This piece wouldn't have been called 'Nature isn't Mathematical' if it weren't for the many other similar titles which I'd seen. For example, 'Everything in the Universe Is Made of Math – Including You', 'What's the Universe Made Of? Math, Says Scientist', 'Mathematics - The Language of the Universe', etc. Indeed from Kurt Gödel to today's Max Tegmark (who says that the "physical universe is mathematics in a well-defined sense”) and ontic structural realism, mathematical Platonism (or derivations thereof) seem to be in the air. Though that's if being a realist about mathematics (or numbers) is the same as being a realist about the mathematical reality of... reality.
The first thing to say is that the claim that 'nature is mathematical' hardly make sense. It's not even that it's true or false. Taken literally, it seems almost meaningless. So perhaps it's all about how we should interpret such a claim.
Some of the applications of Gödel's theorems to physics, for example, simply don't seem to make sense. They verge on being Rylian “category mistakes”. This bewilderment is brought about in full awareness of the fact that there would hardly be physics without mathematics. Indeed that's literally the case with quantum physics and everything which goes with it.
However, when John D. Barrow asks us whether or not “the operations of Nature may include such a non-finite system of axioms” (as well as when he replies to his own question by saying that “nature is consistent and complete but cannot be captured by a finite set of axioms”), it can still be a philosophical struggle to see the connection between mathematical systems or Godel's metamathematics and reality/the world.
Strictly speaking, Nature isn't any “mathematical system of axioms” and it doesn't even “include” such things. Mathematics is applied to Nature or it's used to describe Nature. Sure, ontic structural realists and other structural realists (in the philosophy of physics) would say that this distinction (i.e., between maths and physics) hardly makes sense when it comes to physics generally and it doesn't make any sense at all when it comes to quantum physics. However, surely there's still a distinction to be made here.
Similarly, Nature is neither consistent/complete nor not consistent/complete. It's what's applied to - or used to describe - nature that's complete/consistent. Again, certain physicists and philosophers of science may think that this distinction is hopelessly naive. Yet surely it's still a distinction worth making.
This phenomenon is even encountered in contemporary philosophy of logic.
The philosopher of logic and logician Graham Priest, for example, mentions the world (or “reality”) when he talks of consistency and inconsistency. When discussing the virtue of simplicity he asks the following question:
“If there is some reason for supposing that reality is, quite generally, very consistent – say some sort of transcendental argument – then inconsistency is clearly a negative criterion. If not, then perhaps not.”
As it is, how can the world be either inconsistent or consistent?
What we say about the world (whether in science, philosophy, mathematics, logic, fiction, etc.) may well be consistent or inconsistent (we may also say - with Spinoza later - that the world isn't “beautiful” or “ugly”). However, surely the world itself can neither be consistent nor inconsistent.
Thus within Graham Priest's logical and dialetheic context, claims of Nature's consistency or inconsistency don't seem to make sense. That must surely also mean that inconsistency is neither a “negative criterion” (as Priest puts it) nor a positive criterion when it comes to Nature itself.
Spinoza vs. Anthropocentrism or Anthropomorphism
What some philosophers of science and physicists are doing seems to contravene Baruch Spinoza's words of warning about having an anthropocentric or anthropomorphic (though that word is usually applied to non-human animals) view of Nature. Spinoza's philosophical point is that Nature can only... well, be. Thus:
“I would warn you that I do not attribute to nature either beauty or deformity, order or confusion. Only in relation to our imagination can things be called beautiful or ugly, well-ordered or confused.”
Spinoza says Nature simply is. All the rest is simply (in contemporary parlance) human psychological projection.
Thus there's even a temptation to contradict Galileo's well-known claim about Nature. Thus:
“Nature is written in that great book which ever is before our eyes -- I mean the universe -- but we cannot understand it if we do not first learn the language and grasp the symbols in which it is written. The book is written in mathematical language, and the symbols are triangles, circles and other geometrical figures, without whose help it is impossible to comprehend a single word of it; without which one wanders in vain through a dark labyrinth.”
Surely we must say that Nature's book isn't written in the language of mathematics. We can say that Nature's book can be written in the language mathematics. Indeed it often is written in the language of mathematics. Though Nature's book is not itself mathematical because that book - in a strong sense - didn't even exist until human beings began to write (some of) it.
Yet perhaps I'm doing Galileo a disservice here because he does say that “we cannot understand [Nature] if we do not first learn the language and grasp the symbols in which it is written”. Galileo is talking about understanding Nature here – not just Nature as it is in itself.
Nonetheless, Galileo also says that the the “book is written in mathematical language”. Thus he's talking about Nature as it is in itself being mathematical. He's not even saying that mathematics is required to understand Nature. There is, therefore, an ambivalence between the idea that Nature itself is mathematical and the idea that mathematics is required to understand Nature.
The Mathematical Description of Disorder
Another point worth making is that if mathematics can explain or describe random events or chaotic 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, at the very same time, 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 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 even contradictory – symmetries in the same phenomenon.
In a similar way, some of the mathematical studies of Bela Bartok's late string quartets have found mathematical patterns and symmetries which the composer was almost certainly unaware of. (See this example; though it's not the one I'm familiar with.) True, Bartok was indeed aware of the “golden ratio” and other mathematically formalisable aspects of his and other composers' music. Nonetheless, the analyses I'm referring aren't really formal in nature. They're more like micro-analyses of the notes; and they serve, I believe, little purpose. Now there can indeed be interesting formal aspects and symmetries in music which composers aren't themselves aware of. Yet, at the same time, a mathematician may still gratuitously apply numbers to specific passages of music in the same way he could do so the same to my mess-of-cards.
*) Kitty Ferguson, in her book The Fire in the Equations, writes:
*) Kitty Ferguson, in her book The Fire in the Equations, writes:
"It is very surprising to find, then, that the chorales Bach wrote himself contain quite a few parallel fifths. The rule forbidding them is a rule which Bach himself violated without the least compunction; in fact, it's doubtful whether he was even aware such a rule might exist." (238)