On 5/29/2012 2:22 PM, Jesse Mazer wrote:
On Sun, May 27, 2012 at 2:51 PM, Aleksandr Lokshin
<aaloks...@gmail.com <mailto:aaloks...@gmail.com>> wrote:
To make the general idea more clear , suppose we are proving the well-
known formula S = ah/2 for the area of a triangle. Our proof will
necessarily begin as follows:
“Let us consider AN ARBITRARY triangle…” Here we obviously apply the
operator of the free will choice which cannot be replaced by the
random choice. In fact, let us imagine that our proof begins in such a
way : “Let us consider A RANDOMLY SELECTED triangle…” Surely, such a
beginning will not lead us to the desired proof. The formula obtained
for a randomly selected triangle is not necessarily valid for all
triangles!
The notion of "choosing" isn't actually important--if a proof says
something like "pick an arbitrary member of the set X, and you will
find it obeys Y", this is equivalent to the statement "every member of
the set X obeys Y". In formal logic this would be expressed in terms
of the upside-down A symbol that represents "universal quantification"
in a given "universe of discourse" such as the set of all triangles (
http://en.wikipedia.org/wiki/Universal_quantification ). In fact, in
proofs like this one typically *doesn't* imagine choosing any specific
triangle, one just thinks about properties that would apply to every
member of the set and thus every "arbitrary member", like the property
of having three sides or or the property of having its angles add up
to 180 degrees in the case of a triangle obeying Euclidean axioms. And
note that any mathematical proof can be expressed in a formal symbolic
way using logical symbols/rules as well as some symbols/rules specific
to the domain of mathematics under consideration (see
http://en.wikipedia.org/wiki/Formal_proof ), and in this form the
proof will often contain the universal quantification symbol, but
there is no separate symbol corresponding to the notion of "pick an
arbitrary member of the set".
On the other hand when proving the formula S=ab/2, obviously, it is
impossible to consider all the triangles simultaneously.
Why not? One can consider the properties that all these triangles are
defined to share, and then show that these properties, along with the
axioms of geometry, can be used to derive some other properties they
will all share.
Thus the
operator of the free will choice must be used inevitably.
More widely, let us consider a variable x which is running about a
sphere of radius 1. Let us pose a question: what does x denote?
Clearly,
a) x does not denote an object,
b) x does not denote a multitude,
c) x does not denote a physical process.
In my opinion, x denotes the free will choice which the reader of the
mathematical text must do. So, the notion of a variable inevitably is
based on the notion of the free will.
If it really depended on free choice, then you would have no way of
being sure that just because *your* choice obeyed a certain rule,
every other possible choice of examples from the same set would do so
as well.
Jesse
--
Hi Jesse,
Would it be correct to think of "arbitrary" as used here as meaning
" some y subset Y identified by some function i or mapping j that is not
a subset (or faithfully represented) in X, yet x => y : x /subset X"?
The "choice" of a basis of a linear space comes to mind. The idea is
that one it is not necessary to specify the method of identification ab
initio <http://en.wikipedia.org/wiki/Ab_initio>.
--
Onward!
Stephen
"Nature, to be commanded, must be obeyed."
~ Francis Bacon
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