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.
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.
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.
Set Theory and foundations of
mathematics - Foundations of physics