Some comments on math communication

My friend Julian Chadez mentioned to me (from a conversation with our friend Semon Rezchikov) the following two things that are vital to progress and quite taxing to constantly keep in mind.

  1. Never be ashamed of not knowing something.
  2. Never shame others for not knowing something.

I read Bill Thurston’s On Proof and Progress this morning. I figured I’d share a few things I’ve learned this summer about the sociology and psychology of being a part of the mathematical community, on the off chance that you might find it encouraging or helpful.

0. Communicating in a pedagogically correct manner Continue reading Some comments on math communication

Spf $E^*[[x]]$: Your walk through a flower garden

Inspired by the extraordinary expository style of Dr. Kazuya Kato, I’ve started reading parts of a (translated) Japanese children’s book when I’m stuck on a tough paper or concept — revisiting the concept with such a dreamlike world in mind usually unfolds an illustrative perspective. A misty world which begs to be put into firm ground via prolonged formal and concrete afterthought.

He embraces that teaching can be poetic and tantalizing, providing not a definition but a deep and creative hint that causes an exploratory shift in perspective, allowing you to walk down the path to the conclusion yourself. I wanted to try to exposit with this philosophy: confusion is expected and encouraged as impetus for reaching understanding. With that in mind, step into your flower garden.


Planted in a line of earth ($\text{Spec }R$)
there are flowers, $C$, whose heads are smooth projective genus 1 curves
with stems that can retract into the ground,
s.t. the flower meets the earth at one point (a marked point).

11270061_10205668389181654_555974065_n Continue reading Spf $E^*[[x]]$: Your walk through a flower garden

Staring.

And I sit here, undeserving,
in a small flat
staring out the window, dear.
Thinking of patterns, and sometimes of regrets.
Staring at symbols, trying to see what the author saw.
Trying to be them, just for a second,
to glimpse the beauty they’d uncovered
by sitting in a small flat,
staring out of their window
thinking of patterns, and sometimes of regrets.

How to: Learn a New Discipline

tl;dr In order to build your knowledge base, start with a question you want answered and work back from it.

Competence in many skills > the mastery of only 1 skill.

Competency in a skill leads you to recognize beautiful executions of the skill; this gives you the power to appreciate the beauty hidden in the world around you.

Example: If you study the art of trombone playing, you are more likely to appreciate the talent of a street musician that is playing high notes with rich timbre.

Wide-spread knowledge in both art and STEM reveals connections between seemingly unrelated concepts; connections that others do not see. These connections often lead to valuable and creative solutions. Overspecialization is dangerous.

Fostering your competency in many fields
$\rightarrow$ wide-spread aesthetic appreciation
$\rightarrow$ being a connoisseur of life.

Complement mere competency by studying a few select topics in enough depth to appreciate their deeper beauty and underlying simplicity.

How, you might ask, do you achieve competency in many disciplines?
Would you like to learn [physics||maths||…], but have no idea where to start?

0. Start with a question that you want answered and work back from it.

This is an effective way to enjoyably learn any field!

Once you have an interesting question to motivate you and esoteric terms to guide your reverse-engineering, you have the motivation and a plan to build your knowledge base.

1. Get excited.

After your burning curiosity pushes you past the give-a-damn point, you give a damn about the basics of the fields that hold the answer to your question.

Having this excitement transforms the drudgery of simply-worded beginner books into a treasure hunt for the missing puzzle pieces you need to understand your interesting question!

2. Explain what you’ve learned.

Explain concepts to yourself &| to those willing to listen

Understanding of a concept and the ability to explain the concept well go hand in hand. Explaining what you’ve learned will reveal the holes in your knowledge base that might otherwise go undetected! Fill these holes.

Thank you, Matthew Lynn, for leading our interesting discussion to cover this topic!

Too Many Ideas: Avoiding “Ooh Shiny” Syndrome

It is often the affliction of creative people that we have too many ideas, and think of time as our most precious resource.

The solution?

0. Mission-Based Motivation

Reframing your situation is the key to making work play. Convincing yourself that your current project is “shiny” can be done by finding the one answer to Why are you doing what you do? [Ryan Lelek].

This must be one answer: what’s yours?

{Some examples to get you started:
To make an impact? To learn? To innovate? To solve an unsolved puzzle?}

1. Write It Down

Leaving half-finished unconnected ideas in your mental cache detracts from your usable RAM. The fear that your new idea will be lost will pollute your focus on completing your current project.

I personally keep 4 notebooks:

  1. Main composition book for brainstorming.
  2. Personal journal in a plain text document.
  3. Pocket sized Field Notes book for ideas that occur at inconvenient times.
  4. Todos in workflowy.com
Free yourself from fear, and add to cached ideas when you have improvements. It is an amazing feeling to search through an old notebook and find that you’ve already worked out the answer to a current problem!

2. Set False Deadlines

We feel the need to autograph our work with excellence, which leads us to waste our time on insignificant details and feeling like our work is never fully completed (even if it is by other’s standards).

As Donald Knuth says, “The root of all evil is premature optimization.”

For drawing, I set a time limit of 2.5 hours. I set a time limit of May 3rd to submit my nonprovisional patent on my latest wheelchair modules.

These artificial deadlines allow you to feel like you’ve “finished” a project to the best of your ability, and move on to fully devote yourself to the next project.

Worry not, my friends. These 3 large changes may take time to become integrated into the busy lifestyle of an active creative.

However, that hard work is worth it! I’ve found that implementing these 3 changes allows me to satisfy my obsessive drive to finish what I’ve started whilst laying the groundwork for my future projects.

Hopefully, they will also help you!