Eugenia Cheng: How to Bake Pi
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Andrius Kulikauskas
Dec 6, 2016, 3:09:11 PM12/6/16
to mathf...@googlegroups.com
Hi, I'm on my third attempt trying to make sense of category theory.
It's a very abstract way of looking at all of math in terms of "objects"
(like sets, groups, vectors spaces, etc.), "arrows" between them (also
known as morphisms), "functors" that map categories (objects & arrows)
into other categories (objects & arrows), "natural transformations" that
map functors into functors, and so on.
I've been watching a series of 10 minute video lessons on that by the
"Catsters".
https://www.youtube.com/user/TheCatsters
They were done eight years ago by two friends, both personable. Eugenia
Chang is delightful in her over-the-top enthusiasm and dynamic humor.
I'm happy that she's really blossomed, both as a high powered category
theorist but also as a champion for popularizing math. I appreciate her
very helpful talk on the the periodic table in higher order category theory:
https://www.youtube.com/watch?v=lJGUMlgCxz8
Then I learned that she had a great 5-minute television appearance on
Stephen Colbert's Late Show:
https://www.youtube.com/watch?v=mA402F5K47o
She talked about her new book which links math and cooking:
"How to Bake Pi: An Edible Exploration of Mathematics"
https://www.amazon.com/How-Bake-Pi-Exploration-Mathematics/dp/0465097677
There's a very nice article about her in the New York Times:
http://www.nytimes.com/2016/05/03/science/eugenia-cheng-math-how-to-bake-pi.html
And she gave a nice Ted-x talk:
https://www.youtube.com/watch?v=CfdFw3hXkf0
She's currently at the Chicago Art Institute as a Resident scientist.
As for myself, I've been studying a lot of math. I'm starting to think
that behind all of math there must be a set of ways by which we create
abstractions. Basically, if we have several examples, (it's not clear
if one example is enough), then we can identify abstract qualities that
they all satisfy. Then we can discard the examples, think in terms of
the abstract qualities, and define abstract entities that satisfy them.
And then we can continue this process in various (but I think, quite
limited) ways. The upshot is that abstraction can grow and grow.
Whereas we have a belief that abstraction will help us get to the root
of unity. Instead, we have all of this math that gets generated,
increasingly abstract and difficult to follow.
It's a bit like my experience as a database developer. On the one hand,
there's a belief that if I think through a system that is general
enough, then it will serve for all possible applications. But actually
that approach tends towards "overengineering". Instead, it's probably
best to find a balance where, say, half of the energy goes into
designing the overall system and half of the energy gets spent on
dealing with exceptions.
I believe that if we thinking about what is going on here, then we can
actually describe exactly how this dilemma unfolds. And we can define it
rigorously and apply it practically by choosing an optimal point between
overengineering and underengineering. In math, that would mean finding
contexts that are not too specific and not too general. Of course, math
education can serve our entire lives or just the moment at hand, and it
is an art to balance the two.
In summary, in order to encompass all of math and discover its unity, I
think I need to watch how math discovery unfolds, especially through
abstraction, and show how that process accounts for all of math.
Andrius
Andrius Kulikauskas
m...@ms.lt
+370 607 27 665
It's a very abstract way of looking at all of math in terms of "objects"
(like sets, groups, vectors spaces, etc.), "arrows" between them (also
known as morphisms), "functors" that map categories (objects & arrows)
into other categories (objects & arrows), "natural transformations" that
map functors into functors, and so on.
I've been watching a series of 10 minute video lessons on that by the
"Catsters".
https://www.youtube.com/user/TheCatsters
They were done eight years ago by two friends, both personable. Eugenia
Chang is delightful in her over-the-top enthusiasm and dynamic humor.
I'm happy that she's really blossomed, both as a high powered category
theorist but also as a champion for popularizing math. I appreciate her
very helpful talk on the the periodic table in higher order category theory:
https://www.youtube.com/watch?v=lJGUMlgCxz8
Then I learned that she had a great 5-minute television appearance on
Stephen Colbert's Late Show:
https://www.youtube.com/watch?v=mA402F5K47o
She talked about her new book which links math and cooking:
"How to Bake Pi: An Edible Exploration of Mathematics"
https://www.amazon.com/How-Bake-Pi-Exploration-Mathematics/dp/0465097677
There's a very nice article about her in the New York Times:
http://www.nytimes.com/2016/05/03/science/eugenia-cheng-math-how-to-bake-pi.html
And she gave a nice Ted-x talk:
https://www.youtube.com/watch?v=CfdFw3hXkf0
She's currently at the Chicago Art Institute as a Resident scientist.
As for myself, I've been studying a lot of math. I'm starting to think
that behind all of math there must be a set of ways by which we create
abstractions. Basically, if we have several examples, (it's not clear
if one example is enough), then we can identify abstract qualities that
they all satisfy. Then we can discard the examples, think in terms of
the abstract qualities, and define abstract entities that satisfy them.
And then we can continue this process in various (but I think, quite
limited) ways. The upshot is that abstraction can grow and grow.
Whereas we have a belief that abstraction will help us get to the root
of unity. Instead, we have all of this math that gets generated,
increasingly abstract and difficult to follow.
It's a bit like my experience as a database developer. On the one hand,
there's a belief that if I think through a system that is general
enough, then it will serve for all possible applications. But actually
that approach tends towards "overengineering". Instead, it's probably
best to find a balance where, say, half of the energy goes into
designing the overall system and half of the energy gets spent on
dealing with exceptions.
I believe that if we thinking about what is going on here, then we can
actually describe exactly how this dilemma unfolds. And we can define it
rigorously and apply it practically by choosing an optimal point between
overengineering and underengineering. In math, that would mean finding
contexts that are not too specific and not too general. Of course, math
education can serve our entire lives or just the moment at hand, and it
is an art to balance the two.
In summary, in order to encompass all of math and discover its unity, I
think I need to watch how math discovery unfolds, especially through
abstraction, and show how that process accounts for all of math.
Andrius
Andrius Kulikauskas
m...@ms.lt
+370 607 27 665
Bradford Hansen-Smith
Dec 7, 2016, 8:18:36 PM12/7/16
to mathf...@googlegroups.com
Hi Andrius,
You have said:
Whereas we have a belief that abstraction will help us get to the root of unity. Instead, we have all of this math that gets generated, increasingly abstract and difficult to follow.
As I understand there is no "root to unity." Unity is of itself, always has been, always will be. It cannot be achieved by adding units together or constructing abstracted logic systems based on generalizations that come from experience; The sustainable container for the experience of evolutionary creation can only be unity, anything less, it is a unit. Confusion increases in all areas due to the influence of mathematical logic.
In summary, in order to encompass all of math and discover its unity, I think I need to watch how math discovery unfolds, especially through abstraction, and show how that process accounts for all of math.
As Eugenia Cheng points out, math is one of many logic systems attempting to understand our experience, it is everywhere. Generalizations start with experience so it would seem reasonable to observe how experiences unfold from within unity before abstracting them.
To "watch how math discovery unfolds" requires us to look at our experience. Starting with a ball as a form of complete unity. Rather than cutting into and destroying it; maybe to compress it to a circle disk, without loss of unity, observe what is generated by folding the circle to actually see what is being unfolded. Twenty-seven years of experience tells me all the abstractions and generalization of math that I understand are relationships inherent in the folds of the circle. If we can get away from thinking about the circle as a picture of a concept about generalizations and begin to accept that unity is to be realized not achieved, then maybe in folding the circle we can see beyond the generalizations to patterns of structural movement that are unfolding.
To "watch how math discovery unfolds" requires us to look at our experience. Starting with a ball as a form of complete unity. Rather than cutting into and destroying it; maybe to compress it to a circle disk, without loss of unity, observe what is generated by folding the circle to actually see what is being unfolded. Twenty-seven years of experience tells me all the abstractions and generalization of math that I understand are relationships inherent in the folds of the circle. If we can get away from thinking about the circle as a picture of a concept about generalizations and begin to accept that unity is to be realized not achieved, then maybe in folding the circle we can see beyond the generalizations to patterns of structural movement that are unfolding.
Brad
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Joseph Austin
Dec 21, 2016, 3:37:33 PM12/21/16
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In Computer Science, we have the notion of "information".
An item of information is basically the answer to a yes/no question: a binary digit, or "bit".
[Technically, we must weight this by the relative probability of "yes" vs "no" answers.]
So, in those terms, what is "abstraction"?
it's the set of whatevers that give a "yes" answer to a question for which other whatevers give a "no" answer.
Two classes can be distinguished with one question question.
Four classes require two questions; e.g. "Bigger than a breadbox? Alive?"
Eight objects require three; in general, N objects: log 2(N).
The proverbial "twenty questions" can distinguish 2^20 = over 1 million classes,
well above the number of words in the English language dictionary.
That's a lot of abstractions!
Math can be a fun game.
It can also be useful.
it becomes useful when the "abstractions" and operation on them can be related to classes of real things.
Let's get real.
Joe Austin
An item of information is basically the answer to a yes/no question: a binary digit, or "bit".
[Technically, we must weight this by the relative probability of "yes" vs "no" answers.]
So, in those terms, what is "abstraction"?
it's the set of whatevers that give a "yes" answer to a question for which other whatevers give a "no" answer.
Two classes can be distinguished with one question question.
Four classes require two questions; e.g. "Bigger than a breadbox? Alive?"
Eight objects require three; in general, N objects: log 2(N).
The proverbial "twenty questions" can distinguish 2^20 = over 1 million classes,
well above the number of words in the English language dictionary.
That's a lot of abstractions!
Math can be a fun game.
It can also be useful.
it becomes useful when the "abstractions" and operation on them can be related to classes of real things.
Let's get real.
Joe Austin
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