Physics is more fundamental than maths.
And, boring blog-posts [still] have to be written about this sheer mundane topic.
But permit me to do so.
[To those who yawn already: I promise to write a [for you] more interesting post within a span of just about a month or two, or so.]
But, yes, physics is more fundamental than maths.
“Elementary,” Sherlock [of the Holmes family] would have said, upon being probed to explain the reasons behind the title assertion.
Actually, the operative word here is not “elementary,” but a bit more philosophical in nature. [In particular, it is metaphysical and epistemological in nature, including logical.]
In short, the reason behind the title-assertion itself is, how to put it, “fundamental.”
The fundamental truth that proper metaphysics teaches is, to cut a long story short, that the what precedes the how.
A higher-level but still damn very fundamental truth that proper epistemology teaches is, to cut a long story short, abstractions are abstractions, formed from [the perceptually evident] concretes.
A slightly even higher-level but very closely associated truth which—you guessed it right—also still very fundamental, is that one looks for completeness of propositions, of assertions, in proofs, etc.
With that being said, if you are still with me, let me illustrate the core of my argument. (Note, it’s merely an illustration, not the actual argument—the actual one will have to be grounded in philosophy, and couched in terms of philosophy of physics. Both are corrupt today, and at least for today, I don’t want to get into either. But I can, and will, illustrate what I have in mind, via a couple of examples.)
First, observe that mathematics cannot describe, not even in terms of principles, the entirety of the physical world.
There are any number of ways to verify the assertion. (To verify is not to prove, but the said activity can be helpful for proofs i.e. for the process of proving.) For instance, consider the following facts: Discoveries (and not just inventions) are possible. No theory of mathematical physics has ever been without some empirically determined constants. People look for a suitable mathematical model, e.g. whether it should be a linear or nonlinear theory, accurate to what order—the first, second or higher order, etc. (The idea of the differential order itself is a dead give-away that a completed work of maths cannot hope to cover describing the entirety of the physical world; the power series is infinite in the number of its terms).
Second (and this part might seem to ensure logical completeness) is this: Mathematics not is only capable of describing a non-physical, non-real, purely imaginary world, but it is exceedingly easy to do that.
In evidence (i.e. of verification), refer to my gravatar icon (which appears in the title-bar of your browser when you browse this blog). It is actually the result of a simulation. The problem was that of the ideal fluid flow (i.e. a potential field) in 2D. Now, observe that no physical object exists in 2D. No object which exists even has an infinitely small thickness let alone a zero thickness. (And there is a difference between the two.) Yet, it is so easy in maths to conceive of such an object. So easy, in fact, that test-cases (or analyses) like that (I mean those in 1D and 2D) are routinely used as “sand-box”es of sort, in engineering. Every one knows (except for physicists and philosophers, of course), that such things are, taken by themselves, unreal. The 1D and 2D models (and often, for that matter, even 3D models) just show some certain similarities to the actual physical behavior, that’s all. But regard them as completely real things in your actual engineering job, and you will soon turn looney. (Many respected physicists, mathematicians, and philosophers in fact are, in terms of their professed convictions, indistinguishable from mere lunatics.)
So there. I presented two complementary aspects. And these two seem to complete the argument—or at least the illustration.
The illustration of the argument seems to be, logically speaking, almost complete.
But how to verify the completeness? … Let me help you out.
Just in case you missed it, here is the summary of the two points I made (but did not prove) above:
(i) Maths by itself (i.e. divorced from or not based on physics) is incapable of describing physical reality in its entirety. In fact, to describe anything concretely real in terms of maths very rapidly gets extraordinarily difficult and very soon collapses into impossible.
(ii) Left to its own devices (i.e. as divorced from or not based on physics), the methods of maths can very easily describe purely imaginary things—things which in principle can have no physical existence.
Elementary, wasn’t it?
Yes, it was.
But then, even while talking about a mere illustration of the real argument, why might have I added the word “almost”—almost as if it were an after-thought?
Completeness requires that I address this part too.
But being too busy for now [affiliations- and accreditations-related work], I would like to leave the answer of that question—and the point of proving the completeness in the “real” complete sense, as an exercise for the reader. [Don’t worry, I will cover it in a short post, sometime in future. [Just remind me, that’s all!]]
A Song I Like:
(Hindi) “woh chaand khilaa, woh taare hanse…”
Singer: Lata Mangeshkar
Lyrics: Hasrat Jaipuri
[PS: Really short of time to add categories and all… But you take care, and bye for now…]