[FOM] Ask Dr. Bill
wwtx at earthlink.net
Wed Mar 15 11:35:15 EST 2006
Gabriel, Your subject line borders, at least to my ear, on insulting.
I hope that you did not intend it to be.
> -3. True or false? A classical existence proof is often a
> good first step towards getting a construction.
I would guess that it sometimes can be. When Bishop began his
project of constructivizing analysis, what he had on hand was
(besides the negative critique of the intuitionists) the body of
classical analysis. Do you think that I am wrong to say that what he
did was to see what and how classical results, by suitable
strengthening of definitions, could be constructivized (perhaps in a
modified form)? Even in cases like the intermediate value theorem,
which is not constructive, the technique used to prove a
constructively valid weaker form is a refinement of Bolzano's
original nested intervals argument.
On the other hand, given a particular real function f, a classical
proof that it is continuous + Heine-Borel will generally not be a big
help in finding a proof that f is continuous in Bishop's sense
(locally uniformly continuous), i.e. in actually constructing the
modulus of local uniform continuity. But, of course, this remark
applies anyway only to the cases in which f has a constructive
modulus of luc.
My last remark is motivated by a consideration that your question may
be ignoring: The agenda of classical mathematics is not restricted to
finding constructive bounds or to proving existential propositions
only when constructive bounds exist.
> -2. True or false? A classical existence proof is often an
> annoying distraction from getting a construction.
Like -3, this is too imprecise to have a truth-value. It is certainly
annoying when, trying to prove 'there exists' constructively, you
realize that you have only proved 'not-not-there exists'. But if one
is used to constructive reasoning, that is not so likely to happen.
But I think my real answer (neither a 'true' nor a 'false') is in my
response to -3. Sometimes the passage from the argument for 'not-not-
there exists' can be replaced by a direct argument for 'there
exists,' while the rest of the classical argument remains intact and
is perfectly constructive. But sometimes it can't. Reference:
Bishop's book on constructive analysis.
> -1. Among classical mathematicians, constructive math is the
> study of what can be proved without omniscience. How does your
> own interpretation of it as a part of classical math differ from
"Among classical mathematicians, constructive math is the
study of what can be proved without omniscience." First of all, I
doubt that you are right about what classical mathematicians would
say. Proof is defined by means of axioms definitions and rules of
inference. This is what is objective and independent of your
intuitions, or whatever, and mine. Classical math admits the logical
axiom 'A or not-A'. You may pars this as an expression of
omniscience---as did Bishop. But I don't see it that way: Doesn't my
view get to count? The talk about omniscience is metaphorical and
perhaps good for pep-talks in favor of constructive math; but what is
its real content? God is omniscient because It can know that P(n)
holds for all natural numbers n, and we are not because we can't. But
constructive mathematics assumes in the same sense that we are able
to know that P(n) holds for all n up to any arbitrary number N, where
P is any decidable property, no matter how complex its definition.
That's still pretty omniscient in my book.
> 0. If classical math didn't exist, would constructive math
> have to invent it?
Perhaps, in a sense, it did. Eighteenth century analysis seems quite
constructive (although not self-consciously so), in the sense that
existence proofs yielded computations. [But I am out on a limb here:
Take this as a conjecture about what a competent historian of the
period would discover.] The first result that I know of for which
this isn't so is Bolzano's proof of the intermediate value theorem in
Of course, his proof was not well-known, but Cauchy's proof some
four years later seems to have been widely known. What is surprising,
looking back at it from here, is how long it took for there to be a
constructive reaction, namely (as far as I know) by Kronecker. [Here
again, a historian of that period might have a different view of
this.] It is also worth mentioning that for him your constructivism
is too profligate. He rejected objects not representable by integers
and properties which are not algorithmic. So it wasn't for him a
matter of rejecting excluded middle; rather mathematics should not be
concerned with propositions for which EM is not provable.
But constructive math (in your sense and self-conscious) would not
only not have to, it indeed would not invent classical math as long
as its aims remained constructive: an existence p[roof should yield
> 1. Why is it important to you that a statement about real
> numbers in constructive math mean the same thing as the formally
> identical statement in classical math?
What is important to me is conceptual clarity. I believe that the
only way to understand the term 'meaning' (of expressions) is their
public meaning, to be understood in terms of how they are correctly
used. (The term 'correctly' here is indeed loaded; but I needn't go
into this now. )The main point is that the locus of meaning cannot be
(as you suggested in an earlier posting) in the mind. For the
question would then be mandatory: Whose mind?
So, its not important to me that a statement about reals mean the
same thing in constructive as in classical math. What I do hold is
that the principles of reasoning about reals in constructive math are
classically valid---once one resolves some conflicts in terminology.
> 2. Why would it be a bad idea for a mathematical community to
> have a paradigm shift in which it replaces its old metaphysics
> with a new one?
I don't care to fish in this pond---or rather ocean.
> 3. Do you believe that classical math provides the big picture
> but that in constructive math we have difficulty seeing the forest
> for the trees?
> 4. Why do you dislike the interpretation of classical math as the
> part of constructive math in which we investigate how it helps to be
I don't really dislike it; it is amusing.
Now let me ask a question: Why is it so important to you to wage war
with classical mathematics? Why should its existence (indeed,
thriving existence) threaten the pursuit of constructive math? The
bible-thumping seems completely inappropriate to me.
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