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
> 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 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 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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