1. This solution is how I solved it.
If you look at problem solving as A* search then the solution is not what was done to solve it. Learning mathematicians are supposed to learn to search trough the space of mathematics by examples of solutions. Each step presented on the blackboard is a step in the right direction, omitting steps in the wrong direction, and backtracking in the initial efforts to solve (or even define) a problem.
2. This idea is abstract and has nothing to do with reality.
Mathematics is a distillation process of abstraction. In this purification process all preconceptions of reality are filtered out, so the result is rigorous and universal. This often leads to a denial of any connection to reality. (A well known counterexamples is Conway's Analysis of go games, leading to the surreal numbers.)
3. This is the only and right way to do it.
You are presented with a single solutions, definitions or axioms, omitting alternative definitions. This get better in higher education (think of how many ways there are to define numbers).
This is not what it looks like doing math.
It leads to frustration in students who learn (wrongly) that math is about doing everything right the first time, intuitive thinking is wrong, and basically only about what you are not allowed to do (as the feedback they get are highlights of mistakes in exams).
Our task should be cleaning that up and reteach exploration, reteach that math - like any universal language - can describe everything however you want, finding solutions is trial and error, intuition is good and fast but fuzzy and can mislead you, to not stop with a solution but explore around it more for better understanding, and while we have good cause to define things as we do, it's not necessarily the only way to do things (looking at you, axiom of choice).
To use an analogy: Otherwise we stick to teaching orientation to scouts by fastest satnav routes.
i think your points 1 2 3 are true in the sense that this is perceived, and that sometimes, the way things are being taught lead to this. but in my mathematics career, as well as in school, i never had the impression that these sentiments were actually the beliefs of teachers.
i also think it is wrong to so confidently state in which way mathematics relates to reality. this is a very nuanced philosophical issue.
also, i don't see how your "counterexample" is one, let alone a well-known one, both things, the way you stated them, are very abstract and not "real world" at all, the way i see it. or are you here still talking about the fact that math gets inspired by things that happen in real life, as opposed to just inventing stuff just out of pure abstract inspiration? no mathematician i have ever met will deny this. and in fact, philosophically, it is an interesting question if it is even possible to separate a human's "purely abstract ideas" from how a human reasons in real life. that goes back to plato.
i also think one should be careful with these strong assignments of terms like "universal language" and "can describe everything however you want".
sorry for the rant. i agree with your sentiments about how we teach and how researches find out things.
He is right in a sense that a lot of abstract concepts, in e.g. combinatorics, can be explained even to middle school students if given as simple examples starting with trivial cases and showing the pattern. How many textbooks do that? Close to zero. How many research notebooks are full of these? I cannot say for sure but I suspect much greater than 0.
I self-taught some math which gave me the benefit of no deadline, and I did not memorize derivations - I learned them. The typical experience is that the derivation given to you is that 'satnav route' and offers little intuition for what it does or how you would have even thought to do it.
The exercise for me was always the same: re-derive the proof by injecting context, motivation and intuition.
Interesting points. Regarding intuition, I highly recommend https://betterexplained.com as a great example of an alternative approach that aligns with your ideas.
> After scratching the itch of knowing that something's been written completely rigorously, it's easy to call a work done when maybe we shouldn't call it done yet. And there's contexts where this is maybe OK, research comes to mind...
It's funny, because all of the advice he gives in the rest of the video would be amazing to apply to research papers as well. I've had people discourage me from writing research papers pedagogically because it is "unusual," which really meant that it would be looked down upon. It's kind of tragic that the norm is to write research papers that strip away the human details of how a definition is motivated or how something was discovered. This kind of information is usually only shared through in-person discussions or occasionally research presentations, but it is often the core knowledge that enables one to do research in a given field.
It's a great video though. He gives a little checklist of ways to check if your work is clear:
- Do definitions have motivating examples?
- Do proofs [or other results] feel rediscoverable?
- Is there personality?
- Are core ideas illustrated with diagrams?
- Is relative importance highlighted?
...
He also gives a beautiful example of how powerful it is to give a motivating example for an abstraction before introducing it. I really got the sense from watching this of what makes him such a great math educator. Highly recommend his YouTube channel 3Blue1Brown for anyone who hasn't heard of him
I think most supervisors try to discourage students from writing in this way because there is a delicate art to it that you are unlikely to be able to meet at that career stage. When I read some of the texts from Martin Hairer and Cedric Villani in my field (or even Riemann's famous paper on the zeta function), you can see that they are motivational and informative but still concise at the same time. Full of personality too, especially Villani. That's a hard balance and it requires exceptional understanding of the topic and the reader.
At the same time, we really should be encouraging it more. I have found that in newer machine learning theory papers (strictly theory, not empirical work), there is something closer to a good balance that is expected even of students.
Including a motivating example is key to this, and should be considered mandatory.
Reviewer 1 will classify half or your analogies as misleading, reviewer 2 will complain about the other half, and anyway the editor will ask to reduce the number of pages before publication.
I recently came to know of the following which others might also find useful;
1) Math Made Visual: Creating Images for Understanding Mathematics by Claudi Alsina and Roger Nelsen - https://bookstore.ams.org/CLRM/28
2) Roger Nelsen has also written other similar works like the Proofs without Words(3-vols), Nuggets of Number Theory: A Visual Approach, Cameos For Calculus: Visualization In The First-year Course, etc. - https://bookstore.ams.org/browse?Author=%22Roger%20B.%20Nels...
This book is not a mathematical text` a la Hilbert and
Cohn-Vossen, not a psychological investigation` a la Hadamard, nor a How-To manual` a la Polya. It is a work of epistemology. But unlike almost all other writing in epistemology of mathematics, it is constrained by results of research in cognitive science and mathematics education. So the book has interdisciplinary roots.
unpopular opinion: imho there is no substitute for deep learning and doing problems. Math is unlike other subjects, where you have to actually do it head on; you cannot just passively consume it like listening to a history podcast or watching a video.
The insane popularity of channels such as 3Blue1Brown suggests that a lot of people do. Learning by watching is many orders more popular than learning by doing. Probably because it makes you feel smart without having to put that knowledge to the test, so it seems like a better way to master new concepts.
There are many folks on HN who watched a YouTube video or read a Quanta article about some mathematical concept and run with that.
> The insane popularity of channels such as 3Blue1Brown suggests that a lot of people do.
I don't think this is true.
First of all, the fact that watching videos is not a sufficient way to learn math does not mean that videos cannot be an excellent supplement for learning math. 3B1B's linear algebra videos for example are wonderful tools for gaining geometric understanding and motivation behind why things are the way the are in linear algebra, but of course they won't teach you how to invert a matrix. (He even says something to this effect explicitly many times throughout the series, if I recall.) Not everyone who watches those videos is engaging shallowly with math just because they're watching videos about it, there's plenty of value for serious students as well.
Your next argument might be that the sheer number of views indicates an unrealistic volume of serious students. This is true, but it still doesn't indicate that people think watching videos is a good way to learn math; in particular, nothing says they're watching those videos because they're trying to learn math. They're beautiful and interesting and well put together videos, and I'm sure many people enjoy watching them passively fully conscious of the fact that they aren't actually deeply learning math. I don't think there's anything wrong with that!
I'm sure there is a subset of viewers who think they're actually learning math by merely watching videos, but I'm willing to bet it's a relatively small fraction.
Personnally, I watch 3Blue1Brown's videos because I find them very enlightening. They're are also very nice to watch and listen to, the production quality is very high. I don't pretend learning a lot of maths by simply watching them but I like watching them anyway for what they are: videos about maths problems and maths concepts.
> Learning by watching is many orders more popular than learning by doing.
But you hardly learn anything from it. You forget. And you don't know how to apply the bits that stick. You only remember some of the words and concepts. You may repeat them to other people. For 99% (or more) of the viewers, that's enough, because there's never a need to apply probability theory or differential calculus. So it is entertainment. Intellectual, more interesting than most other content, sometimes stimulating, but still entertainment. The 1% that does want to apply that knowledge, really needs to practice, though.
a common comment i read is something like “if <youtuber> was my teacher/professor i would have learned so much better!”
and no, i don’t think so. you would have been entertained, but your teacher was making you learn. ability to explain concepts is important but it’s only the first step on the pathway to learning.
Hmm, I'm not so sure. You may have never had a truly horrible math teacher. I distinctly remember learning matrix multiplication in high school for the first time. We were taught the algorithm without any motivation at all. Not a single picture was drawn, she just did some examples and then passed out a giant worksheet. I didn't learn linear algebra in that class. If Grant Sanderson had taught it I'm sure I would have.
I think a lot of people, sadly, imply that they think this, even if they don't explicitly say it.
I recently tried to obliquely bring this up to a friend and he asked out loud, "Sometimes I do wonder if it's better to watch YouTube videos or do problems. I'm not sure."
This has been one piece of acedata representing an aggravating trend in my personal life. It could just be my attracting these types (which I've been fretting about since I'm thinking wtf is wrong with me then), but other examples are family members even expressing this bad habit with something as dumb as Marvel. They read forums about comics without ever reading them and feel possessive enough about the genre to get pissed when I, who...does read comics..., criticize the Disney Marvel movies.
I'm hoping it's just me over indexing for it since it bothers me, but yeah - I have found that there are certainly people who just consume content without practicing or implementing the thing they're supposedly learning about, and then think they've actually learned.
ofc ymmv - it could just be my strange social circles.
"Don't just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis?"
I heard the following nice metaphor from a professor: learning mathematics is like learning to ride a bike, at some point you just need to get on the bike, you don't just study some "theory of bicycles".
I don't think math is that different from other subjects, but it has this advantage that it is a whole world that you can carry with you and experiment, you don't need that many tools for your exploration.
I'm pretty sure the vast majority of people understand linear algebra etc, if they were taught that the reason why linear algebra works is that algebra (algebra over a field) are an approximate isomorphism to physical reality.
From that perspective you do not need contrived examples like using function composition as alternative operator.
It would be simply enough to teach the student that any symbol system following the definition of an algebra over a field has the same properties as e.g. natural numbers with respect to the (possibly physical) composition operator.
If you don't understand this then it's because you don't understand linear algebra.
Consider this, you can write numbers as symbols or as words. There is no single universal language so a number can be represented by many words and mean the same thing.
Now let's ignore the concept of a number to begin with. Let's say numbers were never invented.
Everyone discovered that you can combine symbols a+a=b b+a=c and so on. They realize that the laws relating to how the words can be combined follow certain properties.
They will also notice that marbles under physical composition follow the same rules.
They then discover that the effect of having a marble in one hand and in another can be physically composed into one hand having two marbles.
They then think about giving the marble quantities a different representation, e.g. one marble = 1 finger
They invented an isomorphism.
Transforming the marbles into fingers then adding the fingers and then transforming back to marbles is the same as taking one and one marbles and putting them together.
And that is how linear algebra was invented, long before anyone even know what a mathematical definition was.
Edit: if it's not clear, any physical composition operator can form an algebra over a field,
In a mechanical calculator, the gear movement is the composition operator
In a computer the adder is an electron based composition operator
Hence the reason why it is possible to build useful computers that perform useful calculations relates directly to the fact that there is an isomorphism between the real world state and the computer state
I don’t think that would help at all, to be honest. I think if you teach people that a vector is an arrow, and a matrix is a transformation of the arrow, you’d get much further. 3b1b has a great series on it in fact which I would consider to be the gold standard.
Sorry, long rant incoming. I think there are a lot of things wrong here.
I think you are right where you talk about the "one marble" and "one finger" isomorphism, that makes sense.
Everywhere else, you throw around a lot of terminology that gets very confusing.
> any symbol system following the definition of an algebra over a field has the same properties as e.g. natural numbers with respect to the (possibly physical) composition operator
What do you mean by symbol system? Are you suddenly in the discipline of logic and formal languages here? What are "natural numbers with respect to the composition operator"? what is _the_ composition operator? Function composition? I don't see any K-algebra structure here, please enlighten me. What does it mean to be physical here, referring to the composition operator? An algebra over a field is a pretty fundamental structure, intuitively polynomials over a field with some extra rules. But this is all still very abstract and I don't see how it is immediately apparent to be an "approximate isomorphism to physical reality", which I can only assume to mean what I think it means.
> Let's say numbers were never invented. [...] Everyone discovered that you can combine symbols a+a=b b+a=c and so on.
Whoah, slow down there, you are already assuming a lot. Infix notation, equality, and suggestively using the plus symbol. I would agree with your claim "everyone discovered [...]" when talking about concepts like addition, but you are talking about abstract "combining symbols", which in the way we do it now is a rather modern advancement, especially because you talk about the logic/language theory concept of words.
> Transforming the marbles into fingers then adding the fingers and then transforming back to marbles is the same as taking one and one marbles and putting them together. And that is how linear algebra was invented, long before anyone even know what a mathematical definition was
That is not linear algebra. That is simply group or ring theory, and really only restricted to the integers, really. The integers (or whatever you mean by "linear algebra") were exactly definitionally invented the moment they had a kind of mathematical definition of some sort. I think you are trying to say the concepts were discovered and/or used long before anyone knew about a mathematical definition.
> if it's not clear, any physical composition operator can form an algebra over a field
Please explain this better, I am not sure what you mean by a "physical composition operator". Do you mean that functions that are already K-linear operators over vector spaces form a (non commutative) K-algebra with the function composition operator as multiplication? Sure, that is true, but that already assumes so much structure. I don't think it is clear at all. I am not saying you are wrong here (smart physics concepts stuff around Noether theorem etc. come to mind, but I am not sure), but I don't think this is easy to see intuitively at all, as opposed to marbles and fingers etc.
> In a mechanical calculator, the gear movement is the composition operator. In a computer the adder is an electron based composition operator
Whoah, slow down again, please. What composition operator? There is no linear algebra here. Also, it is all over the integers, if anything, not over a field. Or if it is a field, then it is just approximated with floats, which mathematically amounts to things that are not even a true ring/algebra, because of rounding errors.
> Hence the reason why it is possible to build useful computers that perform useful calculations relates directly to the fact that there is an isomorphism between the real world state and the computer state
You make the conclusion sound so easy, yet you left out all of the important steps. And computers are famously not there as an "isomorphism between the real world state and the computer state", if you talk about physical stuff, as they can only approximate. If you are talking about fingers and marbles, then sure, I guess the computer reflects that.
However, the marbles vs fingers vs the abstract notion of a number have a totally, completely different flavor than physics and linear algebra. The classic undergrad philosophical debate around "are numbers reality?" exactly explains the fact about how the concept of counting is already an abstraction, because we don't care about the physics of the atoms in the marbles or the fingers or whatever. With physics, the separation of math vs physics, or abstraction vs reality, is a bit clearer, because we use math to model/describe/approximate reality as reality unfolds etc.
Mathematicians lie.
The three lies (by omission) are:
1. This solution is how I solved it. If you look at problem solving as A* search then the solution is not what was done to solve it. Learning mathematicians are supposed to learn to search trough the space of mathematics by examples of solutions. Each step presented on the blackboard is a step in the right direction, omitting steps in the wrong direction, and backtracking in the initial efforts to solve (or even define) a problem.
2. This idea is abstract and has nothing to do with reality. Mathematics is a distillation process of abstraction. In this purification process all preconceptions of reality are filtered out, so the result is rigorous and universal. This often leads to a denial of any connection to reality. (A well known counterexamples is Conway's Analysis of go games, leading to the surreal numbers.)
3. This is the only and right way to do it. You are presented with a single solutions, definitions or axioms, omitting alternative definitions. This get better in higher education (think of how many ways there are to define numbers).
This is not what it looks like doing math. It leads to frustration in students who learn (wrongly) that math is about doing everything right the first time, intuitive thinking is wrong, and basically only about what you are not allowed to do (as the feedback they get are highlights of mistakes in exams).
Our task should be cleaning that up and reteach exploration, reteach that math - like any universal language - can describe everything however you want, finding solutions is trial and error, intuition is good and fast but fuzzy and can mislead you, to not stop with a solution but explore around it more for better understanding, and while we have good cause to define things as we do, it's not necessarily the only way to do things (looking at you, axiom of choice).
To use an analogy: Otherwise we stick to teaching orientation to scouts by fastest satnav routes.
i think your points 1 2 3 are true in the sense that this is perceived, and that sometimes, the way things are being taught lead to this. but in my mathematics career, as well as in school, i never had the impression that these sentiments were actually the beliefs of teachers.
i also think it is wrong to so confidently state in which way mathematics relates to reality. this is a very nuanced philosophical issue.
also, i don't see how your "counterexample" is one, let alone a well-known one, both things, the way you stated them, are very abstract and not "real world" at all, the way i see it. or are you here still talking about the fact that math gets inspired by things that happen in real life, as opposed to just inventing stuff just out of pure abstract inspiration? no mathematician i have ever met will deny this. and in fact, philosophically, it is an interesting question if it is even possible to separate a human's "purely abstract ideas" from how a human reasons in real life. that goes back to plato.
i also think one should be careful with these strong assignments of terms like "universal language" and "can describe everything however you want".
sorry for the rant. i agree with your sentiments about how we teach and how researches find out things.
He is right in a sense that a lot of abstract concepts, in e.g. combinatorics, can be explained even to middle school students if given as simple examples starting with trivial cases and showing the pattern. How many textbooks do that? Close to zero. How many research notebooks are full of these? I cannot say for sure but I suspect much greater than 0.
One of the best HM comments I've seen.
I self-taught some math which gave me the benefit of no deadline, and I did not memorize derivations - I learned them. The typical experience is that the derivation given to you is that 'satnav route' and offers little intuition for what it does or how you would have even thought to do it.
The exercise for me was always the same: re-derive the proof by injecting context, motivation and intuition.
Interesting points. Regarding intuition, I highly recommend https://betterexplained.com as a great example of an alternative approach that aligns with your ideas.
> After scratching the itch of knowing that something's been written completely rigorously, it's easy to call a work done when maybe we shouldn't call it done yet. And there's contexts where this is maybe OK, research comes to mind...
It's funny, because all of the advice he gives in the rest of the video would be amazing to apply to research papers as well. I've had people discourage me from writing research papers pedagogically because it is "unusual," which really meant that it would be looked down upon. It's kind of tragic that the norm is to write research papers that strip away the human details of how a definition is motivated or how something was discovered. This kind of information is usually only shared through in-person discussions or occasionally research presentations, but it is often the core knowledge that enables one to do research in a given field.
It's a great video though. He gives a little checklist of ways to check if your work is clear:
- Do definitions have motivating examples?
- Do proofs [or other results] feel rediscoverable?
- Is there personality?
- Are core ideas illustrated with diagrams?
- Is relative importance highlighted?
...
He also gives a beautiful example of how powerful it is to give a motivating example for an abstraction before introducing it. I really got the sense from watching this of what makes him such a great math educator. Highly recommend his YouTube channel 3Blue1Brown for anyone who hasn't heard of him
I think most supervisors try to discourage students from writing in this way because there is a delicate art to it that you are unlikely to be able to meet at that career stage. When I read some of the texts from Martin Hairer and Cedric Villani in my field (or even Riemann's famous paper on the zeta function), you can see that they are motivational and informative but still concise at the same time. Full of personality too, especially Villani. That's a hard balance and it requires exceptional understanding of the topic and the reader.
At the same time, we really should be encouraging it more. I have found that in newer machine learning theory papers (strictly theory, not empirical work), there is something closer to a good balance that is expected even of students.
Including a motivating example is key to this, and should be considered mandatory.
Reviewer 1 will classify half or your analogies as misleading, reviewer 2 will complain about the other half, and anyway the editor will ask to reduce the number of pages before publication.
I recently came to know of the following which others might also find useful;
1) Math Made Visual: Creating Images for Understanding Mathematics by Claudi Alsina and Roger Nelsen - https://bookstore.ams.org/CLRM/28
2) Roger Nelsen has also written other similar works like the Proofs without Words(3-vols), Nuggets of Number Theory: A Visual Approach, Cameos For Calculus: Visualization In The First-year Course, etc. - https://bookstore.ams.org/browse?Author=%22Roger%20B.%20Nels...
Awesome, thanks! See also https://betterexplained.com (no affiliation, just a fan)
For an interesting philosophical/cognitive study see;
Visual Thinking in Mathematics: An epistemological study by Marcus Giaquinto - https://academic.oup.com/book/11260
From the Preface:
This book is not a mathematical text` a la Hilbert and Cohn-Vossen, not a psychological investigation` a la Hadamard, nor a How-To manual` a la Polya. It is a work of epistemology. But unlike almost all other writing in epistemology of mathematics, it is constrained by results of research in cognitive science and mathematics education. So the book has interdisciplinary roots.
Writing "3Blue1Brown" would be more recognizable than "Grant Sanderson" :)
I'd argue he's made himself enough of a public figure that he's earned his name and not just his YouTube channel.
True fans recognized him immediately either way ;)
”The roots of education are bitter, but the fruit is sweet.” — Aristotle
unpopular opinion: imho there is no substitute for deep learning and doing problems. Math is unlike other subjects, where you have to actually do it head on; you cannot just passively consume it like listening to a history podcast or watching a video.
This is the single most popular opinion about learning math.
Learning nearly anything? You can’t learn to drive a car from YouTube. Or paint a house, or wash a dish, or code, or … Learning requires doing.
Who thinks otherwise?
The insane popularity of channels such as 3Blue1Brown suggests that a lot of people do. Learning by watching is many orders more popular than learning by doing. Probably because it makes you feel smart without having to put that knowledge to the test, so it seems like a better way to master new concepts.
There are many folks on HN who watched a YouTube video or read a Quanta article about some mathematical concept and run with that.
> The insane popularity of channels such as 3Blue1Brown suggests that a lot of people do.
I don't think this is true.
First of all, the fact that watching videos is not a sufficient way to learn math does not mean that videos cannot be an excellent supplement for learning math. 3B1B's linear algebra videos for example are wonderful tools for gaining geometric understanding and motivation behind why things are the way the are in linear algebra, but of course they won't teach you how to invert a matrix. (He even says something to this effect explicitly many times throughout the series, if I recall.) Not everyone who watches those videos is engaging shallowly with math just because they're watching videos about it, there's plenty of value for serious students as well.
Your next argument might be that the sheer number of views indicates an unrealistic volume of serious students. This is true, but it still doesn't indicate that people think watching videos is a good way to learn math; in particular, nothing says they're watching those videos because they're trying to learn math. They're beautiful and interesting and well put together videos, and I'm sure many people enjoy watching them passively fully conscious of the fact that they aren't actually deeply learning math. I don't think there's anything wrong with that!
I'm sure there is a subset of viewers who think they're actually learning math by merely watching videos, but I'm willing to bet it's a relatively small fraction.
Personnally, I watch 3Blue1Brown's videos because I find them very enlightening. They're are also very nice to watch and listen to, the production quality is very high. I don't pretend learning a lot of maths by simply watching them but I like watching them anyway for what they are: videos about maths problems and maths concepts.
> Learning by watching is many orders more popular than learning by doing.
But you hardly learn anything from it. You forget. And you don't know how to apply the bits that stick. You only remember some of the words and concepts. You may repeat them to other people. For 99% (or more) of the viewers, that's enough, because there's never a need to apply probability theory or differential calculus. So it is entertainment. Intellectual, more interesting than most other content, sometimes stimulating, but still entertainment. The 1% that does want to apply that knowledge, really needs to practice, though.
a common comment i read is something like “if <youtuber> was my teacher/professor i would have learned so much better!”
and no, i don’t think so. you would have been entertained, but your teacher was making you learn. ability to explain concepts is important but it’s only the first step on the pathway to learning.
Hmm, I'm not so sure. You may have never had a truly horrible math teacher. I distinctly remember learning matrix multiplication in high school for the first time. We were taught the algorithm without any motivation at all. Not a single picture was drawn, she just did some examples and then passed out a giant worksheet. I didn't learn linear algebra in that class. If Grant Sanderson had taught it I'm sure I would have.
I think a lot of people, sadly, imply that they think this, even if they don't explicitly say it.
I recently tried to obliquely bring this up to a friend and he asked out loud, "Sometimes I do wonder if it's better to watch YouTube videos or do problems. I'm not sure."
This has been one piece of acedata representing an aggravating trend in my personal life. It could just be my attracting these types (which I've been fretting about since I'm thinking wtf is wrong with me then), but other examples are family members even expressing this bad habit with something as dumb as Marvel. They read forums about comics without ever reading them and feel possessive enough about the genre to get pissed when I, who...does read comics..., criticize the Disney Marvel movies.
I'm hoping it's just me over indexing for it since it bothers me, but yeah - I have found that there are certainly people who just consume content without practicing or implementing the thing they're supposedly learning about, and then think they've actually learned.
ofc ymmv - it could just be my strange social circles.
Lol yes
"Don't just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis?"
-The goat paul halmos
I heard the following nice metaphor from a professor: learning mathematics is like learning to ride a bike, at some point you just need to get on the bike, you don't just study some "theory of bicycles".
I don't think math is that different from other subjects, but it has this advantage that it is a whole world that you can carry with you and experiment, you don't need that many tools for your exploration.
That's also Engineering in a nutshell. And medicine. And software.. This is not a unique property.
Aren't most hard things like this?
> unpopular opinion: imho there is no substitute for deep learning and doing problems
You are probably correct.
https://en.wikipedia.org/wiki/Deep_learning
Less popular opinion, apparently.
(2023)
I'm pretty sure the vast majority of people understand linear algebra etc, if they were taught that the reason why linear algebra works is that algebra (algebra over a field) are an approximate isomorphism to physical reality.
From that perspective you do not need contrived examples like using function composition as alternative operator.
It would be simply enough to teach the student that any symbol system following the definition of an algebra over a field has the same properties as e.g. natural numbers with respect to the (possibly physical) composition operator.
If you don't understand this then it's because you don't understand linear algebra.
Consider this, you can write numbers as symbols or as words. There is no single universal language so a number can be represented by many words and mean the same thing.
Now let's ignore the concept of a number to begin with. Let's say numbers were never invented.
Everyone discovered that you can combine symbols a+a=b b+a=c and so on. They realize that the laws relating to how the words can be combined follow certain properties.
They will also notice that marbles under physical composition follow the same rules.
They then discover that the effect of having a marble in one hand and in another can be physically composed into one hand having two marbles.
They then think about giving the marble quantities a different representation, e.g. one marble = 1 finger
They invented an isomorphism.
Transforming the marbles into fingers then adding the fingers and then transforming back to marbles is the same as taking one and one marbles and putting them together.
And that is how linear algebra was invented, long before anyone even know what a mathematical definition was.
Edit: if it's not clear, any physical composition operator can form an algebra over a field,
In a mechanical calculator, the gear movement is the composition operator
In a computer the adder is an electron based composition operator
Hence the reason why it is possible to build useful computers that perform useful calculations relates directly to the fact that there is an isomorphism between the real world state and the computer state
I don’t think that would help at all, to be honest. I think if you teach people that a vector is an arrow, and a matrix is a transformation of the arrow, you’d get much further. 3b1b has a great series on it in fact which I would consider to be the gold standard.
Different explanations work better for different people. See for example this post outlining various ways to develop intuition for adding the numbers in a series: https://betterexplained.com/articles/techniques-for-adding-t...
Sorry, long rant incoming. I think there are a lot of things wrong here.
I think you are right where you talk about the "one marble" and "one finger" isomorphism, that makes sense.
Everywhere else, you throw around a lot of terminology that gets very confusing.
> any symbol system following the definition of an algebra over a field has the same properties as e.g. natural numbers with respect to the (possibly physical) composition operator
What do you mean by symbol system? Are you suddenly in the discipline of logic and formal languages here? What are "natural numbers with respect to the composition operator"? what is _the_ composition operator? Function composition? I don't see any K-algebra structure here, please enlighten me. What does it mean to be physical here, referring to the composition operator? An algebra over a field is a pretty fundamental structure, intuitively polynomials over a field with some extra rules. But this is all still very abstract and I don't see how it is immediately apparent to be an "approximate isomorphism to physical reality", which I can only assume to mean what I think it means.
> Let's say numbers were never invented. [...] Everyone discovered that you can combine symbols a+a=b b+a=c and so on.
Whoah, slow down there, you are already assuming a lot. Infix notation, equality, and suggestively using the plus symbol. I would agree with your claim "everyone discovered [...]" when talking about concepts like addition, but you are talking about abstract "combining symbols", which in the way we do it now is a rather modern advancement, especially because you talk about the logic/language theory concept of words.
> Transforming the marbles into fingers then adding the fingers and then transforming back to marbles is the same as taking one and one marbles and putting them together. And that is how linear algebra was invented, long before anyone even know what a mathematical definition was
That is not linear algebra. That is simply group or ring theory, and really only restricted to the integers, really. The integers (or whatever you mean by "linear algebra") were exactly definitionally invented the moment they had a kind of mathematical definition of some sort. I think you are trying to say the concepts were discovered and/or used long before anyone knew about a mathematical definition.
> if it's not clear, any physical composition operator can form an algebra over a field
Please explain this better, I am not sure what you mean by a "physical composition operator". Do you mean that functions that are already K-linear operators over vector spaces form a (non commutative) K-algebra with the function composition operator as multiplication? Sure, that is true, but that already assumes so much structure. I don't think it is clear at all. I am not saying you are wrong here (smart physics concepts stuff around Noether theorem etc. come to mind, but I am not sure), but I don't think this is easy to see intuitively at all, as opposed to marbles and fingers etc.
> In a mechanical calculator, the gear movement is the composition operator. In a computer the adder is an electron based composition operator
Whoah, slow down again, please. What composition operator? There is no linear algebra here. Also, it is all over the integers, if anything, not over a field. Or if it is a field, then it is just approximated with floats, which mathematically amounts to things that are not even a true ring/algebra, because of rounding errors.
> Hence the reason why it is possible to build useful computers that perform useful calculations relates directly to the fact that there is an isomorphism between the real world state and the computer state
You make the conclusion sound so easy, yet you left out all of the important steps. And computers are famously not there as an "isomorphism between the real world state and the computer state", if you talk about physical stuff, as they can only approximate. If you are talking about fingers and marbles, then sure, I guess the computer reflects that.
However, the marbles vs fingers vs the abstract notion of a number have a totally, completely different flavor than physics and linear algebra. The classic undergrad philosophical debate around "are numbers reality?" exactly explains the fact about how the concept of counting is already an abstraction, because we don't care about the physics of the atoms in the marbles or the fingers or whatever. With physics, the separation of math vs physics, or abstraction vs reality, is a bit clearer, because we use math to model/describe/approximate reality as reality unfolds etc.