The
first thing to say is that the claim that “the
universe is mathematical” hardly make sense at a

*prima facie*level. It's not even that it's true or false. So this must surely mean that it's all about how we interpret such a claim.
Despite
saying that, sometimes it's hard to express (or even understand)
precisely what Max Tegmark's actual position is. Can we say that
reality (or the world)

*is*mathematics or mathematical (as in the “is of identity”)? That reality is*made up of*numbers or equations? That reality*instantiates*maths, numbers or equations? Or should we settle for Tegmark's own very radical words? -
“The
Mathematical Universe Hypothesis... at the bottom level, reality is a
mathematical structure, so its parts have no intrinsic properties at
all! In other words, the Mathematical Universe Hypothesis implies
that we live in a

[See later comments on Carlos Rovelli, Lee Smolin and relational theory.]

*relational reality*, in the sense that the properties of the world around us stem not from properties of its ultimate building blocks, but from the relations between these building blocks.”[See later comments on Carlos Rovelli, Lee Smolin and relational theory.]

To
put the case formally and as clearly as possible: Tegmark believes
that physical “existence” and mathematical existence are “

*one the same”*(which is a phrase he often uses)*– they equal one another. More specifically, Tegmark stresses “structures”. Thus if we have a mathematical structure, it must exist physically as well. Or, more strongly, all mathematical structures exist physically.***The Unreasonable Effectiveness of Mathematics**

Tegmark
mentions Eugene
Wigner a couple of times in his book and he's clearly inspired by
his well-known question.

Wigner
once
wrote:

“The
miracle of the appropriateness of the language of mathematics for the
formulation of the laws of physics is a wonderful gift which we
neither understand nor deserve.”

As
for the “miracle” of mathematics and its applications:

“It
is difficult to avoid the impression that a miracle confronts us
here, quite comparable... to the two miracles of laws of nature and
of the human mind's capacity to divine them.”

And
then we have the question which Tegmark quotes
a couple of times
in his book:

“The enormous usefulness of mathematics in the natural sciences is something
bordering on the mysterious and... there is no rational explanation
for it.”

Albert
Einstein also asked the same question

**in the following:**
“How
can it be that mathematics, being after all a product of human
thought which is independent of experience, is so admirably
appropriate to the objects of reality?”

However,
Einstein's
conclusion appears to be radically at odds with both Wigner's and
Tegmark's:

“[...]
In my opinion the answer to this question is, briefly, this: As far
as the laws of mathematics refer to reality, they are not certain;
and as far as they are certain, they do not refer to reality.”

So
Does the “unreasonable effectiveness” of electricity or roads
also “demand an explanation”? That ironic question is asked
because there are indeed explanations of maths effectiveness.
However, I feel that they won't satisfy Max Tegmark (or Eugene Wigner for
that matter).

**The Mathematical Universe Hypothesis**

The
drift of Tegmark's central position is that the only way we can
justify our belief in a mind-independent reality is to accept that it
is mathematical. As it stands, of course, that almost seems like a

*non sequitur*. Tegmark does provide some argument for this position; though very little. Indeed only a small part of his book (*Our**Mathematical Universe: My Quest For the Ultimate Nature of Reality*) is devoted to this central thesis.
Perhaps
Tegmark is not alone in his position. Take
this very Tegmarkian utterance from the physicist Brian
Greene in which he deprecates what Tegmark calls
“baggage”:

"The
deepest description of the universe should not require concepts whose
meaning relies on human experience or interpretation. Reality
transcends our existence and so shouldn't, in any fundamental way,
depend on ideas of our making."

Interestingly
enough, Tegmark's juxtaposition of mathematical realism with metaphysical realism was preempted

**by Hilary Putnam in 1975. Putnam spoke of Wigner's “two miracles”: i.e., the power of mathematical descriptions of the world and the mind's “capacity to divine them”. Putnam concluded that in order to be a metaphysical realist and a believer in mind-independence, one must see the world as mathematical. (This is part of Putnam's “indispensibility argument”, which is not strictly platonic in any way.)**
Tegmark's
position on the mind-independence of reality is different to most
positions advanced by metaphysical realists. Reality is not
mind-independent in the metaphysical realist's sense. It's
independent of human beings (or minds) simply because it's an
abstract mathematical structure. Thus this has little to do with
whether reality is observed; the way it's observed; its verification;
etc. Reality is independent of human beings even if (or when) humans
observe it.

So
let me sum up that in a basic argument:

i)
Mathematics is

*mind-independent*.
ii)
All non-mathematical descriptions of reality are

*mind-dependent*.
iii)
Therefore in order to achieve a true mind-independent description
of reality, one must only use mathematics (or mathematical
structures) to do so.

One
part of Tegmark's argument is that if a mathematical structure is identical (or “equivalent”) to the physical structure it
“models”, then they're

*one and the same thing*. Thus if that's the case (that structure*x*and structure*y*are identical), then it makes little sense to say that x “models” - or is “isomorphic” with -*y*. That is,*x*can't model*y*if*x*and*y*are one and the same thing.
Tegmark
applies what he deems to be true about the isomorphism of two
mathematical structures to the isomorphism between a mathematical
structure and a physical structure. He gives an explicit example:

electric-field
strength = a mathematical structure

Or
in Tegmark's
words:

“'
[If] [t]his electricity-field strength here in physical space
corresponds to this number in the mathematical structure for example,
then our external physical reality meets the definition of being a
mathematical structure – indeed, that same mathematical structure.”

In
any case, if

*x*(a mathematical structure) and*y*(a physical structure) are one and the same thing, then one needs to know how they can have any kind of relation to one another. This truism displays the problem:
(

*x*=*y*) ⊃ (*x*=*x*) & (*y*=*y*)
In
terms of Leibniz's law, that must also mean that everything true of

*x*must also be true of*y*. But can we observe, taste, kick, etc. mathematical structures?
In
addition, can't two structures be identical (if not numerically identical) and yet separate?

All
this is perhaps not the case when it comes to mathematical structures
being compared to other mathematical structures (rather than
something physical). Yet if the physical structure

*is*a mathematical structure, then that qualification doesn't seem to work.
All
this is also problematic in the sense that if we use mathematics to
describe the world, and maths and the world are the same, then we're
essentially either using maths to describe maths or the world to
describe the world.

In
addition, Tegmark's Mathematical Universe Hypothesis (MUH) is
certainly played down by Israel Gelfand when he
writes:

“There
is only one thing which is more unreasonable than the unreasonable
effectiveness of mathematics in physics, and this is the unreasonable
ineffectiveness of mathematics in biology.”

In
the end, Tegmark is telling us what physicists have always believed
about the importance of mathematics when it comes to describing the
world. However, he's adding the perhaps unjustified conclusion that
both “structures” are one and the same thing.

**Platonism/Pythagoreanism**

It's
easy to see that Tegmark's position is pythagorean or platonist in
that it stresses mathematical entities. However, it is platonist (at
the least) in a much stronger sense in that it states that only
mathematical objects or structures exist. (Whether Plato held that
position is for others to decide.) As a consequence of this, Tegmark's
position must also be a form of monism in that literally everything
is deemed to be mathematical. (Just as former monists believed that reality - or
the world - was all “spirit”,
matter, “neutral” or God.)

Tegmark
is at his most platonic and radical in the following passage:

“...
a complete description [of external physical reality] must be devoid
of any human baggage. This means that it must contain no concepts at
all! In other words, it must be a purely mathematical theory, with no
explanations or 'postulates'...”

Yet
how can one even construct a single thought without “no concepts at
all”?

Thus
when Tegmark talks about mathematics, it's clear that he's stressing
that it is in no way “human”. That it's not what he calls
“baggage”. Yet even if
mathematics is an abstract Platonic realm, it is still human beings
that gain access to it. It's still human beings that give such
abstract objects and equations names or symbols. It's still human beings that make use of these abstract entities. And it's still human
beings who may be getting it all wrong.

In
terms of detail, we can ask if Tegmark is correct to say that time
and space are “purely physical objects”. He says the same about
“curvature” and particles.

Thus
when Tegmark says that a dodecahdron
was never created,
he's saying what Plato might have said. Similarly, when he says that
dodecahedron doesn't exist in space or time at all, that too is pure
Plato. To quote
the

*New Scientist*:
“A
dodecahdron was never created, says Max Tegmark of the Massachusets
Institute of Technology. A dodecahedron does not exist in space or
time at all - it exists independently of both. Space and time
themselves are contained within large mathematical structures, he
adds. These structures just exist; they cannot be created or
destroyed.”

Of
course Tegmark adds to Plato when he also argues that such things as “space and time themselves are contained within large mathematical
structures”. The purely platonic statements are easy to grasp (at
least because they've been on the table for two thousand years);
though Tegmark's addition of space and time to the Platonic world is
(as it were) harder to grasp.

Tegmark
also

**says that “the rectangular shape of this book [doesn't] count” when it comes to “geometrical patterns such as circles and triangles” and their being “mathematical”. So why don't “human-made designs” count? In addition, is it really the case that the “trajectory” which results from our “throwing a pebble [and] the beautiful shape [a parabola] that nature [then] makes” is more precise, symmetrical and exact than anything we human beings can construct, as Tegmark claims?**
This
reminds me of problem the mathematician Richard W. Hamming notes when he
writes the following:

“We
select the kind of mathematics to use. Mathematics does not always
work. When we found that scalars did not work for forces, we invented
a new mathematics, vectors. And going further we have invented
tensors... Thus my second explanation is that we select the
mathematics to fit the situation, and it is simply not true that the
same mathematics works every place.”

Thus
non-platonic geometrical patterns are deselected according to
Tegmark's platonic vision. That is, they don't fit the mathematics.
Platonic shapes, patterns, etc., on the other hand, are “select[ed]”
instead and other mathematics is used to “fit” more convenient
“situation[s]”. In that case, those non-Platonic patterns or
shapes aren't in Tegmark's “reality”. Doesn't that create a
problem for the oneness of mathematics and reality? This also means
that the “mathematics at hand does not always work”. At least it
doesn't (if we follow the logic of Richard Hamming) until a new maths
is utilised.

In
addition, if particles, etc. are “mathematical objects”, then
aren't we using mathematics to describe, plot or explain mathematics?
Is mathematics, therefore, describing itself? Is it the case that we
never get out of the circle of maths? Perhaps Tegmark likes that
idea.

**Platonic Structuralism**

Tegmark's
position seems to be a fusion of (ontic)
structural realism

**and mathematical structuralism. Indeed, in one of his notes, he acknowledges John Worrall's***structural realism*thus:
“In
the philosophy literature, John Worrall has coined the term

*structural**realism*as a compromise position between scientific realism and anti-realism; crudely speaking, stating that the fundamental nature of reality is correctly described only by the mathematical or structural content of scientific theories.”
As
for platonic mathematical structuralism, Tegmark's most clear
exposition of his position is the following passage:

“The
notation used to denote the entities and the relations is irrelevant;
the only properties of [for example] integers are those embodied by
the relations between them. That is, we don't invent mathematical structures – we discover them, and invent only the notation for describing them.”

The
platonic part of the above passage is expressed in the final sentence
about our discovery of abstract entities. (That sentence doesn't
logically follow from the proceeding words.) Tegmark also says that a
structure is “a set of abstract
entities” How can that be? Surely entities (or elements) are
parts of structures. The structure may well be derivative of the
relations of the entities (to each other). However, there's still a
distinction to be made here. A set of abstract entities is, well, a
set, not a structure. Unless Tegmark takes sets and structures to be
one and the same thing.

Tegmark
is also both a mathematical structuralist and a mathematical
platonist. Indeed this position exists in mathematical structuralism
itself and it's opposed to Aristotelian mathematical structuralism;
though it's not necessarily identical to Tegmark's own position.

In
terms of the specific platonist position: mathematical structures are
deemed to both abstract and real. This position is classed as

*ante rem*(“before the thing”) structuralism. The platonist position on structures can be characterised as the position that structures exist before they are instantiated in particular “systems”. The Aristotelian position on structures, on the other hand, has it that they don't exist until they are instantiated in systems.
To
explain the platonist position one can use Stewart
Shapiro's own analogy. In his view, mathematical
structures are akin to offices. Different people can work in a
particular office. When one office worker is sacked or leaves, the
office continues to exist. A new person will/can take his role in the
office. Thus offices are like mathematical structures in that
different objects can take a role within a given structure. What
matters is the structure – not the objects within that structure.

Nonetheless,
the people who work in offices are real. The idea of an office which
is divorced from the people who work in it is, of course, an

*abstraction*. Thus one may wonder why the office/structure is deemed to be more ontologically important than the persons/objects which exist in that office/structure. Surely it should be the other way around.
Tegmark
also seems to fuse his position with that of

*relational theory*. Indeed the former (platonic) mathematical structuralism can be seen as being an example of relationism (or vice versa). For example, Tegmark says that
“[t]o
a modern logician, a mathematical structure is precisely this: a set of
abstract entities with relations between them. [He then cites the
integers and “geometric objects” as examples.]”

And,
on page 267, he sounds very much like Lee
Smolin

**and****Carlo Rovelli when he tells us [as quoted at the beginning] that**
“the
Mathematical Universe Hypothesis implies that we live in a

*relational reality*, in the sense that then properties of the world around us stem not from properties of its ultimate building blocks, but from the relations between these building blocks.”**Tegmark's Examples of the Use of Numbers**

Tegmark
keeps on supplying us with examples of what numbers do and what
functions they serve in physics. Yet he does so without going into
much (or even any) detail as to why the world

*itself*is mathematical. We're told about numbers “representing” letters in computers; how pixels are “represented by numbers”; how the “strengths of the electron field and the quark field relate to the number of electrons and quarks at each time and space”; etc. In more detail, Tegmark says
“[that]
there's a bunch of numbers at each point in spacetime is quite deep,
and I think it's telling us something not merely about our

*description*of reality, but about reality itself”.
Yet
Tegmark appears to contradict himself. At one point he
says that

**a field “is just [ ] something represented by numbers at each point in spacetime”. Here we have both the words “something represented”***and*“is just”. More clearly, we have a field which is “represented by numbers” and “just is” - the latter two words implying that*all we have*is numbers. Again, Tegmark says that the magnetic field is “represented” by “three numbers at each point in spacetime”. Yet he doesn't say that the magnetic field*is*a set of numbers (or even a “structure” which includes numbers).
Tegmark
also cites the example of

*y*=*x*^{2}describing the parabola and*x*^{2}^{ }+*y*^{2}*= 1*describing the circle. Yet aren't these*descriptions*of shapes, trajectories, etc., not of numbers or mathematics? Don't the numbers plot the shapes, trajectories, etc, not*equal*them? Yet that's precisely what Tegmark denies.
In
addition, one would be hard pressed to interpret that idea that
“space and time themselves are contained within large mathematical
structures” without additional information.

Tegmark
then indulges in what amounts to number
mysticism. For example, he asks:

**“[W]hy are there 3 dimensions, rather than 4 or 2 or 42?” He also asks: Why are there “exactly 6 kinds of quarks in our Universe?”. But couldn't we just as easily ask:***Why are there 101 dimensions, rather than 4 or 2 or 42?*Similarly:*Why are there 109 kinds of quarks rather than 6?*
What
do these questions so much as mean? What would acceptable answers (to
Tegmark) so much as look like? True, because there are 6 quarks, then
only certain (physical) consequences are allowed to occur. The same
is true of 3 dimensions. But is Tegmark making another point here?

In
addition, is spin or charge really “just
a number”? Tegmark connects charge and spin number to lepton
number; though the latter seems to belong to a different category.
That is, the number of leptons is a very different thing to the
numbers we assign to spin and charge – even if the numbers are the
same. What we have here (as stated) is number mysticism of the most
crude kind. (This kind of number mysticism can be most clearly seen - in my
view - when it comes to correlating the Fibonacci
sequence with aspects of nature; especially in view of the later
Kitty Ferguson quote. See this titillating documentary here.)

Another
point worth making is that if
numbers can

*plot anything*, that's simply because they*can plot everything*. That is, if mathematics can explain or describe random events, chaotic conditions, or dynamical systems (which it can), then it can also explain or describe just about everything.
What
I mean by this is that it's always said that mathematics is perfect
for describing or explaining the symmetrical, ordered and even
“beautiful” aspects of nature. Yet the science writer Kitty
Ferguson throws a spanner in the works in
the following:

“The diamond shapes in a sunflower seed-head [are] 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. Natural crystals are not perfect geometric shapes either.” As for mirror
symmetry, one side of the human face is not the true mirror image of
the other... At the level of elementary particles, we discover a
right- and left-handedness about the universe, slightly favoring the
left. In the early universe there may have been an infinitesimal imbalance between the amount of matter and the amount of antimatter,
an imbalance which has resulted in the universe of matter we see
today.... If someone or something had taken the symmetries found in
physics and 'corrected' them, we and our universe could not exist.”

Ferguson
then carries on with
her theme:

“The
things we build and the art we 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?

Roger
Penrose also makes the point that mathematical descriptions of the
world are often (or always) more “beautiful” than the world
itself.

And
Richard Hamming

**chimes in with the following statement:**
“Humans
see what they look for... our intellectual apparatus is such that
much of what we see comes from the glasses we put on.”

In
terms of my own take on this. If I were to randomly throw an entire
pack of cards on the floor, then that

*mess-of-cards*could still be given a mathematical description. The disordered parts of that mess would be just as amenable to mathematical description as its (accidental) symmetries.
Similarly,
if I were to improvise “freely” on the piano, all the music I
played could still be given a mathematical description. Both the
chaos and the order would be amenable to a mathematical description
and even a mathematical explanation. Indeed a black dot in the Sahara
desert could be described mathematically; as can probabilistic events
at the quantum level. It's possible that mathematicians can find
different – or contradictory – symmetries in the same phenomenon.

Another
problem here is that many philosophers, mathematicians and physicists
have said that mathematical physics doesn't use all of mathematics to
describe the world. That is, there are large chunks of maths which
seemingly don't apply to the world; or, at the least, at present they
don't fulfill a purpose in mathematical physics. Yet if maths and the
world

*are one*, then why are there (to use Roger Penrose's words) “bodies of maths with no**discernible relations to the physical world”? In terms of Penrose's actual examples:**
“Cantor's
theory of the infinite is one noteworthy example... extraordinary
little of it seems to have relevance to the workings of the physical
world as we know it... The same issue arises in relation to...
Godel's famous incompleteness theorem. Also, there are the
wide-ranging and deep ideas of category theory that have yet seen
rather little connection with physics.”

Here
again, according to Tegmark's position, this split can't be real. If
mathematics and the world

*are one*, then it doesn't make sense to say that there are parts of maths that aren't applicable to the world. Having said that, it still seems acceptable to argue that we can have parts of mathematics that aren't applicable to the world and yet still accept that the world is mathematical.**Conclusion**

Is
mathematics seen (if tacitly) as God's language? Did God write the
“book” which Galileo referred to? Indeed is this assumption implicitly behind much of what Tegmark and others argue?

Of
course the precise relation between mathematics and the world (or reality) has been debated for a long time. As we have seen,
this isn't such a big problem for Tegmark for the simple reason that
he believes that mathematics and the world are

*one and the same thing*.
**********************************