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Topic: Non-Euclidean Arithmetic
Replies: 108   Last Post: Sep 13, 2012 3:39 PM

 Messages: [ Previous | Next ]
 kirby urner Posts: 3,690 Registered: 11/29/05
Re: Non-Euclidean Arithmetic
Posted: Sep 4, 2012 11:56 AM

On Tue, Sep 4, 2012 at 3:33 AM, Paul Tanner <upprho@gmail.com> 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

>
> It's necessary and inescapable because the irrationals and especially
> the transcendental irrationals are there, mucking up the repeated
>

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,
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
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."""

http://en.wikipedia.org/wiki/Multiplication

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
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

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

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.

Kirby

Date Subject Author
9/1/12 Jonathan J. Crabtree
9/1/12 Paul A. Tanner III
9/2/12 kirby urner
9/3/12 Paul A. Tanner III
9/3/12 kirby urner
9/3/12 Paul A. Tanner III
9/4/12 kirby urner
9/4/12 Paul A. Tanner III
9/4/12 kirby urner
9/5/12 Paul A. Tanner III
9/5/12 Robert Hansen
9/6/12 kirby urner
9/1/12 kirby urner
9/1/12 Joe Niederberger
9/1/12 Wayne Bishop
9/1/12 Joe Niederberger
9/2/12 Robert Hansen
9/3/12 Paul A. Tanner III
9/3/12 Robert Hansen
9/5/12 Paul A. Tanner III
9/3/12 Joe Niederberger
9/3/12 Robert Hansen
9/5/12 Paul A. Tanner III
9/3/12 Joe Niederberger
9/3/12 Paul A. Tanner III
9/4/12 Joe Niederberger
9/5/12 Paul A. Tanner III
9/5/12 Joe Niederberger
9/5/12 Robert Hansen
9/5/12 Paul A. Tanner III
9/5/12 Joe Niederberger
9/5/12 kirby urner
9/5/12 Joe Niederberger
9/5/12 Robert Hansen
9/5/12 Joe Niederberger
9/6/12 Joe Niederberger
9/8/12 Robert Hansen
9/7/12 Jonathan J. Crabtree
9/8/12 kirby urner
9/8/12 Paul A. Tanner III
9/10/12 kirby urner
9/10/12 Paul A. Tanner III
9/10/12 kirby urner
9/10/12 Paul A. Tanner III
9/10/12 kirby urner
9/8/12 Robert Hansen
9/8/12 kirby urner
9/8/12 Robert Hansen
9/8/12 kirby urner
9/8/12 Joe Niederberger
9/8/12 Jonathan J. Crabtree
9/9/12 kirby urner
9/8/12 Clyde Greeno @ MALEI
9/8/12 Jonathan J. Crabtree
9/8/12 Jonathan J. Crabtree
9/8/12 Joe Niederberger
9/8/12 Joe Niederberger
9/9/12 Paul A. Tanner III
9/9/12 Robert Hansen
9/9/12 Paul A. Tanner III
9/9/12 Robert Hansen
9/9/12 Paul A. Tanner III
9/9/12 Robert Hansen
9/10/12 Paul A. Tanner III
9/10/12 Wayne Bishop
9/10/12 Paul A. Tanner III
9/9/12 Joe Niederberger
9/10/12 Clyde Greeno @ MALEI
9/9/12 Joe Niederberger
9/9/12 Paul A. Tanner III
9/9/12 Wayne Bishop
9/9/12 Paul A. Tanner III
9/10/12 Wayne Bishop
9/10/12 Paul A. Tanner III
9/9/12 Paul A. Tanner III
9/9/12 Joe Niederberger
9/10/12 Clyde Greeno @ MALEI
9/10/12 Joe Niederberger
9/10/12 Paul A. Tanner III
9/10/12 Joe Niederberger
9/10/12 Clyde Greeno @ MALEI
9/10/12 Joe Niederberger
9/11/12 Joe Niederberger
9/11/12 Paul A. Tanner III
9/11/12 kirby urner
9/11/12 Paul A. Tanner III
9/11/12 kirby urner
9/11/12 Paul A. Tanner III
9/11/12 kirby urner
9/12/12 Paul A. Tanner III
9/12/12 kirby urner
9/12/12 Paul A. Tanner III
9/12/12 kirby urner
9/13/12 Paul A. Tanner III
9/13/12 kirby urner
9/13/12 Paul A. Tanner III
9/11/12 Joe Niederberger
9/11/12 Joe Niederberger
9/11/12 kirby urner
9/11/12 Joe Niederberger
9/11/12 Joe Niederberger
9/11/12 Paul A. Tanner III
9/11/12 israeliteknight
9/11/12 Joe Niederberger
9/12/12 Paul A. Tanner III
9/12/12 kirby urner
9/12/12 Paul A. Tanner III
9/12/12 kirby urner
9/11/12 Jonathan J. Crabtree