Hidden Structure

Beauty wilts
and love will flucture
what remains
is hidden structure

blushing primes
sly chiding knots
betwixt our thoughts

lives to create
chipping from
the ob’lisk slate

talk amoungst
our little selves
filling shelves

search we will
and see we’ll not
blindness is
our earthly lot

but the voice
of the void is vast
notice it
through the nets we cast

Some comments on math communication

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

0. Communicating in a pedagogically correct manner

I have spent a large part of the summer learning how to speak to other mathematicians, and the standards generally enforced, and I have much more to learn. It is somewhat complicated to keep in mind what is commonly thought of in various subfields as “easy to understand” or “basic,” and what is “complex” or “impossible to grasp.” You musn’t allow it to affect what you view as natural. Just recognize that what you see as simple is not necessarily so in the eyes of others, and vice versa.

Keep track of what causes people to shut off, what causes people to feel like they are being talked down to; probe to find out what their mental models are and what they think of as important. Teaching in a way that feels collaborative involves a large amount of empathy and a change of language, for example:

  • No: I was explaining this, and your question confused me.
  • Yes: We were doing this, and I got confused.

1. Articulating vague thoughts

It is very important to realize that there are many ways to make precise a vague question (e.g., an undeveloped feeling of connection between two things usually not thought of as connected, an uncertainty about a concept which you are not able to pinpoint).

Spending a large amount of time alone to develop things in your own mental sandbox until they are ready to be translated into words is healthy and natural. But do translate them into words, even if they are not fully formed, s.t. you don’t end up with a theory so removed and technical from the outside that it remains unabsorbed.

Trying to communicate the way you think about an idea in its purest form is not always useful or interesting to the other party, but sometimes it is.

2. Giving a lecture

Giving a good talk and controlling the room are, unfortunately, entirely different skills.

It seems that in a lecture setting, one must abandon hope of communicating formal information, and instead try to communicate key insights and mental models — usually just one or two — and back up philosophical statements with numerous simple examples!

(I gave a talk where each sentence I said aimed to convey things that had originally taken me months to even begin to appreciate. I also let a few people interrupt the flow of the talk in order to heckle wrt small vocabulary differences and technical details. Someone stood up in the middle of my talk and talked for 10 minutes. This is not the way to go.)

3. Building your own mental models

You are going to reinvent things, lots of things. This is good: it is important to practice making original discoveries! If you hold the belief that understanding = you could have invented it yourself, then reinvention is especially encouraging!

Most of my current mental models come from sitting alone and drawing in my notebook what concepts mean to me, the questions they lead to and flowed from, what images they invoke…

It is an intimate act, personally understanding a concept. For me, it involves a lot of doodling and staring out into space.

It also involves a lot of chunking (e.g., a CW-space is an indexed array with attaching maps; a functor is a generalized manifold; approximation of continuous processes via power series; reducing complexity by looking for the base objects and laws which generate the objects you care about).

I started reading histories of mathematics and found that some of the connections and motivations I’d come to myself, and many I hadn’t seen, were the historical reasons for their invention (H-spaces are more general versions of Lie groups, homotopy theory came from complex analysis so what comes from the calculus of variations, etc).

For this reason, there is incredible joy in reading original papers, or in historical and careful recounts of such papers (e.g., Dirichlet’s lectures on Gauss’s Disquisitiones Arithmeticae), as the life of concepts often seems to be lost through a game of paper telephone (the citations of old papers usually don’t convey the interests of the old author).

4. Vocab hunting

This is an incredibly fun and superficially rewarding game: coming across a technical term (e.g., pre-mouse) in a language in which you are not conversant (e.g., model theory), and then “chasing” (via wikipedia links and journal articles) the concept until you find/reformulate the definition into a language you speak. This is best done when you need a pick me up or can’t sleep.

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).


Cutting the flowers off at their stems \(C \xrightarrow{p} \text{Spec }R\)
\(\Rightarrow\) you’re left with the line of earth (\(\text{Spec }R\))


Cutting into the petals a small ring around their stems (formal disk at marked point)

\(\Rightarrow\) you’re left with the remaining (infinitely-layered) base of the flower sitting on top of \(\text{Spec }R\)


Feeling frustrated that you can’t see clearly, you use your hands to move each disk lying flat on the ground to lay on its side, s.t. these bases are now stacked on top of each other like CDs and form a loose layered cylinder, centered around the line of earth.

1st layer = \(\text{Spec }R[x]/x^2\), first infinitesimal neighborhood
1st&2nd layer = \(\text{Spec }R[x]/x^3\), second infinitesimal neighborhood;…


You stare at this line of earth, adorned with (infinitely-layered) flower bases on top forming a layered tube around the line of earth. Looking closer, you see how the layers fit together, \(\text{Spf }R[[x]]\).

(If you’d cut out the disk and forgotten how the layers fit together, you’d find yourself with \(\text{Spec }R[[x]]\) — an awfully boring topology.)

Glance away, toward a different line of earth \(\text{Spec }E^*\)
with (infinitely-layered) flower bases on top, \(CP^\infty_E := \text{Spf }E^*(CP^\infty)\).

Someone already cut the flowers down to their bases, before you had a chance to see them!

Flustered, you remember that \(CP^\infty\) is the colimit of \(CP^n\).

You reach into your pocket for your book-keeping device, and use it to look at the connectivity rings of each \(CP^n\).


Content, you label the layers of the flower base:

1st layer is Spec \(E^*(CP^1)\) = Spec \(E^*[x]/x^2\)
1st&2nd layer is Spec \(E^*(CP^2)\) = Spec \(E^*[x]/x^3\), …


Given the flower bases (\(CP^\infty_E\)), can you tell what flowers were over \(\text{Spec }E^*\)?

That is, is there a group object with a map to \(\text{Spec }E^*\) whose formal completion along its 0-section is \(CP^\infty_E\)?

I thought the answer was no, but I think the answer is instead tautological. There is a H-space flower which we can trim to \(CP^\infty_E\). What is it?

The notation for \(CP^\infty_E\) itself is extremely suggestive.

The H-space in question is \(CP^\infty\), and we’ve \(E\)-localized it!

You have enough to precisely decipher this story, each of your loose ends can be tied.


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!

On Categories and Concepts: Hofstadter Talk

This is a summary of a talk I attended at Stanford by Douglas Hofstadter (well known for his authorship of Gödel, Escher, Bach: An Eternal Golden Braid).

He prefers lecturing without notes. I found it interesting/challenging to sort and summarize the main points of an improvised talk. Enjoy!

The label of a category can be anything from a conjunction to the essence of a situation

A paradigm for the situational label is Danny at the Grand Canyon.

Hofstadter’s family traveled to see the Grand Canyon. As Hofstadter turned his entranced gaze away from the great abyss, he rested his eyes on his son. His 1-year-old son, Danny, sat facing away from the Grand Canyon and staring at ants. He was a child so young that he had no idea of distance. This situation can be generalized to the idea of focusing on what you’re interested in (and are capable of focusing on), harboring little interest in what others consider as gems.

Similarly, idioms are categories:
Left hand doesn’t know what the right hand is doing \(\equiv\) One part of an organization is contradicting the other

Tail wagging a dog \(\equiv\) small things have inordinately large control over a situation.

Conjunctions (logical connectives) are categories:
The conjunction “/” (\(\equiv\) “slash”) is two things that aren’t quite exclusive combined together. Formally, let \(A\) and \(B\) be categories, a slash denotes \({A \cup B : A \cap B \neq \varnothing}\). For example: they’re a bimbo/self-marketing genius.

We may consider ourselves as each having a unique, private repertoire of categories (memories and thoughts)

We are a composition of public and private repertoires. The action of categorization is at the core of cognition. The difference between humans and other animals is the snowballing of categories; we continue to accrete categories to our repertoire through analogy.

These categories begin as a singleton: a set with one member. They evolve and blur into analogies as we add experiences to our private repertoires. This blurring may be coined as “pluralization”.

Reminders are analogies: unconscious thought pushed into consciousness by a situational queue.

The evolution of a category:
\\(\text{Singleton}\rightarrow\text{Superimposed idea}\rightarrow\text{Pluralization}\rightarrow\text{Label}\\)
He defines intelligence as the ability to put one’s finger on the essence of a situation rapidly. In other words, finding propelling analogies quickly.

Circumstances that evoke the choice of a category are extremely subtle

Without getting bogged down in examples, subtlety in is demonstrable in the difference between appropriate use-cases of “go to school” and “go to the school”.

  • “I have to stop by the school today to pick up my spectrometer.”
  • “Ender, you must go to school today.”

Many of cognitive science’s esoterics are bastardized versions of terms inherited from formal logic

In mathematics, predicate logic is an umbrella term for symbolic formal systems, informally, a predicate is a statement that is true or false depending on the values of its elements.

However, predicate calculus is “a general system of logic that accurately expresses a large variety of assertions and modes of reasoning”, capturing the essential logical structure of a complex idea independent of its elements.

The proposition Matvei loves Ubuntu can be represented by a predicate calculus in the form:

[Relationship between elements]([Subject element], [Object element])

However, a predicate calculus does not emit a yes or no. Likewise, category membership is not a boolean. Membership is fuzzy: the strength of membership is on a spectrum including central and peripheral membership of a given category.

Meaning is contextually dependent

The communication of ideas usually takes place through language: a stream of symbols flowing out of the mouth/fingers. Before symbolic exchange generates meaning, the situation must be explored and evaluated in the discourse space: a completely context dependent environment.

Sidenote: Hofstadter realized that meaning had a contextual dependence after he decided that his mathematically based notion that “all that matters in language is truth and falsehood” was incorrect.

Often, meaningful sounding questions are in fact meaningless due to ill-defined terms

When queried: How many languages do you know? He answers, “I’m \(\pi\)lingual”.

\(\pi\)lingual not in the sense that his knowledge of language is transcendental. Instead, \(\pi\) is the result of summing of pieces of languages he knows into blurred fractions.

Before the question is meaningful, the questions What does it mean to know? let alone What does it mean to know a language? must be addressed.

Is it an efficient way to equate the meaning of 2 sentences to represent a network of word-relationships as a weighted graph? Or is this a seemingly logical but meaningless question due to fuzzy definitions?

My question, left unanswered/as an exercise for the reader.

Cognitive Science: An Introduction to the Study of Mind Jay Friedenberg, Gordon Silverman

The Utility of Art vs. STEM

I was recently asked an interesting question in an interview:

You stated in a previous article that you believe math and science are “toolkits” to solving problems in ways that writing and the arts cannot. Can you elaborate on that?

I think science and art are two sides of the same coin. The distinction is quite fuzzy for the fields overlap in a variety of ways that depend on the perception of the viewer. For example, mathematicians find aesthetic beauty in eloquent proofs and concise equations. The main difference between the terms lies in what they contribute to the to the world. Art provides inspiration and science provides understanding and explicit utility. I specify “explicit utility” for implicitly, inspiration provides the driving force for scientific advancement.

Art enables us to describe every emotion and experience known to man, but mathematics enables us to understand the laws that govern everything. Art cannot show us something that is not a human experience, for it is limited by the person who uses it. Mathematics, on the other hand, can show as absolute truth realities too grand to be fully understood by the human mind while science allows us to precisely and repeatedly implement these truths in the physical world.

College, Would You Like Fries With That?

I enjoy songwriting and have recently gotten into playing Third-Wave ska on my trombone. The basic premise of ska is to play an incredibly upbeat tune with shocking/macabre lyrics.

Here are the lyrics to a silly ska song I wrote in class. Note that I’m happy with my university experience; this song presents the point that university isn’t for everyone.

College? Would you like fries with that? by Catherine Ray

You’ll struggle through your classes, and they’ll let you out of school
Clutching a degree you’ve wasted your life to get, you fool
Up next, the corporate overlords will laugh and watch you squirm
They’ll lead you on in interviews, in the end you’ll just be spurned
Run with your tail between your legs,
maybe McDonald’s will accept your unskilled naivete.
Life has revealed its disguise
Yell, “I wish I’d Realized!”

Writing insightful term papers on the Third Reich,
you never thought you’d be one of those guys who quit their work on strike
First you paid thousands just to go to lectures and to class
All of that money and time, for 9 to 5, for a chair and a cubicle, to sit on your ass
The next 20 years consist of watching TV and arduous jobs, paying off your student loans
Working over time for mortgage, so you can pay for your worthless home
After work, go to the mirror.
Just look yourself in the eyes,
Yell. “I wish I’d realized!”

The goal of college isn’t to simply earn a scroll and a title
Get something concrete from it – a job, a pay raise, a promotion
Look back now – Clearly you’ll see, your only profit was debt and negative emotion.
Listen up, my friend Getting some degree
won’t cement your name in most minds, surroundings, nor history
Education is awesome for your outlook on life, real life experience is the fastest way.
The assumption that education must be IRL – Get used to living in meatspace
Be an autodidact, it can’t be all that hard.
Get out your laptop, a pen and some flashcards
Learn a language, free eBooks, a chapter, or if you’re lazy, a word a day
All along there was Wikipedia, free online courses, to your dismay
It’s too late for you – unless, you’re a chrononaut
Life ain’t something you can revise.
Yell. “I wish I’d realized!”

Frodo Baggins Contacts REX Computing

Name: Frodo Baggins

Interested usecases: “Scientific Computing”

Subject: Natural Resource Mining Oppurtunity


Eredor is only safe to drill for short intervals of time due to the current draconian government of the region. Due to this, the main restraint on the flourishing of our company is the lack of rapid data turnaround which is necessary for us to meet our drilling schedules.

We require optimized hardware for data processing and image analysis.

Any initial slowdown in collection of acoustic impedance data (which must be processed into a 3D matrix) delays the beginning of analysis.

Hello Master Baggins,

Thank you for your interest in REX Computing. This is a very interesting area that we are actively exploring and would be interested in working with you to capture this market. In exchange for providing evaluation systems for your use, we would be interested in access to any mithril deposits in order to work on new prototype systems requiring mithril oxide deposition techniques. In general, we believe that our product solutions would be able to accomplish your goals and assist in reconstruction efforts in Middle Earth.

Would you have any potential connections for recruiting in your land? I’ve CC’d our Chief Scientist and ex-Rider of Rohan, who speaks fluent elvish and dwarfish.


Thomas Sohmers
CEO, REX Computing

Hello Master Thomas,

Unfortunately, mithril is not found in the mines of Erebor. The production of all new mithril stopped entirely after the fall of Moria, and this precious metal has become priceless. The only way we can obtain mithril is to melt down existing heirlooms and weapons to use as mithril oxide in your new prototype systems.

Given your need for large amounts of this practically extinct and priceless metal, I propose the following: Our Fellowship is glad to partner with yours in symbiosis if you partner with our company to overthrow the draconian government of Erebor and take back the hoard of the Longbeards: the largest dwarfish treasure hoard in Middle Earth. Your reward will be the doors of Moria which are inscribed with Ithildin, an alloy that contains mithril. We have removed the aforementioned doors and they are kept in an undisclosed location.

If our arrangement is broken, you will have no source of mithril-oxide besides that of our lovely Galadriel’s Nenya. This ring is ordinarily invisible to all but other ringbearers such as myself. I will warn her of your interest and warn you that the ring is unattainable and it is unethical to consider destroying it due to its role in her people’s preservation, protection, and concealment from evil.

On a lighter note, our company partners with teams of dwarfish engineers who would be glad to assist in the research and development of systems that may rid our mining system of inefficiencies to optimize production. All elvish engineers have been called in emergency to assist in the preservation of the kingdom of Lórien, and are unavailable to work on any side-projects at this time. However, I will alert them of your interest and keep you posted on their availability.

Fondly, Frodo