Is "Unreasonable Effectiveness of Mathematics in the Natural Sciences" a tautology?

Question

According to Gelfand,

"Eugene Wigner wrote a famous essay on the unreasonable effectiveness of mathematics in natural sciences. He meant physics, of course. There is only one thing which is more unreasonable than the unreasonable effectiveness of mathematics in physics, and this is the unreasonable ineffectiveness of mathematics in biology."

The highlighed quotes, from Wigner and Gelfand, beg a few questions:

1 What is the subject matter of Physics?

2 What is the subject matter of Biology?

3 What is the difference between the two?

4 What is Mathematics?

Physics deals primarily with universal patterns of behaviour, the ones that are applicable to inanimate objects as well as living things.

Biology deals with living creatures, and behaviours exclusive to the living creatures.

The above begs for some clarity in the distinction between "living creatures" and "inanimate objects". The way I see it, the difference between a rock and a turtle is in that the turtle can suddenly decide to get up and walk somewhere, with no discernible physical cause. That is, the difference between the living creatures and the inanimate objects in that the former ones have a bit of, eh, let's call it "free will", that makes them inherently unpredictable.

Now, for the definition of Mathematics I prefer something like "a study of reproducible mental patterns." That is, it deals with well-behaved, reproducible abstract patterns. Mathematics studies mostly things that are deterministic.

Given the above, wouldn't the highlighted parts of Gelfand quote mean the following:

"effectiveness of a subject (Mathematics) that studies reproducible deterministic abstract patterns in the science (Physics) that studies reproducible deterministic patterns of Nature"

and

"ineffectiveness of a subject (Mathematics) that studies reproducible deterministic abstract patterns in the science (Biology) that studies creatures capable of exhibiting non-deterministic behaviour"

Both statements sound almost like tautologies to me. Is there any gap in the above reasoning?

Answer 1

The reason it is not a tautology is that there is no reason a priori that deterministic patterns should be so easy to find (inverse square law?! How easy is that!), nor that behaving creatures should be composed of parts for which such laws act at such an incredibly low level as to be nearly useless in understanding the whole organism.

To put it another way, we can reliably tell when someone is grumpy, and when an apple is falling. That mathematics is so good at precisely describing the latter and so lousy at describing the former is surprising. (Even if you have to admit that some aspects of flexible goal-directed behavior will be surprising.)

Answer 2

Yes, because mathematics is the second of the three degrees of abstraction:

1. Physics deals with that which is in motion and is material.
2. Mathematics deals with that which is material and is not in motion [∵ mathematical objects do not move or change]
3. Metaphysics deals with that which is not in motion nor is material.

In other words: there is no mathematics without physics since there is "nothing in the intellect that is not first in the senses" (nihil est in intellectu quod non prius in sensu)—all intellectual knowledge, including mathematical knowledge, comes through the physical senses from the physical world.

Think about how you first learned what the number 2 is. You probably took two little physical building blocks, which you knew to be distinct beings, together to form a group; thus, you discovered that two wholes has something to do with the number 2.

Answer 3

Mathematics is the creation of interesting tautologies. It's bounded on one side by uninteresting tautologies, and on the other by the limits of mathematicians and people and things that use math. We use it to start with assumptions and find conclusions. In the sciences, we have mathematical models that, for assumptions, have regularities in the Universe that we've observed, and provide conclusions we can test against the real world.

What we deal with in physics are either simple things (like elementary particles and combinations of a relatively small number of them) or abstractions that serve all the necessary purpose. Dark matter was first noticed because galaxies didn't rotate as predicted, and the predictions were based on very general mass distribution. To an astrophysicist, a certain area of space will have stuff of a certain mass, and it doesn't matter what of that is stars, planets, or hamsters.

Both the rock and the turtle can have complicated internal structures, but nobody really cares about the rock (except geologists) because it's just going to sit there. Everything we're interested in can be expressed by a few quantities defined over the rock's volume. Turtles are more complex, but that's not the big difference. Turtle behavior is a lot more complex than rock behavior. Minor differences in the internal structure of the turtle can result in drastically different behavior (for example, whether that heart thing is beating or not), so we have to take an immense amount of things into account.

So, with physics, the math is usually simple for what we actually care about. With biology, there's a whole lot of complicated interacting systems, and the math gets really really messy. Since we can't handle messy math beyond a certain point, it becomes a lot less useful in describing what happens.

It's quite possible that we could predict turtle behavior from first principles, if we had the quantum wave equations and a sufficiently powerful computer. There are open questions in quantum physics, but we may know enough to predict turtles in principle. We are not going to be able to construct a sufficiently powerful computer in this Universe, either electronically or as a brain in a mathematician.