Nick: I'm a bit confused about what you'd like from this.
Paragraph 1: The observation that there may be Math-envy amongst
programmers.
Paragraph 2: A reference to the prior threads on the philosophy of
math along with a correct observations on silos .. or possibly a depth/
breadth contrast.
Paragraph 3: A nice statement on degree of formalism as pertains to
computing and mathematics.
If you want an agreement to Paragraph 3, I'd have to be in the
negative. I'd also say that Math is beginning to have Computer-Envy,
and to realize their notation is in the same state is Roman Numerals
were before modern positional number formats.
Here's why. After pondering the division between the practice of
mathematicians and computer scientists, I came to the rather obvious
that CS is a branch of Math. I will be somewhat reductionist here by
reducing "computing" to "algorithms".
So I don't have math envy as a CS guy, I just am aware how rich all
the rest of Math is as Glen so nicely pointed out. Your silo
reference is germane here too: like in many disciplines, we tend to
depth rather than breadth.
Now to get into really hot water. Much of the division is the
difference in syntax of mathematics .. called Mathematical Notation
(MN) .. and the syntax of algorithms .. Scripts (or code). This
really is a bear when one tries to cross the boundaries.
It's horrid to express MN in Scripts. The impedance mismatch is
wild. And much of MN is "ambiguous" in the sense that ab + 1 could
mean the variable "ab" incremented by 1, or the product of the
variable "a" and "b" then incremented. Conventions are used for
disambiguation. Math folks are now attempting to reconcile the
differences, but its my guess that 2 centuries on, most of current MN
will have been replaced by something more like APL or J syntax.
There is one other interesting difference, one that Knuth has done a
superb job of discussing: the distinction between the continuous and
discrete. This is not strictly due to limitations in the precision of
computer numbers. It is a mindset. Most Math folks tend to the
"asymptotic" leap: presuming at some point moving from Sigma's to
Integrals is appropriate. Knuth begs to differ, and one of his best
books, Concrete Mathematics, is about the joint of Cont[inuous and
Dis]Crete mathematics.
-- Owen
On Oct 1, 2008, at 11:01 AM, Nicholas Thompson wrote:
> Ever since I first came to Santa Fe, and joined the extensive
> computation
> culture here, I felt I have detected in the software people here
> something
> equivalent to the physics- envy that we psychologists are prone to:
> let's
> call it math-envy. Math-Envy seems to be that while programming is
> subject
> to the vicissitudes of any linguistic enterprise, mathematics
> displays true
> formalism.... "you always know where you stand" in mathematics.
>
> The more I have read ... most recently Rosen, Reuben Hersh, George
> Laykof,
> Monk's biolography of Wittgenstein --- the more it seems that the
> best one
> can say of mathematics is that "If you know where you are standing in
> mathematics, you know where you stand" in mathematics. Take Zero for
> instance, and minus numbers, and roots of minus numbers, etc., etc.
> All of
> these things are metaphoric extensions and, as Laykof points out,
> in fact
> zero is different depending on which of several metaphors one has in
> mind
> when one is using it. Thus, the sense of safety one gets in
> mathematics
> comes from the tendancy of mathematicians to hide out in deep silos,
> rather
> than from a greater power or universality of their inter-silo
> discourse.
> It is the same sense of safety that one gets in any monastery. Or, I
> imagine, that one gets from deep involvement in any programming
> language.
>
> Now, the proposition having been stated so baldly -- and no doubt
> ineptly
> -- by an outsider, I suspect that ALL mathematicians on the list
> will now
> agree that the case has been OVER stated and that, whatever the
> differences
> in degree of formalism within the various forms of mathematics, all
> mathematics is clearer than other forms of argument, such as plain old
> vanilla philosophy, or, say, experiment and proof in psychology.
> Getting
> you all to agree in this way will have been my highest achievement
> of the
> day.
>
> Nick
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