Chapter 1 is brilliant.
I've been shouting from the rooftops for years that math[0] courses need more context. We can prove X, Y, and Z, and this class will teach you that, but the motivating problem that led to our ability to do X, Y, and Z is mentioned only in passing.
We can work something out, and then come back and rework it in more generality, but then that reworking becomes a thing in and of itself. And this is great! Further advances come from doing just this. But for pedagogical purposes, stuff sticks in the human brain so much better if we teach the journey, and not just the destination. I found teaching Calculus I was able to draw in students so much more if I worked in what problems Newton was trying to solve and why. It gave them a story to follow, a reason to learn this stuff.
Kudos to the author for chapter 1 (and probably the rest, but chapter 1 is all I've had time to skim).
[0] And honestly, nearly every subject.
Be careful.
In undergrad, I worked as a math tutor at their tutoring institute. We had two calculus classes - one for engineers/science and the other for business/economics.
If you're into Newtonian stuff or it's relevant to your degree, then your approach is all great. What I consistently saw is people in degrees like biology (or even industrial engineering) had a harder time learning because of those physics applications in the book. They came in with one problem: Having trouble with the calculus. And then they discovered they had two problems: To understand calculus they suddenly had to understand physics as well.
You'd get students who were totally adept at differentiating and integrating, but would struggle with the problems that involved physics. Yes, it is important to be able to translate real world problems to math ones, but it's a bigger problem when you don't care about that particular field.
Same problem with the business students. Since all the tutors were engineering folks, they had trouble tutoring the business students because their textbook was full of examples that required basic finance knowledge (and it really doesn't help that financial quantities are discrete and not continuous, adding to the tutors' confusion).
If you can find an application the student is interested in, then by all means, use that approach. For a general purpose textbook, though, it hinders learning for many students who don't care for the particular choice of application the book decided to use.
I got to see that yesterday with my daughter: her algebra book used PV=nRT as an example of joint proportion and asked some questions about it. She, having never seen the ideal gas law, was quite thrown by it.
That said, after we went through it and had a brief physics lesson, it worked quite well and I'm glad they used the example instead of just making something up -- but it required having a tutor (me) on hand to help make the context make sense.
I wonder how good ChatGPT would be as that tutor. You can ask it to “explain like I’m 12”
It's almost unbelievably good as a tutor. Still, you need to check everything it tells you. Treat verification as part of the lesson.
ChatGPT is an unbelievably bad tutor if what you want a tutorial about is even a little bit obscure (e.g. the answer you want isn't already included in Wikipedia). It just confidently states vaguely plausible sounding made up nonsense, and then when you ask it if it was mistaken it shamelessly makes up different total nonsense, and if you ask it for sources it makes up non-existent sources, and then when you look whatever it was up for yourself you have to spend 2x as long as you originally would have chasing down wrong paths and trying to understand exactly which parts ChatGPT was wrong about (most of them).
And that's assuming you are a very savvy and media literate inquirer with plenty of domain expertise.
In cases where the answer you want was already easily findable, ChatGPT still is wrong about a lot of it, and you could have more easily gotten a (mostly) correct answer by looking at standard sources, or if you want to be more careful tracking down their actually existing cited sources or doing a skim search through the academic literature.
If you ask it something in a topic you are not already an expert about, or if you are e.g. an ordinary high school or college student, you are almost certainly coming away from the conversation with serious misconceptions.
ChatGPT is an unbelievably bad tutor if what you want a tutorial about is even a little bit obscure
That has absolutely not been my experience at all. It's brought me up to speed in areas from ML to advanced DSP that I'd been struggling with for a long time.
How long has it been since you used it, and what did you ask it?
I have tried asking it all sorts of questions about specific obscure word etymologies and translations, obscure people's biographies (ancient and modern), historical events, organizations, academic citations, mathematical definitions and theorems, physical experiments, old machines, native plants, chemical reactions, diseases, engineering methods, ..., and it almost invariably flubs every question I throw at it, sometimes subtly and sometimes quite dramatically, often making up abject nonsense out of whole cloth. As a result I don't bother too much; I've found it to waste more time than it saves. To be fair, the kinds of questions I would want a tool like this to answer are usually ones I would have to spend some time and effort hunting to answer properly, and I'm pretty fast and effective at finding information.
I haven't tried asking too much about questions that I could trivially answer some other way. If what you want to know can be found in any intro undergrad textbook or standard dictionary (or Wikipedia), it's plausible that it would be better able to parrot back more or less the correct thing. But again, I haven't done much of this, preferring to just get hold of the relevant dictionary or textbook and read it directly.
I'll give you an example. I just now asked chatgpt.com what Lexell's theorem is and it says this:
This gets the basic topic right ("is a result in geometry related to spherical triangles", involves area or spherical excess) but everything else about the answer, starting with the mathematician's identity, is completely wrong.
If I tell it that this is incorrect, it repeats a random assortment of other statements, none of which is actually the theorem I am asking about. E.g.
or
If you want to know what Lexell's theorem actually is, you can read the Wikipedia article I wrote last year: https://en.wikipedia.org/wiki/Lexell%27s_theorem
The problem ChatGPT has is that it's not able to just say something true but incomplete such as "I'm not sure what Lexell's theorem is or who Lexell was, but I know the theorem has something to do with spherical trigonometry; maybe it could be found in the more comprehensive books about the subject such as Todhunter & Leathem 1901 or Casey 1889".
Instead it just authoritatively spouts one bit of nonsense after another. (Every topic I have ever tried asking it about in detail is more or less the same.) The incorrect statements range from subtly wrong (e.g. two different things with similar names got conflated and some of the properties of the more common one were incorrectly applied to the other) to complete nonsense (jumbles of technical jargon strung together that are more or less gibberish). It's clear if you read carefully about any technical topic that it doesn't actually understand what it is saying, and is just combining bits of vaguely related material. Answers to technical questions are almost never entirely technically accurate unless you ask a very standard question about a very basic topic.
Anyone using it for any purpose should (a) be already pretty media literate with some domain expertise, and (b) be willing to carefully verify every part of every statement.
Can't argue with that. Your earlier point is the key: "e.g. the answer you want isn't already included in Wikipedia." Anything specialized enough not to be covered by Wikipedia or similar resources -- or where, in your specific example, the topic was only recently added -- is not a good subject for ChatGPT. Not yet, anyway.
Now, pretend you're taking your first linear algebra course, and you don't quite understand the whole determinant thing. Go ask it for help with that, and you will have a very different experience.
In my own case, what opened my eyes was asking it for some insights into computing the Cramer-Rao bound in communications theory. I needed to come up to speed in that area awhile back, but I'm missing some prereqs, so textbook chapters on the topic aren't as helpful as an interactive conversation with an in-person tutor would be. I was blown away at how effective GPT4o was at answering follow-up questions and imparting actionable insights.
A problem, though, is that it is not binary. There is a whole spectrum of nonsense, and if you are not a specialist it is not obvious to figure out the accuracy of the reply. Sometimes by chance you end up asking for something the model knows about for some reason, but very often not. That is the wrong aspect of it. Students might rely on it in their 1st year because it worked a couple of times and then learn a lot of nonsense among the truthy facts LLMs tend to produce.
The main problem is not that they are wrong. It would be simpler if they were. But then, recommending students to use them as tutors is really not a good idea, unless what you want is overconfidently wrong students (I mean more than some of them already are). It’s not random doomsayers saying this; it’s university professors and researchers with advanced knowledge. Exactly the people that should be trusted for this kind of things, more than AI techbros.
Things don't have to be incredibly obscure to make ChatGPT completely flub them (while authoritatively pretending it knows all the answers), they just have to be slightly beyond the most basic details of a common subject discussed at about the undergraduate level. My example before is discussed in at a wide variety of sources over the past 2.5 centuries, including books and papers by several of the most famous mathematicians in history, canonical undergraduate-level spherical trigonometry textbooks from decades ago, and several easy-to-find papers from the past couple decades, including historical and mathematical surveys of the topic. It just doesn't happen to be included in the training data of reddit comments and github commit messages or whatever, because it doesn't get included in intro college courses so nobody is asking for homework help about it.
If you stick to asking single questions like "what is Pythagoras's theorem" or "what is the most common element in the Earth's atmosphere" or "who was the 4th president of the USA" or "what is the word for 'dog' in French", you are fine. But as soon as you start asking questions that require knowledge beyond copy/pasting sections of introductory textbooks, ChatGPT starts making (often significant) errors.
As a different kind of example, I have asked ChatGPT to translate straightforward sentences and gotten back a translation with exactly the opposite meaning intended by the original (as verified by asking a native speaker).
The limits of its knowledge and response style make ChatGPT mostly worthless to me. If something I want to know can be copy/pasted from obvious introductory sources, I can already find it trivially and quickly. And I can't really trust it even for basic routine stuff, because it doesn't link to reliable sources which makes its claims unnecessarily difficult to verify.
It's going to give you the right basic explanation (more or less copy/pasted from some well written textbook), but if you start asking follow-up questions that get more technically involved you are likely to hit serious errors within not too many hops which reveal that it doesn't actually understand what a determinant is, but only knows how to selectively regurgitate/paraphrase from its training corpus.
You can get the same accurate basic explanation by doing a quick search for "determinant" in a few introductory linear algebra textbooks, without really that much more trouble; the overhead of finding sources is small compared to the effort required to read and think about them.
Are you sure it did? Or did it just convince you that you understood it?
If the code I wrote based on my newly-acquired insight works, which it does, that's good enough for me.
Beyond that, there seems to be some kind of religious war in play on this topic, about which I have no opinion... at least, none that would be welcomed here.
ML and DSP are both areas where buggy code seems to work, but actually gives suboptimal performance / wrong results. See: https://karpathy.github.io/2019/04/25/recipe/#2-neural-net-t...
So, I'm a professor, and I have, um, really strong opinions about this. :-) Perhaps too strong and long for the current forum. But I'll see if I can be brief.
ChatGPT is really, really good at providing solid answers of varying levels of detail and complexity to hyper-common questions such as those used in problem sets. This is one part of the skill set of a tutor, and it's a valuable one.
When I interview TAs for my classes, however, I actually put a lot more emphasis on a different skill: The ability to get into a student's head and understand where their conceptual difficulty or misunderstanding is. This is a very different skill, and it's one that ChatGPT isn't as good at, because we've gone from "maximum likelihood answers from questions that are in the middle of the distribution" into a wide range of possible sources of confusion, which the student may lack the words to explain in a precise way.
In the case of my kid, the PV=nRT question manifested as "I don't get it!" (with more exclamation points).
Asking ChatGPT (well, copilot, since I have institutional access to that, but it uses ChatGPT) to help understand the problem: It digressed and introduced Boyle's Law, threw in a new symbol "I" (ok, the 12 year old) had never seen for "proportional to", and ... in some sense just added to the cognitive overload.
The human approach was to ask a question: Have you ever seen this equation before? (No) Oh! Well, let's talk a little about gases..
Now, responding to ChatGPT and asking "No, that didn't help. Please ELI5 instead?" actually produced a much better answer: An analogy using a balloon. Which, amusingly, is exactly how I explained the behavior of gases to her.
But even here, there's a bit of a difference: In explaining it to her, I did so socratically:
"Ok, so imagine a balloon. If you heat the air inside the balloon, what happens?"
"Um, it gets bigger, right?"
"Yup, ..." (and now, knowing that she got that part, we could go on...)
That's something you can absolutely imagine trying to program around an LLM, but it's not a native way of interacting with it.
So ... I'd instead be a little more cautious here and say that ChatGPT potentially provides a really useful piece of what a human tutor offers, but it loses on the interactive exchange that helps much more rapidly zoom in on the source of confusion and correct it. Assuming that it's right.
I think that for a particularly sophisticated consumer, it can be more valuable, but it requires knowing what you don't know, in some sense: The ability to already isolate what you're confused by. Once you know the question to ask, ChatGPT can often provide it -- again, assuming that some quirk of its training or your phrasing doesn't cause it to generate an answer that's wrong in some way.
I wonder if you are getting very different results with Copilot relative to what I've seen lately from ChatGPT4o. It did a pretty good job with the successive "Explain Galois theory like I am 5/16/133" prompts that I tried earlier. Seems likely that the chat system prompt is different from Copilot's, but who knows how.
I will say that I have asked it to solve gas-law problems before, back when the base ChatGPT4 model was in use, and it didn't do so well there. It came up with the correct logic but I found it hard to get it to use consistent units, so the numbers kept coming out wrong.
In general I find it super useful at improving my conceptual understanding of things -- it's at least as good as a good human tutor, at levels up to and including fairly-advanced signal processing topics -- but the details are less reliable. Feeding the same question to Gemini is a good way to double-check the numbers in my experience. If they agree, the numbers are probably right, while if they don't, then it's likely they're both wrong.
Can you ask it to explain galois theory like he's 21?
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"Let’s imagine Évariste Galois survived that fateful duel and continued his work. What would he have done next? If Galois had lived past 21, he would have likely expanded and deepened his revolutionary ideas. By that time, he had already laid down the foundation of what we now call Galois theory, but there were still many directions he could have explored further.
Generalization of Galois Theory: Galois had just scratched the surface of the powerful connection between field theory and group theory. His next logical step might have been to generalize his ideas to more complex situations, such as understanding the symmetries of solutions to equations in multiple variables or extending his work to non-polynomial equations.
Exploration of Solvable Groups: Galois introduced the concept of solvable groups to determine whether a polynomial equation could be solved by radicals (using addition, multiplication, and roots). If he had lived longer, he would likely have delved deeper into classifying different types of groups and exploring their properties. This could have led to a more detailed theory of group structures, potentially anticipating parts of what we now know as modern group theory.
Linking Galois Theory with Other Areas: Galois might have explored connections between his theory and other areas of mathematics, such as geometry, number theory, or even early versions of what we now call topology. For instance, he could have investigated how the symmetries of geometric shapes relate to the symmetries of algebraic equations.
Developing Further Applications: Beyond theory, Galois could have sought applications of his work in more practical areas. Perhaps he would have applied his ideas to solving problems in physics, mechanics, or other emerging fields, where symmetry plays a crucial role.
Inspiring the Next Generation: If Galois had lived, he might have become a prominent teacher and mentor, influencing a new generation of mathematicians. His radical ideas would have challenged and inspired others to push the boundaries of mathematics even further. Galois could have spearheaded a new movement in algebra, similar to how Einstein revolutionized physics.
Refining His Work: Galois' original manuscripts were hastily written under dire circumstances. Given more time, he would likely have polished and clarified his ideas, making them more accessible to other mathematicians of the time. This might have sped up the acceptance and expansion of his ideas in the mathematical community."
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I'd give it a C for effort, I guess. I don't know enough about the topic to ask it for more specific predictions. At the end of the day it can only work with what it was trained with, so any genuine insights it comes up with will be due more to accident than inspiration.
I experienced that first hand as someone who just enjoyed math and had several courses from uni, and tried to help my SO and a few friends which struggled hard with different pre- or entry-level college math courses. They all needed quite different approaches to be able to understand the material.
For one I had to go all the way back and re-learn basic algebra as they had had a poor teacher which hadn't properly taught that. It would manifest in not understanding steps, not being able to solve equations properly and so on.
One really didn't get the visual graph explanation of derivation of composite functions, and instead got it by deriving the formula and using it in several examples. An approach which didn't work with the others as they needed the graph as a reference or motivation.
Was a very interesting experience, and as you say a very different challenge from just knowing the source material well.
Ah yes I know that exact problem. AoPS intro to algebra. That question was great for getting an intuitive sense of proportion, though I do remember one of the sub problems gave me some trouble.
That's the one! It was a very nice example. I suspect for some students they could ignore the physics, but daughter needed to walk through the physical interpretation of the components before getting into the math.
From my perspective as a tutor, it was a good use of time (gotta learn it some day anyway, and it provides useful physical intuition throughout life), but I could see it causing frustration if someone just wanted to learn algebra or didn't have a resource to turn to.
(Love those books. I went and asked all of my colleagues who had won teaching awards, what books they recommended, and all of them said aops)
I owe a great deal of my mathematical maturity to going through nearly every AoPS book published in middle and high school :)
Now imagine you were reading a mathematics textbook, and they use an example from something you really have no interest in (which for me would be finance). It includes terminology you have to learn - terminology that has no bearing on the math.
As discouraging it is to learn math without context, it's even more discouraging to learn it in a context you hate.
I have no interest in finance but I liked this book:
https://nononsense.gumroad.com/l/physicsfromfinance
If someone is having trouble understanding the absolute most basic part of classical mechanics (masses undergoing acceleration) something tells me they're not going to understand Calculus no matter what lens you view it through. It's not even about the equations, it's simply using it as a backstory for the motivation of the mathematics.
This seems absurd. Just because some people will find a particular application less interesting, I don't think the answer is to throw out ALL applications and turn it into a generic boring slog through which no one will be able to see when and how it's useful.
From my experience as a tutor, you are quite wrong in coming to that conclusion. Most people conflate mass and weight all the time, and have a fuzzy understanding of acceleration. It's not because they're thick in the head and incapable, but because they don't care. That doesn't mean they don't care about other applications where math can help.
And standard calculus textbooks go beyond what you are describing. All the ones I encountered would have the integral of force with displacement to get work (which confuses people when they hear it's the same as "energy").
The truth often does seem so.
This is spot on. Everybody has lots of favorite subjects. They're not all the same.
Conflating mass and weight is generally irrelevant in calculus textbooks, since they're generally giving you the mass, and weight doesn't even come up.
And coming in having a fuzzy understanding of acceleration is fine, because calculus is where you learn what acceleration is.
Learning that velocity and acceleration are the first and second derivatives is the most intuitive way to introduce them to anyone.
If you're taking calculus but you don't want to learn what acceleration is, then I don't know what you're even doing. Even if you're doing it for finance or medicine or something, velocity and acceleration are still the most useful and intuitive ways to introduce derivatives.
Most people don’t need or care to learn calculus. They just want to learn how to use calculus
Similar to how most people don’t need or care to learn statistics. They just want to learn how to say statistical words to silence critics of their pseudoscience.
Bingo. I rarely (if ever) used motivating examples from physics as test or homework problems. This was to ground calculus in reality somewhere. This was after years of wondering how to deal with the common student complaint of, "but why, where does this even come from?"
So I started telling them where it comes from.
The alternative is to ground it in philosophy and theoretical mathematics which would be even more abstruse...
You're missing the part that they may have taken zero classical mechanics classes. Even if it's something you learn in week 3 in that class, if you've never taken it you have no idea where those ideas fit in.
Why is someone studying science in college, if they haven't finished high school?
Your mileage may vary, but my industrial engineering program had two semesters of proper physics and two semesters of engineering mechanics.
That's me with statistics. I have 0 interest in gambling and sports, so most of the examples put me off instantly. I'll need to study this topic in the context of biology (bioinformatics), so I'll want to find a good stats learning guide which avoids those topics.
Do they explain anywhere why we consider radicals special and ask these questions about them in the first place? Nobody ever explains that. To my programmer brain, whether I'm solving t^3 = 2 or t^100 + t + 1 = 7 numerically, I have to use an iterative method like Newton's either way. ("But there's a button for Nth root" isn't an argument here - they could've added a more generic button for Newton.) They don't fundamentally seem any different. Why do I care whether they're solvable in terms of radicals? It almost feels as arbitrary as asking whether something is solvable without using the digit 2. I would've thought I'd be more interested in whether they're solvable in some other respect (via iteration, via Newton, with quadratic convergence, or whatever)?
I'd say one of the fundamental lessons of field and Galois theory is that they're not intrinsically special. They're just easier and more appealing to write down. (For the most part, anyway. They're a little bit special in some subspecialties for some technical reasons that are hard to explain.)
One reason to focus on them: when you tell students that x^5 - x - 1 = 0 can't be solved by radicals, that "even if God told you the answer, you would have no way to write it down", this is easy to understand and a powerful motivator for the theory. It's a nice application which is not fundamental, but which definitely shows that the theory has legs.
If you want to know which polynomials are solvable by Newton's method? All of them. It illustrates that Newton's method is extremely useful, but the answer itself is not exactly interesting.
But isn't this also true for generic quadratics/cubics too? Like the solution to x^3-2=0 is cubed_root_of(2), so it seems we can "write it down". But what is the definition of cubed_root_of(2)? Well, it's the positive solution to x^3-2=0...
When I say that a fundamental lesson of field theory is that radicals are not really special, this is what I mean. You are thinking in a more sophisticated way than most newcomers to the subject.
Oh sorry, right.
I feel there is an interesting follow-up problem here. The polynomials x^n+a=0 are used to define the "radicals" which is a family of functions F_n such that F_n(a) = real nth root of a = real solution of x^n + a = 0. Using these radicals you can solve all quadratics, cubics and quintics.
Now take another collection of unsolvable polynomials; your example was x^5 - x - 1 = 0 and maybe parameterize that in some way such that these polynomials are unsolvable. This gives us another family of functions G_n. What if we allow the G_n's to be used in our solutions? Can we solve all quintics this way (for example)?
Nope. Wanna hazard a guess what theory was instrumental in proving that there's no closed form for general quintics?
https://en.wikipedia.org/wiki/Galois_theory#A_non-solvable_q...
He is not wholly wrong, while not solvable in the general case by normal radicals, there is a family of functions, a special "radical" as he said, the Bring radical that solves the quintic generally. Of course as said its not a 5th root, but the solution to a certain family of quintics.
Fascinating, thanks!
I'm not sure I understand your question exactly, but I am fairly certain that not all quintic fields can be solved by the combination of (1) radicals, i.e. taking roots of x^n - 1, and (2) taking roots of x^5 - x - 1. I don't have a proof in mind at the moment, but I speculate it's not too terribly difficult to prove.
If I'm correct, then the proof would almost certainly use Galois theory!
Thanks! I should have made it more explicit that we would need some family of quantic equations, not just x^5 - x - 1. And looks like from another reply the answer is yes? https://en.wikipedia.org/wiki/Bring_radical#Solution_of_the_...
Bring radicals
Awesome reference, thank you!
Superb question! Interestingly, my math professor asked the exact same question after teaching Galois theory and stated that he does not have a good answer himself. Let me try to give sort of an answer. :)
We have fingers. These we can count. This is why we are interested in counting. This is gives us the natural numbers and why we are interested in them. What can we do with natural numbers? Well the basic axioms allow only one thing: Increment them.
Now, it is a natural question to ask what happens when we increment repeatedly. This leads to addition of natural numbers. The next question is to ask is whether we can undo addition. This leads to subtraction. Next, we ask whether all natural numbers can be subtracted. The answer is no. Can we extend the natural numbers such that this is possible? Yes, and in come the integers.
Now, that we have addition. We can ask whether we can repeat it. This leads to multiplication with a natural number. Next, we ask whether we can undo it and get division and rational numbers. We can also ask whether multiplication makes sense when both operands are non-natural.
Now, that we have multiplication, we can ask whether we can repeat it. This gives us the raising to the power of a natural number. Can we undo this? This gives radicals. Can we take the root of any rational number? No, and in come rational field extensions including the complex numbers.
A different train of thought asks what we can do with mixing multiplication and addition. An infinite number of these operations seems strange, so let's just ask what happens when we have finite number. It turns out, no matter how you combine multiplication and addition, you can always rearrange them to get a polynomial. Formulated differently: Every branch-free and loop-free finite program is a polynomial (when disregarding numeric stability). This view as a program is what motivates the study of polynomials.
Now, that we have polynomials, we can ask whether we can undo them. This motivates looking at roots of polynomials.
Now, we have radicals and roots of polynomials. Both motivated independently. It is natural to ask whether both trains of thought lead to the same mathematical object. Galois theory answers this and says no.
This is a somewhat surprising result, because up to now, no matter in which order we asked the questions: Can we repeat? Can we undo? How to enable undo by extension? We always ended up with the same mathematical object. Here this is not the case. This is why the result of Galois theory is so surprising to some.
Slightly off-topic but equally interesting is the question about what happens when we allow loops in our programs with multiplication and addition? i.e. we ask what happens when we mix an infinite number of addition and multiplication. Well, this is somewhat harder to formalize but a natural way to look at it is to say that we have some variable, in the programming sense, that we track in each loop iteration. The values that this variable takes forms a sequence. Now, the question is what will this variable end up being when we iterate very often. This leads to the concept of limit of a sequence.
Sidenote: You can look at the usual mathematical limit notation as a program. The limit sign is the while-condition of the loop and the part that describes the sequence is the body of the loop.
Now that we have limits and rational numbers, we can ask how to extend the rational numbers such that every rational sequence has a limit. This gives us the real numbers.
Now we can ask the question of undoing the limit operation. Here the question is what undoing here actually means. One way to look at it is whether you can find for every limit, i.e., every real number, a multiply-add-loop-program that describes the sequence whose limit was taken. The answer turns out to be no. There is a countable infinite number of programs but uncountably infinite many real numbers. There are way more real numbers than programs. In my opinion this is a way stranger result than that of Galois theory. It turns out, that nearly no real number can be described by a program, or even more generally any textual description. For this reason, in my opinion, real numbers are the strangest construct in all of mathematics.
I hope you found my rambling interesting. I just love to talk about this sort of stuff. :)
Thanks so much, I feel like you're the only one who grasps the crux of my question!
I think this is the bit I'm confused on - we have an operation that is a mixture of two operations, where previously we only looked at pure compositions of operations (let's call this "impure"). Why is it surprising that the inversion of an "impure" operation produces an "impure" value?
It's like saying, if I add x to itself a bunch, I always get a multiple of x. If I do the same thing with y, then I get a multiple of y. But if I add x and y to each other, I might get a prime number! Is that surprising? Mixing is a fundamentally new kind of operation; why would you expect its inversion to be familiar?
What is surprising and what not is always very subjective.
Now that I think more about it, one could argue that everything you can do with inverting radicals can also be done by inverting polynomials. So You could look at radicals as the step after multiplication and at inverting polynomials as the step after radicals. With this may depiction that these are two competing extensions falls apart a bit.
My chain of argumentation was that one could expect that there is a single natural ever growing set of "numbers" starting with the natural numbers, then integers, then rational numbers, then real numbers, culminating in the real complex numbers and ever set is a superset of the previous one. This is the "natural" order in which they are taught in school and somewhat mirrors how they historically were discovered. In retrospect, this is obviously not true. Just look at the existence of rational complex numbers. However, when all you have are natural, integer and rational numbers, it seems like it could be true.
Let me try a different way of explaining why it is surprising to some.
In school, I learned that I can solve quadratic equations by combining the inverse operations of the basic operations that make up the quadratic polynom. This seems natural as it worked for solving the linear equations I had seen so far. Inverse of combination is combination of inverse. At some point the teacher showed the formula for degree three. Cubic radicals appeared. We were overwhelmed by it's size but the basic operations used matched what we expected. The teacher said that degree 4 is even drastically larger with degree 4 radicals and we definitely do not want to see that, which is true. Nothing was said about degree 5 but it felt like it was implied that the pattern continues and the main problem with degree 5 is that our brains are just not able to handle the amount of operations that make up the formula.
Fast forward to university. Now the professor proves in the Galois theory course that, no, it's not that you are too stupid to handle degree 5. It's just that degree 5 cannot be handled this way at all. I am still unsure about whether my teacher in school knew that degree 5 is impossible or just assumed that he too is just too stupid.
I guess this mathematicians must have felt something similar back then. You learn about linear equations. All is easy and works. You learn about quadratics. After mixing in quadratic radicals, all is well again. You try to grasp cubics, and yes, with a lot of work this too can be learned. You think about quartics and after lots and lots of time come to the conclusion that yes it is possible but impossible to master the formula. It feels like the pattern should continue and the reason you don't have a quintic formula with degree 5 radicals is not because it does not exist but because of it's sheer size and just stating it would fill a whole book. Turns out, there is no such book.
Suppose you are a renowned mathematician back then who has failed for years to find a quintic formula. Now this teenager named Évariste comes along and fails too but says that it's not because he's too stupid but because it's impossible. At first, this does sound like an excuse of a lazy student, doesn't it?
Let's say you are not surprised that roots of degree 5 polynoms cannot be computed using just addition, subtraction, multiplication, division, and radicals. Does it surprise you that degree 4 polynoms can? Why does this work for degree 2, 3 and 4 yet fails for 5 and higher? I can see that one can argue that there is no reason to assume that it always works. However, at least learning the fact that it starts failing at degree 5 should be non-intuitive.
Yep!
You could think of it as a proof saying which polynomials are solvable by which algorithms. Solvable by radicals is one class of simpler algorithm and it so happens we have a cute proof as to when it will work or fail.
In a real problem you can have equations with parameters, where coefficients depend on some other parameters. And you should be able to answer questions about the roots depending on parameter values, like 'in which range of parameters p you have real (non-complex) roots'? Having a formula for the roots based on coefficients lets you answer such questions easily. For example, for quadratic equation ax^2 +bx + c = 0 we have D = b^2 - 4ac, and if D < 0 then there is no real (non-complex) roots.
Radicals are a "natural" extension in a certain sense, just as subtraction and division are. They invert an algebraic operation we often encounter when trying to solve equations handed to us by e.g. physics. I find it understandable to want to give them a name.
Why not something like "those things we can solve with Newton"? As you note Newton is broadly applicable; one would hope, given how popular the need to invert an exponent is, that something better (faster, more stable) if more specific than Newton might be created. It is hard to study a desired hypothetical operation without giving it a name.
On a related note, how come we don't all already have the names of the 4th order iterative operation (iterated exponents) and its inverse in our heads? Don't they deserve consideration? Perhaps, but nature doesn't seem to hand us instances of those operations very often. We seemingly don't need them to build a bridge or solve some other common practical engineering problem. I imagine that is why they fail to appear in high school algebra courses.
But regardless of whether you're a programmer, mathematician, machinist, carpenter, or just a kid playing with legos, there's always a good time to be had in the following way: first you look at the most complex problems that you can manage to solve with simple tools; then you ask if your simple tools are indeed the simplest; and then if multiple roughly equivalently simple things are looking tied in this game you've invented then now you get the joy of endlessly arguing about what is most "natural" or "beautiful" or what "simple" even means really. Even when this game seems pretty dumb and arbitrary, you're probably learning a lot, because even when you can't yet define what you mean by "simple" or "natural" or "pretty" it's often still a useful search heuristic.
What can you do with a lot of time and a compass and a ruler? Yes but do we need the "rule" part or only the straight-edge? What can we make with only SKI combinators? Yes but how awkward, I rather prefer BCKW. Who's up for a round of code-golf with weird tiny esolangs? Can we make a zero instruction-set computer? What's the smallest number of tools required to put an engine together? Yes but is it theoretically possible that we might need a different number of different tools to take one apart? Sure but does that really count as a separate tool? And so it goes.. aren't we having fun yet??
Lots of good answers here already, but I can also add my own perspective. Fundamentally, you're right that there isn't really anything "special" about radicals. The reason I personally find the unsolvability of the quintic by radicals interesting is that you can solve quadratics, cubics, and quatrics (that is, polynomial equations of degree 2, 3, and 4) by radicals.
To say the same thing another way: the quadratic formula that you learned in high school has been known in some form for millennia, and in particular you can reduce the question of solving quadratics to the question of finding square roots. So (provided you find solving polynomial equations to be an interesting question) it's fairly natural to ask whether there's an analogous formula for cubics. And it turns out there is! You need both cube roots and square roots, and the formula is longer and uglier, but that's probably not surprising.
Whether you think n'th roots are "intrinsically" more interesting than general polynomial equations or not, this is still a pretty striking pattern, and one might naturally be curious about whether it continues for higher degrees. And I don't know anyone whose first guess would have been "yes, but only one more time, and then for degree 5 and higher it's suddenly impossible"!
They aren't really special except that adjoining the solutions to a radical to a field make the associated group of automorphisms simplify in a nice way. Also at the time tables of radical roots where common technology and Newton's method was not. But fundamentally, we can define and use new functions to solve things pretty easily; if you get down to it cos() and sin() are an example of this happening. All those applied maths weird functions (Bessel, Gamma, etc) are also this. As well, the reason not to just use numeric solutions for everything is that there are structural or dynamic properties of the solutions of equations that you won't be able to understand from a purely numeric perspective.
I think taking a specific un-solvable numeric equation and deriving useful qualitative characteristics is a useful thing to try. You have cool simple results like Lyapunov stability criterion or signs of the eigenvalues around a singularity, and can numerically determine that a system of equations will have such and such long term behavior (or the tests can be inconclusive because the numerical values are just on the threshold between different behaviors.
That's one of the really fun things about taking Galois theory class - you get general results for "all quintics" or "all quadratics" but also you can take specific polynomials and (sometimes) get concrete results (solvable by radicals, but also complex vs. real roots, etc.).
The prime importance of solving by radicals is actually, that it led to the theory of groups! Groups are used in all sorts of places. (One nice pictorial example is fundamental group of a topological space). Just like Complex numbers arose in trying to solve the cubic. Also, the statement of Fermat's Last Theorem doesn't have any applications but its solution led to lot of interesting theory like how ideals get factorized in rings, elliptic curves, Galois representations...
BTW, the same theory can be extended to differential equations and Differential Galois Theory tells you if you can get a solution by composing basic functions along with exponentials.
Historically, radicals can be motivated by looking at people trying to solve linear, then quadratics, medieval duels about cubics and quartics, the futile search for solving quintics etc. Incidentally, quintics and any degree can have a closed form solution using modular functions.
More discussion on the MathOverflow page https://mathoverflow.net/questions/413468/why-do-we-make-suc...
I think these are two different questions: - Why care about radicals? - Why try to solve polynomial equations in terms of radicals?
For the first question:
Taking Nth powers is a fairly basic operation, which occurs all the time in mathematics. Taking Nth roots is simply the inverse operation, so it is fairly natural to be interested in it/having to deal with it.
For the second question:
Let’s pretend for a moment that we didn’t know how the quadratic formula looked like. Could we nevertheless say anything about it?
The quadratic formula is supposed to give us the solutions to the equation a x^2 + b x + c = 0. A special case of this general quadratic equation is x^2 - p = 0. There are two ways of solving this specialized equation: either by taking a square root, giving us the two solutions ±√p, or by using the general quadratic formula (with a = 1, b = 0, c = -p). Both of these approaches need to give us the same results, since they are both correct.
This tells us that if we simplify the quadratic formula with a = 1, b = 0, c = -p, then a square root needs to appear. How can this happen? Well, the most basic guess is that the quadratic formula contained at least one square root to begin with.
Looking at the actual quadratic formula tells us that this guess is correct: the formula uses the four basic arithmetic operations (addition, subtraction, multiplication, division) and a square root.
We can repeat the same thought experiment for cubic equations, and we find that the cubic formula should probably contain third roots. Looking up the formula confirms this suspicion. However, it should be noted that the cubic equation does not only contain third roots, but also square roots.
The situation for the quartic equation is similar: we suspect that the quartic formula contains fourth roots. And thanks to our experience with the cubic formula, we may also suspect that the quartic formula contains third roots and square roots. Looking up the formula, we see that it contains both third roots and square roots, but not (directly) any fourth roots. (Our original idea breaks down a bit because fourth roots can be expressed as iterated square roots. This makes it possible that the general quartic formula does not contain fourth roots, even though its simplified version will contain them.)
So what about a general polynomial equations of degree N >= 5? Our original observation tells us that a solution formula needs to contain some sort of operation(s) that, when the formula is applied to certain special cases, gives us Nth roots. Just as before, the most basic guess is that the formula will contain Kth roots, and the previous examples suggest that one should expect K = 2, ..., N to occur.
Summary: To find a formula for polynomials equations of degree N >= 2, we are forced to use additional operations apart from the four basic arithmetic operations. In certain special cases, these additional operations need to simplify to roots. This suggests using roots in the formula, and the cases N = 2, 3, 4 support this idea.
Heuristically speaking, we are not trying to use roots because we want to, but because they seem to be the bare minimum required to even hope of finding a formula.
You are right in the sense that solvability by radicals has no practical importance, especially when it comes to calculations.
It is just a very classical pure math question, dating back hundreds of years ago. Its solution led to the development of group theory and Galois theory.
Group theory and Galois theory then are foundational in all kinds of areas.
Anyway, so why care about solvability by radicals? To me the only real reason is that it's an interesting and a natural question in mathematics. Is there a general formula to solve polynomials, like the quadratic formula? The answer is no - why? When can we solve a polynomial in radicals and how?
And so on. If you like pure math, you might find solvability by radicals interesting. It's also a good starting point and motivation for learning Galois theory.
Typically when you're solving a polynomial equation in an applied context (programming, engineering, physics, whatever), it's because you have modeled a situation as a polynomial, and the information you really want is squirreled away as the roots of that polynomial. You don't actually care about the exact answer. You've only got k bits of precision anyway, so Newton's method is fine.
But we're not interested in the solution. We don't particularly care that t = approx. 1.016 is a numerical solution.[0] We're not using polynomials to model a situation. In mathematics, we are often studying polynomials as objects in and of themselves, in which case the kind of roots we get tells us something about polynomials work, or we are using polynomials as a lens through which to study something else. In either case it's less about a specific solution, and more about what kind of solution it is and how we got it.
Not to mention, specific polynomials are examples. Instead of t^100 + t + 1 = 7, we're usually looking at something more abstract like at^100 + bt + c.
[0] And in the rare case we actually care about a specific root of a specific polynomial, and approximate numerical solution is often not good enough.
Doesn’t your programmer brain want things to run as fast as possible?
If you have another weapon in your arsenal for solving polynomial equations, you have an extra option for improving performance. As a trivial example, you don’t call your Newton solver to solve a linear equation, as the function call overhead would mean giving up lots of performance.
Also, if you solve an equation not because you want to know the roots of the equation but because you want to know whether they’re different, the numerical approach may be much harder than the analytical one.
That’s fine. For that, you read up on the theory behind various iterative methods.
I think it's quite natural to ask oneself how far one can go applying the same method that is used for solving the equation x² = 4.
I imagine it has something to do with the fact that there is a somewhat simple pen-and-paper procedure, quite similar to long division, for calculating square/cubic/etc. roots to arbitrary precision, digit by digit. So finding roots is, in a sense, an arithmetic operation; and of calculus one better not speak in a polite society.
Hot take:
"18th Century" mathematics was intiuitive and informal, to the point that it was inconsistent.
The 19th and 20th Centuries added rigor and formalism (and elitism) and devalued intuition, to the point that it begame uninterpretable to most.
The 21st Century's major contribution to mathematics (including YouTube! and conversational style writing) was to bring back intuition, with the backing of formal foundations.
Elements is formal, right?
Pretty much. According to someone on Stack Exchange, "The most serious difficulties with Euclid from the modern point of view is that he did not realize that an axiom was needed for congruence of triangles, Euclids proof by superposition is not considered as a valid proof." But making mistakes in a formal treatment of a subject does not negate the fact that it is a formal treatment of the subject IMHO, and AFAICR none of the theorems in Elements is wrong; i.e., Elements is formal enough to have avoided a mistake even though the axioms listed weren't all the axiom that are actually needed to support the theorems.
18th Century European math was much more potent than ancient Greek math, and although parts of it like algebra and geometry were, for a long time, most of it was not understood at a formal or rigorous level for a long time even if we accept the level of rigor found in Elements.
Isn't how formal or rigorous something is just a social convention? Grammer Nazi's used to make online speech be formal with perfect rigor. Isn't it all relative to what your society defines?
No. That's the colloquial definition of formal. In mathematics, the word formal refers to something more specific: one or more statements written using a set of symbols which have fully-defined rules for mechanically transforming them into another form.
A formal proof is then one which proceeds by a series of these mechanical steps beginning with one or more premises and ending with a conclusion (or goal).
Both are a formality based on whats in fashion. I like the Axiom of Choice and not taking Math or words as literal or biblical truth.
If you're a formalist in philosophy of math, then math is neither true nor false, it's merely a bunch of meaningless symbols you transform via mechanical rules.
To an extent. A truly completely formal proof, as in symbol manipulation according to the rules of some formal system, no. It's valid or it isn't.
But no one actually works like this. There are varying degrees of "semiformality" and what is and isn't acceptable is ultimately a convention, and varies between subfields - but even the laxest mathematicians are still about as careful as the most rigorous physicists.
Elements is mostly formal, but it's also concrete and visual.
Euclid developed arithmetic and algebra through constructive geometry, which relies on our visual intuition to solve problems. Non-concrete problems were totally out of scope. Even curved surfaces (denying the parallel postulate) were byond Euclid. Notably, Elements didn't have imaginary or transcendental numbers. Euclid made no attempt to unify line lengths and arc lengths, and had nothing to say about what fills the gaps between the algebraically (geometrically!) constructible numbers.
Elements has pi.*
*It proves the ratio of a circle's area to the square of its diameter is constant.
But doesn't say how large that constant value is.
Did he even know those gaps existed? Euclid lived around 300BC. The problem of squaring the circle had been proposed around two centuries before that (https://en.wikipedia.org/wiki/Anaxagoras#Mathematics), but I don’t think people even considered it to be impossible by that time.
Incidentally, here's one beautiful edition:
https://farside.ph.utexas.edu/Books/Euclid/Elements.pdf
I like your idea.
I was schooled in abstract 20th century math - indeed YouTube is the opposite, and it’s a good thing.
One of my math teachers was once talking to Jean Dieudonné https://en.wikipedia.org/wiki/Jean_Dieudonn%C3%A9
who was part of the Bourbaki group and asked him why on earth he insisted on inflicting raw dry theory to the world with no intuition , when his day job involved drawing ideas all day long !
Edit: interestingly enough, one of my colleagues thinks very strongly that intuition should not be shared, and the path to intuition should be walked by everyone so that they ´ Make their own mental images ´. I guess that there’s a tradeoff between making things accessible, and deeply understood, but I don’t know what to make of his opinion.
Do you know how Dieudonné answered?
On foundations we believe in the reality of mathematics, but of course, when philosophers attack us with their paradoxes, we rush to hide behind formalism and say 'mathematics is just a combination of meaningless symbols,'... Finally we are left in peace to go back to our mathematics and do it as we have always done, with the feeling each mathematician has that he is working with something real. The sensation is probably an illusion, but it is very convenient.
The ‘make thier own mental models’ vs sharing/providing full information is difficult.
‘Make thier own model’ of the domain can lead to deeper understanding but takes time and may lead to different (possibly incorrect) understanding of the issues and complexities. If not reviewed with others.
Providing full information upfront to a person can be quicker but lead to a superficial knowledge.
I think that it comes down to whether that deeper knowledge is directly needed for the main task. Can I get by with an superficial (leaky) abstraction and concentrate on the main job.
If the objective is to advance mathematics instead of making it accessible, then this is a somewhat reasonable position. The mathematical statements that a person can come up with is often a direct product of their mental image. If everyone has the same image, everyone comes up with similar mathematical statements. For this reason you want to avoid that everyone has the same picture. Forcing everyone to start with a clean canvas increases the chance that there is diversity in the images. Maybe someone finds a new image, that leads to new mathematical statements. At least that's the idea. One could also argue that it just leads to blank canvases everywhere.
On the subject of more context in math, I've always wondered if having a grasp of the history of math would be helpful in getting better at solving mathematical problems. i.e. would learning more about how math developed over time, and how people solved important problems in the past, help me in trying to solve some other problem today?
Years ago I bought the 3-volume set "Mathematical Thought from Ancient to Modern Times", but never had the time to get past the first few chapters. I'd be interested in any recommendations for math history tomes like that.
Not a book, but FWIW, I've enjoyed a few videos from Norman Wildberger's "Math History" playlist[0]. Interestingly, he has a unconventional view of infinite processes in mathematics, a point of view that used to be common about a century ago or so.
I'm sure knowing some amount of history is useful, but there must be a limit to how much of it is practically useful though.
https://www.youtube.com/watch?v=dW8Cy6WrO94&list=PL55C7C8378...
You're conflating a few things here.
Constructivists are only interested in constructive proofs: if you want to claim "forall x in X, P(x) is true" then you need to exhibit a particular element of x for which P holds. As a philosophical stance this isn't super rare but I don't know if I would say it's ever been common. As a field of study it's quite valuable.
Finitists go further and refuse to admit any infinite objects at all. This has always been pretty rare, and it's effectively dead now after the failure of Hilbert's program. It turns out you lose a ton of math this way - even statements that superficially appear to deal only with finite objects - including things as elementary as parts of arithmetic. Nonetheless there are still a few serious finitists.
Ultrafinitists refuse to admit any sufficiently large finite objects. So for instance they deny that exponentiation is always well-defined. This is completely unworkable. It's ultrafringe and always has been.
Wildberger is an ultrafinitist.
It's likely: I purposefully stayed loose about the "infinite processes" to avoid going awry. I do however remembered him justifying his views as such though: he's not going into details, but he's making that point here[0] (c. 0:40). I assumed — perhaps wrongfully — that he got those historical "facts" correct.
https://youtu.be/I0JozyxM1M0?si=IFdWcEWNeNKDid7t&t=39
I don’t mean to be pedantic (although it’s in keeping with constructivism) but in the case you describe, you don’t have to provide a particular x but rather you have to provide a function mapping all x in X to P(x). It may very well be that X is uninhabited but this is still a valid constructive proof (anything follows from nothing, after all).
If instead of “for all” you’d said “there exists”, then yes constructivism requires that you deliver the goods you’ve promised.
I've been going through that very video lecture series the last couple of weeks. Good stuff. And in the lectures he mentions a number of books. I looked a few up on Amazon, and then looked at the associated Amazon recommendations, and so far have this small list of books related to Maths history that look worth reading:
There is also a "thing" in mathematics that is sometimes called the "genetic approach" where "genetic" is roughly equivalent to "historical" or maybe "developmental". IOW, a "genetic approach" book teaches a subject by tracing the development of the subject over its history. One popular book in this mold is:Stillwell's book is incredible, it hits the right balance between a textbook and a (advanced) lay person introduction to a huge range of topics.
Wildberger is a crank
A crank who provides hundreds of hours of fairly decent mathematical education content free of charge; it's not because he harbours unusual/fringe opinions that he's altogether worthless…
When I was an undergrad doing the mandatory measure theory course, I stumbled on a super old book(pre-1950 if I remember correctly, library card showed 3 people taking it out in the last 10 years) ,the name of which I forgot, that basically "re-built" the process that Lebesgue/Caratheodory/Riemann/etc followed, the problems they encountered (i.e Vitali set), why Lebesgue measure was the way it was and so on.
I really wish I could remember the name of the book, but it made so much more sense than how even something like Stein Shakarchi or Billingsley, which introduced measures by either simply dumping the Vitali set on you as the main motivation or just not really explaining why stuff like outer measure/inner measure made sense.
I found it helpful in some of my University math classes when I actually took the time to read the biographies that some of the textbooks included, the classes when I skimmed past them I did not remember details of the proof formulae for. But I don't know if this says more about the aid of history to the process of remembering or about my a priori interest in the topic for those particular classes.
For a really good example of integrating the history along with the mathematics, and much more accessible than those math texts, I would recommend "Journey Through Genius" by Dunham[0]. It may be a little dated (published in 1990) and its focus is limited to algebra, geometry, number theory, and the history is perhaps too Western-biased, but it's good and it's short. Its material would make a solid foundation to build on top of because, in addition to the historical context, it shows a lot of the thought process into approaching certain landmark problems.
[0]: https://www.goodreads.com/en/book/show/116185
Boyer/Merzbach "A History of Mathematics" [0] is a tome in that vein. It spends a lot of time discussing the (mind-boggling, to a modern-mathematics-educated reader) methods that ancient peoples used to do real math (e.g. for engineering) as a way to motivate the development of the features of modern symbolic mathematics.
[0] https://www.wiley.com/en-us/A+History+of+Mathematics%2C+3rd+...
When I took real analysis, I didn't get an intuitive sense of limits and all that delta/epsilon stuff. It was too abstract. Strangely enough, when I read David Foster Wallace's "Everything and More" on the history of infinity it all made more sense, because he described all the paradoxes and dead ends that mathematicians had run into before Cauchy and others brought rigor to the problem. Wallace was a postmodern novelist. I don't know why he described infinity so clearly; I didn't enjoy his other work.
Interesting... This ε-δ stuff is needed when you talk about functions, but when you start instead with the limits of numeric sequences, which are easy to grasp, the function limits come easy as well, because of the analogy between the ways these are defined.
It’s not so much that I didn’t understand the principles behind continuity. It’s just that the way my teachers presented the material lacked historical context.
After a BSc in pure math I discovered that I enjoyed applied math and CS much more, which told me that I need concrete examples to understand a theory: if you tell me about abstractions like groups and rings, which took years to establish, I lose interest. Tell me that groups express properties of matrix multiplications, or permutations, or modular arithmetic, and I’ll get it right away.
It’s the way my mind works, but I’m sure I’m not alone, and mathematical pedagogy would benefit from historical context.
The maths appendices in Infinite Jest are about half the book. There's a section written by Hal's friend (?), proving something like the intermediate value theorem where he says something like: we'll use epsilon delta because it's mad fun to say.
Yeah!
You're probably already aware of https://betterexplained.com -- an amazing resource that exemplifies this same mindset.
I was not, thank you for this.
Chapter X is brilliant. Galois was very brilliant. He is always my #1 choice for ever of who, if anyone from history, I would like to have lunch with.
He was introduced to me by my 5th grade calculus teacher Mr Steven Giavant, PhD. Galois bridged several disciplines and invented a new area of math.
Most unfortunately he met his end extremely tragically.
https://en.wikipedia.org/wiki/%C3%89variste_Galois
5th grade calculus!?
A slightly different perspective: It used to be common for professors to teach a lot of history in graduate algebra courses. (Generation X and older.) I've talked to students from a few other schools who experienced this. None of us cared for it at the time. My teacher spent at least two class periods writing down the history on the board. We were all quite bored. But then again, we were in math grad school, so there was no need to try to motivate us.
Absolutely agree. When I was a kid, I took a MOOC (MITx 6.002x) because I was interested in EE. They said calculus was a prerequisite, which I didn't know yet, but I went ahead anyways; every time the course required calculus, I'd go read the relevant chapters in Strang's Calculus. Felt incredibly natural, and I ended up going through the rest of Strang just for fun- probably the best learning experience I've ever had, and I doubt I'd have ever done it without having the MOOC there to motivate the problem.