On 5/30/2012 12:06 AM, meekerdb wrote:
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.
Hi Brent,
Yes, that is a very good point! The axiom of choice is a suspect
here. Banach and Tarsky proved a paradox of the axiom of choice, it is
the "scalar field" of mathematics, IMHO; you can get from it anything
you want.
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".
What does that word mean? Try this from
http://encyclopedia2.thefreedictionary.com/Optimality
*1.* (mathematics) *optimal* - Describes a solution to a problem which
minimises some cost function. Linear programming
<http://encyclopedia2.thefreedictionary.com/Linear+programming> is one
technique used to discover the optimal solution to certain problems.
*2.* (programming) *optimal* - Of code: best or most efficient in
time, space or code size.
Is that helpful?
--
Onward!
Stephen
"Nature, to be commanded, must be obeyed."
~ Francis Bacon
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