http://sites.math.rutgers.edu/~zeilberg/Opinion108.html is interesting On Mon, Jun 5, 2017 at 6:11 PM Henry Baker <hbaker1@pipeline.com> wrote:
Perhaps I'm a bit warped, but having had a fair amount of mathematical logic, what with axioms, theories, models and such, I no longer believe in numbers, infinities, etc., in ways that seem to tie many people up in knots.
There are usually many different models for the same theory, so it becomes kind of pointless to worry about which one is "best", because different models produce different insights. If one person thinks of +oo as the limit "point" of the positive integers, that's great; if someone else wants to deny pointhood to +oo, but rather talk about some sort of a limit, that's also great.
WRT .999... v. 1.000..., there are often congruence classes of representations that can't be distinguished with a certain class of operations -- this happens all the time in computer science when we have to choose a digital representation for an object, and keeping the object in a "canonical" form is (or seems to be) too much work -- e.g., a rational fraction representation that doesn't always "reduce to lowest common terms".
At 07:45 AM 6/5/2017, James Propp wrote:
In recent email conversation with a polite but insistent .999... dissenter, I've been made aware of strands of thought that rejects the mathematical concept of infinity as being unrealistic, incoherent, and repugnant (three different charges requiring three different sorts of defense).
There are books and articles and on-line videos that approach the topic of infinity from different angles, but are there any that address these attacks head-on?
It's possible that what's really being rejected (at least by some dissenters) is not the concept of infinity in isolation but the whole enterprise of making deductions in imaginary worlds (like the world of Euclidean geometry) to draw conclusions that are applicable in our world. A convincing defense of "unrealistic" concepts like a Euclidean point or the set of all counting numbers should in my opinion include an admission that we don't know why the mathematical enterprise has been so successful (the "unreasonable effectiveness" phenomenon).
What's been done along these lines?
One thing that's relevant, sort of, is an essay by Isaac Asimov in which young Isaac gets into an argument with a philosophy professor, challenging him to produce half a piece of chalk and then pointing out that it's probably not exactly HALF. But I think the philosopher was onto something when he got the whole in-class argument started by insisting that mathematicians are idealists because they believe in imaginary numbers. Old Isaac's point in his essaywas that complex numbers, properly understood, are no more fanciful than ordinary fractions; but you can turn this around and say that ordinary fractions, properly understood, are no less fanciful than complex numbers.
Jim Propp
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun