fom
Posts:
1,968
Registered:
12/4/12


Re: some amateurish opinions on CH
Posted:
Apr 11, 2013 4:43 PM


On 4/11/2013 1:48 PM, apoorv wrote: > "> That is Euclid :'every line segment can be extended in either direction' >> Can it be ? Who can verify? > > I don't need to "verify" . Just apply the principle of sufficient > reason to show how various properties fit together (including > infinitude) . > > There's an interesting correspondence between the points of the plane > (unbounded/infinite) and the points of the surface of the sphere > (bounded/finite) > http://www.youtube.com/watch?v=JX3VmDgiFnY > It's absurd to reject one and accept the other . " > Can we subdivide a segment ad infinitum? > A line segment of zero length > Should that be a null segment, or a point? > > > "> Is there always a next cell on the tape? Is the tape infinite? >> Can these questions ever be answered? >> >> >> >>> Within infinity is harmony. >> >> But can something finite really represent the infinite? >> >> apoorv > > Why should we even be able to talk about the infinite if it's > something completely beyond our gasp?Is it some cruel joke form Nature? " > We talk about so many things ,some forming the present reality and some > being possibilities for the future. > Apoorv >
You hit the nail on the head with your statement.
Perception of the future as a collection of possible futures introduces modal thinking and language. Tense logic is represented similarly to modal operators.
Once one begins to deal with possibility, what one can and cannot talk about becomes harder.
Leibniz framed existents in terms of their potential existence (possible worlds) because practical knowledge of the world was insufficient to specify an individuated being. Mathematically, this is later found in Hilbert who reportedly stated something like "that which is not contradictory exists".
A different tradition follows from Newton and his followers because Newton "let the mathematics speak for itself". Ultimately, this led to the foundational crises and interpretation of universal quantifiers as a course of values rather than an arbitrary application.
It is curious that the quantifiers over constructive objects in Markov have the classical feature of being applied to any arbitrarily given constructive object. The practicality that had led to Leibniz' expansive view of existents is the same practicality that leads to a rejection of the courseofvalues interpretation for universal quantifiers in the Russian constructive school.

