Maths describing nature.

[ollypenrice]

Galileo famously said that nature was written in the language of mathematics and it's hard to disagree. I used to beat myself up over why this should be but, more recently, I've come to wonder if there's really anything very surprising about it. My question to those on here who are mathematically competent (as I am not) is, Can maths not describe anything which is consistent?  If it can, is there really anything more to the ability of maths to describe nature than that nature is consistent?

And then, if nature were not consistent, would anything be possible - including thinking about it? Without consistency, would nature not just be noise?

Olly

[JeremyS]

🍿

[vlaiv]

Well, it this is nicely summed up in famous quote:

The least comprehensible thing about universe is that it is comprehensible.

As far as I can tell (and been able to find on google), it was Einstein who said that, and I completely agree.

This goes in line with your assertion Olly - yes, math is language of comprehensible and given the nature is comprehensible - it is no wonder it can be expressed in language of math. However - that really does not explain why it's comprehensible in the first place and the fact that comprehensible universes are rather low in probability in space of all possible universes

[Zermelo]

Some physicists' ideal theory-of-everything is one that can be deduced only from the requirement that it be consistent, which is really quite ambitious.

Eugene Wigner wrote about the "unreasonable" effectiveness of mathematics, in its ability to describe the real world. Some of the early quantum physicists took the view that the mathematics was the reality. More recently, there has been a bit of traffic in the other direction: theories developed specifically for a physics context have contributed to progress in some areas of pure maths.

My hunch is that maths is not itself fundamental to reality, but is so "close" to it that the two are likely in lock-step for all practical purposes.

[Ratlet]

Let's just hope the universe continues to remain consistent...

[andrew s]

Evolution has equipped us with a brain that allows us to make sense of our environment.  We modelled it and noted its regularities it order to survive and prosper. 

It seems that as a consequence, this brain, has been able to generate natural languages, the creative arts and the abstract language of mathematics.

Not all mathematics (as far as we know) is applicable to modeling the physical world and we have not managed good models of all physical phenomena e.g. turbulence. Even some very old areas, e.g. statistical mechanics,  have issues in their foundations. 

Is it too surprising then that the brain that noted and was shaped by the regularities of its environment, codified a way of capturing them and so much more - mathematics.

That we struggle to comprehend the quantum world, so far removed from our experience,  other than than through its mathematical formalism is a deep irony though very understandable. 

Regards Andrew 

Edited July 28, 2023 by andrew s

[ollypenrice]

11 hours ago, vlaiv said:

 

The least comprehensible thing about universe is that it is comprehensible.

 

Tricky. The term 'comprehensible' strikes me as being meaningless without the presence of an entity to do the comprehending. There can be no comprehension in a non-sentient universe.  This brings us to the role of the comprehending entity: what capacity for comprehension does it possess?  Any comprehension the entity achieves of the universe will be a product of both the entity's power of comprehension and of properties of the universe being observed.  This suggests that what is comprehended is not the universe itself but a relationship between the observer and the universe.  If I were to stick to this line of thought I'd have to say that the universe is not comprehensible, only our perception of it is comprehensible.

Olly

[Macavity]

I think consistency is a great thing. My THESIS showed stuff that was consistent with...
But, back then, such phrases were trotted without notable (philosopical?) reflection! 😅
"It never do me no harm"! lol. But: Is it something to do with this?  [See Penrose etc.]

Edited July 29, 2023 by Macavity

[vlaiv]

35 minutes ago, ollypenrice said:

Tricky. The term 'comprehensible' strikes me as being meaningless without the presence of an entity to do the comprehending. There can be no comprehension in a non-sentient universe.  This brings us to the role of the comprehending entity: what capacity for comprehension does it possess?  Any comprehension the entity achieves of the universe will be a product of both the entity's power of comprehension and of properties of the universe being observed.  This suggests that what is comprehended is not the universe itself but a relationship between the observer and the universe.  If I were to stick to this line of thought I'd have to say that the universe is not comprehensible, only our perception of it is comprehensible.

Olly

Well, I think we do need to be careful with our use of the term comprehensible.

I will give you an example of distinction between two things that might give you clue of different meanings of the word.

One is for example - reading the future from chicken intestine. Here we have:

a) pattern

b) someone who recognizes the pattern

c) does interpretation of the pattern (right or wrong)

We do need sentience for that

On the other hand - here is another example:

All powers of 2 written in binary form have same structure - 1 followed by some number of zeros. Here we have

a) pattern

b) no need for recognition or interpretation - pattern just exists and there are relationships behind it that can be comprehended by someone if that someone exists - or not

 

Now, we must be careful with how we use word pattern again. Throw bunch of lego bricks on the floor and you'll be able to spot so many patterns. Here pattern is formed because we assign it some "meaning" - but it is not necessarily "universal" pattern that would exist without us assigning it a meaning.

Strange thing about universe is that it is ordered in such way that universal patterns exist (regardless of the fact if anyone is there to recognize them as patterns or not).

In vast configuration space of possible universes - there is much more those that are chaotic without any rules (the same reason why we have second law of thermodynamics - there are more disordered states of the system then ordered ones).

 

[Nik271]

Once we dig deep down into the foundations of maths there are lots of surpsises:

There is no proof that the maths we all use: arithmetic is consistent. This was discovered by Godel in 1930s and is known as Godel's second incompleteness theorem. 'Consistent' here means non-contradictory: there is no statement P such that both the validity and the falseness of P is provable from the standard set of math axioms.

The first Godel theorem is actually even more depressing: mathematics as we know it is incomplete, that is there always will be a statement X such that neither 'X is true' can be proved nor 'X is false' can be proved. So what can do when we encounter such a statement? We can make a choice and add either 'X is true' or 'X is false' to our axioms of maths and go on. Buth then there will be another statement call it Y such that we can neither prove no disprove the validity of Y and we will face the same dilemma and so on.

This created a big storm within maths at that time because until then many people believed that everything in maths is in principle provable from a fixed set of axioms. Hilbert in particular famously said  'We shall know.' He was wrong.

In this sense maths will never be completely satisfatory model of the universe.

Godel and Einstein became good friends in Princeton, I think I can see why. 

[vlaiv]

5 minutes ago, Nik271 said:

In this sense maths will never be completely satisfatory model of the universe.

Why?

There is no indication that model of universe needs to be of infinite complexity.

Maybe we can prove that subset of math that is needed to model the universe is in fact complete?

[beka]

13 hours ago, ollypenrice said:

Galileo famously said that nature was written in the language of mathematics and it's hard to disagree. I used to beat myself up over why this should be but, more recently, I've come to wonder if there's really anything very surprising about it. My question to those on here who are mathematically competent (as I am not) is, Can maths not describe anything which is consistent?  If it can, is there really anything more to the ability of maths to describe nature than that nature is consistent?

And then, if nature were not consistent, would anything be possible - including thinking about it? Without consistency, would nature not just be noise?

Olly

Hi Olly,

What about chaos theory? Does this fall into "being consitent"?

Cheers

[andrew s]

13 minutes ago, Nik271 said:

completely satisfatory model of the universe.

Mathematical or not a model is satisfactory or not depending on what you want to do with it.

The modern way of looking at physical theories is to consider their range of applicability. No model or theory covers all phenomena in all domains. 

In some areas phyisicts have led mathematician for example the Dirac delta function.

Regards Andrew 

[Nik271]

17 minutes ago, vlaiv said:

Why?

There is no indication that model of universe needs to be of infinite complexity.

Maybe we can prove that subset of math that is needed to model the universe is in fact complete?

Godel's proofs use only the natural numbers to prove both of his theorems. So any theory which includes the familiar properies of natural numbers (addition, multiplication and induction, the so-called Peano axioms) will be incomplete as well.

 

There is the question what is allowed to be called 'a proof'. Godel's theorems and most of maths is concerned with finite proofs, that is a finite series of logical deductions starting from the axioms. If you allow infinite proofs then a lot of these obstacles disappear.

Edited July 29, 2023 by Nik271

[vlaiv]

2 minutes ago, Nik271 said:

Godel's proofs use only the natural numbers to prove both of his theorems. So any theory which includes the familiar properies of natural numbers (addition, multiplication and induction, the so-called Peano axioms) will be incomplete as well.

Yes, but incompleteness means that we can't prove all the theorems. However - we can prove large number of theorems, and if we can prove all the theorems needed to model the universe - I don't see why would incompleteness defined that way pose a problem?

[Ags]

What about transcendental functions and irrational numbers? These are relatively simple and commonplace things that maths appears unable to express?

[vlaiv]

19 minutes ago, Ags said:

What about transcendental functions and irrational numbers? These are relatively simple and commonplace things that maths appears unable to express?

Not sure what you mean by commonplace things and math being unable to express.

They are all well defined within mathematical framework. Sure, they need infinite sums, but we know how to work with those (without actually calculating all the way to infinity and back ).

 

[ollypenrice]

47 minutes ago, beka said:

Hi Olly,

What about chaos theory? Does this fall into "being consitent"?

Cheers

I wonder if this is answered by Andrew S when he says

43 minutes ago, andrew s said:

 

The modern way of looking at physical theories is to consider their range of applicability. No model or theory covers all phenomena in all domains. 

It seems to me, a non-mathematician, that something may be unpredictable without being inconsistent.  The outcome of a chain of events highly sensitive to initial conditions cannot be predicted but, when it has happened, it can be 'reverse engineered' and found not to have violated any existing models.

Turbulence may even go beyond this and, if it does, might we not consider it to be a phenomenon existing as an isolated pocket of incomprehensibility within a wider matrix of comprehensibility?

1 hour ago, vlaiv said:

 

I will give you an example of distinction between two things that might give you clue of different meanings of the word.

 

 

I wondered about the possibility of this distinction while writing my post. I'm still wondering about it! 

Olly

[Nik271]

When we propose to disregard some mathematical statements the truth of which we agree not to investigate then we face the question which are the mathematical statements which actually have 'interpretation' or meaning in the real universe? It gets a bit philosophical from here on.

Ags points out the transcendental numbers, they are of a lot of interest to mathematicians but somewhat hard to find use in the 'real world'

For example it has long been known (= proved) that e and pi are both irrational and even transcendental.  It's still an open question if e+ pi is irrational. I'm sure a big prize awaits those who find the answer to this question. But is this of any use in the real world? Hard to see now, but who knows

I think we can agree that  the arithmetic of natural numbers is important, for example positive integers govern the energy levels of electrons in  shell in the Bohr model of the  atom and thus are key to understanding chemisty. The pereidic table is a consequence of some simple relations concerning integers.

Some irrational numbers appear often in nature for example in the golden ratio, pi and e. I think we can agree these are also important. Suppose we have some very complicated logical statement about real numbers that we can neither prove nor disprove. Does this tell us something about universe?

Godel and some later work by more people tells us the following: there is some polynomial P(x,y,z,..)  with integer coefficents in several variables ( I think 13 variables  is enough) such that we are not  able to prove or disprove if the equation P=0 has integer soluton x,y,z,... or not. So is there some physical interpretation of this polynomial and its zeroes? I have no idea.

[andrew s]

33 minutes ago, ollypenrice said:

It seems to me, a non-mathematician, that something may be unpredictable without being inconsistent. 

Indeed it can, Brownian motion is a classic example, there are many others in classical physics. Also quantum theory is a probabilistic theory and does not (except  in special circumstances) predict specific outcomes but only their relative probabilities. 

Regards Andrew 

[andrew s]

37 minutes ago, Nik271 said:

It's still an open question if e+ pi is irrational

It always seems to me that e^ipi= -1 is pure magic. 

Regards Andrew 

[andrew s]

43 minutes ago, Nik271 said:

When we propose to disregard some mathematical statements the truth of which we agree not to investigate then we face the question which are the mathematical statements which actually have 'interpretation' or meaning in the real universe? It gets a bit philosophical from here on.

I think that's very simple. The ones that allow us to make useful models of some part of the reality we are interested in.

You only need to look around you to see the effectiveness of a pragmatic approach. Our technology and engineering, for example, this forum and the nested technologies it relies on.

Regards Andrew 

[iantaylor2uk]

Complex numbers and complex analysis are rigorous mathematically and also incredibly useful for engineering and mathematics. It is surprising to me that quaternion and octonion analysis is not more developed.

In terms of common tools used in physics, perturbation theory and Feynmann path integrals have not yet been shown to be mathematically rigorous.

I think also in fluid mechanics, it is not known if the Navier Stokes equations have unique solutions and even the Euler equation for fluids has been shown to "blow up" for certain boundary conditions.

Edited July 29, 2023 by iantaylor2uk
Typo

[Zermelo]

30 minutes ago, iantaylor2uk said:

It is surprising to me that quaternion and octonion analysis is not more developed.

I believe Hamilton tried to promote the use of quaternions in physics after he discovered them, and for a period in the 19th century there was some rivalry between advocates for quaternions and those who preferred to use vectors instead. The latter approach was found to be easier for most purposes, but both quaternions and octonions still get used oaccasionally. I came across this paper recently, describing a telescope pointing model based on quaternions.

[Zermelo]

1 hour ago, andrew s said:

It always seems to me that e^ipi= -1 is pure magic. 

Regards Andrew 

Polls taken among mathematicians and scientists tend to agree with you.

I think it is also true that the appreciation of relationships in maths like this one can go through three stages:

(1) the practitioner does not know enough about the context of the statement to feel any surprise (>unremarkable)
(2) the practitioner understands the context, but not enough to understand why the statement is true (>fascinating)
(3) the practitioner understands enough to see why the statement is inevitably true (>obvious)

Lord Kelvin is said to have used, in one of his lectures, the remarkable(?) fact that:

and then commented to his audience, "A mathematician is one to whom this is as obvious as the fact that twice two makes four is to you. Liouville was a mathematician."
Liouville had reached stage 3, at least as far as that integral was concerned. Kelvin's audience were probably at stage 2.

It's interesting that deep connections between apparently distinct areas of maths keep coming up, and even the most capable researchers find themselves, initially, at stage 1:

https://en.wikipedia.org/wiki/Monstrous_moonshine

https://en.wikipedia.org/wiki/Langlands_program