Re: 3-valued logic and Principle of the Excluded Middle
- From: "T.H. Ray" <thray123@xxxxxxx>
- Date: Sun, 28 Sep 2008 09:00:53 EDT
amy666 wrote
ive only read it very fast ( dont have time )
but is this pi hat ( p^ ) the reference to an integer
n wich is the position of the digit sequence in pi of
0123456789 ?
isnt that question answered nowadays by spigot
algoritms , advances in number theory and computer
searches ?
regards
tommy1729
Let's isolate the relevant portion from Brouwer's
lecture:
"Now every construction of a bounded finite nature in a
finite mathematical system can only be attempted in a
finite number of ways, and each attempt proves to be
successful or abortive in a finite number of steps.
We conclude that every assertion of possibility of a
construction of a bounded finite nature in a finite
mathematical system can be judged, so that in these
circumstances applications of the Principle of the
Excluded Third are legitimate.
"But now let us pass to infinite systems and ask for
instance if there exists a natural number n such that
in the decimal expansion of pi the nth, (n+1)th, ...,
(n+8)th and (n+9)th digits form a sequence 0123456789.
This question, relating as it does to a so far not
judgeable assertion, can be answered neither
affirmatively nor negatively. But then, from the
intuitionist point of view, because outside human
thought there are no mathematical truths, the assertion
that in the decimal expansion of pi a sequence 0123456789
either does or does not occur is devoid of sense.
"The aforesaid property, suppositionally assigned to the
number n, is an example of a fleeing property, by which
we understand a property f, which satisfies the following
three requirements:
"(i) for each natural number n it can be decided whether
or not n possesses the property f,
(ii) no way of calculating a natural number n possessing
f is known;
(iii) the assumption that at least one natural number
possesses f is not known to be an absurdity.
"Obviously the fleeing nature of a property is not
necessarily permanent, for a natural number possessing f
might at some time be found, or the absurdity of the
existence of such a natural number might at some time be
proved."
Do you see what Brouwer is saying about the difference
between finite and infinite properties? A number called
"infinity" is absurd. Properties of assumed infinite real
numbers, like the decimal expansion of pi (this is
arbitrary; we might choose sqrt2, e.g., to the same end)
aren't "real" properties until we calculate them by some
algorithm, and these properties may change with further
calculation, or another construction.
As Davis and Hersh explain, suppose we call a possible
sequence of one hundred zeros in the decimal expansion
of pi by the name "pi-hat." "If pi does contain such a
sequence, and it starts at an even number in the
expansion, then pi-hat is greater than pi. If it starts
at an odd number, pi-hat is less than pi. Now let's
compute, not pi, but the difference pi-hat minus pi.
Call this difference Q. Is Q positive, negative or
zero?"
Davis and Hersh continue, "The constructivists' argument
is that none of the three is true. Q _will_ be zero,
positive or negative at such time as someone determines
which of the three is the case; until then, it is none
of the three. Thus, mathematical truth is time-dependent
and is _subjective_, although it does not depend on the
consciousness of any particular live mathematician."
This time-dependency is a very important issue nowadays,
as computing power grows exponentially. You might look
into the work of computer scientists such as Gregory
Chaitin to get an idea of why.
Tom
.
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