Limitation of calculation/mathematical reasoning in search for truth

Question

Eddington’s attempt to determine the values of the fundamental constant is an extreme example.

Schopenhauer was not fond of mathematics. Without denying its practical use, he was convinced that mathematics, though it can produce a quantitative description of the material world, could never provide an understanding of its causal relationships. “Where calculating begins, understanding ends” was his statement on this subject .His aversion to mathematics was shared by other intellectuals of his time who yearned for a direct perception of nature, without the intermediacy of abstract concepts.

Nowadays, we are used to the enormous success of mathematics in describing physical phenomena. However, Schopenhauer was right in a sense; a mathematical description only produces numbers, but not an understanding of the phenomenon that is being observed.

Quantum mechanics is an excellent example of this: it is based entirely on abstract concepts –wave functions, operators, probability amplitudes, spin, etc.—, together with a rigorous mathematical formalism. But although quantum mechanics has proven to be the most accurate description of physical phenomena, any attempt to “explain” it in terms of familiar concepts inevitably leads to paradoxical conclusions. The fact that mathematics is so effective may well be one of the greatest mysteries of modern physics. Such effectiveness is, in fact, quite unreasonable (as Eugene Wigner rightly pointed out).

In any case, we can agree with Schopenhauer that “Where calculating begins, understanding ends”, provided we also realize that calculating can reach unsuspected limits!

If there is limitation then how to overcome it? ( any additional method/procedure like developing intuition, thought experiments etc...)

Answer 1

^{This answers the first version of this question, which asked "Can the laws of physics be deduced from pure mathematical reasoning?" The question has since been inappropriately edited to apparently invalidate this answer.}

No, pure mathematical reasoning does not result in "the laws of physics."

One could never produce a theory of physics without some model of what the math represents. The math needs to be predictive of events in the world.

One cannot deduce without observations the gravitational constant or the speed of light.

Answer 2

Can the laws of physics be deduced from pure mathematical reasoning?

No. A formal theory, if it is meant to represent some aspect of reality, requires that we first give to what we think are our relevant beliefs a formal interpretation. This is not specific to science or mathematics. It applies to every logical reasoning.

For now at least, we wouldn't know where to begin if we had to justify this operation formally.

In practice, people who can agree with each other that a formal interpretation is correct will also be able to agree that the corresponding theory applies. This is how logic works. There is for now at least no prospect of moving beyond this limitation.

So, in a nutshell, science is ultimately grounded in common sense. Literally.

a mathematical description only produces numbers, but not an understanding of the phenomenon that is being observed.

This strikes me as wrong. If you know what your formalism means, the theory, if it is logical, begets comprehension. If you define n as the GDP of the US and m as that of North-Korea, you will definitely understand what the theory says that n > m.

Formal reasoning allows efficient calculation. During calculation, meaning is not only nowhere but irrelevant. So, it is true that “Where calculating begins, understanding ends”. However, at the end of the calculation, meaning can be restored from the formal results, using definitions. So, “Where calculating ends, understanding resumes”.

Again, nothing specific to mathematics. Just how logic works.

The fact that mathematics is so effective may well be one of the greatest mysteries of modern physics.

There is nothing mysterious in the effectiveness of mathematics. Mathematics is merely a specialised extension of natural language, and we have used natural language to describe the world well before we started to develop the mathematical langage. Natural language is itself merely a formal expression of our ideas about the world, ideas which can only be effective because our brain is logical. The effectiveness of mathematics is fundamentally the effectiveness of our logical capacity.

There is again nothing mysterious in the effectiveness of our logical capacity, for it is obviously a product of natural selection. Ineffective logic had long been selected out when humans first walked the Earth. We were just lucky to inherit our logical capacity from our immediate ancestor species when it was already so remarkably . . . fine-tuned.

Answer 3

There is nothing surprising about the general "effectiveness" of applying mathematical models to the world (not just the physical world but also to for instance evolutionary biology or economics). The effectiveness is caused by us making sure the models work well enough for us. Ineffective models will, after a while, usually be abandoned.

Once we create a model (for instance a model of dynamic negotiation in politics, using game theory), we may sometimes notice things (and can come to understand things) which we didn't notice before. These specific discoveries may be surprising -- but again, they are not generally surprising, since the purpose of simplification and formalization is to make it easier to notice relevant relations that might have been overlooked in the past.

Being surprised about the general effectiveness of applying math in physics is -- perhaps -- equivalent to assuming the Anthropic Principle. I regard this (in each of its variations) as a false, fundamentally misleading and useless overgeneralization of the trivial (true) observation that every observation is itself an event in the world. It's useless since it itself is either tautological or metaphysical (it cannot be verified or falsified by experiment, and it doesn't in any way change any physical theory or the process of theory development).

Answer 4

The short answer is no. For the long answer we have to think about what mathematics is and its relation to physics.

David Hilbert, one of the great mathematicians of the 20th C set out a list of questions in 1900 to address what he thought the important questions for maths in the 20C. The sixth quwstion was a proposal that physics ought to be axiomatised. This is an ongoing programme some of which has been massively successful. For example, classical mechanics is axiomatised by symplectic geometry. The outstanding counterexample is QFT which is still resisting axiomatisation today but there are bold proposals to axiomatise the perturbative version, such as Costello's approach through factorisation algebras.

I would argue that the first two examples of a wildly successful axiomatisation of physics is geometry and arithmetic. In the former this is even described by the name used for the discipline: geo-metry or measuring the earth. This is an eminently physical thing. Likewise, the physics of individuals, like the individual trees in a forest, can be counted is an eminently physical thing. The notion of a number is so intuitively obvious that its roots have been lost to time as the etymology of arithmetic is the art of numbers.

So pure mathematics comes from axiomatising the physical concepts we encounter in the nature. So to deduce physics from maths is wrong-headed because these formalisms are there to axiomatise physics in the first place.

Answer 5

Perhaps Schopenhauer can be elaborated:

Geometry stimulates thinking, while calculating (algebra) replaces it
From matheducators SE

As a counter weight to Schopenhauer and the math-SE version there's the Turing award winner Richard Hamming:

The purpose of computing is insight, not numbers