On 5/29/2012 8:47 PM, Stephen P. King wrote:
On 5/29/2012 5:18 PM, Jesse Mazer wrote:
On Tue, May 29, 2012 at 4:38 PM, Aleksandr Lokshin <aaloks...@gmail.com
<mailto:aaloks...@gmail.com>> 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.
He didn't refer to a specific property but to a specific choice of element, which is what
Loskin says entails the magic ability to select one among an infinite number. He
apparently thinks of it like the complement of the axiom of choice: to pick an element you
need to say,"Not this one. Not this one. Not..." an infinite number of times.
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.
??? The function is specified in advance, e.g. "triangles" is a function that picks out
things with three sides meeting pairwise as three vertices. But I have no idea what you
mean by "optimality".
Brent
--
Onward!
Stephen
"Nature, to be commanded, must be obeyed."
~ Francis Bacon
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