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 repertiores. 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 repertiore 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 repitiores. 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.

Sources:
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.


Betrand Russell::> Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as poetry. 

Controlling Fear

Having a reliable method to control your fear and achieve focus quickly is indispensable.  During one sleepless night reading Frank Herbert, I discovered my solution.

A bit of backstory:
During undergrad, it was common before and during tests for me to be unecessarily nervous. This degenerated my ability to focus and fully apply myself to the important task at hand: aceing the test creatively in a reasonable time frame. 

While meditating, one of my many calming methods is the recitation of poems, such as the Jabberwocky, Litany Against Fear, and If. I find that of all poems, the Litany calms me the most quickly.
To immediately calm myself in risky situations, I use the same techniques. My favoured methods are solving simple integrations or reciting the Litany Against Fear
This Litany, from the Original Dune, an “incantation spoken by many highly educated people who faced danger or fear during their everyday lives. The incantation helped focus their minds in times of peril.” 
The content of the Litany:
I must not fear.
Fear is the mind-killer.
Fear is the little-death that brings total obliteration.
I will face my fear.
I will permit it to pass over me and through me.
And when it has gone past I will turn the inner eye to see its path.
Where the fear has gone there will be nothing. 
Only I will remain.
If you aren’t sure how to begin searching for a reliable and personalized method: think of things that make you happy and calm and read while simultaneously reading the Litany aloud to yourself. 

18 General Lessons I Learned in University

Some of these may seem obvious; keeping their importance in mind is not so obvious.

Through my experience as an undergraduate, I’ve found that the following lessons are useful to be aware of. Hopefully, they will help you and those you love avoid unnecessary suffering. Some of these lessons overlap and they are not in order of priority.

I recently received the Thiel fellowship, and wrote a seperate post on additional lessons I’ve learned.

0. Putting as many hours as you can into doing what you love pays off.

The best programmers I know started programming as a hobby first.
The best researchers I know started researching as a hobby first.
The best mathematicians I know started playing with math as a hobby first.
The trend continues and transcends disciplines of study.

1. Don’t lie about your knowledge base.

It’s better to say “I don’t know.” People respect honesty and pure intellectual pursuit/curiosity more than your pride. Plus, you learn more that way!

2. Don’t cut corners.

Build up your knowledge base from understanding.
Put effort into doing (even tedious or arduous work) as well as you can. By doing this you may discover passions you wouldn’t expect!

3. Fix your procrastination habit.

Don’t know how to fix it? Read this.

4. You can do anything but you can’t do everything. 

Don’t start an entirely different project each day.
Good quality work is done by working through the tedium inevitable in the details of a complex project, and thinking of each step as a new project.

Sort out your priorities and keep them in mind.
Good food, exercise, and sleep are foundational to long-term functionality.

5. Create your own motivation.

Do things in chunks, reach the “mindset”, if you hit a wall, come back to it, you will ponder it unconsciously.

You have to assign your own meaning to life – no one will do it for you. The rules and meaning I live by are an improved version of Neil DeGrasse Tyson’s philosophy:

1. Stay healthy.
2. Learn [and make] something new every day.
3. Lessen the suffering of others.

Keep these in mind; these simple rules changed my life for the better.

6. Being a Jack of All Trades is itself a talent.

Limiting yourself to the current ideas of one discipline makes original research unnecessarily difficult. I find original research is really just connecting past ideas into a new idea that is more than the sum of its parts.

My friend Adam Munich wrote this lovely post detailing the dangers of overspecializing. I highly recommend it!

7. Be aware that friendly curiosity occasionally comes off as interrogation.  

I’ve recently begun briefing myself with: “I’m an incredibly curious person, feel free to stop me if I’m asking too many questions.”

8. Don’t fear failure.

Giving up before you begin is far worse than failing.
Don’t waste opportunities due to fear of failure and cling to the theoretical hope offered by the past possibility of success.

9. You can’t please everyone.

You will quickly exhaust yourself trying to conform to the needs of all.
If you are motivated, honest, kind, and aware of (and abide by most) social norms, fearing the opinions of others is largely unnecessary. Many things that work for others may not work for you and vice versa.
Don’t feel obligated to help everyone that requests your help.

10. Have a workspace.

Make yourself a workspace that you can retreat to and feel at home in.
I have some friends who prefer to work on different task types in different places, and others that enjoy using one workspace for most if not all of their work. Find your preference.

11. Accept compliments.

Compliments are mostly beneficial to those who give them.
“Thank you, that’s very kind of you to say” is my favored response, for it is truthful.

12. There is a difference between nice and kind. 

In my opinion, the difference lies in your motivation.
Being kind is honestly (and gently if possible) doing what is best for the other person.
Being nice is saying what they want to hear.

13. Communicating your ideas is an important skill.

Keeping the audience interested during public speaking is is of equal if not more importance to the content of your speech.

14. Credit your collaborators and prior art.

Don’t claim the work of others as your own.
Keep a log of your contributions to a project to avoid IP disputes.

This applies to homework as well: Making use of the internet to research a problem is to be encouraged as there could be hidden treasures of mathematics to be discovered beneath the surface of many of these problems. However, there is a fine line between researching ideas and using the answer you found on another website. If you photocopy a crossword solution then what have you achieved?

15. Stuck? Explain it to someone.

If you are stuck on a problem, explain the problem, then brainstorm or bounce your ideas off of a friend.
Can’t find a friend who will listen? Explain it to yourself out loud or free write.

16. Write it down.

Although possible to keep all of your project plans and todos in mind, a purely mental method is an unnecessary burden. Writing things down significantly lowers my stress level.

17. If you don’t immediately find your passion, fear not.

I switched my major from:

Philosophy/Physics
$\rightarrow$ Psychology/Studio Art
$\rightarrow$ Anthropology/Studio Art
$\rightarrow$ Math
$\rightarrow$ Math/Computer Science
$\rightarrow$ Math/Physics
$\rightarrow$ Electrical Engineering
$\rightarrow$ Computational Physics.

Playing in these disciplines I’ve discovered that what makes me happy is implementing my passion for STEM to enable the disabled. I wonder what you will discover!

The Engineer-Musician

There is a connection between engineering and music. For the sake of simplicity, I will shorten “math, physics, computer science, mechanical & electrical engineering” to “engineering.”

Let’s start with the general question: Are engineering and music the same? 

My immediate thought was that these fields constitute a gradient of knowledge and defining boxes that constitute each field is limiting to the solving of problems and the mastering of skills. I then realized that my immediate thought was an oversimplification and the question was too general to answer meaningfully, thus I refined the question slightly:

Are engineering and music two ways to look at the same thing? 

I answer with a vague hypothesis: these fields are connected by general principles applied in different contexts.

As we know, one can’t prove that something is true purely by example; however, finding examples of these connections is exciting and interesting to think about. An example of evidence that supports this hypothesis is number theory. Number theory is deeply connected to the theory of harmonics. Using mathematics to prove that there exists a deep connection between the structures of engineering and music is almost cheating. Modern math becomes increasingly abstract. Each major concept works with many diverse objects, all sharing some common property. An abstract theory is born from the consequences of this property, which may then be applied to any of the diverse objects.

This led me to think about the connection of general thoughts and actions between the fields, rather than the underlying principles themselves. For example: the connection between completing projects in engineering and playing a musical instrument. Both activities are never-ending fountains of entertainment and learning; there is always room to improve yourself in your understanding and implementation of your knowledge in both cases.

The application of these knowledge bases involve a similar level of 3D spatial reasoning and abstract thinking. These subjects are built on modules and components. The understanding of each piece of the puzzle and how said piece connects and interweaves with the whole picture (which itself is another puzzle piece) is integral to both completing a project on time and understanding a piece of sheet music (especially if it is in an orchestral context).

I will attempt to demonstrate this connection with two specific examples:
learning a complex piece <= learning an instrument
learning to code <= learning a programming language.

(1) Before one learns and reliably performs a piece, they understand the language of music, whether that understanding is purely by ear or complemented by reading sheet music. They learn to play an instrument (in my case, the trombone). This learning forms a dictionary of muscle memory that consists of sheet music/sounds as keys and embouchure/hand position as corresponding entries. After practicing consistently over a long period of time, I began to feel the instrument as an extension of myself, as a voice to communicate with. To learn a complex piece, I practice each measure, and then practice weaving the measures together.

(2) Before one becomes a good programmer, they understand the basics of computer science (what it means to compile/interpret a program, how to create efficient programs, how to write general instead of hardcoded source, how to use existing libraries in your project, the practices of writing readable source/how to document code). They learn the basics of coding: variables and data types, strong vs weak typed languages, assignment vs. equality. They learn the syntax and lexicon of a programming language, and they learn to create code to execute their ideas. This new lexicon leads to a new way of thought, it includes the structures and semantics at a level of abstraction above the basic syntax of the language, this includes the concepts such as instantiation, monads, and verification. Enough practice leads to thinking in code, and intuitively recognizing how to compose an elegant and efficient solution to a computationally intensive problem.

Similarly, I find that composing orchestral music and creating/working on original research use similar structural thought and insight. Enough examples, I think you’ve got the point!

Although I am not a master of engineering nor music, I consider myself an engineer/scientist and an amateur trombonist. I enjoy learning and developing myself in both areas. I personally think of both music and engineering (especially mathematics, programming, and robotics) as collections of 3D structures to play with and put together whilst keeping in mind various properties and limitations.