Mathematics and the real world

nightflyer

Why is mathematics - a language developed through reasoning - so effective when it comes to describing the empirical world? Could either of the following have anything to do with this phenomenon:

- the brains that conduct and observe the empirical experiments are naturally the same type of brains that develop the mathematics: human.

- the empirical world is somehow bound, on a metaphysical plane, to function based on mathematical principles.

I would also like to raise another question, which I have touched upon in previous posts, but would like to discuss in more general terms. Is mathematics a good tool when it comes to describing biological systems, or are they too complex to fit into the models?

Any thoughts, links or other references are much appreciated!

All the best,
Anders.

CaptainQuasar

I think the thing is that mathematics is the easiest language / tool with which to construct objective models, i.e. models that two people, even if they disagree, can examine and reach the same conclusions. The tradition of science has frequently found that constructing a mathematical model of a real-world phenomenon is a powerful tool to accompany experimentation and that these mathematical models are also usually the easiest way to clearly communicate concepts.

Gib Z

Mathematics may potentially explain biological systems, the question is whether or not the mathematics can develop that mathematics =]

HallsofIvy

I once read a response to the idea that science fiction has predicted so much of what has actually "come true"- science fiction has predicted many things, most of which have not come true. But if you predict enough things some of them are bound to come true!

The key point is that there is not just one "mathematics"- there are many different "mathematical theories" (group theory, linear algebra, Euclidean geometry, etc. are all different "mathematical theories"). Mathematical theories are "templates". At the basis of any mathematical theory are "undefined terms" and axioms using those undefined terms. To apply mathematics to a "real" problem, we look for a "template" (a mathematical theory) that "fits". That is, we assign meanings, from the problem, to the undefined terms. If with those values, we can show that the axioms are true, then we know that all theorems derived from the axioms and all "methods" derived from those axioms will work for this problem. Of course, the best we can hope for is that we can show that the axioms are approximately true. Since real problems always involve measurement, which is approximate, we cannot expect them to be exactly true.

oldman

That's why I believe that mathematics, taken as a whole, may be a larger (or perhaps more varied) invented structure than discovered reality itself. There seem to be parts of mathematics which don't correspond to parts of reality, spare mathematics, as it were, from a physicists point of view. Perhaps Hamilton's Quaternions are an example? But then what look like spare parts suddenly turn out to be very useful -- Tensor analysis for General Relativity for example. One never can tell!

As to the original question: , I've set out my ten cents worth in another concurrent thread: (http://www.physicsforums.com/showthread.php?p=1611545)

arildno

Of course, as an addendum to Halls' post, with a "mathematical template", we should ALSO understand that set of rules of inference (i.e type of logic) that allow us from the axioms and undefined terms to deduce what else might belong in the template.

Furthermore, when modelling, we must assume that the world, or the world facet ALSO abides by those rules of "deduction".

CaptainQuasar

That's not entirely true. A great deal of scientific progress has been made with completely incorrect scientific models - corpuscular optics, phlogiston chemistry, the Rutherford atom. Inaccuracy in a model certainly limits its usefulness but by no means does it make the model useless.

For example, QM and GR are very far from being integrated; at least one of them must have major errors or inadequacies. But that doesn't prevent them from being essential to science.

arildno

Hmm..you misunderstood me, possibly because my bad phrasing:

Any prediction made from mathematical modelling "should" be understood to have the form :

"INSOFAR as our rules of inference holds in the world, and the structures in it has been correctly represented in our model, THEN it must hold that we will see...blah-blah".

That is what I meant by "must assume".

CaptainQuasar

Ah, I see. Yes, that seems like it's an assumption behind employing models in science. Though it definitely seems to me that when theorists are having fun they just go nuts and may not be too concerned with that.

arildno

Not too concerned??

That's their main concern!

For example, if the various models within string theory hadn't, say, been able to imply general relativity, no one would have bothered with them.

And if some such model predicted something utterly wrong, for fun's sake that whales tend to fly, particularly on midsummer eve over the Scottish highlands, then that theory would end up in the garbage bin.

What theorists play with are alternate models, trying to deduce divergent predictions of those models, disputes that hopefully can be settled by some experiment.

However, they need to "play" quite a lot with their models as mere mathematical beasts if they are to master them in such a way that they can yield such predictions.

you've got to get to know the beast you're riding on, and then you first must de-focus on where you want it to go.

CaptainQuasar

You're saying that theorists never get overly entranced with the intricacies of the model they're building and veer off down a path that an experimentalist would never look twice at? You're a theorist, aren't you?