[FOM] vagueness in mathematics?
Dennis E. Hamilton
dennis.hamilton at acm.org
Tue Mar 7 14:29:24 EST 2017
I am astonished of the degree to which computer programming and programming-language metaphors infect important distinctions concerning mathematical abstractions. The seizing of previous language for interpretations in contingent reality (especially computer software behavior) strikes me as unfortunate co-option.
My first exposure to matrices (and vectors and arrays), in the 1950s, was in books produced before the ubiquity of computer programming systems and languages.
- There was recognition of the mathematical group properties around matrices and vectors
- The relationships among matrices were descriptive and at the level of functions at the matrix level
- Descriptions would appeal to relationships among elements and functions on them, but it was descriptive
- There were useful notations, such as using super-/sub-scripting. Written linearly, for example,
A[i,.]B[.j] signifying the inner-product of A and B.
I want to recognize the power of this level of abstraction and notation distinct from computational procedures, however much they might be hinted (often computationally-inefficient and numerically-dangerous). What we did with calculators and pencil and paper tabulations (aka arrays below), was procedural, with much error-checking, and there was considerable companion activity required to reliably compute determinates, perform Gaussian elimination, etc. But it was grounded on the mathematical connection, even if the human "computers" of the time were only tangentially aware of it.
These days, we seem to rely on procedural approaches as the learning path toward conception of mathematical objects. I suppose this starts with arithmetic as access to numbers, but with greater difficulty, since there is so much overpowering of essentials by incidentals, however much the incidental is a consequence of computational reality.
With regard to vagueness, one will not have been vague to the computer, even if the definiteness is not recognized and might be surprising. That computers operate in the contingent reality is also important to appreciate and failure is always possible, despite the remarkable performance of present-day computers.
Since mathematicians live in the world, as do the communications of their findings, we are left with an unavoidable dilemma: Having *enough* confidence in the face of uncertain reality such that reliance on mathematical results is of practical value and such that defects, when revealed, are dealt with appropriately.
- Dennis
> -----Original Message-----
> From: fom-bounces at cs.nyu.edu [mailto:fom-bounces at cs.nyu.edu] On Behalf
> Of Annatala Wolf
> Sent: Monday, March 6, 2017 15:53
> To: Foundations of Mathematics <fom at cs.nyu.edu>
> Subject: Re: [FOM] vagueness in mathematics?
>
> On Tue, Feb 28, 2017 at 11:55 AM Jacques Carette <carette at mcmaster.ca
> <mailto:carette at mcmaster.ca> > wrote:
>
>
> It is worth noting that there are similar issues elsewhere. Most
> prominently, in computer science, most people incorrectly identify
> arrays and matrices; but arrays are a method of memory storage,
> while
> matrices are representations of (finite dimensional) linear
> operators
> with respect to a given basis.
>
>
>
>
> I think the simple definition here on arrays is not entirely correct.
> The term "array" refers to more than just a method of memory storage.
> (For sake of argument, I'll gloss over the fact that arrays and the
> interface to arrays is language-dependent, and there are frequently more
> than one implementation of this concept in the same language.)
>
[ ... ]
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