Why does maths describe nature?

[ollypenrice]

In a Radio 4 programme I just heard,John Barrow was asked why maths described nature. He didn't have long and just said that nature had patterns, making science possible, and that maths was about patterns.

I wish he could have been given longer. It set me thinking. I think that the fact

that nature, as Galileo said, is written in the language of mathematics must contain a message. The message may be about maths, or nature, or both, or really about us as observers.

What does anyone else think?

Olly

[brianb]

You can't have an order to anything without mathematics serving as a model. Just remember it is a model, and a necessarily (Godel's Theorem) imperfect one at that.

Mathematics can exist without a physical universe but the contrary isn't true.

[themos]

nature had patterns, making science possible, and that maths was about patterns.

Science is only possible when one has access to lots of events (called "observations" or "experiments"). That's why religious topics cannot be approached by science. That's also why some experiments which detected only one event led to no science (google Cabrera event).

The human brain has a pattern-finding bias. It will even find patterns where there are none. I think we are all aware of this inner drive to say "this is just like that". Mathematics is nothing but a collection of such hunches that have proved useful to survival. Some people take them seriously and end up becoming cranks. However, a minority of such people end up producing something powerful like the wheel or nuclear weapons and these things produce new civilisations.

One fascinating aspect of human mathematical intuition is its unconscious nature. People who have trouble analysing logical syllogisms correctly become very good at it when the test amounts to detecting cheaters. (Evolutionary Psychology Primer by Leda Cosmides and John Tooby)

[themos]

Mathematics can exist without a physical universe

I just can't imagine such a thing.

[ollypenrice]

Okay, I take the point that maths deals in patterns and that nature is sytematic and, therefore accessible to maths. That's what john Barrow was saying. But why does elegant maths so often describe nature best? Why are there not more occasions when maths has to get messy to describe nature? When it does it is usually wrong. Eoicycles and equant points. String theorists beware!

Maybe it is just because nature takes the shortest path. But maybe there is something else lurking in there, too. The maths we know is OUR maths and the universe we observe is OUR universe.

Think of the Ishihara colour test. A normally sighted person sees, say, a letter A in a circle of dots. A colour blind person does not. We, with our system of observation (mainly our senses but also our minds) take a cross section through reality and see our universe. Is it conceivable that another alien observer might take a different cross section and therfore inhabit a different universe? The contentious question would be, Would they have a different mathematics in thier universe? Could they?

What troubles me is that the common denominator betwen maths and nature is us.

The Copernican principle at this point shouts BEWARE! If you are dead confident that maths exists outside us and that nothing about our own view of the universe limits its universality then you have nothing to worry about. I can't say I am as confident as that. I think we shoud always be on the lookout for assumptions we don't realize we have made. I just seem to sense one in the background of the maths-nature accord.

Olly.

[part timer]

The answer is simple.

Mathematics describes nature because that is what the human mind has created it to do. Mathematics is not some sort of fundamental property of the universe. It is the language used to describe the physical world by humans. The logic, both practical and abstract is human.

There is no need to mystify Maths, it is what is and nothing more.

Mathematics is not perfect because we are not perfect. We do not have have the experience to make accurate predictions on the very large or the extremely small and this is why these areas are very much theoretical and unproven where as we do extremely well describing more 'common' physical process. For example we can predict quite complex orbits or even flows within the sun etcetera but to describe the Universe in full we have to use magic energy's and as many infinities as we can find.

Let me just add that this is my opinion and I don't want to start a fight! people do get touchy on this subject.

Luke

[barkis]

Is It not the math's they struggle with in uniting the large and the minute. Perhaps the equasion will be a simple one whenever they do reconcile it.

[mba007]

One could argue that Newton's Laws are elegant, whereas Einstein's relativity isn't... although it depends upon who you are. Same as for cosmological mathematics, it can be messy. Same for stellar structure equations, etc... Having studied astrophysics the maths was only elegant to the maths geniuses who studied with me. The rest of us struggled to find any beauty whatsoever :-)

[themos]

I take the point that maths deals in patterns and that nature is sytematic and, therefore accessible to maths

That's not my point of view. I have no idea if nature is systematic or not (whatever that may mean). What I know is that the human mind can find patterns in nature (but the human mind can find patterns in tea leaves too). Regarding aliens, the proof will be in the eating: if they eat us, they will have better science.

[ollypenrice]

'The answer is simple.

Mathematics describes nature because that is what the human mind has created it to do. '

I'm not at all sure about this. If you consider numbers to be part of nature then yes. If not, though, I would have thought not. Mathematics was fairly well advanced, I thought, when it was first applied to physical nature. Wasn't it the musical scale that first yeilded to mathematical description? However, the idea that our minds might be responsible for both mathematics and nature as we observe (create?) it is exactly what I'm thnking about... so I seem to share a point of view with you to some extent.

'That's not my point of view. I have no idea if nature is systematic or not (whatever that may mean).'

No great depth intended here in my use of systematic. I simply meant that nature usually does today what it does tomorrow and the same cause usually produces the same effect - outside the atom. Maybe 'consistent' would have been a better word.

Some equations are indeed horrendous but isn't this usually where they are having to talk simultaneously about different things? In stellar structure, for example, so many things are involved at the same time. Taken individually are they really that bad? I have only met them at a very amateur level.

I remain astounded that E = MC2. Why is it so simple? Surely it doesn't have to be - or does it?

Olly

[kev 102 42]

look at a bees hive or a sunflower seeds i see math...............kev

[barkis]

The Universe is a Symphony, we just can't read the score.

'Simples'

Ron.

[themos]

If you consider numbers to be part of nature then yes.

Some animals have evolved a "number sense" so that they, for example, can know if some other animal has left an extra chick in the nest. A hypothesis is that all mathematical sense has evolved to detect cheating. My hypothesis is that humans have taken this rudimentary "number sense" and the abstraction power of their linguistic sense and made "abstract mathematics" from it. Well, I say "my hypothesis" but I am sure I probably heard it somewhere.

[ollypenrice]

Some animals have evolved a "number sense" so that they, for example, can know if some other animal has left an extra chick in the nest. A hypothesis is that all mathematical sense has evolved to detect cheating. My hypothesis is that humans have taken this rudimentary "number sense" and the abstraction power of their linguistic sense and made "abstract mathematics" from it. Well, I say "my hypothesis" but I am sure I probably heard it somewhere.

Interesting! I have a similar thought about our sense of beauty - that it is a shortcut for recognizing 'rightness' or 'fitness for purpose'. Which almost brings us full circle to the surprize that elegant smple maths often describes the universe.

As for maths in bee hives, Yes! So do I. But my question is why maths? And why is it so often simple elegant maths?

Olly

[arad85]

I'd say that the reason it's simple is that simpler things are likely to have less entropy. As systems tend to settle in minimum entropy states, simple maths is the most likely to be able to explain things....

[acey]

That's an interesting new version of the second law of thermodynamics.

[part timer]

Simple usually means stable. In general people tend to overcomplicate things. The truth is that a complicated mathematical theory is usually a way of trying to describe something you don't understand.

Sometimes Mathematical logic can make physical sense and help to understand the Physics. It doesn't always work however and that, really, is the point I think I was trying to get across earlier. Maths is not 'the truth' it's just the tool created by some advanced monkeys to solve problems and describe them.

[themos]

But my question is why maths? And why is it so often simple elegant maths?

"Maths" is defined by its use in problem-solving so it becomes "whatever works". "Simple" rules are preferred because it's hard enough to work out a simple rule's implications, a complicated rule would be hopeless.

[ollypenrice]

I'd say that the reason it's simple is that simpler things are likely to have less entropy. As systems tend to settle in minimum entropy states, simple maths is the most likely to be able to explain things....

Good point.

Olly

[brianb]

Just remember that "simple" maths is sometimes the most complex ... how many of you remember the whole semester undergraduate course that's required to prove that 1+1=2 ? If you've been there, nonlinear partial differential equations suddenly start to look rather easy....

[acey]

Systems end up in maximal entropy states - that's the second law of thermodynamics.

[arad85]

Yes, you're right. It's been a long time since I did thermodynamics.... You know what I meant though

[acey]

Actually I don't.

[ollypenrice]

Yes, you're right. It's been a long time since I did thermodynamics.... You know what I meant though

I didn't spot the inversion either (entropy is a funny word for me) but I did know what you meant. Systems shake down into their simplest state and so the maths needed to describe them is thus simplified over what it would have been had the sytem not done so. This seems quite reasonable to me. Indeed it seems like a great idea. I am more than happy to be proved wrong, however.

I know that proving 1 + 1 = 2 is formally complicated but there is another sense in which it can be considered simple - I think...

Olly

[astoc]

If you're going to be completely rational then you can only consider Mathematics (and indeed, science) within the context of human perception. But then, what we perceive is the universe we inhabit. Separation of what we observe from what 'may' exist must be done with care; religion is born from such creative endeavors.

Mathematics, perhaps as its greatest assumption, considers the universe we perceive to be the only one that warrants study. Here, perceive means inward thought and logical analysis as much as it means what we see (I'm talking logically here, not geometrically). With this in mind the patterns that we observe in Mathematics are indeed ingrained on nature. As a common example of this, consider the prime numbers.

The primes (3, 5, 7, 11, ...) are numbers which cannot be forged through multiplication of any other number. On it's own that might not seem impressive, but within the logical framework that we inhabit, it is trivial to prove that there are infinitely many of them, and that all other numbers can be built by unique combinations of the primes. Essentially they are the 'atoms' of Mathematics, yet interestingly they are dispersed throughout the natural numbers seemingly without pattern. In the quest for a pattern, however, a number of remarkable steps have been taken to show that the density of the primes (the number of them you'll find within a given range) increases in a predictable way the higher you go. Without going into too much detail, this pattern exists independent of experimentation or visual observation. It is just there and pervades every corner of the universe we seem to inhabit.

This is just one example, there are a plethora of other far more interesting pure mathematical proven theorems that given my limited experience I am unable to offer at will . One example that comes to mind is that numbers like Pi and e are what we call transcendental - that is they cannot be expressed as the solution to an equation. They exist outside of algebraic reach. Now we don't know many of these transcendental numbers at all, maybe just a handful, yet we can prove that there are far more (when you count them systematically) transcendental numbers than there are algebraic numbers. These numbers are 'out there' yet we can't seem to find them. Isn't it interesting that of the few we do know, one of the describes the ratio of a circle (fundamental in all aspects of physics) and the other the base of the natural logarithms (appears everywhere in physics, is very elegant and 'natural')?

Mathematics certainly does describe patters and properties of the universe that must be adhered to by natural phenomena.

Hope this helps a bit

John