Explanation of Axioms, First Principles, and Assumptions

[SF49erfan]

College sophomore here and just wanted to ask for an explanation of terms that I have come across in my math and philosophy courses thus far, but without direct teaching of them. I've seen these terms used in forum discussions and mentioned off-handedly by profs., but don't have a specific grasp of what they are.

Also, are these terms synonymous? That was the impression I got, but wasn't sure. Appreciate everyone's help here.

Thanks a bunch.

 

[harmonic_lens]

an axiom is something so intuitive to humans that we just assume proof is unnecessary. this may change over time (e.g. in 1400, the world was flat).

a first principle is something that cannot be derived from any other idea. in philosophy, i would venture to say a good example would be the value of human life. we cannot derive why we value life, because that would require knowing its meaning. instead, we just accept it as a first principle from which we base our other deductions.

and lastly, an assumption is any logical deduction made which is inherently reliant upon axioms or first principles. these deductions are assumptions because as humans, we can only assume that the interpretive aspect of our intuition which sees things as self-evident or axiomatic is indeed true (e.g. if i fall from 10 stories, i will die, because i assume that the intuitive learning that occurred in all previous life experience where i have fallen and been hurt is indeed true)

 

[gopher_p]

I get the impression that the words mean something different for philosophers than they do for mathematicians. It's also likely that the meaning changes within those groups depending on the context.

It is my experience that, among mathematicians, an assumption is any statement that is presently accepted, without demonstration, as being true, usually for the express purpose of demonstrating the truth of another statement. Its usage is fairly informal in that most mathematicians (I'm guessing) probably wouldn't be able to give a precise mathematical definition of what they mean, nor would they intend for it to mean anything particularly precise.

For most (I'm guessing) mathematicians, an axiom is kind of like a formal assumption, sometimes even acting as (part of) a definition of terminology. Logicians have a slightly different, more formal take on what an axiom is compared with your "average" working mathematician. But both would consider an axiom to essentially be a "baseline" assumption which, for present purposes, is accepted as unproved and unprovable (well ... the logicians would technically consider an axiom to be trivially proven/provable, but that's neither here nor there).

I have not heard "first principles" thrown around very often among my mathematician friends. I have used it occasionally informally to mean anything "immediately" derivable from the definitions of the terms being used (along with any "sub-definitions"). For example, if you asked me to prove that $\lim_{x\rightarrow 0}\frac{\sin x}{x}=1$ "from first principles", I would ask you what definition of $\sin$ you wanted me to use and would proceed, assuming (there's that word ) basic limit laws (since most are immediately derivable from the definition of the limit), with some sort of $\epsilon-\delta$ proof of the limit. I would not using anything involving the derivative of $\sin$ (such as l'Hopital's Rule) or any other kind of "advanced tech". It's reasonable, though that someone else might interpret the "first principles" request to indicate an even more basic proof than the one I would give. The problem with using that term in a mathematical setting is that the rabbit hole goes VERY deep. True first principles are essentially too much, and then the question turns into which first principles do you mean. Any mathematical question which has a "fair" first-principals answer would basically amount to definition chasing, in my opinion.