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 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.]
[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.
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.
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 = x2 describing the parabola and x2 + y2 = 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.
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.