Is truth a legitimate goal in creating a system of mathematical axioms?

[PseuTonym]

Or is it better to aim merely for consistency?

Presuming that number theory and set theory are agreed to be branches of mathematics, it should be possible to focus on those two branches. I hope that nobody is going to attempt to hijack the thread to demand an explanation of how we determine the truth in geometry, unless there is an explanation of why it is thought that – in every branch of mathematics – our conclusions must be deduced from a list of geometrical axioms.

 

[Vic_Taltrees_UK]

• is there a point in arithmetic if it doesn’t add up?
• I am every inch behind you that we shouldn’t try to fit the whole inside the part (geometry)
• surely consistency speaks its own truth
• in the infinite branches of mathematics, each infinitely unfoldable, the ramifications of truth will prove ever more complementary
• I abhor the kind of trite “syllogisms” things too often get reduced to in these columns
• therefore truth is big
• evidence is evidence
• WOW !!!

 

[Tomdstone]

in every branch of mathematics – our conclusions must be deduced from a list of geometrical axioms.

I didn’t know that in every branch of mathematics the foundational axioms are geometrical?

 

[JapaneseKappa]

I didn’t know that in every branch of mathematics the foundational axioms are geometrical?

For a while people thought that math looked like a tree, where geometry was one of the branches. I think the current view is that all the “branches” are more like trunks; you could theoretically re-derive all of mathematics from just about any mathematical starting point.

 

[Tomdstone]

I think the current view is that all the “branches” are more like trunks; you could theoretically re-derive all of mathematics from just about any mathematical starting point.

What would be an example of a mathematical starting point for deriving the fundamental theorem of Galois theory. the Heine Borel theorem and the Cauchy Schwarz inequality?

 

[JapaneseKappa]

What would be an example of a mathematical starting point for deriving the fundamental theorem of Galois theory. the Heine Borel theorem and the Cauchy Schwarz inequality?

I mean, you can probably get to the Galois theorem and Cauchy–Schwarz inequality pretty directly from algebraic geometry. The Heine Borel theorem probably doesn’t even require the “algebraic” qualifier in front of geometry.

 

[Tomdstone]

I mean, you can probably get to the Galois theorem and Cauchy–Schwarz inequality pretty directly from algebraic geometry. The Heine Borel theorem probably doesn’t even require the “algebraic” qualifier in front of geometry.

Do you claim that the starting point for all of mathematics is algebraic geometry?

 

[JapaneseKappa]

Do you claim that the starting point for all of mathematics is algebraic geometry?

I claimed that there is no such thing as “the” starting point.

 

[Tomdstone]

I think the current view is that all the “branches” are more like trunks; you could theoretically re-derive all of mathematics from just about any mathematical starting point.

I doubt you could derive all mathematics from any starting point. Take for example the starting point of the standard Zermelo–Fraenkel set theory even with the axiom of choice added to it. How do you propose to prove or disprove the continuum hypothesis from that starting point?
Further if you start only with the standard Zermelo–Fraenkel set theory without the axiom of choice, how do you propose to construct a set which is not Lebesgue measurable?
Further, suppose your starting point is a non-measurable set. how do you propose to derive the Gauss Bonnet theorem, the Freudenthal suspension theorem or the Poincare duality theorem using a non-measurable set and nothing else as your starting point?

 

[Tomdstone]

I think the current view is that all the “branches” are more like trunks; you could theoretically re-derive all of mathematics from just about any mathematical starting point.

I doubt you could derive all mathematics from any starting point. Take for example the starting point of the standard Zermelo–Fraenkel set theory even with the axiom of choice added to it. How do you propose to prove or disprove the continuum hypothesis from that starting point?
Further if you start only with the standard Zermelo–Fraenkel set theory without the axiom of choice, how do you propose to construct a set which is not Lebesgue measurable?
Further, suppose your starting point is a non-measurable set. how do you propose to derive the Gauss Bonnet theorem, the Freudenthal suspension theorem or the Poincare duality theorem using a non-measurable set and nothing else as your starting point?
And what would be your starting point to derive the Zermelo Fraenkel set theory?

 

[rossum]

Or is it better to aim merely for consistency?

Consistency. Inconsistent axioms are useless because any statement whatsoever can be proved from them.

It is mathematically possible to derive consistent results from untrue axioms. For example, we know a lot of things about odd perfect numbers, however, no odd perfect number has ever been found. None may exist.

There are different version of the parallel axiom used to derive different geometries. At most one of those version is ‘true’, yet all the ‘false’ versions of that axiom produce a consistent geometry.

In mathematics, truth is derived from the axioms, not the other way round. For example, 1 + 1 = 10 is true given the axioms of binary arithmetic. Given different axioms, decimal arithmetic say, then 1 + 1 = 10 is false.

$0.02

rossum