Again about limted stacks (and other limitations) Message #1 Posted by Nick on 2 Jan 2008, 8:48 a.m.
Hi all, hi Walter and a great new year for all!
Walter, excuse please the late reply. I was away for a couple of days, but your message was indeed very interesting. So I feel that some additional thoughts on the subject might be of some value for a better clarification  or even confusion! ;) Long message follows..
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Nevertheless, one comment: Looking to the market of "scientific" calcs, it seems the big share is not taken by "some device of supreme *mathematical* power" but by a lot of bread and butter calcs, just featuring a more or less complete set of "scientific" operations on top of SQRT. So for most tasks in the real world, "some device of supreme *mathematical* power" seems to be overkill.
Undoubtfully I would also say that still in our days most of the time a scientific calculator is used for just performing quick operations with numbers (other than 0 and 1 ;)). This numbercrunching has been also considered as supreme mathematical power that *had* to be done by mind in times priorly to the dawn of such instruments. Let's keep that in mind for later on.
But on your paragraph above I have to also mention that the idea of the calculator itself as a portable machine that crunches only numbers is outdated. There is no reason to limit the design of such machines to numbercrunching; quite the contrary, and even if we consider market shares and the like! There is also a big number of mathematicians or physicists or astronomers out there who would be definitely very glad to be able to solve some partial differential equation and get a result that is not simply 3.7656789 but some formula with all its variables and parameters, or say, to use a small portable machine for doing theoretical research upon the question about some properties of a ring. Why not? The technology is already there and the theoretical fundaments too.
So perhaps the question boils down to asking if it would be possible to attract a reasonable number of the more "theoretical" professionals around for whom the "result" is not 3.7656789 but, say, some formula? I think that this can be achieved. And I think that this is the only consequent continuation of the steps that HP did in the 70s when the company *did* change the way that calculations with numbers were carried out up to that time. (But I also have to admit that it could be only my own steady denial to accept that the visionary times are over  grandpappy you see ;))
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AFAIK the other grandpappies, we simply wish to show up in the real world with a small and decent (understatement!) updated 42S instead of a big fancy student calc loaded with bells and whistles we'll never blow (again ;))
Well, Walter, it might be true. I notice the same tendencies on myself too. But then... some questions come up!
1) Do we really think that what *we* wish is so important?
2) Do we really think that our (slowly dying out :D ) generation will return perhaps in some kind of a zombie state and build up the customer group of tomorrow? (It's hard, I know, and I can't get comfortable with the answer to this question too ;))
3) It could be said that the same wish for "understatement" existed also in the past, when the grandgrandpappies still wanted their slide rules  couldn't it? So, if things *were* changed at those times, and if *we* were the enthousiasts, the very supporters of those changes, what makes us think that the same changes won't happen again? You know, induction is very fundamental! ;)
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IMHO if the 42S would be reissued with an LCD of nowadays resolution, allowing to show some more lines, and supporting some I/O, this will catch 80% of our wishes. Add beveled keys, a (settable) larger stack size, and an extended allowance for CUSTOM or USER functions, and you'll see the forum dance. Hoping to experience this once before I die,
Your words to HP's ears, and I would be only glad to be able to touch such a reissued 42S even just for minutes before I have to face the supreme faschist, as Paul Erdös used to say ;) In addition I would be even happier to type in, say, some set theory conjecture and follow its logical consequences up to the proof of truth of falsehood, or even of the impossibility of the proof itself! It would be only then that I would say: Burry this machine along with me! ;)
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Addendum: Shouldn't we rather strive for "supreme *mathematical* power" in our brains than in a tool?
This is a very very... dense question, Walter, as it seems to me that the set of answers can only be nondiscrete. (You must already have noticed my "basic instincts" ;)) I think that mathematical power cannot be just the ability to draw roots out of a given number in mind. I also think that the same power cannot be even the ability to solve some given nonlinear system of equations in mind. I think that the particular *case* of some given problem must be embedded into the most general possibility of some solution, and then the question arises: What is the main essence of "supreme methematical power"?
Let's remember again that simple numbercrunching has also been considered as some kind of skill of even "gift". But is it really a gift of mathematical power? This is where my comment from above comes to play. For me "supreme mathematical power" of mind is the ability to draw the most consequences out of the least premises in their mist general form. It is being able to handle symbols and their logical interdependencies. And that's all! So, it is not as important to find out, say, the solutions of some certain problem by hand, but rather to know/understand what the mathematical logic states about the involved steps. Take for example some formally illdefined set and all its consequences as being brought into existence by some "insane" assumptions. Well, is it then so important to be able to carry out all logical steps by hand/mind errorfree? I think not. I think that the perception of the very fundamental concepts is *the* important thing, and also that the handling of the too many "parameters" of the particular problem should be left to the machine  iff it has the power..
As an example, I was just thinking about the principal possibility of a wave function of the general form EXP(a*r)*SIN(b*r), where a,b are just parameters and r is the distance from the origin. What general kinds of potentials would give rise to that? My HP49G+ was able to help a lot  even with all its design flaws ;) The underlying potential  so it said  must be something of the form 1/TAN(X) which is a very mind crushing result considering that the wave function does extend to infinity, and that at the same time the potential goes to infinity at x=pi . The HP49G+ *was* indeed able to generate that kind of irresistible wish to know more... Isn't that great?
And still further on! The very flaws of design of the CAS of that small thingie are a blessing in disguise! Being equiped with >ALG and similar commands, one can (mis)use the existing "flaws" as the very definitions, for example, for investigating sets themselves! And much more! One can also (mis)use the existing behavior of the CAS in order to examine the CAS (in its very essence) itself. Now, knowing that any CAS can only be intriguingly connected to the universal findings of people like for example Turing or even Cantor, how could we say that dealing *with* and *about* the CAS is not dealing with mathematics itself? This is my question and this is also my hope for that hypothetical portable device of "supreme mathematical power", which I still am sure that I will be able to touch before I face the supreme fasHewchist  you know, the guy that keeps the book of theorems as a secret! ;)
I know that I often tend to be too... theoretical, too progressive, or perhaps even an ignorant of real existing market needs. I can only repeat that Bill Hewlett thought differently when it came to the decision about the HP35 that started it all, or am I just too mistaken?
If we take into consideration all the above, can any stack or similar limitation be reasonable for what is (hopefully) to come yet? I wish not.
Cheers to all, thanks for all the inspiring thoughts and messages, and have yet another great year  even as a grandpappy! ;)
Nick
Edited: 5 Jan 2008, 5:34 a.m. after one or more responses were posted
