Thanks,
This does indeed clarify the subject and puts it in a perspective
that I feel that I can understand as much as possible without working through
the intricacies of the proof. m.a.
----- Original Message -----
From: Bruno Marchal
To: [email protected]
Sent: Sunday, September 13, 2009 1:12 PM
Subject: Re: Yablo, Quine and Carnap on ontology
Marty,
Could you please clarify to a non-mathematician why the
principle of excluded middle is so central to your thesis (hopefully without
using acronyms like AUDA, UD etc.).
Without the excluded middle (A or not A), or without classical logic, it is
harder to prove non constructive result. In theoretical artificial
intelligence, or in computational learning theory, but also in many place in
mathematics, it happens that we can prove, when using classical logic, the
existence of some objects, for example machines with some interesting property,
and this without being able to exhibit them.
In my preceding post on the square root of two, I have illustrated such a non
constructive existence proof. The problem consisted in deciding if there exist
a couple of irrational numbers x and y such that x^y is rational.
And by appying the excluded middle, in this case by admitting that a number
is either rational or is not rational, I was able to show that sqrt(2)^sqrt(2)
was a solution, OR that (sqrt(2)^sqrt(2))^sqrt(2) was a solution. This, for a
realist solves the existence problem, despite we don't know yet which solution
it is. Such an OR is called non construcrtive. You know that the suspect is
Alfred or Arthur, but you don't know which one. Such information are useful
though.
Many modern schools of philosophy reject the idea. Thanks,
Classical logic is the good idea, imo, for the explorer of the unknown, who
is not afraid of its ignorance.
Abandoning the excluded middle is very nice to modelize or analyse the logic
of construction, or of self-expansion.
Classical logic can actually help to exhibit the multiple splendors of such
logic, even, more so when assuming explicitly Church thesis, or some
intuitionist version of Church thesis. It is a very rich subject.
Now there are Billions (actually an infinity) of ways to weaken classical
logic. When it is use in context related to "real problem", I have no issue.
When we will arrive to Church thesis (after Cantor theorem), you will see
that it needs the excluded middle principe to make sense.
Few scientists doubt it, and virtually none doubt it for arithmetic. It is
the idea that a well defined number property applied on a well defined number
is either true or false. The property being defined with addition and
multiplication symbols.
I hope this help. Soon, you will get new illustration of the importance of
the excluded middle.
I could also explain that classical logic is far more easy than non classical
logic, where you have no more truth table, and except some philosopher are
virtually known by no one, as far as practice is taken into account.
Technically, UDA stands up with many weakening of classical logics, but it
makes the math harder, and given that the arithmetical hypostases justifies the
points of view by what is technically equivalent weakening of classical logics,
it confuses the picture.
To a non mathematician, I would say that classical logic is the most suited
for comparing the many non classical internal views of universal machines. I
would add it helps to take into account our ignorance. A simpler answer is that
without it I have no Church thesis in its usual classical sense.
Bruno
----- Original Message -----
From: Bruno Marchal
To: [email protected]
Sent: Sunday, September 13, 2009 4:02 AM
Subject: Re: Yablo, Quine and Carnap on ontology
Given that I am using "Platonic" in the sense of the theologian, and not
in the larger sense of the mathematician, it would be nice to cooperate a
little bit on the vocabulary so as not confusing the mind of the reader.
I am commited to the use of the excluded middle in arithmetic, that's all.
Once you accept the excluded middle
principle, like most mathematicians, you discover there is a
"universe" full of living things there, developing complex views.
Bruno
http://iridia.ulb.ac.be/~marchal/
http://iridia.ulb.ac.be/~marchal/
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