In a simple sense, mathematics may be viewed as an extremely useful tool. The English cosmologist, theoretical physicist and mathematician John D. Barrow, however, holds a slightly different view. That is, he posits the view known as Pythagoreanism, summing up the relationship between mathematics and physics in the following way:

“By translating the actual into the numerical we have found the secret to the structure and workings of the Universe.”

So the Universe and its parts are assigned numbers… Or are described by numbers… Or are captured by numbers… Or are explained by numbers… Or are (to use Barrow’s own words) translated into numbers.

But what does all that actually mean?

Sure, if Pythagoreanism holds at least some water, then it’s no wonder that so many people have also believed that through maths (as Barrow puts it) “we have found the secret to the structure and workings of the Universe”. But even here there’s a non-Pythagorean (as it were) remainder. After all, maths finds the secretof things which already exist — i.e., the “structure and workings” of the world. It isn’t actually being argued that these structures and workings are literally maths. The world is not itself maths.

… Or is it?

To the Pythagorean, the world and its parts are actually mathematical. This means that it isn’t that maths is simply helpful for describing the world — the world itself is mathematical. Indeed one must take this literally. Here’s Barrow again on the Pythagorean position:

“[The Pythagoreans] maintained ‘that things themselves are numbers’ and these numbers were the most basic constituents of reality.”

And Barrow then becomes ever clearer when he continues in the following manner:

“What is peculiar about this view is that it regards numbers as being an immanent property of things; that is, number are ‘in’ things and cannot be separated or distinguished from them in any way.”

Moreover:

“It is not that objects merely posses certain properties which can be described by mathematical formulae. Everything, from the Universe as a whole, to each and every one of its parts, was number through and through.”

It’s hard to grasp what the phrase “things themselves are numbers” even means. Can we say that reality and its parts are mathematics (as in the “is of identity”)? That reality and its parts are literally made up of numbers or equations? That reality and its parts somehow instantiate maths, numbers or equations?

And what does it mean to say that “numbers are in things”? Indeed there’s a problem here. If things are literally numbers, then how can numbers also be “in” things? In other words, how can numbers be in themselves? That would be like saying that cats “are in” cats; or that a dog is in the very same dog.

The Unreasonable Effectiveness of Mathematics

The important distinction here is that this isn’t about the “unreasonable effectiveness of mathematics in the natural sciences”. (In which we note the miraculous fact that maths is often a perfect tool for describing the world and then wonder why that is the case.) This is about (to use Barrow’s words again) “the Universe as a whole, [and]each and every one of its parts [being] number through and through”.

Thus description and being are two very different things.

Another way to describe the maths-world relation is to say that (as Barrow again does) that

“mathematics has proved itself a reliable guide to the world in which we live and of which we are a part”.

Yet here again, saying that mathematics is “ reliable guide to the world” is no more of a Pythagorean statement than saying that maths describes the world.

To be clear. A map of Essex is not actually Essex. The set of rules for chess is not an actual game of chess. (Though each game of chess is always — as it were — a variation on the rules of chess.) So it may well be the case that if we didn’t have a map of Essex, then we’d get lost in that county. Similarly, if we didn’t have maths, then we wouldn’t have a reliable guide to the world.

But what of that “miracle” that is math and its relation to the world?

The Hungarian-American theoretical physicist Eugene Wigner (famously) put it this way:

“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. 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. 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 was similarly perturbed when he wrote:

“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 own conclusion appears to be radically at odds with Wigner’s when he continues with the following words:

“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.”

Despite those sceptical remarks from Einstein, let’s get back to the being I mentioned a while back.

The physicist and cosmologist Max Tegmark puts the contemporary case for (Pythagorean) being in the following very concrete example:

“[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.”

To spell out the above.

Max Tegmark isn’t saying that maths is perfect for describing the “electricity-field strength” in a particular “physical space”. He’s saying that the electricity-field strength is a “mathematical structure”. That is, the maths we use to describe the electricity field is one and the same thing as the electricity field. Thus, if that’s the case, the “miracle of mathematics” is hardly a surprise! That’s because it’s essentially a situation in which that maths is describing maths. And if maths is describing maths, then the word “describing” is surely not apt word to use in the first place.

In any case, Tegmark gives us more detail on his position when he tells us 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 in the above. At one point he says that a field “is just [ ] something represented by numbers at each point in spacetime”. So here we have the two words “something represented”. Yet elsewhere Tegmark says that the field “is just” (or just is) a mathematical structure— the latter two words implying that all we have is number. To repeat: Tegmark says that the field is “represented” by “three numbers at each point in spacetime”. Yet he doesn’t (in this passage at least) say that the field is a set of numbers (or even a “structure” which includes numbers).

So perhaps there’s a difference between saying that “things themselves are numbers” (as the Pythagoreans did) and saying that the world is mathematical. (I may be drowning in a sea of grammar here.) The latter may simply state that the world exhibits features which are best expressed (or described) by mathematics. The former, on the other hand, says that the world literally is mathematics. But (as already stated) it’s hard to grasp what that even means.