Although a simple EE, I learned Galois Theory in college (coincidentally also in Edinburgh, although <other-university>). In 4th/final/senior year there were various elective classes including Advanced Mathematics which I chose as a kind of masochistic challenge. The class was very small, and it turned out taught by a "real mathematician" who commuted from the Mathematics department every day. Even though I've had a great deal of mathematics education I think this was the only time the teacher was someone who did mathematics all-in (as in he created new mathematics, published papers etc.) as opposed to someone who had the job of teaching some field (sic) in mathematics.
He taught Galois Theory using its application to coding theory for worked examples. That class was something of a turning point in my life to be honest.
I'd never think of constructing a heptagon again, for example. Definitely avoided Duels, and Montparnasse. Ok joking aside, it caused the proverbial lightbulb to turn on in my brain, and helped tremendously in my career later when I ran into folks trying to seem smart because they understood ECC or ZKPs.
It was like the extreme opposite of those people who say "I never used a single thing I learned in college".
> Ok joking aside, it caused the proverbial lightbulb to turn on in my brain, and helped tremendously in my career later when I ran into folks trying to seem smart because they understood ECC or ZKPs.
Presumably it helped to know that there was something beyond these folks' knowledge, but did it help in any more direct way?
The main benefit was to do with how I conceived mathematics as a subject:
In the before times, it appeared to be an infinite linear journey with subjects already studied in the set of "understood" things, and everything else in the "hard" set of things. Proceeding on the journey, subjects are expected to become harder and harder until eventually you're defeated and have to stop trying to learn more mathematics.
After that class, my conception of mathematics was of an extensively cross-linked tree of subjects where seemingly unrelated fields connect to each other, and where although the number of fields is large, it is not infinite, and with effort and time everything can be understood if required.
To be fair, that was the stated objective of the guy teaching the class. On day 1, he said "my objective is to teach you enough that you can understand whatever mathematics you need to understand in the future" and darn it he pretty much succeeded.
In summary, more of an attitude than specific knowledge of anything in particular.
Interesting question, and I don't remember. This was 40 years ago. I did find the final exam paper recently and marveled that at one time I was able to ace it. Notable in that there was a typo in one question that rendered it unsolvable. I was able to gain credit nevertheless by showing it to be unsolvable. I think that's the only case I remember where an exam had been printed with a major mistake.
Galois Theory by Ian Stewart is an excellent book indeed! I've got a hard copy lying at home that I am currently reading slowly page by page. I am planning to host book club meetups with this book later this year, perhaps during the winter if I am able to make good progress with this book.
In the meantime, if there is someone here who is interested in reading this type of books together and share updates with each other, I'd like to invite you to the IRC and Matrix channel named #bitwise [1][2] (the IRC and Matrix channels are bridged together, so you could join either one of them). The channel consists of some HN users as well as some users from other channels like ##math, ##physics, #cs, etc. It serves as an online space to share updates about mathematics and computation books you are reading and discuss their content.
Thanks for this! I tried joining a few other channels on Libera like #robotics and ##typetheory but they’re kinda dead. Usenet Newsgroups are also all spam. Will check it out.
I can agree with that, this is the primary textbook for the Open University's (excellent) Galois Theory MSc course (https://www.open.ac.uk/postgraduate/modules/m838. I really enjoyed my time on that course.
I was also very interested in reading about the the original papers prior to the various advancements in mathematical thinking and notation that tend to reframe how the theory is taught today. For that I highly recommend Peter Neumanns "The mathematical writings of Evariste Galois" (https://ems.press/books/hem/102) – it has the french side by side with a direct English translation along with notes explaining the context and possible thought process (it also served as a fun way to read some more French whilst I was trying to learn the language).
I hope it's OK to hijack the thread a bit: could you please tell me whether you'd recommend the Open University?
I think I might be interested in one of their Mathematics MSc. I don't know anyone personally who's gone so I'd love to hear what you thought of the experience please.
Yes, I would certainly recommend them if you are personally motivated. I've previously studied at Imperial for my BEng and then did a Mathematics BSc and then followed up with an MSc with the OU and certainly as a mature student, I have loved my time with the OU.
You get out of it very much what you put in. The materials on the undergrad side are excellent, at the MSc level you will basically be given a list of textbook resources, the odd OU prepared summary material and then a few online video tutorials (live) and then you get on with it.
There are forums and you have direct contact with your tutor though speaking myself, I never really engaged with them. You'll have a few "TMAs" to do (tutor marked assignments) and that's where you'll get direct feedback on your approach and questions but you are encouraged to message tutors if you need some guidance, and when I have I have always had a good experience.
My email is in my bio if you want to reach out, I'd be happy to send you some of the M838 course notes for you to get an idea of what you can expect.
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).
> 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.
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.
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.
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?
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.
> The “possible error surface” is large, logical (as opposed to syntactic), and very tricky to unit test. For example, perhaps you forgot to flip your labels when you left-right flipped the image during data augmentation. Your net can still (shockingly) work pretty well because your network can internally learn to detect flipped images and then it left-right flips its predictions. Or maybe your autoregressive model accidentally takes the thing it’s trying to predict as an input due to an off-by-one bug. Or you tried to clip your gradients but instead […]
> Therefore, your misconfigured neural net will throw exceptions only if you’re lucky; Most of the time it will train but silently work a bit worse.
Actually Karpathy is a good example to cite. I took a few months off last year and went through his "Zero to hero" videos among other things, following along to reimplement his examples in C++ as an introductory learning exercise. I spent a lot of time going back and forth with ChatGPT to understand various aspects of backpropagation through operations including matmuls and softmax. I ended up well ahead of where I would otherwise have been, starting out as a rank noob.
Look: again, this is some kind of religious thing where a lot of people with vested interests (e.g., professors) are trying to plug the proverbial dyke. Just how much water there is on the other side remains to be seen. But finding ways to trip up a language model by challenging its math skills isn't the flex a lot of you folks think it is... and when you discourage students from taking advantage of every tool available to them, you aren't doing them the favor you think you are. AI got a hell of a lot smarter over the past few years, along with many people who have found ways to use it effectively. Did you?
With regard to being fooled by buggy code or being satisfied with mistaken understanding, you don't know me from Adam, but if you did you'd give me a little more credit than that.
I'm not a professor and I don't have any vested interest in ChatGPT being good or bad. It just isn't currently useful for me, so I don't use it. In my experience so far it's basically always a waste of my time, but I haven't really put in that much work to find places where it isn't.
It's not a religious thing. If it suddenly becomes significantly better at answering nontrivial questions and stops confidently making up nonsense, I might use it more.
You are obviously experienced and have knowledge of advanced abstract topics.
For you using ChatGPT as a NLP and flawed search mechanism is fine and even more efficient than some alternatives.
Advocating that it would be just as useful and manageable by inexperienced young students with far less context in their minds is disingenuous at best.
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:
> Lexell's theorem is a result in geometry related to spherical triangles. Named after the mathematician Michel Léonard Jean Leclerc, known as Lexell, it states: ¶ In a spherical triangle, if the sum of the angles is greater than π radians (or 180 degrees), then the spherical excess (the amount by which the sum of the angles exceeds π) is equal to the area of the spherical triangle on a unit sphere. ¶ In simpler terms, for a spherical triangle, the difference between the sum of its angles and π radians (180 degrees) gives the area of the triangle when the sphere is of unit radius. This theorem is fundamental in spherical geometry and helps relate angular measurements directly to areas on a sphere.
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.
> [...] In a spherical triangle, if you have a spherical triangle with vertices A, B, and C, and the sides of the triangle are a, b, and c (measured in radians), then: ¶ cos(a)cos(b) + sin(a)sin(b)cos(C) = cos(c). [...]
or
> [...] In a spherical polyhedron, the sum of the angles at each vertex is equal to 2π radians minus the sum of the interior angles of the faces meeting at that vertex. [...]
> every spherical triangle with the same surface area on a fixed base has its apex on a small circle, called Lexell's circle or Lexell's locus, passing through each of the two points antipodal to the two base vertices.
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.
We could probably find a middle ground for agreement if we said, "Don't use current-gen LLMs as a tutor in fields where the answer can't be checked easily."
So... advanced math? Maybe not such a good idea, at least for independent study where you don't have access to TAs or profs.
I do think there's a lot of value in the ELI5 sense, though. Someone who spends time asking ChatGPT4 about Galois theory may not come away with the skills to actually pass a math test. But if they pursue the conversation, they will absolutely come away with a good understanding of the fundamentals, even with minimal prior knowledge.
Programming? Absolutely. You were going to test that code anyway, weren't you?
Planning and specification stages for a complex, expensive, or long-term project? Not without extreme care.
Generating articles on quantum gravity for Social Text? Hell yeah.
A statement I would support is: "Don't use LLMs, for anything where correctness or accuracy matters, period, and make sure you carefully check every statement they make against some more reliable source before relying on it. If you use LLMs for any purpose, make sure you have a good understanding of their limitations, some relevant domain experience, and are willing to accept that the output may be wrong in a wide variety of ways from subtle to total."
There are many uses where accuracy may not matter: loose machine translation to get a basic sense of what topic some text is about; good-enough OCR or text to speech to make a keyword index for searching; generation of acceptably buggy code to do some basic data formatting for a non-essential purpose; low-fidelity summarization of long texts you don't have time to read; ... (or more ethically questionably, machine generating mediocre advertising copy / routine newspaper stories / professional correspondence / school essays / astroturf propaganda on social media / ...)
But "tutoring naïve students" seems currently like a poor use case. It would be better to spend some time teaching those students to better find and critically examine other information sources, so they can effectively solve their own problems.
Again, it's not only old theorems where LLMs make up nonsense, but also (examples I personally tried) etymologies, native plants, diseases, translations, biographies of moderately well known people, historical events, machines, engineering methods, chemical reactions, software APIs, ...
Other people have complained about LLMs making stuff up about pop culture topics like songs, movies, and sports.
> good understanding of the fundamentals
This does not seem likely in general. But it would be worth doing some formal study.
> Anything specialized enough not to be covered by Wikipedia or similar resources [...] is not a good subject for ChatGPT.
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. Lexell's theorem, to take my previous example, is discussed in 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 the mid 20th century, 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. Even published work by professionals often contains factual errors, but when you read them you can judge their name/reputation, look at any cited sources, compare claims from one source to another, and so on. But if ChatGPT tells you something, you have no idea if it read it on a conspiracist blog, found it in the canonical survey paper about the topic, or just made it up.
> Go ask it for help [understanding determinants], and you will have a very different experience.
It's going to give you the right basic explanation (more or less copy/pasted from some well written textbook or website), 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 (and routinely picks the wrong source to paraphrase or mashes up two unrelated topics).
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 using the free version?
GPT 4 Turbo (which is paid) gives this:
> Lexell's theorem is a result in geometry related to triangles and circles. Named after the mathematician Anders Johan Lexell, the theorem describes a special relationship between a triangle and a circle inscribed in one of its angles. Here's the theorem:
Given a triangle \(ABC\) and a circle that passes through \(B\) and \(C\) and is tangent to one of the sides of the angle at \(A\) (say \(AB\)), the theorem states that the circle's other tangent point with \(AB\) will lie on the circumcircle of triangle \(ABC\).
In other words, if you have a circle that touches two sides of a triangle and passes through the other two vertices, the point where the circle touches the third side externally will always lie on the triangle’s circumcircle. This theorem is useful in solving various geometric problems involving circles and triangles.
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.
> The ability to get into a student's head and understand where their conceptual difficulty or misunderstanding is.
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.
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.
"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."
----------------
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.
This is like a forester extending their work to non-forests. The person can learn to do other things, but those things aren't in any way an extension of forestry.
> Exploration of Solvable Groups […] Linking Galois Theory with Other Areas
This doesn't say anything.
> Perhaps he would have applied his ideas to solving problems in physics, mechanics, or other emerging fields, where symmetry plays a crucial role.
Still isn't saying anything, but if I pretend this has meaning: he was born about a century early for that.
> he might have become a prominent teacher and mentor, influencing a new generation of mathematicians.
He's far more likely to have been a political revolutionary. By the time of his death, academia had excluded him about as much as was possible.
> Given more time, he would likely have polished and clarified his ideas, making them more accessible to other mathematicians of the time.
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)
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.
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.
> 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.
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.
> something tells me they're not going to understand Calculus no matter what lens you view it through.
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").
> Most people conflate mass and weight all the time, and have a fuzzy understanding of acceleration.
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.
> And coming in having a fuzzy understanding of acceleration is fine, because calculus is where you learn what acceleration is.
Really? Because I learned what acceleration was a few years before I'd been introduced to calculus.
> 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.
Your statement highlights very well the point I'm trying to make.
> 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.
I didn't point it out in my earlier comments, but I learned basic calculus in the 10th grade by two very simple (non-physics) concepts:
The derivative gives you the slope of the tangent (and I had already been taught a year prior that the slope of the tangent is the "point" rate of change). We'd already studied the relevancy to physics (or other applications) of getting the slope of the tangent in prior years (in physics courses, which is where one should be introduced to it). So I definitely did not need a math textbook to give me context.
And the integral gives you the area under the curve. I did not even need a physics application, as I'd done years of geometry up to that point to understand the concept of "area". Again, the application to things like energy was appropriately left to a further physics course.
BTW, I never said providing context in a math book is a bad idea - just that it'll help some people and hurt some people by the book's choice of context. If I were doing 1:1 tutoring, I would definitely try to provide context from the real world. The difference is that I can try to identify the relevant context for the particular student.
That alone was sufficient in motivating me to learn more. Of course, you can go from there to computing volumes, etc.
You're speaking from the perspective of how you happened to learn things.
>> 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.
> Your statement highlights very well the point I'm trying to make.
But you're missing the point I'm making. Which is that sometimes there is a simply a clearest way to explain a subject regardless of what a student is interested in.
Saying you want to learn calculus but you're not interested in acceleration is like saying you want to learn 20th-century European History but you're not interested in WWII.
I've done my share of teaching. I greatly appreciate that you need to make things relevant to students. But at the same time, you just have to teach what the thing is, using the time-tested analogies that actually work to educate students.
If a student doesn't want to learn calculus because they have no interest in what acceleration is, then I don't think they want to learn calculus at all.
I'm now learning German in Duolingo, and I noticed a correlation between how hard it is for me to learn one more pack of a vocabulary and how this vocabulary is relevant for me.
There is nothing that I cannot understand, but when some words do not relate to my needs I need to make a conscious effort to learn them. When the words are the ones I'd happily use, I'll learn them easily without any effort, I just need to spend enough time with Duolingo.
I believe that people who do not care about physics are having the very same issues with calculus that was explained with references to physics. Probably they can overcome this difficulties, like I can overcome my difficulties with uninteresting words, but it means they need to spend more effort and more time to get the same result.
> It's not even about the equations, it's simply using it as a backstory for the motivation of the mathematics.
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?"
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.
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.
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.
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.
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.
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.
> 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.
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:
Mathematics and Its History (Undergraduate Texts in Mathematics) 3rd ed. 2010 Edition by John Stillwell
The History of the Calculus and Its Conceptual Development (Dover Books on Mathematics) by Carl B. Boyer
A History of Mathematics by Carl B. Boyer
A Concise History of Mathematics: Fourth Revised Edition (Dover Books on Mathematics)A Concise History of Mathematics: Fourth Revised Edition (Dover Books on Mathematics) by Dirk J. Struik
Introduction to the Foundations of Mathematics: Second Edition (Dover Books on Mathematics) Second Edition by Raymond L. Wilder
Mathematical Thought from Ancient to Modern Times by Morris Kline (3 volume set)
The Calculus Wars: Newton, Leibniz, and the Greatest Mathematical Clash of All Time by Jason Socrates Bardi
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:
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.
> 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
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.
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.
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 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.
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.
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.
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)?
> To my programmer brain, whether I'm solving t^3 = 2 or t^100 + t + 1 = 7 numerically
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.
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.
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.).
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.
> "even if God told you the answer, you would have no way to write it down"
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.
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)?
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!
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.
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.
> 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,
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??
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.
> 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.
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.
> 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)
That’s fine. For that, you read up on the theory behind various iterative methods.
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.
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.
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!
> 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.
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.
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.
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.
> I've been shouting from the rooftops for years that math[0] courses need more context.
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.
"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.
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.
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.
> 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.
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 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.
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).
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.
> Euclid […] had nothing to say about what fills the gaps between the algebraically (geometrically!) constructible numbers
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.
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.
Galois theory is the explanation and apex of theoretical math that you can motivate and talk about at a dinner table with people that don't even like math, lol.
Start with the quadratic formula, everyone seems to have some recollection of this. Talk about solving for x in polynomials. Then discuss if you can always solve for x, and what does that even mean. If you graph a polynomial it crosses the x-axis so there's a solution for x, but does that mean you can solve for it in a formula (this alludes to the fundamental theorem of algebra that every polynomial of degree n has n solutions in the complex numbers)?
It's tough to get the idea of solution by radicals and how that relates to what it means to have a formula for x in terms of the coefficients of the polynomial.
Anyways, the punchline is that there's no formula for x using basic arithmetic operations up to taking radicals, where the formula is in terms of the coefficients of the polynomial for a general degree 5 or higher polynomial. Galois theory proves this.
Galois is credited with this because it took a lot of imagination to think about how to formulate and prove that there is no formula. What does it mean to not have a formula? How do you formulate it properly and then prove it?
This isn't quite right -- Abel proved that there's no quintic formula before Galois came along. Galois theory gives a whole lot more insight, lets you understand why some quintics do have solutions in radicals, etc., but Galois doesn't (or at least shouldn't) get credited for proving that there isn't a quintic formula, because he wasn't the first to do that.
Don't let the truth get in the way of a good story! hahahaha
But yeah you're right
edit: i don't recall Abel's proof, but Galois reformulation of what it means to be solvable by radicals, introducing the permutation group of the roots is the big thing in my mind.
In lay terms the best I can say is that for n greater than or equal to 5, the set of all possible permutations of n things is complicated. For n less than 5 the set of all possible permutations of n things is simple just because n is small. That's what leads to there being general formulas for n = 2, 3, 4.
Galois translated whether a polynomial has a solution for x in terms of the coefficients using algebraic operations up to using radicals into a property of the group of permutations of the roots of the polynomial. The property of the group is whether the group is solvable. For n greater than or equal to 5, the general permutation group on n objects is not solvable but for n less than 5 is is. There just are not that many permutation groups for n = 2, 3, and 4 objects and all these permutation groups are solvable. Generically a group is not solvable and so we see this with larger n.
I will always remember Galois theory as the punchline to my Abstract Algebra courses in college. Galois was a brilliant math mind, and I'm curious what else he would have contributed had he not died at 20 in a duel.
Great to see this course material public. It's a real missed opportunity though to not mention that Galois wrote a lot of it down staying up all night before being shot.
That's a common myth. See this paper referenced in the wikipedia article:
Rothman, Tony (1982). "Genius and Biographers: The Fictionalization of Evariste Galois". The American Mathematical Monthly. 89 (2): 84–106. doi:10.2307/2320923. JSTOR 2320923
Quite the clickbait. You can only access it from the pay site, or unless you can get a school library to access it, which I will do. Only the first page is available free.
I'd be genuinely curious to have a chat with similar minded people. They must see the world quite differently to be able to fork a new path in hard maths mostly on their own ..
A few years ago, I led a study group through A Book of Abstract Algebra by Charles C Pinter. It culminated in Galois Theory and was one of the best books I've ever used in a math study group.
I have fond memories of reading this book in college. I enjoyed it immensely.
I also remember reading the section at the back of the book about Galois. There was also an entertaining section about the history of solving the roots of polynomial equations and in particular solving equations of arbitrary order.
For non-math people, is this "simple Wikipedia" article about right? I've always seen Galois theory listed in mathematics courses and wondered what it is, speaking as a humble engineer.
https://simple.m.wikipedia.org/wiki/Galois_theory
Yes and no, it's a bit too simplistic and doesn't explain the actual "why" of Galois Theory, just the how. The brilliant insight Galois figured out is that there is a fundamental connection between fields and groups, but that is just the "technique" with which he solved the problem. The "why even bother" is a bit more complex but simply put Galois wanted to establish a criterion to determine what polynomials are solvable or unsolvable in which fields. E.g. we know x^2=-1 is solvable in C with x=i but not in real numbers. Can we generalize that proof to such a degree that we can mechanistically run it for arbitrary polynomials in arbitrary fields?
What it states is correct and it gives you a good overview over what you do in a Galois theory course. It does, however, not give you an idea of why this is interesting. When just reading that article one might get the idea that some mathematicians just had too much free time.
I tried to motivate the questions leading to Galois Theory in https://news.ycombinator.com/item?id=41258726 in a way that is hopefully accessible to more down-to-earth programmers and engineers.
I should probably add why I think the motivation is so important here. For pure engineers, numbers are a tool. They ask: What can I build with numbers? Pure mathematicians ask a different question. They are interested in the limits of numbers. They ask: What can I not build with numbers? Studying these two questions is deeply related but also a constant source of frustration for engineers taking math courses designed for mathematicians by mathematicians.
Galois theory, is a theory of "no". It ultimately serves to answer several "Can I build this?" questions with no. This makes it very interesting to pure mathematicians. However, for pure engineers that are looking for numeric machine parts that can be assembled in other useful ways to actually build something... Galois theory can be quite disappointing.
My second semester of algebra had a section on Galois theory and I remember thinking it was abstract nonsense and I didn't get it. I'm actually interested in going through this to see if my perspective has changed.
It's all abstract nonsense until it starts having practical results lol :)
The main motivation for Galois theory was proving the insolvability of the quintic. For those not aware, there is a general formula solving the quadratic equation (i.e. solving ax^2 + bx + c = 0). That formula has been known for millenia. With effort, mathematicians found a formula for the cubic (i.e. solving ax^3 + bx^2 + cx + d = 0), and even the quartic (order 4 polynomials). But no one was able to come up with a closed-form solution to the quintic. Galois and Abel eventually proved that a quntic formula DOES NOT EXIST. At least, it cannot be expressed in terms of addition, subtraction, multiplication, division, exponents and roots. You can even identify specific equations that have roots that cannot be expressed in those forms, for example x^5 - x + 1.
I took an entire course on it that went through the proof. It's actually very interesting. It sets up this deep correspondence between groups and fields, to the point that any theory about groups can be translated into a theory about fields and vice versa. And it provides this extremely powerful set of tools for analyzing symmetries. The actual proof is actually really anticlimactic. You have all these deep proofs about the structures of roots of polynomial equations, and at the very end you just see that the structures of symmetries of certain polynomials of degree 5 (like x^5 - x + 1) don't follow the same symmetries that the elementary mathematical operations have. Literally, the field of solutions to that polynomial doesn't map to a solvable group: https://en.wikipedia.org/wiki/Solvable_group
In almost the same breadth, it also proves that it is impossible to trisect an angle using just a compass and straightedge, a problem that had been puzzling mathematicians for millennia. It's actually almost disappointing: we spent then entire course just defining groups and fields and field extensions and all this other "abstract nonsense". And once all of those definitions are out of the way, the proof of the insolvability of the quintic takes 10 minutes, same for the proof of impossibility of trisecting an angle.
Did the solution taking 10 minutes make it seem like it was all just semantics from old faulty definitions?
How do you personally imagine trisecting an angle now? Is it possible to describe your new intuition of the impossibility in different human understandable terms that are also geometric? Impossible things are weird conversation subjects.
> Did the solution taking 10 minutes make it seem like it was all just semantics from old faulty definitions?
I recall that I, and the rest of the class, were very suspicious of the proof. The proof took maybe 10 minutes, but it probably took another 10-15 minutes for the professor to convince us there wasn't a logical error in the given proof. Though the situation was kind of the opposite of what you would thing. We understood field extensions and the symmetries of the roots of polynomials really well. What took convincing was that any formula using addition, subtraction, multiplication, division, exponents, and rational roots would always give you a field extension that mapped to a "solvable group". The proof is essentially:
1. Any field extension of a number constructed using those mathematical operations must map to a solvable group.
2. For every group there exists a corresponding field extension (this is a consequence of the fundamental theorem of Galois theory).
3. There exist groups that are not solvable.
4. Therefore, there are polynomials with roots that can't be constructed from the elementary mathematical operations.
Basically the entire course is dedicated to laying out part 3, and the part we were suspicious about was part 1.
The one thing that is interesting about the proof is that it is actually partially constructive. Because there is no general quintic formula, but there are some quintics that are solvable. For instance, x^5 - 1 clearly has root x=1. And Galois theory allows you to tell the difference between those that are solvable and those that are not. It allows you to take any polynomial and calculate the group of symmetries of those roots. If that group is solvable, then all of the roots can be defined in terms of elementary operations. If not, at least one of the roots cannot.
> How do you personally imagine trisecting an angle now? Is it possible to describe your new intuition of the impossibility in different human understandable terms that are also geometric?
So the trisection proof I don't remember as well, but looking it up it isn't very geometric. It essentially proves that trisecting an angle with a compass and straight edge is equivalent to solving certain polynomial equations with certain operations, and goes into algebra.
That said, Galois theory itself feels very "geometric" in the roughest sense of the term. Fundamentally, it's about classifying the symmetries of an object.
The reason that the proof isn't geometric, is that the algebriac proof is a proof that Euclidean geometry is incomplete. How can you use a language (any given language!) to express the idea that the selfsame language is incapable of expressing a certain concept?
You can draw a picture of trisecting an angle using an ruler (with cube-root markings) or an Archimedian sprial, which are clearly more powerful than purely Euclidean geometery, but how can you draw a picture of it being impossible without something like this?
How do you draw a picture of something that doesn't exist?
2. For every group there exists a corresponding field extension (this is a consequence of the fundamental theorem of Galois theory).
Just a nit, but when talking about extensions of Q, this is called the Inverse Galois Problem and it is still an open problem.
That said, you don’t actually need this strong of a statement to show general insolvability of the quintic. Rather you just need to exhibit a single extension of Q with non-solvable Galois group. I believe adjoining the roots of something like x^5+x+2 suffices.
Your story makes me picture Geometric Algebra, defining Complex Numbers or Quaternions and multiplication on them, and their symmetries and elegant combinations. Ty for sharing your math memories. Makes me wonder if Galois theory can determine valid or invalid imaginary number combinations / systems.
It solves a problem that was puzzlimg mathematicians for millennia - I wouldn't call that a footnote. You may not consider pure math applications interesting, but that doesn't make them unimportant.
Galois theory is big today because it provides a connection between a ring theory and field theory. This has huge applications for other branches of modern math - for example, number theory. But that always culminates in a pure math application, which makes it a footnote, I guess.
On the other hand, getting yourself shot in the gut 152 years before the invention of the Bogotá bag might be a textbook example of the english chengyu "book smart life stupid"?
There are a few interesting places where Galois Theory touches on compilation/programming.
Abstract interpretation models a potentially infinite set of program behaviors onto a simpler (often finite) model that is (soundly) approximate and easier to reason about (via Galois connections); here the analogy is to Galois Theory connecting infinite fields with finite groups. I often think about this when working on Value Numbering for instance.
Also (perhaps a bit of stretch) it's interesting to think of extending a computational domain (say integers) with additional values (say an error value) as a kind of field extension, and as with field extensions, sometimes (perhaps unexpectedly) complications arise (eg loss of unique factorization :: LLVM's poison & undef, or NaNs).
To extend on this, while abstract interpretation may sound a bit "abstract" (pun not intended), it is the basis for many techniques for software verification and compiler optimizations. At its core it basically allows you to soundly approximate the set of reachable states in a program, which in turn can be used to check that no "bad" state can be reached (for software verification) or that e.g. some checks are useless and can be removed (for compiler optimizations). Other applications include the new borrow checker for the Rust programming language, which is built on various dataflow passes to determine at each program point which variable is "live" and/or "borrowed".
> But then you realize something genuinely weird: There’s nothing you can do to distinguish i from −i.
Relatedly, to this day I still don't know how distinguish a left-handed coordinate system from the right-handed one purely algebraically. Is the basis [(1,0,0), (0,1,0), (0,0,1)] left- or right-handed? I don't know without a picture! Does anyone?
No, it's impossible, even in principle, to answer that question. You can draw a picture for either answer.
"Left" and "right" are a dipole. Neither one can exist without the other, and they are symmetric. It's the same issue as we have with the conjugates discussed in Galois Theory.
In fact, in an algebraic (non-ordered/arithmetic/analytic) perspective, it's misleading to use the symbols + and - to label the conjugates in field extensions like sqrt2 and i. Left and Right are better names than + and - for those conjugate pairs. Only when we impose an arithmetic ordering (which is not needed in the theory of algebraic equalities) is it meaningful to use + and -: -sqrt(x) < 0 < +sqrt(x), where x is a positive real number.
(and when x is a negative real number, we immediately see the problem with - again: -i and i are not separable via ordering with respect to 0.)
The sentence you quote actually gives you the answer to your question but it is not completely obvious why. :)
The complex numbers are essentially the theory of 2D-space. You are asking about 3D-space. The statement that you quoted tells you that you cannot distinguish between up and down in theory.
Now, 2D-space is part of every 3D-space. There are multiple ways to see this. The easiest is to just drop the z-coordinate.
Suppose you could distinguish left-handed and right-handed in 3D-space. In this case, you would have a way to distinguish up and down in the embedded 2D-space. However, you cannot do this distinction in 2D-space and therefore you cannot distinguish left-handed and right-handed in 3D-space.
It's 2 cosets, one is arbitrarily left handed and the other arbitrarily right handed. If you are in an orientable space :) if not, then there's no global concept of left or right. a
Danny O’Brien’s blog post A Touch of the Galois is my favorite writing on Galois,
> Flunked two colleges, fought to restore the Republic, imprisoned in the Bastille, and managed to scribble down the thoughts that would lead to several major fields of mathematics, before dying in a duel — either romantic or political — at the age of twenty.
There is a nice topological proof which gives a more direct and visual understanding what solving by radicals means. It is quite short but might take some time to absorb the concepts.
I watched one of the videos, titled "Galois groups, intuitively".[1] It is about 18 minutes long and gave me a somewhat understandable overview what Galois groups are (not the theory yet!).
I suppose if you have the time to spare, at least it should give you an idea if the topic is of interest and whether you need to remind yourself of some mathematical concepts to be able to study rest of the material, or if it is too elementary for you and a shorter treatise from elsewhere would be preferable.
(I actually thought I had learned something about Galois theory in University algebra, but either I hadn't, or I've forgotten more than I wanted to admit. Which is to say, if you watch the video, your mileage may vary!)
Someone could tell you that Galois theory is the study of field extensions and the structure of their automorphism groups but is that really going to help?
The fundamental theorem of Galois theory reduces certain problems in field theory (like finding roots of polynomials) to group theory (counting permutations and symmetries), which makes them simpler and easier to understand and solve (or prove non-solvable). Galois theory is the proof and applications of this theorem, and related topics. Key to this theory is being more methodical about extending the rational numbers into the real numbers, by introducing new numbers one at a time, instead of all at once.
One of the immediate discoveries in beginning this study is the fact that in many common cases you cannot add just one numbers one at a time, but must add 2 or more numbers at once. These sets of numbers are called conjugates, and have the interesting property that even though you can prove how many must exist and that they are distinct from each other, they are otherwise identical except in the arbitray names you give them.
We need the Toki Pona of Math. I hope Geometry one day becomes the foundation of Math again. Anyone can participate in Geometry just with a stick or VR headset.
Sometimes when you move a shape around in front of you, you end up with the same shape. Maths people call this symmetry, and have lots of names for different ways you can get back to the same shape.
For instance, if you flip a square around you get back the same square. This is called reflective symmetry.
If you spin a triangle around you sometimes end up with the same triangle. This one is rotational symmetry.
Galois spent a lot of time thinking about numbers instead of shapes. What he realised is that when you add and multiply numbers in lots of different ways, you sometimes end up with the same number at the end. And sometimes different numbers, when added and multiplied in the same way, also give you the same number at the end.
For instance, if you take 1, multiply it by itself and subtract 1, you get 1x1-1=0.
If you do the same with -1, you get (-1)x(-1)-1=0. A different number, using the same pattern, gives us the same result.
What we're seeing here is that there are some symmetries in numbers, not just shapes! Galois theory is all about the nitty gritty of how these number symmetries work, how to find them, and how to use them to do interesting mathematics.
I don't think this is Galois theory. Galois theory is about "The Fundamental Theorem of Galois Theory", which states that there is a nice mapping from field-extensions to group-extensions, where the resulting groups are usually finite. When the resulting groups are finite (as they usually are), many problems involving field extensions can be solved using brute-force search.
Galois fields happen to be something else named in honour of Galois.
That quote about equations with rational numbers was from here. Galois didn't have a computer of course. Rational Numbers and trisecting an angle sound related.
The quote is followed immediately by this: "It extends naturally to equations with coefficients *in any field*, but this will not be considered in the simple examples below." Emphasis on 'in any field' is mine. Among the other fields that can be considered include the Galois fields, which are another name for finite fields. (There are also infinite fields other than the rationals, so 'in any field' does not just mean Galois fields/finite fields.) https://en.wikipedia.org/wiki/Finite_field
Galois fields have nothing to do with being able to represent rational numbers in a computer: elements of a finite field aren't even rational numbers.
I can see how the idea extends naturally and also how it doesn't extend naturally. Thanks for explaining, I wonder what Galois would do if alive today with computers.
Leibniz does get a mention, but it'll already take a while to explain these two, so I'll leave his full reconciliation as an exercise for the reader ;)
Every mathematical monad comes from an adjunction, which unfortunately implies they're not as windowless as Leibniz would like? (or does it merely imply that in the Leibnizian setting the structure is always external and never internal?)
Can we make a mathematical monad out of the adjunction presented above? Let g(C) be the minimum G and c(G) be the maximum C in the model above, then we certainly have g(C) = g(c(g(C))) and c(G) = c(g(c(G))), which formally suggest we may be able to do something.
Exercise: does the triple (T, μ, η) exist? what about (G, δ, ϵ) in the other direction? If they do exist, what are they, for Aquinas, Leibniz, and Spinoza?
They don't need to be reconciled, but they do differ, which suggests we might be able to find a framework which reconciles them.
GaloisSpinozaAquinas in Google yields (for me at least) the following HN thread: https://news.ycombinator.com/item?id=39885475 , in which an attempt at "Algebraic Theology" not only provides such a subsuming (thanks to Galois Theory) framework, answering that particular question in the affirmative, but also raises many other questions which might be amusing to pursue. (I could summarise those Q's in this thread, if any of you all are more interested than the downvoters were)
[I will summarise that thread further up in this one, but as it was months ago it may take me an hour or two to page everything back in]
Aquinas, Leibniz, and Spinoza all agree that there is a God (G) that created[1] a Creation (C) which we are a part of.
One major way Aquinas (and Leibniz) differ from Spinoza is in how determinate C may be. Spinoza says C is determined by G; Aquinas says there are many possible C's for any given G. (Leibniz splits the difference and says there are many possible C's, but in our particular case, our G has created the best[2] possible C.)
The reconciliation: let G and C be in an adjoint relationship, such that we have functions picking out the maximum C any given G may create, and the minimum G that can create any given C.
Now, if you are Aquinas, G is omnipotent, and hence has the possibility to create other C's, but our C is the maximum[3] one.
On the other hand, if you are Spinoza, G determines C[4], so it is trivially maximal. (The maximum of a singleton being the unique element)
Does that make sense?
Question: do there exist Gods that are incapable of creating any creation, or Creations that are impossible for any god to create? If so, need we replace "maximum" and "minimum" above by LUB and GLB? What other situations (eg. gods or creations being only domains rather than lattices) would also require further abstraction?
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[0] my apologies for any non-standard notation. I tried to find a survey paper on "Algebraic Theology" so I could follow the existing notation, but failed to find any concrete instances in this (currently only platonic?) field.
[1] Spinoza is accused of "pantheism": the heresy of identifying God and the Creation. Reading him according to a Galois-theoretic model, he would be innocent of this heresy, for when he says "God, or, the Universe", he is simply using metonymy, for in his model God and Creation are dual, so (being in a 1:1 relationship) one determines the other. Note that in general, not only are duals not identical, they're not even isomorphic.
[2] but cf Voltaire, Candide (1759)
[3] if you are Leibniz, the order in which it is maximal corresponds to the traditional "worst" "better" "best" order. I don't know Leibniz well enough to say if he had a total, or only partial, order in mind; presumably in his model if there are several maximal "best" creations they would all be isomorphic?
Exercise for the reader: work out the Leibnizian metaphysical adjunction.
[4] turning this arrow around, C determines G, which explains why Einstein would say he believed in the "God of Spinoza", and chose to base his research by thinking about C, unlike the medieval colleagues of Aquinas, who spent a lot of time and effort trying to work the arrow in the other direction, hoping to come to conclusions about C in starting by thinking about G.
> The Galois theory normally taught in graduate-level algebra courses ... involves a Galois connection between the intermediate fields of a Galois extension and the subgroups of the corresponding Galois group.
Instead of speaking informally, of abuse of Galois Theory, I should have spoken formally, of abstraction of Galois Theory.
He taught Galois Theory using its application to coding theory for worked examples. That class was something of a turning point in my life to be honest. I'd never think of constructing a heptagon again, for example. Definitely avoided Duels, and Montparnasse. Ok joking aside, it caused the proverbial lightbulb to turn on in my brain, and helped tremendously in my career later when I ran into folks trying to seem smart because they understood ECC or ZKPs. It was like the extreme opposite of those people who say "I never used a single thing I learned in college".