|
|
Re: Non-Euclidean Arithmetic
Posted:
Sep 4, 2012 12:00 AM
|
|
On Mon, Sep 3, 2012 at 1:43 PM, Paul Tanner <upprho@gmail.com> wrote:
<< snip >>
> A field is a ringoid. If we are dealing with two binary operations on
> a set such that we have a distributive property, then a ringoid is the
> most general case.
>
I suppose I'm less interested in the general case for K12 STEM.
Groups are fun and we can study them easily, especially finite ones,
with permutations. The software may be provided as scaffolding i.e.
when we first start multiplying permutations, we might explain the
code was written by their fellow students but at a more advanced
level.
The school has a software library (shared servers) and students get to
reuse what other students have written over the years, even as they
role their own. The idea of open sharing is important to this network
of schools. Students appreciate what other students have done.
Alumni still access the servers. Sporting events, theatrical
productions, cooking shows... these get edited and saved, becoming raw
materials for other video projects (such as the video year book, and
the school's "TV station" -- could be streaming, so no broadcast
license required, same with "radio").
Given permutations, we have cyclic notation to talk about (no high IQ
needed, not super advanced nor tricky), which lets a permutation be
represented in a compact form.
The J language has a specific operator for generating that, which I
think is pretty interesting (J is an interesting language, connected
to lots of mathematically mature literature).
> A group is a groupoid (or a magma if you wish that term - I prefer
> "groupoid" since that is the more universal usage [the term "magma" is
> the one those of the Bourbaki school use]). If we are dealing with one
> binary operation, then a groupoid is the most general case.
>
I wouldn't mind mentioning these things, but I'm not big fan of the
Bourbaki project (its empty formalism) and think a lot of details can
be bleeped over (merely hinted at) at this level.
Group Theory has some interesting parts though. Again, RSA is a goal
and we get to that through Fermat's Little Theorem and Euler's
generalization.
Random links to other writings (of mine) on this topic:
http://old.nabble.com/Signuture-Lesson-Plans-(intro)-td22927296.html
http://grokbase.com/t/python/edu-sig/095wcb3rtf/poking-around-in-py3k-recycling-old-algebra
http://worldgame.blogspot.com/2009/01/short-list.html
> I prefer to deal in the more general case - I like to "weed out"
> unnecessary algebraic properties as much as possible, to try to see
> what is generally going on.
>
My goal is to scatter key concepts on the terrain that have a tendency
to hook up with one another, send out tendrils and form networks.
Networks "connect around in all circumferential directions" meaning
not many if any dead ends. The topic selection is self-reinforcing,
to the point of self booting and self sustaining.
The goal is to cycle through these key topic areas many times, but
coming from different angles, giving a sense of the lay of the land.
We are not trying to over-specialize. On the contrary, a goal here is
to make K-12 less narrow, but without eliminating the depth dimension.
One way we do this that's critical: make other objects besides
numbers participate in operations such as addition and multiplication.
Do not confine the idea of a mathematical operation to "numbers" in
any set. Not N, Z, Q, R or C.
Yes, show numbers in action and cover them, but don't limit yourself to numbers.
Permutations, as I've been defining them, operate as a group with
binary multiplication, and yet they are not numbers. Often they're
presented in terms of numbers but they don't have to be. The
advantage of using letters is each permutation can then be "a
scramble" for plaintext using those same letters, a substitution code.
P(m) --> e where m is plaintext and e is "encrypted". This opens up
playing with phrases in math class. Yay.
> Since the discussion is about one binary operation in terms of another
> binary operation being repeated via a distributive property, it's
> clear that we need two and not just one binary operation - a group or
> groupoid (recall again they require only one binary operation) won't
> help us. So we need to be a context of a field or most generally a
> ringoid.
>
That's not how I see the discussion. The discussion is about how to
help people conceptualize about addition and multiplication, as
operations. Sometimes we want to play up "repeated addition", other
times we want to play it down. I'm showing how to do either.
To play it down, but without trying to discourage or end its use, we
can look at sets / situations wherein multiplication is defined and
not addition.
Since both addition and multiplication are defined for the same set in
some cases, we can still talk about "multiplication" in general terms.
But we can also show how these operations may not both be present or,
if they are, they're not linked in a way that makes "repetition"
imaginable or useful.
That's not an irrelevancy. On the contrary, that's making the point
that they're different.
>>
>>>>
>> It may seem like radical surgery to you but I assure you the avid
>> teachers of the "repeated addition" meme took this step long ago.
>
> I'm talking about how it must seem to the average kid being hit with
> this. If you think that having to break up one of the factors before
> being able to apply the repeated addition model on two factors neither
> of which is one of the two original factors is not for the average kid
> a radical departure from being able to apply it directly to the two
> original factors, then be my guest.
>
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.
So there's no need to just stick with integers. (1/2)(1/3) is like
"adding 1/2 of 1/3" whereas (5/2)(1/3) is adding (1/2) of (1/3) to
itself 5 times.
>> As
>> I was saying above, even your "scaling" which you present in cartoon
>> form *in contrast* to repeated addition, is just more repeated
>> addition (of some unit of vector / length / distance).
>
> For purposes of calculation, that is true, and Devlin also I think
> would agree. But this is about whether we should try to get them to
> see that with multiplication, there is fundamentally something more
> going on than merely the calculative part of repeated addition, to see
> that multiplication is in fact a different animal an addition.
> Especially as we jump to reals, specifically the irrationals and most
> especially the transcendental irrationals.
I think just visiting realms wherein multiplication is defined but
addition is not is a good way to break the hold of the "repeated
addition" meme, which is this case means merely to show it has limited
applicability.
>
>> You have an ungodly huge Avogadro Number of atoms and you divvy them
>> so that 3.14159... of them (some trillions) get to be a "unit" in some
>> way, with respect to some multiplier. Numbers like pi and e are just
>> fractions that "never end" in this vague hand-wavy extension of Q to
>> R.
>
> This does not appreciate at all what an irrational - especially a
> transcendental irrational - is.
>
I don't think that's necessary to focus on in the case of
multiplication as repeated addition.
We can say we have pi% (pi percent) of the balls, meaning 3.14159...% of them.
That's pi% = pi * (1/100). it doesn't matter, in this context, that
we approach a limit without being able to terminate the decimal
expression (for pi). That's moving to a new topic, whereas when it
comes to multiplication as repeated addition, we find that Reals (R)
and Rationals {Q} don't present significantly different conceptual
challenges.
When I took honors calculus with Thurston (who recently died), I
remember he giving us grades like pi and pi/2 on tests. Yes, a bit
whimsical.
> I'd say that if there is any isolation going on here, it would be in
> people being isolated from understanding just how fundamentally
> different an animal an irrational number is from a rational number,
> and this especially applies to transcendental irrationals, which are
> so fundamentally different they are fundamentally different even in
> comparison to algebraic irrationals.
>
I think you're changing the subject, getting off onto something else.
True, there are important differences between p/q rationals, members
of Q, and numbers with no p/q expression (p, q both in Z).
But when it comes to what is addition, and what is multiplication, I
don't think the differences between Q and R are really that important.
Rationals
can have billions of digits. In terms of actual number crunching, no
one uses non-terminating decimals i.e. Reals in the sense of
infinitely precise numbers have no application in computing. 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
Anyway, to wander off into the differences between Q and R is somewhat
of a rhetorical fallacy if you're saying the similarities between Q
and R, when it comes to multiplication as repeated addition, somehow
should not be raised.
Differences to not make similarities go away. Multiplying to
rationals and multiplying two reals is not an appreciably different
process, just as working
with lengths doesn't change just because the diagonal of a unit square
is somewhere between 1.41 and 1.42.
> Not only that, to see more of just how fundamentally different a
> transcendental irrational actually is, the number of elements in the
> set of all reals being an uncountable infinity and the set of all
> reals being a continuum is due to the number of elements in the set of
> all transcendental irrationals being an uncountable infinity. The
> number of elements in the set of all algebraic numbers, which is the
> union of the set of all rational numbers (all of which are algebraic)
> and the set of all algebraic irrationals, is merely a countable
> infinity, which means that this union cannot be a continuum. (A set is
> a continuum only if it contains an uncountable infinity of elements.)
>
I see you wandering off into Cantorism but failing to make a point, as
the differences between Q and R are not at issue here. Rather,
when it comes to addition and multiplication, we don't have to switch
analogies on the basis of those differences. We have an
opportunity to dwell on their sameness. They're all subsets of C
after all. I think we should jump to C sooner.
Remember, I'm customizing for STEM, which is very concerned with
measure and discreteness. The uncertainty inherent in measurement is
a focus. The finitude of computing, its dependence on memory as well
as energy, is a focus i.e. it takes real time and real energy to do a
computation, i.e. we're interested in computing as physical processes.
Back to multiplication, another use of the symbol is in functional
composition. You can see this in terms of group properties in that
the identity function does
nothing to the output i.e. x = f(x). An inverse function may be a
relation, so we maybe want to limit ourselves to bijections for the
moment i.e. one-to-one functions.
h = f * g means h is a function h(g(x)) where x is whatever inputs
(might well not be numbers -- important for use to get away from
numbers as the only objects
upon which operations are performed (a mind-numbing restriction in
most of K-12 which must be removed lest students be mentally shackled
beyond repair).
So we have an inverse function such that if inverse(f) = g, then g * f
= identity function.
So we can talk in terms of group theory if we like (CAIN), about a
world in which there is no corresponding meaning for addition such
that g * f = f + f + f...
g times. Makes no sense. Doesn't matter than we could define f + g
as f(x) + g(x). That gets us no closer to defining * in terms of
repeated +.
sqrt(log(x)) is not a result of adding the log function to itself sqrt
times etc.
The point being: if we want to weaken the hold of "multiplication as
repeated addition", we can do so any time, by expanding our idea of
what "to multiply" means. And we really need to expand the idea of
multiplication.
The STEM course I'm currently teaching (which I didn't write) does
this, using Composable as a type of object, and having two objects of
the Composable type compose functions in response to the
multiplication operator. Students have to override __mul__ to define
multiplication, and __pow__ to define power. The functions themselves
may be non-numeric, such as reversing strings of letters. Compose a
reverser with a reverser and you get an identity function that leaves
the string as it was. (f * f)(s) = h(s) = f(f(s)) = s where s is some
string object like "ABCD". We don't call it "math" though as in STEM
it's not important to distinguish S from T from E from M. There's no
need to sort topics like that anymore. That's work for luddites, as
long as they keep finding funding for themselves.
Kirby
|
|
