Multiplication, transformation and preservation

[Andrius Kulikauskas]

Joe,

Thank you for your thoughts on multiplication. They were helpful and
timely as I prepared my talk:
http://www.ms.lt/sodas/Book/DiscoveryInMathematics
"Discovery in Mathematics: A System of Deep Structure"

Please see my diagram and discussion at the end. It's based on Maria
Droujkova's and her Natural Math team's work on the many different
meanings of ordinary multiplication.

Also, please see my discussion of the four different geometries. I know
nothing about them but my approach is able to help bring out their
importance, though. Symplectic geometry preserves "areas" which is
probably the same as what you mean by "volumes", yes?

I look forward to learning and thinking a lot more about these subjects.

Thank you and all for your insights!

Andrius

Andrius Kulikauskas
m...@ms.lt
+370 607 27 665

-----------------------------------

Kirby et al,
The more I think about "multiplication", the more I think the term is
being abused.

What we call "scalar" multiplication seems to capture the essence of one
understanding: multiplication as repeated addition, the Peano definition.
Physically, this often take the form of a "production" equation: rate *
time = product; e.g. 60 miles per hour times 4 hours = 240 miles.

(Which, BTW, is the operation of extracting the distance component from
a speed, not constructing a composite of distance and time.)

But then there are all sorts of other multiplications.

My question: what do you call "volume conserving" transformations"? Is
there a "branch" of mathematics devoted to their study?
Clearly translations and rotations fall in this category, but only
certain kinds of "stretching".
But more generally, consider the shapes assumed by a blob of viscous
liquid (of uniform density) tumbling through space or distributing
itself over an uneven surface,

My intuition is that "volume" is conserved by nature, even
pseudo-volumes composed of multiple types such as kilogram meters or
volt amperes.

And taking a cue from Geometric Algebra, "rotations" are not so much
"multiplications" as "additions" of arcs.

As for addition, we see in "vector addition" and more generally in
molecular structure,
that physical "additions" include a component of direction as well as a
component of magnitude, so we might call this "construction".

So we have a "constructive" operation of addition, extended to scalar
multiplication;
And a "transformative" operation traditionally called "multiplication"
but which I would rename as, well, "transformation".
So my "new math" would have operators of "construction" and
"transformation", which both can be considered forms of "arrangement",
which probably ends up in group theory!.

If I were a mathematician, I would probably be able to develop such
ideas into a "grand unification of mathematics"!

But for now, I'm still wrestling with the idea that the exponentiation
operator has two inverses
and yields (at least) two new kinds of irrational numbers: radical and
complex.

Complex seems to come from the dimensionality-increasing nature of
"multiplication",
and radical from the notion that "root taking" should be closed and
single-valued, i.e. all n factors of x^n equal.

[Eureka! In my own mind, I had never crystallized the idea into such
statements until just now!]

Coupled with the idea that the algebraic sign of the integers is just
the 1D version of direction,
we have the concept that "numbers"--or more generally, quantities--have
both a magnitude (arithmetic) and a directional (geometric) component.
So we need to put Euclid and Peano together!

But back to multiplication.
As I have been saying, I think we have been looking at "multiplication"
thru the wrong end of the telescope.
I say "nature" starts with multi-dimensional hyper-volumes, and we
humans observe and compute the projections.

Which led to my original question: is there a mathematics of "volume
conserving" (constant integral) transformations?

Joe Austin

[Joseph Austin]

Andrius, It will take me a while to digest all that!

Re: "volume", the more correct term would be "hyper-volume", but for physical reasons I'm focusing on 3-space-1-time dimensions,
or whatever describes the "natural world".
But since you bring it up, "area" in Hamiltonian phase-space (position,momentum) is indeed a conserved quantity of motion.

The sort of "product" that occurs most often in physics is one that produces a "hyper-volume" that is conserved.
Indeed, "conservation" or "invariance" seems to be the fundamental concept underlying what it means to be a "law" of physics:
to understand "motion" or "change" is to understand what stays the same vs. what varies,
and how the "variables" are constrained by the "constants".

Joe

[kirby urner]

On Tue, Jun 21, 2016 at 9:19 AM, Joseph Austin <drtec...@gmail.com> wrote:

Andrius, It will take me a while to digest all that!

Re: "volume", the more correct term would be "hyper-volume", but for physical reasons I'm focusing on 3-space-1-time dimensions, 

or whatever describes the "natural world".

I've been meaning to mention this other difference between the M (maths) and the S (science).

Science uses mathematics to record instrument measures, such as ruler distances, in a precise and savable format suitable for sharing with others.  Without standardizing on the digits 0-F, our global civilization would be untenable.  Then science uses these measures to construct explanatory and predictive models that ideally guide in the construction of new apparatus and measuring devices.

Mathematics, however, is more likely to use math as like a music notation to control instruments, to play them, but not with an eye to making empirical measurements of the surroundings.  Mathematicians are allowed to simply "make noise" and not talk about the real world at all.  They get paid anyway.  In the physics department, if you're not talking about the real world, they say "why don't you get a job with the mathematicians down the hall?"

In STEM we're wanting to keep them both, and in pushing M towards Art and Art History (The Tesseract, the Time Machine, and the Tetrahedron is an art history story as well as a math story), I'm opening the door to a purely meaningless arrangement of polyhedrons in a kind of mental geometry that's as yet un-applied. 

We care neither about crystals nor carbon crystals in particular (diamonds) when just looking at the pretty patterns on the screen.  We care not for gold.  Math is not mere Mineralogy, praise Allah.  Mineralogy is more a meeting ground twixt E and S.  Their perceived relative value, as well as physical properties, should enter in:  our bridge to Economics, the engineering of coinage (bitcoin, regular coin...).

When talking curriculum, it's all about ordering (also known as "sequence").  I harp on spiraling because we want and need repetition but not mindless review of the same material.  We want what we've learned to build, to carry us to a next level in some way.  That's what spiraling is about, versus just going around and around and never getting anywhere. 

In the bliss of childhood, when we're physically growing and need lots of outdoor time to practice motor skills, we allow a brand of M that's closer to Music, in that we let children indulge in pure pattern and creative play, oblivious to any "meaning" they might have to be making, if they want an A grade, and enough of a personal workspace to say they're not homeless.  The Waldorf School curriculum is especially big on this.

 

But since you bring it up, "area" in Hamiltonian phase-space (position,momentum)  is indeed a conserved quantity of motion.

The difference between the Lagrangian and the Hamiltonian is a college-level topic, still K-16 though:
http://worldgame.blogspot.com/2016/06/math-summit-revisited.html
 

The sort of "product" that occurs most often in physics is one that produces a "hyper-volume" that is conserved.

However mathematically this may not always be the same algebra.  We're projecting ahead that Clifford's could be "the one", but what's conserved in quantum mechanics (quantum number) is tracked using different equations that used to look at pool balls on a frictionless pool table.  There's no unifying notion of "product" we can express as a single algorithm.

 

Indeed, "conservation" or "invariance" seems to be the fundamental concept underlying what it means to be a "law" of physics:

"Exceptionless" is another characteristic of generalized principles.  Kind of like "fixed price" in economics i.e. you don't sway gravity with a little cash under the table.
 

to understand "motion" or "change" is to understand what stays the same vs. what varies,
and how the "variables" are constrained by the "constants".

Joe

Well put.

Kirby

 

> On Jun 21, 2016, at 9:26 AM, Andrius Kulikauskas <m...@ms.lt> wrote:
>
> Joe,
>
> Thank you for your thoughts on multiplication.  They were helpful and timely as I prepared my talk:
> http://www.ms.lt/sodas/Book/DiscoveryInMathematics
> "Discovery in Mathematics: A System of Deep Structure"
>
> Please see my diagram and discussion at the end.  It's based on Maria Droujkova's and her Natural Math team's work on the many different meanings of ordinary multiplication.
>
> Also, please see my discussion of the four different geometries.  I know nothing about them but my approach is able to help bring out their importance, though.  Symplectic geometry preserves "areas" which is probably the same as what you mean by "volumes", yes?
>
> I look forward to learning and thinking a lot more about these subjects.
>
> Thank you and all for your insights!
>
> Andrius
>
> Andrius Kulikauskas
> m...@ms.lt
> +370 607 27 665
>
> -

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[kirby urner]

ADDENDUM:

We care neither about crystals nor carbon crystals in particular (diamonds) when just looking at the pretty patterns on the screen.  We care not for gold.  Math is not mere Mineralogy, praise Allah.  Mineralogy is more a meeting ground twixt E and S.  Their perceived relative value, as well as physical properties, should enter in:  our bridge to Economics, the engineering of coinage (bitcoin, regular coin...).

By "Their perceived relative value" I mean of the minerals.  I've been going to a sequence of lectures on minerology at the Pauling House, very appropriately as the structure and make-up of molecules is what Linus was into really deeply.

http://controlroom.blogspot.com/2016/03/boot-camp-continues.html

It's hard to discuss minerals and gemology without discussing the axes of scarcity and aesthetic appeal.  Very pretty gem stones that are very hard to find, fetch a pretty penny.  Some minerals actually get to serve as pennies once properly coined.

So in that sense we bridge to Economics.

When just flying around in the static lattices we call crystals, looking at the various molecules, their inter-distances and angles, we're not too concerned with time.  However once stress and strain enters the picture and we want to see a delta, such as the impact of a shock, well the time interval needs to enter it i.e. action (mvd).

http://mybizmo.blogspot.com/2016/06/first-person-calculus.html

Kirby