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So, You Want To Study Mathematics?

Wow, wonderful post covering what one needs to get started in Math!

Math has a bad reputation. I believe it is the most beautiful form of human thought, but most people see it as a sterile rule system made to punish students. This disconnect is due to the way our schooling system approaches math — it focuses on the mechanics and not the why. While the basics are necessary, they can hide the deeper beauty that math contains. Luckily, it is easier than ever to get an amazing mathematics education on your own thanks to the internet.

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Before I begin my subject recommendations, there are some things to keep in mind. These pieces of advice come from my own experience with 10+ years of self-studying math and working as a college math teacher. I have provided sources where relevant if you want to see why I recommend these practices.

• First, accept that this will be hard. Math is a new language that has to be mastered. It is much less flexible and more demanding than any spoken language. Expect to run into roadblocks and to have to push yourself. Some ideas will take a long time to get comfortable with; you are literally forming new connections in your brain as you study math!
• The best way to study anything is with consistency, not 3-hour marathon sessions once a month. Your brain works better on 15 solid minutes a day. My rule of thumb is to never miss two days in a row. Life gets in the way, and I can’t always make every day work, but I go out of my way to never let that happen twice.
• I remind my students about this all the time: the only way to learn math is to actually do problems. Following along with a video is an important step, but you do not understand something until you can solve the problems on your own. My recommended resources are largely based on the quality of their problems.
• Studying math can be lonely, but it doesn’t have to be. Find someone who wants to learn the same content and work together. This will help you both with accountability and increase your understanding as you talk through the content with each other. Maybe we can create an online community of people learning different topics: any interest?
• AI can be an excellent tool for self-studying math, if used correctly. There is no substitute for a good teacher, but AI can give intuitive explanations and help check your work. Resist the temptation to use it to overcome every obstacle you face, but it can be wonderful for checking your work or offering hints when there aren’t answers provided.
• Interested in one-on-one tutoring and guidance for your studies? Send me a message; maybe I can help
• Take breaks! Schools know this and have many breaks throughout a semester and even longer breaks between semesters. If an idea isn’t sticking, sometimes you need to walk away and do something else for it to click. Let your subconscious tackle it for a while. Rest is just as important as the work.
• And most importantly, have fun! You are doing this for you. Discipline can only take you so far; remember why you are self-studying math in the first place.

Part of the problem with standard college mathematics curriculum is that it heavily emphasizes calculus. I love teaching calculus, and I think the subject is really fun, but this sequence is based on outdated priorities left over from the space race. Calculus certainly has its place, but it is not the only subject to study.

Meta Value - 55

White Swans are more worrisome than Black Swans. 

Even fundamental; the mere existence of Swans should teach us a thing or two about risk.

Wisdom Of Yvon Chouinard On Living With Intent

“There is no adventure without risk,” Yvon said. “You get something important from committing to whatever you do. And if you don’t risk your neck, you haven’t made a big enough commitment. We climb for our own pleasure or to become a better person; otherwise it’s just a sport. But climbing is really a passion. And the commitment isn’t random. Every climber I know has had a fucked-up experience with their father or almost-father. Mine was an alcoholic. I was never close to him. It was a traumatic household. I spent a lot of my life just getting away from that. And I was never on the typical career path where you go to college and you learn something, then you go work at that and you know what’s expected of you. I had to find my own way of dealing with life. Climbing, for me and my friends, it saved our lives. It gave us a purpose even though it was a meaningless purpose. Every time we’d go out to do something, we gained confidence in ourselves.”

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“I’ve been a student of Zen Buddhism for a lot of my life,” Chouinard said, “and it’s taught me that with business as with everything else, if you do everything the right way, profits will come. So you don’t focus on profit. You focus on intent. Climbing is the same way. So is cooking. Simplicity of intent forces creativity, and creativity is where we find pleasure. I did a little film about this eighty- year- old guy in Italy who fishes with a big old bamboo rod and makes his own horsehair lines and ties his own tiny flies by hand. He doesn’t use anything fancy, but you can’t out fish this guy.

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I’ve always believed that you achieve perfection through simplicity. It’s so easy to complicate your life. It’s so much harder to simplify it. That’s why I don’t have a Yvon Chouinard computer. I don’t have a cell phone, either. I don’t have any of that shit ’cause I don’t want it. Just say no. And I’ve applied that principle in all kinds of places. Once, I had the idea to kayak the Gros Ventre River without a paddle. It was pretty serious water. My friend lost his boat, and it ended up all the way down in the Snake River. But I had learned how to roll a kayak upright with just my hands—in case I lost my paddle—and it gave me this wild idea to try the whole river that way. A paddle is a powerful thing. You can see a rock coming and just do one deep stroke to get around it. Without a paddle, you have to look much farther ahead and prepare. The boat becomes like a pair of skis that you have put on edge to turn. I found that I had to sit way down low in the boat and turn by putting the boat on edge with my hands. I had been kayaking all over South America and New Guinea and doing first descents, but that was the day I really learned to kayak.

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In fact, right now, we’re trying to get Patagonia back to that initial intention. Every company needs a revolution every ten years and we’re going through a big one right now. Because a while back, I was sitting at my desk and a bunch of people came roaring in and said, ‘Guess what? We just hit a billion dollars in sales!’ And I just went, ‘Fuck.’ I was so bummed out because I knew that running a billion-dollar company was a lot different from running a little climbing hardware company and that we could easily become the bureaucracy, the behemoth, that we didn’t want to be. And in fact, it was turning out that way.

I had a CEO at the time who was opening up new accounts in general sporting goods stores and department stores. We were making a 20 percent net profit but selling out—just like The North Face did when they hired Kim Kardashian to do a skiwear line. Because they exceeded the size they should be. So we cut way back. We killed all our department store accounts. And that’s about living within our limits. And I mean, what happens when you’re free soloing and you exceed your limits? You die. That’s what we were facing. I want Patagonia to be a company that grows maybe 5 percent a year. Maximum. Just to keep up with inflation and make a decent profit and have a good culture. Keep a good lifestyle. And I don’t see any examples of anyone else who has done that. Name one company that’s purposely restricted its own growth. But that’s what Patagonia is about now. We’ve been in business fifty- four years. So what are the next fifty years going to be about? I have four thousand employees around the world. They’re all dependent on the company’s survival and yet our products are getting knocked off by everybody else.

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I don’t think you can be an excellent company or make excellent products by getting bigger and bigger. You lose control. The average age of an American corporation before it goes belly- up is less than eighteen years. If we decided, at Patagonia, to start making watches and sunglasses, that would be suicide for what we wanna do. There’s a limit to everything. I don’t care what it is. Everybody dies, no exceptions. Every empire collapses, usually after 200 to 250 years.

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In 2022, Yvon gave Patagonia away—the whole company, I mean. He gave it to a nonprofit foundation that he created with one goal: environmental conservation.

“I gave Patagonia away three or four years ago because I didn’t see any other way,” he said. “My kids didn’t want to inherit the company. They have simple lives like I do. I didn’t want to take Patagonia public because we’d end up killing the company’s values. I didn’t want to bring in an Yvon Chouinard investor or a partner, because those relationships always fall apart, just like most marriages. And I didn’t want to sell for the same reason—there was no way to sell and still be sure that the values endured.”

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“But giving it away worked out great,” Yvon told me. “You’ve seen my lifestyle. I don’t need any money. You’ve seen my place in Jackson. I’ve been in that house for fifty years, and half the electrical wiring is burned out.”

- Excerpts from the new book Rules to Live By: How Extraordinary Lives Define What Really Matters by Jimmy Chin (profiles the founder of Patagonia Yvon Chouinard)

We Asked 100 Scientists About Unsolved Problems In Their Fields...

Epistemological Modesty, my friend. Epistemological Modesty! 

Beautiful list and stupid me, I thought we knew the answers to a few of these questions and some, I didn't even know it was a "question". 

There are so many unknown questions about animals and biology in general than physics. I am glad that many minds are working on these. 

• How did the first Indigenous people enter the Americas? - Michael Waters, geoarchaeologist, Center for the Study of the First Americans, Texas A&M University

• A million years ago, how did earlier human species colonize remote and previously uninhabited islands in Southeast Asia? - Adam Brumm, archaeologist, Griffith University, Australia

• What kind of information — about people, places and other dogs can dogs acquire through smell - Alexandra Horowitz, cognitive scientist, Dog Cognition Lab, Barnard College

• How do birds migrate with such energy efficiency? - David Lentink, engineer and biologist, University of Groningen, the Netherlands

• How many undiscovered animal species live in the deep sea, and do they matter? - Lisa Levin, marine ecologist, Scripps Institution of Oceanography, University of California, San Diego

• Which other animals dream, and how similar are their dreams to ours? - Benjamin Scott, neuroscientist, Boston University

• Why do animals that fly live longer? - Elinor Karlsson, vertebrate geneticist, UMass Chan Medical School and Broad Institute

• Why are humans the only species with a chin? - Daniel Lieberman, evolutionary biologist, Harvard University

• How did the building blocks of life, like proteins and nucleic acid, first organize into a living cell? - Steve Brusatte, paleontologist, University of Edinburgh

• When does a cell become destined to develop into cancer? - Dr. Kimryn Rathmell, physician-scientist, The Ohio State University Comprehensive Cancer Center, The Arthur G. James Cancer Hospital

• Blue whales are the biggest animals ever to exist on Earth. Where are their babies born? - Helen Scales, marine biologist

• Most cancers are not genetic, so why do they occur? - Dr. Mehra Golshan, cancer surgeon, Yale School of Medicine

• Why does the immune system in some people, and not others, control cancers and viruses? - Dr. Sharon Lewin, infectious diseases physician, the Peter Doherty Institute for Infection and Immunity, University of Melbourne, Australia

• Why do the viruses that cause colds and flu spread so much more easily in winter? - Linsey Marr, environmental engineer, Virginia Tech

• Can we learn what emotions animals are experiencing? - Tom Smulders, evolutionary neuroscientist, Newcastle University, England

• Are we alone in the universe? - Robert Hazen, staff scientist, Carnegie Institution for Science

Food Is Important Than Moronic AI & Dreams Of Utopia

The bigger fear is what scientists call “multiple breadbasket failures” – that is, several of the world’s great growing regions failing at once. The journalist George Monbiot tells me that globalisation has allowed us to make up for a bad harvest in one region with imports from another. This year, military activity has at times reduced Russia and Ukraine’s Black Sea grain export capacity by 97 per cent, and the Iran war has throttled Hormuz – the conduit for a third of the world’s urea, a fertiliser. El Niño is bearing down on India’s rice, Australia’s wheat and southern Africa’s maize.

India, at least, has a buffer. The monsoon season was drier than expected, but Delhi is sitting on a record strategic reserve of 59.6 million tonnes of rice, and has so far resisted repeating the 2023 export ban that pushed world rice prices up by more than a fifth. Britain? “We used to have strategic food reserves in this country,” Monbiot says. “ But we left it all to the private sector who have now moved to this just-in-time production and delivery model.” The Department for Environment, Food and Rural Affairs estimates that the UK imports roughly 40 per cent of its food.

And what does come from our own fields is coming under increasing pressure. This summer’s drought reduced expected wheat yields by a seventh. The Met Office says El Niño is likely to lead to a wet and stormy autumn, and a colder than usual winter. During the last El Niño winter, in 2023, fields were so sodden that the following year’s wheat harvest fell by a fifth.

Monbiot’s deeper worry is that the food system now looks like the banks before the financial crash: concentrated, efficient and without flexibility. In 2008, governments rescued the banks by issuing future money. “If the food system starts to go down,” he says, “you can’t bail it out by issuing future food.” Food security and self-sufficiency, at a governmental and household level, is one of the central issues of our uncertain times.

But there are two things I can say for certain: next year is going to be hotter. And your big shop, dearer.

- More Here

Meta Values - 54

Living through something is not the same as learning from it, and learning from it is not the same as consciously extracting and communicating the wisdom it contains.

Many don't learn even from painful experiences.

Learning without experiencing firsthand and by using wisdom from past humans is absolutely free and will save one’s life. 

This is "magic" behind the long survival of organic life on this planet.

 

How Did the English Build Such Beautiful Villages?

The materials tell another part of the story, because before railways and modern transportation made standardized construction products available across the country, builders relied heavily upon whatever stone, clay, timber, flint, straw, and other materials could be obtained locally.

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This created harmony almost automatically, because two cottages could be built a century apart by different craftsmen, with different windows and different proportions, while the shared material immediately connected them to one another and to the surrounding countryside.

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Stone cottages, rendered façades, timber-framed buildings, walls, gates, gardens, and small shops all sit close to the road, creating a street whose shape is defined by architecture rather than by the engineering requirements of traffic.

The buildings turn corners naturally, doors face the pavement, rooflines overlap, windows remain close enough to be noticed by someone walking past, and walls continue the line of the street wherever the houses pull away from it.

Walking through Lacock therefore feels very different from moving through a settlement where buildings sit behind large setbacks and roads occupy most of the space, because here the lane remains intimate enough for architecture to stay within the pedestrian’s field of vision.

The street becomes an outdoor room formed from cottages, walls, trees, hedges, and gardens.

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The English also became exceptionally good at using vegetation as architecture, with hedges forming walls, trees creating canopies, climbing plants softening cottages, and garden gates turning the transition between public lane and private home into a small architectural event.

A stone wall might continue from the side of a cottage before gradually disappearing beneath ivy, then become a hedge, then open through a wooden gate into a garden, allowing constructed and natural forms to flow into one another without a sharp boundary.

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The village never needs a dramatic ending because architecture simply gives way to landscape.

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William Morris, who knew and loved these places, put the deeper principle simply:

“Simplicity of life, even the barest, is not a misery, but the very foundation of refinement: a sanded floor and whitewashed walls, and the green trees, and flowery meads, and living waters outside; or a grimy palace amid the smoke with a regiment of housemaids always working to smear the dirt together so that it may be unnoticed; which, think you, is the most refined, the most fit for a gentleman of those two dwellings? … Now if the public knew anything of art, that is excellence in things made by man, they would not abide the shams of it; and if the real thing were not to be had, they would learn to do without… Have nothing in your houses that you do not know to be useful, or believe to be beautiful.”

And this may be the simplest answer to how the English built such beautiful villages, because they built churches where people worshipped, pubs where people gathered, cottages from the materials beneath their feet, walls around gardens, bridges across streams, and lanes along routes people already traveled, then allowed generation after generation to add another layer without completely erasing the one that came before.

- More Here

Beauty Of Probabilia - A Good sceptic Has More Than Doubt & Less Than Certainty

How tightly should you hold your beliefs? You face the question several times a week without noticing. Hold them too loosely and you don’t stand for anything, hold them too tightly and you become the sort of person no argument can reach. Most of us swing between the two and have no vocabulary for the ground in between. A Roman lawyer worked one out, under pressure far worse than anything we are likely to face, and it is still the best answer available.

The time is early 45 BCE, the place a villa near Lucrine Lake, northwest of Naples, in southern Italy. One of the most fascinating and controversial figures in the entirety of Western history, Marcus Tullius Cicero, is finishing writing a book that will be known as the Academica, where he explains the nature of sceptical philosophy and why it is so crucial not just to the advancement of human knowledge, but to living a good life.

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Which is why he is worth your time now. What follows is partly the story of a man, but mostly a usable method: a way of holding beliefs that is neither the performed humility of the professional doubter nor the armoured certainty of the person who has ‘done their own research’. Cicero had a name for it – probabilia – and I think you will find it very useful any time you have to assess the validity of a notion and make a decision about it.

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The difference shows in practice: a genuine sceptic, confronted with compelling new evidence, updates – whereas the contrarian does not: they shift goalposts, raise new objections, or ignore what they cannot explain, all while citing studies selectively and appealing to the handful of dissenting experts cherrypicking the evidence. Their doubt is not the productive Socratic doubt that opens enquiry but a rhetorical shield that forecloses it. These self-styled sceptics are in fact dogmatists, having reached their conclusions in advance and thereafter simply engaging in rationalisations.

On the other side, a certain strand of scientism has overcorrected by demanding certainty as the price of admission for any belief worth holding, deploying ‘follow the science’ as a conversation-stopper rather than an invitation to enquiry. That is not how real science works either.

Both pathologies make the same mistake: they treat doubt and certainty as the only two options. Neither leaves room for what we actually need: calibrated confidence, the ability to say This is the best available account, I will act on it, and I remain open to being wrong. What is missing, then, is a philosophy of provisional but committed belief, a philosophy Cicero already worked out, in Latin, 2,000 years ago.

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Cicero’s own formulation is plain. In the Academica, he separates his school from the dogmatists in a single sentence: others declare that certain things are settled and fixed, whereas ‘we hold many things to be probabilia, which we can easily follow but can scarcely affirm.’ Follow easily, affirm rarely. The entire method is encapsulated by that asymmetry.

The term probabilia does not mean ‘probable’ in the modern, quantitative statistical sense, but something closer to ‘worthy of approval’: a claim that has withstood scrutiny, coheres with the available evidence, and a reasonable person would assent to while remaining open to revision. It is enough to act on.

Nor is ‘probabilia’ a single, undifferentiated category, and this is the important part usually left out. Carneades graded impressions (ie, preliminary judgments) by how much scrutiny they had survived. At the first level, an impression simply strikes you as a bit more likely to be true than not. At the second level, it rises to being persuasive. At the third, it is ‘undiverted’, meaning that nothing else you believe conflicts with it. At the fourth and highest level, it is also thoroughly examined, essentially the result of scientific investigation. Trivial matters need only the first or second grade; a medical decision or a criminal verdict demands the third or fourth. What Cicero inherited and deployed, then, was not a generic mood of open-mindedness but a graded, checkable procedure.

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Cicero is explicit in the Academica: the wise person acts on the best available view, not because they are certain it is true, but because suspending all judgment would make life impossible. We are not the kind of people who think nothing is true, say the Academics; we are the people who say that some falsehoods are mixed in with truths, so similar to them that we have no sure criterion for judging, so we can assess only provisionally.

Strip away the Latin and the toga, and the Academic method comes down to a handful of habits, all of which can be practised, starting today:

• Say how confident you are, not just what you think. ‘I believe X’ is nearly useless. ‘I believe X, at about 7 out of 10, and here is what is holding back the other 3’ gives your interlocutor something to work with and gives you somewhere to move in the light of new evidence or better arguments.
• Grade your beliefs by how hard they have been tested. Most of what any of us believes sits at Carneades’ second level: it merely struck us as persuasive. Reserve confident action for what has been through the third and fourth levels.
• Separate your values from your empirical beliefs. That children should not go hungry is a value and may be held firmly. That a particular school-meals programme reduces hunger is an empirical claim, to be held only as firmly as the evidence warrants. Conflating the two is what makes people defend failing policies as though their integrity depended on it.
• State in advance what would change your mind. If nothing would, you do not have a belief, you have an identity.
• Argue the other side properly. The Academy’s signature exercise was arguing in utramque partem, on both sides of a question. Carneades notoriously did it in Rome in 155 BCE, defending justice one day and demolishing it the next. The point was not cynicism but that you do not truly know what you believe until you have built the strongest case against it yourself.
• Act anyway. The sceptic who treats uncertainty as a reason to postpone action has missed the point entirely. Uncertainty is the permanent condition of humanity, deciding effectively within that is a skill we can all benefit from.
• Treat being wrong as data rather than disgrace. If you follow these sceptical rules of thumb, you will change your mind from time to time – or at least ought to. The question is whether you experience that as humiliation or as the method working.

None of this was merely theoretical for Cicero. It informed every major decision of his life, including the most contested one.

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The letters he exchanged with his friend Atticus in the months before his death are the most instructive documents Cicero left. Knowing that Mark Antony wanted him dead, he debated whether to flee, whether to return to Rome, whether to trust the young Octavian (the future emperor Augustus): reasoning under uncertainty, weighing probabilities, deciding anyway. When he returned and resumed attacking Antony in the discourses known as the Philippics, he knew he might be signing his death warrant, but the probable good outweighed the probable cost. He was murdered on 7 December 43 BCE. His last words, to the legionnaire sent to kill him, were: ‘I am stopping here. Come over, soldier. There is nothing right about what you are doing, but at least make sure you cut off my head properly.’

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Second, democracy requires the willingness to be persuaded. Deliberation is possible only if citizens hold their political views as probabilia rather than as dogma, and the current political crisis is partly an epistemological one.

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The well-examined life is one of continuous calibration and revision: strong-enough commitments to act with integrity, loose enough to update when the evidence changes.

This is countercultural in an age that prizes authenticity in the sense of having a fixed self, and urges us to own our identity. Cicero would have found that to be self-congratulatory. He changed his mind about friends, alliances and tactical judgments, and worried about it, as his letters make plain. But developing a crystallised self that never updates is not integrity: it is rigidity dressed up as character. The Ciceronian person is recognisable by a willingness to be embarrassed by earlier certainties, and by treating that embarrassment not as failure but as evidence that enquiry is working.

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The genuine sceptic is not the person who doubts everything, nor the person who demands proof before believing anything. It is the person who has learned to hold their best views with exactly the right strength of grip: firmly enough to act on them, loosely enough to let go when the evidence turns. That is what probabilia means. It is harder than dogmatism and harder than nihilistic doubt, but Cicero thought it the only honest stance available to a finite human being. He was probably right.

- More Here

Meta Values - 53

Early in life, visit all of these places (maybe schools should take kids as part of mandatory yearly trip):

• Slaughter House
• Hospital ICU 
• Hospice 
• Prison
• Cemetery and Cremation Center
• Poor neighborhoods
• Farms
• Water Processing Centers
• Aftermath of a natural disaster

Most of the above would cost zero dollars but will instill priceless wisdom and gratitude for life. 

The second and multiple order effects would be even more priceless - elimination of pain and suffering of animals, humans, food/water waste, and nature sans Anthropocene.