On Tue, Sep 4, 2012 at 3:33 AM, Paul Tanner <firstname.lastname@example.org> wrote:
>> WIth the average kid, I can say (1/2)(1/3) is like starting with half >> a third or 1/6, whereas (5/2)(1/3) is like 5(1/2)(1/3) or 5(1/6) i.e. >> it's like 5 bucket fulls of marbles that are each 1/6 full. Since we >> did (1/2)(1/3) earlier, it wasn't hard to turn (a/b)(c/d) into >> (1/b)(c/d) first, and then multiply by a. (1/b) is taking an "nth" of >> something (a fraction) whereas a(1/b)(c/d) is to repeatedly add >> (c/bd). That's not stretching the meaning of repeated addition, just >> breaking it down into steps. > > This most certainly stretching the original meaning. The original > meaning is simply to take one of the given factors and add it to > itself a number of times equal to 1 less than the other factor. We > simply cannot do this with non-integer rationals. For them, we first > have to derive two new factors to do what I just said. >
And what I'm saying is the "repeated addition" meme easily covers Q and R. You disagree. I don't find your arguments persuasive.
>> I don't think that's necessary to focus on in the case of >> multiplication as repeated addition. > > It's necessary and inescapable because the irrationals and especially > the transcendental irrationals are there, mucking up the repeated > addition works. >
I don't think so. 3 * pi = pi + pi + pi. (1/3) * pi = adding a 3rd of pi to 0 one time (not "repeated" addition, but sets the stage for (2/3)*pi = pi * (1/3 + 1/3).
Once you accept that repeated addition works in Q (which you don't, you restrict it to Z integers whereas advocates of repeated addition don't necessarily follow that restriction -- something you apparently can't or won't understand) it's easy to generalize to R, regardless of the fact that Q is but a subset of R.
> Again, this does not appreciate just how different the irrationals and > especially the transcendental irrationals are from the rationals. The > irrationals are not merely repeating decimals like some rationals are > in their decimal expansion.
I don't think it's important or necessary to dwell on these differences when talking about multiplication as repeated addition, but our divergence comes earlier in you don't think Q is covered at all.
You say only Z is covered and we need to switch to "scaling" for Q.
That's your own peculiar / idiosyncratic way of thinking which I happen not to share. I am aware of many who do not think as you do.
"""Because the result of scaling by whole numbers can be thought of as consisting of some number of copies of the original, whole-number products greater than 1 can be computed by repeated addition; for example, 3 multiplied by 4 (often said as "3 times 4") can be calculated by adding 4 copies of 3 together.... Multiplication of rational numbers (fractions) and real numbers is defined by systematic generalization of this basic idea."""
That's Wikipedia. I assume it represents a broad consensus. Notice "Multiplication of Q and R is defined by systematic generalization of this basic idea." (paraphrase). That's what I've been focusing on: the generalization. You haven't been able to follow the generalization, as you are stuck in Z, don't even make it to R.
> No irrational has an infinitely recurring > finite block of digits in its infinite decimal expansion. Every
You have resorted to lecturing us about "what everybody knows" I'm afraid. This does not help make your point.
> irrationals are not even algebraic - meaning almost all of them are > transcendental, where the conceptual difference between algebraic and > non-algebraic numbers is absolutely fundamental. >
Again: even if there's a fundamental and important difference between A and B, it's not a problem to dwell on the similarities of A and B when it comes to making generalization X.
I agree with this author:
""" Now, the only problem with defining multiplication as repeated addition comes when the definition of addition is incomplete. Anyone saying that multiplication as a scaling factor and multiplication as repeated addition are not the same thing seems to be making an arbitrary point. Why limit the definition of addition? It can be easily connected to scale because it is connected to scale. And whoever decided that fractions and negative numbers are counterexamples to the repeated addition explanation unfortunately does not seem to understand either. """
>> I think you're changing the subject, getting off onto something else. >> > > No I'm not. See the below. >
> The differences are all-important if you are tying to salvage the > (false) claim that multiplication is repeated addition in even the > reals.
Will you argue that scaling as an analogy or mental model works with R, but repeated addition does not?
You can scale a length e by pi (make it roughly 3 times longer).
I'm arguing, like many writers, that scaling and repeated addition may be seen as analogous / isomorphic.
You can use volume: (4/5) * pi is like adding pi/5 to itself 4 times and pi/5 is like taking a fifth of a large number like 314159 (adding that to 0 to start). You can add less than 1 of a quantity.
(2/3)(5) requires visualizing 5 as something that can be "scaled" in the sense of taking some balls from a bucket, but not all of them. Or it's 5 copies of (2/3) much as 5 * pi is. The point is we can associate pi (irrational) with a distance and this is done all the time (or don't you believe in circles?).
This is about developing useful mental pictures, analogies, that help with understanding.
I'm saying the "repeated addition" meme can be extended to Q and from Q to R. I'm also saying I'm not the only one who takes this position.
You're saying the "repeated addition" meme starts to break down after Z, maybe even after N (natural numbers) and that the differences between Q and R are fatal to this meme if we get as far as Q. That seems to be the core of our disagreement.
Since I don't think it's at all important that we agree (we often don't), I am going to happily settle for a concise description of our differences.
> In Q, you can at least break down one of the factors to obtain > two new factors to apply the repeated addition model on. In R - namely > the irrationals, especially the transcendental irrationals, which make > up almost all the reals as I point out below - you can't do that at > all. And then there are the absolutely fundamental differences I > talked about above between the rationals and irrationals. Devlin said > and argued what he said and argued in part because of these > differences. >
I'm not overly impressed with Devlin's position or his arguments in this case.
Certainly your arguments have not been persuasive.
>> Reals in the sense of >> infinitely precise numbers have no application in computing. >> > > And that's where things go wrong when trying to funnel everything in > mathematics through a computer science bottleneck. So much of the > class of all objects studied by professional mathematicians just > cannot fit. >
I prefer to think of different brands / branches of mathematics. Discrete math is its own thing. If we decide we don't need the real numbers at all (just arbitrary precision ones), even theoretically, within a given domain, that's what's called a namespace or language game. It's not leaving mathematics, it's carving out a space within mathematics.
To set off "computer science" and "professional mathematicians" is a cultural move, one could say administrative, and it's far from a brilliant move to make. I consider it antiquarian, parochial and basically ignorant. You've seen me writing to Hansen about that. He suffers from the same malady in my book: a desire to categorize and separate such that "mathematics" and "computer science" are separated. Thats why STEM is potentially a breath of fresh air: we don't indulge in such mindless distinction-making in STEM (we're not forced to do so).
>> We use >> floating point or extended precision decimal instead. >> >> Some mathematicians want to cast all mathematics in terms of rationals >> or at least tilt in that direction. >> >> Example: http://en.wikipedia.org/wiki/Rational_trigonometry >> > > "Some" meaning "one and only one person who rejects most of > mathematics who's going nowhere with this."
No, that's not what "some" means, though it's interesting to see the spin you add. You're perhaps uncomfortable with divergence within mathematics.
> > (I checked out what's being said about him. It's not pretty.) >
You looked for gossip apparently.
I was actually looking for another mathematician I know about, at University of Pennsylvania I think it is. I'll try to dig up the name by writing to my friend Chris in Philadelphia. Doesn't matter. The point is there's still ferment and change in mathematics. Doing more with rationals, now that we have the ability to do so (thanks to extended precision), is something of a trend.
> It's not appreciably different in computer science since you are not > really dealing with irrationals at all in the first place - you are > dealing only with rational approximations. >
There is no need to section off "computer science". There are philosophies of mathematics, schools of thought.
> The point is that Devlin is right when he says that it is not the case > that multiplication in the reals is repeated addition, irrespective of > the fact that from using certain algebraic properties like the > distributive property we can derive models of multiplication as > repeated addition. I went over that before in my post > > http://mathforum.org/kb/message.jspa?messageID=7883395 >
And my point is his rhetoric is adequately countered by other writers.
The repeated addition meme may be usefully generalized over both Q and R. Wikipedia agrees with my position in saying "repeated addition" may be generalized in this way.
But then there are other memes, other ways to conceptualize multiplication, and I'm all for using those *as well*. I haven't been dragged into an either/or position as so many have.
> in this thread. That is, he's right that merely being able to derive > multiplication as repeated addition in the reals - as I did in that > post - does not mean that repeated addition is what multiplication is > in the reals. That is, the derivation is purely an artifact of the > algebraic properties, and has nothing to do with what multiplication > is. In the reals, it is as fundamental and elemental as addition.
As most writers realize, what's interesting is not what multiplication "is" but what mental models are worth cultivating versus which will prove misleading.
What I have been showing is "repeated addition" will take us a long way, if we extend it to include what others call "scaling" -- we're repeatedly adding length.
Adding 1/2 of length or volume pi to 0 is the same as multiplying pi by 1/2. The use of the preposition "of" is important. "Add 1/2 a cup to the mixing bowl" -- ideas about volume enter in (no reason to just stay with length).
I think this accurately characterizes our disagreement. I'm not interested in resolving the disagreement, just being clear on what it is. Your position is a dramatic foil for mine, provides contrast.