Get thee behind me, Plato

The blogosphere, even the tiny little bit of it that I know anything about, has a habit of summoning up strange coincidences between things so, following EM Forster’s maxim “only connect”, I thought I’d spend a lazy saturday lunchtime trying to draw a couple of them together.

A few days ago I posted what was intended to be a fun little item about the wave-particle duality in quantum mechanics. Basically, what I was trying to say is that there’s no real problem about thinking of an electron as behaving sometimes like a wave and sometimes like a particle because, in reality (whatever that is), it is neither. “Particle” and “wave” are useful abstractions but they are not in an exact one-to-one correspondence with natural phenomena.

Before going on I should point out that the vast majority of physicists are well away of the distinction between, say,  the “theoretical” electron and whatever the “real thing” is. We physicists tend to live in theory space rather than in the real world, so we tend to teach physics by developing the formal mathematical properties of the “electron” (or “electric field”) or whatever, and working out what experimental consequences these entail in certain situations. Generally speaking, the theory works so well in practice that we often talk about the theoretical electron that exists in the realm of mathematics and the electron-in-itself as if they are one and the same thing. As long as this is just a pragmatic shorthand, it’s fine. However, I think we need to be careful to keep this sort of language under control. Pushing theoretical ideas out into the ontological domain is a dangerous game. Physics – especially quantum physics – is best understood as a branch of epistemology. What is known? is safer ground than what is there?

Anyway, my  little  piece sparked a number of interesting comments on Reddit, including a thread that went along the lines “of course an electron is neither a particle nor a wave,  it’s actually  a spin-1/2 projective representation of the Lorentz Group on a Hilbert space”. That description, involving more sophisticated mathematical concepts than those involved in bog-standard quantum mechanics, undoubtedly provides a more complete account of natural phenomena associated with the electrons and electrical fields, but I’ll stick to my guns and maintain that it still introduces a deep confusion to assert that the electron “is” something mathematical, whether that’s a “spin-1/2 projective representation” or a complex function or anything else.  That’s saying something physical is a mathematical. Both entities have some sort of existence, of course, but not the same sort, and the one cannot “be” the other. “Certain aspects of an electron’s behaviour can be described by certain mathematical structures” is as I’m  prepared to go.

Pushing deeper than quantum mechanics, into the realm of quantum field theory, there was the following contribution:

The electron field is a quantum field as described in quantum field theories. A quantum field covers all space time and in each point the quantum field is in some state, it could be the ground state or it could be an excitation above the ground state. The excitations of the electron field are the so-called electrons. The mathematical object that describes the electron field possesses, amongst others, certain properties that deal with transformations of the space-time coordinates. If, when performing a transformation of the space-time coordinates, the mathematical object changes in such a way that is compatible with the physics of the quantum field, then one says that the mathematical object of the field (also called field) is represented by a spin 1/2 (in the electron case) representation of a certain group of transformations (the Poincaré group, in this example).I understand your quibbling, it seems natural to think that “spin 1/2″ is a property of the mathematical tool to describe something, not the something itself. If you press on with that distinction however, you should be utterly puzzled of why physics should follow, step by step, the path led by mathematics.

For example, one speaks about the ¨invariance under the local action of the group SU(3)” as a fundamental property of the fields that feel the nuclear strong force. This has two implications, the mathematical object that represents quarks must have 3 ¨strong¨ degrees of freedom (the so-called color) and there must be 32-1 = 8 carriers of the force (the gluons) because the group of transformations in a SU(N) group has N2-1 generators. And this is precisely what is observed.

So an extremely abstract mathematical principle correctly accounts for the dynamics of an inmensely large quantity of phenomena. Why does then physics follow the derivations of mathematics if its true nature is somewhat different?

No doubt this line of reasoning is why so many theoretical physicists seem to adopt a view of the world that regards mathematical theories as being, as it were,  “built into” nature rather than being things we humans invented to describe nature. This is a form of Platonic realism.

I’m no expert on matters philosophical, but I’d say that I find this stance very difficult to understand, although I am prepared to go part of the way. I used to work in a Mathematics department many years ago and one of the questions that came up at coffee time occasionally was “Is mathematics invented or discovered?”. In my experience, pure mathematicians always answered “discovered” while others (especially astronomers, said “invented”). For what it’s worth, I think mathematics is a bit of both. Of course we can invent mathematical objects, endow them with certain attributes and proscribe rules for manipulating them and combining them with other entities. However, once invented anything that is worked out from them is “discovered”. In fact, one could argue that all mathematical theorems etc arising within such a system are simply tautological expressions of the rules you started with.

Of course physicists use mathematics to construct models that describe natural phenomena. Here the process is different from mathematical discovery as what we’re trying to do is work out which, if any, of the possible theories is actually the one that accounts best for whatever empirical data we have. While it’s true that this programme requires us to accept that there are natural phenomena that can be described in mathematical terms, I do not accept that it requires us to accept that nature “is” mathematical. It requires that there be some sort of law governing some  of aspects of nature’s behaviour but not that such laws account for everything.

Of course, mathematical ideas have been extremely successful in helping physicists build new physical descriptions of reality. On the other hand, however, there is a great deal of mathematical formalism that is is not useful in this way.  Physicists have had to select those mathematical object that we can use to represent natural phenomena, like selecting words from a dictionary. The fact that we can assemble a sentence using words from the Oxford English Dictionary that conveys some information about something we see doesn’t not mean that what we see “is” English. A whole load of grammatically correct sentences can be constructed that don’t make any sense in terms of observable reality, just as there is a great deal of mathematics that is internally self-consistent but makes no contact with physics.

Moreover, to the person whose quote I commented on above, I’d agree that the properties of the SU(3) gauge group have indeed accounted for many phenomena associated with the strong interaction, which is why the standard model of particle physics contains 8 gluons and quarks carrying a three-fold colour charge as described by quantum chromodynamics. Leaving aside the fact that QCD is such a terribly difficult theory to work with – in practice it involves  nightmarish lattice calculations on a scale to make even the most diehard enthusiast cringe –  what I would ask is whether this  description in any case sufficient for us to assert that it describes “true nature”?  Many physicists will no doubt disagree with me, but I don’t think so. It’s a map, not the territory.

So why am I boring you all with this rambling dissertation? Well, it  brings me to my other post – about Stephen Hawking’s comments about God. I don’t want to go over that issue again – frankly, I was bored with it before I’d finished writing my own blog post  – but it does relate to the bee that I often find in my bonnet about the tendency of many modern theoretical physicists to assign the wrong category of existence to their mathematical ideas. The prime example that springs to my mind is the multiverse. I can tolerate  certain versions of the multiverse idea, in fact. What I can’t swallow, however is the identification of the possible landscape of string theory vacua – essentially a huge set of possible solutions of a complicated set of mathematical equations – with a realised set of “parallel universes”. That particular ontological step just seems absurd to me.

I’m just about done, but one more thing I’d say to finish with is concerns the (admittedly overused) metaphor of maps and territories. Maps are undoubtedly useful in helping us find our way around, but we have to remember that there are always things that aren’t on the map at all. If we rely too heavily on one, we might miss something of great interest that the cartographer didn’t think important. Likewise if we fool ourselves into thinking our descriptions of nature are so complete that they “are” all that nature is, then we might miss the road to a better understanding.


16 Responses to “Get thee behind me, Plato”

  1. Anton Garrett Says:

    Nice post Peter. The reason that everything blew up wiith quantum mechanics is that it departed from all previous theories in no longer having a 1:1 correspondence between things out there and symbols in the mathematical formalism. Some of us still see this as a Bad Thing and hanker after hidden variable theories, to which quantum theory is a statistical average. Frankly I dislike the adjective ‘hidden,’ for if you go down this route then you can see their influence very clearly – not hidden at all – when two systems with identical wavefunctions behave differently, as in a Stern-Gerlach apparatus. The difficulty is that John Bell taught us these hidden variables would have to be acausal. (They are also nonlocal, but too much has been made of that – nonlocality has been around since Newtonian gravity, as ‘action at a distance,’ although quantum nonlocality does not fade with distance.)

    I’m not sure I agree that “Physicists have had to select those mathematical object that we can use to represent natural phenomena, like selecting words from a dictionary.” Until roughly the 20th century, mathematics was developed FOR physics, ie physics spurred the development of mathematics, often by the same people (Newton, par excellence). So there was not much ‘other’ mathematics. Only fairly recently in historical terms has there been maths on the shelf for physicists to pick up as needed. You have generously let me use your blog to suggest that the older way might have been better…


    • telescoper Says:

      Anton, Yes I should have probably have said “have to” rather than “have had to” because it’s definitely a more modern tendency for maths to have as little as possible to do with anything other than mathematics. It is true that string theory has “inspired” people to develop new kinds of mathematics, but in my book string theory isn’t physics and won’t be unless and until it actually becomes testable.

    • Anton Garrett Says:

      As an addendum: acausality is unavoidable in physics anyway, in (for example) John Wheeler’s “delayed choice” experiments regarding whether to measure a *wave* interference pattern, or which slit the *particle* went through, AFTER the the energy has passed the location of the screen.

      So lete us reject indeterminism and embrace acausality as the explanation why we can presently predict some things only statistically at quantum level. After all, if you accept acausality then you might be able to construct a time machine to get the info you need in order to predict deterministically. (I don’t necessarily mean a Tardis, but some physical channel for storing and conveying information.)

      At the moment we know more about what the hidden variables aren’t than what they are:

      They are nonlocal, with distance (space?) irrelevant
      They are acausal (with time irrelevant?)

      NB Since nature is nonlocal and acausal it is remarkable that we can predict anything at all. This provides a very strong constraint on such theories.

      The hidden variables have to be building blocks for such ‘fundamental’ quantities as angular momentum, since the 3-particle Bell set-ups(GHZ; see also Lucien Hardy’s work) indicate that it is a contradiction to consider an electron as having x,y and z components of angular momentum simultaneously.


      The aausality must not be such as to permit paradoxes.

      Statistical averaging must reprodiuce the predictions of quantum mechanics.

      There is enough here for a research program. What I regret is that too many excellent physicists are brainwashed into denying the existence of hidden variables, so that almost nobody is working along these lines. This is a lot more fundamental than string theory.


  2. Great post. I completely agree.

    Here’s a helpful analogy. Suppose I construct a doll house that is a model of a real house. It is obvious that the doll house is not the same thing as the real house. And indeed, even if I’m very careful about constructing the doll house, it will almost certainly be an approximation to the real house and will never have a perfect 1-to-1 mapping to the real house.

    For some reason, people are tempted to treat mathematics differently, but I don’t believe it is fundamentally any different. No matter how accurate your mathematical model is, it is a model. It may or may not be 1-to-1 with the real thing, but almost certainly it is not 1-to-1. Modeling an electron as “a spin-1/2 projective representation of the Lorentz Group on a Hilbert space” might be extremely accurate, but probably not perfect. Suppose one day we discover, through experiment, some subtle properties of electrons we have until now missed. We would be forced to humbly update our models no matter how convinced we were we had been right.

    I can use the same analogy to argue against mathematical Platonism. Some doll houses correspond to real houses. But I can also construct a fantasy doll house, with a space-dock, a bat-cave, and an Olympic-sized swimming pool. Is there necessarily a house in the real world that corresponds to my doll house? Of course not. Same thing with mathematics. Some mathematical models, like doll houses, are only good at modeling themselves.

  3. Addendum: It’s interesting to note that in order to illustrate my point, I did not construct a fantasy doll house. I constructed a theoretical fantasy doll house.

  4. Great post.

    My understanding has always been that mathematics is a tool, and from a physics point of view, crafted by physicists to better understand the physical world. This tool can be fine tuned, or be used ad crudely as desired.

    At the moment physics uses second-order differential equations to describe motion etc. but who is to say that this is the only method of describing the Newton’s laws of motion? This could be the limiting factor with a greater M theory.

    It was shown that magnetic and electric waves are linked, but does the use of calculus limit the tying up of gravity to such phenomena? We know it exists since we can measure it, but how can we best describe it on paper?

    Often, equations are encountered with a real and imaginary part, however the real part can describe a phenomenon perfectly well on its own. The imaginary part can be seen as a redundancy of the tool that is mathematics.

    • Complex numbers are undoubtedly useful in physics but we should always remember that imaginary numbers are not real. In fact, I seem to remember hearing Steve Gull give a talk with precisely that title some years ago!

    • Anton Garrett Says:

      Complex numbers are really only a superbly economical form of 2-dimensional vector analysis (including calculus). We can do better not by junking threm but by working out how to extend that economy to higher dimensions. The good news is that this has been done: it is Clifford algebra. No more need for tensors in one problem, spinors in another, quaternions in a third, twistors in a fourth, regular vector calculus in others (how to generalise the cross product to higher than 3 dimensions), etc etc. It’s all unified, with added value, in Clifford algebra. Best book: Clifford Algebra To Geometric Calculus, by David Hestenes and Garret Sobczyk.


  5. There is nothing special about math in this respect. The only way we ever deal with the universe is by some kind of map or model. You think color is real? The specific color you experience is a complex product of the light and the way you process the information.

    Look at the argument over whether Pluto is a planet or not. Well “planet” is just a word that can mean whatever we want it to mean. Whatever definition we chose for it is just an element in a model. The argument actually has no content as it is usually presented.

    As I said before you should be careful how seriously you take your ontological commitments. That is as true of stars and planets as it is for waves and particles.

  6. It’s interesting, though, that there is a process by which theoretical concepts become “real”. No doubt five hundred years ago the planets seemed unimaginably distant so the assertion that their existence and properties were “real” would have been absurd to many. A hundred years ago it would have been the same for electrons. Now we are happy to talk about them as if they are real.

    I don’t think there’s anything really deep about this, it’s just that the predictions we have made have been so successful so often that we physicists, being pragmatic people, just stop questioning the philosophical issues as much. In otherwords, we have such good maps that we tend to work with them as much as possible, without constantly looking at precisely how they correspond to the philosophical landscape. It’s a form of dead reckoning – fine if you know what you’re doing, but dangerous if you don’t.

  7. Carl Jung had a long association with the Nobel laureate
    physicist, Professor W. Pauli. Their final conclusions were
    published under title, “atom and archetype” – letters,
    Their claim is that the “natural numbers” are a tangible
    connection between the spheres of matter and psyche.
    Jung’s comments, in part:

    Since the remotest times men have used number to establish meaningful coincidences, that is, coincidences that can be interpreted.

    There is something peculiar, one might even say mysterious about numbers. They have never been entirely robbed of their numinous aura. If, so a textbook of mathematics tell us, a group of objects is deprived of every single one of its properties or characteristics, there still remains, at the end, its number, which seems to indicate that number is something irreducible.

    The sequence of natural numbers turns out to be unexpectedly more than a mere stringing together of identical units; it contains the whole of mathematics and everything yet to be discovered in this field.

    Number, therefore, is in one sense an unpredictable entity.

    It is generally believed that numbers were invented, or thought out by man, and are therefore nothing but concepts of quantities containing nothing that was not previously put into them by the human intellect. But it is equally possible that numbers were found or discovered..
    In that case they are not only concepts but something more-autonomous entities which somehow contain more than just quantities.

    Unlike concepts, they are based not on any conditions – but on the quality of being themselves, on a “so-ness” that cannot be expressed by an intellectual concept.

    Under these conditions they might easily be endowed with qualities that have still to be discovered. I must confess that I incline to the view that numbers were as much found as invented, and that in consequence they possess a relative autonomy analogous to that of the archetypes.

    They would then have in common with the latter, the quality of being pre-existent to consciousness, and hence, on occasion, of conditioning it, rather than being conditioned by it.

    In quantum physics, natural numbers are considered to be the ultimate structural element of being. Pauli….

    Ref: On the nature of transcendence, acausal connections
    in the space-time continuum, Pauli (1952)

    Synchronicity-an acausal connecting principle, Jung….

    Number and Time, M.L. von Franz

    numomathematics, T. Delaurence

    • Anton Garrett Says:

      I don’t particularly wish to dispute whether numbers are more than descriptors. But they are certainly not material/physical, and their role *in physics* is purely as description. I disagree wtih Pauli’s 1952 quote above, and I wonder what evidence he brings to bear for this claim.

      The notion of abstract number, of pure 1,2,3 as what 1,2,3 boats in a harbour, 1,2,3 sheep in a field, etc etec have in common is due to the ancient Greeks. It is one of the great conceptual leaps that fed the birth of mathematics. While I stand in awe of the beauty of number theory, I do not wish to return to the Dark Ages that preceded it.

      Jung: “The sequence of natural numbers turns out to… contain the whole of mathematics”. Sorry, but I don’t think that topology has anything to do with number.


  8. Steve Jones Says:

    I must confess I’m at a loss with regards this whole discussion.

    I can’t help thinking that for several of the commenters here the best part of Copernicus’ De revolutionibus is the (in)famous forward by Andreas Osiander were he points out that the hypothesis “that the earth moves” merely provides a model

    “consistent with the observations”

    and it is

    “not put forward to convince anyone that are true, but merely to provide a reliable basis for computation”

    I don’t know…I’m off to scratch my head and wonder what is “real” and indeed what is real (with and without scare quotes).


    • Yes, at least today it seems strange to think of the heliocentric theory
      as merely a hypothesis to aid calculation and not as a theory of reality.

      However, consider quantum mechanics. In the old quantum theory,
      Bohr’s original atomic model gave the correct results, even though we
      now know it is wrong. However, I don’t think that this is what Osiander
      meant (I think he was trying to avoid being killed by the Church). We
      now have several different interpretations, or models, of QM which provide
      reliable a basis for computation, and give the same results. Which, if
      any, is true?

  9. Todd Laurence Says:

    This, to my mind explains the difference between
    coincidences and acausal events, i.e., “acts of creation.”

    Example 1.

    Angelo Gallina, a 78-year-old retired railroad machinist from Belmont, cheerfully admitted he has bought $20 worth of lottery tickets every day since the lottery started in 1985.

    For the SuperLotto Plus drawings, he and his 65-year-old wife choose their numbers by shaking a $10 plastic gadget they purchased at a local drugstore. The gadget is full of tiny numbered balls that fall into slots, providing lottery players with lucky numbers to bet on.

    So the Gallinas shook it up last month, and out came 10, 41, 7, 47, 21 and mega 4. That was the winning combination for the Nov. 20, 2002 draw.

    Meanwhile, their Fantasy Five combination of 15, 18, 26, 35 and 37 came in, too.

    For the record, the odds of winning the SuperLotto Plus are 1 in 41.4 million and the odds of winning the Fantasy Five are 1 in 576,000. Multiplying those numbers yields 1 in 24 trillion.

    Example 2.

    I came across the Chaldean number/alphabet in the
    book, “star signs.” I kept a copy on my nightstand,
    thinking it might be useful somehow.

    About four months later I had a very unique dream.

    “In a classroom, a teacher, (woman) is talking about
    astronomy. The scene changes to a field, where two
    bears are romping and a duck is flying overhead.”

    I decided to try and interpret this dream:

    Story continues here:

    “such is the nature of reality, that anyone can
    experience that which is least understood.” TDL

  10. […] when a wave-function collapses? These questions take us into philosophical territory that I have set foot in already; the difficult relationship between epistemological and ontological uses of probability […]

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