deductive reasoning and axioms

[abskebabs]

Hi everybody. I remembered when having a discussion today with my driving instructor about centripetal forces, mechanics and gravity; that I stumbled upon the realisation that whereas I know the maths regarding these subjects, I cannot justify it or gain a true understanding from first prinicples or axioms. For example, I know that centripetal accelleration is equal to [math] \frac{v^2}{r}[/math], but I cannot derive or prove it from axioms/first principles. To be honest I do not know any either.

 

I would love to be able to refurnish up my understanding from the ground up, to put it metaphorically. Thereby I have 2 favours to kindly ask of you. Could someone please tell me how, or show me how I could derive this expression for centripetal acceleratuion?

 

Secondly and more importantly, could someone tell me what the basic or founding axioms of mathematics were, from which the following theorems were deductively reasoned. Am I wrong in thinking these axioms could be so neatly comprised, and do you think it is important to know them? Also is this kind of reasoning taught in Maths courses and related subjects?

 

I would be very grateful for replies:-)

[matt grime]

There is no (finite) set of axioms from which all of mathematics is deduced. The axioms you choose either explcitly or implicitly are those which seem most reasonable for the task at hand.

[abskebabs]

There is no (finite) set of axioms from which all of mathematics is deduced. The axioms you choose either explcitly or implicitly are those which seem most reasonable for the task at hand.

I see... are there any that are important, and would be useful learning in your opinion, or at least being aware of?

[matt grime]

Depends what you want to do with it.

 

The standard axioms of set theory are ZF (Zermelo Frankel), and you might want to throw in the continuum hypothesis, the generalized continuum hypothesis and definitely the axiom of choice, though there is no real need to do this in any depth. Just be aware of the fact that it is forbidden to have a set of sets that (do not) contain themselves (Barber's paradox or Russell's paradox), e.g. so that you're happy with the idea the class of all sets is not a set.

 

You also ought to know the axioms of the three geometries (Euclidean, hyperbolic, and spherical).

 

It might be a good idea to get to grips with the fact that

 

1. axioms are not self evidently true; they are not in themselves true at all, nor are they false, they are just strings of symbols.

 

2. it is models of axioms that count - this is where we actually do maths.