[FOM] re harvey re my "effective number theorists" (II)
friedman at math.ohio-state.edu
Sun Apr 16 02:24:42 EDT 2006
On 4/15/06 11:33 PM, "Gabriel Stolzenberg" <gstolzen at math.bu.edu> wrote:
>>> However, it remains to be checked whether number theorists mean
>>> the same thing by these words that Harvey does. They are, after
>>> all, classical mathematicians and "effective algorithm" is a term
>>> used by classical mathematicians not constructivists.
>> I don't know what you have in mind for a possible difference between
>> how I use the relevant words and how number theorists use the relevant
>> words. I would like to hear.
> Sorry, I was trying to be tactful and ended up being misleading.
> They may indeed use words like 'algorithm' a different way, e.g.,
> as in classical recursion theory (although I think this is unlikely).
> But the main difference I have in mind is that you call something a
> construction (say, a bound or algorithm) only if it is one and you say
> that there is none only if there is none---whereas I would need to
> know more before I could be confident that number theorists do too.
The examples that you go on to give are in contexts where the objects are
infinitary. Here one has to be very careful about the meaning of algorithm,
and there are major alternative notions, and major confusions.
The context that I have been discussing, and that you have been reacting to,
is that of Pi02 and Pi03 statements, where the objects involved are all
finitary, and no such confusions arise.
Of course, I would be quite interested in hearing about any surprising
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