Reply to Lee Smolin's fqxi essay: A naturalist account of the limited, and hence reasonable, effectiveness of mathematics in physics

This is the copy of the comments I already posted in the fqxi comment thread of his article. We can see that most people there just behaved as mindless zombies deciding to give him good marks just because of the stupid rumor that Smolin was supposed to be a great physicist, and because he claimed to defend the view they like, no matter how crappy his arguments really were, nor how solid could be the proposed refutation of them, which nobody ever started trying to understand or reply to.

Hello. You come to propose a conception of things coherent with naturalism. Great ! I stand for the opposite view ;-)

I actually never found a formulation of naturalism that seemed coherent, as it seems to me logically impossible, somehow already in principle, and then even more with quantum physics. So I am very curious when I see such a proposition announced ! For now most of the essays I reviewed here in support of such a view seemed to be amateur-level. I was full of hope for discussions to become at last serious and challenging, at the first sight of an essay with this purpose by a reputed physicist coming to the list.

One thing I was puzzled with when reading some naturalist views, is how they dismiss any idea of considering consciousness as fundamental, by calling this an "explanation by a mystery" and thus no explanation at all. Indeed it may look like this, in the sense that consciousness escapes all mathematical description. So if your condition to call something "non-mysterious" is to have a mathematical, deterministic description of it then indeed consciousness is "mysterious" in this sense. Which does not mean that noting can be said about it (as I did express some important features of consciousness for its connection with physics). However, on their side they claim to explain everything as "Nature". But what the heck do they mean by "nature", and, in lack of a clear definition for this kind of stuff and its working principles, how is an "explanation" of the world by an undefined "nature" assumed to be primary, be any less mysterious than the view taking consciousness as primary ?

I once saw an "argument" that if a miracle is real then by definition it must be part of nature because nature is "all what exists" so that nothing can be meaningfully called "supernatural". Then well, if "all what exists" is the definition of "nature" then it makes naturalism tautological, but no more informative. To be informative we need to specify what kind of stuff is "nature" supposed to be. It seems supposed to mean "physical stuff". Well if we were in the 19th century, and still with General Relativity, it could indeed look like there was such a thing as "physical stuff" that the universe could be made of. However, quantum physics broke that.

Namely, an important question I would have, is whether "nature" is supposed to be finitely or infinitely complex, or maybe just locally finitely complex, in case it could be considered locally (which you seem to reject as you seem to favor non-locality in interpretations of quantum physics). So for example if it is locally finitely complex but not locally causal then, finally, it is infinitely complex if the universe is infinite (in hope that the dependence of local stuff on the rest of the universe converges). Quantum physics makes the physical world locally finitely complex indeed. I consider consciousness infinitely complex. But if "nature" was physical and infinitely complex, how could it have definite causalities that depend on infinitely complex stuff ? Bohmian mechanics describes things as infinitely complex, but I suspect its laws to diverge when considered in their globality.

Here are points of interest I found in your article:

"The effectiveness of mathematics in physics is in [Platonism] mysterious because proponents of this view have failed to explain both how there could be such a correspondence and how we, as beings trapped in time bound physical reality, can have certain knowledge of the hypothesized separate realm of mathematical reality."

What failure to explain ???? I do not see the slightest problem here: it is a one-way dependence. Anything that exists must be coherent with itself, so that whenever we can discern mathematical structures somewhere, they have to be coherent with themselves, thus obey the laws of coherence which are the mathematical theorems. So it is "affected" by the mathematical world, but does not affect it in return (nothing can change the facts of what is coherent and what isn't). It is possible for mathematical structures to be more or less involved by contingent (non-mathematical) realities.

"if you believe that the ultimate goal of physics is to discover a mathematical object, O, which is in perfect correspondence with nature, such that every true fact about the universe, or its history, is isomorphic to a true fact about O, then you are also not a naturalist because you not only believe in the existence of something which is not part of nature, you believe that everything that is true about nature is explained by a true fact about something which exists apart from nature. You are instead a kind of mystic, believing in the prophetic power of the study of something which exists outside of time and apart from nature."

All right, so this means naturalism rejects any possibility to describe nature in mathematical terms. In this case, nature escapes any rigorous mathematical description and is therefore assumed to be fundamentally "mysterious". Like consciousness in my view.

You wrote: "Mathematics thus has no prophetic role in physics, which would allow us an end run around the hard slog of hypothesizing physical principles and theories and testing their consequences against experiment". Then you "hypothesize two principles which we take to define temporal naturalism". Are these two principles not supposed to have any prophetic role in physics, that would allow you an end run around the hard slog of hypothesizing physical principles and theories and testing their consequences against experiment? Because in the rest of your essay I did not find any big care to test these principles against experiment, or against the body of modern science which sums up so many experiments already done, in the sense of a possible challenge to the truth of your principles.

"All that exists is part of a single, causally connected universe. The universe and its history have no copies, and are not part of any ensemble."

Right. I would qualify the spiritual multiverse (where souls can migrate between universes) in these very terms, though the connections between parts (universes) can sometimes be poor.

"There is no other mode of existence, in particular neither a Platonic realm of mathematical objects nor an ensemble of possible worlds exist apart from the single universe." And why not ? You seem to have quite a faith in this negation.

"All that is real or true is such within a moment, which is one of a succession of moments"

You already multiply the modes of existence, between past, present and future existences, and where the time-status of the existence of any particular event... depends on time. So you admit multiple possible modes of existence, but you deny the possibility for still another mode of existence than these (the mathematical existence).

"The activity of time is a process by which novel events are generated out of a presently existing, thick set of present events. "

How thick is the set of present events, and how do you measure this thickness, both in space and time dimensions ? My view of the spiritual reality would be similar except that I take all past events as still presently existing and indestructible, and from which novel events are generated.

" we adopt a strong form of Einstein’s principle of no unreciprocated action according to which there can be no entity A which plays a role in explaining an event B, that cannot itself be influenced by prior physical events."

That is quite an assumption, of trying to generalize a principle far beyond the form in which it was initially considered and justified by experiment ! But is it really just a plausible strengthening of a well-defined principle, or rather an endless multiplication of fanciful assumptions only superficially similar to the initially successful version ? Something like justifying philosophical relativism as "a strong form" of the Special Relativity principle.

Of course you cannot understand the possible relation between mathematical and physical realities if you exclude by principle the possibility of one-way influences, and by "satisfying explanation" you mean "explanation that agrees with this principle", assumed to have such a prophetic role in physics, that it allows you, in your own words, "an end run around the hard slog of hypothesizing physical principles and theories and testing their consequences against experiment". By the way, how do you apply this principle to the dependence between past and future ? How can the past affect the future without being affected by it in return ?

You wrote " Among the things that violate a strict definition of naturalism are (...) absolute, timeless laws," yet you defend the view of "the singular universe" which seems to fit absolute timeless laws. It seems quite hard for these laws to vary inside the same universe, both theoretically (the formal rigidity of the physical laws that do not easily let coherent ways to glue together parts of space-time that do not obey the same laws) and as we did not see them vary, but it would be much easier between different universes. Don't you see it hard to reconcile both principles of uniqueness of the universe and contingency of the laws ?

In trying to argue against the idea of preexistence of mathematical realities, you mention a wide spectrum of things ranging from the somewhat mathematical to the non-mathematical. Your argument seems to be that since you can find some (non-mathematical) things that do not preexist some act of creation and you can also go "continuously" from these to mathematical systems, you conclude that mathematical systems do not preexist some act of creation either. However I see this fallacious : just because you want to believe that different stuff are the same kind and you can look for intermediates between them, and pretend you find some which make the spectrum continuous, does not mean that they are really of the same kind. Discontinuities in this range can be found, that can justify to not put all these things in the same category.

Rules of poetry also implicitly require sentences to be meaningful and appropriate for poetry, a condition which cannot be mathematically defined. So the complete expression of its rules may depend on time (as language and cultural context evolve, modifying the condition of meaningfulness of sentences), thus making this incomparable to the case of mathematical systems.

For example, chess is an exact problem, but the rules of chess are rather complex and arbitrary, so that it is just one game in a range of billions of possible games with a similar degree of complexity of their rules. Civilizations on independent planets have only a very small probability of having the same game of that complexity level becoming popular. Still, from a mathematical viewpoint, this game exists as a game among others, just like any number between 1 and 1010 exists as a number among others in this range, no matter that it has only a very small chance of being picked up by a particular person who is choosing a number at random in this range. The only thing in chess which is not strictly of this kind (of existing in the abstract but having a very small chance of being picked up), is not the game itself but the names and pictures of the pieces involved.

Question : if Chess does not exist before a civilization "invents" it, then, did any number between 1 and, say, 1015, remain non-existing until someone uttered it ? You seem to not adopt that view, however, in the sense that you admit that all possibilities inside an axiomatic system exist as soon as the rules of the system were fixed. So, as soon as we have a theory of arithmetic, all natural numbers must exist. More precisely, at least the standard ones, and even more precisely those lower than a number we can tell, such as for example, all numbers between 1 and 1015. This makes your concept of existence of an object independent of the degree of conscious awareness of people towards this object, unlike the rules of chess, whose heavy "existence" in this world above other possible games of similar complexity, actually consists in the conscious attention of people towards it. As explained in my essay, I hold conscious awareness as forming the other component of existence aside mathematical existence, that is where "novelty" as we know it resides (the act of becoming aware of a mathematical object that mathematically existed, but that one did not think of before).

In biology, things are picked up in a landscape of possibilities that is explosively huge because of the high complexity of everything there. So it would be completely impossible for someone to enumerate all possibilities one by one. But then what ? If that was a reason to deny the preexistence of possibilities not yet picked up, should we also claim that most numbers between 1 and 101015 are non-existing just because nobody ever paid attention to them ? If we recognize the existence of all these numbers just because we have a theory of arithmetic for them, no matter our concrete inability to enumerate them all, then we should also recognize the existence of all biological possibilities because we have laws of physics which, in principle, determine this landscape of possibilities.

Now about axiomatic systems, and the idea that the whole infinity of truths from an axiomatic system are being born at the time when the particular axiomatic system is being uttered. I'm sorry but this is so ridiculous to draw the line of existence here (I was tempted to say it is one of the most ridiculous places to draw the line, however I'm not here to try arguing that a less ridiculous defense of naturalism is otherwise possible, either). Because, as is well-known in mathematical logic but as you may have missed if you are ignorant in this field (since you admitted that you only recently happened to accidentally discover that a respectable account of a philosophy of mathematics also needs to tell something about the rules of proof, while it might have been better if you went as far as caring to seriously inform yourself on the core concepts and works actually done by specialists of this well-established field of mathematical knowledge, instead of just assuming that, just because you are a renowned physicist and famous blogger, your random baseless speculations on the foundations of maths should be seen just as plausible as anything else), there is a well-known general concept of axiomatic systems and their logical consequences, whose rules are universal and independent of the particular axiomatic system. Somehow you even also implicitly admitted yourself the Platonic existence of this universal system with its absolute concept of proof, that you awkwardly tried to condone and reduce to some pragmatic stuff.

But, since, in fact, these universal rules of the game of writing axiomatic systems and deducing their logical consequences have been discovered (or "evoked" if you prefer), according to your philosophy, this automatically gives existence to the whole of mathematics, with the totality of possible axiomatic systems and all their consequences. Bingo ! The whole truth of mathematical Platonism is now accomplished.

Indeed, in case you didn't know, we can easily write down a computer program whose function is to automatically enumerate all possible axiomatic systems one by one, only restricting the possibility for particular axiomatic systems to be included there by the practical limits of computer resources. (We can also enumerate all algorithmically enumerable infinite axiomatic systems by automatically generating and emulating all programs able to generate axioms).

If on the other hand we considered particular axiomatic systems as not yet created as long as they are not actually uttered by a computer, but created when they are uttered, a problem would be, just uttering is not enough. If a program utters an axiomatic system, it is not yet really an axiomatic system that is uttered as long as it is not functionally used in the intended way, otherwise there would be no objective truth on which axiomatic system was really uttered at at time (it all depends, for example, whether a given logical symbol is interpreted as meaning "and" or "or", just like uttering "1464" remains ambiguous on which number this chain of symbols is supposed to represent, unless we specify some conventions on how numbers are denoted). However it is just a matter of adding one more piece of software and a lot of computer power, for a program of automatic generation of axiomatic systems to also actually give their full meanings to these axiomatic systems, by starting to deduce all logical consequences of these systems in parallel. Then, is it that latter piece of software which, when put in conjunction with the utterance of each axiomatic system, provides these uttered axiomatic systems their actual existence with all their truths ?

You call "mystical" the belief in the independent existence of mathematical entities. You point out that they "add nothing and explain nothing". Well, I do not see the idea of independent existence of mathematical entities as trying to add or explain anything, as if it was any kind or addition or speculation. It is not. Mathematical facts are necessary facts. I cannot see any sense in which the truth of 2+2=4 can be said to be or have been "non-existing" at any time. It is the belief in the possibility of non-existence of such truths, that I would call a mystification.

What is the problem ? You have the problem that you think that whenever such ideas are raised, it "involves us in a pile of questions that, unlike questions about mathematics, cannot be answered by rational argument from public evidence."

Which questions ? I looked at the questions you listed on page 5, and sorry, this is just laughable. You call these "questions" ? Well of course it is always possible to feel uncomfortable with any idea or any truth, by the sickness of reacting to them by asking tons of "questions" which may be naively thought of as legitimate but which are in fact senseless, just a psychological reaction of inventing problems where there is no problem, because the truth that is seem "problematic" was not grasped in the correct manner. Such reactions are frequent in the crackpot world. For example those who cannot accept relativity theory may ask questions such as "What causes the slowdown of time ?" "What causes the contraction of length ?". On other topics, one can ask "What is an electric charge", "what is a number", "how dense is a black hole", "what happened before the big bang", "what is a specie", trying (as I saw science philosophers do) to make sense of "structural realism" so as to define what is the reality of the structures that are studied by biology and other sciences; and wonderiong a long time about whether light and other quantum substances must be "explained" as waves or as made of particles.

Example: "If the FAS existed prior or timelessly, what brought it into existence?". Well, nothing, why ? If it existed timelessly then there is no need of any such thing as an event of bringing it into existence. It would only be needed under the assumption of existence of a previous time when that FAS did not exist. But the idea of such a time is a belief I would call a deep, crazy mystification. There never was a need of any physical event to create an FAS because there never was in the first place any physical time when it did not exist and remained to be created. As simple as that.

"How can something exist and not be made of matter?"

Well, and how can matter exist and not be made of something else ?

You choose to call "mystification" the belief of existence of something else than matter. But, well, can we reject as "mystification" the belief of existence of anything at all ? Of course not, as we are aware of our own existence. So we can only reject a belief in the existence of some specific kind of things in favor of that of another kind. The question is to know which are the kinds of things that exist. The only mystification would be to misattribute our existential beliefs in ways not supported by evidence. Our own existence, as conscious beings, is something clear, that cannot be denied. The existence of mathematical truths is also clear as we can study and understand them. But the existence of matter, what the heck is that ? We cannot access it, all we have is sensations about it. These sensations naively suggest to the layman a real presence of material things by means of their coherence (logical patterns). These patterns can be described mathematically. But when analyzed in details, we discover quantum physics, which strongly indicates that material things do not really exist at a fundamental level, but are created by our conscious perceptions of them. Indeed: for example I even heard in this debate on interpretations of quantum physics, all of whose participants are hardcore materialists, a report that many physicists tend to dismiss the reality of the wavefunction, and at the same time hold that "nothing else is real", which would imply that "nothing [exists] at all" (since they did not make the step of admitting another kind of fundamental reality). So I'm not inventing the idea that quantum physics denies the existence of matter, even materialist physicists somehow acknowledge it.

So we have evidence (or at least strong indications from experience) that matter is not real. Now if a belief in the existence of something we clearly see (mathematical truths) is "mystification", then, how can we call the hard unshakable belief which you expressed in your text, that only one kind of things that we cannot see (matter) exists while other kinds of things which we clearly perceive (our own self and mathematical truths) don't, in spite of the evidence from modern physics that matter is not real ? Maybe "total insanity", why not ?

Now about your page 7.

You wrote : "the answer to Wigner’s question is that mathematics is reasonably effective in physics, which is to say that, where ever it is effective, there is reason for it". This claims comes as logically deduced from your philosophy, in the traditional way of philosophers, that is, as a pure theoretical (but not even so carefully logical) blind guess, that proudly comes as self-sufficient reasoning with no need to check it against any review of how things were observed to be : here, the measure of how mathematics was found to be effective in reality. Indeed, where is your review of these observations ? Instead of observing or checking anything, you satisfy yourself to prophesy: "There will never be discovered a mathematical object whose study can render unnecessary the experimental study of nature". Still you are coming with philosophical principles whose study seem to suffice for you to deduce in the abstract how effective should math be to the study of nature. Just like usual (bad) philosophers, your confidence in your principles makes you see unnecessary not only to abstain concluding and humbly consider to wait and see what future discoveries may show (maybe giving your claims a status of falsifiable predictions to be tested and eventually refuted), but you also see it unnecessary to check their compatibility with the present record of the state of things actually found by modern science: whether the effectiveness of mathematics that was actually "observed" by the development of modern science fits these expectations of effectiveness you are presenting. Does the self-evidence of your principles and prophecies carry sufficient logical or metaphysical reliability to give you such a faith in their truth that this confidence can legitimately supersede for a rational mind, any concern for experimental check, any verification against any past or future research, such as a search for a counter example to your claims (some mathematical object that might be successful enough to make some experiments unnecessary) ?

Actually, theoretical physics happened to be so successful that, well of course there is still some place for experiments, but this place is now quite reduced either to very complex (macroscopic) systems (where computations would be too complex for our supercomputers, so that the studied properties are only consequences of known laws in principle but not in practically computable ways), or to the case of extreme conditions that are very hard to explore (with particle accelerators, some subtle aspects of astronomy and cosmology to analyze the properties of dark matter... not mentioning the mind/brain interaction that I expect, as I explained in my essay, to involve subtle processes, linked to the nature of quantum measurement, beyond established mathematical physics, that have not yet been well investigated); in many other cases, such as gravitation, theory suffices. Fortunately indeed we do not need to send hundreds of probes in space all over again for each space exploration mission until finding out by chance which trajectory may actually lead to the desired destination.

After this, in guise of illustration of your belief, you give examples from modern physics, so as to make it look as if your principles were not pure abstract principles disconnected from modern science, but compatible with it, or even supported by it. I am deeply amazed at what a badly distorted report you manage to make of how things go in modern physics, so as to make it look as if it supports your philosophy. This is so ridiculous, and just the same style of absurd distortions and misinterpretations of modern physics as what is usual from the part of cranks who claim to refute Special Relativity by criticizing Einstein's book and finding a "new explanation" for the Michelson-Morley experiment (or rather an old one, always the same : a "mechanically explained" Lorentz contraction of moving things and absolute slowdown of clocks with respect to an absolutely still ether), or who similarly "explain" quantum physics by classical waves, or who claim there must be a local realistic deterministic explanation of quantum randomness because they believe that any randomness must hide such a determination (assuming that physicists just did not try to look for one but lazily and dogmatically preferred to "shut up and calculate") and they did not learn about the logical and experimental arguments against it.

You see "a large degree of arbitrariness" in mathematical physics. Of course there is some arbitrariness in the list of particles in the Standard Model and the values of all constants there as we know them (about 20), but this is nevertheless often qualified by many physicists as quite elegant as compared to the amount of observations this theory explains, far from "a large degree of arbitrariness" as you say. The Higgs boson, like many other particles (such as antiparticles), was predicted before being observed.

You wrote "In most cases the equation describing the law could be complicated by the addition of extra terms, consistent with the symmetries and principles expressed, whose effects are merely too small to measure [by] given state of the art technology. These “correction terms” may be ignored because they don’t measurably affect the predictions, but only complicate the analysis". Sorry I do not see well what kind of example you are thinking about here.

On the contrary, I see in most cases that such "correction terms" you mention, such as "correcting" classical mechanics by Special Relativity and then General Relativity, indeed complicate the work of numerical computation of results with "additional terms" from the viewpoint of numerical analysis, however the corresponding theoretical picture is simplified instead. What they actually reflect is a more unified, simple and elegant theory. They are not arbitrarily added for complications, but they come as more or less theoretical necessities. Indeed I explained in my web site how, for example, Special Relativity is simpler as a theory than Galilean space-time. Consequently, Relativistic mechanics is also simpler than classical mechanics, as it comes from a simple principle (the least action principle, more elegantly applied to the space-time of Minkowski than it was to the Galilean space-time) and unifies all conserved quantities (mass, energy, momentum, angular momentum, center of mass) in a unique mathematical object (an antisymmetric tensor in the 5-dimensional vector space associated with the 4-dimensional affine space-time). General Relativity is very elegant too, should I develop this point ?

"every one of the famous equations we use is merely the simplest of a bundle of possible forms of the laws". Please list 10 possible non-equivalent theories of speed and movement that behave approximately the same in many practical cases of experiments, and among which Galilean space-time and Special relativity are non-remarkable particular possibilities. If for any reason you do not like this example, please do a similar thing for other problems such as electromagnetism, gravitation, quantum physics or whatever. You say the only advantage of admitted versions is their simplicity just because it is convenient for us ? But how to explain that in so many cases of theories, among all possible alternatives, there happens to be one that is both extremely simpler than any alternative that can be thought of, and extremely well-verified by observation, with no need of correction by any alternative (no arbitrary complication that we may naturally think of for the sake of complication rather than for the sake of elegance, ever turns out to be better verified, as far as I know) ? Or do you claim this is not the case ?

So I'm sorry but this is bullshit : "Often we assert that the right one is the simplest, evoking a necessarily mystical faith in “the simplicity of nature.” The problem is that it never turns out to be the case that the simplest version of a law is the right one". First, we do not assert by faith that the right one is the simplest we think of. Instead, we conclude it as we verified it by observation. Second, when we had a seemingly simple equation which worked (such as Newton's law of gravitation), the new one that turns out to be more correct to replace it (General Relativity), turns out to be conceptually simpler (more elegant) than the first one; only it was not thought of at first because it is a more subtle, sublime kind of mathematics that requires some familiarity with high mathematics to be grasped. Finally thus, it remains true that the right one is the simplest, except only that we did not know at first the theory which turned out to be both simplest and better verified.

You gave another example : "Maxwell’s equations received corrections that describe light scattering from light-a quantum effect that could have been modelled-but never anticipated-by Maxwell". This example is supposed to illustrate your claim of possible complications and "under-determination" of laws among multiple possibilities. It doesn't. The truth is that these corrections by light scattering from light are not an option among alternative possibilities, but a logically necessary consequence of inserting electromagnetism in the framework of quantum field theory. Of course Maxwell could never anticipate it because quantum theory was not known at that time, but this impossibility to anticipate it before the birth of quantum physics is completely irrelevant here. It does not change the fact that this effect is a necessary consequence of quantum physics. This quantum physics had to be introduced for very different (and necessary) reasons than looking for corrections to electromagnetism. There is no logical possibility for this "correction" of electromagnetism to be not there with its exact necessary amplitude as soon as we live in a quantum world with all its other, more direct consequences (such as the stability of atoms). There is no trace of any "radical under-determinacy" here. To take a related example, consider the measure of the anomalous magnetic moment of the electron, where the calculation as logically determined from theory was verified by observation to an amazing degree of accuracy. We did not need to adjust anything in the theory to put it in agreement with this observation.

To complete my criticism, in reply to the last 2 pages of the essay, while I replied to the previous pages earlier (see my previous replies above): why I see this essay a rather laughable illusion of argument for naturalism, not worth being taken seriously by any scientifically educated person, at the antipodes of the above expressed beliefs by some who lazily enjoy the claim that arguments for naturalism are given, as, just like in religious apologetics, they love to dream in the existence of arguments to validate their belief, but are too lazy or incompetent to think logically about which argument can be actually valid. They dream it would be able to convince some platonists ? Of course it cannot. It can only convince those who are already convinced.
"There are four of these core concepts: number, geometry, algebra and logic."
This description looks as if there was nothing more interesting in the maths of theoretical physics, than school-level mathematics. As if the school-level concepts already gave the essence of all the main mathematical ideas needed in physics. They don't. Very far from it. Just the fact that some of the high-level maths used in theoretical physics (tensors, spinors) can be called "algebra", and that gauge theories can be called "geometry", does not mean that they are as boring as school math. And Fourier transforms, which are essential to quantum physics, clearly do not enter these school-level categories.

Finally, this "argument" is here to be praised and high rated by the public, for the precise reason I gave in my review of this contest: "Obscurantism = Deny the amazing efficiency of mathematics observed in physics; stay ignorant about it. Such people usually hate mathematics because they cannot understand it, so they need pseudo-arguments to feel proud of their ignorance."

This way of pretending that theoretical physics is just as boring and conceptually down-to-earth as school math, so as to make ignorant people feel proud and sufficient of the boring little school math which is all they know, can be a good way to be popular indeed. But it is just an expression of ignorance (may it be true ignorance or pretense of it, doing as if the wonderful stuff of theoretical physics was not there). To see how wrong is this view, see the section "Arguments for Mathematical Platonism" of my review, and the 4 essays I referenced there, which develop the observation of how amazing is the mathematical understanding of physics.

Now the last page : "we still have to explain why mathematics is so effective in physics. It will be sufficient to..." (just blindly pretend that there is nothing remarkable about the effectiveness of maths in physics). Well, just like so many other naturalist essays, the main idea there is to believe that the connection between maths and physics is best explained by pretending that it does not exist, i.e. that there is nothing remarkable about it, that it is nothing else than an illusory impression from what would just be the remarkable efficiency of the naturally evolved human brain to understand mathematics, together with the fact that it should be possible to mathematically analyze anything that happens because finding mathematical structures in anything is what the scientific activity is about, and the reader is not supposed to have any imagination to figure out anything else than this which the remarkable connection between maths and physics might be about. Well, if that was all what the connection between maths and physics was about, why would anyone have come to declare amazement at this connection in the first place ? It would have been simply stupid to do so.

Now the closing "examples". When it was first announced on page 1 that "There indeed may be properties enjoyed by physical reality which have no counterpart in mathematics. I will mention two below", I expected (not paying attention to the restrictive "may be") that the examples would come to make a point showing that such things actually exist, giving good reasons to see physical reality and maths as different. I expected these to be scientifically well-founded, such as reports of scientifically well-established facts. I had one particular example in mind, which I expected to be given in the list : the wave-function collapse, that is found physically real but does not admit any coherent mathematical description.

But it turns out that the given 2 examples of differences between mathematical and physical reality are very disappointing. They are not reports of any scientifically well-established facts. They are only examples of the author's fanciful assumptions introduced earlier in the essay. And not only this, but they are purely metaphysical assumptions, where by "metaphysical" I mean what logical positivism (which is the usually good scientific methodology) dismisses as senseless : it is neither logically well-structured, nor intended as a reference to any possible observational verification.

In reply to the first example "In the real universe it is always some present moment, which is one of a succession of moments. Properties off mathematical objects, once evoked, are true independent of time.": in the details of the sentence, the comparison is unfair between the "real universe" and "mathematical objects", as the difference that is presented does not come from the difference between reality and mathematics, but between a universe and an object inside it. If we reverse the correspondence, comparing between a mathematical universe and a physical object, the stated difference remains between a universe and an object, no matter which one is mathematical.

Indeed, in my study of the detailed properties of the foundations of maths, I showed that the universe of set theory is not fixed but expands in time. During this expansion, its properties never stop evolving, as established by the truth undefinability theorem.

As for "In the real universe it is always some present moment", it still begs for a specification of the mathematical shape of the present moment : in which direction does it slice space-time ? What determines the choice of this direction ? Does it span the whole universe ? I have the same "problem" with my own interpretation of quantum physics, except that I clearly admit that the real answer is in a metaphysical reality that escapes the laws of physics. And finally, as I asked earlier : how thick is the slice of the present ? do events vanish into non-existence as soon as they are past, only remaining temporarily real in the form of a destructible memory ? In my view they don't (the past reality keeps eternally existing as a past reality).

"The universe exists apart from being evoked by the human imagination, while mathematical objects do not exist before and apart from being evoked by human imagination." Did the universe exist before the Big Bang occurred ?

Now coming back to my wonder, of : why did he not give the example of the wave-function collapse as a difference between reality and mathematics ? Well, it may be because his work on the foundations of quantum physics is precisely about believing hard in the possibility, and actively searching for, a mathematical description of the wave-function collapse. Since, no matter the pretense to believe in the metaphysical or any conceptual differences between maths and physics, the fact is, in which sense can anyone conceive of a naturalistic explanation of the wave-function collapse (or generally, any naturalistic law of physics), if not in the sense that it is expressible as a deterministic law ? Which, of course... ultimately has to take the form of a mathematical equation in order for it to be a deterministic law at all (no matter his insistence, in some other articles, on the difference between linearity and non-linearity : this does not constitute any essential difference in the sense of the fundamental difference between mathematical and non-mathematical laws or realities).
More observations with references, of why Smolin and others are crackpot.

Back to: Set Theory and foundations of mathematics - Foundations of physics