On 5/29/2012 4:38 PM, Aleksandr Lokshin wrote:
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 simultaneously* considering then all their
common properties at a single really existing object. Therefore two
different mathematicians will necessarily obtain the same result.
Hi Aleksandr,
This makes mathematicians very special as they are able to escape
the bounds of resources that all non-mathematicians are subject to. How
would a mathematican prove to a layman that he truly has such powers?
Would they tend to beincorrigible
<http://www.merriam-webster.com/dictionary/incorrigible>? :-P
On Wed, May 30, 2012 at 12:13 AM, Jesse Mazer <laserma...@gmail.com
<mailto:laserma...@gmail.com>> wrote:
On Tue, May 29, 2012 at 3:01 PM, Aleksandr Lokshin
<aaloks...@gmail.com <mailto: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"?
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Onward!
Stephen
"Nature, to be commanded, must be obeyed."
~ Francis Bacon
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