Axioms & Faith: What's the Difference?

[disregardthat]

In which way? They seem fine to me.

The axioms as premises are fine in themselves, but it is their mathematical equivalence I protest.

Axiom 1 is analogical to an axiom which states the mathematical existence of a mathematical object in a non-constructive way. That is absurd and contradictory to me. I don't buy into the game in which mathematics is the meaningless play with symbols, and mathematical models is detached from this. For me, mathematics is what we call mathematical models where statements makes sense.

 

[dx]

Pick up any high school geometry book. It should start with a discussion of points and lines. It will say explicitly that they are undefined. Then come the axioms. They do not define points and lines either, but they do rule out some commonplace misconceptions such as marks made with a pencil on a piece of paper. The problem with trying to define things is that they must be defined in terms of other things. These other things need to be defined and so on ad infinitum. So to cut the knot, Euclid just gave up and made them abstract.

As to proving theorems about things not defined, you use the axioms (these are theorems not proven). So let's not define foo and bar, and let's accept two axioms:

Axiom 1: There is a foo.
Axiom 2: Every foo is a bar.

Theorem 1: There is a bar.
Can you prove this theorem even though foo and bar are not defined? Note that the axioms do not define foo and bar either, but they do rule out that either of them could be Santa Claus (apologies to our Christian friends). What's more, the axioms might be vacuous. That is, nothing in the real world nor in the world of ideas, satisfies them.

If you take your two axioms to be the definition of a mathematical structure, then yes, those axioms do define 'foo' and 'bar', in the sense that any property of these elements, as they occur in this structure, is implicitly contained in the axioms.

A given mathematical structure, such as Euclidean geometry, can be axiomatized in many ways, but for a set of propositions to be considered axioms, they must completely define the structure. All properties of the concept of 'line' are implicitly contained in the axioms, and therefore the axioms implicitly define the idea of 'line', and also implicitly define the whole scheme of Euclidean geometry. Here, by 'define', I mean mathematically defined.

The reason I brought this up in the first place was to show that the Euclidean axioms about lines and points are not to be thought of as 'self-evident truths', or 'things accepted on faith'. The axioms are simply to be thought of as the definition of these concepts (implicit definitions, to be precise). This removes the mystery surrounding the foundations of geometry, where people originally thought that the axioms are somehow a prori "truths", and the reason for this being that they could not distinguish between the mathematical/conceptual scheme of Euclidean geometry, which by itself is physically vacuous, and their intuitive and tacit conversion of this into a physical theory connected with their experience.

 

[Hurkyl]

At this point, I think you're debating over your favorite meaning of an English word, rather than over mathematics.

 

[Hurkyl]

All that is required is that they be consistent and independent.

Actually, neither of those is required of axioms either: first is merely a desirable property, and it is often useful to violate the second.