First things first.
This essay may appear to advance two mutually-contradictory positions. On the one hand, it argues that without the mathematics (or, more correctly, without the mathematical formalism/s), there would be no quantum world — or at least no quantum mechanics. Yet, on the other hand, this essay also argues against Pythagoreanism — at least as it applies to this specific issue.
The main anti-Pythagorean argument in the following is that the world (or Nature) isn’t literally mathematics (whatever that may mean) or “made up” of numbers. It’s simply that, in quantum mechanics at the very least, without the mathematics, we’d have (almost) nothing.
Pythagoreanism: Things are Numbers
Pythagoreans believe that the world literally is mathematical. Or, perhaps more accurately, they believe that the world literally is (without the suffix “cal”) mathematics. I make this either/or distinction because if a physicist argues that “the world is mathematical”, then that may only mean that the world can be accurately — even very accurately — described by mathematics. The Pythagorean, however, states that “things are numbers”. Such a person therefore establishes a literal identity between the maths and the world (or parts thereof).
Yet we don’t need to accept the latter.
Having said all the above, it may well be the case that this essay does indeed advance a Pythagorean position — at least when it comes to quantum mechanics. That’s because it’s sometimes hard to tell what the Pythagorean position actually is. For one, it’s hard to make sense of the locution that the world is mathematical or that it’s made of numbers.
So the question which must now be asked is this:
What is it for “things” to be “numbers”?
To repeat: the Pythagorean position isn’t to only to argue that mathematics can describe (or model) things — it’s to argue that things literally are numbers. But what does that actually mean? And, as a consequence of that, it can now be asked if the statement “All things are numbers” is to be taken poetically or literally. Taken literally, it hardly makes sense. Taken poetically, it still requires much interpretation.
One interpretation of the Pythagorean position is that if things are numbers, then it’s no surprise that — for example — string theory is on top of things when it comes to describing reality. What I mean by that is this:
i) If things are numbers,
ii) and numbers are also used to describe (or model) things (which are numbers),
iii) then numbers are describing (or modelling) numbers.
That would mean that we never escape from numbers. Who knows, perhaps that’s precisely the result which Pythagoreans want!
To change tack a little.
The physicist John Archibald Wheeler (1911 — 2008) provided the best riposte to Pythagoreanism in physics. (I’m not entirely sure if this was his intention.)
It’s often been said that Wheeler used to write many arcane equations on the blackboard and stand back and say to his students:
“Now I’ll clap my hands and a universe will spring into existence.”
According to Pythagoreans, however, the equations are the universe.
And, after that comment, Steven Hawking (1942–2018) trumped Wheeler with an even better-known quote. He wrote:
“Even if there is only one possible unified theory, it is just a set of rules and equations. What is it that breathes fire into the equations and makes a universe for them to describe?”
The American science writer Kitty Ferguson (1941-) offered a (possible) Pythagorean answer to Hawking’s question. She suggested the possibility that “it might be that the equations are the fire”. Alternatively, could Hawking himself have been “suggesting that the laws have a life or creative force of their own?”. Again, is it that the equations are the fire?
So what, exactly, does “breathe[] fire into the equation [to] make a world”?
John D. Barrow
Mathematics is an extremely useful tool. The English cosmologist, theoretical physicist and mathematician John D. Barrow (1952–2020), however, went one step beyond that truism. Barrow actually put his — arguably — Pythagorean position in the following way:
“By translating the actual into the numerical we have found the secret to the structure and workings of the Universe.”
Of course almost everyone can happily accept the Universe and its parts are assigned numbers… Or are described by numbers… Or are captured by numbers… Or are explained by numbers… Or are (to use Barrow’s own words) translated into numbers. The thing is, that’s not actually a Pythagorean position.
So, as a consequence of all the words above, it’s no wonder that so many people have believed that through maths (as Barrow puts it) “we have found the secret to the structure and workings of the Universe”. Yet even here there must be a non-Pythagorean (as it were) remainder. What I mean by this is that maths finds the secret of things which already and separately exist — in this case, the “structure and workings” of the world. Surely it doesn’t also need to be argued that these structures and workings are literally mathematics (or literally numbers).
… Or perhaps it does.
To repeat: to the Pythagorean, the world and its parts are actually mathematical. This means that it isn’t that maths is simply helpful for describing the world — the world itself is mathematical. Indeed one must take this literally. Here’s Barrow again on the Pythagorean position:
“[The Pythagoreans] maintained ‘that things themselves are numbers’ and these numbers were the most basic constituents of reality.”
Barrow then became ever clearer when he continued in the following manner:
“What is peculiar about this view is that it regards numbers as being an immanent property of things; that is, number are ‘in’ things and cannot be separated or distinguished from them in any way.”
Moreover:
“It is not that objects merely posses certain properties which can be described by mathematical formulae. Everything, from the Universe as a whole, to each and every one of its parts, was number through and through.”
As stated earlier, it’s hard to grasp what the sentence “things themselves are numbers” even means. Can we really argue that reality and its parts are mathematics (as in the “is of identity”)? Can we really argue that reality and its parts are literally made up of numbers or equations? And can we even argue that reality and its parts somehow instantiate maths, numbers or equations?
Max Tegmark
The physicist and cosmologist Max Tegmark (1967-) also puts the contemporary case for Pythagoreanism 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 passage above.
Max Tegmark isn’t simply arguing that mathematics is perfect for describing the “electricity-field strength” in a particular “physical space”. He’s arguing that the electricity-field strength is a “mathematical structure”. That is, the mathematics we use to describe the electricity field is one and the same thing as the electricity field. Thus, if that’s really the case, then the so-called “miracle of mathematics” is hardly a surprise! And that’s because — as already stated above — we essentially have a situation in which maths is describing maths. And if maths is describing maths, then the word “describing” is surely not the right 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”.
It can be argued that Tegmark contradicts himself in the above.
At one point Tegmark argues that a field “is just [ ] something represented by numbers at each point in spacetime”. Note here that we have the two words “something [my italics] represented”. Yet elsewhere Tegmark also argues that the field “is just” (or just is) a mathematical structure — the latter two words implying that all we have is number. To repeat: Tegmark argues that the field is “represented” by “three numbers at each point in spacetime”. Yet he doesn’t (in this passage at least) also say that the field is a set of numbers (or even a “structure” which includes numbers).
So perhaps there’s a difference between arguing that (as the original Pythagoreans did) “things themselves are numbers” and arguing 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, states that the world literally is mathematics.
Now take the case of string theory.
Michio Kaku and String Theory
Not only is string theory seemingly more dependent on mathematics than all the other areas of physics (though, of course, that can be debated), it seems that some physicists even see string theory as being a “branch of pure mathematics”.
The string theorist Michio Kaku (1947-), for example, doesn’t hide from this when he quotes a “Harvard physicist” saying as much. In Kaku’s own words:
“One Harvard physicist has sneered that string theory is not really a branch of physics at all, but actually a branch of pure mathematics, or philosophy, if not religion.”
After Kaku puts the Pythagorean position, he then quotes Albert Einstein (i.e., as backup) stating the following:
“‘I am convinced that we can discover by means of purely mathematical construction the concepts and the laws… which furnish the key to the understanding of natural phenomena.’”
Einstein went deeper when he added these words:
“‘Experience may suggest the appropriate mathematical concepts, but they most certainly cannot be deduced from it.’”
Now all the words above do indeed sound Pythagorean (or, more broadly, Rationalist) — at least on the surface. And Einstein seems to more or less come clean about this in his final sentence. Thus:
“‘In a certain sense, therefore, I hold it true that pure thought can grasp reality, as the ancients dreamed.’”
The strange thing is that Kaku also seems to offer us a mutually-contradictory account (in his book Beyond Einstein) of Einstein’s position on mathematical physics. Kaku writes the following words:
“Einstein revealed a clue to the way he arrived at his great discoveries: he thought in physical pictures. The mathematics, no matter how abstract or complex, always came later, mainly as a tool by which to translate these physical pictures into a precise language.”
Indeed elsewhere in the same book Kaku also writes:
“[] Feynman, and other great scientists []thinks in terms of pictures that express the essential physical concept. The math comes later.”
If we return to Kaku’s own position.
As for the charge (if it is an charge) of Pythagoreanism against Kaku, don’t take my word for it: take the words of the man himself. Firstly Kaku lays out the essential Pythagorean position in this way:
“Not surprisingly, the Pythagoreans’ motto was ‘All things are numbers.’ Originally, they were so pleased with this result that they dared to apply these laws of harmony to the entire universe.”
Then Kaku continues by arguing that “with string theory” what we have is “physicists [] going back to the Pythagorean dream”.
Quantum Mechanics
Philip Ball
The science writer Philip Ball (1962-) argues (in his book Beyond Weird: Why Everything You Thought You Knew about Quantum Physics Is Different) that the mathematics of quantum mechanics “doesn’t say anything about the ‘real world’”. Many physicists — and some philosophers — have also echoed that sentiment. Yet that may appear to be an odd position. It’s odd because if the mathematics of quantum mechanics is extraordinarily successful when it comes to predictions, applications, engineering, technology and whatnot, then (perhaps almost by definition) surely it simply must be about the real world…
Yet what work is the word “real” actually doing here? Does it imply that we (or the maths) must mirror the world? But how does — or would — that work? And even if the maths perfectly describes physical phenomena in terms of their magnitudes, values, strengths/charges, velocities, spatial dimensions/positions, etc., then is all that actually a case of mirroring the world itself? Surely if the maths of quantum mechanics mirrored the world, then it would look — and even be — the same as that world. In that case, what purpose would such mirroring actually serve? (Think here of the often-made claim: The best model of x is x itself.)
So it can be argued that maths can’t — literally — mirror the world.
Despite saying that, one thing is still certainly the case.
As stated in the introduction, without the maths, we’d have almost (or even literally) nothing to say about the quantum world — real or otherwise. When it comes to the quantum world, the usual (as it were) means of ownership aren’t available to us. That is, we can’t observe, feel, smell or (often) even imagine the quantum world. Thus the maths is all we’ve got.
All this is excellently expressed in the following passage from the science writer John Horgan (1953-):
“[M]athematics helps physicists definite what is otherwise undefinable. A quark is a purely mathematical construct. It has no meaning apart from its mathematical definition. The properties of quarks — charm, colour, strangeness — are mathematical properties that have no analogue in the macroscopic world we inhabit.”
Thus if maths is all we’ve got, then it’s not really a surprise that many physicists (i.e., the more philosophical ones) argue that quantum mechanics doesn’t really say anything about the real world. (This has been said since Niels Bohr in the 1920s.) Or, at the very least, everything important — or even relevant — that’s said about the quantum world is said by the maths.
So when Philip Ball also writes that Richard Feynman could only do “quantum theory” (i.e., the maths), then that’s not a surprise. That’s because it can be argued that the maths is all we’ve got and all Feynman had. Indeed when we stray beyond the maths into interpretation, then we (perhaps by definition) can’t help but get things wrong.
Or at least that’s one (sceptical) scenario we must consider.
Again, it’s not a surprise that — even — Feynman didn’t “know what the maths means”. That may be because the words what the maths means are — almost — meaningless. At the very least, there’s a hint here that we can’t go beyond the maths. Yet it’s still the case that so many philosophers, and a somewhat lesser number of physicists, believe that the maths is only second best to something far… deeper.
John Gribbin
The British science writer and astrophysicist John Gribbin (1946-) appears to agree with these conclusions. That is, as a consequence of much of what’s been said above, it can be concluded that all the imagery, picture painting, metaphors, analogies, etc. we find in the popular accounts — and even the technical interpretations — of the quantum world are simply (to use Gribbin’s own words)
“crutches to help us imagine what is going on at the quantum level and to make testable predictions”.
Indeed Gribbin also believes that
“none of [the quantum mechanical interpretations] is anything other than a conceptual model designed to help our understanding of quantum phenomena”.
Indeed Gribbin also talks about the interpretations of quantum mechanics:
“I stress, again, that all such interpretations are myths, They are not, any of them, uniquely ‘the truth’; rather, they are all ‘real’, even where they disagree with one another.”
Many will read Gribbin’s words as being very radical — and even deflationary — when it comes to quantum mechanics. Yet, despite all the above, Gribbin also happily acknowledges that all these quantum interpreters genuinely believe that their very own interpretations are true. He writes:
“[T]he interpreters and their followers will each tell you that their own favoured interpretation is the one true faith, and all those who follow other faiths are heretics.”
And that passage comes straight after Gribbin had told us that
“[a]t the level of equations, none of these interpretations is better than any other”.
Thus, logically, “none of the interpretations is worse than any of the others, mathematically speaking”. That said, all this hinges on precisely how we’re supposed to take the phrases “at the level of equations” and “mathematically speaking”.
Gribbin also becomes very psychological (or aesthetic) when he concludes (as the very end of one of his books) that we are
“free to choose whichever one gives you most comfort, and ignore the rest”.
Again, there may well be an argument that all the interpretations of quantum mechanics are superfluous when it comes to predictions, tests, experiments, technology, etc. However, that certainly doesn’t mean that all these interpretations are “equally good”. They may all be equally bad in the sense that they don’t make the slightest bit of difference when it comes to to mathematical theory, predictions, (quantum) technology, etc. However, are they all equally good in literally every other respect?
Despite all Gribbin’s words above, he still stresses the importance of what he calls a “physical model” of “mathematical concepts”. He writes (in his Schrodinger’s Kittens and the Search for Reality) that “a strong operational axiom” tells us that
“literally every version of mathematical concepts has a physical model somewhere, and the clever physicist should be advised to deliberately and routinely seek out, as part of his activity, physical models of already discovered mathematical structures”.
Yet even in Gribbin’s case and on a final quasi-Pythagorean reading, it’s still clear that a “mathematical concept” comes first and only then is a physical model found to square with it.
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