Re: Why does Cantor a target for cranks?

From: "Peter Webb" <webbfamily-diespam...@xxxxxxxxxxxxxxx>
Wildberger's hyperbole of speech notwithstanding, his
argument is not particularly cranky.
http://web.maths.unsw.edu.au/~norman/views2.htm

I'm looking at that now. It looks rather cranky to me, but also
rather interesting and insightful, a mix of the two. As I start to
respond here, I'm at the point where he points out (not his words
here, my paraphrasing:) that in category theory we side-step
Russel's paradox by dealing with categories which are not
fullfledged sets, so we don't have *all* finite groups for example
collected into a single set, rather we have a category of finite
groups which is basically a schemata for dealing with such things.
So-far that's all standard. Now he applies the same logic to the
integers. We don't really need *all* the integers collected into a
set. It suffices, for purposes of arithmetic, that we have a
category of integers, with general rules for working with them. In
that way we eliminate the need for infinite sets. That's an
interesting point of view, IMO. An alternative somewhere between
his view and the standard view of Peano is my algorithmic view,
that if we have a computing process that generates a stream of
output data in a deterministic way, and that program could in
principle run forever, always generating more and more data without
bound, then it's meaningful for such a specific process (algorithm)
to talk of the totality of all data it will ever generate if
allowed to run forever. So we don't exclude infinite sets, but we
include them only as a way of talking about specific unbounded data
output from algorithms.

Now he gets into the rational numbers. To a mathematician, a
rational number is an equivalence class of pairs of integers. To a
child a rational number is just a particular fraction. Long ago I
realized that mathematics is essentially "systemic ambiguity",
whereby we conflate various things that are really different, but
we treat them as if they were not different, creating ambiguities
as to what we are really talking about, but we do conflation a
systemic way. For example, we work with specific fractions, but
whenever we have two fractions n/d and m/e where d*m = n*e we
conflate the two fractions as if they were the same. Thus 1/2 and
2/4 are conflated as if the same fraction. We make it clear in each
field of math which such ambiguities are in effect. For example, in
data processing, we have the fraction 1/2, or we have the fraction
2/4, and we have to make it clear which we have at any time,
especially if we ask the object what its numerator is and later ask
what its denominator is. We would like the two answers to be
consistent. We wouldn't like the fraction object to tell us at one
moment the numerator was 1 and at the next moment the denomiator
was 4, because the fraction changed from 1/2 to 2/4 between the two
calls. Likewise in elementary education about skills with
fractions, we need to talk about each particular fraction
separately, as the student works with them, showing how to
manipulate one to become the other, but keeping track at each
moment of what we have now. But in other parts of math, we
generally treat 1/2 and 2/4 as equivalent. Likewise in group
theory, we may talk about specific groups, where the additive group
containing the integers and the multiplicative group containing all
powers of 5 are different groups, but they are isomorphic so as
abstract groups they are the same. Likewise when constructing the
real numbers, we build objects that are Dedekind cuts or
equivalence classes of Cauchy sequences or equivalence classes of
nested-interval sequences, all of which are *different* objects,
even though they are isomorphic in arithmetical structure. And we
define an embedding of the rational numbers within each, a 1-1
mapping from the original rational numbers we started with into
each of those real-number systems. The rational images in each
system are *not* the same object as the actual rational number we
started. The rational numbers are *not* a subset of any of the
three defined systems of real numbers. But once all the properties
have been verified, we stop talking about the three constructions
of reals, instead we talk about the abstract reals which are
ambiguously whichever of the three we want, and we stop talking
about the original rationals and each of the three rational-real
embeddings, instead we talk about the abstract rationals which are
ambiguously any of the four different actual rational systems.

Actually my original example of "systemic ambiguity" dealt with
simple integers as an abstraction of counted sets of specific
objects. For example, I have two ears, two eyes, two hands, two
feet, two nostrils, and from those examples and many more we have
the abstract notion of a set containing two objects. Notationally,
we can write the number 2 in ink or pencil or felt-tip-pen or chalk
or type it into a computer, we can write it in Arabic numerals or
in English or any of several other languages, or in Roman numerals.
We can write it in binary. And we can write it large or small, with
any color of pen or chalk, any font or style, bold or italic. Yet
all these different ways to express 2 or II or zwei, and the
cardinality of all those various sets of two ears or two hands, are
conflated together into a single concept of the number 2. The key
in math, whether conflating various numbers 2 into a single
abstract number 2, or conflating various groups of order 6 into the
abstract symmetry group on three objects and various other groups
also of order 6 into the abstract cyclic group of order 6, is to
conflate only things which have properties in common, state clearly
what those common properties are, not try to derive anything that
is specific to some members of the category but not others and
claim it applies to them all, and to be consistent in sticking with
a particular category throughout a derivation/proof/problem instead
of switching definitions midway.

This critical issue of describing the points on the continuum should
have a strong connection with notions of computability, ...

OK, here he passes moral judgement on math, which IMO is wrong.
He's basically stating his religious position here. Everything
before this point was his interpretation of how math works, with
some jibes at some usual practices, but no pronouncements of a
moral/religious nature, until here.

Even the `computable real numbers' are quite misunderstood. Most
mathematicians reading this paper suffer from the impression that the
`computable real numbers' are countable, and that they are not
complete. As I mention in my recent book, this is quite wrong. Think
clearly about the subject for a few days, and you will see that the
computable real numbers are not countable , and are complete.

Long ago I came to a similar conclusion: The computable numbers are
what I called "sub-countable", a subsequence of an enumeration, but
since membership in the subsequence isn't decideable (per a
variation of Turing's famous theorem that no Turing machine can
decide the halting problem), there's no way to enumerate that
subsequence. The subsequence must remain a theoretical possibility,
an idea that some Turing machines produce acceptable output (a
neverending sequence of digits, or an explicit HALT at some point
after which we cite an infinite sequence of zero digits) and others
don't (they keep running but never produce another digit after some
point, leaving us hanging forever as to whether they will ever
produce another digit or gracefully HALT), and there's no way to
know *all* that will produce acceptale output and *all* that won't,
so there's no way to eliminate exactly the ones that won't and
thereby enumerate exactly the sequence that will. We have to
"believe" that some machines produce acceptable output and some
don't, and that these two possibilities exhaust the space of all
Turing machines of the sometimes-produce-output-digit class, and
that for each such machine it's an unchangeable fact whether it
does or does not, so that it's well-defined whether such a machine
is in the sub-sequence that generates computable reals or not. But
we can't actually produce that definition of exactly which are in
the sub-sequence and which aren't without that leap of faith to
just assume the sub-sequence is well-defined.

So I agree with Wildberger that the computable reals can't be
enumerated, but agree with Cantor that per an oracle that decides
Turing-machine halting etc. such a sub-sequence would exist, but
disagree on a very subtle point which I won't discuss here.

Now as to Wildberger's claim that the computable reals are complete:
<http://en.wikipedia.org/wiki/Computable_number>
... A computable Dedekind cut is a computable
function D\; which when provided with a rational number r as input
returns D(r)=true\; or D(r)=false\; , satisfying ...
A real number is computable if and only if there is a computable
Dedekind cut D converging to it.
So in that sense, Wildberger is correct, the computable reals are
complete, but:
... the set of computable numbers is not closed under basic
operations such as taking the supremum of a bounded sequence, ...
In that sense Wildberger is not correct. I'm confused. Help!

<http://en.wikipedia.org/wiki/Complete_space>
... a metric space M is said to be complete (or
Cauchy) if every Cauchy sequence of points in M has a limit that is
also in M.
Somehow I thought there was an equivalent definition in terms of
Dedekind cuts instead of Cauchy sequences, but I don't see it on
that WikiPedia page. Am I mistaken, or is that page missing it?

Thinking about the difference: Given a computable nested-interval
sequence (with each endpoint rational, and interval diameter
approaching zero), that defines or determines a computable Dedekind
cut (decision function on rationals), because for any rational not
exactly at the cut point we simply run the nested-interval process
long enough to exclude the rational one way or the other, which is
all we need for definining the computable Dedekind cut in the case
of non-rational cut point. So in all such cases the computable
nested-interval sequence directly defines the computable Dedekind
cut. For rational cut point, we simply use the natural algorithm
for testing whether a rational number is strictly less than the cut
rational or not, where not includes a surefire test for equality.
In that case the computable nested-interval sequence doesn't define
the rational number, but in theory it "determines" it, and in fact
is *is* computable (every rational is computable).

But given a computable Cauchy sequence, there might not be any
computable nested-interval sequence or computable Dedekind cut.
With just a Cauchy sequence, there's no way to know how long to run
it to be sure we know whether it'll converge above or below any
given rational number we're trying to test. Is that why computable
real numbers are computable Dedekind complete but not computable
Cauchy complete? So the truth of Wildberger's claim depends on
whether he's talking about Dedekind or Cauchy completeness?

Back to Wildberger Web page... He talks about integers which can
actually be expressed by finite expressions small enough to fit
within the physical Universe. He mentions some really big numbers
that are individually expressible such as by using exponents of
exponents of exponents, but where the totality of all the smaller
numbers are too immense to explicitly express *each* of them
without amiguity using the entire Universe as RAM. Hence:
... Conclusion: The vast
majority of numbers less than $c$ cannot be written down in our
universe. These numbers are completely inaccessible to us, and always
will be. But $c$ can be written down in one line. Numbers `close' to
$c$ in the sense of having expressions that are not all that different
from that of $c$ form little `islands of simplicity' in a sea of
overwhelming complexity.
OK, for example googleplex+1 is easy to express. Now the real fun:
It follows that long before you get to $w,$ you are going to reach
numbers whose prime factorizations are impossible, since some of the
factors, if they existed, would require more room to write down than
$w.$
Indeed. I have that problem when I look at the Mersenne primes and
factorizations of the Mersenne non-primes. For example, writing
down 2**37 - 1 in that form is easy, and stating it's not a prime
is easy, but writing down the actual prime factorization of it
takes a bit more work. Now suppose that 2**32582657 - 1 is a prime,
but that 2**(2**32582657 - 1) - 1 is not a prime. I rather doubt
it's physically possible to write down the prime factorization of
that non-prime, or even just its one largest prime factor.

Still, it's a "convenient fiction" to imagine that the entire set
of integers exists as a set, as the entire possible output of a
computing algorithm that enumerates the integers, a lot more
convenent than "re-inventing the wheel" for each specific finite
set of integers by re-deriving all the math that works for that
finite set and noting the exceptions which don't work there even if
they *would* work if only we had more RAM available. Better to do
the math in the "convenient fiction" system that has nice uniform
properties, then reduce it to a finite amount of RAM at the last
moment when necessary for some practical application. So while I'm
basically a "computationalist", I'm not a "finitist". Still the
finitist viewpoint of Wildberger is interesting and enlightening.

Twentieth century physicists have
learnt to disregard `concepts' which are not measurable or observable
in some form or another, and we mathematicians ought to be equally
skeptical.
Good point, and worth debating how it should be applied in math.

Now back to the newsgroup article:
(Snip some quotes from the Wildberger's Web page, some I already
quoted above, some I didn't quote.)

doesn't this seem at least a little bit "cranky" to you?
Computable reals as uncountable?
(I partly quoted that.)

More precisely, as not computably-enumerable. Nothing cranky about
that. Everybody since Turing knows that fact!

Mathematics as a secret cult?
(I didn't quote that, see previous article, or Web page, if curious.)

Yeah, that's rather cranky.
.

Relevant Pages

  • Re: Arithmetic on computable reals
    ... |It is fairly obvious that all rationals are D-computable. ... separately to show that it holds for irrationals. ... then, is keep computing terms of the sequence r1,r2,r3,... ... Then apply the equivalence of D-computability with computability. ...
    (sci.math)
  • Re: Arithmetic on computable reals
    ... The computable reals are ... a vector space over the rationals with a countable algebraic ... and its unique representation as a linear combination ... of the particular sum we are dealing with by computing the ...
    (sci.math)
  • Re: The real numbers, and general comments
    ... > that r<s by computing them precisely enough. ... > of computable sequences of rationals. ... > sequence of computable reals is always a limit of a computable ...
    (sci.math)
  • Re: Range#member? Oddity
    ... and that's not possible for ranges of real numbers. ... Being enumerable means the possibility of a "succ" operation, and therefore of traversing the whole set with "each" (which may take infinite time). ... only rational numbers and rationals are countably infinite ... work with a set of rational numbers - in computing terms you can certainly ...
    (comp.lang.ruby)
  • Re: Groebner bases over the integers
    ... computing over the rationals would make it ... want to simplify a formula ... this would mean, however, that I have to carry out the reduction really ...
    (sci.math.symbolic)