Date: Nov 28, 2012 11:45 AM
Author: Joe Niederberger
Subject: Re: Some important demonstrations on negative numbers
>When I work out a complicated problem or proof, entirely symbolically, never once referring to or thinking of something "concrete", then how can I be using common sense?
You are not - but that's not where people start from, using formal rules that way. What they do before that stage is still math. Searching for better ways of grounding concepts is a worthy mathematical activity.
Witness Richard Dedekind's essays.
1. "Formal thought", "reasoned thought", etc. are ill-defined as used by you to denote "mathematical thinking". "Formal" is a very good word to describe some kinds of mathematical thinking, as in formal logic and algebra. Its because one works via rules, according to the form of propositions and sentences and expressions.
It goes back to syllogisms. All toves are slithy. X is a tove, therefore, ...don't need to even know what a tove is to know that it really is slithy.
2. Its true that we want students to become accustomed to working this way (in certain narrow disciplines, such as H.S. algebra,) but that is after a time. Approaching any new area of math after even that still benefits from being grounded, as much as possible. Much later "being grounded" may mean motivating a new branch of study (topology) in other branches (geometery, analysis).
3. In the history of math, we find some pioneers using "formal" methods to advance, develop and incorporate new notions long before better intuitive ways of "making sense" out of them were developed (geometric interpretations of complex numbers.) This is also very true of negative numbers.
There is no need to expect everyone to travel in the footsteps of the pioneers.
Let's look at negative numbers again. "Formally" we may write -1 < 0, and -1/1 = 1/-1. This perplexes mathematicians for a long time:
One can say "just accept it, because it makes other things work out nicely". That's the "formal" approach.
Or one can attempt to understand it another way that grounds it in the everyday. Its not hard, but depending on ones deeply engrained prejudices they may never accept something simple I'm about to say now: Negative numbers (and complex) have two distinct notions embedded in one number (if I can ever get to three I'll have Catholic theology down.) Quantity (or magnitude,) and direction (or polarity.)
Furthermore, we have to re-arrange how we think about symbols like "<", ">" in this new system. "<" has to be re-interpreted as a directional or ordering indicator (as does "-") but *not* a "size" or "quantity" comparison. Persisting with the terminology "less than, greater than, bigger, smaller" is really not helpful, although it has unfortunately stuck. Here is what is leads to:
"Antoine Arnauld argued against negative numbers by using proportions; to say that the ratio of -1 to 1 is the same as the ratio of 1 to -1 is absurd, since, 'How could a smaller be to a greater as a greater is to a smaller?'"
Leibniz agrees by the way (Oooooh! Hate that)
"but said the since the form of such proportions is correct, one could still calculate with them."
There's your purely formal approach.
The confusion here is entirely brought on by the engrained thinking that x < y "means" x is a quantity that is smaller, lesser than y. This kind of teaching and thinking persists everywhere today. Here's an alternative - once we get to negative numbers, we reinterpret. When we compare two numbers, we think separately of magnitude and sign. The rule that for any negative number x and positive y, x < y; now that's the thing that gets the "convention" treatment. Saying "smaller" is discouraged ;) Suppose someone says, I lost $100 million dollars, but I don't worry, its a really, really small amount!
So, how we ground our understandings of these basic things is of immense importance in how we see things later, even if later we are working the purely formally.
If children were taught that negative numbers involve two parts, and magnitude and a sign, and that x < y no longer "means" x is lesser or smaller than y, but its merely a way of ordering them on a line together. If that could be done consistently, then I think they'd be in better shape later to see complex numbers again as "numbers with parts" Now we have a "direction" can range freely in 2 dimensions. (Note the conflict between this "direction" and the earlier "direction" of "sign". Some physical interpretations again help clarify. Perhaps it would be better to start off calling +,- signs "polarity", but I don't know if that would be helpful at that age. And of course the use of +,- persists in complex numbers alongside everything else.)
I am not saying that there is a unique way to ground any particular math concept. There are usually multiple. I see people get hung up looking for the "correct" one.
Some are potentially wrong and hurtful (such as "smaller than".)
I do believe though, that people who never learn to really accept complex numbers as "real" also never learned to see integers as number comprising two separable notions.
Back to negative numbers though, every common sense illustration will necessarily distinguish clearly the magnitude from the "direction", "polarity" or sign. That's that's needed. If you don't like my poker example, find something else, but make sure is clearly distinguishes magnitude from something that serves as a bi-polar attribute.
"I am not suggesting that it is mere coincidence that the concrete world is held accountable to the same mathematics and logic we strive to understand so deeply. If the world operated according to some other set of principles then I am sure that those would be the principles that we would strive to understand so deeply. But our understanding of those principles is not grounded in our common sense perception of the world."
Again I must disagree, syllogism, algebra and the rest of formal manipulation of symbols according to rules that I like to call simply "logic", or perhaps more comprehensively, "computation" absolutely did arise from noticing how the merely formal could faithfully adhere and correspond to the concrete. So it is "grounded" in that sense. There continue to be arguments of how best to ground the formalities. That people can take off and fly while no longer paying much attention to the ground is great, its the very nature of formal manipulation to not be tied to the concrete at every turn.
Its simply a mystery why we have this incredible correspondence between the concrete world and the games we spin out of formal symbol manipulation. I have my own ideas on why this may be so, that can be discussed another day.