[Most of the quotes in this piece are taken from Lee Smolin's Time Reborn: From the Crisis in Physics to the Future of the Universe, which was published in 2013.]
This piece focuses on Lee Smolin's position on what he takes to be Platonism in (mathematical) physics. Smolin's words are also used as a springboard for discussing other issues and positions (including my own) within this general debate.
Firstly, Platonism in physics is tackled as it was explicitly stated by the physicists John Wheeler and Stephen Hawking. Max Tegmark (as a Platonist) is also featured. The position advanced by Tegmark is that mathematics can perfectly describe the world/reality because the world/reality is itself mathematical. Wheeler and Hawking argued against such a position (or at least they appeared to).
Then there's a section on a position best described as “the-map-is-not-the-territory”. This too inevitably focuses on Platonism in physics. It also asks the question as to how, exactly, (mathematical) models relate to the world/reality.
There's also discussion of the relation between mathematical objects and mathematical concepts as this is brought out within the specific context of Platonism in physics.
An old problem is then discussed: the precise relation between our world and the Platonic world. The issue of (as it were) “causal closure” was the traditional focus of this particular debate; though other aspects are tackled in the following.
Finally, mathematical structuralism - and how it relates to Platonism in physics - is discussed. This leads naturally on to the final section which discusses what Smolin calls “intrinsic essences” (or what philosophers call “intrinsic properties”).
Lee Smolin puts a psychological and sociological slant on the issue of Platonism in physics when he discusses the personal motivations of Platonic philosophers and mathematicians. He writes:
“Does the seeking of mathematical knowledge make one a kind of priest, with special access to an extraordinary form of knowledge?”
It can safely be said that this was true of Pythagoras, Plato and their followers. Whether or not it's also true of an everyday mathematician or philosopher ensconced in a university department in Nottingham or Oxford, I don't know. Having said that, Smolin does speak about a friend of his in this respect. Smolin tells us that he “sometimes wonder[s] if his belief in truths beyond the ken of humans contributes to his happiness at being human”.
In any case, it's probably best to leave the personal psychologies of Platonists there. After all, if Smolin argues that Platonists are Platonists for reasons of personal psychology, then Platonists can also argue that Smolin is an anti-Platonist for reasons of personal psychology. And where does that get us?
Fire In the Equations
The physicist John Archibald Wheeler provided the most powerful riposte to Platonism in physics. In an oft-quoted story, we're told 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 Max Tegmark and others, however, the equations are the universe - at least in a manner of speaking - and perhaps not even in a manner of speaking! (More of which later.)
Then Steven Hawking (in his A Brief History of Time) nearly 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 science writer Kitty Ferguson (in her The Fire in the Equations) offers a possible Platonist answer to Hawking's question by saying 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”?
The theoretical physicist Lee Smolin, on the other hand, explains why the idea that “mathematics is prior to nature” is unsupportable. He writes:
“Math in reality comes after nature. It has no generative power.”
More philosophically, Smolin continues when he says that “in mathematics conclusions are forced by logical implication, whereas in nature events are generated by causal processes in time”.
The Platonist will simply now say that mathematics fully captures those “causal processes”. Or, in Max Tegmark's case, the argument is that the maths and the causal processes are one and the same thing.
More relevantly to the position of people like Tegmark, Smolin says that
“logical relations can model aspects of causal processes, but they're not identical to causal processes”.
What's more, “[l]ogic is not the mirror of causality”.
Yet according to Tegmark:
i) Because the models of causal processes are identical to those processes,
ii) then they must be one and the same thing.
More precisely, Tegmark's argument is as follows:
i) If a mathematical structure is identical (or “equivalent”) to the physical structure it “models”,
ii) then the mathematical structure and the physical structure must be one and the same thing.
Thus if that's the case (i.e., that structure x and structure y are identical), then it makes little sense to say that x “models” (or is “isomorphic with”) y. That is, x can't model y if x and y are one and the same thing.
So Tegmark also applies what he deems to be true about the identity of two mathematical structures to the identity of a mathematical structure and a physical structure. He offers us an explicit example of this:
electric-field strength = a mathematical structure
Or 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.”
In any case:
i) If x (a mathematical structure) and y (a physical structure) are one and the same thing,
ii) then one needs to know how they can have any kind of relation at all to one another. [Gottlob Frege's “Evening Star” and “Morning Star” story may work here.]
In terms of Leibniz's law (Smolin is a big fan of Leibniz and frequently mentions him), that must also mean that everything true of x must also be true of y. But can we observe, taste, kick, etc. mathematical structures? (Yes, if they're identical to physical structures!) In addition, can't two structures be identical and yet separate (i.e., not numerically identical)? Well, not according to Smolin's Leibniz.
All this is perhaps easier to accept when it comes to mathematical structures being compared to other mathematical structures (rather than to something physical). Yet if the physical structure is a mathematical structure, then that qualification doesn't seem to work either.
All this is also problematic in the following sense:
i) If we use mathematics to describe the world,
ii) and maths and the world are the same thing,
iii) then we're essentially either using maths to describe maths or using the world to describe the world.
What's more, maths can't be the “mirror” of anything in nature if the two are identical in the first place. In other words, any mathematical models which are said to “perfectly capture nature” (or causality) can only do so because nature (or causality) is already mathematical. If that weren't the case, then no perfect modelling (or perfectly precise equations) could exist. Thus, again, that perfect symmetry (or isomorphism) can only be explained (according to Tegmark) if nature and maths are one and the same thing.
A sharp and to-the-point anti-Platonist position is also put by the science writer, Philip Ball. He writes:
“... equations purportedly about physical reality are, without interpretation, just marks on paper”.
In other words, what exactly (as Hawking put it above) “breathes fire into the equation [to] make a world”?
The Philip Ball quote above also highlights two problems.
i) The fact that we can make mistakes about physical reality.
ii) That even if the equations are about physical reality, they're not one and the same as physical reality.
Indeed, even Ball's “interpretation” won't make the equations equal physical reality.
So let's go all the way back to Galileo (as Smolin himself does).
Surely we must say that “Nature's book” isn't written in the language of mathematics. We can say that Nature's book can be written in the language mathematics. Indeed it often is written in the language of mathematics. Though Nature's book is not itself mathematical because that book - in a strong sense - didn't even exist until human beings began to write (some of) it.
Perhaps I'm doing Galileo a disservice because he did say that
“we cannot understand [Nature] if we do not first learn the language and grasp the symbols in which it is written”.
Yet Galileo was talking about our understanding of Nature here - not just Nature as it is “in itself”.
Nonetheless, Galileo also said that the the “book is written in mathematical language”. So was he also talking about Nature as it is in itself being mathematical? Perhaps Galileo wasn't only saying that mathematics is required to understand Nature. There is, therefore, an ambivalence here between the idea that Nature itself is mathematical and the idea that mathematics is required to understand Nature.
Sure, ontic structural realists and other structural realists (in the philosophy of physics) would say that this distinction (i.e., between maths and the world) hardly makes sense when it comes to physics generally - and it doesn't make any sense at all when it comes to quantum physics. Nonetheless, surely there's still a distinction to be made here.
The Map is Not the Territory
Philip Ball (who's just been quoted) puts the main problem of Platonism perfectly when he says that
“[i]t's not surprising, the, that some scientists want to make maths itself the ultimate reality, a kind of numinous fabric from which all else emerges”.
Thus, in more concrete terms, such mathematical Platonists fail to see that the “[r]elationships between numbers are no substitute” for the world/reality. Indeed, adds Ball, “[s]cience deserves more than that”.
This is the mistaking-the-map-for-the-territory problem. As the semanticist Alfred Korzybski once put it:
“A map is not the territory it represents, but, if correct, it has a similar structure to the territory, which accounts for its usefulness.”
Indeed we can take this further and say that “all models are wrong”.
This the-map-is-not-the-territory idea is put by Smolin himself when he tells us that “[m]athematics is one language of science”. In other words, the maths (in mathematical physics) isn't self-subsistent: it needs to be tied to reality: it isn't reality itself. Thus,
“[maths] application to science is based on an identification between results of mathematical calculations and experimental results, and since the experiments take place outside mathematics, in the real world, the link between the two must be stated in ordinary language”.
More directly, Smolin tells us that
“the pragmatist will insist that the mathematical representation of a motion as a curve [for example] does not imply that the motion is in any way identical to the representation”.
“By succumbing to the temptation to conflate the representation with the reality and identify the graph of the records of the motion with the motion itself...”
Then Smolin tells us about one Platonist (or Tegmarkian) conclusion to all this. He writes:
“Once you commit this fallacy [i.e., of mistaking the map for the territory], you're free to fantasise about the universe being nothing but mathematics.”
Finally, Smolin puts his particular slant on the importance of time in all of this. He writes:
“The very fact that the motion takes place in time whereas the mathematical representation is timeless means they aren't the same thing.”
How Can Maths Model Nature?
To put it at its most simple and - perhaps - extreme. The Platonic mistake is to move from the fact that mathematics can be (almost) perfect for describing or modelling the world to the conclusion that the world must therefore be intrinsically mathematical itself. Smolin captures this position when he discusses the work of Isaac Newton. According to Smolin, Newton's world was
“infused with divinity, because timeless mathematics was at the heart of everything that moved, on Earth and in the sky”.
Slightly earlier, Smolin had also written that
“[w]hen Galileo discovered that falling bodies are described by a simple mathematical curve, he captured an aspect of the divine”.
We can of course ask if Galileo thought in these terms himself: even if only at the subconscious level. However, would that even matter to Smolin's take on this?
In any case, is mathematics “at the heart of everything that move[s]” or is it simply a tool for description or modelling? Max Tegmark (again) may argue the following:
i) If mathematics is “at the heart of everything that moves”,
ii) and it's also a perfect tool for description and modelling,
iii) then in what sense is the world not itself mathematical?
Indeed Smolin himself goes way beyond Galileo and Newton and says that “the whole history of the world” [in general relativity] is “represented by a mathematical object”.
Now if we turn to quantum mechanics and the words of Philip Ball, he says that superposition is “considered only as an abstract mathematical thing”. It's also the case the the/a wavefunction is also a “mathematical object”.
Now if we turn to quantum mechanics and the words of Philip Ball, he says that superposition is “considered only as an abstract mathematical thing”. It's also the case the the/a wavefunction is also a “mathematical object”.
If we get back to mathematical models.
It was said earlier that mathematics can describe (or even perfectly model) nature and that the physicists who aren't Platonists have no problem with this. How could they? Indeed Smolin himself tells us that “[i]t's impossible to state these laws [i.e., Newton's laws] without mathematics”. This is often said about quantum mechanics. Yet Smolin is going beyond that and saying that it's also true “the first two of Newton's laws”. More specifically, Smolin says that “[a] straight line is an ideal mathematical concept”. That is, “it lives not in our world but in the Platonic world of ideal curves”.
In terms of “acceleration” and the “rate of change of velocity” (to take just two examples), it was the case that “Newton needed to invent a whole new branch of mathematics: the calculus” in order to “describe it adequately”. But here again we mustn't conflate the maths with what the maths describes (or models).
Philip Ball (again) puts this position as it applies specifically to Hilbert space. He tells us that “a Hilbert space is a construct – a piece of maths, not a place”. He then quotes the physicist Asher Peres stating the following:
“The simple and obvious truth is that quantum phenomena do not occur in a Hilbert space. They occur in a laboratory.”
Ball also mentions Max Tegmark's position. He writes:
“If the Many Worlds are in some sense 'in' Hilbert space, then we are saying that the equations are more 'real' than what we perceive: as Tegmark puts it, 'equations are ultimately more fundamental than words' (an idea curiously resistant to being expressed without words). Belief in the MWI seems to demand that we regards the maths of quantum theory as somehow a fabric of reality.”
Mathematical Objects and Mathematical Concepts
Smolin has a problem with such mathematical objects. He (implicitly) argues against this Platonic position when he says (in a note) that
“[m]athematicians like to speak of curves, numbers, and so forth as mathematical 'objects', which implies a kind of existence”.
However, it's fairly clear that Smolin has a problem with this position. He says that you may want to call these “mathematical objects” by the name of “concepts”. That, on one interpretation, surely takes mathematical objects out of the Platonic world and places them in the realm of human minds. (Except for the fact the concepts too can be seen as “abstract objects”.)
Stephen Hawking (for one) certainly didn't believe that maths and nature are one - and he too talked about “concepts”. He once wrote that “mental concepts are the only reality we can know”. Furthermore he stated: “There is no model-independent test of reality.”
This seems to mean that Hawking went further than simply saying that mathematics describes (or perfectly models) nature. After all, he stresses the importance of “mental concepts”. However, it can still be said that the models of physics are of course mathematical and accurate. Thus even if we require mental concepts to get at these mathematical models, the models can still perfectly capture “reality”. So whichever way we interpret Hawking's words, he certainly doesn't seem to put a Platonic position (or replicate Tegmark's stance) on mathematical physics.
Smolin himself distinguishes mental concepts from mathematical objects when (in a note) he writes:
“If you aren't comfortable adopting a radical philosophical position [i.e., of believing in mathematical objects] by a habit of language, you might want to call them [mathematical objects] concepts instead.”
In that passage Smolin doesn't seem to explicitly commit himself to mathematical concepts (rather than mathematical objects); though elsewhere he is more explicit when he also talks about “inventing” (i.e., not “discovering”) mathematical objects. It's also interesting to note that Smolin puts a Wittgensteinian position. Wittgenstein, for example, once wrote that “a cloud of philosophy condenses into a drop of grammar”. Smolin, on the other hand, talks about “adopting a radical philosophical position [because of] a habit of language”.
In any case, Smolin defines a “mathematical object” thus:
“Mathematical objects are constituted out of pure thought. We don't discover the parabolas in the world, we invent them. A parabola or a circle or a straight line is an idea. It must be formulated and then captured in a definition... Once we have the concept, we can reason directly from the definition of a curve to its properties.”
Of course there are a couple of words in the passage above which a Platonist may have a problem with. Firstly, the word “invent” (as in “we invent [mathematical objects]”. And then there's the use of the word “concept” (i.e., rather than “object”). In Fregean style, we can have a “concept of an object”. Thus an abstract mathematical object can generate (as it were) various mental concepts. In terms of “[o]nce we have the concept”, then certain things logically and objectively follow from that concept. So it's the philosophical nature of the concept which raises questions.
How Do We Get to the Platonic Realm?
Even if the Platonic mathematical realm does indeed exist, then it' still the case that we still need to gain (causal) access to it. This is a problem that's often been commented upon. Smolin himself puts it this way:
“One question that Jim [a friend of Smolin] and other Platonists admit is hard for them to answer is how we human beings, who live bounded in time, in contact only with other things similarly bounded, can have definite knowledge of the timeless realm of mathematics.”
Plato himself answered Smolin's question when he argued that we have “intuitive” (or even genetic) access to this abstract realm from birth. (He elaborated on this in his slave boy story.)This doesn't seem to solve the problem of causal access to a Platonic realm. Thus, as a addendum to this argument about causal access (or the “causal closure” of both the human world and the Platonic world), Smolin says that “[b]ecause we have no physical access to the imagined timeless world, sooner or later we'll find ourselves just making stuff up”. In other words, even if the Platonic realm does exist and we can also gain access to it, that doesn't mean that we can't get things wrong or make mistakes about it.
Smolin himself says that “[w]e get the truths of mathematics by reasoning, but can we really be sure our reasoning is correct?”. What's more:
“Occasionally errors are discovered in the proofs published in textbooks, so it's likely that errors remain.”
I suppose Plato himself might have argued that we can't get things wrong because our intuition somehow guarantees access to the truths found in this realm. Or, more correctly, if we use our reason (or intuition) correctly (as Descartes also argued), then we simply can't go wrong.
So now here's Smolin quoting Roger Penrose (who's a personal friend of Smolin) putting the Platonic/Cartesian position just mentioned:
“You're certainly sure that one plus one equals two. That's a fact about the mathematical world that you can grasp in your intuition and be sure of. So one-plus-one-equals-two is, by itself, evidence enough that reason can transcend time. How about two plus two equals four? You're sure about that, too! Now, how about five plus five equals ten? You have no doubts, do you? So there are a very large number of facts about the timeless realm of mathematics that you're confident you know?”
It's of course the case that many philosophers and mathematicians will say that one doesn't need a “timeless realm” to explain all that's argued in Penrose's words above. It can, for example, be given a Wittgensteinian explanation in terms of rules and our knowledge of the rules. Our “intuitive grasp” (as it's sometimes put) of basic arithmetic can also be partly explained by cognitive scientists, evolutionary psychologists or philosophers.
It's also interesting that Penrose gives basic arithmetical examples as demonstrations of our Platonic intuition. So what about higher or more complex maths? Do mathematicians have immediate intuitions about such equations or do they need to work at them? And if they do need to work at them, then surely intuition must have a minimal role to play.
In one of his notes, Smolin gives another argument as to why the Platonic realm and the human realm can never be split asunder. He writes:
“It's also not quite true to say that the truths of mathematics are outside time, since, as human beings, our perception and thoughts take place at specific moments in time – and among the things we think about are mathematical objects.”
The Platonist would say that Smolin is conflating the Platonic realm with the fact that we can gain access to that realm. That is, one realm can still be abstract and timeless even if we concrete and time-bound human beings can gain access to it.
But here we have a analogue of the mind-body problem. That is, what is the precise relation between the time-bound and concrete world and the timeless and abstract world? Smolin himself explains the Platonic position in terms of human psychology. He continues:
“It's just that those mathematical objects don't seem to have any existence in time themselves. They are not born, they do not change, they simply are.”
Smolin uses the word “seem” in the above (as in “seem to have any existence in time”). That implies that what seems to be the case may not actually be the case. Yet Smolin does then say that mathematical objects “are not born, they do not change, they simply are”. Here he may simply be putting the position of the Platonist. Again, even if mathematical objects aren't born, we still need to explain our access to them and acknowledge the possibility of getting things wrong about them - even systematically getting things wrong!
Interestingly, Smolin offers us a kind of “conventionalist” middle way when he states that
“[w]e invent the curves and numbers of mathematics, but once we have invented them we cannot alter them”.
A Platonist would have a profound problem with the word “invent”. However, even though we may indeed invent numbers (or functions), once they're invented or created, then they become (as it were) de facto Platonic objects. That is, they're then set in stone and other things must necessarily follow from them. This is something that a philosopher like the late Wittgenstein might have happily accepted. That is, that rules and symbol-use themselves create the “objectivity” (or at least the “intersubjectivity”) of maths - and also, perhaps, even the timelessness of mathematics.
Interestingly enough, Smolin puts his anti-Platonist position by adopting the position of mathematical structuralism. (There are also types of mathematical structuralism which are Platonist - see here.) Firstly (in a note) he expresses the essence (as it were) of mathematical structuralism when he says that “relationships are exactly what mathematics expresses”. He then makes the ontological point that
“[n]umbers have no intrinsic essence, nor do points in space; they are defined entirely by their place in a system of numbers or points – all of whose properties have to do with their relationships to other numbers or points”.
Moreover, “[t]hese relationships are entailed by the axioms that define a mathematical system”. It can be said that Platonists believe that numbers do have an "intrinsic essence". In other words, a system doesn't gain its nature because of the relations between numbers: the relations between numbers are parasitical on the nature of numbers themselves. After all, the following can be argued:
i) If numbers didn't have an intrinsic essence,
ii) then they couldn't engender the precise relations to other numbers which they have in each system.
i) If numbers have intrinsic essences,
ii) then those essences can't be dependent on the systems to which they belong (or, indeed, to any system).
iii) Therefore those intrinsic essences must come before all systems of relations.
Of course the obvious point to put against that position was put by Paul Benacerraf in 1965. The French philosopher wrote:
“For arithmetical purposes the properties of numbers which do not stem from the relations they bear to one another in virtue of being arranged in a progression are of no consequence whatsoever. But it would be only these properties that would single out a number as this object or that.”
In simple terms, we can say that the number 1 is (partly) defined by being the successor of 0 in the structure determined by a theory of natural numbers. In turn, all other numbers are defined by their respective places in the number line. So, again, it can of course be said that the “essence” of, say, the number 2 is that it comes after 1 and before 3. But surely then its intrinsic essence is determined by its relations to 1, 3 and to other numbers. Perhaps, then, relations and numbers are two sides of the same coin. Having said that, it's still hard to understand what the intrinsic essence of a number could be when that essence is taken separately to that number's relations to other numbers, functions, etc.
Of course this foray into the philosophy of mathematics completely ignores the precise relation between mathematical structuralism and the world. Despite saying that, Smolin does make an explicit philosophical commitment to mathematical structuralism. He writes:
“If there's more to matter than relationships and interactions, it is beyond mathematics.”
Thus Smolin firstly began by articulating the/a position of mathematical structuralism and ends up stating a position that's very close to ontic structural realism. However, the ontic structural realist argues that there's no “beyond mathematics” – or at least that there's nothing beyond the “relationships and interactions” of physics which are described by mathematics. Yet Smolin himself appears to leave it open that there may well actually be a beyond mathematics. And elsewhere in his writings Smolin seems to state that there are “intrinsic properties” (qualia, etc.) beyond mathematics and even beyond physics itself.
Intrinsic Essences, Qualia, Etc.
Smolin makes it explicit that he (at the very least) acknowledges the possibility of “intrinsic properties” as they occur in both minds (i.e., qualia) and in inanimate objects. For example, he writes:
“We don't know what a rock really is, or an atom, or an electron. We can only observe how they interact with other things and thereby describe their relational properties. The external properties are those that science can capture and describe – through interactions, in terms of relationships.”
The passage above might well have been written by someone like David Chalmers or Philip Goff – both of whom are advocates of panpsychism. In the case of panpsychists, the “what is” (or “what it is like to be”) of a rock can be explained by referring to its experiences (or to its “proto-experiences”). These experiences are therefore the “intrinsic essences” (to use Smolin's own term) of rocks for panpsychists (if not for Smolin himself). Clearly, according to the passage above, philosophical relationalism (or relationalism in physics itself – which Smolin thoroughly endorses) doesn't capture these intrinsic properties.
So it's no surprise that Smolin continues on the theme of intrinsic essence. He writes:
“The internal aspect is the intrinsic essence; it is the reality that is not expressible in the language of interactions and relations.”
We can of course ask why Smolin accepts the very existence (or reality) of an “internal aspect” of anything when many philosophers and other physicists reject this idea.
What's more, Smolin ties all this to both consciousness and qualia. Firstly he writes the following:
“What's missing when we describe a color as a wavelength of light or as certain neurons lighting up in the brain is the essence of the experience of perceiving red. Philosophers give these essences a name: qualia.”
Again (like Smolin's “intrinsic aspect” earlier), why does Smolin need to use the somewhat archaic word “essence” (archaic at least according to certain philosophers) at all? Why believe in essences?
Finally, Smolin writes:
“Consciousness, whatever it is, is an aspect of the intrinsic essence of brains.”
So clearly Smolin has been reading some contemporary (analytic) philosophers. It's just a little odd that he begins with the words “consciousness, whatever it is”; and then goes on to tell us exactly what it is: “the intrinsic essence of brains”.