Janet, Pamela, I share this letter I wrote because it reflects my
educational approach and because nutrition is a practical subject, too.
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.
I'm very grateful. I also appreciate how many supportive people you've
attracted here at
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:
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
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:
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
your vision of math that is "natural" like spoken language:
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:
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:
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
* 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
* 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
* 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
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
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 plytelės (1/6 of a 100 gm bar or half a
Bacon: 2420 calories 16 gabalų (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
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:
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
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
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.
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:
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
If the model above makes sense we could view Mathematics in a different
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,