> About a week ago, David Fowler posted some comments including these:
>
>> "Knuth has argued that algorithmatics and mathematics [that word needs
>> much refining; for the moment I'll just put a small m], are different
>> things, and those who are good or able in one may not be good at the
>> other; see
>> D.E. Knuth, Algorithmic thinking and mathematical thinking, American
>> Mathematical Monthly 9 (1985) 171-181.
>>
>> "Generalising this, I think of Mathematics in the wide sense, big M,
>> as a great multidimensional cloud, in which the cluster of
>> Euclidean-and-later-type deductive procedures are a long way from the
>> cluster of algorithmatics...."
>
> Now Knuth treats "algorithms" broadly as "the whole range of concepts
> dealing with well-defined processes," while agreeing that others define
> them only as "methods for the solution of particular problems."
> At a minimum, we must realize that the tendency to think of an
> algorithm as necessarily numerical is just a historical accident.
I completely agree, and my earlier posting was wrong in its emphasis of
that feature and should be refined, but I still think that there is a
different kind of attitude to proof between 'mathematics' (small m) and
algorithmatics. Two examples:
* If I remember correctly, in what Knuth describes as his very limited and
nonscientific search (he looked a p100 in a dozen books he had on his
shelves), the bit of 'mathematics' which Knuth found that was closest to
algorithmatics was a proof by Bishop in his book on Constructive Analysis.
But mathematicians seem uneasy with or patronising about that approach and
kind of proof.
* I quoted computers in terms of the renaissance of algorithmatics today,
but I suspect that a relatively small amount of that algorithmatics is
numerical. And the proofs that are deployed in these kinds on
algorithmatics don't look very 'mathematical', or sometimes even don't look
like 'proofs', to mathematicians.
I would argue (and even have!) the other way round: that 'mathematics' has
always gained a tremendous amount from numerical calculations. To restrict
it to the turn of the 16th century and thereafter, see how Viete, Stevin,
Harriot, ...Newton, ..., Legendre, Euler, Gauss, ... (would anyone like to
add more names?) were calculators, at least at some crucial points in their
development.
> If we agree then that a specified process for producing
> a solution for a specified type of mathematical problem must count
> as an algorithm, we can observe that algorithms abound in classical
> Greek mathematics -- and are also always accompanied by proofs of
> their validity. For instance, Euclid, Elements I.9 is "To bisect a
> given rectilinear angle." A specific procedure for constructing
> a line is laid down, and it is then proved that the line so
> constructed does indeed bisect the angle.
But what I am trying to illustrate is that there are different kinds of
things which constitute proof in different parts of this cloud of
Mathematics, big M, so it might not be sufficient to say 'Here's a
procedure and here's its proof'.
David Fowler