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: everything-list.domain.name.hidden
> 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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Received on Sun Sep 13 2009 - 19:12:17 PDT