In
his book,

__, Lee Smolin puts the case against string theory when he says that__*The Trouble With Physics*
“we must conclude
that string theory has nothing to do with nature, because every
single one of [the theories that are known to exist] disagrees with
experimental data”.

It must also be said that elsewhere Smolin concedes that string
theory does indeed have its own predictions. However, “[t]he few
clean predictions it does make have already been made by other
well-accepted theories”. Not only that. When string theorists say
that there are “extra dimensions” and that “all forces are
united into one force, and that there is super-symmetry”, Smolin
also states that all these “are independent of string theory”.
Thus “finding evidence for any one of them does not prove that
string theory is true”. Nonetheless,

“[i]f we learn
that there is no super-symmetry or no higher dimensions or no
unification of all the forces, then string theory is false”.

All
these factors – one must suppose - are within the domain of the
evidential, experimental and predictive.
Thus they run free of the unique claims of string theory. In other
words, when string theory offers us something new or original, it's
that which disagrees with experimental data.

More
specifically and on the subject of this piece, Smolin explicitly
comments on the

*maths/reality*connection when discussing Albert Einstein and the German physicist, Hermann Weyl. He quotes Einstein's response to one of Weyl's theories:
“'Apart from the
[lack of] agreement with reality, it is in any case a superb
intellectual performance.'”

This
clearly shows that there can often be (at least in principle) a complete disjunction
between mathematics (or mathematical physics) and reality. This isn't a
surprise because that link has often been either underplayed or
completely rejected (by mathematicians and philosophers) throughout
history. The quote above also stresses the mathematical

*virtuosity-for-its-own-sake*problem which has particularly beset string theory.
Smolin
again puts the case for this maths/reality disjunction
(this time with direct reference to string theory) in the following:

“The feeling was
that there could be only one consistent theory that unified all of
physics, and since string theory appeared to do that,

*it had to be right*. That was the stuff of Galileo. Mathematics now sufficed to explore the laws of nature. We had entered the period of postmodern physics.”*Prima facie,*it can also be inferred that not only can there be a maths/reality disjunction - there can also be a

*unified theory/reality*disjunction too. In other words, unification – or at least the intellectual

*desire*for unification – might sometimes have run completely free of reality. Particularly this may still be the case if reality is messy, random, chaotic, non-symmetrical, ultra-complex, dynamical, etc. In other words, wouldn't a unified theory be difficult – even impossible – if the world were truly messy?

Even
before string theory, many writers have commented on the physicists'
“rage
for order”.

__This is Kitty Ferguson__on this theme:
“The diamond
shapes in a sunflower seed-head were lop-sided. One had to give
tree-trunks the benefit of the doubt in most cases to call them
cylinders. The earth bulges and is not a perfect sphere. As for
mirror symmetry, one side of the human face is not the true mirror
image of the other.”

She
goes on to say:

“The things we
build and the art we create exhibit much more geometry and symmetry
than we can find in nature. Are we bettering nature, imposing
rationality on a less rational universe....?”

Of
course none of Ferguson's examples above are from the world of
physics; though physics is of course indirectly connected to every
example she cites. It can also be said,

*prima facie*, that symmetry and order are perhaps more likely in the world of physics than in the world of “medium-sized dry goods” (to use J.L. Austin's phrase). Symmetry (even “supersymmetry”), particularly, is a very important aspect of string theory.
However,
physics doesn't fair very well either. As Smolin himself points out
in the following:

“... we had
learned that nature lacked a certain symmetry – that of parity
between left and right. Specifically, all neutrinos are what is
referred to as left-handed (that is, the direction of their spin is
always opposite to that of their linear momentum). This means that if
you look at the world in a mirror [as mentioned by Ferguson above] ,
you will see a false world – one in which neutrinos are
right-handed. So the world seen in a mirror is not a possible world.”

Thus
can't that “bettering [of] nature” fixation also be applied to string
theory and even other theories in physics? Is string theory
attempting to “exhibit more geometry and symmetry than we can find
in nature”? Is string theory also “imposing rationality on a less
rational universe”?

This

*thirst for order*appeals to mathematicians. It also appeals to mathematical physicists. And it especially appeals to string theorists.
It's
also the case that string theory satisfies (or at least attempts to)
the

*mathematical*side of mathematical physics. Some experts think it does so with flying colours. Indeed string theory's mathematical virtuosity and dexterity is what appeals to many people – especially those experts who have less knowledge of physics than they have of mathematics.
So
it can be concluded that maths can run free of physical reality. Or
at least there has been a long debate about this. On the one hand,
philosophers and mathematicians - dating back to the ancient Greeks -
have stressed the tight link between maths and physical reality.
Other philosophers and mathematicians, on the other hand, have denied
the importance - or even relevance - of that link.

**Maths and Reality Disjoined**

It's
easy to see how the maths of string theory can run free of reality.
However, how would non-experts know if that were the case? It may
also be the case that string theorists have no sure (i.e.,
philosophical or non-mathematical) mechanisms which can show them
that they're disappearing up their own mathematical backsides.

As
Smolin puts it, the maths of a particular string theorist may simply
be his personal “toy”. He tells us that such a person (though in
this instance he's not talking about a string theorist) “may simply
have invented a theoretical toy that has nothing to do with nature”.

The
string theorist will of course say that his theory

*does indeed have something “to do with nature”*. How would we know that this is the case? The string theorist will also say that his mathematical theories describe reality or – at the very least - the way that reality*may be*. Again, how would we know? We'd need to be in Sherlock Holmes's situation in which we literally have to go through the entirety of the string theorist's mathematics and thus replicate his own labour and results. To re-quote William Poundstone's passage (already quoted in Part One of this piece):
“... William
Shanks, a mathematician of our fair island, has recently computed

*pi*to 707 decimal places. It took him twenty years. His result filled a whole page with quite senseless, random numbers. Should anyone doubt Mr. Shank's result, he would have to budget an equal amount of time and duplicate his work. In that case also, verifying the answer would be precisely as difficult as coming up with the answer in the first place – the very antithesis of an 'obvious solution.'....”
In
terms of the dearth
of experiments, observations and predictions in string theory, that's
presumably why some string theories leave us in a position in which
“there is no evidence either way”. And if there's no evidence
either way, then, again, where does that leave us? Of course a string
theorist may hint at evidence. He may even suggest actual or possible
experiments. (That is, he may say that in

*such-and-such*a situation one would observe*this-and-that*.)
Smolin
doesn't seem to have a problem with this kind of talk about
hypotheticals/conditionals (at least when talking about “doubly special relativity”). When referring to such a theory, Smolin
writes:

“The idea is an
elegant one.... We don't know whether it describes nature, but we
know enough about it to know that it could do.”

Could
we say the same about string theory – or about a particular
statement within string theory?

Thus let's
take it as given that a particular string theory is “elegant”. In
that case, we still wouldn't know that “it describes nature”.
Would we know that “it could do”? We can conclude by saying
that there's one thing which is the case:

*mathematical elegance alone will never be enough.***Does Maths Correspond With Reality?**

Smolin
recalls a conversation he had with João Magueijo (the Portuguese
cosmologist and theoretical physicist) in which he seems to say that
there should never be an absolute disjunction
between maths and reality. Smolin writes:

“João said that
everything to do with physics could be understood without the fancy
mathematics. Giovanni argued that it was easy to talk nonsense about
these theories if you weren't careful about which mathematical
expressions corresponded to things that could be measured.”

This
depends on what that

*understanding*of physics amounts to. It can be argued that there is a*kind of*understanding without the maths; though not a*full*understanding. (This is parallel to the use of images, metaphors and analogies in quantum mechanics.)
Nonetheless,
the link between “mathematical expressions” and the demand that
they “correspond to things that could be measured” seems to be
very strong. So is that meant in the same sense that each word in a
natural-language sentence must correspond with a

*thing*? In terms of language, this is obviously problematic from a philosophical point of view. Some words don't correspond with*things*. Indeed some words don't even correspond with*anything*– not even with abstract entities. In addition, a word gets it meaning (or role) from its context within a sentence (or statement). Perhaps this is also true of the mathematical expressions in a mathematical theory (or statement). Again (as with a sentence), why should*all*mathematical expressions correspond to anything at all? The mathematical expressions in*pure maths*don't need to correspond with anything (which is a point that Wittgenstein and the constructivists often stressed). However, surely the same can't be be said of the maths in mathematical physics or string theory.
The
other point (in reference to the quote above about “fancy
mathematics”) is that even if mathematical expression

*x*corresponds to thing*y*, and then that thing*y*is “measured”, then isn't measuring itself a mathematical process or function?
The
astrophysicist and science

**writer, John Gribbin, also seems to put a similar point. He does so, however, in terms of what he calls a “physical model” of “mathematical concepts”. He says (in his**__) that “a strong operational axiom” tells us that__*Schrodinger's Kittens and the Search for Reality*
“literally every
version of mathematical concepts has a physical model somewhere, and
the clever physicist should be advised to deliberately and routinely
seek out, as part of his activity, physical models of already
discovered mathematical structures”.

Yet even in Gribbin's case, it's still clear that a “mathematical
concept” comes first and only then is a “physical model” found
to square with it.

**Many Theories about**

**x**
One
problem with the possible divorce between maths and reality is that
many theories – perhaps an infinite amount of them – can be
constructed precisely because of that freedom. More specifically,
many theories can be formulated to deal with the very same evidence,
data or experiment. This is certainly true of string theory (or
string theories in the plural).

There
are many problem with this multiplicity of theories and their
possible parallel lack of
a connection to reality. Smolin tells us (though specifically about
“unified-field theories”) that “[r]ather
than being hard to find, unified-field theories were a dime a dozen”.
What's more, “[t]here were many different ways to achieve them and
no reason to choose one over another”.

Nonetheless,
there may not be a huge problem with many theories fighting for the
same space. It may even be the case that all these rival theories are
in fact “complementary” (or at least some may be). Why assume
that the different theories about the same

*x*must mean that not all these theories can be useful? Indeed why assume that only one can be true or correct? Moreover, why should a multiplicity of theories about a given*x*hint at a break from reality? Why not be a pluralist about such theories?
John
Gribbin (again) is such a pluralist (as are many scientists and
philosophers). Gribbin, in the following, specifically talks about
quantum theories:

“I stress, again,
that

*all*such interpretations are myths, crutches to help us imagine what is going on at the quantum level and to make testable predictions. They are not, any of them, uniquely 'the truth'; rather, they are*all*'real', even where they disagree with one another.”
This
is heavy stuff. Of course what's true of quantum
theories/interpretations (or “myths”) may not be true when it
comes to mathematical theories or theories in other disciplines.
However, the above is certainly problematic for string theory for the
simple reason that Gribbin talks about such theories/interpretations
making “testable predictions”.

**Philosophy of Mathematical String Theory**

One
can of course say that the maths of string theory must have a
connection to reality for simple Aristotelian reasons. As Aristotle
himself put it (in
his

*Metaphysics)*when talking about the Pythagoreans:
“All is number.
The principles of mathematics are the principles of all things.”

In
other other words, if the principles, theories or statements of
mathematical string theory are true/correct/accurate; then, by
Aristotelian definition, they must match “all things” -

*they must match reality*.
There's
a distinction, however, to be made here.

It
can be said - in loyalty to Aristotle - that the maths in string
theory can't contradict anything in reality and reality can't
contradict anything in mathematical string theory. But does that
also mean that the maths of string theory must also

*describe*or*explain*given aspects of reality - or any part of reality? Nothing in string theory may contradict the “principles of all things”; yet a given string theory may not have anything to do with any aspect of reality.
In
that sense, what's true of maths generally is passed over to
mathematical string theory. That is, if the maths in string theory is
true/correct/accurate, then nothing in it can contradict reality.
Still, string theory may not describe (or explain) any aspect of
reality. More specifically, it may not describe such things as extra
dimensions,

**branes,****multiverses and even strings simply because such things don't exist in reality.**
We
can also bring in Galileo here. Galileo famously claimed that “the
book of nature
is written in the language of mathematics”. Thus, again, if the
maths of string theory is true/correct/accurate; then – by
definition - it can't contradict anything in nature or reality.
Still, it may not describe (or explain) anything in nature or reality either.

We
can go further here and say that a “false
dichotomy”
has been set up with this talk of

*maths or reality*. It may even be the case that*maths and reality*is a false conjunction.
What
I mean by all this is that many people
say that “physics is maths”. (Though not that “maths is
physics”.) Therefore can't we also say that reality/nature is maths
(along with with Galileo)? And if that's true, if the maths in string
theory is true/correct/accurate, then it simply must be about
reality. Though, again, this distinction must be made here:

i) String theory does not
contradict reality.

ii) Does string theory actually describe or explain reality?

In
criticising what can be seen as string theory's over-dependence and
over-reliance on maths, we also face another problem.

Physics
is nearly always utterly dependent on mathematics. The science
writer, John Horgan (in his

__), puts it this way:__*The End Of Science*
“Numerical models
work better in some cases than in others. They work particularly well
in astronomy and particle physics, because the relevant objects and
forces conform to their mathematical definitions so precisely.”

Horgan
goes further than that. He says that many of the entities, fields and
forces
of
physics are entirely mathematical in
nature.

Firstly
he tells us that “mathematics helps physicists definite what is
otherwise undefinable”. Then he cites

__an example__:
“A quark is a
purely mathematical construct. It has no meaning apart from its
mathematical definition. The properties of quarks – charm, colour,
strangeness – are mathematical properties that have no analogue in
the macroscopic world we inhabit.”

This
isn't necessarily to say that quarks are nothing other than maths or
the numbers which express their nature. (Otherwise why use the words
“their nature”?) It's to say that we couldn't say much – or
indeed anything – about quarks without the requisite mathematics.

This
may also mean that mathematical physics not only describes nature: it
also – in some sense - creates it. This may also mean that the
entities, strings, dimensions, etc. which string theorists postulate
may make much more sense than we think. In other words, if we've no
grasp of reality that's separable from the maths of mathematical
physics, then how can we even so much as question what string
theorists are saying? As with quantum mechanics, we may have no
independent ground to stand on.

Thus
surely we can follow on from all that and question if there really is
something beyond the mathematical formulations which express the
nature of strings and quarks. In other words, what's left after we
take the maths away?

*Nothing*or*just a little something*“we know not what”? What can be said without the maths? And what if what's said without the maths is highly misleading in that's it's purely analogical, metaphorical or imagistic in nature? Indeed perhaps we would be better off – in both string theory and quantum mechanics - without the analogies, images and picture-painting.**A Digression: Wilder on Constructivist Maths**

We
can compare string theory with mathematical constructivism in the
sense that the latter is deemed to be unchained from reality and the
former may well be too.

The
constructivist theory of mathematics (which includes mathematical

*intuitionism*) has it that rather than maths being chained to reality, it is actually free to journey wherever it likes. (This parallels - at least to some extent - “free logic”.) That's because mathematics is said - by constructivists - to be a human construction and thus each “mathematical concept” is an individual mental construction.
The
American mathematician and anthropologist, R.L. Wilder (1896
to 1982),
saw this mathematical freedom in terms of developments which came to
fruition in the 19th century. In his book,

__, he wrote:__*Introduction to the Foundations of Mathematics*
“Following the
19th century developments, the mathematical world came to feel that
it was no longer restrained by the world of reality, but that it
could create mathematical concepts without the restrictions that
might be imposed by either the world of experience or an ideal world
to whose nature one was committed to limited discoveries. ”

What
we have here is the ancient battle between what we can call
“empirical maths” (which can, of course, be seen as an oxymoron) and “Platonic maths”.

In
the Platonic scheme, to quote Wilder again, mathematics was seen as
“a description of an ideal world of concepts existing over and
above the so-called real world”.

^{1}
The
question is whether or not the maths in string theory is also (at
least partly) Platonic or constructivist (if only by
default)
in nature. (The large distinctions which can be made between
mathematical constructivism and mathematical Platonism don't matter
in this precise context.)

Wilder's
own position is very different from both Platonism and
constructivism. He offered us what may be called a

*conventionalist*- or even an anthropological/“materialist” - view of mathematics. He wrote:
“Mathematics
derives its concepts initially from the existing world of reality and
uses them as a way of dealing with this reality…”

The
traditional view was that it's indeed the case that mathematics can
be used “as a way of dealing with” reality (though not
necessarily that it also “derived its concepts… from the existing
world of reality”). In the case of string theory, the primary
question here is about its descriptions or explanations of reality; not whether maths of string theory is
“derived” from reality. However, if mathematical concepts are
derived from reality (if only initially), then it's not surprising
that they can also be applied to - or deal with - that reality.

Wilder
then explicitly puts the anti-Platonic or constructivist position. He
says that the concepts of mathematics

“were [in the 19th
century onward] no longer embodiment of an independently existing
realms of ideas, having an existence before and after the fact of
their discovery, but only of a world of concepts continually under
construction and having no existence until conceived in the minds of
the mathematicians who created them”.

More
relevantly, we can rewrite the passage above in the following manner:

Mathematics is no
longer the embodiment of an independently existing realms of ideas,
having an existence before and after the fact of their discovery, but
only of a world of concepts continually under construction and having
no existence until conceived in the minds of the mathematicians who
created them.

The
above only stresses the case of mathematical string theory being
opposed to mathematical Platonism (with its abstract world full of
abstract Forms). String theory may also run free of concrete reality
too. Again, to rewrite another passage from Wilder above:

String theorists
came to feel that string theory was no longer restrained by the world
of reality, but that it could create mathematical concepts without
the restrictions that might be imposed by the world of experience.

*************************

**Note**

**1**This freedom from “the world of experience” (or from physical reality) may make one think in terms of a Platonic conception of mathematics. However, if one is a Platonist, one must be equally committed to Plato’s “ideal world” which would inevitably “limit [one’s] discoveries” (to use phrases from Wilder). It's no wonder that philosophers have said that Platonic conception of mathematics is also implicitly - or even explicitly - committed to a correspondence theory of both numbers and equations. That is, mathematicians must be both committed to - and make their numbers and equations correspond with - the abstract mathematical entities in Plato’s abstract world. As one can see, this is just as much of a limitation as making one’s mathematics abide by the dictates of experience or the physical reality. It was no wonder, then, that some mathematicians rejected the infinite, never mind Georg Cantor’s infinite infinities. They did so because they believed that there are no actual infinities in the physical world or in the world of experience... at least not until 20th century physics.

***) See my 'The Scientific Problem with Panpsychism & String Theory (With Lee Smolin) (1)'.**

**To follow: 'Experiments and Predictions', 'What is a Theory?', 'The Aesthetics of Theory-Choice'.**

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