Is Circular Reasoning or Undefined Terms Better for Axiomatic Foundations?

[madness]

Which do you consider to be a better foundation for some system of (axiomatic) beliefs: a circular argument, or one with undefined terms? Take Newton's laws for example, we can either say "an inertial frame is one in which Newton's laws hold, and Newton's laws hold in an inertial frame", or we can leave an inertial frame undefined. I remember reading that the mathematician Weyl went back to redo much of the logician Frege's work, because he was unhappy with his use of recursive logic as a foundation. Any thoughts?

 

[0xDEADBEEF]

If given the choice circular logic still enforces more consistency. A system with undefined entities becomes meaningless very fast. Because as you add more and more properties to your ill defined entity, and you only make one mistake and add something contradicting, then you are likely arguing about the contents of an empty set. And those have amazing properties :)
They are even and odd at the same time. Smaller and bigger then 1. Contradict fermats last theorem...

 

[kote]

Which do you consider to be a better foundation for some system of (axiomatic) beliefs: a circular argument, or one with undefined terms? Take Newton's laws for example, we can either say "an inertial frame is one in which Newton's laws hold, and Newton's laws hold in an inertial frame", or we can leave an inertial frame undefined. I remember reading that the mathematician Weyl went back to redo much of the logician Frege's work, because he was unhappy with his use of recursive logic as a foundation. Any thoughts?

Aren't axioms simply well defined assumptions? I don't know if I'd call them circular, just assumed. I'm not sure what you'd call undefined axioms. You can't create well-formed formulas without defined logical rules. I'm not sure what good a logic is without well-formed formulas.

Did you mean to mention physics or is this a purely formal question? If you start asking about modeling the real world using logic you get into other issues like intentionality.

 

[madness]

Hmmm I don't think I expressed myself very clearly in my original post. As far as I can see, any rigourous logical system will need some fundamental undefined entities, or it will need entities defined in terms of each other (ie circular). For example, we might take numbers to be undefined. If we keep asking "and what is this?" on and on, we should eventually reach an undefined entity or start a circular loop. I used an example from physics because it was in my head at the time, no other reason.

 

[apeiron]

All modelling has this internally developed dynamic. But then there are two kinds of "circularity".

There is the single scale tautological loop. This means that and that means this. You do go round in a symmetric loop not getting anywhere.

The alternative and more philosophical is a loop with hierarchical scale. So a dichotomy in which the local and global emerge as opposing "ends" of the "circular" interaction.

This is why you have maths based on integers and axioms, physics bases on initial conditions and fundamental laws. Knowledge or modelling as an interaction between local and global scales.

 

[madness]

Yes I understand that either basic undefined entities or circular definitions are always necessary, this is the point of my post. I was asking whether basic undefined entities (maybe numbers in arithmetic or points in geometry) are preferable to entities defined in terms of each other (which would be circular in any closed system). As I said earlier, the mathematician Weyl went back to redo a lot of Frege's foundational work on logic, as he was unhappy with his recursive definitions. It seems as though Frege preferred circular logic, whereas Weyl preferred undefined entities.

 

[kote]

Yes I understand that either basic undefined entities or circular definitions are always necessary, this is the point of my post. I was asking whether basic undefined entities (maybe numbers in arithmetic or points in geometry) are preferable to entities defined in terms of each other (which would be circular in any closed system). As I said earlier, the mathematician Weyl went back to redo a lot of Frege's foundational work on logic, as he was unhappy with his recursive definitions. It seems as though Frege preferred circular logic, whereas Weyl preferred undefined entities.

I don't think I'm sure what you mean by entities. In any case, basic (circular?) axioms are always described in a meta-language outside of the system. This meta-language is undefined within the system. When Euler defines a line as a length without breadth, neither length nor breadth are defined. You can always replace the "defined" axioms of a system with the undefined words used to write them.

By minimizing the number of axioms we minimize the number of undefined terms. I guess what I'm saying is circularly defined axioms can be shown to be undefined axioms. I don't know that there's a difference.

(Thanks for the good question... it made me go back through my logic textbook )

 

[madness]

Ok so in the example of a line you could say that length and breadth are undefined entities. You talk about language - the dictionary is basically a huge circularly defined system, all the words being defined in terms of each other. Of course you also have an intuitive understanding of what a "lion" is for example, which would correspond to the basic undefined entity. I think you are probably right that the two ways of formulating the system are really the same.

 

[apeiron]

Again, I would argue that there is a systematic circularity that is the basis of both good philosophy and ordinary thought.

The assumption here is that a good definition of some axiomatic entity needs to come about as a construction. A lion is...insert a list of atomistic properties that add up to create the cannonical lion. In the same way, we would want to construct a line via a succession of points (the classical Euclid approach to the axioms of geometry anyway).

But the alternative way of creating these local thought objects is via top-down constraint. A lion is what remains once all these non-lion like properties are subtracted away. And Euler's definition of a line would have the same top-down constraint approach.

It is important to understand that what I am saying is that both construction and constraint are employed to establish meanings - it is an interaction across scale.

Where things get tricky of course are where we are trying to "localise" the most global or universal kinds of "objects". Defining things like truth, beauty, good.

A lion is easy to localise. It is constrained by a more general sense of cat (which is not-dog, not-goat - though a toddler may take a while to get this). And then even more generally by being not-inanimate, not-safe, etc.

But the kind of most general metaphysical objects of thought must arise as complementary pairs - as dichotomies or antimonies. By mutual constraint where each arises as being completely not what the other is.

So change and stasis. Each is defined entirely as being what the other is not. Same with discrete and continuous or substance and form. There is a circularity, a mutually creative interaction, by which each is defined.