[GW-C] Re:numbers [was Re: understanding probability]

[Lennart Sorensen]

On Mon, Aug 12, 2013 at 03:54:05PM -0400, D. Hugh Redelmeier wrote:
> As explicitly as I can say it:
> 
> - what does "arithmetic" mean to you?
> 
> - what does "number" mean to you?
> 
> Surely your answer should be with respect to an algebraic ofject
> having elements and operators on those elements.  Just talking about
> operators and not elements or elements and not operators cannot be
> rigorous enough to make your meaning clear and specific.

Well it does all come down to your defined system in the end.

> Letters?  Do you mean formal variables, as in polynomials?  Or as
> elements in an algebra?

I was thinking of them as values, nor variables.

> The brains behind 754, William Kahan, certainly had clear (and
> complicated) ideas behind these infinities.  He had plenty of uses for
> them.  I've always thought that he wanted too many properties from
> numerical hardware and elementary functions, overconstraining the
> implementation.  But his talks are pretty entertaining and convincing.
> 
> Amazing (to me) story about Kahan.  He showed that the original
> IBM/360 floating point hardware specifications yielded significantly
> lower precision than could be achieved.  IBM took his advice and
> changed the specs and retrofitted a "guard digit" into every /360 in
> the field!  This was not a trivial reflash-the-firmware kind of thing.
> It must have cost a fortune.

Well there probably weren't that many machines, and if they were
microcoded then if that microcode was changeable, then it probably
wouldn't be that costly.  Even if it involved hardware swaps, the
development cost for the machines was probably much larger than the
actual chip manufacturing costs, so again perhaps not that bad.

> If I remember correctly, Kahan wanted infinities so that he could
> safely write strraight-line code for (for example) elementary
> operations.  Eliminating conditionals makes the code simpler and
> faster.

That sounds fair enough.

> Are the elments of Z mod (2) numbers?  Or are the elements the same as
> (isomorphic to) Heads and Tails, or Naughts and Crosses, or...
> 
> It's not reasonable to say that a number is anything written down
> using only digits.

Depends on what you consider digits.  After all hex uses a-f as digits
too but perfectly well defined meanings.

Of course as long as your operations are defined, your numbers could
be anything.

-- 
Len Sorensen
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