On 5/30/2012 4:45 AM, Bruno Marchal wrote:
On 30 May 2012, at 08:12, Stephen P. King wrote:
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.
Banach and Tarski proved an amazing theorem with the axiom of choice,
but it is not a paradox, in the sense that it contradicts nothing, and
you can't get anything from it.
Bruno
The axiom of choice allows for violation of conservation laws, if
it where to be a physical law.
--
Onward!
Stephen
"Nature, to be commanded, must be obeyed."
~ Francis Bacon
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