Stephen Montgomery-Smith wrote: > firstname.lastname@example.org wrote: > > Stephen Montgomery-Smith <email@example.com> wrote: > > > >>guenther vonKnakspot wrote: > > > > > >>>I agree with you in that it is not a crackpot view that "mathematics > >>>should try to constrain itself to physical reality". It is however a > >>>very deplorable one, comparable to an intelligent person intentionally > >>>constraining himself to the intellectual level of his environment. > > > > > >>I answered this elsewhere. I think my statement has been > >>overinterpreted. What I meant was that mathematicians should use > >>reality as a guide. Otherwise they may fall into the trap of discussing > >>things that future generations will compare to discussions about how > >>many angels can fit onto a pinhead. > > > > > > But opinions of reality change as well. When people were > > considering the question of how many angels can fit onto a pinhead, > > angels were considered a reality. The underlying question is, > > how many infinitely small objects can fit into an infinitely > > small space. The "reality" of infinitely small objects > > is unknown. Electrons are currently believed to be point > > particles, but that may change someday. >
One difference between beliefs about electrons and beliefs about angels is how such questions are to be resolved. I don't think Aquinas (see below) was particularly /illogical/ in his /reasoning/ about angels; but his source of "experimental data" was the Bible and the lives of the saints, as opposed to real-world observations. So he wasn't doing "science" as we now know it, but neither was he simply engaging in poetic whimsy.
But to return to the current discussion, there is a (to me, obvious) qualitative difference between asserting that electrons or angels are actual entities "in the real world" having particular properties; and asserting that there "exists" a well-ordering of the real numbers.
IMHO, asserting that "angels exist" is /always/ a statement about the real world; whereas a claim that "the set of naturals as defined by (the usual formulation) exists" is more in the nature of an agreement that allows a mathematical argument to go forward, in spite of the different philosophical positions readers of such an argument might take regarding what the natural numbers "really are".
> Yes. I wish I knew what this "angels on a pinhead" argument was really > about. I know that us moderns like to laugh at the old theologians for > discussing this, but I wonder if I am only getting half the story, and > in fact there are some very real issues at stake here. >
Apparantly, it's a matter of how many angels can dance on the /point/ of a very fine /needle/. Which makes more, less, or the same amount of sense to consider, depending on your tastes.
At any rate, according to the above link, the reference is an Enlightenment Era satirical jab at Thomas Aquinas, who instead inquired whether several angels could be in the same place at once (his answer was no).
Gotta love that Aquinas guy:
"That the saints may enjoy their beatitude and the grace of God more abundantly they are permitted to see the punishment of the damned in hell" - Thomas Aquinas
Sounds a bit like the allure of reality-based TV shows.