i) Introduction ii) Maths and Reality Disjoined iii) Does Maths Correspond With Reality? iv) Many Theories About x v) Philosophy of Mathematical String theory
In his book The Trouble With Physics Lee Smolin puts the case against string theory when he says that
“we must conclude that string theory has nothing to do with nature, because every single one of [the theories that are known to exist] disagrees with experimental data”.
It must also be said that elsewhere Smolin concedes that string theory does indeed have its own predictions. However, “[t]he few clean predictions it does make have already been made by other well-accepted theories”. Not only that. When string theorists say that there are “extra dimensions” and that “all forces are united into one force, and that there is super-symmetry”, Smolin also states that all these “are independent of string theory”. Thus “finding evidence for any one of them does not prove that string theory is true”. Nonetheless,
“[i]f we learn that there is no super-symmetry or no higher dimensions or no unification of all the forces, then string theory is false”.
All these factors — one must suppose — are within the domain of the evidential, experimental and predictive. Thus they run free of the unique claims of string theory. In other words, when string theory offers us something new or original, it’s that which disagrees with experimental data.
More specifically and on the subject of this piece, Smolin explicitly comments on the maths/reality connection when discussing Albert Einstein and the German physicist, Hermann Weyl. He quotes Einstein’s response to one of Weyl’s theories:
“‘Apart from the [lack of] agreement with reality, it is in any case a superb intellectual performance.’”
This clearly shows that there can often be (at least in principle) a complete disjunction between mathematics and reality. This isn’t a surprise because that link has often been either underplayed or completely rejected (by mathematicians and philosophers) throughout history. The quote above also stresses the mathematical virtuosity-for-its-own-sake problem which has particularly beset string theory.
Smolin again puts the case for this maths/reality disjunction (this time with direct reference to string theory) in the following:
“The feeling was that there could be only one consistent theory that unified all of physics, and since string theory appeared to do that, it had to be right. That was the stuff of Galileo. Mathematics now sufficed to explore the laws of nature. We had entered the period of postmodern physics.”
Prima facie, it can also be inferred that not only can there be a mathematics/reality disjunction — there can also be a unified theory/reality disjunction too. In other words, unification — or at least the intellectual desire for unification — might sometimes have run completely free of reality. Particularly this may still be the case if reality is messy, random, chaotic, non-symmetrical, ultra-complex, dynamical, etc. In other words, wouldn’t a unified theory be difficult — even impossible — if the world were truly messy?
Even before string theory, many writers have commented on the physicists’ “rage for order”. This is Kitty Ferguson on this theme:
“The diamond shapes in a sunflower seed-head were lop-sided. One had to give tree-trunks the benefit of the doubt in most cases to call them cylinders. The earth bulges and is not a perfect sphere. As for mirror symmetry, one side of the human face is not the true mirror image of the other.”
She goes on to say:
“The things we build and the art we create exhibit much more geometry and symmetry than we can find in nature. Are we bettering nature, imposing rationality on a less rational universe?”
Of course none of Ferguson’s examples above are from the world of physics; though physics is of course indirectly connected to every example she cites. It can also be said, prima facie, that symmetry and order are perhaps more likely in the world of physics than in the world of “medium-sized dry goods” (to use J.L. Austin’s phrase). Symmetry (even “supersymmetry”), particularly, is a very important aspect of string theory.
However, physics doesn’t fair very well either. As Smolin himself points out in the following:
“[W]e had learned that nature lacked a certain symmetry — that of parity between left and right. Specifically, all neutrinos are what is referred to as left-handed (that is, the direction of their spin is always opposite to that of their linear momentum). This means that if you look at the world in a mirror [as mentioned by Ferguson above] , you will see a false world — one in which neutrinos are right-handed. So the world seen in a mirror is not a possible world.”
Thus can’t that “bettering [of] nature” fixation also be applied to string theory and even other theories in physics? Is string theory attempting to “exhibit more geometry and symmetry than we can find in nature”? Is string theory also “imposing rationality on a less rational universe”?
This thirst for order appeals to mathematicians. It also appeals to mathematical physicists. And it especially appeals to string theorists.
It’s also the case that string theory satisfies (or at least attempts to) the mathematical side of mathematical physics. Some experts think it does so with flying colours. Indeed string theory’s mathematical virtuosity and dexterity is what appeals to many people — especially those experts who have less knowledge of physics than they have of mathematics.
So it can be concluded that maths can run free of physical reality. Or at least there’s been a long debate about this. On the one hand, philosophers and mathematicians — dating back to the ancient Greeks — have stressed the tight link between maths and physical reality. Other philosophers and mathematicians, on the other hand, have denied the importance — or even relevance — of that link.
Mathematics and Reality Disjoined
It’s easy to see how the maths of string theory can run free of reality. However, how would non-experts know if that were the case? It may also be the case that string theorists have no sure (i.e., philosophical or non-mathematical) mechanisms which can show them that they’re disappearing up their own mathematical backsides. As Smolin puts it, the maths of a particular string theorist may simply be his personal “toy”. He tells us that such a person (though in this instance he’s not talking about a string theorist) “may simply have invented a theoretical toy that has nothing to do with nature”. The string theorist will of course say that his theory does indeed have something “to do with nature”. How would we know that this is the case? The string theorist will also say that his mathematical theories describe reality or — at the very least — the way that reality may be. Again, how would we know? We’d need to be in Sherlock Holmes’s situation in which we literally have to go through the entirety of the string theorist’s mathematics and thus replicate his own labour and results. To re-quote William Poundstone’s passage (already quoted in Part One of this piece):
“William Shanks, a mathematician of our fair island, has recently computed pi to 707 decimal places. It took him twenty years. His result filled a whole page with quite senseless, random numbers. Should anyone doubt Mr. Shank’s result, he would have to budget an equal amount of time and duplicate his work. In that case also, verifying the answer would be precisely as difficult as coming up with the answer in the first place — the very antithesis of an ‘obvious solution.’….”
In terms of the dearth of experiments, observations and predictions in string theory, that’s presumably why some string theories leave us in a position in which “there is no evidence either way”. And if there’s no evidence either way, then, again, where does that leave us? Of course a string theorist may hint at evidence. He may even suggest actual or possible experiments. That is, he may say that in such-and-such a situation one would observe this-and-that.
Smolin doesn’t seem to have a problem with this kind of talk about hypotheticals/conditionals (at least when talking about “doubly special relativity”). When referring to such a theory, Smolin writes:
“The idea is an elegant one…. We don’t know whether it describes nature, but we know enough about it to know that it could do.”
Could we say the same about string theory — or about a particular statement within string theory?
Thus let’s take it as given that a particular string theory is “elegant”. In that case, we still wouldn’t know that “it describes nature”. Would we know that “it could do”? We can conclude by saying that there’s one thing which is the case: mathematical elegance alone will never be enough.
Does Maths Correspond With Reality?
Smolin recalls a conversation he had with João Magueijo (the Portuguese cosmologist and theoretical physicist) in which he seems to say that there should never be an absolute disjunction between maths and reality. Smolin writes:
“João said that everything to do with physics could be understood without the fancy mathematics. Giovanni argued that it was easy to talk nonsense about these theories if you weren’t careful about which mathematical expressions corresponded to things that could be measured.”
This depends on what that understanding of physics amounts to. It can be argued that there is a kind of understanding without the maths; though not a full understanding. (This is parallel to the use of images, metaphors and analogies in quantum mechanics.)
Nonetheless, the link between “mathematical expressions” and the demand that they “correspond to things that could be measured” seems to be very strong. So is that meant in the same sense that each word in a natural-language sentence must correspond with a thing? In terms of language, this is obviously problematic from a philosophical point of view. Some words don’t correspond with things. Indeed some words don’t even correspond with anything — not even with abstract entities. In addition, a word gets it meaning (or role) from its context within a sentence (or statement). Perhaps this is also true of the mathematical expressions in a mathematical theory (or statement). Again (as with a sentence), why should all mathematical expressions correspond to anything at all? The mathematical expressions in pure maths don’t need to correspond with anything (which is a point that Wittgenstein, the constructivists and formalists often stressed). However, surely the same can’t be be said of the maths in mathematical physics or string theory.
The other point (in reference to the quote above about “fancy mathematics”) is that even if mathematical expression x corresponds to thing y, and then that thing y is “measured”, then isn’t measuring itself a mathematical process or function?
The astrophysicist and science writer, John Gribbin, also seems to put a similar point. He does so, however, in terms of what he calls a “physical model” of “mathematical concepts”. He says (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, it’s still clear that a “mathematical concept” comes first and only then is a “physical model” found to square with it.
Many Theories about x
One problem with the possible divorce between maths and reality is that many theories — perhaps an infinite amount of them — can be constructed precisely because of that freedom. More specifically, many theories can be formulated to deal with the very same evidence, data or experiment. This is certainly true of string theory (or string theories in the plural).
There are many problem with this multiplicity of theories and their possible parallel lack of a connection to reality. Smolin tells us (though he's talking specifically about “unified-field theories”) that “[r]ather than being hard to find, unified-field theories were a dime a dozen”. What’s more, “[t]here were many different ways to achieve them and no reason to choose one over another”.
Nonetheless, there may not be a huge problem with many theories fighting for the same space. It may even be the case that all these rival theories are in fact “complementary” (or at least some may be). Why assume that the different theories about the same x must mean that not all these theories can be useful? Indeed why assume that only one can be true or correct? Moreover, why should a multiplicity of theories about a given x hint at a break from reality? Why not be a pluralist about such theories?
John Gribbin (again) is such a pluralist (as are many scientists and philosophers). Gribbin, in the following, specifically talks about quantum theories:
“I stress, again, that all such interpretations are myths, crutches to help us imagine what is going on at the quantum level and to make testable predictions. They are not, any of them, uniquely ‘the truth’; rather, they are all ‘real’, even where they disagree with one another.”
This is heavy stuff. Of course what’s true of quantum interpretations (or “myths”) may not be true when it comes to mathematical theories or theories in other disciplines. However, the above is certainly problematic for string theory for the simple reason that Gribbin talks about such theories making “testable predictions”.
The Philosophy of Mathematical String Theory
One can of course say that the maths of string theory must have a connection to reality for simple Aristotelian reasons. As Aristotle himself put it (in his Metaphysics) when talking about the Pythagoreans:
“All is number. The principles of mathematics are the principles of all things.”
In other other words, if the principles, theories or statements of mathematical string theory are true/correct/accurate; then, by Aristotelian definition, they must match “all things” — they must match reality.
There’s a distinction, however, to be made here.
It can be said — in loyalty to Aristotle — that the maths in string theory can’t contradict anything in reality and reality can’t contradict anything in mathematical string theory. But does that also mean that the maths of string theory must also describe or explain given aspects of reality — or any part of reality? Nothing in string theory may contradict the “principles of all things”; yet a given string theory may not have anything to do with any aspect of reality.
In that sense, what’s true of maths generally is passed over to mathematical string theory. That is, if the maths in string theory is true/correct/accurate, then nothing in it can contradict reality. Still, string theory may not describe (or explain) any aspect of reality. More specifically, it may not describe such things as extra dimensions, branes, the multiverse and even strings because such things simply don’t exist in reality.
We can also bring in Galileo here.
Galileo famously claimed that “the book of nature is written in the language of mathematics”. Thus, again, if the maths of string theory is true/correct/accurate; then — by definition — it can’t contradict anything in nature or reality. Still, it may not describe (or explain) anything in nature or reality either.
We can go further here and say that a “false dichotomy” has been set up with this talk of maths or reality. It may even be the case that maths and reality is a false conjunction. What I mean by all this is that many people say that “physics is maths”. (Though not that “maths is physics”.) Therefore can’t we also say that reality/nature is maths (along with with Galileo)? And if the maths in string theory is true/correct/accurate, then it simply must be about reality. Though, again, this distinction must be made here:
1) String theory does not contradict reality. 2) Does string theory actually describe or explain reality?
In criticising what can be seen as string theory’s over-dependence and over-reliance on maths, we also face another problem. Physics is nearly always utterly dependent on mathematics. The science writer John Horgan (in his The End Of Science) puts it this way:
“Numerical models work better in some cases than in others. They work particularly well in astronomy and particle physics, because the relevant objects and forces conform to their mathematical definitions so precisely.”
Horgan goes further than that. He says that many of the entities, fields and forces of physics are entirely mathematical in nature. Firstly he tells us that “mathematics helps physicists define what is otherwise undefinable”. Then he cites an example:
“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.”
This isn’t necessarily to say that quarks are nothing other than maths or the numbers which express their nature. (Otherwise why use the words “their nature”?) It’s to say that we couldn’t say much — or indeed anything — about quarks without the requisite mathematics. This may also mean that mathematical physics not only describes nature: it also — in some sense — creates it. This may also mean that the entities, strings, dimensions, etc. which string theorists postulate may make much more sense than we think. In other words, if we’ve no grasp of reality that’s separable from the maths of mathematical physics, then how can we even so much as question what string theorists are saying? As with quantum mechanics, we may have no independent ground to stand on.
Thus surely we can follow on from all that and question if there really is something beyond the mathematical formulations which express the nature of strings and quarks. In other words, what’s left after we take the maths away? Nothing or just a little something “we know not what”? What can be said without the maths? And what if what’s said without the maths is highly misleading in that’s it’s purely analogical, metaphorical or imagistic in nature? Indeed perhaps we may be better off — in both string theory and quantum mechanics — without the analogies, images and picture-painting.