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Re: A Sum-like notation for iteration

  • To: mathgroup at smc.vnet.net
  • Subject: [mg102415] Re: A Sum-like notation for iteration
  • From: magma <maderri2 at gmail.com>
  • Date: Sat, 8 Aug 2009 04:40:02 -0400 (EDT)
  • References: <h5bk7g$hn3$1@smc.vnet.net> <h5ebii$1rc$1@smc.vnet.net>

Being neither Greek nor Trojan, but rather a Roman, I shall try to
bring a Pax Romana between the 2 arguing parties :-)

1- The Theorema project is a great Mathematica package about which I would
like to see much more discussion. Indeed the logicographic symbols
technology is an excellent example of the kind of sophisticated and
innovative notation that is possible to achieve in Mathematica.
As Carlos correctly points out, they invented a notation where no
standard notation existed before. So why not for Table too?

2-Carlos, can we see what you have done - "up to 3 iterators" - so
far ? Please post the code. I always enjoy reading some nice notation-
tweaking before dinner.
The Notation package has some limitations as David Park first pointed
out to me 1.5 years ago. It's OK for little things, but the real
connoisseurs do not use it, preferring to sculpt their notations from
virgin Boxes. Of course "boxing" is not for the puny, so beginners
should stick to the Notation Package until their Mathematica muscles are
strong enough to enter the ring.

3- But then...why stop at Table? The Underoverscript notation could be
used also for other "iterative" constructions. For example the Plot-
type of things.

4-  "Timeo Danaos et dona ferentis" (Latin: I fear the Greeks even
bearing gifts) says Lacoon in Virgil's Aeneid, but pfalloon (who
rhymes with Lacoon :-)) does not need to be so cautious in this case.
Introducing an extra 2D notation for Table & Co. through the gates of
Mathematica, will not bring down its powerful walls.
Users who do not like it , will simply avoid it in input, plodding
along with the usual 1D notation, while the more visually driven ones
will boldly press another button in a palette to create the 2D
template.
The (unevaluated) output could be 1D or 2D as is now with Sum (though,
it seems to me that unevaluated Sums are always displayed 2D,
regardless of how you input them.
see for ex.
Sum[f[i], {i, 1, Infinity}]
but there might be other situations where the output is 1D)

Quote: how do we format something like Table[f[x], {x, 1, 100, 2}]?
Or,
worse, Table[f[x], {x, {1,2,3,4,5}}]? Whatever solution is chosen, the
point is that it is no longer "intuitive", at least not in the sense
that one could readily guess what it should look like.

5- Sum and Table use the standard Mathematica iteration specification.
If we look carefully, we will see that Sum offers a 2D notation only
for some simple iterator forms, the most general being {i,imin,imax}.
For more complicated type of iteration - for example with a step size
- there is no 2D notation supported by Sum. So the same limitation
could apply to Table

Quote:The whole point of using 2D inputs (for powers,
fractions, named special functions, sums, products, and so on) is that
they should be general constructs recognized universally.

6- The brain of T-Rex had special receptors for movement detection so
they could capture hiding preys, the brain of the Mathematician has
special receptors for "form" detection, so they can capture hiding
symmetries.
There is hardly any "form" in 1D. So Mathematicians created 2D "forms"
to please their minds. That is the whole point of 2D notations.
So there is no point in denying them this whole point :-)

Summing up point 1....6  (no pun intended....really!): a more visually
appealing notation for "iterative" constructs like  Table, Plot, ect
is certainly welcomed. It does not need to interfere with the 20 year
old habits of the "flatliners" and it can and should be developed
right away without waiting for the uncertain approval/implementation
of the Royal House.



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