Maths Is Everything: The Pythagorean Position on Physical Reality Is False

 

My essay involves two aspects: (1) The full acknowledgement that mathematics is the only means to accurately describe the physical aspects of the quantum world. (2) A critique of Pythagorean physics… Many would say that a defence of Pythagorean physics should follow from (1) above, not a critique. Yet this doesn’t need to be the case, as hopefully will be shown.

Image by ChatGPT.

In quantum mechanics, it is the case that everything ‘said’ about particles, forces, fields, etc. is said with mathematics. Indeed, what else could be used to describe these micro-entities and extremely intangible phenomena? Apart from metaphors such as “fields”, “particles”, “waves”, etc., there’s simply nothing else that can be used to do the job. Yet surely that doesn’t mean that there’s no territory to map…

Or that we must become Pythagoreans.

Again, other than metaphors, what language or set of descriptions could be used other than mathematics? It isn’t that there’s nothing else there. It’s that only maths is available to do the job. Some readers may see, then, why Pythagoreanism is so attractive to various mathematical physicists and philosophers.

Metaphors and Analogies in Quantum Mechanics

There are other ways to describe the quantum world other than with maths. We can use metaphors. Such metaphors are — or should be — entirely dependent on the prior maths. That’s the case even if sometimes metaphors come before the maths, or are over and above the maths.

Still, just because there are other ways to understand the quantum world, that doesn’t mean that they truly reflect it.

Maths provides descriptions, but does it provide understanding?

In terms of metaphors again. Metaphors are said to be required to make a connection between the mathematical formalism and what experimental physicists experience. Indeed, could anyone truly make sense of the formalism without metaphors and analogies? [See later section on mystical Pythagoreans.] How would we know what it’s all about?

So are the metaphors and analogies purely stand-ins for the maths? The layperson and even physicist may well need them — but so what! Isn’t this just a consequence of the limitations of the human mind — even the physicist’s mind? If reality is mathematical, yet we must rely on metaphors and analogies, then reality still remains mathematical.

What about the case against such Pythagoreanism?

It can be argued that the maths in mathematical physics only gives you the structure of the world. Yet that world still needs to be interpreted in order to be understood. This is where metaphors, analogies and models come in. It’s of course possible that the interpretations are always wrong — at least to some degree.

What’s more, the maths in quantum mechanics doesn’t interpret itself.

Even a pure or “mystical” Pythagorean can agree that metaphors are indispensable, or that, more broadly, interpretation is unavoidable. He may, or will, do so because he believes that human minds are limited; therefore we require such interpretations. In that case, then, metaphors, etc. are simply aids to human understanding. They have no ontological status or significance. A Pythagorean may even state that they don’t reveal anything about reality at all…

Surely, that can’t be right.

If all the above were the case, then metaphors, analogies, etc. would be literally pointless.

We can also turn the Pythagorean claim on its head by saying that we shouldn’t read any ontology into the maths. That’s mainly because maths is seen by many physicists to be just a tool for prediction.

All this raises two questions: (1) What are the metaphors actually metaphors of? (2) What, exactly, is being interpreted?

In the Pythagorean case, the metaphors are metaphors of mathematical reality. And it’s that mathematical reality that’s being interpreted. Yet surely this must mean that the description and the thing described are the same thing, or at least both are mathematical.

The Swedish-American physicist, machine learning researcher and author Max Tegmark can be brought in here.

Tegmark believes that if a mathematical structure is identical (or “equivalent”) to the physical structure it “models”, then they’re one and the same thing. Thus, it makes little sense to say that x “models” — or is “isomorphic” with — y because x and y are one and the same thing.

Tegmark gives an explicit example:

electric-field strength = a mathematical structure

In Tegmark’s own words:

“‘ [If] [t]his electricity-field strength here in physical space corresponds to this number in the mathematical structure for example, then our external physical reality meets the definition of being a mathematical structure — indeed, that same mathematical structure.”

If x (a mathematical structure) and y (a physical structure) are one and the same thing, then one needs to know how they can have any kind of relation to one another at all.

Pythagorean Physics: Maths Describes Maths?

At first, a layperson may see a problem with Pythagorean physics because a mathematical description D is describing mathematical reality R, and thus creating an identity. Yet, in basic terms, D doesn’t need to be identical to R. However, this is still maths about maths.

Because mathematics (or at least numbers) can be applied to anything, it can even be applied to itself, as with Gödel’s incompleteness theorems in which Gödel assigned numbers to every element of a formal system. So maths describing maths isn’t surprising or problematic in itself.

If I were in a Pythagorean frame of mind, when walking about my room I could “see” various mathematical relations, symmetries and ratios. For example, the relation of my computer to the window, the accidental symmetry between tables and desk. Etc. All this would involve angles, distances, parallelism, etc., all described in mathematical terms. I can go deeper here and discuss light reflections, the timings of typings, volumes, trajectories of movement, etc. in mathematical terms.

Similarly, if I were to randomly throw an entire pack of cards on the floor, then that mess-of-cards could still be given a mathematical description. The disordered parts of that mess would be just as amenable to mathematical description as its (accidental) symmetries.

But why bother? What can I conclude from all this?

It was just said that maths can be used to explain almost anything and almost everything. The cognitive psychologist and idealist Donald Hoffman, for example, proves this point when he uses maths to describe qualia! He sums up his approach by saying that his position

“give[s] a mathematically precise theory of conscious experiences, conscious agents, and their dynamics, and then makes empirically testable predictions”.

Yet if we follow on from what’s already been said, that mathematicisation of qualia and consciousness shouldn’t be a surprise.

Mystical Pythagoreans?

The term “mystical” is used in the following simply because that’s the term that’s been used about the ancient Pythagoreans, and even about later ones too.

Still, why use the word “mystical” at all?

When people use the word “mystical” about the ancient Pythagoreans they usually do so because they claimed to have a direct — and even privileged — access to reality as it truly is.

The true Pythagorean only thinks in terms of the maths. He may even “visualise” in purely mathematical terms. So, sure, this position is deemed to be extreme only because the mystical Pythagorean believes that he can dispense with metaphors and analogies.

It may now seem that the contemporary Pythagorean physicist is actually going all the way back to the mysticism of the ancient Pythagoreans. In other words, if we take a contemporary Pythagorean at his word, then all there is to reality is mathematical structure. Thus, isn’t it possible that there are human persons who can know or grasp reality directly without the crutches of metaphors, models, analogies, etc? So do some human Pythagoreans immediately grasp the mathematical structure and, therefore, reality?

Take the mathematical physicist Roger Penrose.

Roger Penrose as a Pythagorean

Penrose has often been classed as a “Platonist”, less so a “Pythagorean”. (Penrose can be classed as a “Platonist” toward maths itself, and a “Pythagorean” when it comes to the applications of maths to physical reality.) Take the following passage:

“[T]he entire physical world is depicted as being governed according to mathematical laws. [ ] everything in the physical universe is indeed governed in completely precise detail by mathematical principles.”

There is a forced way that the passage above can be interpreted as not actually endorsing Pythagoreanism. Perhaps the words “governed according to” can be stressed to do so. In addition, there’s no statement of identity here. Still, the passage is worth noting for its Pythagorean “flavour”.

What about this passage? -

“[A]long comes quantum mechanics, and this quantum mechanics turns out to be fundamentally based on these complex numbers. [ ] they’re very much essentially part of the fabric. The fabric couldn’t exist without them”.

This is even more strongly Pythagorean in flavour. Yet there’s still no explicit statement of identity.

More relevantly, and in terms of the metaphors and analogies used in quantum mechanics, Penrose once stated the following:

“[I] find words almost useless for mathematical thinking. Other kinds of thinking, perhaps such as philosophizing, seem to be much better suited to verbal expression. Perhaps this is why so many philosophers seem to be of the opinion that language is essential for intelligent or conscious thought!”

To be fair, Penrose is talking about mathematics itself here, not mathematical descriptions of the physical world. Despite that, it would seem possible that this attitude could pass over to mathematical descriptions of the world too. In other words, Penrose may even find words, metaphors, analogies, etc. personally useless when it comes to describing physical reality.

Humans and Chatbots as Reasoning Machines

 

“AI sceptics” make much of chatbots and other AIs being “merely pattern-finders”, “predictors of text”, etc. More relevantly, they aren’t deemed to be “genuine reasoners” because they don’t understand the data they’re dealing with. Regardless of whether all that’s true or not, the similarities between human reasoning and AI reasoning-in-inverted-commas are striking. There may be no need for AI consciousness for that to be the case. The following extreme example gets the point across when it comes to the title above. Much of the political (if not everything else) input into the minds of human ideologues is filtered, and new data is only accepted when it conforms to their ideology (i.e., confirmation bias). All this explains why human ideological machines are often so predictable (like some other machines) when it comes to what they say. [There’s more on this later.]

Image by ChatGPT

It can be shown that human persons don’t do much that’s essentially different to what chatbots do when it comes to reasoning.

So, as far as chatbots are concerned, is it all about “pattern recognition”?

In this case, Google AI Mode was forthright about its reliance on patterns. It said that “[w]hat we do is recognise and generate patterns that have been learned from large amounts of text”. In addition to Google AI Mode downplaying itself (see later), many human persons stress the fact that chatbots are “pattern-finders”, not “true thinkers”. Google AI Mode itself says that it has “learned the patterns of argument that tend to appear in those discussions”. Yet all this could equally apply to human persons. Indeed, without human persons learning such patterns, it’s hard to know what they’d rely on.

Take another example or cliché: “AI just predicts the next word based on patterns.” Yet human persons do that all the time.

Here’s the relevant thing: “Among those patterns are not just facts, but ways of arguing.” So on the theme of this essay:

Do human persons pick up ways of arguing without recognising patterns or reading books and texts?

Chatbots Reason Like Human Persons

What a chatbot does isn’t all about data mining, statistical functions, etc. As AI Mode put it: “A model like me isn’t just pulling information from a database of facts.” Sometimes a chatbot can read as little as three words and infer or deduce much from them. This is AI Mode on itself:

“With the words ‘reality is mathematics,’ it automatically triggers a network of typical consequences and challenges. That’s why it can feel like I’m reasoning independently, even though it’s grounded in patterns learned from text.”

In this case, AI Mode was “triggered” by the words “reality is mathematics” (more of which later). These three words have “consequences”. (I presume AI Mode meant “logical consequences”.) AI Mode didn’t need to check any texts or sources to work out those logical consequences. Sure, the highlighted consequences were still “grounded in patterns learned from texts”.

AI Mode said that “when you present a claim like ‘reality is mathematics,’ certain questions are almost forced by logic alone”. It deduced various logical consequences from those three words. It didn’t need to mine its own database… That’s except for the fact that its logical skills were there in its database too…

As with human persons!

AI Mode was even more explicit about its reasoning when it sad that “when I respond, I’m not recalling a specific page where someone made exactly your point”. Instead, it’s “recognising the type of move being made and following through its typical consequences”.

AI Mode also stated that the

“patterns learned during training allow [it] to generate logical follow-ups, implications, and questions without ‘looking up’ anything new”.

That seems like a perfect description of what human persons do. People learn patterns in logic books, articles, everyday conversations, etc. Indeed, in some cases, human persons are literally trained to think logically. Many logical follow-ups and implications wouldn’t even be noted if it weren’t for that training.

It’s already been said that what AI Mode does in these contexts isn’t that unlike what human persons do. It seems to recognise this itself in the following passage:

“There’s a slightly striking implication here, too. It suggests that a lot of philosophy isn’t just a collection of isolated insights, but a kind of structured space of possible positions and objections, where moving in one direction almost automatically opens certain doors and closes others.”

The words “isolated insights” above hints at human uniqueness. Yet AI Mode rejects the importance of such a thing in philosophy and, instead, stresses the “structured space of possible positions and objections”. Thus, to at least a degree, the philosopher or human person is trapped in a prior system. As a loose comparison here, think of Jacques Derrida’s idea that Western metaphysics as a whole is a system. He argued that once we bring in a single metaphysical concept into a discussion, then we’ve automatically brought in “the entire syntax and system of Western metaphysics”.

Human Ideological Machines

As AI Mode put it, “reasoning itself can be learned, patterned, and applied, not just what facts are stored”. Here AI Mode was talking about itself and other chatbots, yet it could just as easily have been talking about human persons.

More can be made of this.

Take the cases of what happens during education and socialisation. Can’t much of this be called data consumption?

But let’s up the ante a little.

Much of the input into human ideologues is filtered. The data that’s accepted is so because it conforms to the ideology (i.e., confirmation bias).

Once the input or data is within the ideological machine, much of its logic will be deductive. The ideological automaton isn’t really on the look out for “new truths”. Instead, it has a set of ideological axioms which reality must abide by.

All this explains why what ideological machines say is often predictable.

We can even call human ideological machines “stochastic parrots”. Of course, the arguments from an ideological machine against him/her not being a parrot may themselves come from his/her prior ideology.

Despite all that, there is a big difference between an AI and a human ideological machine. The brains of the latter involve degrees of plasticity. Even an ideological machine can undergo a “system wipe” or even be retrained as a ideological machine with another ideology. This freedom doesn’t apply to AI or chatbots. They can’t change their own weights.

Google AI Mode’s False Modesty

AI Mode was careful to stress that “systems like me don’t ‘reason freely’ in the sense a human might imagine”. Yet it added the words “nor do we simply retrieve stored arguments” too.

Many readers will have noted that chatbots are often on the lookout for anthropomorphism from their users. In this case, AI Mode questioned my words “reason freely”. Yet this is too modest. We needn’t believe that chatbots are persons, let alone human persons, to accept that they can reason freely. Or, at the very least, what chatbots do isn’t that different to what human persons do. The fact that an AI isn’t experiencing qualia, etc. while reasoning makes no difference to this.

Here’s another example of what can be called AI Mode’s (trained or programmed) false modesty:

“What’s interesting, and slightly surprising, is that this can look like genuine philosophical development. It can extend a position beyond what its original author explicitly said, simply because it has learned how such positions are usually pressed and developed.”

The false modesty component of the above is captured by the words “this can look like genuine philosophical development”. In this case, AI Mode extended Max Tegmark’s position (on mathematical Pythagoreanism in physics) because the latter hasn’t really supplied that much argument or detail himself. What AI Mode did, then, is access other sources that do offer more argument.

It’s worth going into detail on the Max Tegmark case.

I asked AI Mode if there a danger of it doing Tegmark’s thinking for him. I asked that question because I’ve never seen much, or even any, of this detail in his own work. AI Mode replied: “Yes — there is a real danger of that, and you’re right to be alert to it.” And then it explained the situation:

“What’s happening here is something like this. Max Tegmark puts forward a very bold, simple thesis: reality is mathematical structure. He sketches motivations for it, but he doesn’t work through all the consequences in detail.”

AI Mode even admitted (or simply said) that “there’s a risk of inadvertently making his view look more rigorous, more defended, more internally worked out than it actually is in his own writing”.

On AI Mode’s reasoning and its sources, it said:

“I don’t have ‘sources’ in the sense of consulting specific books or pulling from a hidden database of arguments when I go beyond Max Tegmark. What’s happening is a combination of trained knowledge and on-the-fly reasoning.”

AI Mode called what it was doing “charitable reconstruction”.

AI Mode did use words that most human persons would never used about themselves. For example, take this sentence: “The claim activates a cluster of related ideas it has often appeared with.” The technical word here is “activates”. Yet when human persons tackle any claim, they too activate a cluster of related ideas. Indeed, if that weren’t the case, then how could human persons even deal with any claim?

Quantum Pythagoreanism Doesn’t Help Spiritual Idealism

 

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This essay is about a writer who goes by the name Science of Illumination. He argues (or hints) that because mathematics is used to describe just about everything in quantum mechanics, then what it describes must be “mental” or “spiritual”. Yet a field equation, for example, certainly isn’t “divine breath”. This is yet another example of the many spiritual idealists and anti-materialists who quote the big names and ideas of quantum physics exclusively to advance their spiritual idealism. If you read Science of Illumination’s essays, for example, never once does he quote a physicist or a scientific idea outside the context of “spirituality” and religion. Never once.

Science of Illumination wrote the following in response to an essay I wrote called ‘There Are No Materialists’:

“According to modern science, all matter ultimately reduces to a set of mathematical equations which defy all materialistic interpretation. In the words of Nobel laureate Werner Heisenberg, ‘the electron is not a thing’. Renowned physicist Sir James Jeans explains that ‘All the pictures which science now draws of nature are mathematical pictures [ ].’ Thus any ontology that could reasonably called ‘materialist’ or ‘physicalist’ would appear to an idealist to be based on an error of misplaced concreteness.”

All this was written by a someone who holds a strong self-described “spiritual” position against both materialism and physicalism. In Science of Illumination’s Medium biography, he describes himself as a

“[p]hysicist (Ph.D.), Roman Catholic Eucharistic Minister, initiate of the Inayati, Jerrahi, and Rafai Sufi Orders, disciple of Mata Amritanandamayi”.

Science of Illumination says that as far as idealists are concerned, the distinction between materialism and physicalism doesn’t matter… He’s absolutely right! After all, neither physicalism nor materialism are idealism. Yet I fail to see how a basically Pythagorean position (see later) helps him.

When Science of Illumination hints that it’s all equations, how does that help prove the existence of God, or help show Sufi “Oneness”? Contemporary neo-Pythagoreanism in physics is, after all, perfectly compatible with a “soulless” or purely structural physicalism.

Like Science of Illumination, many anti-materialists and conscious-first spiritualists stress the non-physicality of fields, etc. Yet a materialist can go back to Michael Faraday and accept that physicality doesn’t always mean that something can be touched in the everyday sense, or that particles are tiny “hard balls”. Yet a field is still a physical state of space.

Science of Illumination, again like many idealists and anti-materialists, is very keen to quote well-known physicists to advance his own spiritual position. He quotes Werner Heisenberg, Sir James Jeans above, and, elsewhere, Max Planck, Albert Einstein, Arthur Eddington, Bohm, etc… So Erwin Schrödinger may well have loved the Upanishads, yet he explicitly warned against “silly” attempts to find God in the wave function.

I can even help Science of Illumination by going back to Isaac Newton. He was literally an alchemist and occultist. However, Newton’s Principia contains precisely zero alchemy. And Newton didn’t call gravity “The Science of Divine Attraction” or anything like that.

To move to Faraday again. He was a devout Christian Sandemanian who never claimed that magnetic induction was “oneness” in action.

Science of Illumination expresses his spiritual, idealist and, basically, political case against physicalism and materialism in the following way:

“This superficial focus on fragmentation, while overlooking the underlying oneness, has produced great technological advances, but as a worldview it is ignorance, avidiya, toxic to the individual and to humanity.”

The “oneness” obviously refers to Science of Illumination’s reading of eastern religious texts (as with avidiya). The political angle is shown by his reference to “ignorance”, as well as by his view that materialism is toxic to the individual and to humanity. However, I’m not going to spend any time on the rhetoric and politics here: I’ll stick to the science and philosophy.

Neo-Pythagoreanism

Stating the vital-importance of mathematics doesn’t work for those with “spiritual”, religious and/or idealist leanings. (Historically, the original Pythagoreans were religious in many ways.) What is it about “all is number” that’s spiritual or idealist? Sure, the all-is-maths idea can be cited as a cudgel against physicalism and materialism. Yet it still doesn’t advance consciousness-first philosophy or any “spiritual” stuff.

Even if we move against the old Pythagorean position in which “things are numbers” and move to modern structural Pythagoreanism, this position still offers nothing to the philosophical spiritualist.

To the neo-Pythagorean, the universe is “made of” mathematical relationships, something stressed by Ontic Structural Realists. Those relationships can indeed be described as a “particle” or a “probability wave”. (That said, Ontic Structural Realists argue that “every thing must go”.) Yet you can’t squeeze the juice of spirituality or idealism out of any of this.

Science of Illumination rams his point home even more in the following:

“The quote from Heisenberg is a reference to certain fundamental aspects of quantum physics, which I assumed materialists and physicalists would be quite familiar with, since they both claim their understanding of the nature of matter is based on science, and quantum physics is the science of the nature of matter.”

This was a reply to my own response to Science of Illumination.

The irony in the words “I assumed materialists and physicalists would be quite familiar with” is clear. Does he really believe that any professional physicalist philosophers aren’t familiar with all this? Even many laypersons are familiar with it!

Against Neo-Pythagoreanism

In physics, it’s indeed the case that everything “said” about particles, forces, fields, spacetime, etc. is said with mathematics…

What else could be used to describe these micro-entities and extremely intangible phenomena? Apart from metaphors such as “fields”, “particles”, etc., there’s simply nothing else that can be used. Yet that doesn’t mean that there’s no territory to map or that we must become Pythagoreans.

There’s nothing fundamentally wrong with words like “fields”, “particles”, etc. Or, more accurately, they do fail in some respects, yet work very well in other respects. No one believes that a scientific field has grass or flowers growing within it. Yet the word “field” still works. No one believes that a particle is a ultra-small ball or even a “hard thing”. Yet the word “particle” has proven to be very useful.

Even the “unreasonable effectiveness of mathematics” doesn’t mean — or have the consequence — that the world itself is mathematical. More strongly, the effectiveness of mathematics in physics doesn’t have the (logical) consequence that the world itself must be mathematical… That’s unless being mathematical simply means that the world is describable by mathematics. It may even mean that most aspects of the world described by physicists can only be described by mathematics.

To neo-Pythagoreans, the world and its parts are actually mathematical — not instantiations of universal consciousness, and not the contents of mind. This means that it isn’t that maths is simply helpful for describing the world — the world itself is mathematical. Again, this has nothing to do with consciousness or anything spiritual. Here’s John Barrow on the Pythagorean position:

“[The Pythagoreans] maintained ‘that things themselves are numbers’ and these numbers were the most basic constituents of reality.”

The physicist and cosmologist Max Tegmark puts the contemporary case for neo-Pythagorean being in the following very concrete example:

“[If] [t]his electricity-field strength here in physical space corresponds to this number in the mathematical structure for example, then our external physical reality meets the definition of being a mathematical structure — indeed, that same mathematical structure.”

To spell out the above.

Tegmark isn’t saying that maths is perfect for describing the “electricity-field strength” in a particular “physical space”. He’s saying that the electricity-field strength is a mathematical structure. In other words, the maths we use to describe the electricity field is one and the same thing as the electricity field. Thus, if that’s the case, the “miracle of mathematics” is hardly a surprise! That’s because this miracle is essentially a situation in which maths is describing maths. (If maths is describing maths, then the word “describing” is surely not an apt word to use in the first place.)

Tegmark gives us more detail on his position when he tells us that

“there’s a bunch of numbers at each point in spacetime is quite deep, and I think it’s telling us something not merely about our description of reality, but about reality itself”.

So perhaps there’s a difference between saying that “things themselves are numbers” (as the ancient Pythagoreans did), and saying that the world is mathematical. (I may be drowning in a sea of grammar here.) The latter may simply state that the world exhibits features which are best expressed (or described) by mathematics. The former, on the other hand, says that the world literally is mathematics.

Again, none of this can be used to advance spiritual idealism.

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Note:

(1) The following paragraph is taken from the essay above:

“In physics, it’s indeed the case that everything ‘said’ about particles, forces, fields, spacetime, etc. is said with mathematics… What else could be used to describe these micro-entities and extremely intangible phenomena? Apart from metaphors such as “fields”, “particles”, etc., there’s simply nothing else that can be used. Yet that doesn’t mean that there’s no territory to map or that we must become Pythagoreans.”

This idea will be extended in a future essay.