Existence of Axioms

Question

I understand that Mathematics is built off of a set of axioms but how do we know that such Axioms do exist? How can we say that we must have these certain axioms?

I thought about this for a while and was wondering if there is a clear answer (in a book or common knowledge to mathematicians).

Answer 1

Some useful comments from :

• Ethan Bloch, Proofs and Fundamentals : A First Course in Abstract Mathematics (2nd ed - 2011);

[page 47] Today, virtually all branches of pure mathematics are based on axiomatic systems, and work in pure mathematics involves the construction of rigorous proofs for new theorems. Much of the great mathematics of the past has been recast with a precision missing from its original treatment. [...]

Mathematics has moved over time in the direction of ever greater rigor, though why that has happened is a question we leave to historians of mathematics to explain. We can, nonetheless, articulate a number of reasons why mathematicians today use proofs. The main reason, of course, is to be sure that something is true. Contrary to popular misconception, mathematics is not a formal game in which we derive theorems from arbitrarily chosen axioms. Rather, we discuss various types of mathematical objects, some geometric (for example, circles), some algebraic (for example, polynomials), some analytic (for example, derivatives) and the like. To understand these objects fully, we need to use both intuition and rigor. Our intuition tells us what is important, what we think might be true, what to try next and so forth. Unfortunately, mathematical objects are often so complicated or abstract that our intuition at times fails, even for the most experienced mathematicians. We use rigorous proofs to verify that a given statement that appears intuitively true is indeed true.

[page 49] Ultimately, a mathematical proof is a convincing argument that starts from the premises, and logically deduces the desired conclusion. [...] What is it that we prove in mathematics? We prove statements, which are usually called theorems, propositions, lemmas, corollaries and exercises. There is not much difference between these types of statements; all need proofs.

[page 50] Theorems are not proved in a vacuum. To prove one theorem, we usually need to use various relevant definitions, and theorems that have already been proved. If we do not want to keep going backwards infinitely, we need to start with some objects that we use without definition, as well as some facts about these objects that are assumed without proof. Such facts are called axioms, and a body of knowledge that can be derived from a set of axioms is called an axiomatic system [emphasis added].

[page 91] A completely rigorous treatment of mathematics, it might seem, would require us to define every term and prove every statement we encounter. However, unless we want to engage in circular reasoning, or have an argument that goes backwards infinitely far, we have to choose some place as a logical starting point, and then do everything else on the basis of this starting point.

[page 92] A common misconception is that mathematicians spend their time writing down arbitrary collections of axioms, and then playing with them to see what they can deduce from each collection. Mathematics (at least of the pure variety) is then thought to be a kind of formal, abstract game with no purpose other than the fun of playing it (others might phrase it less kindly). In fact, nothing could be further from the truth. Not only would arbitrarily chosen axioms quite likely be contradictory, but, no less important, they would not describe anything of interest. The various axiomatic schemes used in modern mathematics, in such areas as group theory, linear algebra and topology, were arrived at only after long periods of study, involving many concrete examples and much trial and error. [...] The point of axiomatic systems is to rigorize various parts of mathematics that are otherwise of interest, for either historical or applied reasons.

the common basis for all systems of axioms used in contemporary mathematics, which is set theory. Though of surprisingly recent vintage, having been developed by Georg Cantor in the late nineteenth century, set theory has become widely accepted among mathematicians as the starting place for rigorous athematics. We will take an intuitive approach to set theory (often referred to as “naive set theory”), but then build on it rigorously. Set theory itself can be done axiomatically, though doing so is non-trivial, and there are a number of different approaches that are used.

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For a very useful overview on the process of "discovery" of set theory axioms, see at least :

• Dedekind's Contributions to the Foundations of Mathematics

• The Early Development of Set Theory

• Zermelo's Axiomatization of Set Theory.

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For an illuminating discussion regarding the growth of mathematical knowledge (and the fact that mathematical knowledge does not start "from scratch" with some axioms) see :

• Imre Lakatos, Proofs and Refutations : The Logic of Mathematical Discovery (1976, Cambridge U.P.).

Answer 2

A concrete example of the process of axiomatization...

For millennia, humans have been counting things, recording their number, adding and subtracting them, etc. They were even able prove quite sophisticated theorems establishing various properties of numbers without the use of any formally stated axioms.

To introduce more rigour into the proofs of these theorems, it was felt that mathematicians should agree on a short list the essential properties of number from which all other known properties could be derived using only the rules of logic, e.g. that every number has a unique successor. These essential properties or axioms were originally identified by Peano and Dedekind in the late 19th century. For the first time, it was possible, for example, to actually prove that addition was associative.