On the unreasonable effectiveness of mathematics in physics

[Magical Realist]

Question: "Anyone who has studied physics will quickly see how fantastically successful mathematics is at describing the universe. The famous physicist Richard Feynman said in his book "the character of physical law" (pg.39):

'But what turns out to be true is that the more we investigate, the more laws we find, and the deeper we penetrate nature, the more this disease persists. Every one of our laws is a purely mathematical statement in rather complex and abstruse mathematics... It gets more and more abstruse and more and more difficult as we go on. Why? I have not the slightest idea...'

What does this imply about the nature of reality? And what can we say about this increasing complexity as we penetrate deeper?"

https://philosophy.stackexchange.com/que...e-universe

It seems to me that mathematics only works in physics to the degree that it formulates law-like predictable events in the universe based on its description of abstract quantitatively-interrelated patterns. But that's it. It doesn't apply to any sorts of events or aspects of reality that doesn't conform to some generalizable pattern of happening. It's like a map, which only represents the territory in terms of roads and cities and states and the distances between them but doesn't represent the territory in terms of its varying terrain or any of its geological or botanical or topographical features. For that we have travel brochures. Trying to understand reality based solely on mathematics is thus like trying to figure out who a person was based only on their skeleton:

"Remember that we sometimes demand explanations for the sake not of their content, but of their form. Our requirement is an architectural one; the explanation a kind of sham corbel that supports nothing."— Ludwig Wittgenstein, Philosophical Investigations

[C C]

The Applicability of Mathematics in Physics
https://iep.utm.edu/app-math-physics/

EXCERPT (history): One of the first schools of thought to deal with the relationship between mathematics and the physical world, in the Western tradition, was the Pythagoreans who were active from the 6th century BC. They believed that numbers were somehow imbued with the divine. These numbers create physical objects and express the harmony of existence. Pythagoreans even incorporated some aspects of aesthetics in their philosophy by stating relationships between lengths of strings and sounds. Western musical scales come from these ideas.

For the Pythagoreans, the applicability of mathematics in natural sciences is simple. Since the universe is made out of numbers, it makes sense the physical world is controlled by mathematics. They believed that with pure thought about abstract numbers, one can come to understand the physical world. This unseen, pure realm cannot be accessed by the physical senses. The idea of such a pure realm came to have a deep and profound influence on Western philosophy and religion.

Platonism took the Pythagorean idea of the pure realm as its central dogma. To Platonists, the physical world was no longer imbued with the divine. Rather, the physical universe was simply a bad imitation of the perfect realm. This realm contains all numbers and all the relationships between numbers. In fact, it contains all of mathematics that ever existed and ever will exist. This Platonic heaven also contains all true physical laws that ever existed and perfect notions of truth, justice, and beauty. Their main point is to accept that the physical world is unimportant, and the only true existence is the Platonic realm accessed through the human mind.

Galileo perhaps provided the most famous “solution” to the applicability problem, following a version of Platonism. According to Galileo, the universe is written in a mathematical language so it should not be a surprise that empirical investigation of the universe (in physics for instance) is filled with mathematical statements:

Philosophy is written in that great book which continually lies open before us (I mean the Universe). But one cannot understand this book until one has learned to understand the language and to know the letters in which it is written. It is written in the language of mathematics, and the letters are triangles, circles and other geometric figures. Without these means it is impossible for mankind to understand a single word; without these means there is only vain stumbling in a dark labyrinth (Galileo 1623, p. 171).

According to Galileo, while the universe is fundamentally mathematical, senses mask the mathematical essence of the universe. So, he divided the qualities of the objects into two categories: primary and secondary. The primary qualities, such as size and shape, are mathematical while the secondary qualities such as color, taste and scent are not. But one should not be deceived by the secondary qualities into forgetting the mathematical essence of the universe. In Galileo’s own immutable words:

Whenever I conceive any … corporeal substance, I immediately … think of it as … having this or that shape; as being large or small … and in some specific place at any given time; as being in motion or at rest; as touching or not touching some other body; and as being one in number, or few, or many. From these conditions I cannot separate such a substance by any stretch of my imagination. But that it must be white or red, bitter or sweet, noisy or silent, and of sweet or foul odor, my mind does not feel compelled … Without the senses … reason … would probably never arrive at qualities like these. Hence I think that tastes, odors, colors, and so on are no more than mere names so far as the object in which we place them is concerned, and that they reside only in the consciousness. Hence if the living creature were removed, all these qualities would be … annihilated (Galilei 1957, p.274).

One can think of Galileo as an advocate of Platonism, according to which the true reality is perfect and mathematical, but sensory experience gives us an imperfect image of this otherwise perfect reality. The same idea is almost ubiquitous in anti-Aristotelian natural philosophy, and it is formulated by Bacon (1620) in Novum Organon and by Descartes in The World (ca.1630) and Principles (1644). Descartes surely took it from Galileo.

Modern Platonism (Roger Penrose): https://iep.utm.edu/app-math-physics/#SH3b

Modern Pythagoreanism (Max Tegmark): https://iep.utm.edu/app-math-physics/#SH3c

[Magical Realist]

One can't overlook the reliance of mathematical formulae of physical laws on language itself to interpret it as such. Without the sense-making structure of theories these abstract equations would have no meaning. Positing them as somehow just existing out there on their own in some eternal state is IMO a case of reification--of mistaking mere abstractions of symbols and logic for concrete in-themselves entities. All they represent are certain patterns that recur given certain ideal states and conditions. It is this inherent ideality of mathematics that is its flaw. Reality as given rarely presents us with instances of mathematical formulae operating smoothly and absolutely.

"The mathematicians and physics men
Have their mythology; they work alongside the truth,
Never touching it; their equations are false
But the things work.
Or, when gross error appears,
They invent new ones; they drop the theory of waves
In universal ether and imagine curved space."-----Robinson Jeffers