematical structures are only defined up to isomorphism, so that on the MUH we can’t avoid the answer that you are all the isomorphic mathematical structures that correspond to your conscious experience. And so the question, “which universe are we in?”, will be answered by determining which mathematical structures contain sub-structures that define us. That, again, can be done purely a priori (except perhaps the bit where we connect mathematical descriptions of brains to conscious experiences, which possibly can’t be done at all).

You could, of course, add something to the mathematical structure to validate indexical discriminations, though that would take some working out. You could also add immaterial souls into your ontology and give them a special relationship with one particular mathematical structure among many. Then you could have a place for empirical science and avoid strict rationalism: empirical science is for finding out which mathematical structure our souls are specially related to. Sometimes I think this is roughly the picture that Descartes had in mind.