On 5/29/2012 5:18 PM, Jesse Mazer wrote:
On Tue, May 29, 2012 at 4:38 PM, Aleksandr Lokshin
<[email protected] <mailto:[email protected]>> 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.
That's just about the number of distinct "chunks" of information you
can hold in working memory, so that you can name the distinctive
features of each one after they are removed from your sense experience
(see http://www.intropsych.com/ch06_memory/magical_number_seven.html
). But I'm not talking about actually visualizing each and every
member of an infinite set, such that I am aware of the distinctive
features of each one which differentiate them from the others. I'm
talking about a more abstract understanding that a certain property
applies to every member, perhaps simply by definition (for example,
triangles are defined to be three-sided, so three-sidedness is
obviously one of the common properties of the set of all triangles).
Do you think it's impossible to have an abstract understanding that a
large (perhaps infinite) set of objects all share a particular property?
A single finite and faithful (to within the finite margin of error)
representation of "triangle" works given that definition. This is there
nominalism and universalism come to blows....
Your remarkable objection that "*if two mathematicians consider
two different arbitrary objects they will obtain different
results"* demonstrates that you are not a mathematician.
Huh? I didn't write the phrase you put in quotes, nor imply that this
was how *I* thought mathematicians actually operated--I was just
saying that *you* seemed to be suggesting that mathematicians could
only prove things by making specific choices of examples to consider,
using their free will. If that's not what you were suggesting, please
clarify (and note that I did ask if this is what you meant in my
previous post, rather than just assuming it...I then went on to make
the conditional statement that IF that was indeed what you meant, THEN
you should find it impossible to explain how mathematicians could be
confident that a theorem could not be falsified by a new choice of
example. But of course I might be misunderstanding your argument,
that's why I asked if my reading was correct.)
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.
So you agree mathematicians don't have to make an actual choice of a
specific element to consider? Then how is free will supposed to be
relevant if there is no actual choice whatsoever being made?
--
Why do you keep insisting on a "specific" property to the "choice"
while being shown that the a priori "specificity" itself that is
prohibited by the definition. The point is is that what ever the choice
is, there are ab initio alternatives that are not exactly known to be
optimal solutions to some criterion and some not-specified-in-advance
function that "picks" one.
--
Onward!
Stephen
"Nature, to be commanded, must be obeyed."
~ Francis Bacon
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