On 12/5/2013 4:09 PM, Zeit Geist wrote: > On Thursday, December 5, 2013 1:42:31 PM UTC-7, fom wrote: >> On 12/5/2013 2:28 PM, Michael F. Stemper wrote: >>> On 12/05/2013 01:04 PM, Zeit Geist wrote: > >>>> Be careful here, the "arbitrarily chose" sequence does Not need to be >>>> a "random" sequence. >>>> It is just a sequence of Real Numbers who only properties is that it >>>> IS a sequence of Real Numbers that "supposedly" contains all Real >>>> Numbers. >> >>> There is no need for the assumption that it contains all reals. We can >>> prove that no sequence of reals contains all of them without having to >>> first assume that it does. >> >> I had to think about that. >> Cantor's proof had been directed to an audience >> who thought of "infinity" as a monolithic concept. >> Proving that any list asserting to put the set >> of reals in correspondence with the naturals was, >> in fact, not a complete list demonstrated that >> "infinity" could be viewed as a plural notion subject >> to logical analysis. >> >> Your statement reflects Hilbert's formalism. In his >> papers he specifically mentions how the ideal formal >> axioms relegate the class-based constructions of >> Dedekind and Cantor to definable structures within >> the theory. Thus, they lose the import they have >> outside of a formalist theory. > > I agree. I consider myself to be a Platonic-Formalist. > It is possible for any Consistent Formal System to Exist ( in "Reality" ), because the Forms of that Formal System Exist ( in our Minds ). > > However, it is Not the Duty of the Mathematicians to find any import of their Objects outside of the Theory. Today the Mathematician is Not Neccesarily the Scientist and vise versa. That is why Not All Mathematics applies directly to Science. >
The problem for me is that much of current mathematics seems to have tangled the different approaches to foundational thought. I had expected to learn about the continuum hypothesis many years ago. Instead, I woke up one day believing it to be true.
In spite of my currently deteriorated skills, I had found mathematics fairly straightforward. But, since that time I am always faced with sorting through statements and proofs with the shadow context of how it might apply to an independent question.
That makes things a little harder. :-)
These days, I am very conscious of the background materials and have to try to keep them distinct. I only recently discovered a little fact concerning my personal assumptions. Consequently, I find myself extremely grateful for the work of Freidman and Simpson in reverse mathematics. I need to learn more about what they have done.