On Tue, May 29, 2012 at 3:01 PM, Aleksandr Lokshin <aaloks...@gmail.com>wrote:
> <<*The notion of "choosing" isn't actually important--if a proof says > something like "pick an arbitrary member of the set X, and you will find it > obeys Y", this is equivalent to the statement "every member of the set X > obeys Y"*>> > No, the logical operator "every" contains the free will choice inside of > it. I do insist that one cannot consider an infinite set of onjects > simultaneously! > Why do you think we can't do so in the way I suggested earlier, by considering common properties they are all defined to have, like the fact that each triangle consists of three straight edges joined at three vertices? If I construct a proof showing that, if I take some general properties as starting points, I can then derive some other general properties (like the fact that the angles add up to 180), where in such a proof have I considered any specific triangle? Do you think mathematicians actually have to pick specific examples (like a triangle with sides of specific lengths) in order to verify that a proof is correct? If they did choose specific examples, and only verified that it worked for those specific examples, how would they be able to achieve perfect confidence that it would be impossible to choose a *different* example that violated the rule? If you prove something is true for an "arbitrarily chosen member" of the set, this implies that in a scenario where someone other than you is doing the choosing, you should be totally confident in advance that the proof will apply to whatever choice they make. If the set they are choosing from is infinitely large, how could you have such perfect confidence prior to actually learning of their choice, without considering shared properties of "an infinite set of objects simultaneously"? -- You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to email@example.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.