My apologies Harv for not giving fuller consideratuion to your arguments.
H: "MT2: "2+2=4", if, and only if, "2+2=4" actually expresses 2+2=4 as a mathematical abstract reality, and 2+2=4 actually obtains in this mathematical abstract reality."
Essentially you are saying "the rules of the game say: "2 + 2 = 4" must match the statement "2 + 2 = 4" (which looks like rules of a language?)
H: "So, MT2 seems to be getting closer to a proper mathematical definition of truth, but we still haven't defined what is meant by 'mathematical abstract reality' and what it means to express or obtain such in a mathematical abstract reality. Here is the real ball of wax. However we go about specifying this we are bound to be in for questions which we cannot properly answer. For example:
"object X lies on plane P", if, and only if, "object X lies on plane P" actually expresses an object X lies on plane P as a mathematical abstract reality, and object X lies on plane P obtains in this mathematical abstract reality. "
"We have no way to specify what it means for an abstract object to 'lie' on a plane."
We do? Chris Langan's "question and answer together". In "math world" we say "2 + 2 = 4" for each case? From a free association perspective; "object " "lie" and "plane" need only meet in a certain order?
H: " Does this mean they touch? If they touch, then what touches?"
"Touch" is not necessary; they need only be defined as MEETING within a certain context which might be quite generally defined for math purposes. This allows transferability of concept between more specified geometries.
H: "The object and plane are abstract and defining something as touching has no clear descriptive meaning."
And note that given your decision to seek a mathematical context; you have already excluded use of categories of more descriptive nature in your scenario?
H: "Similarly, where does this occur? In mathematical abstract reality?"
You could define "touch" for that mathworld.
H: " Who's mathematical abstract reality? Is everyone's reality the same as everyone else's abstact reality?"
Two meet: they are different; but one meeting.......
H: "This might seem unfair to throw these kind of issues in the middle of a simple mathematical attempt to setup an abstract situation, however this is the result of trying to establish a definition of mathematical truth if one chooses to do so."
You appear to have substituted "corespondence with the facts" with "correspondence with rules of mathworld".
H: "I believe Hilbert was one of the first wellknown mathematicians to recognize the folly of defining certain undefined terms, and in his axiomization of geometry unlike Euclid he didn't bother with the definitions of all of his terms. Rather, he simply supplied such terms as undefined. Instead, he focused on the conditions on how to treat and build on those terms, and modern axiomization was born."
So he kept definitions broad enough to allow concept transferability among different more specified contexts? But in so far as he had defined his rules; these became axioms?
H: " Truth in mathematics remains undefined, but the conditions of truth (truth functionals) is of course the heart and soul of mathematics and logic."
I do not think so: a minimaly defined rule is still defined. Truth needs no definition by math; truth is what/ who exists. In math as elsewhere; as you judge, so you are judged.
I presume by "truth functionals" you mean "structures in keeping with not contradicting the rules upon which they are built"?
Alan
