Originally posted by Coletti
Although you may "sort of" object to my use of the term faith, it means nothing more than something believed without proof - or assumed true.

I have never heard of a mathematician claim that 2+2 = 4 is anything but true.

Could you explain what a rational world-view is, and why one is to be predicated on faith?

You'd have failed math in Babylon.

Originally posted by royalchicken
[b]But you are still presupposing the axioms are correct.

No I'm not. The word 'correct' has no meaning at all in this sort of reasoning except 'consistent with the axioms'. I can choose whatever axioms I find interesting, and they define the notion of correctness.

In Euclidean geometry, I don't assume the axioms are 'true' other t ...[text shortened]... in the trivial sense that they are consistent with themselves.

Axioms are not propositions.[/b]

Axioms are propositions by definition. Trivial or non-trivial, they must be assumed true for that system because they define that system. The system is deduced from the axioms. And to do this, the axioms must be assumed true.

There is no escape - resistance if futile! 😀

Originally posted by Coletti
The system is deduced from the axioms. And to do this, the axioms must be assumed true.

There is no escape - resistance if futile! 😀

Axioms are propositions by definition.

It's not really important, but I've always seen 'proposition' as a synonym for 'theorem'.

Trivial or non-trivial, they must be assumed true for that system because they define that system.

Axioms are 'assumed true' in the sense that if A is an axiom and T a theorem true within the axiom system containing A, then A --> T is noncontradictory.

We need make no assumption that the axioms are 'true' outside the system, however, so saying 'the axioms must be assumed true' is vague, because you can't assume something is true in a system defined by it, because it is the only standard of truth that exists.

Originally posted by royalchicken
I'd like you to clarify what you mean by 'true'. When I assume an axiom is true, what properties am I ascribing to it?

What do I mean by true?

How about: conforms to other true propositions or is not contradictory or contrary to other true propositions - and does not logically infer contradictions or contraries.

To determine truth, one must systematize the relationships between terms. To have a system, one must first assume some propositions are true - these are by definition axioms.

World-view truths should be coherent and comprehensive to describe the world. Mathematical truths are formally true - coherent within that system of mathematics. Although mathematical systems can be coherent with the world, it is not necessary (hyperbolic geometry does not compare well with our perceptions of reality).

World-view systems are not the world itself - it is a description of the world.

Originally posted by Coletti
What do I mean by true?

How about: conforms to other true propositions or is not contradictory or contrary to other true propositions - and does not logically infer contradictions or contraries.

To determine truth, one must systematize the relationships between terms. To have a system, one must first assume some propositions are true - these are by d ...[text shortened]... of reality).

World-view systems are not the world itself - it is a description of the world.

'Conforms to other true propositions'? Let's make this somewhat concrete.

Suppose I want to do some Euclidean geometry. I assume that the First Postulate is true. Please list the things I have assumed about the First Postulate in doing this.

Originally posted by royalchicken
[b]Axioms are propositions by definition.

It's not really important, but I've always seen 'proposition' as a synonym for 'theorem'.

Trivial or non-trivial, they must be assumed true for that system because they define that system.

Axioms are 'assumed true' in the sense that if A is an axiom and T a theorem true within the ax ...[text shortened]... mething is true in a system defined by it, because it is the only standard of truth that exists.[/b]

I agreed with you all the way up to the last statement - "you can't assume something is true in a system defined by it, because it is the only standard of truth that exists."

You must assume the axioms are true within the system to have any meaning. But you don't "prove" axioms withing the system they infer. I think that is what you are getting at. To prove an axiom within the system they infer would be circular - for you've already assumed it is true to begin with.

Originally posted by royalchicken
'Conforms to other true propositions'? Let's make this somewhat concrete.

Suppose I want to do some Euclidean geometry. I assume that the First Postulate is true. Please list the things I have assumed about the First Postulate in doing this.

By assuming the First Postulate - you are saying that it does not, and will not infer, any statements that contradict any other postulates (axioms). True is defined in relationship to other propositions or axioms or postulates.

Originally posted by Coletti
By assuming the First Postulate - you are saying that it does not, and will not infer, any statements that contradict any other postulates (axioms). True is defined in relationship to other propositions or axioms or postulates.

Can I construct a consistent system with exactly one axiom?

Originally posted by Coletti
I agreed with you all the way up to the last statement - "you can't assume something is true in a system defined by it, because it is the only standard of truth that exists."

You must assume the axioms are true within the system to have any meaning. But you don't "prove" axioms withing the system they infer. I think that is what you are g ...[text shortened]... the system they infer would be circular - for you've already assumed it is true to begin with.

I'm not claiming that axioms can be proved at all -- that would violate the definition. I'm claiming that when we construct axiom systems, external notions of 'true' or 'correct' have nothing to do with it.

Originally posted by Bosse de Nage
Could you explain what a rational world-view is, and why one is to be predicated on faith?

You'd have failed math in Babylon.

A rational world-view is a one the presupposes the laws of logic. There are "irrational" world-views that deny the validity of logic and reason, but these are self-refuting and incoherent.

To have a rational world-view requires a theory of knowledge. How do we know what we know. But a theory of knowledge can not be proven logically prior because that would mean you already know something before you have proven you can know something. So the theory of knowledge must be assumed true. And assuming this is true requires you to believe something without logical proof from prior truth, i.e. it requires a faith.

Originally posted by royalchicken
I'm not claiming that axioms can be proved at all -- that would violate the definition. I'm claiming that when we construct axiom systems, external notions of 'true' or 'correct' have nothing to do with it.

What are external notions of truth? Do you mean what is true external to the system has no logical bearing on the truth within the system - I agree. Do you mean true itself has a different meaning - I disagree. True is always with regard to the system that the proposition or axiom is part of.

Originally posted by Coletti
What are external notions of truth? Do you mean what is true external to the system has no logical bearing on the truth within the system - I agree. Do you mean true itself has a different meaning - I disagree. True is always with regard to the system that the proposition or axiom is part of.

Exactly. That's why it doesn't get us anywhere to say we 'assume axioms are true'. Instead, we simply state them; they define the notion of truth, and we don't conclude anything about anything outside the system (in a purely axiomatic argument) from what we prove within the system. When we state our axioms, we are (with some Goedelian ambiguity) defining truth, not assuming it. We're committing acts of God, not acts of faith.

Originally posted by royalchicken
Exactly. That's why it doesn't get us anywhere to say we 'assume axioms are true'. Instead, we simply state them; they define the notion of truth, and we don't conclude anything about anything outside the system (in a purely axiomatic argument) from what we prove within the system. When we state our axioms, we are (with some Goedelian ambiguity) defining truth, not assuming it. We're committing acts of God, not acts of faith.

Assume them true or state them - it's the same thing. To state them is to assume them true. The notion or concept of true is not different from one system to another - the 'assignment' and/or proof of the true/false value of statements is systematic and internal - the the concept of 'true' is external. We can not make any sense of systems if we don't have a prior notion of what true means. You don't define 'true' with a system. The concept of 'true' is really an axiom of all coherent systems. It is the 'self-evident' law of contradiction that makes the concept of true mean something.

How do you check a system for consistency or lack of contradiction without a prior concept of what 'true' means? You can not do it. We do not define "truth" by stating axioms, we are merely defining what statements are true within that system as a consequence of first assuming or assigning a 'true' state propositions to be the axioms of the system. We can not determine what's true within or systems if we do not assume something is true to start with.

But it is true we are committing acts of God when we define new systems in mathematics. This is especially clear when we create geometric worlds that have no "real world" correspondence. I find it fascinating that we can create geometries that are entirely coherent and internally consistent but do not look like anything we normally perceive around us. I think it reflects the image of God in man - our ability to create abstract worlds.

Originally posted by Coletti
Assume them true or state them - it's the same thing. To state them is to assume them true. The notion or concept of true is not different from one system to another - the 'assignment' and/or proof of the true/false value of statements is systematic and internal - the the concept of 'true' is external. We can not make any sense of systems if we don' ...[text shortened]... around us. I think it reflects the image of God in man - our ability to create abstract worlds.

The 'self-evident' law of non-contradiction is itself an axiom in the first-order predicate calculus; logic itself is an axiom system, so you still haven't given a 'prior concept' of what 'true' means that is not a consequence of some axiom-system. Truth (in axiomatic reasoning) is always relative to the axioms used (in practice, our axiom-systems almost always include a first-order predicate calculus, so in practice, but not in principle, your comment is correct), and this is by definition sufficient for constructing axiom systems and proving theorems within them. I don't see the faith; we could pick our axioms at random if we liked.

Specifically, we could choose P and ~P as our entire system of axioms, with no others, and since this axiom system does not include the axioms of a first-order predicate calculus, you can't cry foul at the contradiction, because there is no rule prohibiting it. It's still an axiom system though, and at least two statements, the axioms themselves, are true.

Originally posted by Bosse de Nage
I can't get into the spirit of this thread. Will somebody please give me a preposterous axiom?

Preposterous Axiom: Coletti knows what he is talking about when it comes to logic and mathematics.