From "Teaching Myself Calculus at Sixty-Five/I was never a good math student, but I was determined to penetrate the mysteries of mathematics" by Alec Wilkinson, who got a book out of this project — “A Divine Language: Learning Algebra, Geometry, and Calculus at the Edge of Old Age.”
July 9, 2022
"What did I learn?... That mathematics is both real and not real. Like novelists and musicians, mathematicians produce thought objects..."
"... that have no presence in the physical world. (Anna Karenina is no more actual than a thought about Anna Karenina.) Like other artists, mathematicians also have the run of a world that others hardly or only rarely visit. For mathematicians, though, this territory has more rules than it does for others. Also, what is different for mathematicians is that all of them agree about the contents of that world, so far as they are acquainted with them, and all mathematicians see the same objects within it, even though the objects are notional. No one’s version, so long as it is accurate, is more correct than someone else’s. Parts of this world are densely inhabited, and parts are hardly settled. Parts have been visited by only a few people, and parts are unknown, like the dark places on a medieval map. The known parts are ephemeral, but also concrete for being true, and more reliable and everlasting than any object in the physical world.... An imaginary world’s being infallible is very strange. This spectral quality is bewildering, even to mathematicians. The mathematician John Conway once said, 'It’s quite astonishing, and I still don’t understand it, despite having been a mathematician all my life. How can things be there without actually being there?'"
From "Teaching Myself Calculus at Sixty-Five/I was never a good math student, but I was determined to penetrate the mysteries of mathematics" by Alec Wilkinson, who got a book out of this project — “A Divine Language: Learning Algebra, Geometry, and Calculus at the Edge of Old Age.”
From "Teaching Myself Calculus at Sixty-Five/I was never a good math student, but I was determined to penetrate the mysteries of mathematics" by Alec Wilkinson, who got a book out of this project — “A Divine Language: Learning Algebra, Geometry, and Calculus at the Edge of Old Age.”
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48 comments:
Herman Wouk interviewed Richard Feynman when he was writing his WWII novels. As they ended the interview, Feynman asked him if he knew Calculus. When he said he didn't, Feynman said, "You should. It is the language God speaks."
Whether or not math is real is an actively debated topic. The extreme position (which many scientists appear to hold) is that math doesn't just describe nature, but actually IS nature.
I have loved spending my retirement learning the math I never really learned in my working years, when I had to limit myself to that which I needed for my work. I am especially interested in general relativity. Am eagerly awaiting delivery of this book.
As a theater tech, I have to deal with math a lot.
Back in the early-1970s, I built a Radio Shack binary counter. When I showed it to people, they would usually say, "Why would anybody need to count in binary?" Then, in the late 1980s, companies started making moving lights that required you to count in binary to set the dip switches for the fixture's address. As the old saying goes, "There are 10 kinds of people: those who can count in binary and those who can't."
" by Alec Wilkinson, who got a book out of this project — “A Divine Language: Learning Algebra, Geometry, and Calculus at the Edge of Old Age.”"
Darn. Another book I should buy.
"Two objects and two objects are always four objects."
I can't believe this falsehood appeared in The New Yorker. Everyone knows that the correct answer is 5. And that men can give birth to babies. And a sitting Supreme Court Justice can't define what is a woman.
"What did I learn?... That mathematics is both real and not real. Like novelists and musicians, mathematicians produce thought objects that have no presence in the physical world.“
I’m not so sure about that. When Dirac wrote down the fundamental equation of quantum electrodynamics, he saw that it required the existence of antiparticles, in particular positive electrons. No such thing had ever been seen or even conceived of before, and Oppenheimer thought that this proved that Dirac was wrong. When some years later the positron was discovered, it had exactly the properties predicted by the Dirac equation. Positrons exist because math says they have to. The plethora of new elementary particles discovered in the 60s only began to make sense when it was realized that they are a reification of group theory, a branch of pure mathematics.
A mathematician can invent anything and is responsible for nothing.
Nonetheless, the elegance of calculus seems undeniable.
Even though I'm good at it, I never understood the enthusiasm for math. But to each his own. Personally, I find it like "How many angels can dance on a pin?" philosophy, a little goes a long ways, since it has no real connection to the real world. Unless you have to use it for some real life function. Which the vast majority of us do NOT. I had to take a year of HS Geometry and got a A, but what use is it? I've never met a single Hypotenuse in real life.
You don't even have to know how to add and subtract anymore. just plug the numbers in your Iphone or computer app and boom you got the answer. Remember all that fraction crap you learned in Elemtary School? Multiplying the Denominator by the Numerator? All completely worthless now.
Yes, math is fascinating and his title (Divine Language) points to one reason why. Perhaps Plato and the many, many commentaries dealing with his theories will be next on Wilkinson's list of subjects to explore.
While the poetry of his write-up has some appeal, for me the trouble begins with the notion that mathematical objects "have no presence in the physical world." That's a conventional view, to the effect that mathematics is a self-referential language formed from axioms, definitions and the like, and is concerned with what can be derived from them using agreed-upon formal methods. (It has its roots in a Platonic notion that the physical world is but the shadow, leaved with a bit of Wittgenstein's theory of language more generally.) And that description may well coincide with how it's practiced by professional mathematicians. But if that's your understanding of mathematics, it's very hard to account for the fact that mathematics has been so successful in describing, indeed leading the way in understanding, the natural world. I think anyone subscribing to the 'self-referential language' view of mathematics ends up explaining its utility in understanding the world either in an extreme Kantian way (the order of the natural world is a construct of how we observe it) or, more simply, as just a matter of dumb luck. Neither view being particularly satisfying, you get back to that notion of "divine language," rationality being found at the very core of existence.
Much to think about. I think I'll give his book a read and see what he does with all of this.
I was just reading an article about mathematicians criticizing what they see as an undue emphasis on calculus for advanced high school math. They propose more focus on statistics and other advanced computation as the foundation for college study.
Lautreamont
Geometry! Grand trinity! Luminous triangle! He who has not known you is a dolt! ... In bygone days as in modern times, more than one great mind saw its genius awe-stricken on contemplating your symbolic figures traced upon fiery paper and living with a latent breath like so many mysterious signs not understood by the vulgar and profane, signs merely the brilliant revelation of eternal axioms and hieroglyphics pre-existent to the universe, and which will outlast it.
I can relate to this in a small way. When I was an undergraduate, my university required me, as a humanities major, to have so many credit hours of hard science (I think they call it STEM now), plus 2 years of a foreign language and a bunch of credits in physical education. My how times have changed!
I was a mediocre math student in high school, but I decided in college to take a 5 credit hour course in analytical mathematics, which was basically an intense pre-calculus course, covering advanced algebra, geometry, trigonometry, and some other things. Classes were every day, Monday through Friday, for 50 minutes each. I threw myself into the work because I knew I could never get behind, as each new day built on the preceding day's study. I ended up loving the class because of how logical and precise it was, compared to the humanities, and I ended up with an A.
My wife is a computer scientist. She occasionally reads mathematical textbooks for enjoyment. She recently finished a textbook on Differential Equations. When I looked at the book it might as well have been written in Sanskrit.
Hey, watch it with that 'old age' crap...
With differential equations, you can rule the world.
I was told there would be no math.
Back in the Silurian, when I could spend my lunch hour browsing among the thousands of magazines and periodicals the library subscribed to, I came across "The American Atheist," which I think was a monthly. It was a product of La O'Hair and her good son, and was a very interesting item for a lot of reasons, one being that they published columns and contributions from atheists of most political persuasions--from traditional conservatism sans religion on the right, to their own brand of Neostalinism. Big Tent Atheism. (She was herself as racist as the day is long, but nobody's perfect.)
Anyway, one contributor's self description was a quotation from one of her parochial school teachers, to the effect that "Miss Soandso is impervious to the truths of revealed religion and higher mathematics alike." Sounded like my kind of gal.
One of Mencken's odder prejudices was that mathematicians just sit around and make stuff up, the more opaque the better. Thought the whole field was a scam, no better than theology. Go figure.
@Original Mike
Differential forms are neat.
The thing about modern mathematics is that you spend years learning a language that hardly anyone speaks, and even mathematicians concentrating on different parts of the subject can barely speak to each other. The age of the universal mathematician passed long ago.
Calculus is not deep enough. Measure Theory is the real deal.
Why are we discussing this when we have been told maths are racist.
math doesn't just describe nature, but actually IS nature.
Feynman said..
Physics is just applied Mathematics.
Chemistry is just applied Physics.
Biology is just applied Chemistry.
Psychology is just applied Biology.
Sociology is just applied Psychology.
Economics is just applied Sociology.
Politics is just applied Economics.
Partial differential equations show how there can be a heartbeat when there is no heart.
keywords: autocatalytic reactions, dissipative structures
This is great. Inspirational. Because of life trauma, etc. I didn't do well in high school, bad grades more from doing very little homework. Did really well in geometry with very advanced proofs requiring dozens of steps, but just didn't have the momentum. My favorite class was physics, and one of the few I tried in, but because I didn't keep up with my math progress, it wasn't something I could pursue in college.
I have long thought if ever I get time and space in life to just study something for its own sake again, I'd love to teach myself more advanced math. This gets me to thinking it's never too late.
I have been on a similar quest since the beginning of last year. My math education in high school was limited, mostly because I was a lazy SOB. I took all the classes, 2 years of algebra, a year of geometry, and a year of trigonometry, but I didn't really do more than the minimum to get the A in the class. I paid for this laziness when I took Calculus I and II my freshman year of college- I struggled to not fail because my math background was much weaker than pretty much all the other people in the class, and that struggle made me even more apathetic at the time- I started not going to class etc. I think I ended up with a B the first semester and C- the second, and the latter was probably a gift from a kind professor.
That Summer between my freshman year and my sophomore year, I went back to the beginning to teach myself higher mathematics. My father at some point around the time of my birth had purchased almost all the Schaum Outline series on math that existed at the time- Plane Geometry, College Algebra, Trigonometry, Calculus, Differential Equations, Advanced Calculus, etc.- he at least 20 of them. Now, my father never had the time to actually work through them- he was busy feeding his growing family, but he kept them, and even gave them to me when I was around 8 or 9 years old, but I didn't take the hint. So, in the Summer of 1985, I started over with Plane Geometry, working every single problem in the book, then moved on to College Algebra, Trigonometry etc. I would finish one about every 3 months or so (I had jobs and other classes that took up more of my time). I eventually finished all the ones I listed above before I started graduate school 3 years later. I had planned to continue on, but I never did- I spent my full time studying chemistry at that point and never went back. Continued in next comment.
So, at the beginning of last year, I realized I had more time on my hands than I was using in any productive way, so I made a New Years resolution to pick up studying math again since I loved the first time. I decided to simply start where I had done so 36 years earlier with the exact same Schaum's Outline series which I had kept all those years, including my actual worksheets, too. I am working on Calculus right now, but doing so with an actual textbook that goes a bit further in detail and theory, but supplementing it with the Schaum's book. It has been interesting comparing my work today with that of 35 years ago. I often make the exact same initial mistakes and approaches to solving problems. My level of understanding today is definitely deeper than it was then, and part of that is the internet where I can go read about anything related I want about the subjects I am working at at the moment. I spend about 4 hours a day working on math- it went slower at first, but once I got into it, it wasn't a problem.
Now, unlike the author of the blog post piece, I have an aptitude for math- it has always been easy for me- what wasn't easy growing up was having the discipline to actually work at and not skip on the details that lead to deep understanding. I would say this to anyone who has a child who shows any math gifts (or any other types of gifts) to get them a tutor early, like at 8 or 9 years old at the latest to teach them how to learn on their own. I could have done all of this by the time I was 8 or 9 years old, and I wasted an embarrassing amount of time as preteen and teenager doing things other than learning and teaching myself how to self-direct my education.
"When Dirac wrote down the fundamental equation of quantum electrodynamics, he saw that it required the existence of antiparticles, in particular positive electrons."
As he himself admitted, Dirac actually thought the positive particle in his theory was the proton, which was the only positive particle known at the time. A boo-boo on par with Einstein's introduction of the cosmological constant for the "wrong reason". It was Weyl who called him in this; 'Hey Paul! You're own theory requires the particle to have the same mass as the electron.'
Mathematics is the study of what could be true. That is, what would not be self-contradictory.
Binary is great! Most people think that all you can count to on your fingers is ten (theoretically, twenty also using the toes). But in binary — using the simple finger up = 1, finger down = 0 convention, one can count up to 32 on 1 hand alone, and 1,024 using both hands.
I've used the method for many years for things like counting stretches of rest in orchestra or paces while walking (1,000 two-legged paces = 1 Roman mile). I know it so well that pretty much my fingers can do the progression of counting up with no need for mental attention (except when rolling over to the other hand, say).
Lately, though, I realized that I could count up — but not down. So I've been practicing that. Now I can also do an automatic countdown in binary on my fingers from 32 on down to 0 employing only one hand.
Rcocean wrote:
I've never met a single hypotenuse in real life.
A colleague of mine attended a conference at which one of the attendees remarked: "I'm here because I'm the only one in my office who can tell the difference between a hypothesis, a hypotenuse, and a hippopotamus."
Imaginary words and imaginary concepts aren't all that useful. But "imaginary numbers" ARE useful.
An "imaginary" number includes an impossible concept; the square root of -1. A "complex" number is partly real (numbers you can calculate with) and partly "imaginary", and the mathematics that describe a LOT of real-word concepts are "complex". For example, radio wave propagation.
Blogger Joe Smith said..."Hey, watch it with that 'old age' crap..."
He said, at the EDGE of old age.
It's a fair cop.
Readering said...
"I was just reading an article about mathematicians criticizing what they see as an undue emphasis on calculus for advanced high school math. They propose more focus on statistics and other advanced computation as the foundation for college study."
If you to analyze something use statistics. If you want to build something use calculous.
"If you to analyze something use statistics. If you want to build something use calculous."
:-)
In my experience, all math up to Analysis is just climbing the mountain. At that point you overlook the vista and need to specialize into your field, where math produces concrete results. You need to concern yourself with whether math is real or not real only in complex analysis.
Trig was awful, and statistics in college was pure hell. The rest wasn't so bad, but I wouldn't want to go through it all again. Numbers don't have much of a personality for me. I know that for some people 1729 is radically different from 1728 or 1730 -- maybe it's more playful or kind to its fellow numbers or maybe it's a real hell-raiser. More power to such people, but I'm not one of them.
Feynman said..
Physics is just applied Mathematics.
Chemistry is just applied Physics.
Biology is just applied Chemistry.
Psychology is just applied Biology.
Sociology is just applied Psychology.
Economics is just applied Sociology.
Politics is just applied Economics.
That is exactly what a mathematician (or a physicist) would say, but at each level new complexities are added that can't be completely explained by the prior, more fundamental science. Otherwise, non-mathematicians would be out of work and the world would be a very dull place for we innumerates.
They propose more focus on statistics and other advanced computation
That's been going on for ages. The idea is that statistics would be more useful, and that might motivate the student. I think statistics is actually more difficult than calculus, students have trouble computing probabilities, there are many distributions, and the null hypothesis is rather subtle. Like biology, there are a lot of specifics to learn and that isn't everyone's cup of tea. The up side is that with bootstrap methods the more theoretical parts can be replaced by simple computation.
Calculus is a decent approximation of whatever language God speaks, but we know that there are limits to what we can measure so we will likely never know the actual language of God, but I would not be surprised if it were not differentiable.
As a practical matter, a deep understanding of probability and statistics would serve most people better than a similar knowledge of calculus.
>>The extreme position (which many scientists appear to hold) is that math doesn't just describe nature, but actually IS nature.
Calling Pythagoras! Pythagoras, please answer on the white courtesy shellphone!
There was supposedly a Pythagorean cult in southern Italy around the sixth century BC that believed in the primacy of numbers as the foundation of the universe. IIRC, there's a story that the discovery that the square root of two is irrational (a fairly simple proof) caused Pythagoras to commit suicide. What would he have made of the much later proofs that e (a number he could not even have conceived) and pi (a number he was certainly familiar with) are not only irrational but "transcendental" (i.e., are not the solutions to any polynomial with rational whatchamacallits)? Let alone the "imaginary" numbers mentioned by Ken Mitchell?
It’s a slightly long and not very interesting story, but I wouldn’t be where I am today (and would almost certainly be in a worse economic position) if St. Ignatius, the premier Jesuit high school in Chicago (shout out to Dave Begley), survivor of the fire (which started half a mile away in Mrs. O’Leary’s barn a year or so after the school was originally built but moved in the opposite direction), and origin of Loyola University In Chicago) had not instituted a math contest for 7th graders when, coincidentally I was in 7th grade.
I’m a bit rusty these days, but math was my thing in high school and college, along with western Indo-European languages. Would not have balked at Sanskrit but never had the time or opportunity to pursue it. The Sanskrit speakers were great linguists but, ironically, some developments in Sanskrit obscured aspects of the original l/E language that were clearer in some of the western European languages.
It’s no great shakes but, for the last couple of years, I have been doing the daily challenges at Brilliant.org, which are math challenges of varying difficulty. Looking into their more advanced/paid offerings might be an option for those who want to explore the world of mathematics.
Finally, in my opinion, Euler’s identity is the most beautiful thing in the whole wide world. It unites in a simple equation the most fundamental numbers and operators in all of mathematics. There are two slightly different versions, and I’m somewhat torn about which I prefer.
--gpm
Whether we like it or not, my fellow humanities majors*, math (and Calculus in particular) does a much better and more accurate job of describing the world we live and move about in than any political construct or modern philosophical school of thought.
Feynman was dead on about Calculus, and cargo-cult science as well.
Just sayin'...
*OK, apologies to Bill Clinton for misstating his original line...
Engineering students don't have to worry about all this. They have to take 2 years of Calculus to meet regs per BET. So I don't really care what the other guys and girls do.
Let me recommend "Div, Grad, Curl, and All That". Multivariable calculus for anyone who stopped math after high school.
You can't learn statistics before you learn calculus, honestly, you can't learn it before you learn Lebesgue integration and measure theory. That is why there are no undergrad statistics programs.
The Pythagorean theorem can exist independently of Pythagoras, but Anna Karenina cannot exist independently of Tolstoy.....I was able to master, after much struggle, long division. After that I was lost....I take comfort, however, in the knowledge that at least part of the chaos of universe is capable of analysis using fixed, immutable principles that describes the properties of perfect bodies. I don't suppose this is an argument for the existence of God, but, on the other hand, it doesn't refute his existence. I have heard that quantum physics shows that dice play God with the universe. That's less comforting. Well, it's a big cosmos, and there's room for all types of eternal truths.
If I understand it correctly (and the probability is very high that I don't), Bitcoin is based on the agreement of people who understand math that the owners of each bitcoin have invested enough energy to power supercomputers to solve the equation which entitles them to ownership.
This fundamental agreement about wholly abstract concepts would seem to be a very shaky foundation for storing, investing and exchanging wealth, except that it is really no more abstract (and apparently far more rigorous) than the basis for all non-barter "money".
Most "money" rests on the foundation that at some level people trust other people. Which is perhaps why things are the way they are. Maybe relying on math is better.
Plato was unavailable for comment. Just think, with classical western education being abandoned, you can pass off all kinds of very old ideas as your own with no consequences.
"You can't learn statistics before you learn calculus, honestly, you can't learn it before you learn Lebesgue integration and measure theory."
That is true if you mean high-powered rigorous statistical theory. But only high-powered rigorous people need to do that. Normal people need to know the basics of how statistics relates to the real world. That there is a lot of randomness to the world but a lot of that randomness has patterns--and those patterns can often be calculated, e.g., how likely you are to toss five heads in a row or how common a straight flush is. That how much you trust statistical manipulation depends on how good the underlying data is. And as important, why you think it's legitimate to manipulate the data the way you do, e.g., how you calculate "the cost of living" or "the rate of inflation".
Call it Statistics for Citizens and have it in high school.
Blogger Fred Drinkwater said..."Let me recommend "Div, Grad, Curl, and All That". Multivariable calculus for anyone who stopped math after high school."
Yep, good book.
---That how much you trust statistical manipulation depends on how good the underlying data is.
Like 81 million. Very poor data quality, wouldn't you agree, Roger?
The article lost me in the top paragraph. First he hopes to get a book out of it; then he mentions casually that he had cheated his way to passing math grades in high school. I skimmed to the end, but there's really nothing that a person who seems proud of these facts has to teach me, was my reaction. No one else mentions it, no one found it bothersome?
Even so, I have really enjoyed reading these comments. Love all the math people and their super knowledge and frames of reference. One of the better post threads lately.
Feynman's chain of "is just applied" demonstrates a lack of humanity and perspective.
Reductionism on steroids.
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