It is impossible to consider common properties of elements of an infinite
set since, as is known from psycology, a man can consider no more than 7
objects simultaneously. Therefore consideration of such objects as a
multitude of triangles seems to be impossible. Nevertheless we consider
such multitudes and obtain results which seem to be true. The method we
employ is comsideration of a very specific "*single but arbitrary*" object.
Your remarkable objection that "*if two mathematicians consider two
different arbitrary objects they will obtain different results"* demonstrates
that you are not a mathematician. Arbitrary element is not an object, it is
a mental but non-physical process which* enables one to do a physically
impossible thing : to observe an infinite set of objects
then all their common properties at a single really existing object.
Therefore two different mathematicians will necessarily obtain the same
On Wed, May 30, 2012 at 12:13 AM, Jesse Mazer <laserma...@gmail.com> wrote:
> 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
> 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
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