tensors...

[kirby urner]

Hey Andrius, whaddya think of this Youtube
introducing tensors:

https://youtu.be/f5liqUk0ZTw

I like his "welcome to my humble abode"

backdrop, which is pretty standard in

"Youtube World" (a backdrop that's

homey-domestic).

He shows us the humble components he'll

be using to trigger our imaginations, but

then pulls out that giant XYZ thing of only

positive basis vectors.

He'll need at least one more vector -(i + j + k)
to span the rest of his room without rotation,
at which point he might want to adjust the
inter-angles to form a more canonical caltrop).

Kirby

[Joseph Austin]

I remember my first encounter with vectors, in a math context{

"A quantity with magnitude and direction".

But at the time, I didn't have any intuition for directed quantities.

It was only when I got to physics, and concepts like velocity and force,

that I grasped the significanceco-variant of "direction", 

and in particular, that arithmetic of vectors required taking the direction into account in determining the sum,

giving results different than the simple sum of magnitudes.

This encounter with tensors left me with the same "huh"? reaction as my first encounter with vectors.

Wouldn't it be better to first illustrate physical "tensor" quantities and the "natural arithmetic" of such quantities

before explaining that they can be described by sets of co-variant and contra-variant components?

Wouldn't all math be better taught by illustrating physical quantities and their physical relationships and "arithmetics"

before introducing the abstracted symbolism and operators?

After all, much of math was developed as an abstraction of physical relationships.

As an aside, has anyone considered Hestenes' Geometric Algebra as an organizing principle for quantitative arithmetic?

Joe Austin

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[Andrius Kulikauskas]

Kirby, Bradford,

Thank you for your great letters, for sharing your intuition.

Bradford, I appreciate your ideas on wholeness. I look forward to
folding more circles. I'm curious if you know of architect Christopher
Alexander and his books on "A Pattern Language" and "The Timeless Way of
Building", "The Nature of Order". "The Timeless Way of Building"
includes a very poetic description of wholeness as "the quality without
a name". "The Nature of Order" includes his thoughts on wholeness
preserving transformations.

Kirby, thank you for telling me about Neo4J graph database software. It
may be perfect. I've downloaded it and have started working with it.

Kirby, thank you for finding that video on tensors. Overall, it's very
helpful to see these subjects from a variety of points of view. I also
like your letter about making math results known. It's why I enjoy
reading Wikipedia very much.

However, there's part of me that thinks that in Math there's a lot of
bad ("unnatural") ways to think about things and not so many good
("natural") ones. Sure, there's a lot to be gained by finding your own
way and it's even necessary for learning. But in the end, the goal is
to find and enjoy the nice, simple, elegant, informative, inspiring ways
to think about a subject or problem.

The 12 minute video "What's a tensor?" devotes only the last couple of
minutes to tensors. And basically he just shows a 3x3 cube. I think
it's completely misleading. Then in his last two sentences he explains
that the reason tensors are important is because the basis doesn't
matter - the changes in the bases and in the scalars cancel each other
out. In other words, his approach (scalars and unit vectors) is
precisely the irrelevant one for understanding why tensors are so important.

Personally, it's very disturbing to learn from somebody who leads you
down a path that makes it harder to understand what it's all about. For
example, he says that a tensor is a generalization of a vector. But I
think that's a bad way to think about it. Instead, I would say that a
tensor is a generalization of a matrix, or better yet, a linear
transformation.

The wrong way to think about tensors is that they are "multidimensional
arrays" of numbers, for example, a 3-dimensional box of numbers. The
reason this is wrong is because it doesn't tell you everything you need
to know. It's like defining "fractions" as "pairs of numbers" like 3
and 5. That's not helpful because it's crucial to know which is the
numerator and which is the denominator. Does the fraction "eat" 5 and
"spit out" 3? Or does it "eat" 3 and "spit out" 5? That's the
difference between 3/5 and 5/3. And in the end, the numbers could be
6/10 and 10/6, which is to say, it's not about 3 and 5. Similarly, for
that 3-dimensional box (a tensor of order 3) it is crucial to know what
those numbers are doing. Is the tensor:
* eating 3 vectors and spitting out a scalar?
* eating 2 vectors and spitting out 1 vector?
* eating 1 vector and spitting out 2 vectors?
* eating a scalar and spitting out 3 vectors?

Similarly, a tensor of order 2 could be:
* eating 2 vectors and spitting out a scalar (as with an inner product)
* eating 1 vector and spitting out 1 vector (as with a linear
transformation)
* eating a scalar and spitting out 2 vectors (as with a "bivector", an
oriented parallelogram whose two sides are given by vectors)
Each of the above tensors/operations can be written out in coordinates
as a 2-dimensional square of numbers, in other words, a "matrix". But
the interpretation of that matrix is in each case, qualitatively,
mathematically, absolutely, incomparably different.

Now imagine not being told of these differences... and trying to figure
it out on your own... with the "help" of the misinformation...

But what I'm learning, without any teacher :( , is that what's crucial
is that the tensor is "eating" M vectors and "spitting out" N vectors.
It is generalizing the situation where a linear transformation (a
matrix) is "eating" (multiplying) a column vector and "spitting out" a
row vector. So that's one crucial idea.

But the other crucial idea is that there is a symmetry, a "duality",
between the vectors "eaten" and the vectors "spit out". They divide up
the total dimensions into two spaces, one being built "bottom up" and
the other being built "top down".

It is late so I will write more another day.

A final point to add is to say that no definition of "tensor" is helpful
unless it can explain what we mean by "tensor product".

I'm writing about what I haven't yet learned myself. So it's very
helpful to get to interact with you.

Thank you, Kirby, Bradford!

Andrius

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

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[Maria Droujkova]

Please be reminded that this community is for sharing things we DO like, not for sharing things we DON'T like. Different people like different things. Please share what inspires you and what you find useful. 

Your moderator,

Dr. Maria Droujkova

NaturalMath.com
Make math your own, to make your own math!

-- .- - ....

On Mon, Apr 25, 2016 at 5:34 PM, Andrius Kulikauskas <m...@ms.lt> wrote:

The 12 minute video "What's a tensor?" devotes only the last couple of minutes to tensors.  

--

 

[kirby urner]

On Mon, Apr 25, 2016 at 5:30 PM, Joseph Austin <drtec...@gmail.com> wrote:

I remember my first encounter with vectors, in a math context{

"A quantity with magnitude and direction".

But at the time, I didn't have any intuition for directed quantities.

I see us teaching the same XYZ coordinate system over and over in different ways.

Going directly to "arrow numbers" and adding them tip to tail, lets us introduce vectors the first time, in the context of building 3D shapes, like in CAD.

We used to have "mechanical drawing" along the vocational non-pre-college track.  Now we have machines controlling 3D printers, making shapes on screen.

This is where my homage to the ray tracer comes in, an earlier posting to mathfuture:

http://tinyurl.com/hfg2sc3

 

It was only when I got to physics, and concepts like velocity and force,

that I grasped the significanceco-variant of "direction", 

and in particular, that arithmetic of vectors required taking the direction into account in determining the sum,

giving results different than the simple sum of magnitudes.

If your only goal is to define the 12 corners, 20 faces and 30 edges of the icosahedron, you'll still want to know about vectors.

They're simply "pointers to points". 

Assume vectors always originate at the origin, also the center of the icosahedron. 

None of the 30 edges are vectors, they're all line segments defined by two vectors to the end points.  Segment = (V0, V1).  I've seen linear algebra textbooks that use this approach.

http://www.amazon.com/Elementary-Linear-Algebra-Stewart-Venit/dp/0534951902

The goal is to have our machines make drawings, including plots. 

Have a vector tip to trace a sine wave.  Not that hard.

http://www.4dsolutions.net/ocn/numeracy1.html  (I even do quaternions here, which have a vector part).

Then if you want said icosahedron to start spinning around an axis, you'll need a rotation matrix (or quaternions).

http://www.3dgep.com/understanding-quaternions/  (used in game engines -- or "models" we might call 'em).

Rotation matrices are an IB level topic (International Baccalaureate, offered in some US high schools).  Peter Farrell is into 'em.

A big insight comes with seeing complex "monads" such as matrices as "just like numbers" i.e. "things to be added and multiplied".

Lambda calc develops our abstract algebra sense, that many sets might feature operations, including sets of non-numeric things.

If the computer language allows operator overloading, so much the better, as then we can see + and * used with vectors, matrices, natural numbers, even polynomials.

 

This encounter with tensors left me with the same "huh"? reaction as my first encounter with vectors.

So far I'm not seeing tensors as K-12, just 13-16 in S (Science) i.e. I'm agreeing with you we should move to physics, once 3D CAD construction, 3D printing, ray tracing have been covered.

Vectors are 9-12 with "arrow numbers" even earlier.

I brought up tensors in part because Andrius was saying he was looking into them.

 

Wouldn't it be better to first illustrate physical "tensor" quantities and the "natural arithmetic" of such quantities before explaining that they can be described by sets of co-variant and contra-variant components?

Wouldn't all math be better taught by illustrating physical quantities and their physical relationships and "arithmetics" before introducing the abstracted symbolism and operators?

After all, much of math was developed as an abstraction of physical relationships.

Agreeing.
 

As an aside, has anyone considered Hestenes' Geometric Algebra as an organizing principle for quantitative arithmetic?

Joe Austin

You'll maybe see I asked Andrius about Grassmann Universal Algebra, inspiring to Clifford of Clifford Alegebra fame. 

http://tinyurl.com/hlkve2k

This is the stuff Hestenes is focused on.

Kirby

[Bradford Hansen-Smith]

 Andrius, I did read "The Timeless Way of Building" and "The Nature of Order" years ago when beginning to investigate geometry. Yes he is both poetic and practical in his approach. Thank you for reminding me, his writing helped move me forward. 

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[kirby urner]

On Mon, Apr 25, 2016 at 2:34 PM, Andrius Kulikauskas <m...@ms.lt> wrote:

< ... >
 

Is the tensor:
* eating 3 vectors and spitting out a scalar?
* eating 2 vectors and spitting out 1 vector?
* eating 1 vector and spitting out 2 vectors?
* eating a scalar and spitting out 3 vectors?

Thanks, interesting. 

This goes straight to computer languages which pre-define what type of output to expect from what inputs.

scalar eat(v0, v1, v2): { }  # <--- eating 3 vectors, spitting out a scalar

vector eat(v0, v1): { }    # <--- eating 2 vectors, returning a vector

pair eat(v0): { }  # <-- eating 1 vector, returning a pair of vectors
...

and so on, where pair is 2 vectors returned as a single object (I'm staying with the convention that what's returned is a single thing (which may in turn be a collection).

What goes inside the { } (the "mustaches") are the statements needed to do the digesting (after eating), so whatever thing of whatever type (written ahead of eat( )) might be returned.

Not every language requires declaring the type of object returned.

< ... >

I'm writing about what I haven't yet learned myself.  So it's very helpful to get to interact with you.

Thank you, Kirby, Bradford!

Andrius

If you come across what you consider an ideal Youtube on tensors, I'm all ears and eyes.

I like discussions that get away from 100% chalk and talk format, i.e. the classical lecture hall with sliding boards, that's OK sometimes, but what about cartoons?

Khan Academy is a step away from chalk and talk (no chalk, more colored pens).  I haven't checked his tensor stuff yet, but I plan to.

https://youtu.be/8vBfTyBPu-4?list=PLA3C55EC6DEB58621

What I especially enjoyed was Dan's willingness to use props, but I understand your qualms.  His video goes with some book neither of us have seen.

Anyway, his giant XYZ octant, a product of our culture, got me musing about spanning space, how his "x-hat", "y-hat" and "z-hat" (three basis vectors) will not get outside the all-positive / non-negative octant  (+, +, +) unless we allow them to change direction.  We need those negative basis vectors -- but do we need them all?

Changing a vectors direction (as distinct from changing its length, or from adding vectors) is what I call "rotation" (change in direction of any kind).  Without rotation, three basis vectors from the origin in directions X+, Y+, and Z+, cannot reach all points on a sphere around said origin, even with addition.

The standard response is we're free to multiply by -1 to reverse direction, so that's scalar multiplication too, not rotation.  But I'd say "mirroring" is fundamentally a different operation from growing and shrinking in the same direction.

These are the kinds of naive questions / concerns of a newbie (but not a child).

Obviously my questions / concerns are somewhat off topic relative to tensors. 

I'm interested in a non-XYZ coordinate system, more than in XYZ in more dimensions.

Kirby

<< ... >>

[Andrius Kulikauskas]

Maria, Thank you for keeping me on topic and with the right attitude.
Andrius

[Bradford Hansen-Smith]

Is the tensor:
* eating 3 vectors and spitting out a scalar?
* eating 2 vectors and spitting out 1 vector?
* eating 1 vector and spitting out 2 vectors?
* eating a scalar and spitting out 3 vectors?

Eating vectors is a delightful way to describe the transforming process of folding and unfolding the tetrahedron from the triangle net (a 2-frequency equilateral triangle.) Folding the net bringing 3 sets of 2 edges together forming 3 vectors forming a scalar (we can call tetrahedron.) See it as joining 2 of 4 vectors make one vector (3 times.) Opening the tetrahedron by removing 2 vectors from one vertex leaves 2 vectors, (multiplying number of vertex points 3 times is 6 points in an octahedron patterned arrangement.) This relates to what is shown using the cube as illustration. I use words as a place to jump into the abstractions through observational understanding.

A tetrahedron defined by four vectors from a central vertex conversely suggest a fully defined tetrahedron from a single external vertex using 3 tensors measured to angles less than 90 degrees. Another tetrahedral arrangement would be 2 vectors and 4 points at 90 degrees without regard to surface area or rotation. So "is the tensor:... " I would have to say yes, as you would have it.

 

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[Joseph Austin]

On Apr 25, 2016, at 10:17 PM, kirby urner <kirby...@gmail.com> wrote:

If your only goal is to define the 12 corners, 20 faces and 30 edges of the icosahedron, you'll still want to know about vectors.

They're simply "pointers to points".  

Assume vectors always originate at the origin, also the center of the icosahedron.  

None of the 30 edges are vectors, they're all line segments defined by two vectors to the end points.  Segment = (V0, V1).  I've seen linear algebra textbooks that use this approach.

Kirby,

Thinking as a physicist, I'm uncomfortable representing points as vectors.

In physics, velocity (vector) and position (point) are independent.

Vectors do not have "position" and points do not have "direction".

We can see the dichotomy clearly in terms of time:

"Two o'clock" (instant, point) is not the same as "two hours" (duration, vector).

From this perspective, the edges of the icosahedron are more "vector-like" than the vertices.

We define a vector, not as a set of numbers, 

but as a mathematical object that obeys certain axioms and operations,

e.g. vector addition, scalar multiplication, inner product, etc.

Though it may be true that a vector can be described (relative to a given coordinate system) as a set of three numbers,

any set of three numbers does not constitute a vector.

But there is a kind of "dual" relationship between points and lines or "vectors".

When teaching object-oriented programming, I explored this dual concept with overloaded operators:

point + vector = point;

point - point = vector;

vector + vector = vector;

(Or replace "point" and "vector" with "instant" and "duration" for time arithmetic.)

But what is "point + point"?  What is  "two o'clock + three o'clock"?  Can we define a "difference" but not a "sum"?

Now consider products:

We have scalar product: number * vector = vector.

But what is vector * vector = ??? [In physics, we have dot and cross products, 

although cross products are not "invariant" so are considered "cheating";

as BTW is treating an area as a normal vector--these only work in 3D, but since Einstein we regard space-time as at least 4D].

More consistently, we have inner and outer product:

vector IP vector = scalar

vector OP vector = matrix, or an object of higher order.

The led me to an unsettling conclusion:

Physical "multiplication" is not closed!

This challenges the standard way of teaching arithmetic that treats numbers as a "field".

Which leads me to suspect that much of math is "degenerate",

that is, several essentially different physical concepts have been "abstracted" into too few mathematical concepts,

resulting in confusion for those trying to understand mathematics in relation to reality.

For example, consider "count".  We can count apples, we can count votes, and we can "count" Halloween candy.

Now suppose several siblings "count" and compare their number of halloween treats.

Does the child with the highest "count" have the "most" candy?

Why do we place so much emphasis on "counting" and practically ignore the "types" of things counted?

I'd favor abandoning "number" theory and replace it with, say, "quantity" theory, where a quantity consisted not only of a magnitude (number),

but a "type" or "unit of measure".  In physics, we learn to carry the "units" along with the arithmetic and algebra. 

Wouldn't this be a good discipline to follow from the beginning?

Joe Austin

[kirby urner]

On Wed, Apr 27, 2016 at 4:54 PM, Joseph Austin <drtec...@gmail.com> wrote:

On Apr 25, 2016, at 10:17 PM, kirby urner <kirby...@gmail.com> wrote:

If your only goal is to define the 12 corners, 20 faces and 30 edges of the icosahedron, you'll still want to know about vectors.

They're simply "pointers to points".  

Assume vectors always originate at the origin, also the center of the icosahedron.  

None of the 30 edges are vectors, they're all line segments defined by two vectors to the end points.  Segment = (V0, V1).  I've seen linear algebra textbooks that use this approach.

Kirby,

Thinking as a physicist, I'm uncomfortable representing points as vectors.

In mathematics there's no time dimension necessarily.  There may be, but there needn't be.  Vectors without any time dimension tend to be purely spatial and so "point to points". 

They're not points, they're vectors.  But they point to points, just like a laser pointer makes a dot.
 

In physics, velocity (vector) and position (point) are independent.

In mathematics we have no velocity necessarily. 

Magnitude may simply represent "distance from the origin" (but then you need direction too, to pick out a particular star in the sky).
 

Vectors do not have "position" and points do not have "direction".

Agreed.

In my picture the vectors all radiate from an origin. 

However when we add them, we may interimly translate one (without rotating it) such that its tail begins at the tip of the other.  I know you know the drill.

That's vector addition.  Then there's scaling. 

In ordinary algebra to flip a vector 180 degrees is called "multiplying by negative one" but then a vector is a lot like a "ray" in geometry and why should it want to flip that easily?  Did we even define the negative numbers yet? 

Do we really need a "number line" of "real numbers" to add "arrow numbers"?  I might just need some straight sticks. 

Painters were using XY grids to fill their canvases with perspective drawings long before the latest formulations regard what "real numbers" really are.

One may use an XY or XYZ grid without knowing exactly what "real numbers" are (few people do -- a very technical subject).

 

We can see the dichotomy clearly in terms of time:

"Two o'clock" (instant, point) is not the same as "two hours" (duration, vector).

From this perspective, the edges of the icosahedron are more "vector-like" than the vertices.

As long as we're each clear about the namespace being entered, I see no problems.

In mine, which is supplemented by a certain "stickworks.py", the vectors all originate at the origin.

They have the usual Gibbs-Heaviside operations, of dot and cross product, convert to spherical.

I even convert them to "quadrays" in some implementations, a 4-tuple representation, not Cartesian.

So in this namespace, if you want just a line segment, you may define it with two vectors to its end points.

You're saying that's not the namespace you'd design.  Such concepts are not quite to your taste.

Fair enough.

I cited a linear algebra book that I know takes the same approach I do, just to underline its legit. 

I'm sure yours is too.

 

We define a vector, not as a set of numbers, 

but as a mathematical object that obeys certain axioms and operations,

e.g. vector addition, scalar multiplication, inner product, etc.

The vector is pointing to the right, let us say. 

Growing and shrinking is what we call scaling, but the direction does not budge. 

I can rotate the arrow around its tail to have it form an angle with what was previously its direction. 

The picture basically becomes one of two vectors, which we may say is "the same vector" before and after rotation.  We see a V, however wide.

We may rotate a vector all the way around to point in the opposite direction from previously.

One might choose to define scaling as "only in the direction a vector is already pointing" and ditch the negative numbers, given we have rotation as well (why have both?).

We're used to "absolute value" |v| as being "always positive" as it is. 

Lets make scaling positive i.e. no direction changing allowed, and then either allow or disallow rotation as an optional axiom (permitted move).  What would that look like?  One could explore.

Putting the XYZ positive basis vectors in such an axiomatic space, I notice rotation is required to reach all points around the origin. 

Otherwise the three basis vectors are "stuck" pointing only to points in the "all positive" octant, given only scalar multiplication (no negatives) and vector addition.

Translation, scaling and rotation tend to be the "big three" introduced in computer graphics explanations.  I think that's a good place to start.  As soon as we have that icosahedron, we spin it around opposite (A) vertexes (B) edge centers (C) face centers to get a network of 31 great circles.  The cuboctahedron gives us 25 great circles.  We have published pictures in our texts and websites.

 

Though it may be true that a vector can be described (relative to a given coordinate system) as a set of three numbers,

any set of three numbers does not constitute a vector.

True enough.

Plus in quadrays we use 4-tuples such as (2, 1, 1, 0) and don't need negatives at all.

 

But there is a kind of "dual" relationship between points and lines or "vectors".

When teaching object-oriented programming, I explored this dual concept with overloaded operators:

point + vector = point;

point - point = vector;

vector + vector = vector;

(Or replace "point" and "vector" with "instant" and "duration" for time arithmetic.)

What language did you use?  Not all of them permit overloading operators.  C++ yes.  Java no.

 

But what is "point + point"?  What is  "two o'clock + three o'clock"?  Can we define a "difference" but not a "sum"?

Now consider products:

We have scalar product: number * vector = vector.

But what is vector * vector = ??? [In physics, we have dot and cross products, 

although cross products are not "invariant" so are considered "cheating";

as BTW is treating an area as a normal vector--these only work in 3D, but since Einstein we regard space-time as at least 4D].

I think vector * vector could be accommodated with quaternions, which have a vector part.  Make the scalar part 1?  I did the code for that in Java in the 1990s at some point.

But it's all in the design of the language games, which add up to some namespace, and whether that namespace gels.

Some board games are more fun than others.  Some stories have thicker plots.

Only some axiomatic systems appear to "gel" to make a "world" worth exploring and developing within.
 

More consistently, we have inner and outer product:

vector IP vector = scalar

vector OP vector = matrix, or an object of higher order.

The led me to an unsettling conclusion:

Physical "multiplication" is not closed!

This challenges the standard way of teaching arithmetic that treats numbers as a "field".

I'm not sure what's meant by "physical" multiplication -- as opposed to what other kind?  Metaphysical?
 

Which leads me to suspect that much of math is "degenerate",

that is, several essentially different physical concepts have been "abstracted" into too few mathematical concepts,

resulting in confusion for those trying to understand mathematics in relation to reality.

Yes, there are many partially overlapping namespaces, as vectors are "applied" meaning "worked in" to various knowledge domains.

Mathematicians hope to keep all the terms so general that it won't matter what the application is, what the "units" are, but that's a pipe dream to some extent. 

Applications weight the abstract terms. 

We back propagate from the field, influencing how they're presented.

 

For example, consider "count".  We can count apples, we can count votes, and we can "count" Halloween candy.

Now suppose several siblings "count" and compare their number of halloween treats.

Does the child with the highest "count" have the "most" candy?

Why do we place so much emphasis on "counting" and practically ignore the "types" of things counted?

Agreed.  If someone has all yucky marshmallow candies that's like having a *negative number* of candies.

 

I'd favor abandoning "number" theory and replace it with, say, "quantity" theory, where a quantity consisted not only of a magnitude (number),

but a "type" or "unit of measure".  In physics, we learn to carry the "units" along with the arithmetic and algebra. 

Wouldn't this be a good discipline to follow from the beginning?

Joe Austin

Sounds to me like some of the computer languages are doing just that, combining quantities with units and keeping careful track of those units.

As for whether to call Number Theory something else, all I can say is memes sometimes have a lot of inertia, meaning if they've been saying "Number Theory" for ages, it's hard to change that overnight. 

I'd say that sort of inertia a "physical reality", i.e. things being the way they are, and hard to change (no "turning on a dime") is no joke. 

Our disciplines are not necessarily optimized nor best nor how they had to be.  They're just how it is, for now.

The idea that there's a way to "start over from scratch" is tempting though, and to some extent I undertake such work in collaboratively carving out some space in the sun for that volumes table I favor (A,B,T and E modules 'n all that).

I encourage you to keep thinking about alternative definitions and axioms for growing a wide variety of systems, some differing from their neighbors in just one respect, others more alien.

That's a type of creativity worth encouraging don't you think?

Kirby

[Joseph Austin]

On Apr 28, 2016, at 2:15 AM, kirby urner <kirby...@gmail.com> wrote:

'm not sure what's meant by "physical" multiplication -- as opposed to what other kind?  Metaphysical?

Kirby,

What I'm referring to is the kind of indicated products one sees in physics formulas, such as:

Einstein's equations:  E = m * c^2; 

or  the gas law: P * V = n * R * T;

What got be started was trying to imagine a "square second".

The example I usually give is:

What is 2 apples times 3 oranges?

Answer:  6 "pears"  (pairs).

Understand I came by mathematical "understanding" by way of physics, then by computer science (Turing machine, recursive functions).

To the extend I "understand" math, I understand it as axiom systems.

But the morphism between the traditional axiom systems and the physical notions I have from science I find lacking.

So I'm suggesting that we (I) need to re-think traditional mathematics into a form that more intuitively matches the kinds of quantitative relationships--and as well the space-time or geometric or structural relationships--that occur in the "real world."  In the real world, we don't have dimensionless points, we have bodies extended in three spatial dimensions and also moving in time.  We have atoms and molecules "dancing" with each other in a community ball, "changing partners" and creating dynamic patterns we call "life".  Or similar patterns we call "the economy" or "elementary particle physics" or "cosmology" or "the ecosystem".

I'm suggesting that, instead of teachers searching for real-world examples of traditional axiom systems, 

we ought instead to be searching for, and teaching our students to build, axiom systems for real-world examples.

We need a way to "mathematically" model the oxidation of hydrocarbons or the long-term effects of consumer credit interest rates on a community.

BTW, it did occur to me that your "vector" system for the icosahedron makes sense if all "vectors" intersect at the "center" (which you conveniently place at the origin).

Doesn't this then become something like an "eigenvector field" for a solid body?  Not only the vertices, but also all "points" on the edges and surfaces correspond to "vectors" in this space, and scalar multiplication of the vectors is a similarity transformation in conventional geometry.

But I still feel there is something missing from the theory: the "connection" between the vectors.

There is a difference between a "rectangle" and two orthogonal velocities, such as a boat rowing across a river.

And the difference is the "connection".  How does that figure into the axiom system?

Joe.

[Joseph Austin]

On Apr 28, 2016, at 2:15 AM, kirby urner <kirby...@gmail.com> wrote:

What language did you use?  Not all of them permit overloading operators.  C++ yes.  Java no.

Yes, C++.

But even C++ doesn't go far enough.

I need something like "a circle of friends" so a specific set of classes can share each other's private data.

I had considered recasting mathematics into "object-oriented" form, 

replacing "binary" operators with "unary" operators with parameters,

and "numbers" with "state variables."

Then I would end up with something more like LISP operating on lists as unary representations of numbers.

For example, see:

http://drtechdaddy.com/2013/06/11/replicount-programmed-in-scheme/

Joe

[kirby urner]

On Thu, Apr 28, 2016 at 7:00 AM, Joseph Austin <drtec...@gmail.com> wrote:

On Apr 28, 2016, at 2:15 AM, kirby urner <kirby...@gmail.com> wrote:

'm not sure what's meant by "physical" multiplication -- as opposed to what other kind?  Metaphysical?

Kirby,

What I'm referring to is the kind of indicated products one sees in physics formulas, such as:

Einstein's equations:  E = m * c^2; 

or  the gas law: P * V = n * R * T;

What got be started was trying to imagine a "square second".

The example I usually give is:

What is 2 apples times 3 oranges?

Answer:  6 "pears"  (pairs).

Understand I came by mathematical "understanding" by way of physics, then by computer science (Turing machine, recursive functions).

To the extend I "understand" math, I understand it as axiom systems.

Yes, you are exploring the all-important interface between everyday reality and fully abstracted symbols, which have no time dimension, and therefore no energetic significance (except to use them burns calories).

An axiomatic system, such as chess, has these timeless knight moves defined, but exactly when or where the knight moves, who moves it, what it's made from (if anything)... these are "additional dimensions" which the chess notations try to remain innocent of. 

But in practice the notation is "applied" e.g. used to record actual games between chess players.  The time dimension creeps in, just as it does in applied math.

Euclidean geometry does not want to know about Planet Earth and its "quantities" or its "timeline", because it wants to apply equally well to a fictional planet in Oz Universe perhaps with an entirely different physics (aka "game engine").

That being said, these distinctions are cultural and perhaps anchored in the subcultures we associate with Platonism and the Pythagoreans. 

Both wanted to abstract number from any physical units, creating what you're calling "axiomatic systems".

 

But the morphism between the traditional axiom systems and the physical notions I have from science I find lacking.

Yes, important point.  "Where does the rubber meet the road?"  You're finding *too little* friction (they don't meet).

 

So I'm suggesting that we (I) need to re-think traditional mathematics into a form that more intuitively matches the kinds of quantitative relationships--and as well the space-time or geometric or structural relationships--that occur in the "real world."  In the real world, we don't have dimensionless points, we have bodies extended in three spatial dimensions and also moving in time.  We have atoms and molecules "dancing" with each other in a community ball, "changing partners" and creating dynamic patterns we call "life".  Or similar patterns we call "the economy" or "elementary particle physics" or "cosmology" or "the ecosystem".

[ emphasis added -- see [2]]

Yes, all good.

And my point earlier was there's real metaphysical inertia to entrenched concepts, so if we're co-strategizing at all, I have to put on my PR / Madison Avenue hat and ask "how are you ever gonna get people to quit smoking what they're smoking?"

Lots of campaigns have proved effective (just chew nicotine gum instead?) but then the cig companies are notoriously successful at recruiting new generations of smoker, as the older cohort dies off.  Their campaigns are effective also, countering ours.

So when you start questioning fundamentals and how we teach them, I have to ask about your army, your generals, your command structure and so on.  Actually, I just freely speculate and don't interrogate, as you're not in my chain of command, so no rank enters in.

In my case, I fly the flag of Lambda Calculus and I'm targeting the status quo in very specific places (in rotten parts).  My network includes a lot of published authors who already know the basics of what we're up to.  The planning started decades before I was born.

 

I'm suggesting that, instead of teachers searching for real-world examples of traditional axiom systems, 

we ought instead to be searching for, and teaching our students to build, axiom systems for real-world examples.

In the humanities we go a long way with definitions.  We must separate definitions from axioms at some point. 

Saying space is "three dimensional" and that these dimensions are "independent" such that when you turn 90 degrees (not just 89 degrees) around a street corner you're in a "whole new dimension!" (wow, amazing!), that's all definitional groundwork, not yet axiomatic in flavor.[2]

Meaningful terms are being injected into a namespace, some gas has been turned on (I know that sounds sinister, but let me remind you that even oxygen, necessary for life but not in pure form, is a piped gas). 

In the humanities, we'll keep injecting meaningful terms until the cows come home, without ever getting to a first axiom -- maddening to some mathematicians I realize. 

But then we have grammar (strict and not strict) and those rules of grammar could be our generative axioms (as long as it's grammatical, you're allowed to publish it, even call it a "theorem" if you like [1]).

 

We need a way to "mathematically" model the oxidation of hydrocarbons or the long-term effects of consumer credit interest rates on a community.

BTW, it did occur to me that your "vector" system for the icosahedron makes sense if all "vectors" intersect at the "center" (which you conveniently place at the origin).

I wouldn't put it in quotes.  My vector space is the strict vector space of linear algebra with spanning basis vectors.  100% traditional linear algebra.  Time dimension optional, strictly speaking.  There's never a need for "velocity" in linear algebra.  That's just one more application.

But then in addition to playing by the rules, I'll occasionally throw in the monkey-wrench of 4-tuples and "basis rays" that don't obey the same axioms, given there's no "reversing direction" by means of mere scalars in some games (variants -- some might say mutants, or Martian?).

 

Doesn't this then become something like an "eigenvector field" for a solid body?  Not only the vertices, but also all "points" on the edges and surfaces correspond to "vectors" in this space, and scalar multiplication of the vectors is a similarity transformation in conventional geometry.

If we define eigenvector as a vector which merely scales (i.e. linearly changes) when a transformation is applied, then yes.  When the icosahedron grows and shrinks (scales) it does so along 12 vectors. 

In stickworks.py, I've overloaded the multiplication operator so that 3 * icosa expands it linearly by a factor of 3, meaning area goes up 9 times, and volume 27 times (I do those multiplications automatically as well).[3]

Again, my next move is probably to spin the thing around its 31 axes.  Here's a published picture any one of my students will have seen a hundred times (this is 1970s stuff):

http://www.rwgrayprojects.com/synergetics/s04/figs/f5730b.html


 

But I still feel there is something missing from the theory: the "connection" between the vectors.

There is a difference between a "rectangle" and two orthogonal velocities, such as a boat rowing across a river.

And the difference is the "connection".  How does that figure into the axiom system?

Joe.

You are asking philosophical questions that point to the heart of western civilization.

I'm part of an advancing army of Lambda Calculus teachers that plans on rescuing mathematics from becoming a train wreck thanks to no one in politics having enough training or background to think coherently, and in the US, just about everyone is a politician, right up to the dean of the college.

I don't care who wins the US presidential election; our army is unstoppable (we've already won, so it's mainly a matter of defending our position at this point).
 

That probably sounds pretty wild but I'm not trying to rename Number Theory or upset the Linear Algebra people by force feeding them a time dimensions.  Mathematicians are notoriously averse to working with time.  Remember:  Coxeter.4D != Einstein.4D (!= for "not equal") i.e. just because we're doing multi-dimensional geometry doesn't mean any of those dimensions have a temporal dimension (or they all do, equally, and so time factors out, doesn't matter).

Kirby

[1]  grammatically correct humanities papers, hit reload to keep generating a next one:
http://www.elsewhere.org/journal/pomo/  (theorems from axioms!)

[2]  By the way, I don't feel obligated to buy that space is "three dimensional" since the "minimum box" is clearly the four-faceted tetrahedron.  Volume says "four" more loudly than "three" when I look at a four-cornered, four faceted shape. 

It's when I look at the regular hexahedron (aka "qyoob") that I might become fixated on 90 degree angles as definitional of dimension, but why would I look at one of those? They're not topological primitves and have many awkward properties such as when you cut parallel to a face, they're no longer self-similar.

[3]  http://www.4dsolutions.net/ocn/stickworks.html

[kirby urner]

You've clearly been mulling these things over a long time.  Interesting stuff.

I do consider operator overloading a brilliant feature.  My considerations about inertia enter in again regarding whether we need to deprecate "A * B" versus "(* A B)" i.e. how much is this an either/or issue.  The fact that Python can be turned into a LISP (called Hy) simply be changing the parser, suggests we not worry about one edging out the other.  https://en.wikipedia.org/wiki/Hy

Caveat:  I mentioned oxygen being necessary for life in my previous post, but of course I was just talking about a subset of life forms.  I think oxygen molecules feature in all life forms that we know of, but it's not necessarily introduced as a breathed gas as not all life forms "breathe".

Some people think those extremophiles that love deep ocean thermal vents were the source of all our DNA on this planet.

http://www.space.com/19439-origin-life-earth-hydrothermal-vents.html

Kirby

[Joseph Austin]

On Apr 28, 2016, at 1:13 PM, kirby urner <kirby...@gmail.com> wrote:

Both wanted to abstract number from any physical units, creating what you're calling "axiomatic systems".

Ah, perhaps a philosophical position exalting "number" to mystical importance,

or a rejection of the material world in preference to the "spiritual" or ideal.

But my point isn't that we have "real" dimensions of "space" or "time" or whatever, but that we have different kinds of units.

And the interpretation of the dimensions isn't so important as the idea that "abstracting" an n-dimensional object to a zero-dimensional point abstracts away many of the degrees of freedom and and possible differences (suppose not all dimensions are "the same").

Imagine a "geometry" in which "point" is replaced by a differential "volume" element can be shrunk as small as we please but never thereby converted to a zero-dimension entity, meaning it always has an orientation wrt axes or other points.  Or perhaps even a "quantum" math in which there is some "smallest" volume but it can be shaped in a variety of ways.

It had been suggested to me that the "primitive" operators of arithmetic ought to be difference and ratio (both of which have a "direction", i.e. anti-commute) instead of sum and product.  We know that we need to expand our concept of number to include the additive and multiplicative inverses of whole numbers, so why start where we can't stop?  

From my cursory examination of Geometric Algebra, I was struck with the idea that perhaps there is an entire "polynomial" of quantities of successive dimensions, and inter-dimensional operations upon them, sort of a "super-closure" of a system of "quantitative objects" of different types.

So I'm saying, to fully represent "reality", we might re-introduce the "units" (and "space" or structure) back into the axioms about numbers.

Perhaps the computer will facilitate such exploration.  We are moving toward languages such as javascript in which the "type" is carried with the data instead of being built in to the operators and variables. So in essence, all operators are overloaded.  

And it seems one can also do OOP in Scheme.

Guess it's time to stop speculating and start exploring.

Joe

[Ted Kosan]

Joe wrote:

> But the morphism between the traditional axiom systems and the physical
> notions I have from science I find lacking.
>
> So I'm suggesting that we (I) need to re-think traditional mathematics into
> a form that more intuitively matches the kinds of quantitative
> relationships--and as well the space-time or geometric or structural
> relationships--that occur in the "real world." In the real world, we don't
> have dimensionless points, we have bodies extended in three spatial
> dimensions and also moving in time. We have atoms and molecules "dancing"
> with each other in a community ball, "changing partners" and creating
> dynamic patterns we call "life". Or similar patterns we call "the economy"
> or "elementary particle physics" or "cosmology" or "the ecosystem".
>
> I'm suggesting that, instead of teachers searching for real-world examples
> of traditional axiom systems,
> we ought instead to be searching for, and teaching our students to build,
> axiom systems for real-world examples.

It seems to me the need you are talking about here is handled in
symbolic logic by "interpretations":

https://en.wikipedia.org/wiki/Interpretation_%28logic%29

A knowledge of interpretations would enable students to understand how
any given formal language can be used in an unlimited number of
applications.

Ted

[Ted Kosan]

Joe wrote:

> I had considered recasting mathematics into "object-oriented" form,
> replacing "binary" operators with "unary" operators with parameters,

Most CASs enable the user to create new operators as needed. The
following example shows how to create a new operator named "+#" which
can handle adding integers and lists in MathPiper:

-----------------
Create a new operator named "+#". All of MathPiper's procedures and
operators are contained in a "rulebase" (which is a database for
transformation rules.)

In> RulebaseHoldArguments("+#", ["lhs", "rhs"]);
Result: True



Make the operator an infix operator with precedence 70.

In> Infix("+#", 70);
Result: True



Evaluate an expression that contains the operator with operands that
are Integers. Notice that the expression is returned unchanged because
no transformation rules have been placed into the +# operator's
rulebase yet.

In> 1 +# 2
Result: 1 +# 2



Create a transformation rule that matches when both of the operator's
operands are integers.

In> x_Integer? +# y_Integer? <-- AddN(x,y);
Result: True

In> 1 +# 2
Result: 3


The operator <-- is used to create global transformation rules. The
left side of the <-- rule is called its "head", and the right side of
the <-- rule is called its "body". The variable names 'x' and 'y' are
arbitrary. During pattern matching, the _ applies the single argument
predicate procedure that is to its immediate right (Integer? in this
case) to the subexpression that is assigned to the variable which is
to its immediate left. If the predicate procedure returns True,
pattern matching of other parts of the expression continues. If none
of the predicates in the head return False, and if the rest of the
expression matches the head, then the transformation rule replaces the
matched subexpression with the result of evaluating the rule's body.



Currently, the +# operator does not have a rule that matches when its
operands are lists.

In> [2,3] +# [1,1]
Result: [2,3] +# [1,1]



Creating a rule that handles lists is straightforward.

In> x_List? +# y_List? <-- Map("+", [x, y])
Result: True

In> [2,3] +# [1,1]
Result: [3,4]


Ted

[kirby urner]

On Thu, Apr 28, 2016 at 2:34 PM, Joseph Austin <drtec...@gmail.com> wrote:

On Apr 28, 2016, at 1:13 PM, kirby urner <kirby...@gmail.com> wrote:

Both wanted to abstract number from any physical units, creating what you're calling "axiomatic systems".

Ah, perhaps a philosophical position exalting "number" to mystical importance,

or a rejection of the material world in preference to the "spiritual" or ideal.

Yeah, stuff like that. 

I'm not saying rejection of the material world in favor of the Platonic one of ideal forms was my idea.  People were thinking that way when I got here. 

Seems to me that imaginary objects are less perfect in missing that one all-important attribute:  energetic existence.  Isn't reality more perfect that what's merely idealized?  What's so great about a "perfect circle" if it can't even make the first cut?

Mainly I think mathematicians wanna be free to explore their multi-dimensional vector spaces *without* thinking they need to know any physics.  Constraints of the physics engine kind are too fettering to the unfettered mathematical imagination.  That's what I gather anyway.

 

But my point isn't that we have "real" dimensions of "space" or "time" or whatever, but that we have different kinds of units.

Indeed.  Many.  But isn't there a recognition of some as more "basic" or "fundamental" though?

https://en.wikipedia.org/wiki/International_System_of_Units

Some quantities are considered irreducible to others yes?

 

And the interpretation of the dimensions isn't so important as the idea that "abstracting" an n-dimensional object to a zero-dimensional point abstracts away many of the degrees of freedom and and possible differences (suppose not all dimensions are "the same").

Yes it does.

If you're modeling the real world, then the possibility of choosing a wrong model is worrisome.

As you know, a lot of mathematics is not concerned with modeling reality.  The model is an end in itself.

Take the icosahedron for example.  Is it a model of something?  Of what, besides the icosahedron? 

I suppose one could say "it models the viral sheath for the RNA coil inside" (capsomeres typically form an icosahedron in a generic virus) but that's collapsing something general to something quite specific. 

Mathematicians like to go the other way:  from specific to general.

I'm just observing, not weighing in.  I see a kind of tug-o-war going on.  Mathematicians are always trying to shake the mud of their shoes and not get their hands dirty.  Engineers keep getting covered in engine grease.

 

Imagine a "geometry" in which "point" is replaced by a differential "volume" element can be shrunk as small as we please but never thereby converted to a zero-dimension entity, meaning it always has an orientation wrt axes or other points.  

That's what I do with a tetrahedron (any four non-coplanar points, volume / edges implied).  That's my symbol for an irreducible volume, a home base shape.  The sphere is actually more complicated as a cloud of points equidistant from a common center.

Karl Menger called what you're proposing a "geometry of lumps" as we never get to zero-dimensional, only the dimensionality of an irreducible lump.  Res extensa.  Clay.  Substance in pure principle.  He wasn't making the link to a tetrahedron at all, when publishing in that anthology [1], that identification would come later.

 

Or perhaps even a "quantum" math in which there is some "smallest" volume but it can be shaped in a variety of ways.

I like it.  You're coming up with your own definitions and axioms.  Perhaps a world of theorems will gel.

 

It had been suggested to me that the "primitive" operators of arithmetic ought to be difference and ratio (both of which have a "direction", i.e. anti-commute) instead of sum and product.  We know that we need to expand our concept of number to include the additive and multiplicative inverses of whole numbers, so why start where we can't stop?  

I wonder if it's either / or.  Maybe once you have the one, you always have the other.

In most maths, subtracting is adding the additive inverse of, while dividing is multiplying by the multiplicative inverse of. 

So if you have the unary operation of "inverse" well-defined, with respect to both addition and multiplication, you're set to both subtract and divide. 

As you well know, the inverse of x with respect to addition, when added to x, gives the additive identity, ditto for the inverse of x w/r to multiplication (we get the multiplicative identity whenever two inverses of each other get multiplied).

I understand your concerns about "closure".  The bookkeeping of physics plays close attention to units, e.g. action (as in Planck's) is in units of pd (momentum times distance) whereas "frequency" (such as Hertz) is the reciprocal of a time interval, 1/t. 

E = hf i.e. Energy = action multiplied by frequency or pd / t = p(d/t) = pv = mvv, which does indeed look like units of Energy, so it all it works out.  So far it looks like we have closure. 

What happens to "h-bar" if we go with tau instead of pi I wonder.  Do we redefine it?

 

From my cursory examination of Geometric Algebra, I was struck with the idea that perhaps there is an entire "polynomial" of quantities of successive dimensions, and inter-dimensional operations upon them, sort of a "super-closure" of a system of "quantitative objects" of different types.

So I'm saying, to fully represent "reality", we might re-introduce the "units" (and "space" or structure) back into the axioms about numbers.

Perhaps the computer will facilitate such exploration.  We are moving toward languages such as javascript in which the "type" is carried with the data instead of being built in to the operators and variables. So in essence, all operators are overloaded.  

I agree with you here.  The objects keep track of their own units.  It's much harder to shake them off.
 

And it seems one can also do OOP in Scheme.

Yes, apparently.  Javascript, is a moving target, and will have syntax more like Python's soon, if it doesn't already (depends on which transpiler you use).
 

Guess it's time to stop speculating and start exploring.

Joe

Sounds like a plan.

Kirby

[1]  'Modern Geometry and the Theory of Relativity', in Albert Einstein: Philosopher-Scientist , The Library of Living Philosophers VII, edited by P. A. Schilpp, Evanston, Illinois, pp. 459-474.

[kirby urner]

On Thu, Apr 28, 2016 at 5:34 PM, Ted Kosan <ted....@gmail.com> wrote:

Joe wrote:

> I had considered recasting mathematics into "object-oriented" form,
> replacing "binary" operators with "unary" operators with parameters,

Most CASs enable the user to create new operators as needed. The
following example shows how to create a new operator named "+#" which
can handle adding integers and lists in MathPiper:

 

Ted's remarks remind me that I just learned a new unary operator recently.

We all know the unary operator 5! ("factorial") as in 5*4*3*2.

I just learned the N#, the "primorial" means the product of all primes up to the Nth prime.

https://oeis.org/A002110
http://primes.utm.edu/glossary/page.php?sort=Primorial [1]

Where I got it from was 49 minutes into this Youtube:
https://youtu.be/pp06oGD4m00

Kirby

[1]  quoting from the 2nd source:

It is customary to only apply the notation p# to primes p, but some authors will apply it to any positive real number (e.g., 10.72# = 2^.3^.5^.7 = 210). When viewed this way, the function log(x#) is Tschebycheff's function, and the prime number theorem is equivalent to the expression

log x# ~ x,

(i.e., (log x#)/x approaches 1 as x approaches infinity.)

[Ted Kosan]

Kirby wrote:

> And my point earlier was there's real metaphysical inertia to entrenched
> concepts, so if we're co-strategizing at all, I have to put on my PR /
> Madison Avenue hat and ask "how are you ever gonna get people to quit
> smoking what they're smoking?"
>
> Lots of campaigns have proved effective (just chew nicotine gum instead?)
> but then the cig companies are notoriously successful at recruiting new
> generations of smoker, as the older cohort dies off. Their campaigns are
> effective also, countering ours.
>
> So when you start questioning fundamentals and how we teach them, I have to
> ask about your army, your generals, your command structure and so on.
> Actually, I just freely speculate and don't interrogate, as you're not in my
> chain of command, so no rank enters in.

I think the most promising initial group of people to target this
disruptive movement at are average computer programmers who have poor
math skills (which are due a defective mathematics education). Some of
the reasons I think this are as follows:

- Their poor math skills are a constant source of pain, so they are
very motivated to find ways to reduce or eliminate this pain.
- Programmers are efficient and constant learners, so if they can be
convinced that learning something new will reduce their pain, they
will do what it takes to learn it.
- Programmers are very influential because they are the main creators
of the 21st century.
- Programmers have an enormous capacity for disruption. For example,
most large 21st century disruptions were started by programmers
(Google, Facebook, Apple, Amazon, Uber, Airbnb, etc.)

How can this movement be started? I think all it would take would be
to get an article on the front page of Hacker News that explains to
programmers who are bad at math that it is not their fault, and that
their problem can be fixed easier than they think. I have been
preparing an article like this for a long time. Here are some
candidate titles I have come up with for the article:

"
You can't hate math if you were never taught math

You can’t be bad at math if you were never taught math

You’re bad at math because you were never taught math

You’re afraid of math because you were never taught math

You were taught the laws of algebra, but were you not taught the rules
of algebra

Your math teachers were clueless"


Ted

[kirby urner]

On Thu, Apr 28, 2016 at 6:44 PM, Ted Kosan <ted....@gmail.com> wrote:

<< SNIP >>
 

I think the most promising initial group of people to target this
disruptive movement at are average computer programmers who have poor
math skills (which are due a defective mathematics education).

Apropos of what you're saying, I recommend this panel discussion of

statisticians and data scientists, asking themselves 'what's the difference?'

(between these two disciplines) and where are we going with education.

https://youtu.be/C1zMUjHOLr4
Data Science and Statistics: different worlds?

From the panelists, we learn that many with programming skills are

indeed attracted to data science but may feel intimidated or scared off

by the statistics aspect, with statistics categorized as a type of math.

The consensus that seems to develop is like what we say here:  "make

math your own".  Hands-on projects with early rewards incentivize going

forward. 

Trying to master all the theory first, before tackling challenges, is a sure
way to get discouraged and burn out.  Curricula which insist on this

approach are up against that that do not.  Programmers vote with their

feet.

Here's another good panel regarding Big Data / Machine Learning etc.:

https://youtu.be/czLI3oLDe8M
Deep Learning: Intelligence from Big Data

Bookmark for later maybe?

Coming from a code school perspective, I'm seeing lots of these

schools offering Machine Learning type courses.  Let me find a

couple examples:

https://www.codeschool.com/blog/2016/02/26/machine-learning-let-your-code-learn-from-text/
http://www.dataschool.io/15-hours-of-expert-machine-learning-videos/
https://www.udemy.com/learn-hadoop-mapreduce-and-bigdata-from-scratch/

I think math teachers might also want to seize the initiative and

insist on professional development courses that are (a) more

future-oriented and (b) empowering of these same teachers

vis-a-vis developing their own curriculum.

Reformers maybe learned their lesson from the failure of New Math

over the long haul.  It was pushed as a top-down program, sourced

by the University of Chicago and other places.  "GNU math" in contrast

is more of a community-driven ("crowd-sourced") phenomenon. 

Yes, we have individual stars, leaders, organizers.  But we work

together more than we try to shut each other out.

MathPiper, in being GNU licensed, is already a contribution to a

more community-driven approach.  But when do math teachers

get any time to catch up?

When I say "my army has won" I'm mostly thinking of the rhetoric

associated with the open source movement.  For a long time, the

tongue in cheek goal was "world domination", and then the message

became "OK we've done that, the world economy now runs on

open source (very much a part of the mix), so now what?" 

http://ecx.images-amazon.com/images/I/41YmeR7lMCL.jpg
http://imaginaction.pbworks.com/f/World%20Domination.jpg

In other words, the goal of world domination has been achieved

already.  It's a way of working, not an ideology so much (although

it does come with specific ideas about what constitutes "open").

How might we bring that open source spirit into the schools more?

Kirby

 

 

[Joseph Austin]

On Apr 28, 2016, at 9:44 PM, Ted Kosan <ted....@gmail.com> wrote:

I think the most promising initial group of people to target this
disruptive movement at are average computer programmers who have poor
math skills (which are due a defective mathematics education).

I've taught programming to students with poor "math skills" by using non-numerical applications and languages

(Scratch, Alice).

But I'd argue that programmers actually have very well-developed "math" skills, just not in the areas of math typically taught.

Surely it takes more "reasoning" skill to develop a working program than to prove a typical "math" theorem.

(The exception may be Advanced Calculus, but then I am not persuaded that the axiom set offered is complete or the inference rules used valid.)

And as you suggest, it may well be that successful programmers have math "intelligence" superior to that of their math teachers.

If we can show programmers that what they do is "math", then they may feel confident to investigate other areas of "math."

And if we can't convince programmers that what they do is math, how will we ever convince mathematicians?

Of course, the "mathematical" nature of programming is obscured by notations that don't look much like math.

If only we could write programs in Greek instead of the Latin alphabet :)

Joe Austin

[Joseph Austin]

On Apr 28, 2016, at 8:39 PM, kirby urner <kirby...@gmail.com> wrote:

If you're modeling the real world, then the possibility of choosing a wrong model is worrisome.

As you know, a lot of mathematics is not concerned with modeling reality.  The model is an end in itself.

Take the icosahedron for example.  Is it a model of something?  Of what, besides the icosahedron?  

Yes, but aren't we searching for a "math track" for kids who aren't on the "math" track?

Shouldn't we be thinking "application first"?

(BTW, I understand the Greeks understood the icosahedron as the form of atoms of water.)

Which of course underscores that our models are just conjectures, but we do attempt to chose models

whose mathematical properties correspond to physical behavior in an intuitive way.

[Joseph Austin]

Speaking of tensors...

I did some searching and found a discussion in the Feynman Lectures on Physics

http://www.feynmanlectures.caltech.edu/II_31.html

Perhaps it's an over-simplification, but I gather a rank-2 tensor is something like an anisotropic version of a vector.

Some physical examples of Tensors:

The mathematics of tensors is particularly useful for describing properties of substances which vary in direction

Tensors are referred to by their "rank" which is a description of the tensor's dimension. 

A zero rank tensor is a scalar, 

A first rank tensor is a vector; a one-dimensional array of numbers. 

A second rank tensor looks like a typical square matrix. 

An example of a physical tensor is the moment of inertia;

for an arbitrarily shaped object, the moment of inertia depends on its orientation with respect to the axis of rotation.

Stress, strain, thermal conductivity, magnetic susceptibility and electrical permittivity are all second rank tensors.

The refractive index in any given direction through the crystal is governed by the dielectric constant Kij which is a tensor. 

A third rank tensor would look like a three-dimensional matrix; a cube of numbers. 

Piezoelectricity is described by a third rank tensor. 

A fourth rank tensor is a four-dimensional array of numbers. 

The elasticity of single crystals is described by a fourth rank tensor.

Joe Austin

[kirby urner]

On Fri, Apr 29, 2016 at 8:36 AM Joseph Austin <drtec...@gmail.com> wrote:

 

And as you suggest, it may well be that successful programmers have math "intelligence" superior to that of their math teachers.

We need to have that not matter too much.  Teachers have skills

having to do with human interaction, the ability to communicate

clearly.  A lot of supposedly brilliant math whizzes don't have these

skills as well-developed, so it's a matter of complementing skills.

If a teacher also has a high degree of math intelligence, not always

measured in degrees, so much the better.

 

If we can show programmers that what they do is "math", then they may feel confident to investigate other areas of "math."

Who will connect these dots for them?  I wrote this essay a generation ago:

http://www.4dsolutions.net/ocn/overcome.html

Capital Sigma and capital Pi (the Greek letters) are just control

structures, "while loops" if you will. 

But what teachers explain that?
 

 

And if we can't convince programmers that what they do is math, how will we ever convince mathematicians?

Of course, the "mathematical" nature of programming is obscured by notations that don't look much like math.

If only we could write programs in Greek instead of the Latin alphabet :)

Joe Austin

Right.

When Iverson's APL came out, wanna be "high priests" flocked

to it as it turned out to be great for obfuscating and hiding behind

code. 

What better way to get a cushy job on Wall Street than to become
the indispensable king of an inscrutable code pile in a language
that's uber-hard to decipher unless one wrote it oneself, and even
then...

CTOs got wise to this downside of over-specialization after awhile

and the open source movement is in many ways about keeping

geeks and their companies accountable.  "Show us your APIs and

documentation at least" say the Wall Street investors, "or how do

we know you're not just a hollow shell?" 

Companies that want to be black boxes and hide in the shadows
of secrecy tend to be only half of what they say they are, if that
much.

Successful companies cultivate an in-house culture of andragogy

meaning peers and colleagues teach each other, the opposite of

engaging in obfuscation. 

Teaching one's peers ends up meaning devoting a significant portion
of one's work to keeping an "intelligent layman" in the loop i.e. if you
can't clue your above average smart cookie, then do you really have
a business model (i.e. a model business)?

Is the core problem that math teachers who do connect the dots

tend to end up in IT themselves?  I don't see anyone besides me

addressing the economics of the situation.  As soon as high school

math teachers are computer literate, they go for the higher paying

job (in aggregate, I'm not saying there aren't exceptions).

Kirby

[kirby urner]

On Fri, Apr 29, 2016 at 9:29 AM, Joseph Austin <drtec...@gmail.com> wrote:

On Apr 28, 2016, at 8:39 PM, kirby urner <kirby...@gmail.com> wrote:

If you're modeling the real world, then the possibility of choosing a wrong model is worrisome.

As you know, a lot of mathematics is not concerned with modeling reality.  The model is an end in itself.

Take the icosahedron for example.  Is it a model of something?  Of what, besides the icosahedron?  

Yes, but aren't we searching for a "math track" for kids who aren't on the "math" track?

Shouldn't we be thinking "application first"?

I'm all for application first.  The icosahedron may be transformed into a globe. Globes display global data.  No classroom should be without one.

Unfolded, you can make it look like this:

https://flic.kr/p/bCuaRv

 

(BTW, I understand the Greeks understood the icosahedron as the form of atoms of water.)

Yes, they associated Platonic shapes with the four elements:  earth, air, fire, water.

The pentagonal dodecahedron was supposedly a symbol of the whole cosmos.

http://www.4dsolutions.net/satacad/martianmath/mm11.html

These days we know the Platonic shapes occur naturally at very small scales in the biological world.  The virus tends to be an icosahedron.

http://www.4dsolutions.net/satacad/martianmath/mm30.html

 

Which of course underscores that our models are just conjectures, but we do attempt to chose models

whose mathematical properties correspond to physical behavior in an intuitive way.

In my book, if you're a in a school that won't connect these dots for you in any way:

*  icosahedron to world map

*  icosahedron to virus

then you're likely in the infamous school-to-prison pipeline, a kind of underground railroad in reverse that helps drive our secret economy (one of penning people for pay).

https://www.aclu.org/fact-sheet/what-school-prison-pipeline

Kirby

[Ted Kosan]

Kirby wrote:

> Apropos of what you're saying, I recommend this panel discussion of
> statisticians and data scientists, asking themselves 'what's the
> difference?'
> (between these two disciplines) and where are we going with education.
>
> https://youtu.be/C1zMUjHOLr4
> Data Science and Statistics: different worlds?
>
> From the panelists, we learn that many with programming skills are
> indeed attracted to data science but may feel intimidated or scared off
> by the statistics aspect, with statistics categorized as a type of math.

This was an excellent discussion. One thing I gathered from it is if
traditional statistic teachers don't modernize their approach ASAP,
they are in danger of having their teaching duties taken over by
people who understand statistics and computer programming. I think
this danger exists for all math teachers.

> I think math teachers might also want to seize the initiative and
> insist on professional development courses that are (a) more
> future-oriented and (b) empowering of these same teachers
> vis-a-vis developing their own curriculum.

In another post you had this brilliant insight:

"As soon as high school math teachers are computer literate, they go
for the higher paying job (in aggregate, I'm not saying there aren't
exceptions)."

I think most math teachers who were interested in learning how to
program computers have already done so, and (as you said) have moved
on to better paying jobs. For the remaining math teachers who have
not learned computer programming yet, it is probably too late to do
so. For years I have tried to figure out if the math/programming
teachers of the future would be math teachers who learned how to
program or programmers who learned how to teach math. Your insightful
post, and the discussion you linked to above, have made it clear to me
that it is programmers who will learn how to teach math who will be
the math/programming teachers of the future.

Ted

[Ted Kosan]

Joe wrote:

> Of course, the "mathematical" nature of programming is obscured by notations
> that don't look much like math.
> If only we could write programs in Greek instead of the Latin alphabet :)

In> RulebaseHoldArguments("Σ", ["rhs"]);
Result: True

In> Prefix("Σ", 1);
Result: True

In> Σ [1,2,3]
Result: Σ [1,2,3]

In> Σ x_List? <-- Sum(x)
Result: True

In> Σ [1,2,3]
Result: 6


I think I could also get Σ [α,β,γ] to work too with a bit of tinkering :-)

Ted

[Joseph Austin]

On Apr 29, 2016, at 12:40 PM, kirby urner <kirby...@gmail.com> wrote:

Is the core problem that math teachers who do connect the dots

tend to end up in IT themselves?  I don't see anyone besides me

addressing the economics of the situation.  As soon as high school

math teachers are computer literate, they go for the higher paying

job (in aggregate, I'm not saying there aren't exceptions).

I'd say a solution to this problem is the flip side of a solution to the "continuing education" problem:

find a way to enlist practicing professionals to teach part time.

I know this works at the college level--there is a burgeoning industry of "after-hours" accredited degree programs

that hire faculty from local industry.

The school's overhead is low (no dorms or football games); 

the faulty think of it as "community service" as much as "extra income."

Joe Austin

[kirby urner]

On Sat, Apr 30, 2016 at 4:01 AM, Joseph Austin <drtec...@gmail.com> wrote:

On Apr 29, 2016, at 12:40 PM, kirby urner <kirby...@gmail.com> wrote:

Is the core problem that math teachers who do connect the dots

tend to end up in IT themselves?  I don't see anyone besides me

addressing the economics of the situation.  As soon as high school

math teachers are computer literate, they go for the higher paying

job (in aggregate, I'm not saying there aren't exceptions).

I'd say a solution to this problem is the flip side of a solution to the "continuing education" problem:

find a way to enlist practicing professionals to teach part time.

I'm wondering to what extent you think Youtube and the like are filling that niche.

How many math teachers ever assign viewing Youtubes X, Y and Z for homework
I wonder.

Assigning homework that depends on Internet access begs the question of where
"home" (as in homework) might be.

I saw that story on CBS News of the school district that sends buses to park in the
trailer parks where kids live, and act as routers for WiFi.

http://www.cbsnews.com/news/california-coachella-valley-school-district-closes-digital-divide-with-wifi-on-school-buses/

True story:  Portland Public Schools (PPS) experimented with busing only its TAG
(talented and gifted) program kids to a building on Marine Drive, where working
professionals (myself included) were given classrooms full of like fifth graders. 

They had Apple 2es (right, this was quite a long time ago) and could play computer
games, which is what they mostly elected to do, Oregon Trail especially.  SimCity
was another favorite, though students did not necessarily want to accept the goal
of building a thriving city.  Some just wanted to trigger disasters and then bulldoze

the whole place.

[ Imagine a game like chess where the goal is to help both kings reach the other side

legally i.e. without ever entering check.  This is to be accomplished with no pieces

taken by either side.  That's not chess, but it borrows the rules for moving. ]

In my big picture view, the focus needs to shift to whether any given student has a
"study bubble" meaning a safe place to just sit and watch Youtubes without a lot of
noise and interruptions.  That might mean wearing noise canceling headphones
if working in a "study hall" setting. 

Everyone who works at Weeble (a Y Combinator startup) gets noise canceling
headphones i.e. this practice of paying close attention to "ergonomics" is percolating
through the private sector.

What I imagine as superior to big study halls (like big libraries that permit talking)

would be more like office cubicles and sure, why not occupy many floors.  You

always have a space to get back to.  In some models, you can sleep there too.

It's called a dorm. 

But the study space and bedroom or camp site need not be collapsed. 

I call this a "personal workspace" (PWS), which is more than just the

virtual desktop, though that's part of it.

http://worldgame.blogspot.com/2015/07/pws-personal-workspace.html

 

I know this works at the college level--there is a burgeoning industry of "after-hours" accredited degree programs

that hire faculty from local industry.

In Portland we have Saturday Academy, which lets adults in many walks of

life share their skills and knowledge with self-selected students, who tend to

come from privileged backgrounds with resources to invest in after-hours

education. 

I've done lots of field testing of my curriculum ideas in these classes.  That's
where I created a special case implementation of Martian Math.

http://www.4dsolutions.net/satacad/martianmath/toc.html

Speaking of Martian Math, I use that to cover Futuristic Math i.e. math

looking ahead in the time dimension. 

Of course no one really knows what that means, but that isn't the point.
 
It's not about "knowing the future" as much as "thinking about the future"
as our one place to invest, which involves "risk based thinking". 

We can't afford to wait for AI to replace common sense right?  We need
sound judgement right away.

Many professions focus on the future, such as urban and regional planning,

architecture, any kind of scheduling or calendar-based activity.  Travel

agents... (functions more people are performing for themselves, given

Expedia and so on).

Neolithic Math (could be called something else, they all could), looks
back along the timeline.  We appreciate that math evolves, morphs,

changes with the times. 

Ralph Abraham (UCLA) was very keen to instill this sense of a timeline,
of history, per our Oregon Math Summit in 1995 (Sir Roger Penrose
also present -- I ran the microphone for his keynote).

http://grunch.net/archives/130

The Martian-Neolithic axis is the Future-Past axis. 

Then going horizontally, cutting across contemporaneous ecosystems,
I call out Supermarket and Casino Math as two more flavors. 

The first is about planting, harvesting and distributing, and is where
we might use databases + JavaScript. 

The latter, Casino Math, is about risk-taking e.g. trying new things,
cultivating new habits. 

We learn to think about the risks involved in *not* changing our ways
as well. 

"Doing nothing" may be the strategy most likely to fail, whereas other
times its a viable option.

Here's a Chicago area math teacher, speaking at a Skepticon who
dives into both Casino Math and Supermarket Math for raw material. 

He goes through a long poker example, with a Youtube excerpt
showing very high stakes play in Las Vegas, and then later goes
into a long discussion of a Supermarket checkout line. 

https://youtu.be/udAucVkzmSA

(Poker:  slide to about 22 mins to land in the middle of that example)

(Supermarket:  about 42 minutes in -- close to the end)

I've done some of my longest workshops (e.g. Python for Teachers)
in the Chicago area (e.g. in 2009 at some Hyatt near O'Hare). 

http://worldgame.blogspot.com/2009/03/urner-workshop.html

I wasn't even sharing about my Heuristics for Teachers back then.

http://wikieducator.org/Digital_Math

(11,552 views as of this morning)

The school's overhead is low (no dorms or football games); 

the faulty think of it as "community service" as much as "extra income."

Joe Austin

Yeah, that sounds like Saturday Academy all right.

I've wondered about companies sending vans around and whisking

self-selected students off site to educational facilities set up elsewhere.

Michigan has those Nexus Academies but I have little insight into

how they're doing.  I'll miss the US Distance Learning conference in

St. Louis this year (I went last year).

Kirby

[kirby urner]

In Python:


def Σ (*seq):
    return sum(seq)
   
def Π (*seq):
    product = 1
    for term in seq:
        product *= term
    return product
   
print ( Σ (1,2,3,4,5) )
print ( Π (1,2,3,4,5) )

OUTPUT:

15
120

[Joseph Austin]

On Apr 29, 2016, at 12:40 PM, kirby urner <kirby...@gmail.com> wrote:

Is the core problem that math teachers who do connect the dots

tend to end up in IT themselves?  I don't see anyone besides me

addressing the economics of the situation.  As soon as high school

math teachers are computer literate, they go for the higher paying

job (in aggregate, I'm not saying there aren't exceptions).

[This may be duplicate: I tried replying before but it may have been lost.]

I'd suggest that a solution to the "talent drain" problem may be similar to the solution to reform the

school-work-retirement paradigm, namely, replace a sequential track with parallel tracks.

So young people would not be "student then worker" but "student and worker" concurrently throughout their lives.

Similarly, we could employ professionals as "parallel time" teachers.

This pattern already works at the college level: there is a burgeoning industry of "after-hours" colleges 

offering accredited degree programs in fields such as  Business and IT.

They hire local professionals as teachers. The professionals are motivated as much by a desire to offer "community service" as for extra money,

and also to keep a foot in academia. 

Joe 

[Joseph Austin]

On Apr 29, 2016, at 12:40 PM, kirby urner <kirby...@gmail.com> wrote:

Is the core problem that math teachers who do connect the dots

tend to end up in IT themselves?  I don't see anyone besides me

addressing the economics of the situation.  As soon as high school

math teachers are computer literate, they go for the higher paying

job (in aggregate, I'm not saying there aren't exceptions).

I'd say a solution to this problem is the flip side of a solution to the "continuing education" problem:

find a way to enlist practicing professionals to teach part time.

I know this works at the college level--there is a burgeoning industry of "after-hours" accredited degree programs

that hire faculty from local industry.

[kirby urner]

On Sat, Apr 30, 2016 at 9:07 AM, kirby urner <kirby...@gmail.com> wrote:

<< EDIT >>
 

In my big picture view, the focus needs to shift to whether any given student has a
"study bubble" meaning a safe place to just sit and watch Youtubes without a lot of
noise and interruptions.  That might mean wearing noise canceling headphones
if working in a "study hall" setting. 

This TED talk by Susan Cain is apropos:

https://youtu.be/c0KYU2j0TM4

Some people require alone time to function and in high density cities, let alone

schools, this may become a real challenge.

In focusing on the need for a secure PWS for each K-16 student, I'm inviting

cross-generation alliances.

Ted, I listened to that radiolab about the surveillance drones (right, they used

airplanes in the story).

If students have school-issued devices that are running the right apps, it'll

be much easier to track them through the cell tower network than from some

eye in the sky.

You may object that criminals aren't going to be using any Uncle Sam smartphones

to start with [1], and I'd agree, but then we're talking about the student population

here, and what kid would turn down a free cell phone?  Even if you have a cell of

your own, the school one will have desirable features.  No one said two phones

was verboten.

Kirby

[1]  http://insights.cermacademy.com/2016/04/133-thoghts-smart-phones-privacy/

[Joseph Austin]

Ted, Do you have a more complete tutorial for creating new operators?
Following the example below, I tried to create and operator "\/" but ran into problems:

In> RulebaseHoldArguments("\/",["lhs","rhs"] );
Result: Exception
Exception: expected 2 arguments, got 1. Starting at index 1.

In> RulebaseHoldArguments("\/#", ["lhs", "rhs"]);
Result: True

In> Infix("\/#",50);
Result: True

In> [2,3] \/# [4,5]
Result:
Exception: Error encountered during parsing: Error parsing expression near token ***( \/# )***. Starting at index 6.

BTW, the documentation of Manipulate in the .904 doc tab didn't say anything about Rulebase, or Tabtitle or Font for that matter.
Joe

[Ted Kosan]

Joe wrote:

> Ted, Do you have a more complete tutorial for creating new operators?

Currently, the best sources of information for creating new operators
are the .mpw files in the library, and asking on this list.

> Following the example below, I tried to create and operator "\/" but ran into problems:
>
> In> RulebaseHoldArguments("\/",["lhs","rhs"] );
> Result: Exception
> Exception: expected 2 arguments, got 1. Starting at index 1.
>
> In> RulebaseHoldArguments("\/#", ["lhs", "rhs"]);
> Result: True
>
> In> Infix("\/#",50);
> Result: True
>
> In> [2,3] \/# [4,5]
> Result:
> Exception: Error encountered during parsing: Error parsing expression near token ***( \/# )***. Starting at index 6.

In MathPiper, the backslash character is the escape character in
strings. Try this code:

In> Infix("\\/",50);
Result: True

In> RulebaseHoldArguments("\\/",["lhs","rhs"] );

Result: True

In> [2,3] \/ [4,5]
Result: [2,3] \/ [4,5]

> BTW, the documentation of Manipulate in the .904 doc tab didn't say anything about Rulebase, or Tabtitle or Font for that matter.

The "Manipulate" procedure is experimental and still undergoing heavy
development. I usually don't document new features of a procedure
until I am fairly sure they make sense to support. At this point I am
not even sure if I want to use the name "Manipulate" for this
procedure as its final name. For now, the best "documentation" for
this procedure is to read the source code and to ask questions about
it on this list :-)

Ted