My Calculus book from back in the day (Calculus Second Edition Michael Spivak) starts out by stating 12 basic properties of numbers which he labels P1-P12. He states:

"Most of this chapter has been an attempt to present convincing evidence that P1-P12 are indeed basic properties which we should assume in order to deduce other familiar properties of numbers."

I had always taken this to mean that those properties were axioms upon which (with the help of a few other properties) he was going to build up the whole of Calculus.

However, a google search for such a proof pops up the following link: https://proofwiki.org/wiki/Real_Addition_is_Commutative

Which purports to be be a proof of the Commutative Property of Addition of the Reals.

I then begin to question whether I even know what an Axiom is and look it up in Wikipedia arriving at: https://en.wikipedia.org/wiki/Axiom

Which immediately provides $a+b=b+a$ as an example of a non-logical axiom for which the terms axiom, postulate or assumption are interchangeable.

  • $\begingroup$ It depends on the author. Spivak takes it as an axiom. Other formalizations do not need it as an axiom. $\endgroup$
    – Git Gud
    Aug 3 '15 at 19:51
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    $\begingroup$ it depends on whether you accept the axioms about the reals or construct the reals from other structures like the natural numbers. $\endgroup$
    – user251257
    Aug 3 '15 at 19:52
  • $\begingroup$ This fact is proven in the course of any one of the usual contructions of $\mathbb R$. $\endgroup$ Aug 3 '15 at 19:58
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    $\begingroup$ Spivak's aim is to provide an introduction to Calculus, not to settle the foundations, so he takes the properties of the system of real numbers as axioms to be able to dive into his subject. $\endgroup$
    – Bernard
    Aug 3 '15 at 20:01
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    $\begingroup$ You can see : Ethan Bloch, The real numbers and real analysis (2011) for a clear exposition of the two approaches : the construction and the axiomatic one. $\endgroup$ Aug 3 '15 at 20:02

In a sense, a postulate means here is where I want to start. If you start at a more elementary level, then your postulates may become theorems. The field postulates for real numbers are a good example of that. In (usually) Foundation of Mathematics, the basic properties of the integers, rationals, and reals are all derived from basic set theory.

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    $\begingroup$ I somehow missed that important distinction 25 years ago. Now, I'm wondering what the true foundational axioms of calculus are... $\endgroup$
    – aepryus
    Aug 3 '15 at 20:16
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    $\begingroup$ @aepryus. I've always told my students that the closer you look at something, the fuzzier it gets. That wil happen to you if you try to find where calculus starts. A good compromise might be to find a good, senior level, real analysis book. For some perspective, you may also want to look into non-standard analysis, which uses a much denser superset of the real number line to construct calculus with. $\endgroup$ Aug 3 '15 at 21:52

I think what you are asking about is not only unclear but illogical, sorry if that is a little harsh. It is not sensible to question things we should already inherently understand, and which everyone inherently sees by instinct.

There exists a philosophy about the whole of mathematics that it is only based on axioms, postulates, etc. We don't really KNOW anything, we just assume a few axioms to arrive at the conclusion of the matter, and that serves all practical purposes. This is a constructivist argument and sort of assumes that understanding is inherently relative.

I believe what the author is saying, and what is true of axiomatic systems, is that one has to check the system by making sure it doesn't contradict itself, become too specific to contain all of the subject matter at hand (too many axioms), or too general to be cohesive, or maybe express certain arguments (too few axioms).

So he is trying to "axiomatise" the system of numbers the book will use, so that it will have a logical base and not be inherently flawed, or otherwise unintelligable. So although he may not be trying to deal with the most basic level, he is trying to convey the information as logically as possible.

There will always be people trying to provide the most technical answer, but if a basic assumption makes sense to you, just stick with that and move on.


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