I first learned about manifolds through Introduction to Smooth Manifolds by John M. Lee. The book is dense but beautifully structured, guiding you from basic topology to smooth maps and tangent spaces with clear logic. It demands focus, yet every definition builds toward a deeper picture of how geometry works beneath the surface. Highly recommended.
It's truly the best book on Smooth Manifolds, though if you'd like a gentler approach which is still useful, then I suggest Loring Tu's books. Lee's Topological Manifolds book is also very nice. His newest edition of the Riemannian manifolds book requires selective reading or it'll slow you down.
That's a great suggestion. I actually started with Topological Manifolds before moving on to Introduction to Smooth Manifolds and it really helped build a solid foundation.
I havent read Loring Tus books before but let me look at them since I have been wanting to revisit the topic with a clearer and more relaxed approach.
This is a very informative article about the history of manifolds and their significance. Don’t let the title fool you into this being just a definition.
It’s actually much more well written than the majority or articles we usually come across.
And they have a RSS feed, although it's a bit tricky to figure out, since the relevant header tag for that is set up incorrectly, pointing to a useless empty "comments" feed even from their main page. The actual feed for articles is https://www.quantamagazine.org/feed/
I'm always surprised more people don't know about Quanta. Seems like it's currently the best science journalism out there, and IMO a very strong candidate for the single best place on the internet that's not crowd-sourced. The mixture of original art and technical diagrams is outstanding. Podcast is pretty good too, but I do wish they'd expand it to have someone with a good voice reading all the articles.
Besides not treating readers like idiots, they take themselves seriously, hire smart people, tell good stories but aren't afraid to stay technical, and simply skip all the clickbait garbage. Right now from the Scientific American front page: "Type 1 Diabetes science is having a moment". Or from Nature: "'Biotech Barbie' says ..". Granted I cherry-picked these offensive headlines pandering to facebook/twitter from many other options that might be legitimately interesting reads, but on Quanta there's also no paywalls, no cookie pop-ups, no thinly-veiled political rage-baiting either
Quanta is amazing because it doesn't have to worry about money. It's a publication run by the Simons Foundation, funded with the proceeds of the wildly successful RenTec hedge fund. So they get pretty much full editorial control.
For other publications they are beholden to people who haven't figured out ad-block, and your bar needs to be pretty low to capture that revenue.
Quanta’s greatest strength is that it doesn’t pretend to be clever. Many tech publications write as if they’re showing off, and you just end up feeling tired after reading them.
> Many tech publications write as if they’re showing off, and you just end up feeling tired after reading them.
I like this honestly because this shows that I learned something intelligent. On the other hand, if I don't feel exhausted after reading, it is a strong sign that the article was below my intellectual capacity, i.e. I would have loved it if I could have learned more.
Often, if the concept is presented in a more complex way the reason is that the author wants to emphasize and explain how the concept relates in a non-trivial way to some other deep concept; thus you learn a lot more than when the author explains things in the most simple (and shallow) way.
I agree. I find their articles very enjoyable. And even though they stay technical, they don’t descend into becoming a technical journal. The content is still accessible to a non-expert like me.
It's because of their Simons Foundation support, but not only because of that. I mean, I invite anyone to name another billionaire pet project of comparable quality.
Agreed. I'm not a mathematician - and to me a manifold is more familar in the context of engines. But I found both the text and the diagrams very useful.
I learned about Calabi Yau manifolds a long time ago and have forgotten most of the details, but I still remember how hard the topic felt. A Calabi Yau manifold is a special kind of geometric space that is smooth curved and very symmetrical. You can think of it as a shape that looks flat when you zoom in close but can twist and fold in complex ways when you look at the whole thing.
What makes Calabi Yau manifolds special is that their curvature balances out perfectly so the space does not stretch or shrink overall.
In physics especially in string theory Calabi Yau manifolds are used to describe extra hidden dimensions of the universe beyond the three we can see. The shape of a Calabi Yau manifold affects how particles and forces behave which is why both mathematicians and physicists study them.
I rarely see manifolds applied directly to cartographic map projections, which I've read about a bit, though the latter seem like just one instance of the former. Does anyone know why cartographers don't use manifolds, or mathematicians don't apply them to cartography? (Have I just overlooked it?)
One reason is that it would be like hanging a picture using a sledgehammer. If you're just studying various ways of unwrapping a sphere, the (very deep) theory of manifolds is not necessary. I'm not a cartographer but I would assume they care mostly about how space is distorted in the projection, and have developed appropriate ways of dealing with that already.
Another is that when working with manifolds, you usually don't get a set of global coordinates. Manifolds are defined by various local coordinate charts. A smooth manifold just means that you can change coordinates in a smooth (differentiable) way, but that doesn't mean two people on opposite sides of the manifold will agree on their coordinate system. On a sphere or circle, you can get an "almost global" coordinate system by removing the line or point where the coordinates would be ambiguous.
I'm not very well versed in the history, but the study of cartography certainly predates the modern idea of an abstract manifold. In fact, the modern view was born in an effort to unify a lot of classical ideas from the study of calculus on spheres etc.
This reminds me of how physicists will define a tensor. So a second rank tensor is the object that transforms according as second rank tensor when the basis (or coordinates) changes. You might find it circular reasoning but it is not, This transformation property is what distinguishes tensors (of any rank) from mere arrays of numbers.
Looking at things from abstract view does allow us not to worry about how we visualize the geometry which is actually hard and sometimes counter intuitive.
This is a tendency among physicists that I find a bit painful when reading their explanations: focusing on how things transform between coordinate systems rather than on the coordinate-independent things that are described by those coordinates. I get that these transformation properties are important for doing actual calculations, but I think they tend to obfuscate explanations.
In special relativity, for example, a huge amount of attention is typically given to the Lorenz transformations required when coordinates change. However, the (Minkowski) space that is the setting for special relativity is well defined without reference to any particular coordinate system, as an affine space with a particular (pseudo-)metric. It's not conceptually very complicated, and I never properly understood special relativity until I saw it explained in those terms in the amazing book Special Relativity in General Frames by Eric Gourgoulhon.
For tensors, the basis-independent notion is a multilinear map from a selection of vectors in a vector space and forms (covectors) in its dual space to a real number. The transformation properties drop out of that, and I find it much more comfortable mentally to have that basis-independent idea there, rather than just coordinate representations and transformations between them.
I think _Spacetime Physics_ takes roughly the same approach (they call it “the invariant interval”), but with much less mathematical sophistication required.
One of the worst examples is Weinberg’s book on GR, which I found nearly unreadable due to the morass of coordinates/indices. So much more painful to learn from than Wald or other mathematically modern treatments of GR.
I agree that focusing on Lorentz transformations is the wrong way to approach thinking about special relativity. But It might be the right way to teach it to physics students.
The issue is the level of mathematical sophistication one has when a certain concept is introduced. That often defines or at least heavily influences how one thinks about it forever.
The basics of special relativity came up in my first year of university, and the rest didn't really get focused on until my second year.
The first time around I was still encountering linear algebra and vector spaces, while for the second I was a lot more comfortable deriving things myself just given something like the Minkowski "inner product".
(As an aside: I really love abstract index notation for dealing with tensors)
> The issue is the level of mathematical sophistication one has when a certain concept is introduced. That often defines or at least heavily influences how one thinks about it forever.
That was one of the most interesting things of my EE/CS dual-degree and the exact concept you're describing has stuck with me for a very long time... and very much influences how I teach things when I'm in that role.
EE taught basic linear algebra in 1st year as a necessity. We didn't understand how or why anything worked, we were just taught how to turn the crank and get answers out. Eigenvectors, determinants, Gauss-Jordan elimination, Cramer's rule, etc. weren't taught with any kind of theoretical underpinnings. My CS degree required me to take an upper years linear algebra course from the math department; after taking that, my EE skills improved dramatically.
CS taught algorithms early and often. EE didn't really touch on them at all, except when a specific one was needed to solve a specific problem. I remember sitting in a 4th year Digital Communications course where we were learning about Viterbi decoders. The professor was having a hard time explaining it by drawing a lattice and showing how you do the computations, the students were completely lost. My friend and I were looking at what was going on and both had this lightbulb moment at the same time. "Oh, this is just a dynamic programming problem."
EE taught us way more calculus than CS did. In a CS systems modelling course we were learning about continuous-time and discrete-time state-space models. Most of the students were having a super hard time with dx/dt = A*x (x as a real vector, A as a matrix)... which makes sense since they'd only ever done single-variable calculus. The prof taught some specific technique that applied to a specific form of the problem and that was enough for students to be able to turn the crank, but no one understood why it worked.
Yeah, I had a slightly odd introduction to these things as I studied joint honours maths and physics. That meant both that I had a bit more mathematical maturity than most of the physics students and that I was being taught the more rigorous underpinnings of the maths while it was being (ab)used in all sorts of cavalier ways in physics. I liked the subject matter of physics more, but I greatly preferred the intellectual rigour of the maths.
Eric Gourgoulhon is a product of the French education system, and I often think I would have done better studying there than in the UK.
I had started in a theoretical physics degree which was jointly taught by the maths and physics department. By my final year I had changed into an ostensibly pure maths degree, although I did it mainly to take more advanced theoretical/mathematical physics courses (which were taught by the maths department), and avoid having to do any lab work—a torsion pendulum experiment was my final straw on that one, I don't know what caused it to fuck up, but fuck that.
In the end I took on more TP courses than the TP students, nearly burnt out by the end of the year, and... didn't exactly come out with the best exam results.
I found the physicist definition of a tensor is actually more confusing, because you are faced with these definitions how to transform these objects, but you never are really explained where does it all come from. While the mathematical definition through differential forms, co-vectors, while being longer actually explains these objects better.
> You might find it circular reasoning but it is not
Um, yes it is. "A foo is an object that transforms as a foo" is a circular definition because it refers to the thing being defined in the definition. That is what "circular definition" means.
A manifold is a surface that you can put a cd shaped object on in any place on the surface, you can change the radius of the cd but it has to have some radius above 0.
In particular, consider two intersecting planes. You can put all the discs you like on that surface, but it's not a manifold because on the line of intersection it's not locally R2.
I always found interesting that the English mathematical terminology has two different names for "stuff that locally looks like R^n" (manifold) and "stuff that is the zero locus of a polynomial" (variety). Other languages use the same word for both, adding maybe an adjective to specify which one is meant if not clear from the context. In Italian for example they're both "varietà"
Man, I wish that the modern internet -- and great stuff like this -- had been around when I took GR way back when. My math chops were never good enough to /really/ get it and there were so many concepts (like this one) that were just symbols to me.
Its unfortunately all too common for Physics/Math to be taught in that way (extremely technical, memorizing or knowing equations and derivations). The best teachers would always give a ton of context as to why and how these came about.
I just looked it up because I was interested in their etymologies, but it seems that the words actually have the same (Old English/Germanic) root: essentially a portmanteau of "many" + "fold."
I thought those are ethymologically about the thin-walled containment of a volumetric interior space where said space is connected to only specific ports/holes, and is often but not necessarily mandatory intertwined with a second such containment for a second space (intake+exhaust).
This has always caused me trouble when learning new concepts. A name for something will be given (e.g. manifold) and it sounds very much like something that I've come across before (e.g. a manifold in an engine) - and that then gets cemented in my brain as a relationship which I find extremely difficult to shake - and it makes understanding the new concept very challenging. More often than not the etymology of the term is not provided with the concept - not entirely unreasonable, but also not helpful for me personally.
It becomes a bigger problem when the etymology is actually a chain of almost arbitrary naming decisions - how far back do I go?!
> They’re as fundamental to mathematics as the alphabet is to language. “If I know Cyrillic, do I know Russian?” said Fabrizio Bianchi (opens a new tab), a mathematician at the University of Pisa in Italy. “No. But try to learn Russian without learning Cyrillic.”
Something's gone badly wrong here. "Without learning Cyrillic" is the normal way to learn Russian. Pick a slightly less prominent language and 100% of learners will do it without learning anything about the writing system.
I thought the same - many languages don't have a writing system and children learn without being able to write. But that's beside the point; the point is just as valid even if the analogy is poor.
Stand at one of the poles. Walk to the equator, turn 90 degrees. Walk 1/4 the way around the equator, turn 90 degrees again. Then walk back to the pole. A triangle with sum 270 degrees!
I first learned about manifolds through Introduction to Smooth Manifolds by John M. Lee. The book is dense but beautifully structured, guiding you from basic topology to smooth maps and tangent spaces with clear logic. It demands focus, yet every definition builds toward a deeper picture of how geometry works beneath the surface. Highly recommended.
It's truly the best book on Smooth Manifolds, though if you'd like a gentler approach which is still useful, then I suggest Loring Tu's books. Lee's Topological Manifolds book is also very nice. His newest edition of the Riemannian manifolds book requires selective reading or it'll slow you down.
That's a great suggestion. I actually started with Topological Manifolds before moving on to Introduction to Smooth Manifolds and it really helped build a solid foundation.
I havent read Loring Tus books before but let me look at them since I have been wanting to revisit the topic with a clearer and more relaxed approach.
This is a very informative article about the history of manifolds and their significance. Don’t let the title fool you into this being just a definition.
It’s actually much more well written than the majority or articles we usually come across.
And they have a RSS feed, although it's a bit tricky to figure out, since the relevant header tag for that is set up incorrectly, pointing to a useless empty "comments" feed even from their main page. The actual feed for articles is https://www.quantamagazine.org/feed/
Nice find, thank you. Your sleuthing is appreciated.
Oh dope. Added to my feedbin!
I'm always surprised more people don't know about Quanta. Seems like it's currently the best science journalism out there, and IMO a very strong candidate for the single best place on the internet that's not crowd-sourced. The mixture of original art and technical diagrams is outstanding. Podcast is pretty good too, but I do wish they'd expand it to have someone with a good voice reading all the articles.
Besides not treating readers like idiots, they take themselves seriously, hire smart people, tell good stories but aren't afraid to stay technical, and simply skip all the clickbait garbage. Right now from the Scientific American front page: "Type 1 Diabetes science is having a moment". Or from Nature: "'Biotech Barbie' says ..". Granted I cherry-picked these offensive headlines pandering to facebook/twitter from many other options that might be legitimately interesting reads, but on Quanta there's also no paywalls, no cookie pop-ups, no thinly-veiled political rage-baiting either
Quanta is amazing because it doesn't have to worry about money. It's a publication run by the Simons Foundation, funded with the proceeds of the wildly successful RenTec hedge fund. So they get pretty much full editorial control.
For other publications they are beholden to people who haven't figured out ad-block, and your bar needs to be pretty low to capture that revenue.
Remarkably, they don't even ask for money anywhere on the site. Now that is a rare thing on the modern internet, especially for high quality writing.
Quanta’s greatest strength is that it doesn’t pretend to be clever. Many tech publications write as if they’re showing off, and you just end up feeling tired after reading them.
> Many tech publications write as if they’re showing off, and you just end up feeling tired after reading them.
I like this honestly because this shows that I learned something intelligent. On the other hand, if I don't feel exhausted after reading, it is a strong sign that the article was below my intellectual capacity, i.e. I would have loved it if I could have learned more.
Seems superficial. If a simple concept is presented in a complex way what did you actually learn?
Often, if the concept is presented in a more complex way the reason is that the author wants to emphasize and explain how the concept relates in a non-trivial way to some other deep concept; thus you learn a lot more than when the author explains things in the most simple (and shallow) way.
I agree. I find their articles very enjoyable. And even though they stay technical, they don’t descend into becoming a technical journal. The content is still accessible to a non-expert like me.
It's because of their Simons Foundation support, but not only because of that. I mean, I invite anyone to name another billionaire pet project of comparable quality.
Agreed. I'm not a mathematician - and to me a manifold is more familar in the context of engines. But I found both the text and the diagrams very useful.
I was reading a book on string theory and I remember the Calabi–Yau manifold
https://en.wikipedia.org/wiki/Calabi%E2%80%93Yau_manifold
I'm not going to pretend to understand it all but they do make pretty pictures!
https://www.google.com/search?q=calabi+yau+manifold+images
I learned about Calabi Yau manifolds a long time ago and have forgotten most of the details, but I still remember how hard the topic felt. A Calabi Yau manifold is a special kind of geometric space that is smooth curved and very symmetrical. You can think of it as a shape that looks flat when you zoom in close but can twist and fold in complex ways when you look at the whole thing.
What makes Calabi Yau manifolds special is that their curvature balances out perfectly so the space does not stretch or shrink overall.
In physics especially in string theory Calabi Yau manifolds are used to describe extra hidden dimensions of the universe beyond the three we can see. The shape of a Calabi Yau manifold affects how particles and forces behave which is why both mathematicians and physicists study them.
>their curvature balances out perfectly so the space does not stretch or shrink overall
Could you elaborate a bit on this? I find it fascinating. Thanks.
>The shape of a Calabi Yau manifold affects how particles and forces behave [...]
Do you know if there's any experimental evidence of this?
I rarely see manifolds applied directly to cartographic map projections, which I've read about a bit, though the latter seem like just one instance of the former. Does anyone know why cartographers don't use manifolds, or mathematicians don't apply them to cartography? (Have I just overlooked it?)
One reason is that it would be like hanging a picture using a sledgehammer. If you're just studying various ways of unwrapping a sphere, the (very deep) theory of manifolds is not necessary. I'm not a cartographer but I would assume they care mostly about how space is distorted in the projection, and have developed appropriate ways of dealing with that already.
Another is that when working with manifolds, you usually don't get a set of global coordinates. Manifolds are defined by various local coordinate charts. A smooth manifold just means that you can change coordinates in a smooth (differentiable) way, but that doesn't mean two people on opposite sides of the manifold will agree on their coordinate system. On a sphere or circle, you can get an "almost global" coordinate system by removing the line or point where the coordinates would be ambiguous.
I'm not very well versed in the history, but the study of cartography certainly predates the modern idea of an abstract manifold. In fact, the modern view was born in an effort to unify a lot of classical ideas from the study of calculus on spheres etc.
This reminds me of how physicists will define a tensor. So a second rank tensor is the object that transforms according as second rank tensor when the basis (or coordinates) changes. You might find it circular reasoning but it is not, This transformation property is what distinguishes tensors (of any rank) from mere arrays of numbers.
Looking at things from abstract view does allow us not to worry about how we visualize the geometry which is actually hard and sometimes counter intuitive.
This is a tendency among physicists that I find a bit painful when reading their explanations: focusing on how things transform between coordinate systems rather than on the coordinate-independent things that are described by those coordinates. I get that these transformation properties are important for doing actual calculations, but I think they tend to obfuscate explanations.
In special relativity, for example, a huge amount of attention is typically given to the Lorenz transformations required when coordinates change. However, the (Minkowski) space that is the setting for special relativity is well defined without reference to any particular coordinate system, as an affine space with a particular (pseudo-)metric. It's not conceptually very complicated, and I never properly understood special relativity until I saw it explained in those terms in the amazing book Special Relativity in General Frames by Eric Gourgoulhon.
For tensors, the basis-independent notion is a multilinear map from a selection of vectors in a vector space and forms (covectors) in its dual space to a real number. The transformation properties drop out of that, and I find it much more comfortable mentally to have that basis-independent idea there, rather than just coordinate representations and transformations between them.
I think _Spacetime Physics_ takes roughly the same approach (they call it “the invariant interval”), but with much less mathematical sophistication required.
One of the worst examples is Weinberg’s book on GR, which I found nearly unreadable due to the morass of coordinates/indices. So much more painful to learn from than Wald or other mathematically modern treatments of GR.
I agree that focusing on Lorentz transformations is the wrong way to approach thinking about special relativity. But It might be the right way to teach it to physics students.
The issue is the level of mathematical sophistication one has when a certain concept is introduced. That often defines or at least heavily influences how one thinks about it forever.
The basics of special relativity came up in my first year of university, and the rest didn't really get focused on until my second year.
The first time around I was still encountering linear algebra and vector spaces, while for the second I was a lot more comfortable deriving things myself just given something like the Minkowski "inner product".
(As an aside: I really love abstract index notation for dealing with tensors)
> The issue is the level of mathematical sophistication one has when a certain concept is introduced. That often defines or at least heavily influences how one thinks about it forever.
That was one of the most interesting things of my EE/CS dual-degree and the exact concept you're describing has stuck with me for a very long time... and very much influences how I teach things when I'm in that role.
EE taught basic linear algebra in 1st year as a necessity. We didn't understand how or why anything worked, we were just taught how to turn the crank and get answers out. Eigenvectors, determinants, Gauss-Jordan elimination, Cramer's rule, etc. weren't taught with any kind of theoretical underpinnings. My CS degree required me to take an upper years linear algebra course from the math department; after taking that, my EE skills improved dramatically.
CS taught algorithms early and often. EE didn't really touch on them at all, except when a specific one was needed to solve a specific problem. I remember sitting in a 4th year Digital Communications course where we were learning about Viterbi decoders. The professor was having a hard time explaining it by drawing a lattice and showing how you do the computations, the students were completely lost. My friend and I were looking at what was going on and both had this lightbulb moment at the same time. "Oh, this is just a dynamic programming problem."
EE taught us way more calculus than CS did. In a CS systems modelling course we were learning about continuous-time and discrete-time state-space models. Most of the students were having a super hard time with dx/dt = A*x (x as a real vector, A as a matrix)... which makes sense since they'd only ever done single-variable calculus. The prof taught some specific technique that applied to a specific form of the problem and that was enough for students to be able to turn the crank, but no one understood why it worked.
Yeah, I had a slightly odd introduction to these things as I studied joint honours maths and physics. That meant both that I had a bit more mathematical maturity than most of the physics students and that I was being taught the more rigorous underpinnings of the maths while it was being (ab)used in all sorts of cavalier ways in physics. I liked the subject matter of physics more, but I greatly preferred the intellectual rigour of the maths.
Eric Gourgoulhon is a product of the French education system, and I often think I would have done better studying there than in the UK.
Mine was similar actually, just in Ireland.
I had started in a theoretical physics degree which was jointly taught by the maths and physics department. By my final year I had changed into an ostensibly pure maths degree, although I did it mainly to take more advanced theoretical/mathematical physics courses (which were taught by the maths department), and avoid having to do any lab work—a torsion pendulum experiment was my final straw on that one, I don't know what caused it to fuck up, but fuck that.
In the end I took on more TP courses than the TP students, nearly burnt out by the end of the year, and... didn't exactly come out with the best exam results.
Thanks for the book recommendation.
I found the physicist definition of a tensor is actually more confusing, because you are faced with these definitions how to transform these objects, but you never are really explained where does it all come from. While the mathematical definition through differential forms, co-vectors, while being longer actually explains these objects better.
> You might find it circular reasoning but it is not
Um, yes it is. "A foo is an object that transforms as a foo" is a circular definition because it refers to the thing being defined in the definition. That is what "circular definition" means.
Lobachevsky... "the analytic and algebraic topology of locally Euclidean metrizations of infinitely differentiable Riemannian manifolds"
There's antimony, arsenic, aluminum, selenium…
боже мой.
A manifold is a surface that you can put a cd shaped object on in any place on the surface, you can change the radius of the cd but it has to have some radius above 0.
Including the hole at the center of the CD?
Nicely done!
Initially I recoiled at the thought of the stiffness of the CD, but of course your absolutely right, at least for 2d manifolds.
> you can put a cd shaped object on
You're thinking of open sets.
In particular, consider two intersecting planes. You can put all the discs you like on that surface, but it's not a manifold because on the line of intersection it's not locally R2.
I always found interesting that the English mathematical terminology has two different names for "stuff that locally looks like R^n" (manifold) and "stuff that is the zero locus of a polynomial" (variety). Other languages use the same word for both, adding maybe an adjective to specify which one is meant if not clear from the context. In Italian for example they're both "varietà"
FTA
> The term “manifold” comes from Riemann’s Mannigfaltigkeit, which is German for “variety” or “multiplicity.”
In English, not all varieties are manifolds, see forex https://math.stackexchange.com/a/9017/120475
This is not really something limited to mathematics.
Man, I wish that the modern internet -- and great stuff like this -- had been around when I took GR way back when. My math chops were never good enough to /really/ get it and there were so many concepts (like this one) that were just symbols to me.
Its unfortunately all too common for Physics/Math to be taught in that way (extremely technical, memorizing or knowing equations and derivations). The best teachers would always give a ton of context as to why and how these came about.
This is such a well written article and the author is such a good communicator. Looks like they've written a book as well called Mapmatics:
[0] https://www.paulinarowinska.com/about-me
A very tight poker player
Funny how a car manifold is also a mathematical manifold but the word seems to come from different roots.
I just looked it up because I was interested in their etymologies, but it seems that the words actually have the same (Old English/Germanic) root: essentially a portmanteau of "many" + "fold."
I thought those are ethymologically about the thin-walled containment of a volumetric interior space where said space is connected to only specific ports/holes, and is often but not necessarily mandatory intertwined with a second such containment for a second space (intake+exhaust).
This has always caused me trouble when learning new concepts. A name for something will be given (e.g. manifold) and it sounds very much like something that I've come across before (e.g. a manifold in an engine) - and that then gets cemented in my brain as a relationship which I find extremely difficult to shake - and it makes understanding the new concept very challenging. More often than not the etymology of the term is not provided with the concept - not entirely unreasonable, but also not helpful for me personally.
It becomes a bigger problem when the etymology is actually a chain of almost arbitrary naming decisions - how far back do I go?!
On many occasions in my mathematics education I was able to figure out and use a concept based solely on its name. (e.g. Feynman path integral)
Names are important.
A $1,500 trip to the mechanic
That's why I clicked the title...thought for sure I was getting some engine knowledge
> They’re as fundamental to mathematics as the alphabet is to language. “If I know Cyrillic, do I know Russian?” said Fabrizio Bianchi (opens a new tab), a mathematician at the University of Pisa in Italy. “No. But try to learn Russian without learning Cyrillic.”
Something's gone badly wrong here. "Without learning Cyrillic" is the normal way to learn Russian. Pick a slightly less prominent language and 100% of learners will do it without learning anything about the writing system.
I thought the same - many languages don't have a writing system and children learn without being able to write. But that's beside the point; the point is just as valid even if the analogy is poor.
Stand at one of the poles. Walk to the equator, turn 90 degrees. Walk 1/4 the way around the equator, turn 90 degrees again. Then walk back to the pole. A triangle with sum 270 degrees!