The Relationship Between Math and Physics

Science
Philosophy

(Antoine Suarez) #1

I am a Swiss quantum physicist and philosopher working on the foundation of quantum physics, anthropology, and questions at the boundary of science and theology.
As a quantum physicist I have proposed and realized experiments on quantum nonlocality which have contributed to the insight that quantum correlations come from outside space-time: “not all that matters for physical phenomena is contained in space-time”.

Mathematics is a basic ingredient of the world we live in. Therefore proving theorems give you new knowledge about the world and allow you to better predict the phenomena.

If I understand well you like Josh are a computational biologist. In your work you are doing nothing other than reproduce computations that are going on in the world and shape it.


Side Comments on Suarez and Swamidass
(Dr. Patrick Trischitta) #2

I too have studied the recent experiments in quantum nonlocality. There are several physicists here who can add much to the discussion. Can we explore your insight that quantum correlations come from outside spacetime? And “not all that matters for physical phenomena is contained in spacetime?”


(John Harshman) #3

Sounds like some variety of Platonism. I would disagree. We use mathematics to describe and model the world. To the extent that our models are accurate, they describe the world correctly. We determine the extent to which out models are accurate by testing them against observation. I don’t see physicists as any different in this respect from biologists.

I would not describe myself as a computational biologist, though I use computers. I’m an avian molecular phylogeneticist.


(Antoine Suarez) #4

With pleasure Patrick.

You find my insight developed in this paper:
https://arxiv.org/abs/1712.06448

Which “recent experiments in quantum nonlocality have you studied”?


(Antoine Suarez) #5

We can do this because the world is shaped mathematically by an omniscient mind (God) who can encompass all the mathematical truth. Not that God needs calculation to make the universe. God rather uses mathematics because He is kind to us and wants to shape a world we can calculate, predict and live in. This is the ordinary world, where for instance the sun follows the trajectory we are used to and can predict with high accurateness. And this is the reason why we can explain the world with mathematics so amazingly well.

The theorems of Gödel and Turing prove that human minds will never encompass all the mathematical truth contained in arithmetics. Accordingly our theories will always be incomplete. Through observation we approach more and more the mathematics God is actually using, we read in God’s mind.

I agree. Think for instance what a magnificent mathematical (geometrical) structure the double helix is!

You can use computers to better describe the world because the world itself is sort of computation. Paraphrasing Moses Mendelssohn: The world is computable because it is computed.


(S. Joshua Swamidass) #6

Do the other @physicists agree?


(John Harshman) #7

That’s a very confident assertion, but how do you know it’s true?


(Antoine Suarez) #8

Mathematics cannot be a product of human mind:
At any time T of history there will be mathematical questions that no human mind can answer with the methods available at time T (Theorems of Gödel and Turing).

So for instance at present no human mind can answer whether or not there is a largest perfect number. Nonetheless the answer to this question exist: it is either YES or NO.

This means that the answers to mathematical questions exist in an omniscient mind before we humans discover a method to get the answer.

So the world can magnificently be described by mathematical means because the omniscient mind makes the world mathematically in order we can calculate it, develop technologies, and live comfortably.


(Neil Rickert) #9

No. It only means that humans can imagine such omniscient minds. But what we can imagine need not necessarily exist.

But the world cannot be described mathematically.

Yes, we can use mathematics in our descriptions. But that’s only because we constructed our own mathematical way of describing.


(Dr. Patrick Trischitta) #10

This sounds like the ramblings of one man’s imaginative (not omniscient) mind. Mathematics is an invention of man. Mathematics is applied by man to solve problems and to explore how the world/universe works. Like science, mathematics is neutral on the existence of God or not.


(Antoine Suarez) #11

Numbers are not simply something we imagine: Numbers exist.
Numbers are mental entities, not objects we access with our senses.

So numbers and answers to questions about numbers are contents of some mind.

Consider now the question of whether or not there is a largest perfect number.
The answer to this question does exist for sure: it is either YES or NO.
Nonetheless for the time being it is contained in NO human mind.

Therefore it is contained in a non-human mind.

Mathematics is “an invention of man” in the sense that refers to truths we discover by thinking. However we are not the authors of these truths.

If Mathematics can be used “to explore how the world/universe works”, this means that the world/universe works mathematically and to this extent the world can be described mathematically.

We construct certainly our own mathematical way of describing, but the fact that this way fits the way the world works means that the world is constructed mathematically by a non-human mind in order we humans can discover how it works.


(Neil Rickert) #12

As a mathematical fictionalist, I disagree :zipper_mouth_face:

That’s not actually obvious, once we allow non-standard arithmetic (non-standard models of the Peano axioms).

I don’t agree at all.

We make mathematical models of the world (or of aspects of the world). And those mathematical models work mathematically. But it does not follow that the world works mathematically.


(Dr. Patrick Trischitta) #13

We’re not? Who is the author(s) of these truths? For example who authorized that 1 +1 = 2 ?


(John Harshman) #14

It seems odd for a person so versed in mathematics to produce such a non sequitur argument. Mathematics is something we make up. Its frequent correspondence to the observed world is interesting, and there are several ways in which it might be explained. One of them is that we come up with initial axioms by abstraction from observation. But the existence of unproven (and even unprovable) theorems is a simple consequence of any symbolic system (for the unprovable bit, one complex enough to incorporate arithmetic). No Platonic ideals in the mind of God are necessary as a backstop; it’s just a consequence. Gödel’s proof would in fact argue against your claim, not for it.


(Antoine Suarez) #15

Nice question Patrick!

1+1=2 is a true statement about the operation of counting.

Counting is a basic operation allowing us to establish consistently interpersonal relationships, as for instance by means of contracts, registers, bank-accounts, etc. Counting is a mental operation although to perform it we can help us with calculating tools, as for instance an abacus.

We discover arithmetical truths by performing such operations. So we can say that arithmetics refers to truths about the way we operate, which happens necessarily step by step .

In this sense the criticism of Platonism is justified: Numbers are not perfect archetypes that exist in a realm separated from our everyday world.

Are we the authors of these truths regarding counting operations?

On the one hand it seems we are the authors of arithmetical truths because to become aware of them we ourselves have first to assemble conveniently a finite number of items to produce a theorem or an algorithm.

So for instance Euclid arranged a theorem to prove that there are infinitely many prime numbers.

On the other hand, to date there is no theorem allowing us to decide whether or not there are infinitely many perfect numbers.

What is more, there are theorems (Gödel and Turing) proving that:

Even if at a future time T we find a method to decide the above question about perfect numbers, there will be other similar questions regarding natural numbers we cannot decide with all the methods available at time T.

These theorems are a consequence of the very fact that to count we have to proceed step by step , and show that we never will get hold at once of the whole truth regarding the way our mind functions.

One can better understand the nature of Arithmetics considering a tiling puzzle : We are not the authors of the picture that appears when the puzzle is complete. However, for the picture to appear, we have to find the way of assembling fittingly the interlocking and tessellating pieces.

Something similar holds for Arithmetics: We discover the arithmetical truths, even if to discover them we have to be creative and invent convenient assemblies of operations.

Actually what the theorems of Gödel and Turing show is that Arithmetics contains an infinite number of puzzle games we humans never will encompass once and for all!


(Antoine Suarez) #16

Our mathematical models can efficiently describe the world because the world is written in mathematical language, as Galileo Galilei rightly acknowledged:

La filosofia è scritta in questo grandissimo libro che continuamente ci sta aperto innanzi a gli occhi (io dico l’universo), ma non si può intendere se prima non s’impara a intender la lingua, e conoscer i caratteri, ne’ quali è scritto. Egli è scritto in lingua matematica (Il Saggiatore, 1623).


(John Harshman) #17

Sorry, but the world isn’t written and it isn’t in a language. It’s a physical thing. Hey, philosophers: would you describe this as Platonism, or neoPlatonism, or something? And is it possible that Galileo was being metaphorical there?


(Neil Rickert) #18

We are going to disagree on this.


(Antoine Suarez) #19

Here a quote by Eugene Wigner, quantum physicist and Nobel Prize laureate (1963):

“The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it” [Eugene Wigner, "The Unreasonable Effectiveness of Mathematics in the Natural Sciences," in Communications in Pure and Applied Mathematics , vol. 13, No. I (February 1960). New York: John Wiley & Sons, Inc. Copyright © 1960 by John Wiley & Sons, Inc.]

The world is made in a way that we can calculate and predict it. This is a wonderful gift we should be grateful for.

Kurt Gödel himself defended mathematical Platonism. Would you also consider this “odd for a person so versed in mathematics”?

Anyway, on my part I endorse neither “Platonic ideals” nor “Neoplatonic ones”. My position is a bit subtler. Remember the metaphor with the “tilling puzzle” in my previous answer to Patrick:

We “make up” mathematics in the sense that we combine conveniently the mental operations to the end of discovering the theorem.

The existence of unproven provable theorems is a consequence of the fact that the basic operation of our mind to act fittingly in space-time is counting, and to count we proceed step by step. Se we build algorithms that allow us to solve partial puzzles in Arithmetics. but we will never have an algorithm capable of solving any arithmetical puzzle: This was the dream of David Hilbert, and this dream was proved wrong by Kurt Gödel and Alan Turing.

The consequence of this is that we will never possess all the richness hidden in our mind. In other words, our mind itself is a wonderful gift.

So we have three wonderful gifts:

  • The mathematical theorems we discover.

  • The way our mind works to discover them.

  • The appropriateness of these theorems to describe the world.

This conclusion is inescapable.
You can stop here if you want: I will not push you to go beyond.
However, I cannot stop here: I am tremendously grateful for these gifts, and feel the necessity to find out who is making me so happy and thank him for it.


(John Harshman) #20

Argument by credentialism; not really valid. And how do we know Wigner intended that remark literally?

Again, argument by credentialism.

It’s more escapable than you might imagine. All you’re doing here is asserting that mathematics is a gift and then concluding on that basis that someone had to give it to us. You forgot the part where you actually show that it’s a gift.