As I believe I had stated earlier, we might go back a step or so and ask:
"Why do we need the multiplication operation at all?"
Just the concepts of 'zero' (including the concept of 'null'), '1' and 'addition' have ultimately enabled the development of ALL of math - including multiplication (see below).
> > (DS): But I certainly wouldn't mind being proven wrong. > I don't believe you are being proven *wrong* - at least, that is not my intention at all - and I don't believe we need to think of proving you *wrong*. (I don't know about Professor Wayne Bishop's intentions).
I DO believe you need to think a little more deeply about your assertion.
The underlying issue is, we do find it useful (and often even NECESSARY) in real life to 'cluster together' various collections of 'elements' and 'operations' and call that clustering together of ideas 'something-or-the-other', e.g., 'calculus', 'analysis', 'topology', 'Galois theory', and so on and so forth. That 'clustering together' seems to add some *value* to our ideas: I believe that - at this stage of intellectual evolution - the value of this 'clustering together' is not adequately understood by most of us. (There may well be a few more 'operations of the mind' involved; I don't know about these).
It should certainly prove useful to stimulate (and perhaps even excite) students by showing them how extremely complex structures are built from fundamentally very 'simple' concepts. I don't know if your website quite does that: you probably need to test it out on a sufficiently large number of students, not on Math-teach participants.
But I think it requires a little something more to grab the student's interest and imagination and 'turn him/her onto math'. Generations of teachers using traditional teaching methods in math have thus far succeeded - in great measure - mainly in turning the vast majority of students 'off math', alas. How to convince students at large that math can be as beautiful and exciting as, say, literature or music or dancing (or even the opposite sex)? This may well be, I believe, the major unresolved problem of math education.
(In regard to some statements in the above paragraph, Robert Hansen [RH] might try to understand that I am NOT suggesting that "teachers make students hate math" [or some such ridiculous idea that he has lately come out with]. What I'm claiming is something entirely different - if he is able to understand the distinction).
I agree with you, Mr Sauter, that the 'school educational system' is ripe for (as you claim) "a math education breakthrough, or revolution, even". I personally believe that breakthrough could come about through an in-depth study of 'systems', particularly our educational systems: I don't believe your website quite does the trick.
In regard to some of the issues that arise from your assertion about 'math':
Not long ago, Andrew Wiles had proved "Fermat's Last Theorem", using (if I'm not mistaken), some pretty sophisticated math called "modular forms", "semistable elliptic curves" and "Ribet's Theorem". (This took generations of some very brilliant people - including several geniuses, as a matter of fact - some 350 years to accomplish).
[By the way, I have seen Andrew Wiles' proof, but have not understood it - though I do understand '0', '1' and 'addition' fairly well - Assertion 'A'] .
Even more recently, Grigori Perelman proved the 'Poincare Conjecture', which is a theorem about "the characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dimensional space" (from Wikipedia). The conjecture states "Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere". This, by the way, took over 100 years to accomplish - and a lot of great mathematicians had tried and given up, before Perelman understood how to use some available ideas (conceived by others) to 'put together' a rather 'miraculous' proof, involving those above-noted ideas along with some of his own.
[By the way, I have seen reconstructions and elaborations of Perelman's proof, but have not understood them - though I do understand '0', '1' and 'addition' fairly well - call this Assertion 'B'].
I believe it can be demonstrated that both these signal achievements of the human intellect (Andrew Wiles' and Perelman's) developed from the simple concepts of '0', '1' and 'addition'. (I personally would hesitate to try to demonstrate this - though the assertion is probably much simpler to demonstrate than either Fermat's Last Theorem or the Poincare Conjecture).
I do believe Assertions 'A' and 'B' above seem to indicate that math, as it has developed over the centuries, has come to contain something more than the 'elements' ('0', '1', and 'addition'). [Call this 'C'].
[By the way, I note that I had originally asserted 'A' and 'B' above specifically to try and prevent Professor Wayne Bishop from making some truly ridiculous assertions, something he is wont to do - to the effect that I am comparing myself with Grigori Perelman, Andrew Wiles, and etc. I then found that I was able to make Assertion 'C', which serves another purpose entirely!
[Of course, Professor Bishop has been responsible for some extremely foolish assertions and slogans, for instance his most famous slogan is the quite delusional
"BLOW UP THE SCHOOLS OF EDUCATION!" -
this was suggested to him, I understand, by someone called Reid Lyon, reading research expert, and he took it up with great gusto for a while].
Further, in another field, ALL of physics - including quantum physics and string theory' - derives from those simple concepts of '0', '1' and 'addition', put together with what people have 'observed' (over thousands of years) about our physical world.
Likewise, in another field entirely, Tolstoy's 'War and Peace' is *only* a collection of words built from the Russian alphabet using some 'rules of (the Russian) language - of course, Tolstoy's imagination also came into play. JD Salinger's "Catcher in the Rye" is likewise only a collection of letters in the English alphabet (along with some of Salinger's imagination)
Is there anything in your argument other than what I've suggested above? If so, I don't see it.
GSC Donald Sauter posted Aug 27, 2013 7:07 PM (http://mathforum.org/kb/thread.jspa?threadID=2592235): > Thanks, for your attention, Mr. Chandy. I think the > problem here is my biplane brain compared to your > starship brain. :-) I've often wondered if maybe > the "deliberativeness" (shall we say) of mine affords > me a better than average peak at the wheels turning > in there. And as hard and as carefully as I have > looked, I have never observed anything other than > basic addition and addition facts called into play to > work through every ("numbery") math problem I've ever > met, 1st-grade through grad school. But I certainly > wouldn't mind being proven wrong. All it would take > is to present a problem from one of the "numbery" > maths (arithmetic, algebra, analytical geometry, > trigonometry, probability, statistics, calculus, for > example) that one would not naturally solve by a > sequence of fundamental additions and/or > multiplications. > > Can anyone come up with a "textbook" problem that > shoots my assertion down? > > Anyone looking in at this stage might be interesting > in reading my web page on the subject: > > http://www.donaldsauter.com/all-of-math.htm > > Since it's just electrons, I'll copy a relevant > paragraph here that appears in my pages devoted to > basic addition and multiplication: > > ds> I'll devote a web page to it one day, but think > of math as being divided into two "rooms" - the > addition room and the multiplication room. Each room > has its own set of tools, many of which correspond > to, but are not identical with, tools in the other > room. For example, each room has a "do nothing" > number called the "identity element". In the addition > room that number is 0; in the multiplication room > it's 1. You must always be totally aware of which > room you're operating in, addition or multiplication. > No matter how complicated a math problem gets, your > brain only ever operates in one room at one time. At > any moment you are doing either addition or > multiplication, nimbly stepping back and forth > between the rooms as needed. And, to repeat myself, > no matter how complicated the addition or > multiplication gets, your brain only processes a pair > of single digits at a time. In a nutshell: all of > math is just a big mixture of single-digit addition > and single-digit multiplication.! > [end quote] > > > Which sort of leads to the point of all this, that > students would be very well served--that there might > be a math education breakthrough, or revolution, > even--if they were shown how all of math is just > basic addition and basic multiplication, and were > trained to see the distinction between the two in > crystal clarity, and to only ever "do" just one of > them at a time. > > Donald Sauter