"Feeling" the relation of math to the real world

[la6ki]

I am not a mathematician but, as such, I think I have a pretty good background in mathematics. I have a good understanding and experience with calculus, differential equations, linear algebra, and probability theory. I also have interest in abstract algebra concepts, though I wouldn't say I am much experienced there.

What I think I lack, in comparison to mathematicians, is a deeper feeling for the big picture. I am not even sure how to formulate my question properly, so I'll just share my raw reasoning and hope that my problem will be understood from it.

Mathematics is used (or can potentially be used) in almost any field. It can give intuition about almost any concept and help with the solution of almost any real world problem. However, how much of the mathematical answers we get depend on the axioms we start with?

Say we discover that a particular system's behavior is completely governed by a set of differential equations. We happen to be able to solve the system analytically and are now able to predict the evolution of the system. How confident are we in the predictions of our model? Does the fact that every single mathematical concept has been ultimately derived from an axiomatic system (like ZFC) introduce any uncertainty in our expectations for the future of the system, assuming no errors were made in the process of solving the system? Do mathematicians "remember" the primary assumptions on which the truth of all those statements rely? Or do mathematical axioms have a different role in mathematics and don't really have any real world implications? Does choosing a different axiomatic system ever lead to any change in any expectations regarding any real world event?

 

[Simon Bridge]

I am not a mathematician but, as such, I think I have a pretty good background in mathematics. I have a good understanding and experience with calculus, differential equations, linear algebra, and probability theory. I also have interest in abstract algebra concepts, though I wouldn't say I am much experienced there.

What I think I lack, in comparison to mathematicians, is a deeper feeling for the big picture.

In other words - you don't actually have a good understanding of the subjects above. If you did, you'd have a feel for them.
But maybe I have misunderstood ...

I am not even sure how to formulate my question properly, so I'll just share my raw reasoning and hope that my problem will be understood from it.

Mathematics is used (or can potentially be used) in almost any field. It can give intuition about almost any concept and help with the solution of almost any real world problem. However, how much of the mathematical answers we get depend on the axioms we start with?

All of them.
In terms of real world models, the model is only as good as the data ... garbage in, garbage out.

Say we discover that a particular system's behavior is completely governed by a set of differential equations. We happen to be able to solve the system analytically and are now able to predict the evolution of the system. How confident are we in the predictions of our model?

That depends on how rigorously the model has been tested - specifically, on the number and cleverness of the attempts to demonstrate that the model is false.

What you are thinking of is the central issue in "empiricism". This is what you want to look up, you should also attempt a course in the "philosophy of science", something else to look up.
Unfortunately, as philosophy, it's off topic here.
I'm going to stick to science as it is practised.

Does the fact that every single mathematical concept has been ultimately derived from an axiomatic system (like ZFC) introduce any uncertainty in our expectations for the future of the system, assuming no errors were made in the process of solving the system?

No. The uncertainty comes from Nature. The data we use to make the predictions from will have statistical variation which makes the prediction uncertain.
Think of the maths as a language - we can use language to describe anything - including false things. A completely correct sentence can still be wrong.
The accuracy of a statement about Nature must be determined by comparing the statement against Nature.

Do mathematicians "remember" the primary assumptions on which the truth of all those statements rely?

The formal mathematics education follows a process of building the subject from basic axioms step-by-step. It's all formal theorems and lemmas and proofs for ages.
Mind you, maybe it is better later ... I never got farther.

The idea is that you should not use a mathematical concept that you are not confident someone has earlier proved ... usually you, in class. Every now and again someone attempts to overturn an earlier proof, and so break all the maths that is based on it.

Sometimes a statement is taken to be true without proof (on the hope that a proof will be forthcoming later) and it's consequences are explored ... I think Fermat's last theorem got used like that before it was proved.

But the short answer is "yes" - mathematicians "remember every base axiom" in the sense that they are following from very clever people who proved them.

Or do mathematical axioms have a different role in mathematics and don't really have any real world implications? Does choosing a different axiomatic system ever lead to any change in any expectations regarding any real world event?

The formalisms of mathematics have little to do with what we are pleased to think of as "the real world", and deliberately so. In my dealings with mathematicians I am constantly cautioned to make the problem as abstract as possible and not tell them anything of what the various symbols mean scientifically. In this way mathematicians can happily talk about an esoteric geometry on n dimensions while physicsist must work with the 3+1 space-time. What mathematicians end up with does not have to be "real" ...

If we change the axioms, we do, indeed, get something different.
Look up "non-Euclidean geometry" to see what happens when you write off one of Euclids axioms (I think it's the one about parallel lines intersecting or something.)

 

[S.G. Janssens]

Mathematics is used (or can potentially be used) in almost any field. It can give intuition about almost any concept and help with the solution of almost any real world problem. However, how much of the mathematical answers we get depend on the axioms we start with?

Say we discover that a particular system's behavior is completely governed by a set of differential equations. We happen to be able to solve the system analytically and are now able to predict the evolution of the system. How confident are we in the predictions of our model? Does the fact that every single mathematical concept has been ultimately derived from an axiomatic system (like ZFC) introduce any uncertainty in our expectations for the future of the system, assuming no errors were made in the process of solving the system? Do mathematicians "remember" the primary assumptions on which the truth of all those statements rely? Or do mathematical axioms have a different role in mathematics and don't really have any real world implications? Does choosing a different axiomatic system ever lead to any change in any expectations regarding any real world event?

I think it is important to agree in advance about what exactly is meant by an axiom, also see: A confusion about axioms and models

(To further complicate matters, the term "model" in the title of that topic should not be understood in the usual physical sense.)

 

[la6ki]

Hi guys and thank you for your replies. I think my worries of not being able to ask my question properly are somewhat confirmed. I'll try to ask 2 additional questions that may serve a better role:

1. Are there any physical theories which (ultimately) rely on mathematical axioms? For instance, would it matter if one of the ZFC axioms somehow turned out to not be true?

2. Is it at all possible for such an axiom to not be true. For instance, one of the ZFC axioms postulates the existence of the empty set. What would it even mean for the empty set to not exist?

 

[jedishrfu]

There is a NOVA show on Math and the Real World that may be of interest to you:



You can look at math as a kind of game with a given collection of rules (axioms). The rules themselves are neither true nor false. However to play the game (doing proofs, solving problems...) you must abide by them to do things correctly.

However when we apply the rules to the real world then we can state whether they are true or not based on physical experiments. If the rules don't agree with the experiment then we search for a collection of rules that do.

 

[chiro]

Mathematicians take intuition and attempt to clarify it and make it consistent.

If the intuition turns out to be wrong then the mathematics with its existing framework of consistency fixes the intuition.

I would say that over the past century or so a lot of the focus is trying to make sense of infinity in a number of ways.

It kind of took off with Cantors work and with Hilbert spaces and Operational Algebra's it is getting more involved. Calculus by the way involves infinities and a lot of work was done to test the intuition of this and make it grounded but the type of variation is a lot higher now when you extend things to many variables - and this century has seen a lot of work in understanding the mathematics of many variables - including infinitely many ones.

Probability had Kolmogorov who formed his axioms and that was the thing that really corrected a lot of bad intuition for probabilities.

The same has and is being done in fields like logic, (partial) differential equations and analysis of stochastic (random) variables and relations.

You start with intuition - which you get from studying existing mathematics and using combinations of other senses and then you see if you can clarify and/or build on it.