educational approach and because nutrition is a practical subject, too.

Andrius, ms@...

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

Maria Droujkova,

Thank you for your reply to Joaquin Carbonara, which I share below.

Joaquin and I were fellow graduate students at UCSD (in San Diego).

He's now a professor at Buffalo State. Joaquin's vision of a "language

of angels" is very dear to me, and I'll end with my thoughts on that.

Maria, your letter to me is among the very best I've ever received.

http://groups.google.com/group/mathfuture/msg/ec0b818ef0ca73af?

I'm very grateful. I also appreciate how many supportive people you've

attracted here at

http://groups.google.com/group/mathfuture/

Yes, I should think about an hourly service, say. That's a good thing

to think about and try. But the next month or so I'm doing what I can

to avoid bankruptcy. I do happen to know some people of means. I write

about my goals and my scenarios here:

http://groups.yahoo.com/group/minciu_sodas_en/message/7346

For example, I know Stephen Wolfram and he could fund me if he thought

it was worthwhile, but I think especially, if others thought it was

worthwhile.

I'm very encouraged by response here in the Math community. I've better

organized my notes for my Math book for adult self-learners:

http://www.gospelmath.com/Math/DeepIdeas

I'll keep working on that further and I'd like to link that with my

quest to share "ways of figuring things out". I'm trying to work

especially in areas where I'm getting response and I can do something

useful. Maria, I looked at your website: http://www.naturalmath.com and

your vision of math that is "natural" like spoken language:

http://www.naturalmath.com/site-pages/about-us.html

Please keep letting us know how we can support your vision!

I'm sorting through responses and thinking what "formats" are relevant

for presenting math, including games and other activities:

http://www.gospelmath.com/Math/Formats

I think it's fantastic how Linda Fahlberg-Stojanovska, Dani Novak, Sue

Van Hattum, Cooper Macbeth and others documented how they solved the

triangle counting problem:

http://mathmamawrites.blogspot.com/2011/02/tin-ceilings-triangles-and-loving-math.html

It's a beautiful problem and it probably has some deep ideas and

practical relevance (say, in quantum physics). It's the kind of problem

that Polya might take on. Yet, it still lets one think that math is a

world onto its own, a bit disconnected in the sense of figuring things

out. I'll share a different problem that shows what I'm striving for,

which helps me grasp what is "natural", what is central to life. I say

this because my math education, looking back, was completely contrived

and tended to destroy intuition. Maybe you'll know more such problems.

A few days ago I made a breakthrough in my efforts to lose weight. I'm

6 foot 2 inch (185 cm) and weight 190 lbs. (86 kg). I'd like to weight

180 lbs. I was almost there, but the last two months I gained some

weight. I didn't understand why because I'm not eating all that much.

Over the last 12 years I had gained some weight, up to 225 pounds, and

maybe never seemed overweight, but it wasn't right for me. Here's some

things I inferred over the years. Right or wrong, but they are all

mathematical notions. They all are based on mathematical models.

* If I exercise, say an hour a day, this helps me lose weight.

* Or it doesn't, in which case there must be something wrong with my

eating habits.

* I eat too much because... I live a stressful life; I live an irregular

life; I travel and eat restaurant food; I try to save money and eat free

food when it's available.

* I once convinced myself to be completely disinterested in food, it

tasting good, etc. and that was effective, but not a long term solution.

* I lost a bit of weight by living in the countryside; eating all

natural foods; having less stress; exercising.

* I stopped sweetening my tea. I lived in a cold, abandoned community

center and was drinking a lot of tea.

* I don't add salt to my food. There's enough salt in the food that we

eat.

* I stopped consuming white sugar. Over time, I started thinking of it

as a poison, an addictive substance. Somebody said that at my friend's

house many years ago and emphasized that foods (like carrots) taste

naturally sweet when you stop using sugar, and that stuck with me.

* As a child, when I learned about vegetarianism, I decided not to have

moral issues about eating meat, because I thought that would be unfair

to humanity as a whole. About twenty years ago, a friend of mine

started an exotic grain business and it made sense to me that grains

should be the center of a meal, but especially, that I should eat meat

infrequently, say once a week, or as an accent. About five years ago,

when I saw a video about the factory farm industry, I decided to eat

meat infrequently, not as a staple.

* As a child, I didn't like fruit or vegetables, but I've tried to

expand my food choices and I make a conscious effort to eat fruits and

vegetables.

* I lost weight by reading a book that inspired me to reexamine all my

notions and distinguish strictly between "eating" (as needed) and

"overeating" (all 'eating' that is not for the purpose of eating). I

had been drinking milk by the liter because it was sold by the liter. I

had been drinking lots of milk because I thought it was healthy for me.

(Maybe true, but not a good reason if I care about my weight!) I had

been eating desserts. Etc. As my grandmother says, "Better to undereat

than to overeat."

* Last year I lost a lot of weight by living in a house with tens of

thousands of cockroaches, so that everything had to be washed

immediately; there was no room to put anything in the refrigerator, so I

didn't buy much food; I worked at a job without breaks; I was eating

extra "Meals on Wheels" which are portion-sized for seniors; I was in

love; I wasn't measuring my weight, just pinching my side.

* I started to appreciate that oatmeal releases energy evenly and for a

long time, as opposed to pasta, which spikes.

All of these behaviors imply simple (and profound) math models (adding,

subtracting, fractions, proportions...)

But recently I gained more than 5 pounds, despite my efforts. This

week, for the first time, I used Wolfram Alpha

http://www.wolframalpha.com to study the calorie content of foods. It

really changed my thinking about food. I had thought, in a primitive

way, that my weight gain or loss each day was based on the "amount" of

food that I eat (say, the "weight") minus any extra that I lost due to

exercise. My research taught me differently!

I researched the foods I eat and asked, "How many calories in a pound?"

Then I defined "serving size" as 100 calories and asked, "What does one

serving size look like?" And if I burn 2,000 calories per day, that is

20 serving-sizes combined, what does that look like? Here are the foods

and, in parentheses, what a "interchangeable serving size" looks like.

Lettuce: 1 pound = 68 calories. (1 serving size = 100 calories = 1 head

of lettuce)

Strawberries: 1 pound = 145 calories. (1 serving size = 20 strawberries)

Carrots: 1 pound = 172 calories (7 carrots)

Brussel sprouts: 195 calories (12 brussel sprouts)

Apples: 227 calories (100 calories = 1 or 2 apples)

Potatoes or Sweet potatoes: 391 calories (1 potato)

Avocado: 676 calories (1/3 of an avocado)

Eggs: 718 calories (2 eggs)

Tortillas: 800 calories (2 small corn tortillas)

Ice cream: 822 calories (1/2 small 1/2 cup scoop)

Beef: 1069 calories (1/2 hamburger patty)

Bagels: 1245 calories (1/2 a plain bagel)

Cheese: 1326 calories (2 slices)

Pasta: 1680 calories (1/4 of what I eat for a meal)

Sugar: 1712 calories (2 tablespoons)

Cheerios: 1740 calories (2/3 cup)

Oatmeal: 1742 calories (1/2 cup)

Parmesan cheese: 1755 calories (5 tablespoons)

Chocolate: 2377 calories 4 plytelÄ—s (1/6 of a 100 gm bar or half a

candy bar)

Bacon: 2420 calories 16 gabalÅ³ (2/3 of a slice)

Peanuts: 2540 calories (16 peanuts)

Peanut butter: 2683 calories (1 tablespoon)

Almonds: 2698 calories (12 almonds)

Butter: 3252 calories (1 tablespoon or 1 tab)

Olive oil: 4010 calories (1 tablespoon)

This information helped me rethink my mathematical model. From the

point of view of calories, 1 head of lettuce or 1 apple or 1 potato is

the same as 1 tablespoon of butter or olive oil or peanut butter. I

wasn't eating "much", but I was eating too much of high calorie foods,

in particular, peanut butter, olive oil and Parmesan cheese. For

example, when I used to go to work, I would take 2 bagels with cheese

and 2 apples and that was my food for the day. Now, I have 3 hours of

breaks and so I can easily eat 3 bagels on which I put a lot of peanut

butter, maybe 3 tablespoons on each. That's 9 tablespoons of peanut

butter which is 900 calories! That one ingredient can throw my whole

diet out of whack! I then learned from my housemate that prisoners use

peanut butter to build up their body weight. So I'm eliminating all of

the "dense" foods from my diet, namely, olive oil, peanut butter,

Parmesan cheese. I'll reintroduce them when I'm at my desired weight.

I also learned that I lose or gain 1 pound of body weight for each 2,500

calories that I "save" or "spend" in my intake budget over time. My

daily intake budget is 2,000 calories and if I exercise moderately for a

half hour or hour that may be an extra 200 or 400 calories. This all

makes it clear to me that it's not about the physical "amount" or

"weight" of the food, but it's about the amount of "potential energy"

stored in the food as "calories" that will get stored in my body as

extra fat.

The data helps me retrain my perceptions. When I eat ice cream with

almonds, I'll realize that the almonds may contribute almost as much as

the ice cream. Or if I eat a hamburger, I'll realize that the bun may

contribute as much as the meat. Salad dressing has many more calories

than the salad. This all goes counter to my instinct that it's the

"weight" or "mass" that matters. It's not the weight that goes into and

out of my body, but rather, the calories of energy that get stored in my

body and either burned or kept as fat.

This is a modest but practical example of mathematical thinking. It

illustrates many "ways of figuring things out". It shows that different

mathematical models lead to different kinds of behavior. Each model is

valid in some way but breaks down. My new model will break down, too.

It would be great to express these as "games", but I suppose the games

should allow us to "play" different models as different strategies.

I've organized my notes for my math book:

http://www.gospelmath.com/Math/DeepIdeas

and I'm concluding that "counting" is the stitching together of our many

faculties for numeric intuition. We have very different mental

faculties for grasping 0 or 1 or 2-4 or 7-8 or dozens or hundreds or

thousands or millions, etc. That is the "natural" way to experience

numbers and is reflected in our language. What is admirably "contrived"

is to unify, sew together, all of these intuitions into one system, as

if a number 1,286,587 meant anything to us, as opposed to 1,286,588.

What we've done is to map all of these faculties onto one big scale.

And then we can change our units and change how we perceive something.

We can switch faculties by reexpressing in different units, by comparing

and contrasting, by thinking in different scales. But we should, I'm

concluding, always be a bit suspect of this. I look forward to studying

the many different "paradoxes" in math, science, economics. I'm

hypothesizing that they may all relate to these issues that what we've

come to believe is a whole cloth is actually stitched together from

different faculties, so that the discrete and the continuous clash, for

example. I think back to when I first learned, in differential

geometry, that a sphere is a bunch of local "open neighborhoods" cut and

stitched together in any which way, and that seemed so crazy and

unnatural and uninteresting, and it turned me off the subject. But

perhaps it's honest, in a sense.

Maria, you see why I think that a short book of essays like this, well

thought out and well-organized, might be profoundly meaningful to adults

(including high school teachers) who've churned through the school

system and have no understand what that was all about, and nobody even

to ask. I feel that I'm still learning the basics. I see that

"counting" is extremely sophisticated. It includes the notion of

"addition" and I think that the "addition rule" is what stitches the

integers together as we "extend the domain" to include dozens, hundreds,

thousands, millions, gazillions, in each case readjusting the number

system in ways we don't make explicit. Multiplication is an accelerated

addition that is key to that readjustment, to the switching from one

intuitive faculty (say 3 "thousands") to a very different one ("three"

1,000s). I'm appreciating that division is the opposite of counting

(dividing by 1/2 is "counting by" halves) and that division and

fractions are assuming a "whole" which is not natural to the external

world, but is natural to our internal world. Thus "division" is

modeling "counting", internalizing it, abstracting it, switching us over

from "counting" external things to "counting by" and so counting our

internal count. This sounds abstract but with enough good examples I

hope it will become very concrete and we'll say, "Wow! Math used to be

taught in such a crazy way, like teaching children dead languages,

without teaching them any living ones."

Maria, Joaquin, I like the vision of a "language of angels" and I

believe that we can and do experience it. It's why I focus on "deep

ideas" and the "classic problems" that illustrate them. I resist

learning anything unless I can incorporate it into my total knowledge,

unless I can understand why I'm learning, why it's interesting and so

on. I'm in control of my own mind. I create my own "private language"

which is meaningful. Joaquin, you know of my quest to "know

everything", but you may not know that I completed it! with my video

summary: http://www.youtube.com/watch?v=ArN-YbPlf8M Hopefully, it

sounds like a language of angels, because it's 27 years of work. It's

how I imagine what everything looks like if God seated me on his lap. I

see the big picture, but I'm nearsighted and so I don't know all of the

details at once, yet if I wanted to, I could venture out and get the

answer regarding any detail and come back to view the big picture. I've

found the big picture, I believe.

Partly it's a matter, as Maria writes, of taking the time to learn

everything. In some sense, there's no short cut to that, but in another

sense, thinking deeply is qualitatively different than thinking

superficially. I saw a quote today that as we learn we can also think

about our learning. And so we develop a different relationship with our

thinking and we operate on a different level, qualitatively.

For example, most people aren't able to address a doubt such as "How do

I know I'm not a robot?" and so they ignore such questions and go back

to sleep. But I learned to respond to them with counterquestions, such

as "Would it make any difference?" I may or may not be a robot; all of

my knowledge may be suspect; and yet I can rescue myself by turning the

question around and saying: You tell me, what is a robot! And then I'll

tell you whether or not I am one. Do robots bleed? Do they care? Are

they moral? Maybe then I am a robot. Maybe it's just semantic. Or

maybe there's a real difference and we can check.

I found a system of such pairs:

* Do I truly like this? How does it seem to me?

* Do I truly need this? What else should I be doing?

* Is this truly real? Would it make any difference?

* Is this truly problematic? What do I have control over?

* Is this truly reasonable? Am I able to consider the question?

* Is this truly wrong? Is this the way things should be?

* Am I anxious? Am I doing anything about this?

and, standing on its own, What do I truly want?

These counterquestions, to my knowledge, express what it means to be

"intelligent". They feel to me like a language of angels.

Joaquin, in 1989, after years of work on "divisions of everything", I

experienced a coming together that was like a mental orgasm. It was

like oxygen bubbles (or hormones) bursting upon my brain. I realized

that there was a cyclic structure to the 8 divisions, so that operations

+1, +2, +3 made sense as I went around, moving from division to

division, as if on a clock. So many things made sense. Too many things

made sense. It went on for an hour and I felt strange for a few days.

That's not optimal.

Maria notes that knowledge can't be compressed from one form to another

in a way that would be reversible. But that's a model that can break

down in a few ways. It assumes that knowledge is like that "weight"

that I thought I was eating and passing, as opposed to a "calorie" that

we accept, then use or store. Instead:

* It may be that all of our deepest knowledge we already know, so we

simply need to reference it. For example, when I see a homeless person,

I know from experience that "my help could make things worse", but I

know prior to this that "I should help those who need help" and the

latter can't be learned because there is no way to illustrate "should".

I believe that babies in the womb think abstractly and we turn away from

abstract thinking only because it's not relevant in society.

* Knowledge also leverages context. I ask, what is 10 + 4 and people

say 14, but I tell them it is 2, because 10 + 4 = 2 on a clock, and I am

right and they are wrong because I know the context. And it's generally

not possible to write down all of the context because you would have to

know the context for that, too. Or the same DNA is in your toe and in

your heart but the cells develop very differently because they are in

very different contexts.

* The so-called knowledge may be mostly garbage and best not learned, or

if learned, then from a completely different angle.

* This means that the so-called knowledge may be "best" learned and

stored in a much longer, decompressed form. I believe it's much more

intelligent and "angelic" to start with "a right triangle is two copies

of itself" and express that by drawing an altitude and, if desired, from

that, deriving a**2 + b**2 = c**2, then elevating the latter to

something overly important.

Angelic knowledge will feel heterogeneous. Some truths ("love God",

"love your neighbor as yourself") will resonate profoundly across all

knowledge. Other facts (like a**2 + b**2 = c**2) will feel a bit

obscure. Some day we'll look at math as a discipline where we

understand its basic idea (I suppose something to do with models of

systems (like real life) and where they hold and where they break

down). We'll know exactly how many branches of math there are and how

they are related. We'll know how to generate all of the deep ideas of

math, and how to illustrate them with just the right problems. We'll

know what is central and peripheral, what is natural and contrived.

We'll know how much beauty or utility there is in a math problem. And

we'll be able to communicate the essence of that in a slim 150 page

paperback, like Euclid's Elements or "How to Lie With Statistics" or

"How to Solve It". As things stand, students are punished with

thousands of pages of textbook and school work which is all homogeneous

so that no single page is more or less important than any other.

Maria, Joaquin and all, Thank you for being so supportive! I appreciate

all manner of response, including:

* What kind of math problems are natural like the dieting problem

* How might we communicate them in useful ways

* How might we relate them to "ways of figuring things out"

* How to support each other's efforts

* Who might be interested in our work together

* How to best engage them so they might support us, including with paid

work or sponsorship

Just as an example, there is a website http://www.sharecare.com where

doctors answer people's questions. I'm thinking of analyzing and

organizing their answers to note how the doctors (or researchers) figure

things out. This could relate to math. If there was some interest in

this or any project, then that would help me ask for funding, which we

could then share, as relevant.

Thank you,

Andrius

Andrius Kulikauskas

http://www.selflearners.net

ms@...

(773) 306-3807

Twitter: @selflearners

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

There are two known laws of nature that make the answer to this question

"No." One comes from computer science, and another from pedagogy. It is

an excellent question to ask, say, as a group exercise in computer science!

The computer science topic is called "repeated compression" and there is

a lot of theory done on it. Basically, the compressed file must contain

no less information than the original, and the compression function must

be reversible. There are quite a few articles on it.

The pedagogical (or psychological, or biological) reason for this

impossibility is the way knowledge is formed. It involves pathways in

neurons, building up metaphors, making conceptual bridges TO meaning.

Understanding is a path, and it has to be walked in time - lived.

The best explanation for the computer science part that I know, also

quite rude, comes from "Good math, bad math" blog:

http://scientopia.org/blogs/goodmath/2009/03/08/i-get-mail-iterative-compression/

Cheers,

MariaD

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

Hi Andrius,

We have not talked for a long time. I took a look at you MathFuture

page. May be you could be interested in

an idea I had recently. It came to me by happenstance, while talking to

some friends that are science historians.

Consider the amount A(L) of "real" and "important" and "deep" meaning

communicated by a

language L. For instance, there could be a language L_i (may be just

noise) that is really meaningless, so A(L_i) ~ 0.

QUESTION: Is there a "language of angels" L_a such that A(L_a) is so big

that a few minutes of listening

to it communicates more than, say, all that you learn in college in 4

years?

If the model above makes sense we could view Mathematics in a different

way:

Mathematics is not just a meaningless exercise in first predicate

calculus, but rather one step in the search for

and angelic language.

Question: Has Mathematics succeeded in this search (even in a small

way)? What are the limits

of Mathematics in the search for and angelic language?

Best regards and best wishes for all your endeavors,

joaquin