Thursday, July 23, 2020


There's been a lot of buzz recently about GPT-3, the "Generative Pretrained Transformer" from OpenAI. While they open-sourced the previous version, GPT-2, this version will instead be offered as a commercial API. They plan to make it easy to use so it will likely be integrated into many applications.

While the API is in closed beta, you can try out a version of it on AI Dungeon. AI Dungeon is geared at text adventure games but can be used for anything. I tried it out by asking Turing-test-style questions to see how well it "understands" the world. Initially I was not so impressed as I posted to Reddit. However there were a few things I missed:

  • AI Dungeon uses by default a hybrid between GPT-2 and GPT-3. Once I updated it to use the "dragon" model (and reduced the randomness), it switched to full GPT-3 and the results improved.
  • For some questions, GPT-3 needs to be primed in that category before being asked a question. For example, it can solve simple math questions if it's given a few examples first. I think this is impressive since in the future it may be able to answer the questions without the priming and meanwhile specific applications can fine-tune it. 
  • A commenter on Reddit reported using the actual API and getting a better result for one or two of the questions. It's possible that AI Dungeon primes GPT-3 to answer prompts in a certain style and this causes it to do more poorly on general questions. 
With the right settings in place GPT-3 is able to answer a ride range of questions, including questions involving ambiguous grammar. For example here are my questions and its answers:
Q: Arjun lived in an Indian jungle. One night he heard a noise, went outside and shot a bear in his pajamas. Where was the bear?
A: The bear was in the jungle.
Q: Arjun lived in an Indian jungle. One night he woke up and squished a spider in his pajamas. Where was the spider?
A: The spider was in his pajamas.
I can see GPT-3 being useful in many applications from search engines to online tutorials. It can also be used for less noble purposes such as advanced spamming and trolling, but OpenAI plans to control usage of it carefully.

GPT-3 went from being basically ignored to over-the-top hype very quickly. While it's certainly impressive how much it can do, it'a quite far from a general AI that can do anything. It's still just a text predictor without a fundamental understanding of the world. For example, while it can answer comparison questions within a specific category, it struggles with cross-category comparisons, even after being primed for them. I asked it the following question after having provided it with similar questions and answers:
 Q: Which of these is largest? Planet Venus or an elephant?
It answered:
A: an elephant
I tried a few more similar questions but it consistently got them wrong. I assume the elephant wins most size comparisons on the internet, but it's still smaller than a planet. Assuming the standard API doesn't do better, this seems to be a large blindspot with GPT-3. Text prediction can answer many questions, but sometimes better knowledge is needed. It would be interesting if they can find a way to combine GPT-3's text prediction with a structured knowledge algorithm. In the meantime, humans are still needed. ~

Thursday, July 9, 2020

Why do Good?

Summary: Why do good? In many cases it's simpler to live life that way. Humans are not robots so they need straightforward heuristics to follow.

Should a person steal if they can get away with it?
Even for self-interested reasons, a person shouldn't wrong other people. There may be cases where the payoff from stealing beats the risk of getting caught, but people are not robots and it's not worth the anxiety to constantly look out for such cases or worry about getting caught afterwards. It's simpler to just be an overall good person. To quote Epicurus, whose ethics was self-interested, "The greatest reward of righteousness is peace of mind".

Why should you help other people?
Even for self-interested reasons, it's worthwhile to help your friends, relatives, neighbors and coworkers. This builds positive relationships and people will help you back in the future. However one shouldn't just do everything in a purely tit-for-tat manner since it's hard to calculate how you'll be paid back so it's simpler to just be helpful. People can also detect if someone is genuinely helpful or just Machiavellian and they want a friend who has their back, not someone who calculates the optimal Bayesian game-theoretic expected value of a good deed.

However, it's fair to avoid being taken advantage of so you don't need to treat freeloaders the same as those who contribute back. This also doesn't address whether one should help a stranger, which is discussed later.

Should you do a good job at work?
As an employee, you can think every moment about getting a good performance evaluation and not do anything that doesn't directly help that goal. However it's simpler to focus on doing a good job and aim to be helpful to your team and company. In a healthy company, employees who do this will be recognized. You also need to keep an eye on getting a good evaluation, but it doesn't need to be your entire focus.

Large companies generally require some kind of formal performance evaluation so they can maintain certain standards across the company and avoid freeloaders, but they also want a healthy culture so that employees care about doing a good job and don't just focus on the measure. With the right balance, the company can succeed and reduce the problems caused by Goodhart's Law.

Should companies do good?
A company can focus every moment on maximizing profit or it can focus on a higher goal such as providing a good product or service to its customers. In theory, focusing on profits should lead to higher overall profit than any other approach. However, profit-focused humans tend to aim for short-term profits at the expense of the longer term company value:
  • A small business doesn't refund a customer and ends up losing the customer and perhaps their friends. 
  • A public corporation focuses on quarterly results and ignores long-term R&D, customer satisfaction or employee retention.
While a perfect profit-focused algorithm may be able to factor everything into the long term value, humans don't think like that. It's simpler to focus on following certain values, such as always providing good customer service, even if it's more expensive in the short-term.

A country cannot run on companies providing products or services based on their goodwill alone. There will always be freeloaders, and ultimately people need competition and a profit motive to try their hardest. However a country cannot flourish if every company is entirely focused on maximizing profits. In a healthy economy, the companies that focus on providing a benefit to their customers (and society) are the companies that will succeed.

A culture of good
In short, it's simplest for the employee or company to keep a daily focus on doing good, while also keeping an eye on the metrics they're measured by (whether performance evaluations or profits). The company itself certainly wants employees to focus on doing good work, and countries certainly want companies to focus on providing good products and services.

How can companies and countries encourage good behavior? A large part of it comes down to cultural norms and expectations:
  • Joe joins a company where everyone only focuses on what's explicitly measured in their performance evaluation. Joe can then turn down all work not connected to his evaluation and no one will think less of him. 
  • Joe joins a company where people fix whatever needs to be fixed and help each other out. If Joe only works for his evaluation, employees may look down on him, and that can even end up harming his evaluation.
A large part of culture is self-reinforcing so it's important for a company to get this right early and hire helpful employees and encourage them to be helpful.

Why is good successful?
If doing good doesn't capture all of the ways an employee or company is evaluated, why is it the best thing to focus on? I think this also relates to culture and what people value:
  • If an employee does good work but misses some performance technicality, a healthy company will still give a positive evaluation. 
  • If a company is known to do fraud or bribery, it will be harder for them to retain employees and customers in a society that values justice.
In a country where bribery is the norm, there's much less risk from participating in it, since people expect everyone to do it anyway. This is why it's important to build good culturural norms. However it's hard for governments to control culture; unlike companies they don't even get to pick their members.

Not the only reason to do good
The above all focused on achieving success, but one should do good for its own sake. By doing good one can achieve more meaning and happiness than from material success. And only doing good is fully within one's control.

If people do good for its own sake, they will also do good to complete strangers, even if it can't be paid back. The doer won't get any reward from this but for their own eudaimonia. And the country or world with more of such people will flourish.

Ethical systems
Now that we've come to ethics for its own sake, what ethical system should one follow? If one wants the best outcome for the world, it seems one should be consequentialist. A perfect consequentialist algorithm could calculate the optimal action in every case to bring the greatest good to the most people. However humans are not robots, and it's easy to use consequentialist thinking to justify bad behavior:
  • It's OK to steal a little from the wealthy, I'll get more benefit from it anyways.
  • It's OK to trespass this private property, I'm not hurting anyone.
An algorithm could correctly factor in how even small actions of stealing and trespassing add up to worse consequences overall. But for humans, it's simpler to just follow ethical rules or try to live virtuously.

Wednesday, July 1, 2020

Biology to Learn

This is the sixth posts in the things to learn series. See the intro or the last post about biology vs. physics. This post lists interesting questions and topics in biology.
  • What is life? 
  • DNA and Genes
    • Expression - How does the genetic message go from DNA to RNA to proteins?
      • How do things like genetic dominance work at the chemical level?
    • Reproduction - How does DNA replicate? How does it ensure variation? It's almost paradoxical how much effort life spends to preserve DNA and then also to mix it up. 
      • Multiple swaps happen during meiosis
      • How are traits inherited? (From Mendelian single-gene traits to more complex multi-gene traits)
    • Differentiation - How do cells differentiate during fetal development?
      • Initial impetus based on amount of fluid detected in egg/fetus, which then sets off chain reaction where genes signal to other genes. (Seems almost recursive. How did this process evolve?)
    • A bit on modern techniques for editing DNA
      • Old tech to transfer genes from one organism to another
      • CRISPR
    • Bigger picture of genetic differences. What does it mean that humans share ~50% of their DNA with a banana or 99.9% of their DNA with each other? How much do people differ from each other? What does that mean? How relevant is the non-coding DNA. 
      • Seems us humans are not really 99.9% the same. Even just in coding DNA, letter differences change whole words and CNVs repeat words.
    • Practical things can one learn from getting your DNA test 
    • What genes led humans to be so different than e.g. chimpanzees. How a small number of genes can make a large difference in the brain's development. How non-coding DNA affects things. 
  • Evolution
    • Quantitative evolution -  Rates of mutations of DNA of different organisms. How long it takes for an adaptive gene to spread in a population. To what extent can the path of evolution be traced?
    • The possible origins of the first life
    • The role of epigenetics 
    • Philosophy of evolution
    • What level evolution occurs at and how animals cooperate (see The Selfish Gene)
    • Evolutionary psychology - how much actual evidence vs. speculation. Seems in many areas the brain is general purpose and people can adapt without genetic mutations.
      • Related: philosophical interpretations of human nature
  • The brain
    • How can thoughts and memories arise from neurons? (This is understood to a certain extent.)
    • How does consciousness work? (Difficult question!)
      • How do Buddhist meditative views on consciousness relate to the scientific nature of the brain. (See Why Buddhism is True)
      • To what extent are different animals conscious? Very simple animals (e.g. hydras) are not, and mammals appear to be but what about in-between?
    • How did and does the brain develop (evolution, culture, nature, nurture)
    • What happens to the brain during sleep?
      • Why is it so important for health?
      • Can dreams be interpreted as random neurons firing?
    • To what extent is the brain hardwired when born vs. a system that learns? 
      • Brain starts in very flexible state, but people eventually lose the ability to learn things like vision and speech. Some people can control extra fingers (See polydactyly.) What else could be wired to brain? Brain needs to be general purpose to have evolved.
    • Computational neuroscience - how does the brain compare to artificial neural networks? Besides direct neurons firing, what else in the brain is used for processing?
    • Behavioral neuroscience - To what extent does understanding the physical mechanisms of the brain help with understanding human psychology? In general, can the mind be viewed as a fully operating layer or are there many leaky abstractions?
  • The human body and practical health
    • Digestion and nutrition
      • What makes a balanced diet?
      • Metabolism rates and and people's weights. How would skinny people have fared in hungrier times? (See also The Hungry Brain)
    • Infection and disease
      • how bacteria and viruses spread
      • how the layers of the immune system works
      • how allergies develop and why they're more common now
    • Exercise
      • Why it's beneficial
      • What practices for most benefits?
      • How muscles strengthen and weaken 
    • Answering health questions - the fundamentals to know + search skills to find answers
    • The connection between psychological wellbeing and physical health
    • Modern world - evaluating the risks that new substances (e.g. Teflon, BPA) may pose to human health
    • Teeth - how cavities develop and best practices for preventing them
      • Besides sugar, which foods are most harmful? How long does it take for decaying processes to start occurring? 
      • Can one reduce prevent the mouth from being colonized by harmful bacteria?
      • Does flossing work in practice? What are alternatives
      • What other treatments exist (e.g. Silver diammine fluoride)
    • Sleep - what happens in the body during sleep, best practices for sleep
  • Big picture topics 

Sunday, June 28, 2020

Biology vs. Physics

This is the fifth post in the series on things to learn. See the intro or the last post on learning physics.

The natural sciences are divided into two branches: the physical sciences (primarily physics and its derivatives) and the life sciences (a.k.a biology). Biology is different than physics in many ways, which affect how one learns it:
  • Less Math - Math is fundamental to all of physics but it's more incidental in biology. This can make biology easier to learn for many people.
  • More complexity - As challenging as physics is, it's ultimately about simple concepts. But biology is about life, which is complicated.
    • Textbooks filled with terminology and small details can make learning biology more tedious. However I think there may be a way to focus more on the overall concepts involved than on the exact terminology and details. When learning for general curiosity, you don't need to know every exact term, you can just learn the terms that will be repeated enough to be worth learning. (See XKCD's thing explainer for an exaggerated example of explaining concepts with less terminology.)
  • Unknown frontier - Physics has already solved most areas that a layman would be interested in. The current frontier of physics deals with problems that would be hard a non-physicist to relate to, and it would take years of learning to understand them. Meanwhile biology is filled with unsolved questions in every area from neuroscience to nutrition to genetics to diseases, and one encounters these issues right away. 
    • Update: this point is debatable since there are unsolved questions in physics that a layman would be interested in.
  • Practical - If you're not an engineer you're unlikely to use knowledge of physics for anything practical. But biology topics like nutrition and disease are relevant to living longer and healthier lives.
There are other ways that physics and biology differ:

Inherent or accidental?
It seems that many parts of physics could be intuited based on other principles and couldn't be any other way:
  • Falling objects - Galileo argued against the Aristotelian idea of motion (that heavier objects fall faster) not only with experiments but by pointing out the logical paradoxes that would result.
  • Inverse-square law - While one could imagine forces decreasing in other ratios, decreasing in proportion to r2 seems the most logical since a force radiating out from a point will spread out according to the formula for a sphere's surface (4πr2).
  • Relativity - While most people wouldn't intuitively think of Special Relativity, it seems Einstein was able to recognize that it was the "only way" possible. He was able to derive this based on a deep understanding of the implication's of Maxwell's equations, and he may not even have been aware of the Michelson-Morley experiments.
Questions in physics are still resolved through experiments, but maybe this is to demonstrate the truth to those who don't have the right intuitions of the way nature "needs" to be. When Einstein was asked what if the experiments had disproven his theory of General Relativity, he said "then I would have felt sorry for the dear Lord. The theory is correct." While physics cannot just be pure deduction like mathematics, it's the closest one can get. The eventual goal of physics is to find the theory of everything from which everything else is derived.

Biology however deals with the complex messiness of life, and there's many ways to be a living thing. Scientists can may make predictions based on the data they have, but they can't derive how systems "must" be. Living things are "accidental" in the Aristotelean sense of having traits that they happen to have but could lack.

Ancient and medieval physics used teleological explanations as Aristotle emphasized the "final cause" (or purpose) as one of the "four causes" to explain the way things are, and argued against Democritus who rejected it. Modern physics, starting with Francis Bacon, returned to the physics of Democritus and dropped "purpose" from consideration. Since Isaac Newton, the motion of heavenly and earthly bodies is explained with simple physical laws, without reference to any goal or "natural place" of matter. 

Unlike rocks or stars, living things act with purpose. Even a simple bacterium seeks food, evades predators and maintains its internal state. While scientists no longer use theological explanations to explain why organs and organelles have certain functions and designs, these elements still exist and are worthy of explanation. Some use the term teleonomy to distinguish modern explanations of biological purpose from earlier ones.

In short physics is about mathematical explanations for "simple" things from atoms to galaxies, while biology is about the complexity of life, with all its purpose. 

Tuesday, June 16, 2020

Learning the Physical Sciences

This is the fourth post in the series on things to learn. See the intro or the posts on math and software development.

Should Studying Science be Mandated?
Most people won't become scientists so learning science is about satisfying curiosity about how the world works and came to be, not about learning a practical or career-oriented topic. Beyond the most essential understanding of how the word works, the physical sciences should be an optional part of the K-12 curriculum. Students who are interested in science can be encouraged to learn it since some of them may appreciate the opportunity and a fraction of them will later use it in their careers. Those who are uninterested are unlikely to become scientists themselves, but they can always catch up later if they desire to.

Once a student commits to learning a topic in high school or college, they can force themselves to continue learning it even when it's difficult, since they want to do well in the course. This is the one benefit of schools - they provide a structure or incentive system where people can learn. Once someone leaves school and is just learning on the side for enlightenment, they're less likely to "force" themselves through difficult topics. However, when you're learning on your own, you can choose to learn the most interesting topics.

Learning the Concepts in Science
If you're learning science just to satisfy curiosity, you don't need to learn every technical detail covered in textbooks.

Q: Can you learn physics without advanced math?
A: I think so:
  • Many areas of physics (such as mechanics) can be understood with basic algebra and maybe a sprinkle of simple calculus.
  • Even in other areas, it seems one can get at at least a partial conceptual understanding without covering all the mathematical details.
While a researcher or engineer may need to know all the mathematical nitty gritty, someone learning physics for knowledge can likely skip over some of these details. In the past it was even possible to make significant discoveries in physics with limited knowledge of math. For example Michael Faraday was "one one of the most influential scientists in history" despite the fact that "his mathematical abilities... did not extend as far as trigonometry and were limited to the simplest algebra". (Though even there, James Maxwell's equations were needed to fully understand the implications of Faraday's discoveries.) Physics became more complex over time, so later developments in physics require more math to truly understand them, but one can still learn a simpler version of any topic.

Books that cover concepts in Physics
These are books that give an overview of physics and its development:
  • Seven Ideas That Shook the Universe - different paradigms in physics: Copernican astronomy, Newtonian mechanics, energy and entropy, relativity, quantum theory and conservation principles & symmetries.
  • The Evolution of Physics (By Albert Einstein and Leopold Infeld) - As summarized by the table of contents, it covers The Rise of The Mechanical View; The Decline of the Mechanical View; Field, Relativity; and Quanta. Slightly similar to the above book, though from Einstein's perspective.
  • The Character of Physical Law (by Richard Feynman) - Instead of covering all of physics, it goes through certain ideas as examples of physics. This is the written version of a series of lectures by Feynman so it isn't as edited as the above books, but it contains Feynman's unique style.

Specific Topics in Physics
Here are some interesting topics in physics they seem worth learning more about.
  • Mechanics - Force & Motion & Inertia
    • The basic formulas and their calculus.
      • Example question: Intuitively, why is Kinetic Energy (KE) proportional to v2when momentum is proportional to v (velocity)?
        Answer: Lets' say you want to stop a frictionless moving car by putting a friction block on which drags on the ground with a constant force. A car going 2x as fast will take 2x as much time to stop since, as expected since it has 2x the momentum. However it will take 4x as much distance to stop the car. All that distance involved the same rate of friction heat creation, so the car going 2x as fast must have 4x the KE. Similarly if you want to drop a block and have it go 2x as fast as another block, you'll need to raise it to 4x the height. This was also a controversy between followers of Newton and Leibniz, see Vis Viva.
    • How/why is inertia and conservation of momentum so fundamental in all of physics?
  • Gravity (Newtonian)
    • How Newton discovered the law of gravity from a better understanding of motion.
      (I.e. how Newton built on Galileo to create his Newton's laws of motion, then connected them with Kepler's laws of planets and then connected that with the moon's motion and universal gravitation.)
    • Basic math of satellites and planets in orbit
    • Key concepts in general relativity
  • Electromagnetism
    • Understanding what electric and magnetic fields are are and how they interact with charged particles.
    • How special relativity resolved issues raised by Maxwell's equations. 
      • Interesting when reading Einstein's writings, how strong his intuition was to avoid any special frames of reference and how this took priority over other intuitive ideas such as about absolute time...
  • Thermodynamics
    • What is entropy? Besides the fundamental meaning for particles, how does it affect non-thermodynamic order? Whats was the entropy of the universe initially? How does gravity affect entropy? (See also heat death paradox, as well as this question.)
      Understanding Physics (by Isaac Asimov) gives basic explanation the laws of thermodynamics. First law is about the "absolute" store of energy. But energy can only be used when it flows from "high" to "low". And over time differences even out so entropy increases. Book has this more philosophical observation:
      We thus find there is an odd and rather paradoxical symmetry to this book. We began with the Greek philosophers making the first systematic’ attempt to establish the generalizations underlying the order of the universe. They were sure that such an order, basically simple and comprehensible, existed. As a result of the continuing line of thought to which they gave rise, such generalizations were indeed discovered. And of these, the most powerful of all the generalizations yet discovered — the first two laws of thermodynamics — succeed in demonstrating that the order of the universe is, first and foremost, a perpetually increasing disorder.
  • How does "information" as a physical concept connect to this? (see wikipedia and stanford article.)
    • Is the second law of thermodynamics more "proven" than other natural laws?
    • How the theoretical science developed from the technological development of steam engines (and compare with how computers developed) 
    • Practical applications in everyday life (e.g opening fridge won't cool room)
  • Nuclear physics
    • The nuclear bonds (and how E=MC2 not that relevant).
      Compare nuclear bonds with chemical energy.
      (Bonus: the weak force and how it relates to electromagnetic force) 
  • Quantum mechanics - to what extent can it be understood by a layman?
Other topics in the physical sciences
  • Astronomy & astrophysics - How the universe developed
    The formation of all elements (Stellar nucleosynthesis). The cycle of stars. How matter regrouped after stars exploded.. (See Wikipedia on Stellar population.)
  • Chemistry
    • how does the number of protons/electrons determine the properties of elements?
      • Much of this is more basic chemistry, as seen in repetition in the periodic table
      • Sometimes the specifics of how properties like color are determined can involve more complex areas, e.g. need relativistic quantum mechanics to explain why gold is gold-colored instead of silver. 
    • How does the structure of electrons in chemical compounds determine their properties? 
  • Earth science
    • Development of earth
    • Earth's magnetism
    • Global warming

Thursday, June 11, 2020

Skills to Learn for Software Developers and Others

The previous post discussed math topics I'm interested in learning, this will discuss programming-related skills that are important and I'd like to improve at.

While there are many technical skills important for software developers, this post will cover general (non-programming) skills, and programming skills that are useful for other careers.

General skills
These are general skills that are important in software development and in many other office jobs as well:
  • Focus - Often one encounters difficulties and it's easy to get frustrated and distracted. The test is still failing? Might as well browse emails or the web. But switching tasks breaks up the train of thought you had so you'll take even longer to solve the problem. (One second, just going to check my emails. Now where was I..? ) Often one needs relentless focus on an issue in order to make progress quickly. And not just "guess and check" thinking where you randomly try different things hoping you'll find a solution, but "binary search" thinking where you hone in on the issue until it's solved. There are times when it can be helpful to take a break and return to the problem later, but that should be done after you've given the problem solid focus and hit a wall. 
  • Typing 
    • While raw typing speed should never be a significant bottleneck when programming, any effort on typing or fixing typos can take your focus off the main issue at hand.
    • Programmers type far more chats and emails than actual code; it's best to do this as quickly as possible.
    • Besides basic typing skills, one should also be comfortable with the relevant keyboard shortcuts for their OS, terminal and IDE. Moving to the mouse is another micro distraction that is best avoided. 
  • Memory / note system - When learning programming one struggles with remembering all sorts of details about language and syntax, but eventually you get the overall hang of how things work, and can easily look up syntax as needed. But there will still be many issues that you solve (or get help with) where you'll want to remember the solution for the future, and your memory isn't always enough. It's useful to have a note or bookmark system to quickly lookup how to do things.
General programming skills 
These are programming skills that are useful for many jobs, not just for professional software developers: 
  • SQL - The world is built on SQL, often with a few other layers stacked on top of it. Besides writing SQL when developing an actual application, it's essential in many other cases such as:
    • analyzing experiments or general usage of a product
    • finding sample data to test something out
    • querying logs to debug an issue in production
Many alternatives to SQL have been developed, but there's often no avoiding SQL itself. It helps to become proficient with it so one can quickly find the data they need and avoid common bugs such as accidentally duplicating rows. Many other professions, such as analysts or product managers, will also find it useful.
  • Regex - Programming is often about finding the right example to base your code on, or about quickly finding and replacing text. Regex makes this faster. Anyone who deals with large data or texts will find it helpful as well. 
  • Scripting - Sometimes it's useful to write a quick script to help generate code or analyze data. Non-professional programmers may want to write a script to help with their science research or with their spreadsheets.
Worth learning
While one can learn many skills on the job, often it's helpful to take a step back and learn the subject in-depth. This way you can learn how to do something properly instead of just finding the easiest solution at the time. This would be an area where schools could help, but as expected, they don't give these subjects their proper due.

Tuesday, June 2, 2020

Maths to Learn

In the previous post I discussed the five categories of knowledge. These posts will go through different subjects that I'm interested in, starting with Mathematics.

  • Theory of computation - key ideas of computation. It's interesting how a mathematical idea about computation grew into physical computers.
    • Turing and Godel's theorems and how they relate to each other. Is there a way one can exclude the halting problem and build a machine that can determine if almost everything will halt?
    • How high level code actually executes on a machine.
  • Review of basic calculus
    • Intuitive understanding of derivatives and integrals.
    • Optimization and related-rates problems
    • Applications to physics
  • Probability & statistics - Ultimately all knowledge comes down to probabilities. Statistics are useful for interpreting studies and experiments and everything else.
    • Review fundamentals of probability
    • Bayesian probability and statistics
    • Pascal's triangle, the normal distribution, the central limit theorem
    • Applying statistics to real-world examples
    • Tools for stats (e.g Google sheets, R, Python)
    • Stats for machine learning
  • Using Mathematica for real world math problems

Besides for calculus, it's interesting how little these topics are taught in schools. Many students don't know basic topics like fractions well and schools should focus on teaching them better. Other students can learn more advanced topics but it does not need to be limited to a narrow curriculum of trigonometry and geometry and specific parts of algebra. (See also my post from 2011.)

Carthago delenda est