Maths describing nature.

[saac]

The physical side of nature certainly appears to have an underlying consistency that mathematics describe beautifully but I am not sure the same can be said of biology. 

Jim 

[Zermelo]

4 hours ago, Ags said:

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

Arguably the single most useful development in mathematics, for modelling the real world, was the invention of calculus, based on the properties of a (continuous) real number system. The continuity leads to irrationals, transcendentals, and more, which some mathematicians have found objectionable. But the usefulness of calculus in science has tended to outweigh the objections from a minority of mathematicians - the universe seems to behave as if it were continuous, whether it is or not. Recent speculation that space and time might be quantized, and not infinitely divisible, would give support to the skeptics, but they seem to be entertained only by a small minority.

[andrew s]

14 minutes ago, saac said:

The physical side of nature certainly appears to have an underlying consistency that mathematics describe beautifully but I am not sure the same can be said of biology. 

Jim 

I feel there is consistency but as complexity grows our ability to master it seems to rest on being able to "course grain" it, to find a scale that loses the complexity while retaining the key characteristics .

For example treating a gas as a whole with PV = RT rather than with the kinetics of the underlying  molecules. 

While we can get away with our point or spherical cow for its bulk motion it does not seem possible to model grass into milk in the same way!

In reality we tend to resort to simulation for large complex systems but again the basic units must be simple. 

Recent advances in AI are making progress but I fear they provide little additional insight.

Some things remain too complex to model at a level we can grasp. Whole biological entities being one example. 

Regards Andrew 

 

[saac]

3 hours ago, andrew s said:

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

Regards Andrew 

If I was ever asked to name a favourite equation it would be Euler's formula. He was a fascinating individual; his work to develop Newton's laws of motion to rigid bodies literally underpins mechanical engineering - a patron saint to the Mechanical Engineer.  

Jim 

[andrew s]

7 minutes ago, saac said:

If I was ever asked to name a favourite equation it would be Euler's formula. He was a fascinating individual; his work to develop Newton's laws of motion to rigid bodies literally underpins mechanical engineering - a patron saint to the Mechanical Engineer.  

Jim 

Interesting,  given your heritage I would have gone for Maxwell and his equations as your choice.

Nurture over Nature?

Regards Andrew 

PS for anyone puzzled by Euler's formula Google Argand diagram and see the light.

Edited July 29, 2023 by andrew s

[saac]

43 minutes ago, andrew s said:

Interesting,  given your heritage I would have gone for Maxwell and his equations as your choice.

Nurture over Nature?

Regards Andrew 

PS for anyone puzzled by Euler's formula Google Argand diagram and see the light.

Maxwell is personal hero but his equations make my head hurt  

Jim

[Ags]

A true mathematician would say that the universe describes maths 😀

There can be an infinite number of universes with infinitely different physical laws, but there is only one maths!

[Nik271]

Actually I was trying to say that there is not even one maths, what we have is a consensus maths based on a list of axioms that common sense tells us reflect the real world. But a different universe may require different maths i.e. very different axioms. For example one can describe the real numbers as a purely mathematical object with some  axioms. A universe without real numbers will be something totally unimaginable to us, but still there will be some list of true statements about it some of which can be deduced from others by logical deduction. I would call this list  the maths of that universe.

 

So to return to our universe, we do have the current maths which we believe reflects  the universe  really well. But how well is really well? As we get closer to understanding some singularity point, say a black hole or the origin of the univese itself our maths may turn out to be insufficient. We can fix it by adding some more axioms according to what we discover further about the universe. So even the axioms of maths can evolve!

I admit this has not happened yet in the real world, at least I don't know an instance where real applications have required rewrting of mathematical axioms. But it could happen. This is actually the same issue as the philosophical question, is maths created by us or imposed on us by the universe.

 

 

 

 

Edited July 31, 2023 by Nik271

[andrew s]

28 minutes ago, Nik271 said:

I admit this has not happened yet in the real world, at least I don't know an instance where real applications have required rewrting of mathematical axioms.

It has already happened. Mathematicians were very critical of the idea of the Dirac delta function. It was eventually made "respectable" via the theory of distributions.

Regards Andrew 

Edited July 31, 2023 by andrew s

[Nik271]

Indeed at the time 1930s there was a lot of research into measures motivated both internally within maths and by the need of applications to physics. Dirac's function is an example of a point measure and such measures (=probability distributions) arose from the need of quantum mechanics where the usual notion of eigenvalues of matrices is no longer sufficient. Instead one gets a probability distribution on the real line which is the 'spectrum' of a linear operator.

In this way maths and theoretical physics developed together. I always thought that the main mathematical ideas were created first, in the 1900 by Hilbert, Lebesque and others but I may be wrong. It's probably hard to tell now who was influenced by who.

[George Jones]

On 28/07/2023 at 13:08, ollypenrice said:

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?

I am going to flip this around somewhat. If logic is a branch of mathematics, how is it possible to define "consistent" without mathematics? 😁

[Nik271]

I guess the meaning of consistent  here is 'regular and not random'. Admittedly not clear how to define something which is 'not random' in a deterministic universe. Perhaps not random' should be taken to mean predictable by some rule that we can study, e.g a set of input data and an algorithm deciding when the event occurs. Anyway this is just my best  guess.

[ollypenrice]

3 hours ago, George Jones said:

I am going to flip this around somewhat. If logic is a branch of mathematics, how is it possible to define "consistent" without mathematics? 😁

 

Nice point.  However, might I not place a cube and a sphere on a gentle slope and observe that the cube never rolls down it and the sphere always does? You might reply by asking me how I can define a sphere and a cube without mathematics but could I not just hold them up and say, 'This is a cube and this is a sphere?' And then you might say, 'Always' means 'every time' and how can you discover that it's 'always' without counting the number of times the cube doesn't roll down the slope and compare this with the number of tries.  By Gad, I think you are winning this one!

lly

[Zermelo]

I think we are identifying two different senses of "consistent" here.

(1)  In the purely mathematical/logical sense, a set of (well-formed) statements is consistent if they cannot be used together to derive a contradiction, using the agreed rules of inference for that system.  In particular, when applied to a proposed set of axioms for a formal system, that axiom set is said to be consistent if it is not possible to derive a contradiction, i.e. ⱯP ⌐(P V ⌐P) where P is any well-formed statement.
Godel showed that any system complex enough to contain normal arithmetic must either be inconsistent or incomplete (in the sense that the system must contain well-formed statements that cannot be proved either true or false from its axioms). This effectively torpedoed the attempts of Hilbert and others to prove the consistency of mathematics by describing it from "outside", with metamathematics (I think, hinted at in a previous post on this thread).

(2) In a more general, physical sense, that property of reality by which it is observed to behave in the same way in different times and places, and hence which behaviour science can then render as "universal laws". This is a bit more debatable, for example do we still count the laws affecting cosmology as "consistent" if the value of the cosmological constant or the speed of light change over time?

[ollypenrice]

On 01/08/2023 at 19:11, Zermelo said:

 

(2) In a more general, physical sense, that property of reality by which it is observed to behave in the same way in different times and places, and hence which behaviour science can then render as "universal laws". This is a bit more debatable, for example do we still count the laws affecting cosmology as "consistent" if the value of the cosmological constant or the speed of light change over time?

Albeit rather nervously, I think I'd say 'yes' to this question. There would just be another variable in the equations describing the consistency, no?