"Implicit math" and/or a "science of math"

[Andrius Kulikauskas]

Dear Maria, Joe, Bradford, Michel and all,

I wrote to Gizem Karaali, an editor of the Journal of Humanistic
Mathematics, about my wish to start a conversation on "implicit math"
and/or a "science of math". I was encouraged by her response. With her
permission, I share our letters here. My thought is to discuss how we
might look at math in a new way, not restricting ourselves to using
mathematical methods, but consider what scientific approaches in general
might tell us about math, especially the cognition ("implicit math")
which takes place in our minds when we work on math. Indeed, linguist
Noah Chomsky developed "transformational grammar" to study how our minds
construct sentences, and certainly a similar approach could be
undertaken with regard to mathematical language. I will write
separately about other approaches that I'm taking such as a
phenomenological investigation ("a taxonomy of moods") and an aesthetic
investigation (an art show about geometry). I'm curious how best to
connect with other people. Gizem Karaali agreed that I could submit an
article based on whatever seems most conducive for such a conversation.

Thank you for our ideas and encouragement here!

Andrius Kulikauskas

________________________________________
From: Andrius Kulikauskas [m...@ms.lt]
Sent: Wednesday, June 22, 2016 4:47 PM
To: Gizem Karaali
Subject: How to foster conversation on "implicit math" and/or a "science
of math"?

Dear Gizem Karaali,

I learned of you from the Journal of Humanistic Mathematics website
http://scholarship.claremont.edu/jhm/
and also found your website and see that you are an editor of The
Mathematical Intelligencer
http://pages.pomona.edu/~gk014747/
and that you wrote your thesis on Lie superalgebras.

I've recently written an essay
"Discovery in Mathematics: A System of Deep Structure"
http://www.ms.lt/sodas/Book/DiscoveryInMathematics

If you take a look, you will see that I am trying to show how
philosophical approaches can ground a "science of math", for example, by
analyzing the ways we figure things out in math, and by revealing the
"implicit math" that takes place in our minds and isn't always made
explicit on paper.

You may also appreciate that I'm trying to show that "implicit math" can
actually be quite fruitful in leading to results in explicit math. For
example, I give an interpretation of the -1 simplex as the "center" of
the simplex which generates all of the vertices. And the notion of the
Center and its dual, the Totality, can be seen at work in four different
classical families of polytopes:
* simplexes An have a Center and a Totality
* cross-polytopes Cn have a Center but no Totality
* cubes Bn have a Totality but no Center
* demicubes, "coordinate systems" Dn have neither a Totality nor a Center.
The symmetry groups for these polytopes are the Weyl groups for the root
systems for the four classical Lie algebras.

I would like to participate in online conversation to foster such
"implicit math" and such a "science of math". However, I can't find any
community that would both welcome and understand that. I'm skeptical
about publishing articles because I feel that nobody reads articles and
that they generally don't serve to foster conversation and
collaboration. But you may know better.

Thus I'm writing to you because you might be interested in my work and
my direction and perhaps you might have ideas how to foster such a
conversation. I'm active at the Math Future google group of math
educators which is very supportive but they don't really study advanced
math. I've started participating at Math Overflow but the "ask a
question" format is a bit contrived for conversation, although so far,
so good. I've written some letters to the Foundations of Mathematics
group but half of them get rejected for being out of bounds. I've
created pages at nLab and had them deleted. I've joined John Baez's
Azimuth Project but have been ignored. I can work independently but my
hope is to find one or two people to collaborate with and build a
culture from there.

I appreciate your thoughts.

Andrius

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


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

2016.08.14 20:21, Gizem Karaali:
Dear Andrius Kullauskas,

Thank you for writing. I have kept your message in my inbox for all this
time because this summer has been one full of strange unsettling events
(I am from Turkey and you might have noticed strange news about it in
the last few weeks...) and a lot of work-related deadlines despite it
being summer. However your message was interesting to me and as you
guessed, I am intrigued by the kind of conversation you are proposing.
Though I am not really capable of taking on a new mathematics project,
(I do have too many other folders beckoning), I'd be interested in
trying to understand your perspective and possibly help facilitate your
project.

We have launched the Journal of Humanistic Mathematics to help create
conversation opportunities. Publishing articles or essays is often
one-directional, but occasionally there will be readers who will find
some idea inspiring and then will get in touch. The idea of peer review
can also be helpful as occasionally it will catch mistaken steps and
offer better options.

To me your approach seems like it fits well into the themes of "math as
metaphor" and "math studying itself". Both should be of interest to
others. Would you consider writing an essay or possibly an article that
explains your perspective? If so, submitting papers to JHM is done via
our server at http://scholarship.claremont.edu/jhm/ If not, perhaps
speaking at various math conferences could help you connect with
possible like-minded people.

Thanks for reaching out. I hope you will be able to pursue your
interests further.

Gizem Karaali, PhD
Department of Mathematics, Pomona College
610 N College Avenue, Claremont, CA 91711, USA
http://pages.pomona.edu/~gk014747
Editor, Journal of Humanistic Mathematics
Associate Editor, The Mathematical Intelligencer

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


________________________________________
From: Andrius Kulikauskas [m...@ms.lt]
Sent: Tuesday, August 16, 2016 1:54 PM
To: Gizem Karaali
Subject: Re: How to foster conversation on "implicit math" and/or a
"science of math"?

Dear Gizem Karaali,

Thank you for your letter and encouragement. I appreciate that. It is a
real step towards conversation.

Yes, it could be meaningful to submit an essay or article to your
Journal of Humanistic Mathematics. Perhaps in the spring.

Would you agree that I engage the online math communities where I am
active to say that you have expressed interest that I submit an essay or
article? Then I could ask them for help to present my ideas in ways
that might be most fruitful for collaboration.

The main projects I see now are:
* Distinguishing and relating "implicit math" (actions we perform in our
minds) and "explicit math" (objects which we represent with symbols).
* Proposing ways to study "implicit math" in its many dimensions and
implications including beauty, cognition, creativity, learnability,
etc.. For example, Terrence Tao has distinguished 21 notions of what is
"good" mathematics https://arxiv.org/pdf/math/0702396v1.pdf
* Proposing to try to overview and map out all of the areas of math, how
they depend on each other, and how the mathematical concepts unfold.
For example, one project would be to study the tags at Math Overflow or
Math StackExchange and see what pairs of tags are most common, that is,
create an adjacency map of major topics.
* Proposing to study and relate "implicit math" and "explicit math" by
way of the ways we figure things out in mathematics, as I have done with
my paper, to show that it seems to be possible to catalog and
systematize such ways.
* Proposing to study the emotional side of math, for example, what makes
something beautiful, understandable, frustrating or memorable, first by
collecting many examples and analyzing some simple, fundamental ones.
* Pointing out the significance on these matters of particular math
research questions that these insights may contribute to as well as gain
from. I'm currently very interested in how the "implicit math" in our
minds expresses itself most basically in:
** Symmetry between "zero" and "infinity" as mediated by "one" and how
that symmetry gets broken. I think this is relevant for the "field with
element one" which I imagine in our minds can be taken in different
senses as zero, one and infinity.
** Four geometries (affine, projective, conformal, symplectic), which I
imagine are related to the four classical Lie algebras/groups, or even
simply the integer, rational, real and complex numbers, and also six
fundamental transformations I expect take us from one way of thinking to
another.

Your kind interest would thus foster my efforts. I would gladly point
to articles at your journal and try to engage some of your authors and
see if they might be interested.

Those are my thoughts. I await yours!

Oh, yes, I have paid a bit of attention to the news from Turkey. I once
spent a week in Istanbul trying to organize Islamic independent
thinkers, and later I vacationed for three weeks in Kutahya. I am
heartened by the Turkish people's civic expressions. I don't know what
to say.

Andrius

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


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


2016.08.24 16:52, Gizem Karaali:
All of these paths seem interesting to me. I would be delighted if you
did pursue them as you wish, and you can of course say you are looking
into writing about these for JHM.

Looking forward to hearing from you,

Gizem Karaali, PhD
Department of Mathematics, Pomona College
610 N College Avenue, Claremont, CA 91711, USA
http://pages.pomona.edu/~gk014747
Editor, Journal of Humanistic Mathematics
Associate Editor, The Mathematical Intelligencer



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

________________________________________
From: Andrius Kulikauskas [m...@ms.lt]
Sent: Wednesday, August 24, 2016 7:01 AM
To: Gizem Karaali
Subject: Re: How to foster conversation on "implicit math" and/or a
"science of math"?

Gizem, Excellent. Thank you!

I dare to ask, may I share our letters online at the Math Future group?
And then point to them in the archives? I would find that especially
helpful.

Either way, thank you!

Andrius


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

Sure, no problem

Gizem Karaali, PhD
Department of Mathematics, Pomona College
610 N College Avenue, Claremont, CA 91711, USA
http://pages.pomona.edu/~gk014747
Editor, Journal of Humanistic Mathematics
Associate Editor, The Mathematical Intelligencer

[Joseph Austin]

Andrius,
Go for it.

As I may have expressed before, I believe "math" is about more than number,
specifically it includes "arrangement" and "pattern",
including ideas of symmetry, similarity, and periodicity.
I think we make a mistake in attempting to do everything with "letters" (algebraic)
instead of "pictures" (geometric).

I started exploring such ideas early on but never pursued it.
I'm not sure where the "state of the art" is today, but I think the approach deserves more attention
than it gets in the general curriculum.

Joe Austin

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[Bradford Hansen-Smith]

Joe

 

As I may have expressed before, I believe "math" is about more than number,
specifically it includes "arrangement" and "pattern",
including ideas of symmetry, similarity, and periodicity.
I think we make a mistake in attempting to do everything with "letters" (algebraic)
instead of "pictures" (geometric).

You are right. Yet we need to go beyond pictures to broaden our understanding of math. Pictures are not in and of themselves, they represent something else that is "real". For example: when cutting the picture of the circle from the paper it becomes that dynamic object the image represents. The object holds the image (includes what can be constructed 2-D) and so much more that is lost to the limitation of a picture point of view and to the distortion that occurs in compressing a 3-D form into a 2-D image. Is there any wonder that even the most abstract of math relates back to the physics of experience in the real world; that is where it comes from in the first place. Without know where things come from we do not know what they are or the potential inherently carried in the form.

We draw pictures of circles, we truncate circles into polygons, we cut out squares, then fold amazing things from the square without realizing what might be revealed through folding the circle without truncation. We have become so caught up that we can no longer see our fingers for the keyboard of neural connectors in our mind.

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www.wholemovement.com

[Joseph Austin]

Bradford,

I agree, just as 2D is better than 1D (strings), 3D is better than 2D,

for example, it can depict "arbitrary" pair-connections with out crossing line, which could lead to ambiguity.

Now that we have 3D printing, creating actual 3D models may be practical.

And then there is 3D virtual reality.

But let's see some useful example.

For example, we can depict algebraic formulae as trees,

making "order of operations" explicit and unambiguous.

Joe

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