Primal

Yes reality ultimately determines the truthfulness of these axioms, however I think they are probably known to humans via very strongly wired brain mechanisms.(Knowing such basic truths as the world was external would be essential for our survival.)

But I do not of course mean reality determined this via meaning complete knowledge of reality. I am not saying you must know everything to know something.

John Page

This makes sense in being consistent with how we believe certain things to be self-evident! (Ironically so because it seems self-evident that our minds/brains must be constructed to employ axioms).

Cheers, John

Witt

John: "Would you agree it appears that reality is the judge of the relative effectiveness of a set of axioms? The problem then becomes one of pinning down reality?"

Primal:
Yes reality ultimately determines the truthfulness of these axioms, however I think they are probably known to humans via very strongly wired brain mechanisms.(Knowing such basic truths as the world was external would be essential for our survival.)

I don't aree with either of you guys.
Axioms are either tautologous or contradictory.
If tautologous they claim nothing about the world at all.
Reality is not tautologous.

For example: that there are an infinite number of numbers between any two different numbers is a tautology (a theorem) of real numbers, but, there is no evidence that reality (space-time) is compact in this way.

Witt

John Page

Axioms can be definitions or statements (which can be consider a definition of a state of affairs). Tautologies need not be true e.g. "god exists therefore god exists".

"There exists an objective reality" is an axiom of certain philosophies and this is neither a tautology or contradictory.

Anything can be contradictory if one syas so!

Cheers, John

Bookman

Neither. It is an axiom. It may be accepted or rejected, leading to different results in mathematics.

The most accessible example is the so-called "parallel postulate" of elementary geometry. Accept the parallel postulate as an axiom, and one develops the results of plane geometry. Reject the parallel postulate as an axiom, and one develops some of the many results of non-Euclidian geometry.

Mathematics is a tool for building models, not a tool for discovering heretofore unknown truths about the universe (see Physics). Most mathematicians (not including those operating in the universe of set theory) accept the axiom of choice because it simplifies their work and doesn't harm the model. Set theoreticians find this problem a useful conjecture.

Not really. I'm assuming that this statement is based upon the conclusion about AC which states:
For most mathematicians, this presents no problem. In most mathematical disciplines, the acceptance of existance claims without regard to whether the thing which exists can be determined are accepted if proven.

Bookman

Witt

Witt : Axioms are either tautologous or contradictory.

John: Axioms can be definitions or statements (which can be consider a definition of a state of affairs).

Statements are not considered a definition of states of affairs.

John: Tautologies need not be true e.g. "god exists therefore god exists".

A tautologous falsity, is a contradiction in terms.
All tautologies 'need' to be true.

"god exists therefore god exists", is as tautologous as any statement can be. (p->p), is undeniable.
But there is no inference here.

The word 'therefore' does not apply!

John: "There exists an objective reality" is an axiom of certain philosophies and this is neither a tautology or contradictory.

I don't agree. All statements of existence are tautologous or contradictory, as are statements of identity.
The generality of logic, demands it.

John: Anything can be contradictory if one syas so!

Certainly not...in my world.

Truth is not a matter of whim, is it?

Witt

Luiseach

Maybe not a matter of whim, but perhaps more a matter of consensus?

John Page

I understand. The title was a play on words which you can equate with acceptance and rejection if you wish.
Thanks. However, in interpreting mathematical results I feel its important to understand what can be related directly back to physcial reality and what has a more indirect relationship not. e.g. One can find 3 sheep, you cannot find minus three sheep. You can find a set of things but the set itself can never fully qualify as a member of itself. The notion of minus three sheep can be helpful to shepherds and a set that can belong to itself may be useful in set theory.

Contradictions don't really occur in math because quantities (numbers of things) are homogenous. For logic, however, the interpretation of AC can result in violation of the LOI and Witt has expounded the theorems of logic he thinks are at fault in causing Russell's Antinomy.

Cheers, John

John Page

Just to clarify, I'm not claiming all statements are so. e.g. "This statement says nothing at all about whether anything is true."

How about "The experiment was carried out in total darkness"? Do you consider this qualifies?
Need? All tautologies are true, by definition.
However, reality doesn't. An objective view is not absolutely so unless tested against all conditions in space, time etc. - we end up, therefore, with degrees of subjectivity. Accordingly, all axioms contain or imply definitions, e.g. "All X are Y" defines that at least some Y are X. "Rocks are hard" defines hardness as an essential quality of rocks. I agree with you on statements of identity, though - if they weren't true then they'd be self-contradictory.
No, truth is a matter for the system chosen to determine it (hence the OP).

Cheers, John

Luiseach

I know you were directing this response at someone else, but I just wanted to say I like its clarity.

It clears up a question I have been trying to answer for some time.