Re: A system of deep structure in solving math problems
Andrius Kulikauskas
hundreds of "ways of figuring things out":
http://www.selflearners.net/ways/
including from Math:
http://www.selflearners.net/ways/index.php?d=Math
Edward, I look forward to responding to your letter at Math Future.
Briefly, I focused on the "deep structure" that describes the structure
of our problem solving, as opposed to the structure of the problem
solved. So, for example, the sequence 0,1,2,... is inherent in
mathematical induction. But I don't know of any similar use of the
"real numbers" or the "complex numbers", "Euclidean space" or
"nonEuclidean space" and many other structures.
In another group, I was encouraged to read "Metaphors We Live By", which
I had, so I write about that. My letter expands on how I'm thinking of
deep structure and surface structure. What's exciting about Math is
that it suggests that they are qualitatively the same structure. The
Math structures lurking in our minds when we solve problems are very
familiar as Math structures that we write about on paper.
Maria, The potato slices are a great metaphor!
Andrius, m...@ms.lt
I ordered a copy of "Metaphors We Live By" from the Chicago Public
Library and it came in today. It was originally published in 1980.
Here are the first 4 chapters: http://theliterarylink.com/metaphors.html
I read it in 1994 because I was trying to catalog "the vocabulary of our
imagination". Lakoff and Johnson collect many examples of everyday
language that exhibit the pervasity of metaphorical thinking. They
argue persuasively that sentences such as "Your claims are indefensible"
point to our mind's applying a general framework ARGUMENT IS WAR that we
recognize at work in other unrelated sentences as well, such as "His
criticisms were right on target."
WAR is a deep structure (as I call it, they say "source domain"), a
mental model for thinking about a different domain, ARGUMENT, which in
this case is a surface structure (they say "target domain"). Linguist
Noah Chomsky in the 1950s introduced revolutionary, clever techniques
for exploring the syntax of sentences by which he was able to argue how
surface structure effects (whether or not certain sentences are
considered grammatical) point to deep structure relationships (how
components are organized in the mind, which may not be visible in a
particular sentence).
In mathematics, in my recent letter, I'm describing a very similar
situation, where the mathematics problem on a sheet of paper (for
example, constructing an equilateral triangle) is solved by our minds
playing with a much simpler mathematical structure (a deep structure,
for example, the 4 ways of satisfying or not 2 conditions A and B: {A
and B, A, B, neither A nor B}). We apply the deep structure when we
draw two circles, A and B, centered at the ends of a line segment AB,
and look for the points where the two circles intersect, which lets us
draw a third point C and complete our triangle.
But there are other "ways of figuring things out" which similarly
involve the "modeling" of a surface structure by a deep structure. I
think the following Gamestorming games are all examples of that:
* Campfire learn from stories, share culture
* Graphic Jam visualize abstract concepts, absorb complexity
* History Map appreciate relevance of organization's culture
* Memory Wall appreciate people, foster team
* Visual Glossary create a shared language
* Welcome to My World share our internal map of reality
In mathematics, Paul Zeitz writes about "wishful thinking" which is a
legitimate approach where we tackle a hard problem by considering an
easier version (the latter models the former). Actually, the simpler
problem may be more informative than the harder problem.
I'm collecting ways of figuring things out (462 so far). I've just
redone my website for that. Here's a link to "Apply constructs of
source domain to target domain" (George Lakoff and Mark Johnson)
http://www.selflearners.net/ways/indexnew.php#805
And nearby you'll see other related ways that link deep and surface
structures:
http://www.selflearners.net/ways/indexnew.php#50
such as my own "Center conversation within their expertise", which is to
say, if I talk to an expert about their domain knowledge (surface
knowledge, say, butterflies), I might find insights that are informative
about more deeper, broader questions of life (deep knowledge, say, the
life cycle).
I note that this (modeling surface structure with deep structure) is
only one strategy for "figuring things out". Lakoff and Johnson heavily
rely on a very different strategy, "collecting examples". Instead of
studying isolated examples of metaphor, one sentence at a time, they
consider dozens of examples to show convincingly that "LOVE IS A
JOURNEY" is a construct that cuts across many examples and can be used
effortlessly to create new examples, such as "we're spinning our
wheels". This approach is roughly the same as in the Gamestorming games:
* Affinity Map discover patterns relating categories
* Card Sort categorize
* Post-Up generate all relevant ideas
* Spectrum Mapping expose the range of positions
In math, similarly, there is the "pigeonhole principle" which says that
if you have more pigeons than pigeonholes, then one of the pigeonholes
will have to have two or more pigeons. Which is to say, there are
categories (like WAR, MONEY, JOURNEY) that apply to more than one
instance. (You can have TIME IS MONEY, BANDWIDTH IS MONEY, AFFECTION IS
MONEY, etc.)
Similarly, in 1995, I made a list of all the "source domains" that
Lakoff and Johnson describe in their book. In other words, given
ARGUMENT IS WAR, TIME IS MONEY, LOVE IS A JOURNEY, I made a list of
about 100 deep structures: WAR, MONEY, JOURNEY, etc. that they note.
Then I categorized that 100 and boiled them down to 12. I think I
should redo that and share that. I related that to 12 "topologies",
backdrop canvases for our imagination, concepts like ONE, ALL, MANY that
Kant thought of as "categories", but I think more of as "expectations
for possible worlds" (like one-ness, subject-ness, possibility-ness,
etc.) In other words, I used their evidence to support a completely
different set of conclusions. Instead of saying, "oh, these source
domains must derive from neurology and human evolution", I said, "no, I
think that I can derive them from first principles". My results in
Mathematics are very exciting in that they are absolutely more real and
practical and tangible than what Nunez has done following their approach.
In 2001 or so, I did get to meet George Lakoff at a party. I was
excited to report my results. I must say, unfortunately, he's one of
the most close-minded persons I've ever met. He'd just say, "It's all
been done. It's all decided. We've solved it." This was before
Nunez's book came out. I did buy that book and read parts but (I speak
as a Ph.D. in mathematics) I thought it was lame.
Interestingly, Jeff Hawkins' book "On Intelligence" focuses on the fact
that we have ten times more neural signal projecting outward from our
brain to our senses, then coming inward. He takes this to mean that our
brains are not primarily integrating abstractions from sensory input.
Instead, our brains are projecting a "virtual world" onto our senses,
projecting huge sets of predictions and noting where they are failing.
That's why if we woke up and didn't hear any sound, we'd know that
something is wrong, even though there's no signal, but our projections,
our predictions, that there should be sound, would fail. Pressed
further, I think it's compatible with an idealist viewpoint which says
that all of the important things in life, and perhaps all of life, comes
from the inside, not from the outside, as Lakoff claims.
X writes "we view the world through our perception" and concludes
"reality trumps models". But Stephen Hawking and Leonard Mlodinow, "The
Grand Design" think of "model realism": We question the conventional
concept of reality, posing instead a "model-dependent" theory of reality.
I have a B.A. in physics from the University of Chicago. X, I've never
met a practicing physicist to hold your notions. X, how much science
have you done? But at least you're open minded, compared with Lakoff.
But there's another way of figuring things out, which is "Give X another
chance". Or more generally, "independent trials", "avoid evil". Just
because X has the same "materialist prejudice" over and over again,
doesn't mean he won't think something new, or even grow, or maybe I
myself will finally "get it", what he's trying to say. But don't
approach X the same way every time, vary the trials.
http://www.selflearners.net/ways/edit.php?numeris=48 And thank you, X,
for responding. I appreciate especially your last letter on ...
Gamestorming doesn't seem to have a game like this, but it would be the
kind of game to play if you wanted to know if a business had failed its
mission and should we shut down and start fresh.
What I'm doing this last year is collecting hundreds of ways of figuring
things out. I'm sorting them into categories and shaking out a system.
I've found 24 general strategies that they fall into. I think of them
as 24 "rooms" in a "house of knowledge".
I've set up my website http://www.selflearners.net/ways/ and you can see
that you can choose and compare domains such as Math, Gamestorming, my
own philosophy, my choir director Dee Guyton's view of life (I made a
video interview), and Other. I'd like to document dozens of domains. I
want to find clients or sponsors for that. I ask for help, how to do that?
Andrius
Andrius Kulikauskas
http://www.selflearners.net
m...@ms.lt
(773) 306-3807
Twitter: @selflearners
Edward Cherlin
> After three weeks, I've just finished redoing my website to compare hundreds
> of "ways of figuring things out":
> http://www.selflearners.net/ways/
> including from Math:
> http://www.selflearners.net/ways/index.php?d=Math
>
> Edward, I look forward to responding to your letter at Math Future.
I look forward to continuing that discussion.
> Briefly, I focused on the "deep structure" that describes the structure of
> our problem solving, as opposed to the structure of the problem solved. So,
> for example, the sequence 0,1,2,... is inherent in mathematical induction.
> But I don't know of any similar use of the "real numbers" or the "complex
> numbers", "Euclidean space" or "nonEuclidean space" and many other
> structures.
The defining characteristic of the standard natural numbers is induction.
* 0 is a natural number
* Every natural number has a unique successor that is a natural number.
* Every natural number except 0 has a unique predecessor among the
natural numbers.
* If K is a set such that:
* 0 is in K, and
* for every natural number n, if n is in K, then S(n) is in K,
then K contains every natural number.
A similar construction and a similar inductive condition define the
transfinite ordinals, but with some differences.
If S is a set of ordinals with no largest member, then the union of
its members is a larger ordinal.
This allows us to define the first transfinite ordinal as the order
type of the set of natural numbers. This ordinal is called omega. Then
we get omega + 1, and so on. Taking all countable ordinals together
gives us the first uncountable ordinal, and so on again.
Proofs by transfinite induction are entirely analogous to proofs by
finite induction.
There are proofs that depend on the algebraic or topological nature of
other mathematical structures, where you can't do induction because
the underlying sets are not well-ordered. This is the essence of
category theory and much of model theory. Proving that two seemingly
unrelated, even apparently incompatible objects are equivalent in some
way, or can model each other, is one of the deepest ideas in math.
Examples:
The proof of Fermat's Last Theorem depends on the Taniyama-Shimura
Theorem that all elliptic functions are modular, that is, that there
is a structure-preserving mapping between elliptic functions and
modular forms.
At one time it was thought that there was a vast space of possible
String Theories in physics. It turns out that all of them are maps of
each other.
The fact that each of elliptic/Riemannian, Euclidean, and
hyperbolic/Lobachevskian geometries contain models of each other shows
that all three are equally valid. For example, a Clifford's surface in
Riemannian space and a horosphere in Lobachevskian space both have
locally Euclidean geometry.
Mathematicians lost interest in Brouwer's Intuitionistic logic and set
theory when it was shown that it and the more usual non-constructive
logics and set theories can model each other.
[snip]
> I'm collecting ways of figuring things out (462 so far). I've just redone
> my website for that.
I'm having trouble navigating your site. Where is that collection?
--
Edward Mokurai (默雷/धर्ममेघशब्दगर्ज/دھرممیگھشبدگر ج) Cherlin
Silent Thunder is my name, and Children are my nation.
The Cosmos is my dwelling place, the Truth my destination.
http://www.earthtreasury.org/
Andrius Kulikauskas
http://www.selflearners.net/ways/
http://www.selflearners.net/ways/?d=Math
Please try them again.
I've included excerpts from your letters:
http://www.selflearners.net/ways/?d=Math#17
http://www.selflearners.net/ways/?d=Math#62
http://www.selflearners.net/ways/?d=Math#819
I look forward to thinking more about them.
Thank you also to Tom Wayburn for your quote from Calvin and Hobbes
which I'll include!
Andrius Kulikauskas, m...@ms.lt, http://www.selflearners.net
Edward Cherlin
> Edward, Thank you for your reply! The links are:
> http://www.selflearners.net/ways/
> http://www.selflearners.net/ways/?d=Math
> Please try them again.
Thank you. There is indeed a lot there.
I noticed your short note on the extreme principle, which has far
greater ramifications. To begin with, we have the Calculus of
Variations for solving problems of maximum area, shortest perimeter,
fastest descent (brachistochrone), and the like. Then we have the
Principle of Least Action in physics, and others like it.
I like your cycle of scientific method,
take a stand (hypothesize), follow through (experiment), reflect (conclude)
although I find that there is more to it. It has been pointed out that
a hypothesis must include a model (usually mathematical) and a mapping
between parts of the model (observables) and observations, including
experiments. But that is not enough. We must also think of other
possible models, and design experiments to rule them in or out, and we
must think of every possible experiment that could invalidate our
model. This is the great service that Einstein performed for Quantum
Mechanics, because he disliked it so much. Every time he thought he
had found a contradiction or something nonsensical in the math, the
lab boys verified that it really worked that way in experiments.
Reflection is never conclusion. We know that our models are reasonably
complete and accurate at best in the areas we have been able to
observe, and that every new addition to our senses in improved
scientific instruments, going back to Galileo's first telescope,
reveals surprises like the mountains of the moon, the constancy of the
speed of light (interferometers) or neutrino oscillations (simple but
quite large neutrino detectors).
I'm sure I could add other examples, but I am trying to work on a Web
site tonight in addition to reading e-mail.
At some point you and I need to discuss what form an OER on figuring
things out could take, with extended examples, and hints for
exploration.
> I've included excerpts from your letters:
> http://www.selflearners.net/ways/?d=Math#17
> http://www.selflearners.net/ways/?d=Math#62
> http://www.selflearners.net/ways/?d=Math#819
> I look forward to thinking more about them.
I should give you a more rigorous version of induction, finite and transfinite.
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Edward Cherlin
rejection of rejection is not necessarily affirmation.
And to consider the mapping between methods of problem solving and the
sets of questions that do and do not have correct answers.
The problem of the shape with the shortest perimeter for a given area
has a specific solution, which can be found by several methods.
The problem of the next version of physics does not yet have a correct
answer, but someday it will, and it will be the version after that
does not.
The problem of what is most important to do next inherently does not
have a single right answer, nor even a guaranteed method for
approaching it. It in part depends on Edsger Dijkstra's dictum
Only do what only you can do.
in stark contrast with Kant's dictum
Act in a manner that could be made a rule for everyone.
Neither is inherently correct. For many people, at many times,
discovering that there is no such absolute rule is the most important
thing they could do. But after that, something else becomes most
important.
The Golden Rule in particular does not work. Few people wish to be
treated as I wish to be, and in most cases I must not treat them that
way.
On Sun, Apr 24, 2011 at 15:55, Andrius Kulikauskas <m...@ms.lt> wrote:
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