Re: does sqrt(2) exist in CM?
From: Keith Ramsay (kramsay_at_aol.com)
Date: 02/19/05
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Date: 19 Feb 2005 00:15:42 -0800
examachine@gmail.com wrote:
| Keith Ramsay wrote:
| > examachine@gmail.com wrote:
| > [...]
| > | So, if we eschew all the talk about infinite sets (which may be
| > | confusing in this part of the discussion), there is no reason, I
| > think,
| > | why we should not regard this series itself constructible.
| >
| > What do infinite sets have to do with it? The sequence is
| > a computable sequence of rationals, and there's nothing
| > wrong with it, regardless of one's opinion on infinite sets.
|
| The talk of infinity is getting in the way, I think.
I realize you have this finitist opinion, but is there
evidence for it?
Worry about infinity is cheap. The concept of infinity is
famous for being controversial (at various times in various
ways), which makes it an easy target. But would you claim
that you are doing more than just echoing this familiar
worry?
[...]
| > People have gotten the idea that constructivism has more to
| > do with finitism than it does. Finitistic mathematics is not
| > necessarily constructive, and constructive mathematics is
| > not necessarily finitistic.
|
| I would prefer to see a tangible example for: "finitistic mathematics
| that is not constructive".
In the theory of Thue equations, the existence of an upper
bound on the solutions was first proven by a method that's
acceptable to most or at least to many finitists, but isn't
constructive. One assumes that no such bound exists, and
deduces that there's a sequence x1,x2,x3,...,x_N of
solutions satisfying some inequalities, which then lead to
a contradiction. This is a constructive proof that the set
of solutions is not infinite, but isn't a constructive proof
that the set of solutions is finite, since it doesn't
provide a method which is sure to generate them all. (Such
a method was found later.)
| > Constructivism is not generally
| > motivated by qualms about "actual infinity", and finitists
| > seldom have any problem with the law of excluded middle, which
| > is the most common nonconstructive principle.
|
| But it seems intuitionists are concerned about abstraction of actual
| infinity.
I think it would be useful if you tried being more specific.
Who? Brouwer? I don't remember Brouwer expressing his
thinking in such terms.
When the Soviet Union still existed, I knew a woman who had
an interest in their ethnic groups like the Uzbeks. She said
the Soviets had adopted the strategy of attempting to coopt
nationalist sentiment. They had this way of presenting Uzbek
history to the Uzbeks with some anachronistic elements--
certain groups from past history were portrayed as being kinds
of patriotic proto-socialists.
To me, the way you're talking about constructivism and
intuitionism has a similar element of projecting your own
issues onto a history that doesn't have all that much to
do with them. You're hardly the only person doing this, of
course.
| > | Furthermore, if we also leave "k" unspecified, we have made the
| > | definition for an unbounded set, which would also be acceptable
to
| > some
| > | forms of constructivism I suppose.
| >
| > The claim that this sequence has a limit is nonconstructive.
|
| Who said "there _is_ a number in the limit"?
Chaitin and most other writers on the topic.
Your remark about what is "acceptable" was somewhat
nonspecific, and my point was to underline the distinction
between what is and is not constructive. The one issue that
could be termed a "problem with its not being constructive"
is that the statement that the sequence converges isn't
constructive.
| > The set of rationals in it is the kind of thing that fits
| > neatly into the approaches used by Brouwer, Bishop, Markov,
| > Heyting, Martin-Lof, and all their followers and offshoots
| > that I'm aware of. To refer to these as "some forms" of
| > constructivism is mighty silly.
|
| Of course, it does fit into many approaches. "Some forms" is not a
bad
| expression at all. I mean to say that Omega itself could be regarded
as
| constructible, just like sqrt(2), which it seems some people do not
| regard constructible.
Do we know of any at all? I think it's a view not really
worth taking seriously.
| So, what's wrong with the expression? Should I
| have said "most forms", which of course would be more precise.
It's as if someone were to ask what Christianity, Judaism,
or Islam had to say about eating meat, and I said that it
was "acceptable to many forms of" those religions. It's
correct, but it's misleading by omission. Even saying just
"most forms" would still be needlessly coy. A much better
simple description would be to say that they all accept it
with different restrictions. If I wanted to be more detailed,
I would mention among others the existence of at least one
smallish group that teaches vegetarianism, and those members
of each who are vegetarian for moral reasons. But it still
remains true that each of these religions has not yet
collectively embraced such a position, nor is it one of the
teachings of their respective founders.
I claim that finitist views like yours play a similarly
modest role in constructivism. You can surely find
constructivists who are finitists (well, I assume you can)
just as you can find vegetarian Christians, but there's
even less basis for describing finitism as a natural part
of constructivism than there is for describing vegetarianism
as a natural part of Christianity.
I think finitism has been kept from having much importance
in either constructive mathematics or in mathematics in
general by virtue of its not being a very good idea in the
first place.
| > | [And as usual, once you erase
| > | 'infinity' from the dictionary, everything starts much better
| sense]
| >
| > Quite the contrary. It's more like the newspeak of 1984.
|
| I disagree. Using it on the contrary is like the system of the
| scholastics.
Probably you consider the scholastics to have been speaking
in much less meaningful terms than I do!
I just don't see what kind of lack of "sense" you are
saying can be cured by refraining from using the term
"infinity".
| > What are we supposed to do when someone says that the set
| > of primes is infinite? Gently but firmly insist that he
| > change that to "for every n, there exists a prime >n"?
| > Tell him it's not a set?
|
| I am not certain, but it does not seem a bad option once you accept
| that infinite entities do not exist.
There's nothing wrong with phrasing the infinitude of primes
that way, of course, but to say that one needs to phrase it
that way is to place unnecessary, arbitrary restrictions on
mathematical expression. We know that these are equivalent
ways of saying the same essential thing, so if one is okay
then surely the other is too.
| A proper response would be to say: "the list of primes (with no
| repetition) is unbounded". There is no need to talk about anything
| further than lists in a finite world, you see. This kind of talk fits
| general constructivist philosophy better, "it is not the case that
you
| can ever _build_ such a completed list of primes" is what it means to
a
| constructivist.
I don't think "infinite" is usually defined to mean merely
"not finite" in constructive mathematics. I had thought that
the more positive sense of there existing an infinite sequence
of distinct elements was more usual, but I don't have any
handy references to confirm that.
| There is a semantic change. This statement is
| completely mechanical, and indicates only the mechanical capacity of
a
| finite, mechanical mathematician. (Thus is also compatible with "our"
| consructivist view of computationalism)
|
| > What about Dedekind cuts? They don't exist, because they
| > have infinite sets in them?
|
| I think you understand the implications of rejecting infinity
perfectly
| well, as do others.
|
| > There's no clarification to be gained by ignoring this,
| > one of the main concepts of modern mathematics.
|
| And of obsolete theology and metaphysics, unfortunately.
So? You're going to toss out the bulk of modern mathematics
for the sake of attacking what you consider obsolete
philosophy?
The sort of framework you're describing-- mathematics as
just combinatorics-- avoids the use of such objects as
arbitrary sets of integers. But we know what an arbitrary
set of integers is. It's something of the form {n: P(n)}
where P is some property of integers, where two such sets
are considered equal if the same integers satisfy their
associated properties.
I object to this kind of restriction as much as I do partly
because I suspect the popular belief that constructive
mathematics is some highly restricted form of mathematics
derives partly from people hearing that this or that
basically arbitrary requirement might turn out to be needed
for some mathematics to be constructive-- at least according
to some versions of constructivism. You can enprison your
mathematics this way if you like, but I wish to remind people
that constructivism per se makes no such demand.
Keith Ramsay
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