On 4/9/2013 4:12 PM, Dan wrote: >> So, although I had to start out by thinking >> that CH would decide something, what I have >> learned is that pluralism on these matters >> is far more valuable. > > Heisenberg's matrix mechanics and Schrodinger wave mechanics were a > 'pluralistic interpretation' in the development of quantum mechanics. > Nonetheless, they ultimately proved to be identical . Pluralism has > its value, but pluralism for pluralism's sake is unacceptable in > mathematics . We should always seek out to understand where and why > pluralism arises, and thus construct a 'unified theory' , a higher > vantage point from which all the 'pluralistic interpretations' would > appear as facets, if they cannot be reconciled in themselves.
I take it that is why you suggested looking at the Weaver article. It began, of course, speaking of a larger conceptual framework.
I will look at it again, more closely. I needed to look at New Foundations in order to reply to William Elliot today.
> If we don't understand how to look at a cube, one may see a square, > another , a hexagon . > Practitioners of intuitionist and classical logic can understand one > another and 'translate between languages' even if they do not speak a > common language . So it is with standard and non-standard analysis . > My fear is that we'll never manage to find a clear unambiguous > interpretation for the concept of set, let alone several. >
My path has been very hard.
First, what any mathematical use of a word means outside of any pattern of grammatical usage (thus covering formal and informal) is a strange question. So, everything I look at has been relative to a 'system'.
ZFC is the general system of set theory claimed implicitly by virtue of celebrity, if you understand what I mean by that. So, all of my set theoretic focus has been with that system. That does not mean I have "chosen" it as a "correct" representation.
But, because my sense of what is at issue concerns the use of the sign of equality, my "research" has driven beyond set theory although I would never have expected that it had to.
I will not try to state what I have done beyond an example to give you a sense.
The system described is just a schema in the sense that a system of quantifiers would involve a an amalgam of ortholattices based on the lines forming the affine geometry.
So, I suppose I am "Brouwerian" in the sense of grounding the classical logical negation in a construct of mathematical origin. But, returning to your final remark, I think that this illustrates just how difficult these questions of foundations can become. At least, having put in the time to decide some things for myself, it is easier to consider how other systems interrelate.