Maths notation is needlessly complex [video](aeon.co)
aeon.co
Maths notation is needlessly complex [video]
https://aeon.co/videos/maths-notation-is-needlessly-complex-it-can-and-should-be-better
146 comments
"people naturally think in words, not in spatial relationships"
I think you may be committing the typical mind fallacy. I often find I have to consciously shove words out of my phonological loop so that I can focus on visualizing a concept in order to understand it.
I think you may be committing the typical mind fallacy. I often find I have to consciously shove words out of my phonological loop so that I can focus on visualizing a concept in order to understand it.
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> fancy notation always gets in the way of understanding because people naturally think in words, not in spatial relationships.
I'm pretty sure this is false for me. I think thinking in spacial relationships comes much more naturally to me than words. Words immediately start making my brain hurt. The notation you wrote is incredibly more painful for me to look at and understand than the triangles in the video.
I'm pretty sure this is false for me. I think thinking in spacial relationships comes much more naturally to me than words. Words immediately start making my brain hurt. The notation you wrote is incredibly more painful for me to look at and understand than the triangles in the video.
I didn't even know that powers, logs and roots were that related. My math is bad :( I struggled through some of the Khan Academy but I find it really hard to remember all the rules. The triangles really helped me. But maybe the video was overly simplistic? These triangles (visual items) can be placed in the mind palace. I can picture them now and could probably figure out where the log2(x) fits in.
Let me rephrase that: people naturally communicate using language. In your head you may think in symbols and spatial relationships, but if you want to get a concept out of your head and into someone else's head you have to use language. There's a reason that the symbology in mathematical papers is invariably wrapped in natural language.
I still disagree. I wouldn't have understood half of my abstract algebra classes if the profs had only used words to explain the concepts.
When you are first exposed to the First Isomorphism Theorem for Rings, a thousand words can't do what that simple triangle diagram can.
When you are first exposed to the First Isomorphism Theorem for Rings, a thousand words can't do what that simple triangle diagram can.
This is entirely backwards.
Mathematics is the study of patterns. Humans have amazing powerful visual and spatial reasoning centers in the brain which can process huge amounts of semi-structured information.
Writing words down instead forces everything through low-bandwidth serialized language processing apparatus instead.
Mathematical notation (and computer code, and music notation, and ...) benefit tremendously from even slightly exploiting spatial reasoning skills. But better is some kind of diagram, interactive simulation, or similar.
The fastest mathematical explanation is direct tutorial, person-to-person, where the amount of symbolic formalism can be reduced to a minimum, and hand gestures, chalk drawings, colorful verbal analogies, etc. can convey most of the message. When limited to pure symbolic formalism in a scholarly paper, mathematics teaching and learning is a much harder slog.
The reason formal mathematics uses symbolic formal language as a canonical way of recording results is that it’s easier to figure out how to verify it step-by-step and make sure that the logic is correct.
* * *
I’ll give you an example, from a paragraph a friend of mine wrote several years ago in a blog diary:
(I posit that you could decipher and understand this paragraph at least an order of magnitude faster if I drew a 30 second napkin sketch.)
> The tea room I went to is an “eight tatami design”, a large square space. Each side is two tatami lengths (four widths, two widths and a length) long. The eight tatami are arranged in the following way: from the entrance—which is in a corner, second forward on the left—two tatami extend into the room to meet the opposite wall on their short sides; the tatami touching the opposite wall is met on its right long side by a third tatami’s short side. This third tatami’s short side in turn meets a fourth tatami’s long side, whose two short sides meet the opposite wall and a fifth tatami’s short side. The fifth tatami’s second short side meets the room’s the fourth corner and its long side meets the sixth tatami’s short side. The sixth tatami’s second short side meets the first tatami’s long side. These six tatami form a square with an square empty middle. The empty middle’s sides’ dimensions are all one tatami length long. The middle, then, is filled by two tatami whose orientation is like that of tatami one, two, four and five—short sides to the opposite wall and the wall with the entrance. I tried out alternative arrangements and I might be wrong but I think that this solution (or a rotation of it) is the only possible tatami arrangement for the room’s dimensions that fulfills the rule “four corners must never converge”. Going in a circle around the room, starting at the door and following the direction in the description there, we can number the outer tatami one through six and the two inner tatami seven (left from the entrance) and eight (right from the entrance).
Mathematics is the study of patterns. Humans have amazing powerful visual and spatial reasoning centers in the brain which can process huge amounts of semi-structured information.
Writing words down instead forces everything through low-bandwidth serialized language processing apparatus instead.
Mathematical notation (and computer code, and music notation, and ...) benefit tremendously from even slightly exploiting spatial reasoning skills. But better is some kind of diagram, interactive simulation, or similar.
The fastest mathematical explanation is direct tutorial, person-to-person, where the amount of symbolic formalism can be reduced to a minimum, and hand gestures, chalk drawings, colorful verbal analogies, etc. can convey most of the message. When limited to pure symbolic formalism in a scholarly paper, mathematics teaching and learning is a much harder slog.
The reason formal mathematics uses symbolic formal language as a canonical way of recording results is that it’s easier to figure out how to verify it step-by-step and make sure that the logic is correct.
* * *
I’ll give you an example, from a paragraph a friend of mine wrote several years ago in a blog diary:
(I posit that you could decipher and understand this paragraph at least an order of magnitude faster if I drew a 30 second napkin sketch.)
> The tea room I went to is an “eight tatami design”, a large square space. Each side is two tatami lengths (four widths, two widths and a length) long. The eight tatami are arranged in the following way: from the entrance—which is in a corner, second forward on the left—two tatami extend into the room to meet the opposite wall on their short sides; the tatami touching the opposite wall is met on its right long side by a third tatami’s short side. This third tatami’s short side in turn meets a fourth tatami’s long side, whose two short sides meet the opposite wall and a fifth tatami’s short side. The fifth tatami’s second short side meets the room’s the fourth corner and its long side meets the sixth tatami’s short side. The sixth tatami’s second short side meets the first tatami’s long side. These six tatami form a square with an square empty middle. The empty middle’s sides’ dimensions are all one tatami length long. The middle, then, is filled by two tatami whose orientation is like that of tatami one, two, four and five—short sides to the opposite wall and the wall with the entrance. I tried out alternative arrangements and I might be wrong but I think that this solution (or a rotation of it) is the only possible tatami arrangement for the room’s dimensions that fulfills the rule “four corners must never converge”. Going in a circle around the room, starting at the door and following the direction in the description there, we can number the outer tatami one through six and the two inner tatami seven (left from the entrance) and eight (right from the entrance).
Comparing a less-than-optimal textual description with an optimsed symbolic or digrammatic description proves little.
It helps to know that tatami are rectangles with length 2x width (something a casual and/or non-Japanese reader might reasonably not know).
What's described is eight tatamis (length 2x their width), arranged in a square. Much of the confusion comes from describing each tatami's relation with its neighbors, rather than the space as a whole, and referencing other tatami rather than the room as a whole.
A simpler description:
A square room, four units per side, filled with eight mats, each one unit wide and two units long. Looking away from you, down either wall are two mats placed lenthways. On the near and far walls, a single mat between those, long edge to the wall. In the center square, two mats placed long edges parallel to the side walls.
Or the 3600x3600 diagram in this image: http://www.tatami.com.my/layout.jpg
The problem is your friend's blog description. Not inherent limits of language.
It helps to know that tatami are rectangles with length 2x width (something a casual and/or non-Japanese reader might reasonably not know).
What's described is eight tatamis (length 2x their width), arranged in a square. Much of the confusion comes from describing each tatami's relation with its neighbors, rather than the space as a whole, and referencing other tatami rather than the room as a whole.
A simpler description:
A square room, four units per side, filled with eight mats, each one unit wide and two units long. Looking away from you, down either wall are two mats placed lenthways. On the near and far walls, a single mat between those, long edge to the wall. In the center square, two mats placed long edges parallel to the side walls.
Or the 3600x3600 diagram in this image: http://www.tatami.com.my/layout.jpg
The problem is your friend's blog description. Not inherent limits of language.
I agree my friend’s description is not the best. He was trying to make it clear and unambiguous, and he’s a smart guy, but he hadn’t practiced this before and I’m sure he didn’t think all that deeply about it. It’s just not a priori obvious how to make a clear verbal description of a spatial relationship. A diagram of the same thing takes much less effort to construct.
I don’t think it’s possible to construct a textual description or mathematical formula of a collection of tatami patterns which takes less than 40–60 seconds per pattern to unpack into a clear mental image.
If I spend about 5–10 seconds looking at each of the 14 tatami mat diagrams in your picture, I’m confident I could reconstruct the patterns from memory a few minutes later. I can’t imagine doing the same without spending a significant amount of time on a textual explanation. Additionally, after examining an image with a bunch of tatami patterns for a couple minutes, I could probably tell you whether additional patterns satisfy the criteria to be acceptable or not, without doing any explicit analysis of those criteria. I’d have to think a lot harder if presented only with prose or formula descriptions.
I don’t think it’s possible to construct a textual description or mathematical formula of a collection of tatami patterns which takes less than 40–60 seconds per pattern to unpack into a clear mental image.
If I spend about 5–10 seconds looking at each of the 14 tatami mat diagrams in your picture, I’m confident I could reconstruct the patterns from memory a few minutes later. I can’t imagine doing the same without spending a significant amount of time on a textual explanation. Additionally, after examining an image with a bunch of tatami patterns for a couple minutes, I could probably tell you whether additional patterns satisfy the criteria to be acceptable or not, without doing any explicit analysis of those criteria. I’d have to think a lot harder if presented only with prose or formula descriptions.
My point wasn't that images or a symbolic (non-language) expression cannot be clearer and more accessible than text. It was that the example text you'd presented was exceptionally bad.
Yes, often a picture is the clearest way to express something -- I'd be better off showing you a blurry potato of the Mona Lisa than trying to describe it to you. At the same time, there are circumstances in which images aren't available.
When two people are conversing without having a visual medium -- in the dark, one is blind, over a radio or phone. When the text itself is all you have: in an anthropological or forensic context. And in such cases, with effort and refinement, that text can often be quite clear.
Which gets to the point: a diagram or symbolic notation is the result of effort and refinement. It's a reduction of a situation to its essential parts. It's also possible for such diagrams to be far other than clear or accurate. Look to ancient maps or images of animals or digrams of equipment or mechanisms, or even of the old-school maths notations that current notations have replaced. Albrecht Durer's illustration of a rhinoceros looks very little like a rhinoceros (though he was working from second-hand reports). It is a poor representation. An overly ornate diagram, chart, or illustration communicates poorly often because it seems to have little idea of what it wants to communciate.
If you've ever read or heard Edward Tufte's exposition of visual presentation, he offers many examples, some old, some new, of both good and bad graphics. His own designs are so excellent because they reduce the topic to its very essence. To draw on a reasonably contemporary example, his comparison of what NASA had used to portray O-ring erosion, prior to the Challenger shuttle disaster, and what a simple plot of erosion depth vs. temperature offered.
Which still leaves us with the question of whether or not any textual description of your tatami problem would be clearer than an image. I'm suspecting that one could get quite close.
Yes, often a picture is the clearest way to express something -- I'd be better off showing you a blurry potato of the Mona Lisa than trying to describe it to you. At the same time, there are circumstances in which images aren't available.
When two people are conversing without having a visual medium -- in the dark, one is blind, over a radio or phone. When the text itself is all you have: in an anthropological or forensic context. And in such cases, with effort and refinement, that text can often be quite clear.
Which gets to the point: a diagram or symbolic notation is the result of effort and refinement. It's a reduction of a situation to its essential parts. It's also possible for such diagrams to be far other than clear or accurate. Look to ancient maps or images of animals or digrams of equipment or mechanisms, or even of the old-school maths notations that current notations have replaced. Albrecht Durer's illustration of a rhinoceros looks very little like a rhinoceros (though he was working from second-hand reports). It is a poor representation. An overly ornate diagram, chart, or illustration communicates poorly often because it seems to have little idea of what it wants to communciate.
If you've ever read or heard Edward Tufte's exposition of visual presentation, he offers many examples, some old, some new, of both good and bad graphics. His own designs are so excellent because they reduce the topic to its very essence. To draw on a reasonably contemporary example, his comparison of what NASA had used to portray O-ring erosion, prior to the Challenger shuttle disaster, and what a simple plot of erosion depth vs. temperature offered.
Which still leaves us with the question of whether or not any textual description of your tatami problem would be clearer than an image. I'm suspecting that one could get quite close.
At the point you’re citing Tufte, I’m pretty sure we essentially agree.
Note, this is what I was responding to: “[...] people naturally communicate using language. In your head you may think in symbols and spatial relationships, but if you want to get a concept out of your head and into someone else's head you have to use language.”
Note, this is what I was responding to: “[...] people naturally communicate using language. In your head you may think in symbols and spatial relationships, but if you want to get a concept out of your head and into someone else's head you have to use language.”
I am not sure that this is the clearest possible written explanation - "Each side is two tatami lengths (four widths, two widths and a length) long"?
It is often the case that words + diagrams is the clearest way to explain something, and in most cases where you have a choice, you can choose both. Symbolic expressions fall somewhere between the two, and when used properly, further increase your chances of getting the point across. There's no need to turn every choice into an exclusive dichotomy, regardless of how common that is on HN.
It is often the case that words + diagrams is the clearest way to explain something, and in most cases where you have a choice, you can choose both. Symbolic expressions fall somewhere between the two, and when used properly, further increase your chances of getting the point across. There's no need to turn every choice into an exclusive dichotomy, regardless of how common that is on HN.
"is actually a better impedance match to people's natural mental processes, which involve language"
This is false when applied to all people. And this is the reason you disagree with the demonstration. Not all people think in language. I would wager that only half the population thinks in language.
I think visually. It's work for me to translate my thoughts into words. I have to do a translation process to write this comment. It's easy work, but my natural thoughts are pictures of concepts.
Richard Feynmann liked to talk about this a lot, but the internal mechanisms of thought vary drastically from person to person. Ironically, it seems common for people who think in words to suppose everyone else is that way. I am not sure why.
This is false when applied to all people. And this is the reason you disagree with the demonstration. Not all people think in language. I would wager that only half the population thinks in language.
I think visually. It's work for me to translate my thoughts into words. I have to do a translation process to write this comment. It's easy work, but my natural thoughts are pictures of concepts.
Richard Feynmann liked to talk about this a lot, but the internal mechanisms of thought vary drastically from person to person. Ironically, it seems common for people who think in words to suppose everyone else is that way. I am not sure why.
> I think visually.
I'm pretty sure everyone thinks visually or geometrically or spatially or something like that. Most certainly visually if you're not blind. Few people would find this to be an unsatisfactory proof that the nth triangular number is equal to the total number of pairings of n+1 objects:
http://i.imgur.com/HCfGOYp.gif
Better notation is certainly not worthless, and having a certain degree of symmetry or geometry to our notation is a good thing. I insist, nevertheless, that notation is not the biggest impedance towards conveying ideas.
I'm pretty sure everyone thinks visually or geometrically or spatially or something like that. Most certainly visually if you're not blind. Few people would find this to be an unsatisfactory proof that the nth triangular number is equal to the total number of pairings of n+1 objects:
http://i.imgur.com/HCfGOYp.gif
Better notation is certainly not worthless, and having a certain degree of symmetry or geometry to our notation is a good thing. I insist, nevertheless, that notation is not the biggest impedance towards conveying ideas.
Few people would find this to be an unsatisfactory proof that the nth triangular number is equal to the total number of pairings of n+1 objects: http://i.imgur.com/HCfGOYp.gif
It would be interesting to test this.
I forced myself to stare at that image longer than was comfortable, and I got nothing from it. Then I came back and read your words, then went back to the diagram, and still couldn't figure out what relevance the diagram had. Possibly I'm part of a small minority, possibly you are, or possibly it's 50/50. Without testing, I wouldn't wager a guess which of these is most likely.
You are likely better than me at visually assembling the individual frames into a coherent whole. The sequence forms a 3-dimensional image, but with 2 dimensions of space and 1 dimension of time. I feel my brain struggling (and failing) to project it into 3 spatial dimensions. The frenetic chaos leaves me longing for the serenity of words.
It would be interesting to test this.
I forced myself to stare at that image longer than was comfortable, and I got nothing from it. Then I came back and read your words, then went back to the diagram, and still couldn't figure out what relevance the diagram had. Possibly I'm part of a small minority, possibly you are, or possibly it's 50/50. Without testing, I wouldn't wager a guess which of these is most likely.
You are likely better than me at visually assembling the individual frames into a coherent whole. The sequence forms a 3-dimensional image, but with 2 dimensions of space and 1 dimension of time. I feel my brain struggling (and failing) to project it into 3 spatial dimensions. The frenetic chaos leaves me longing for the serenity of words.
>I'm pretty sure everyone thinks visually or geometrically or spatially or something like that. Most certainly visually if you're not blind.
I don't think visually or geometrically or spatially or something like that.
I'm naturally thinking in terms of words and abstract concepts. Visualizing something in my mind is actual work; you can describe a person to me in minute detail without me ever visualizing what they might look like. My sense of direction sucks, and I typically don't solve problems visually (or geometrically or spatially).
Of course I can still convert between thoughts and images (though conversion between thoughts and words is often easier), and sometimes a problem is easier to explain visually. I understand your proof and find it satisfactory, but it took me some time and I would have prefered a proof by induction.
I don't think visually or geometrically or spatially or something like that.
I'm naturally thinking in terms of words and abstract concepts. Visualizing something in my mind is actual work; you can describe a person to me in minute detail without me ever visualizing what they might look like. My sense of direction sucks, and I typically don't solve problems visually (or geometrically or spatially).
Of course I can still convert between thoughts and images (though conversion between thoughts and words is often easier), and sometimes a problem is easier to explain visually. I understand your proof and find it satisfactory, but it took me some time and I would have prefered a proof by induction.
What about being convinced that local extrema of smooth functions occur when the derivative is zero? You find it easier to work through an epsilon-delta algebraic proof than to look at a graph and see that it's flat wherever there's a local extremum? Do you find graphs to represent data useless and you would rather look at long lists of numbers?
The derivative is the rate of change. If I chose any two points to the left and right to the local extremum, they are non-zero and have different signs. Since there has to be a zero between those two points, the local extremum must have a derivative of zero.
That's the line of reasoning I would go. Looking at a graph, searching for exceptions and concluding that there aren't any is probably as easy in this case. Like probably most people I have a harder time thinking about epsilon-delta proofs.
>Do you find graphs to represent data useless and you would rather look at long lists of numbers?
No, graphs are great. They sacrifice precision to show simple relations between large amounts of numbers, and are really good at that. Numbers a row of numbers can't do that (the closest equivalent to a line graph is to replace each but the first number with their difference to the previous number, but that's still inferior). Natural language is lacking precision at reasonable density for the job.
So graphs are pretty much the only decent tool for their job we have invented so far.
That's the line of reasoning I would go. Looking at a graph, searching for exceptions and concluding that there aren't any is probably as easy in this case. Like probably most people I have a harder time thinking about epsilon-delta proofs.
>Do you find graphs to represent data useless and you would rather look at long lists of numbers?
No, graphs are great. They sacrifice precision to show simple relations between large amounts of numbers, and are really good at that. Numbers a row of numbers can't do that (the closest equivalent to a line graph is to replace each but the first number with their difference to the previous number, but that's still inferior). Natural language is lacking precision at reasonable density for the job.
So graphs are pretty much the only decent tool for their job we have invented so far.
> Few people would find this to be an unsatisfactory proof that the nth triangular number is equal to the total number of pairings of n+1 objects:
I accept it as a proof for the specific case, but it took me quite a while to understand what was going on. I definitely can't tell whether it generalizes to arbitrary sizes this way. Maybe if I had a look at all frames at once I could notice a pattern, but the constant movement is really distracting.
I accept it as a proof for the specific case, but it took me quite a while to understand what was going on. I definitely can't tell whether it generalizes to arbitrary sizes this way. Maybe if I had a look at all frames at once I could notice a pattern, but the constant movement is really distracting.
>It seems common for people who think in words to suppose everyone else is that way. I am not sure why.
Perhaps because that's how you get to know what someone else is thinking? Even when you draw pictures (which is quite common in technical fields) they usually need some explanation.
Perhaps because that's how you get to know what someone else is thinking? Even when you draw pictures (which is quite common in technical fields) they usually need some explanation.
> Not all people think in language
Yes, you're right. But everyone communicates in language. I've updated the original comment.
Yes, you're right. But everyone communicates in language. I've updated the original comment.
In response to your update, I believe that visual methods of communicating are superior to words. Ever hear how a picture says 1000 words?
Is a description of a triangle really superior at communicating the concept when compared to a picture of one?
All that said the understanding of functions you point is helpful. Especially for someone from a computer science background.
Is a description of a triangle really superior at communicating the concept when compared to a picture of one?
All that said the understanding of functions you point is helpful. Especially for someone from a computer science background.
Counterexample:
> In theoretical physics, Feynman diagrams are pictorial representations of the mathematical expressions describing the behavior of subatomic particles. [...] The interaction of sub-atomic particles can be complex and difficult to understand intuitively. Feynman diagrams give a simple visualization of what would otherwise be a rather arcane and abstract formula. As David Kaiser writes, "since the middle of the 20th century, theoretical physicists have increasingly turned to this tool to help them undertake critical calculations", and as such "Feynman diagrams have revolutionized nearly every aspect of theoretical physics".
https://en.wikipedia.org/wiki/Feynman_diagram
----
This is the first example that came to mind, but there are certainly very many others. Because the idea that written flat text is inherently a global maximum for representing/communicating/working with all concepts is absolutely absurd.
This is immediately self-evident if you've ever tried to teach anything complex to anyone who wasn't already a domain expert in an immediately adjacent area.
> In theoretical physics, Feynman diagrams are pictorial representations of the mathematical expressions describing the behavior of subatomic particles. [...] The interaction of sub-atomic particles can be complex and difficult to understand intuitively. Feynman diagrams give a simple visualization of what would otherwise be a rather arcane and abstract formula. As David Kaiser writes, "since the middle of the 20th century, theoretical physicists have increasingly turned to this tool to help them undertake critical calculations", and as such "Feynman diagrams have revolutionized nearly every aspect of theoretical physics".
https://en.wikipedia.org/wiki/Feynman_diagram
----
This is the first example that came to mind, but there are certainly very many others. Because the idea that written flat text is inherently a global maximum for representing/communicating/working with all concepts is absolutely absurd.
This is immediately self-evident if you've ever tried to teach anything complex to anyone who wasn't already a domain expert in an immediately adjacent area.
> This is immediately self-evident if you've ever tried to teach anything complex to anyone who wasn't already a domain expert in an immediately adjacent area.
I am a practising computational scientist, and I work in an interdisciplinary field. I work with the sort of folks you mentioned above daily.
And while I often do not start with an equation, that does not mean they are not useful. I don't think anyone is suggesting that to a non-specialist they are. Typically, I start with thought experiments or plots. People are often visual learners.
However, that does not mean that mathematical notation is a problem. Mathematics is a language, and like any language one must spend years (or in my case, decades) learning how to 'speak' it. While I would not suggest it is inherently a global maximum, I'm skeptical it is not at least a pretty good local max.
I am a practising computational scientist, and I work in an interdisciplinary field. I work with the sort of folks you mentioned above daily.
And while I often do not start with an equation, that does not mean they are not useful. I don't think anyone is suggesting that to a non-specialist they are. Typically, I start with thought experiments or plots. People are often visual learners.
However, that does not mean that mathematical notation is a problem. Mathematics is a language, and like any language one must spend years (or in my case, decades) learning how to 'speak' it. While I would not suggest it is inherently a global maximum, I'm skeptical it is not at least a pretty good local max.
Could this be because as you go, you introduce more dimensions, degrees of freedom into your mental model. To accommodate for this, the brain needs to make room, and time and time again needs to reshuffle to make space for more connections or to eventually collapse them into multiple different models of a lesser degree.
I just read that second order logic can represent all higher order logics. The degree of the logic means the degree of nesting of sets into sets, that are quantified over by the logic expression. If sets of sets of values are all it takes, a two dimensional spatial representation should be enough. So far so good?
The problem with text is that it is largely sequential, i.e. one dimension, left to right.
> I am a practising computational scientist
only practice makes perfect (notice the 2nd C ;-)
I just read that second order logic can represent all higher order logics. The degree of the logic means the degree of nesting of sets into sets, that are quantified over by the logic expression. If sets of sets of values are all it takes, a two dimensional spatial representation should be enough. So far so good?
The problem with text is that it is largely sequential, i.e. one dimension, left to right.
> I am a practising computational scientist
only practice makes perfect (notice the 2nd C ;-)
Diagrams are good for representing and reasoning about physical things, and for mathematical concepts that have straightforward physical models, like numbers. But this is because there's a natural correspondence between these diagrams and the things they represent. There is no natural correspondence between a triangle and logs/exponents/roots beyond that fact that triangles have three sides and three vertices, and the set {log, expt, root} has three members.
> the idea that written flat text is inherently a global maximum
It's quite the opposite, text is a global minimum. Sequential text is best suited to represent some first order logic, a recursive enumeration of nouns. Or better yet a zeroth order logic, a simple enumeration of nouns. That is a desirable logic to work in, because it's complete. But we do need second order logic to ... beats me :)
It's quite the opposite, text is a global minimum. Sequential text is best suited to represent some first order logic, a recursive enumeration of nouns. Or better yet a zeroth order logic, a simple enumeration of nouns. That is a desirable logic to work in, because it's complete. But we do need second order logic to ... beats me :)
This would have made self learning much easier for me. I remember when I was in high school I tried reading "The Road to Reality" by Roger Penrose and the gradient symbol seemed like some magical incantation only sorcerers could understand. If I had been told it was just a simple symbolic manipulation, I probably could have gotten through that section (now I would probably just Google it).
I bet if Professor Penrose rewrote his book in code-language it would be much easier to learn, at least for software engineers!
I bet if Professor Penrose rewrote his book in code-language it would be much easier to learn, at least for software engineers!
"By relieving the brain of all unnecessary work, a good notation sets it free to concentrate on more advanced problems, and, in effect, increases the mental power of the race"
-- Whitehead
-- Whitehead
> not everyone thinks in language. But everyone communicates
> in language. For communicating mathematical concepts,
> language is the best tool we have. pow(3) b*n b⍟x x⍟bPeople need to understand notation is a language. In you three lines of functions, you've used (),= four notational symbols, without which your statement would be uselessly long. People do communicate in languages. Math notations happen to be the native tongue of people using math (they were taught at first in schools).
For people speaking in that different language, the following are the same:
bⁿ=x
b^n=x
pow(b,n)==x
b*n=x
[deleted]
> It's not about the notation at all, it's about understanding that EXPT, SQRT and LOG are functions with a particular relationship to each other.
The notation being proposed is for those three specific functions. It's just replacing the existing notations for those functions with a new notation for those functions. Specifically, the author claims that this notation will make the relationship between those functions clearer.
> because there are a lot of different relationships that are naturally described by putting three things in a triangle.
He's not proposing this as a generic functional notation, but rather as a notation for these three specific functions. So yes, many other functions might benefit from a similar sort of notation, but this proposal isn't for any of them.
> But everyone communicates in language. For communicating mathematical concepts, language is the best tool we have.
When teaching math, teachers use a combination of natural language, formulae, tables, and diagrams. Teaching math without words would be (as you point out) nonsensical; but teaching math only using words would be incredibly hard to understand & follow.
The notation being proposed is for those three specific functions. It's just replacing the existing notations for those functions with a new notation for those functions. Specifically, the author claims that this notation will make the relationship between those functions clearer.
> because there are a lot of different relationships that are naturally described by putting three things in a triangle.
He's not proposing this as a generic functional notation, but rather as a notation for these three specific functions. So yes, many other functions might benefit from a similar sort of notation, but this proposal isn't for any of them.
> But everyone communicates in language. For communicating mathematical concepts, language is the best tool we have.
When teaching math, teachers use a combination of natural language, formulae, tables, and diagrams. Teaching math without words would be (as you point out) nonsensical; but teaching math only using words would be incredibly hard to understand & follow.
Nope... The triangle notation is actually awesome because it does NOT put all the focus on the fact that EXP, SQRT an LOG are functions, and it actually promotes thinking at a higher level than first order functions: it's about an expression than can generate multiple functions or a higher order function.
Let me explain:
Let we define
And from this on you can go much easier to much deeper questions than the ones you get from staring at the graphs of 3 different functions that don't seem to relate much to each other.
Classic mathematical notation is great for doing very basic physics and engineering... but horrible for anything deeper. Only thing is that people doing anything deeper are smart enough to be able to tolerate a few horrible notations and probably even like them because they scare the "peasants" away :)
But we should be prioritising what's best for education, not for the macho-ism of uber-physicists or uber-mathematicians, so go triangles go!
Let me explain:
Let we define
EX(b, e, x) := { pow(b, e) == x }
EX(b, e, *) == pow(b, e)
EX(a, *, b) == log(a, b)
EX(*, e, x) == root(e, x) # or sqrt(x) if e == 2
And from this on you can go much easier to much deeper questions than the ones you get from staring at the graphs of 3 different functions that don't seem to relate much to each other.
Classic mathematical notation is great for doing very basic physics and engineering... but horrible for anything deeper. Only thing is that people doing anything deeper are smart enough to be able to tolerate a few horrible notations and probably even like them because they scare the "peasants" away :)
But we should be prioritising what's best for education, not for the macho-ism of uber-physicists or uber-mathematicians, so go triangles go!
Yup. Frequently, in mathematics, we have some ternary relation r(a, b, c) such that fixing values for any two arguments uniquely determines a corresponding value for the third. This encapsulates not only the relationship between a^b = c, log_a(c) = b, and b-th root of c = a, but also the relationship between a + b = c, c - a = b, and c - b = a, the relationship between a * b = c, c/a = b, and c/b = a (as well as the non-commutative version of this where we have separate notions of left- and right-division), and myriad other such ternary relations.
In all of these cases, a triangular notation is apt for expressing and emphasizing the structure that any two arguments fix the third.
That having been said, that is all the triangular notation does; it does not help with any further structure that may be around for that particular relation (e.g., rules like a^(b * c) = (a^b)^c, which the triangular notation does little to simplify, though perhaps further notational choices could make these also immediate and clear). Still, when that ternary structure is what one cares to emphasize (and why shouldn't it often be of interest?), one should go ahead and emphasize it.
In all of these cases, a triangular notation is apt for expressing and emphasizing the structure that any two arguments fix the third.
That having been said, that is all the triangular notation does; it does not help with any further structure that may be around for that particular relation (e.g., rules like a^(b * c) = (a^b)^c, which the triangular notation does little to simplify, though perhaps further notational choices could make these also immediate and clear). Still, when that ternary structure is what one cares to emphasize (and why shouldn't it often be of interest?), one should go ahead and emphasize it.
> a * b = c, c/a = b, and c/b = a
this readily shows that it's not about triangles, but some sort of binary tree, because / or * do not operate on the same domain. One is a special case of the other, or an extension.
a bifurcation or whatchamacallit.
this readily shows that it's not about triangles, but some sort of binary tree, because / or * do not operate on the same domain. One is a special case of the other, or an extension.
a bifurcation or whatchamacallit.
Huh? I'm having a little bit of difficulty following you.
The triangle just captures that there are three quantities involved (with any two determining the third). I don't know what you mean with references to bifurcations and binary trees and so on.
If the mention of different domains is about the fact that you can't divide by zero, sure; in the same way, we find problems taking the logarithm of zero, or raising zero to negative powers, with uniquely pinning down roots of negative numbers, etc. For now, I am glossing over these things; let us suppose, for example, that in the ternary relation a * b = c, I intend all quantities to be drawn from some multiplicative group (thus, nonzero, and thus, with any two determining the third).
The triangle just captures that there are three quantities involved (with any two determining the third). I don't know what you mean with references to bifurcations and binary trees and so on.
If the mention of different domains is about the fact that you can't divide by zero, sure; in the same way, we find problems taking the logarithm of zero, or raising zero to negative powers, with uniquely pinning down roots of negative numbers, etc. For now, I am glossing over these things; let us suppose, for example, that in the ternary relation a * b = c, I intend all quantities to be drawn from some multiplicative group (thus, nonzero, and thus, with any two determining the third).
I didn't watch the video for several reasons, so if I was off, I'm sorry. From the comments I couldn't deduce what's actually wrong with the notation, or which notation for that matter. Yes, bad HN ettiquette is to respond anyway.
> If the mention of different domains is about the fact that you can't divide by zero
> I intend all quantities to be drawn from some multiplicative group
That's it. Still, I was trying to draw some hierarchical network, where log and root are both inverses of exp.
In the same way, multiplication is a special case of addition. Although It might not have to be, if it's just my preference to look at it that way. It reminds me of the diamond dependency problem (https://en.wikipedia.org/wiki/Diamond_problem#The_diamond_pr...).
Multiplicative groups to me look like a special case, too. The arrow diagrams look like category theory. I on the other hand just talk from intuition and my experience with the elementary functions, the order I learned in school.
EDIT: Subtraction and division as well aren't associative, so do they really form a subgroup? Another problem besides needlessly complicated notation is ambiguous notation. Wikipedia lists two alternatives for multiplicative groups. One is a special case of a ring which does have a null element. Now, in c/b=a, c can be the null element, but then a would be too, so c/a=b is still undefined. I'd guess that holds for the non-commutative version as well.
> If the mention of different domains is about the fact that you can't divide by zero
> I intend all quantities to be drawn from some multiplicative group
That's it. Still, I was trying to draw some hierarchical network, where log and root are both inverses of exp.
In the same way, multiplication is a special case of addition. Although It might not have to be, if it's just my preference to look at it that way. It reminds me of the diamond dependency problem (https://en.wikipedia.org/wiki/Diamond_problem#The_diamond_pr...).
Multiplicative groups to me look like a special case, too. The arrow diagrams look like category theory. I on the other hand just talk from intuition and my experience with the elementary functions, the order I learned in school.
EDIT: Subtraction and division as well aren't associative, so do they really form a subgroup? Another problem besides needlessly complicated notation is ambiguous notation. Wikipedia lists two alternatives for multiplicative groups. One is a special case of a ring which does have a null element. Now, in c/b=a, c can be the null element, but then a would be too, so c/a=b is still undefined. I'd guess that holds for the non-commutative version as well.
If people around here are interested in this idea, I would just like to point out that this is basically what embedding miniKanren (a super-simple logic programming DSL) in your programming language lets you do. You write "give me all q such that 2^q = 8 with an expression like `find q. PowEq(2, q, 8)." Also since the asterisk is sort of "loaded" it may be more helpful notation-wise to use an underscore, PowEq(2, _, 8) standing for the above.
The most obvious problem here is that you have to think about the composability of relations the way you do with functions, because PowEq(3, PowEq(2, _, 8), _) makes sense (find me the number that you get when you raise 3 to the [whatever number you'd have to raise 2 to, to get 8]), but the very similar-looking PowEq(3, PowEq(2, 3, 8), _) is actually a type error, the function is being called on a statement rather than a value.
The most obvious problem here is that you have to think about the composability of relations the way you do with functions, because PowEq(3, PowEq(2, _, 8), _) makes sense (find me the number that you get when you raise 3 to the [whatever number you'd have to raise 2 to, to get 8]), but the very similar-looking PowEq(3, PowEq(2, 3, 8), _) is actually a type error, the function is being called on a statement rather than a value.
I can't quite figure out which side you're arguing here. On the one hand you say that the triangle notation is awesome, but then you go on to use linear strings of symbols in exactly the manner that I was advocating. The whole point of the triangle notation (and all spatial notations in math -- subscripts, superscripts, the various arguments to sums, products, and integrals, etc) is that it's two-dimensional. That makes it really hard to render into ascii, which is one of the many reasons I think it's a bad idea.
I am certainly not arguing against teaching higher level concepts.
I am certainly not arguing against teaching higher level concepts.
> The whole point of the triangle notation is that it's two-dimensional.
Nope. I'm arguing that what the OP calls "triangle notation" is just a particular representation of a cool idea. The idea actually happens to be independent of how you happen to represent it. Yes, it lends itself well to 2D scribling. but the real reason I find it's cool, and useful to more than schoolchildren is that it's a much deeper idea in disguise. I think the author doesn't even realize how awesome is the idea he's promoting and why...
P.S. And about sides: generally, when I enter a 2-sided argument, I pretend to support one but argue for neither or for both at the same time, and from the argument I try to pull out a 3rd side (yup, I love triangles) that no one has yes seen, offering a fresh perspective. You can never have enough opposing points of view :)
Nope. I'm arguing that what the OP calls "triangle notation" is just a particular representation of a cool idea. The idea actually happens to be independent of how you happen to represent it. Yes, it lends itself well to 2D scribling. but the real reason I find it's cool, and useful to more than schoolchildren is that it's a much deeper idea in disguise. I think the author doesn't even realize how awesome is the idea he's promoting and why...
P.S. And about sides: generally, when I enter a 2-sided argument, I pretend to support one but argue for neither or for both at the same time, and from the argument I try to pull out a 3rd side (yup, I love triangles) that no one has yes seen, offering a fresh perspective. You can never have enough opposing points of view :)
We may be in violent agreement here. I agree that the idea being communicated is a cool idea. I just don't think the triangle notation actually contributes much to the effective communication of that idea.
And...
> That makes it really hard to render into ascii, which is one of the many reasons I think it's a bad idea.
...this is one idea that I really dislike. We have touchscreens everywhere, soon 3D glasses everywhere, a whole plethora of input devices, and at some point we'll have usable direct electric bran-computer-interfaces. Why limit to ASCII?! Heck, even 3D is not enough in my mind - you can "easily" imagine RGB color displayed on 3D objects as a secondary attached 3D space, hence boom, you can "easily" (probably not for most, but with enough training doable) visualize 4 (3 space + 1 time) + 3 (r, g, b) = 7 dimensions using sight alone!
Heck even programming languages should move away from 1.5D text. I can't wait to see the first really cool visual syntaxes. I imagine that the Perl 6 equivalent of a visual 2D/3D programming language combined with a more useful input device than a keyboard would look awe inspiring!
Imagine how cool would it be to play with 7D mathematical notations and to what cool insights they will lead :) (though for teaching and publication 2D is probably enough for math as the cool thing about math is still that you can do it with just pen and paper or in you head while using the 3D-processing part of your mind do things like walk on the street...)
> That makes it really hard to render into ascii, which is one of the many reasons I think it's a bad idea.
...this is one idea that I really dislike. We have touchscreens everywhere, soon 3D glasses everywhere, a whole plethora of input devices, and at some point we'll have usable direct electric bran-computer-interfaces. Why limit to ASCII?! Heck, even 3D is not enough in my mind - you can "easily" imagine RGB color displayed on 3D objects as a secondary attached 3D space, hence boom, you can "easily" (probably not for most, but with enough training doable) visualize 4 (3 space + 1 time) + 3 (r, g, b) = 7 dimensions using sight alone!
Heck even programming languages should move away from 1.5D text. I can't wait to see the first really cool visual syntaxes. I imagine that the Perl 6 equivalent of a visual 2D/3D programming language combined with a more useful input device than a keyboard would look awe inspiring!
Imagine how cool would it be to play with 7D mathematical notations and to what cool insights they will lead :) (though for teaching and publication 2D is probably enough for math as the cool thing about math is still that you can do it with just pen and paper or in you head while using the 3D-processing part of your mind do things like walk on the street...)
> Why limit to ASCII?!
Because it's precise and unambiguous, like math should be but often isn't. And it's easy to manipulate with code, so that makes it easy to enlist the help of computers to help with the work.
Because it's precise and unambiguous, like math should be but often isn't. And it's easy to manipulate with code, so that makes it easy to enlist the help of computers to help with the work.
Whatever the relation between them, the fact is the EXP, LOG and ROOT really are distinct functions with their own personalities. EXP grows fast, and is single-valued. LOG and ROOT grow slowly and are multi-valued, but in different ways.
Triangles and the star notation above are useful in cases where the ternary relation is very salient. And that might include teaching, but they are only part of the picture.
Triangles and the star notation above are useful in cases where the ternary relation is very salient. And that might include teaching, but they are only part of the picture.
> people naturally think in words, not in spatial relationships
Not at all. There was a prior post titled "The link between language and cognition is a red herring" (https://aeon.co/ideas/the-link-between-language-and-cognitio...). The brain follows a spatial layout that, as a graph, is at least three-dimesional.
I'd guess that language can be encoded in higher dimensions, too, but I assume that the sequential nature of words in a sentence would require an ordering in a zero order logic to be logically complete. As I've learned from the RegEx discussion about the stackoverflow bug today, backreferences don't have to be done with recursion and that recursion and backtracking have suboptimal space and time complexity (https://news.ycombinator.com/item?id=12131909) (or sumsuch, I'm sure I misunderstood some).
it's easy to transform your examples into one line, in some sort of differential equation. At least representing log and root as inverse(exp(x, n)) for bound x or n.
Not at all. There was a prior post titled "The link between language and cognition is a red herring" (https://aeon.co/ideas/the-link-between-language-and-cognitio...). The brain follows a spatial layout that, as a graph, is at least three-dimesional.
I'd guess that language can be encoded in higher dimensions, too, but I assume that the sequential nature of words in a sentence would require an ordering in a zero order logic to be logically complete. As I've learned from the RegEx discussion about the stackoverflow bug today, backreferences don't have to be done with recursion and that recursion and backtracking have suboptimal space and time complexity (https://news.ycombinator.com/item?id=12131909) (or sumsuch, I'm sure I misunderstood some).
it's easy to transform your examples into one line, in some sort of differential equation. At least representing log and root as inverse(exp(x, n)) for bound x or n.
As Gauss wrote: "In our opinion, however, such
truths should be extracted from notions rather than from notations."
(Nowadays most technical math is written in English, but back in Gauss's time people used Latin. He really wrote: "At nostro quidem iudicio huiusmodi veritates ex notionibus potius quam es notationibus hauriri debebant.")
(Nowadays most technical math is written in English, but back in Gauss's time people used Latin. He really wrote: "At nostro quidem iudicio huiusmodi veritates ex notionibus potius quam es notationibus hauriri debebant.")
The video doesn't give any examples of complex formulae written using their "triangle of power" notation. This may be useful for teaching mathematics, but it's not clear that it scales. Much of the benefit of the notation could come from simply writing x⁽¹/²⁾ instead of √x. (Annoying, unicode doesn't have a superscript slash. "(",")", "=", and all the digits are available in superscript, but not "/".).
A huge hassle in moderately advanced mathematics, where new domain-specific operators are introduced, is ambiguous precedence of operators. There's a tendency to define operators in such a way as to minimize the number of parentheses required for the most popular uses of that operator. Such idioms make formulas hard to read. For an example, watch Andrew Ng's videos on machine learning.
It might be useful to always parenthesize in textbooks. Teach kids to always write "log(n)" instead of "log n". After all, how does "log n × m" parse? Is there an official standard on that? If so, where?
A huge hassle in moderately advanced mathematics, where new domain-specific operators are introduced, is ambiguous precedence of operators. There's a tendency to define operators in such a way as to minimize the number of parentheses required for the most popular uses of that operator. Such idioms make formulas hard to read. For an example, watch Andrew Ng's videos on machine learning.
It might be useful to always parenthesize in textbooks. Teach kids to always write "log(n)" instead of "log n". After all, how does "log n × m" parse? Is there an official standard on that? If so, where?
When I learned the basics of electronics in middle school our teacher (and, actually, every teacher since) explained the relation between resistance, voltage and current using a triangle:
I'm not sure it helped me. Anecdotal evidence: Just now, as I was trying to figure out which letter goes where, I was actually thinking in terms of what's going, as in "if at constant voltage I increase the resistance, the current should drop, ok, so I = U/R?".
So, in conclusion, I don't know which one is "better". Most likely, different people think in different terms and require different methods of learning, so if there's another way of explaining things I think that's good, isn't it?
/U\
/R*I\
I'm not sure it helped me. Anecdotal evidence: Just now, as I was trying to figure out which letter goes where, I was actually thinking in terms of what's going, as in "if at constant voltage I increase the resistance, the current should drop, ok, so I = U/R?".
So, in conclusion, I don't know which one is "better". Most likely, different people think in different terms and require different methods of learning, so if there's another way of explaining things I think that's good, isn't it?
I always pick one at random, U=R/I
You really only have 1/2 chance to make a mistake.
You really only have 1/2 chance to make a mistake.
Most of the way through the video, the narrator is saying things like "the student can easily see that..." No, she can't! Not if 'see' implies understanding. The narrator acknowledges this at the end, but by then, he has passed up the opportunity to show the usefulness of this notation (if that is so) by applying it in more complex situations.
The video is also plagued with distracting visual effects (there are some that are worthwhile, but most are not.)
The video is also plagued with distracting visual effects (there are some that are worthwhile, but most are not.)
While HN has definitely identified the problem with adding yet another notation, I think it's fairly straightforward to to remove the "root" operator. Simply use the exponent notation with a power of "1/3".
This gets directly at the core concept of what a "root" really is, and it's straightforward to manipulate with addition/multiplication of groups of roots using the normal arithmetic methods.
This gets directly at the core concept of what a "root" really is, and it's straightforward to manipulate with addition/multiplication of groups of roots using the normal arithmetic methods.
While I do prefer fractional exponents sometimes, what about the "asymmetry" of imaginary numbers?
There may be a benefit to capturing that sqrt(square(x)) != x
There may be a benefit to capturing that sqrt(square(x)) != x
This seems like a good teaching tool, but seriously would anyone want to write out big equations with the triangle instead of x^n ? Powers are used incredibly often and the standard notation is clearly easier to write in both typing and handwriting, and the same is true for log. And as has been mentioned the n-th root symbol is technically not a true extra notation, as you could always write x^(1/n) - but it does make some equations extra clear because there are so many roots that often need to be taken.
So, don't complain about the notations - the triangle would be far more annoying to write equations with if that's the only thing you use. The notations are like 'helper' functions that make implementing a larger function easier, but also make things less clear to start out with because there are more forms of the same thing - so use the triangle to point out it's all the same thing (or all related, anyhow) but keep the notation all the same.
So, don't complain about the notations - the triangle would be far more annoying to write equations with if that's the only thing you use. The notations are like 'helper' functions that make implementing a larger function easier, but also make things less clear to start out with because there are more forms of the same thing - so use the triangle to point out it's all the same thing (or all related, anyhow) but keep the notation all the same.
- 2^3, log_2(8) ... why do we have 3 different ways ? Because, playing with tautologies is the whole point of mathematics (and language) !
- The new "notation" is essentially expressing the same language in a different alphabet.
I was hoping for some smack talk about partial derivatives (see SICM) or about the absolute proliferation of symbols in Differential Geometry...
- The new "notation" is essentially expressing the same language in a different alphabet.
I was hoping for some smack talk about partial derivatives (see SICM) or about the absolute proliferation of symbols in Differential Geometry...
I strongly need to reread that book.
It's a work in progress https://en.m.wikipedia.org/wiki/Zenzizenzizenzic
I agree that the notation can be a problem, but it has to serve a number of different uses:
1. Communication between mathematicians.
Like coding (or legal communication), the notation needs to be precise and unambiguous or you can cause misunderstandings the derail the point you are trying to make. This does not always make for easy to understand notation.
2. A shorthand for key concepts and relations to other math users.
This is something of a problem that is caused by the fact that mathematicians are often the people who teach the non-mathematicians. Who then in turn use the math or teach it to novices. It's a large hurdle, but then you're left with people who learned one notation as a novice, and are then forced to relearn how to communicate these concepts if you want to participate in the academic conversation.
3. A method of communication between non-mathematicians (like physicists or engineers).
There is actually a fairly large difference between how physicists communicate a key concept in their field, and how a mathematician might communicate the same idea. This isn't a large problem at the start, but then you're left with trying to move the ideas down the chain to novices and you have possibly several competing notations that eventually have to be sorted out. Which is why some notation is carried down and others eventually gets "weeded out".
1. Communication between mathematicians.
Like coding (or legal communication), the notation needs to be precise and unambiguous or you can cause misunderstandings the derail the point you are trying to make. This does not always make for easy to understand notation.
2. A shorthand for key concepts and relations to other math users.
This is something of a problem that is caused by the fact that mathematicians are often the people who teach the non-mathematicians. Who then in turn use the math or teach it to novices. It's a large hurdle, but then you're left with people who learned one notation as a novice, and are then forced to relearn how to communicate these concepts if you want to participate in the academic conversation.
3. A method of communication between non-mathematicians (like physicists or engineers).
There is actually a fairly large difference between how physicists communicate a key concept in their field, and how a mathematician might communicate the same idea. This isn't a large problem at the start, but then you're left with trying to move the ideas down the chain to novices and you have possibly several competing notations that eventually have to be sorted out. Which is why some notation is carried down and others eventually gets "weeded out".
Presumably people have been rightly complaining about this for 100's of years?
I'd say the same is true for musical notation.
Though similarly I doubt anything will ever be done to ameliorate these issues, the bad old system which has accreted over time has a tremendous amount of cultural momentum
This also brings to mind other attempts to clarify notation, such as https://mitpress.mit.edu/sites/default/files/titles/content/... and https://en.wikipedia.org/wiki/Geometric_algebra#Relationship...
Not this crap again.
The notation is the way it is because it works for mathematicians. Works well. It's not that complicated, but it does take some time.
Just learn the god damn notation and quit whining about it. Ugh.
The notation is the way it is because it works for mathematicians. Works well. It's not that complicated, but it does take some time.
Just learn the god damn notation and quit whining about it. Ugh.
I've always viewed it kind of like minimized and obfuscated source code. If you took a program and replaced all of the variables with single (greek) letters and created custom notation for every type of function you would get something that looks a lot like mathematical notation.
People can certainly learn it, but it's hard to argue that it isn't a barrier to entry and has no doubt helped to turn many students off from mathematics.
I mean the primary reason a lot of our notation looks the way it does is so that it can be written compactly (paper was expensive) using quill pens. Is it so crazy to consider a world where mathematical language is more self documenting?
People can certainly learn it, but it's hard to argue that it isn't a barrier to entry and has no doubt helped to turn many students off from mathematics.
I mean the primary reason a lot of our notation looks the way it does is so that it can be written compactly (paper was expensive) using quill pens. Is it so crazy to consider a world where mathematical language is more self documenting?
It's not just math, but every other field as well. Notation has only become more succinct as the price of paper fell, so it's really programming that's held back by legacy limitations and ASCII.
Even in latex or scala you're going to write out the symbol name anyway, for the writer of the symbol there is no significant advantage over raw ascii. Unfamilar readers will have to spend time understand the notation instead of the underlying concept. Familar readers however do get a moderate increase in readability.
Generally in tech we have a lot of domain specific knowledge which means you will find yourself far more often in the position of the writer or unfamilar reader than the familar reader.
Generally in tech we have a lot of domain specific knowledge which means you will find yourself far more often in the position of the writer or unfamilar reader than the familar reader.
I don't like it for a number of reasons.
One relatively simple one is that it doesn't naturally convert to a typed out version. Outside of category theory, where big commutative diagrams really do a ton of work, we should try to avoid introducing too much notation you can't type out, especially at the introductory level. Suppose two kids are trying to study together over facebook chat - how are they going to write these huge triangles out?
Second, it's actually unecessarily complicated. It introduces three concepts (exponentiation, logarithms and roots) as though they were entirely separate. But actually, the easiest way to understand how to work with roots is to just define them in terms of exponentiation. E.g. the nth root of x is just x^(1/n). All of the normal rules for roots follow from the rules for exponentiation (and fractions) immediately. You don't need a third side to the triangle, that's just adding extra complications - all you really need is exponentiation and logarithms and a way of representing that they're inverse operations in a certain sense.
So there's actually a simpler way to express all this in a notation that's much more similar to what we've seen before. The trick is to draw out parallels between familiar operations like multiplication and division. Note: I made this up 10 minutes ago, apologies if there are very similar proposals, it's just really obvious.
Most of the basic arithmetic operations can be written using binary infix operators, e.g. +, * and /. It turns out, exponentiation and logarithms can be too. In fact, if you're typing, exponentiation already is.
Let x raised to the nth power be written as (x ^ n) (note this is essentially exactly the way it's already typed out, only I'm using some unusual extra white space to emphasize that we're treating ^ as an infix operator). It's a little upward arrow that says scale x up by n (exponentially).
And let log base n of x be written as (x v n) or possibly (x \/ n). It's a little downward arrow that says scale x down by n (logarithmically).
This makes the two operations work in a way that's fairly analogous to multiplication and division in a relatively neat way. For example, for positive integers, multiplication can be thought of as (linearly) scaling one number up by another, while division scales it (linearly) downward in an inverse way. As already noted The same holds for these two operators, only the scaling is non-linear.
Lots of familiar relationships carry over, e.g. Note that ((x * n) / n) = x. Similarly ((x ^ n) v n) = x. And where (n * (x / n)) = x, it's also the case that (n ^ (x v n)) = x.
And where multiplication and division interact with addition and subtraction, those operators interact similarly with multiplication and division, e.g. where x * (n + m) = (x * n) + (x * m), similarly x ^ (n * m) = (x ^ n) * (x ^ m). And so on. You can derive all the relationships you need from a very small number of rules that are easy to remember because they're structurally almost exactly like the rules for operations you're familiar with. You already know those when you learn about exponentiation, so you don't have to learn new and weird geometric relationships.
All of those nice properties there follow from the fact that exponentiation is just the next operation in the sequence of hyperoperations (see https://en.wikipedia.org/wiki/Hyperoperation ) after multiplication.
Introducing this weird three place operator actually masks the underlying simplicity of exponentiation and its inverse.
One relatively simple one is that it doesn't naturally convert to a typed out version. Outside of category theory, where big commutative diagrams really do a ton of work, we should try to avoid introducing too much notation you can't type out, especially at the introductory level. Suppose two kids are trying to study together over facebook chat - how are they going to write these huge triangles out?
Second, it's actually unecessarily complicated. It introduces three concepts (exponentiation, logarithms and roots) as though they were entirely separate. But actually, the easiest way to understand how to work with roots is to just define them in terms of exponentiation. E.g. the nth root of x is just x^(1/n). All of the normal rules for roots follow from the rules for exponentiation (and fractions) immediately. You don't need a third side to the triangle, that's just adding extra complications - all you really need is exponentiation and logarithms and a way of representing that they're inverse operations in a certain sense.
So there's actually a simpler way to express all this in a notation that's much more similar to what we've seen before. The trick is to draw out parallels between familiar operations like multiplication and division. Note: I made this up 10 minutes ago, apologies if there are very similar proposals, it's just really obvious.
Most of the basic arithmetic operations can be written using binary infix operators, e.g. +, * and /. It turns out, exponentiation and logarithms can be too. In fact, if you're typing, exponentiation already is.
Let x raised to the nth power be written as (x ^ n) (note this is essentially exactly the way it's already typed out, only I'm using some unusual extra white space to emphasize that we're treating ^ as an infix operator). It's a little upward arrow that says scale x up by n (exponentially).
And let log base n of x be written as (x v n) or possibly (x \/ n). It's a little downward arrow that says scale x down by n (logarithmically).
This makes the two operations work in a way that's fairly analogous to multiplication and division in a relatively neat way. For example, for positive integers, multiplication can be thought of as (linearly) scaling one number up by another, while division scales it (linearly) downward in an inverse way. As already noted The same holds for these two operators, only the scaling is non-linear.
Lots of familiar relationships carry over, e.g. Note that ((x * n) / n) = x. Similarly ((x ^ n) v n) = x. And where (n * (x / n)) = x, it's also the case that (n ^ (x v n)) = x.
And where multiplication and division interact with addition and subtraction, those operators interact similarly with multiplication and division, e.g. where x * (n + m) = (x * n) + (x * m), similarly x ^ (n * m) = (x ^ n) * (x ^ m). And so on. You can derive all the relationships you need from a very small number of rules that are easy to remember because they're structurally almost exactly like the rules for operations you're familiar with. You already know those when you learn about exponentiation, so you don't have to learn new and weird geometric relationships.
All of those nice properties there follow from the fact that exponentiation is just the next operation in the sequence of hyperoperations (see https://en.wikipedia.org/wiki/Hyperoperation ) after multiplication.
Introducing this weird three place operator actually masks the underlying simplicity of exponentiation and its inverse.
"And let log base n of x be written as (x v n) or possibly (x \/ n)."
"Lots of familiar relationships carry over, e.g. Note that ((x * n) / n) = x. Similarly ((x ^ n) v n) = x."
You must mean ((x^n) v x) = n. Note that ^ is not at all a symmetric operator, and neither is v.
"e.g. where x * (n + m) = (x * n) + (x * m), similarly x ^ (n * m) = (x ^ n) * (x ^ m)."
You must mean x ^ (n + m) = (x ^ n) * (x ^ m).
It seems your proposed notation does not make things so clear as that one cannot get lost in it, either.
You must mean ((x^n) v x) = n. Note that ^ is not at all a symmetric operator, and neither is v.
"e.g. where x * (n + m) = (x * n) + (x * m), similarly x ^ (n * m) = (x ^ n) * (x ^ m)."
You must mean x ^ (n + m) = (x ^ n) * (x ^ m).
It seems your proposed notation does not make things so clear as that one cannot get lost in it, either.
Ah bummer. Both of those were actually just transcription errors because I was writing formulas out in the course of a meeting, playing with whether the order of the v operator should be that way or the other way around - I don't have time to give it much thought at the moment but I'm admittedly less convinced I picked the right one.
In the second case, that was a typo. In the first case, what I'd meant to write was: "Note that ((x * n) / x) = n. Similarly, ((x ^ n) v x) = n."
You're right that neither is a symmetric operator but of course neither is division or subtraction. I sort of glossed over that intentionally, but the non-commutativity of the ^ operator would indeed probably be the main place that switching to infix operators could mislead.
I didn't mean to come across as unduly harsh. But, yeah, I think a misleadingly symmetric looking notation for an asymmetric operation can be a source of confusion (this is also a potential problem with the misleadingly symmetric looking triangle of the linked video, and standard notation for subtraction, etc., for that matter [though in this last case, it's been beaten into us with enough familiarity that we all know the deal]).
Actually, I don't like infix notation in general: it leads to unnecessary, distracting questions about operator precedence and so on. I'd rather we all switched to some other notation for writing out even additions and multiplications and such; drawing out the actual tree structure of nested operations, say. (I often feel notation should simply follow the structure of what's being notated, nothing more or less. But, alas, inertia; I can only use the notation I like in the privacy of my own home...)
Actually, I don't like infix notation in general: it leads to unnecessary, distracting questions about operator precedence and so on. I'd rather we all switched to some other notation for writing out even additions and multiplications and such; drawing out the actual tree structure of nested operations, say. (I often feel notation should simply follow the structure of what's being notated, nothing more or less. But, alas, inertia; I can only use the notation I like in the privacy of my own home...)
Infix notation IS funny, though I figured anyone learning arithmetic would be familiar and we're stuck with it - we probably can't convince grade schools to use Polish notation for arithmetic by default.
In general, my attitude is that you should pick a notation that makes all the similarities with things you already know as obvious as possible and then stress the differences. They've already learned about non-symmetric infix operators if they've learned subtraction and division, and the only reason to expect that ^ would be symmetric was by analogy to multiplication, which is easy to clear up. Take it as a lesson that by default you should never assume operations are symmetric, in preparation for linear algebra ;)
"It introduces three concepts (exponentiation, logarithms and roots) as though they were entirely separate."
Hm? This is the exact opposite of what the notation is intended to do. It is intended to emphasize that there is just one underlying ternary relation [with all of these seemingly separate functions just being different ways of solving for one argument to this ternary relation in terms of the other two].
Hm? This is the exact opposite of what the notation is intended to do. It is intended to emphasize that there is just one underlying ternary relation [with all of these seemingly separate functions just being different ways of solving for one argument to this ternary relation in terms of the other two].
I guess what I was trying to say is that treating that as a ternary relation is already overcomplicating the matter. Really, there's no need for nth root notation at all, because we already have fractional exponentiation. Why have a third place on the triangle putting any n there is the same as putting 1/n on the top?
Part of the argument for the triangle is that the relationships you'd have to learn are suggested by simple geometric manipulations.
But why learn whole new geometric manipulations when you've already learned the algebraic manipulations behind multiplication and division? By converting exponentiation and logarithms into infix notation, you do away with one side of the triangle and you can learn the relationships by just thinking "it's exactly like multiplication and division, except [etc]"
Part of the argument for the triangle is that the relationships you'd have to learn are suggested by simple geometric manipulations.
But why learn whole new geometric manipulations when you've already learned the algebraic manipulations behind multiplication and division? By converting exponentiation and logarithms into infix notation, you do away with one side of the triangle and you can learn the relationships by just thinking "it's exactly like multiplication and division, except [etc]"
Incidentally, sure, we could just pick ^ to be our basic operator and define everything else in terms of that. Well, just as well, we could define everything in terms of logarithm. Or everything in terms of roots! In fact, one might ask why you are so happy to define a logarithm operator v to undo ^ in one direction, but do not want ever to define and work with a root operator to undo ^ in the other direction.
Is it because we can get away with saying a^(1/n) instead of n-th root of a? Well, just as well, why should we have a division operator? Instead of a/b, we can say a * b^(-1).
Or perhaps we oughtn't have multiplication, or exponentiation with arbitrary bases. Perhaps we should just have addition, negation, natural logarithm, and the natural exponential. Then we define a * b as exp(ln(a) + ln(b)), define b^c as exp(c * ln(b)), etc.
But, of course, just because F can be expressed in terms of G doesn't mean it's always convenient or pertinent to think of F that way; sometimes it's pertinent to think of F just qua F, as an atomic entity in its own right. And, thus... the proliferation of different functions people talk about.
Though this does mean people then have to keep track of what all these different functions are defined as, and trace back through how that makes them relate. So... I don't know. I don't have all the answers. Yet.
Is it because we can get away with saying a^(1/n) instead of n-th root of a? Well, just as well, why should we have a division operator? Instead of a/b, we can say a * b^(-1).
Or perhaps we oughtn't have multiplication, or exponentiation with arbitrary bases. Perhaps we should just have addition, negation, natural logarithm, and the natural exponential. Then we define a * b as exp(ln(a) + ln(b)), define b^c as exp(c * ln(b)), etc.
But, of course, just because F can be expressed in terms of G doesn't mean it's always convenient or pertinent to think of F that way; sometimes it's pertinent to think of F just qua F, as an atomic entity in its own right. And, thus... the proliferation of different functions people talk about.
Though this does mean people then have to keep track of what all these different functions are defined as, and trace back through how that makes them relate. So... I don't know. I don't have all the answers. Yet.
Why think of ternary relations? Well, if you do not recognize that there is an "Any two variables here determine the third" structure here, you are missing a fruitful insight into the matter. That's why. Sometimes that's a useful perspective, sometimes perhaps less so.
That can be useful not only for exponentiation/logarithm/roots, incidentally. It can be useful to recognize with addition/subtraction/subtraction, or multiplication/division/division, as well. These all have a similar structure: three values in a relationship, such that any two determine the third. The triangle is supposed to help recognize this structure, though it helps with nothing else on top of that.
"By converting exponentiation and logarithms into infix notation, you do away with one side of the triangle and you can learn the relationships by just thinking 'it's exactly like multiplication and division, except [etc]'"
The problem is, there's rather a LOT of "except". Standard * is commutative, while ^ isn't. * is associative, while ^ isn't. * turns additions on either side into additions of the result, while ^ turns multiplications on left into multiplications of the result, but multiplications on the right only into chained operations (i.e., a ^ (b * c) = (a ^ b) ^ c), while additions on the right are turned into multiplications of the result (i.e., a ^ (b + c) = (a ^ b) * (a ^ c)), and additions on the left are turned into binomial theorem expansions!
These are big differences! There are particular similarities too, of course. There's a mix of similarities and lots of differences. What that amounts to pedagogically is... a muddle.
Anyway, the triangle manipulations in the video aren't at all compelling to me, but regardless, the ternary relation structure IS useful to recognize (that any two out of three determine the third, and this is all the various traditional functions do), and again, this is something to be recognized not just for powers, logarithm, etc., but even for our familiar addition/subtraction or multiplication/division problems.
That can be useful not only for exponentiation/logarithm/roots, incidentally. It can be useful to recognize with addition/subtraction/subtraction, or multiplication/division/division, as well. These all have a similar structure: three values in a relationship, such that any two determine the third. The triangle is supposed to help recognize this structure, though it helps with nothing else on top of that.
"By converting exponentiation and logarithms into infix notation, you do away with one side of the triangle and you can learn the relationships by just thinking 'it's exactly like multiplication and division, except [etc]'"
The problem is, there's rather a LOT of "except". Standard * is commutative, while ^ isn't. * is associative, while ^ isn't. * turns additions on either side into additions of the result, while ^ turns multiplications on left into multiplications of the result, but multiplications on the right only into chained operations (i.e., a ^ (b * c) = (a ^ b) ^ c), while additions on the right are turned into multiplications of the result (i.e., a ^ (b + c) = (a ^ b) * (a ^ c)), and additions on the left are turned into binomial theorem expansions!
These are big differences! There are particular similarities too, of course. There's a mix of similarities and lots of differences. What that amounts to pedagogically is... a muddle.
Anyway, the triangle manipulations in the video aren't at all compelling to me, but regardless, the ternary relation structure IS useful to recognize (that any two out of three determine the third, and this is all the various traditional functions do), and again, this is something to be recognized not just for powers, logarithm, etc., but even for our familiar addition/subtraction or multiplication/division problems.
One practical concern I have for this is that you're going to need bigger triangles to avoid things bumping into eachother around the triangle.
I really like the point he's making, just not sure if this triangle notation is the most practical.
I really like the point he's making, just not sure if this triangle notation is the most practical.
Does anyone know the history of this notation? Like were the concepts discovered independently by three different people? Hence there are three ways to notate?
I can't help but wonder if the triangle of power is a Zelda triforce reference.
I'm sure if would capture imaginations if it were called the "triforce" :)
My own personal opinion on this, is that it's all well and good but it's not that students are "Made" to learn three different types of notation but that these are three standard forms that you need to know to work in the mathematical space.
Is there a quick way to use this triangle in LaTeX?
A variation of this should do the trick:
\newcommand[3]{\triangle}{{}_{#1}\overset{#3}{\Delta}_{#2}}
\newcommand[3]{\triangle}{{}_{#1}\overset{#3}{\Delta}_{#2}}
How is learning all the special cases of the triangle manipulation (O-plus when its bottom right and the top is missing but multiply in the bottom left, multiply when the top is given and bottom is missing etc) better than just learning the different notations?
Not to mention the existing mathematics you miss out on by using this.
That being said I think it's a fantastic tool to quickly explain the relationship between the notations (the first half of the video).
Not to mention the existing mathematics you miss out on by using this.
That being said I think it's a fantastic tool to quickly explain the relationship between the notations (the first half of the video).
And we should all learn esperanto because it is so much more rational and regular than English. Somehow it doesn't work that way.
The problem is not that the notation exists and that there are multiple ways to say the same thing, it's that students are forced to memorize all of it for its own sake.
In fact, there aren't just "three" ways to describe 2 * 2 * 2=8, there are infinitely many. Because 2 * 2 * 2=8 shows up in so many different contexts, and notation in each of those contexts highlights a different (hopefully useful) feature for that context, you'll never be able to have "just" one way to say a thing. You can have your favorite, sure, but all notational preferences are aesthetic.
FWIW I think this triangle notation is also misleading in its own way. Students have to memorize arbitrary rules about how "mirroring" the triangle changes the operations at each corner, and whether they actually connect that to the underlying arithmetic operations is just as tenuous as with the classical notation.
You see the video maker say, as an afterthought, that once the students are fluent in this beautiful notation they can go about understanding why the it works, but the same problem as before! They're memorizing arbitrary symbol shuffling, maybe reducing cognitive load but also introducing random extra facts along the way like parallel resistance (which algebra students care about that, again?), and the connection between the true idea and the work they're doing is thin.
In fact, there aren't just "three" ways to describe 2 * 2 * 2=8, there are infinitely many. Because 2 * 2 * 2=8 shows up in so many different contexts, and notation in each of those contexts highlights a different (hopefully useful) feature for that context, you'll never be able to have "just" one way to say a thing. You can have your favorite, sure, but all notational preferences are aesthetic.
FWIW I think this triangle notation is also misleading in its own way. Students have to memorize arbitrary rules about how "mirroring" the triangle changes the operations at each corner, and whether they actually connect that to the underlying arithmetic operations is just as tenuous as with the classical notation.
You see the video maker say, as an afterthought, that once the students are fluent in this beautiful notation they can go about understanding why the it works, but the same problem as before! They're memorizing arbitrary symbol shuffling, maybe reducing cognitive load but also introducing random extra facts along the way like parallel resistance (which algebra students care about that, again?), and the connection between the true idea and the work they're doing is thin.
I agree with everything you say here, except for one remark, which, I'm sorry, is the silliest thing I've read all day:
> all notational preferences are aesthetic
Try multiplying Roman numerals sometime. Or read up on ancient Egyptian fractions. Or learn group theory without Cayley diagrams. Or do algebra on equations written in prose -- prose! -- as was typical everywhere for a thousand years before the Renaissance. And what good are tensors without indices? Or matrices: do matrices have a first and second index, or do they have rows and columns?
Yes, matrix algebra with indices one and two is just as true, but that's the wrong observation. It takes what we already know for granted. In fact what we already know is the destination, and the point of departure is what you look at, or stare at, until you understand matrix algebra.
How does the brain turn markings on paper into something like abstract truth? Nobody knows, but it's silly to say the markings don't matter. The brain is biology, and in biology everything matters to everything else.
> all notational preferences are aesthetic
Try multiplying Roman numerals sometime. Or read up on ancient Egyptian fractions. Or learn group theory without Cayley diagrams. Or do algebra on equations written in prose -- prose! -- as was typical everywhere for a thousand years before the Renaissance. And what good are tensors without indices? Or matrices: do matrices have a first and second index, or do they have rows and columns?
Yes, matrix algebra with indices one and two is just as true, but that's the wrong observation. It takes what we already know for granted. In fact what we already know is the destination, and the point of departure is what you look at, or stare at, until you understand matrix algebra.
How does the brain turn markings on paper into something like abstract truth? Nobody knows, but it's silly to say the markings don't matter. The brain is biology, and in biology everything matters to everything else.
The key word here is preference, which is subjective by definition. Just because certain notations are considered outdated does not mean there is no situation in which they can have merit.
To wit, my preferred method for understanding tensors is without indices, and matrix notation is primarily useful for computers, not humans (in that regard I prefer the coordinate-free perspective). In fact, the insistence that linear algebra must be understood entirely using matrices and rows and columns is a red flag in my book!
I'm not saying that notation is irrelevant, I'm saying it's a matter of perspective. Of course there are notational breakthroughs throughout history. But what makes a breakthrough depends on what problems you're trying to solve, which is largely why computer scientists like indices and algebraic geometers like commutative diagrams. Each is as unwelcome in the wrong domain as roman numerals are in algebra.
To wit, my preferred method for understanding tensors is without indices, and matrix notation is primarily useful for computers, not humans (in that regard I prefer the coordinate-free perspective). In fact, the insistence that linear algebra must be understood entirely using matrices and rows and columns is a red flag in my book!
I'm not saying that notation is irrelevant, I'm saying it's a matter of perspective. Of course there are notational breakthroughs throughout history. But what makes a breakthrough depends on what problems you're trying to solve, which is largely why computer scientists like indices and algebraic geometers like commutative diagrams. Each is as unwelcome in the wrong domain as roman numerals are in algebra.
>FWIW I think this triangle notation is also misleading in its own way. Students have to memorize arbitrary rules about how "mirroring" the triangle changes the operations at each corner, and whether they actually connect that to the underlying arithmetic operations is just as tenuous as with the classical notation.
Yes, this times 10^100.
One thing to remember is that all notation is yes, tradition, but that tradition is somewhat subject to natural selection, which I'd argue is selection for ease of use. This is for example why we use symbolic notation in the first place, because it is more convenient once it's been learned.
I'm not sure how to deal with this. Perhaps one way is to teach the long, tedious way, just do a bazillion multiplications and then say, hey, exponentiation makes it much easier to write! That's usually how we get students to appreciate short-cuts in the first place. Teaching them the short-cut without at least giving them a taste of the internals is usually a recipe for bad abstractions disconnected from the internals.
Yes, this times 10^100.
One thing to remember is that all notation is yes, tradition, but that tradition is somewhat subject to natural selection, which I'd argue is selection for ease of use. This is for example why we use symbolic notation in the first place, because it is more convenient once it's been learned.
I'm not sure how to deal with this. Perhaps one way is to teach the long, tedious way, just do a bazillion multiplications and then say, hey, exponentiation makes it much easier to write! That's usually how we get students to appreciate short-cuts in the first place. Teaching them the short-cut without at least giving them a taste of the internals is usually a recipe for bad abstractions disconnected from the internals.
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i feel like the usual best practice for studying math is to use the same concepts in as many different settings/with as many different terms for the same thing as possible. By this I mean, literally, knowing the subject so well that there's not a context that can confuse you. Notation is no exception to this. If you know 2 ways to correctly solve a problem all the better. If you know 3 that's even better, for instance.
This is futile. It's just another standard. xkcd said it best: https://xkcd.com/927
I suppose what would be really nice is for computerised mathematics to be marked up such that these interchanges can be made automatically, with libraries of rule sets shared online.
You're reading a document and it has some funny triangle thing you don't understand? Click on it to get a menu of alternative representations, and you see it can be swapped to "log" notation. Further, you go to your reader preferences and add a rule "whenever you see this triangle thing, show me it as a log".
A student finds some crusty old document with funny "heart monitor" symbols, clicks on them and finds they're just a particular kind of power triangle. They update their preferences to replace those symbols with power triangles.
Of course, holy wars rage on mailing lists about whether the default rules should convert "tau" to "2pi" or "pi" to "tau/2" ;)
Seriously though, too much time is spent making computerised mathematics look right in PDFs (e.g. TeX), compared to telling the computer precisely what it is/means. Thankfully there are some attempts at this (e.g. OpenMath), but they don't seem to be very widely used.
You're reading a document and it has some funny triangle thing you don't understand? Click on it to get a menu of alternative representations, and you see it can be swapped to "log" notation. Further, you go to your reader preferences and add a rule "whenever you see this triangle thing, show me it as a log".
A student finds some crusty old document with funny "heart monitor" symbols, clicks on them and finds they're just a particular kind of power triangle. They update their preferences to replace those symbols with power triangles.
Of course, holy wars rage on mailing lists about whether the default rules should convert "tau" to "2pi" or "pi" to "tau/2" ;)
Seriously though, too much time is spent making computerised mathematics look right in PDFs (e.g. TeX), compared to telling the computer precisely what it is/means. Thankfully there are some attempts at this (e.g. OpenMath), but they don't seem to be very widely used.
I really need to expand this into a blog post that has been brewing in my head for years, but my overall thesis is that notation is the least difficult part to learn in mathematics, yet because it's the first part that is encountered, it is also the most derided one.
Sure, notation can be better, and maybe this triangle of power is a cute way to make it better. Notation changes and improves all the time, btw. Well, all the time in the mathematical scale of time, which is two or three millenia. In this scale, things like the Greek letter for the ratio of circumference to diameter are remarkably modern, merely 300 years old. Notation for linear algebra is even newer, all from the 20th century.
However, I don't think better notation is where we need to focus most of our efforts in order to make our mathematics easier to understand. Logarithms and square roots are very basic things, and if keeping the mainstream notation for them straight is someone's biggest problem, then there are far bigger things that are likely to be problematic to this individual. If you start reading, say, the following mathematical discussion of neural networks,
http://neuralnetworksanddeeplearning.com/chap1.html#eqtn7
you're baffled because you don't know what those symbols mean, there's likely far deeper things that are unfamiliar, such as differentials, rate of change, derivatives, and the multivariable chain rule. A couple of days ago we had someone come to ##math in Freenode asking for help with this, and I tried, but the guy had never had any calculus training whatsoever. Normally going from no calculus to the multivariable chain rule as applied to differentials or as a best linear approximation takes at least three semesters in university, and I don't think this path to enlightenment could be shortened much more.
I guess I am being very old school and reiterating that the royal road everyone's been looking for for the past couple millenia just doesn't exist.
https://en.wikipedia.org/wiki/Royal_Road#A_metaphorical_.E2....
Sure, notation can be better, and maybe this triangle of power is a cute way to make it better. Notation changes and improves all the time, btw. Well, all the time in the mathematical scale of time, which is two or three millenia. In this scale, things like the Greek letter for the ratio of circumference to diameter are remarkably modern, merely 300 years old. Notation for linear algebra is even newer, all from the 20th century.
However, I don't think better notation is where we need to focus most of our efforts in order to make our mathematics easier to understand. Logarithms and square roots are very basic things, and if keeping the mainstream notation for them straight is someone's biggest problem, then there are far bigger things that are likely to be problematic to this individual. If you start reading, say, the following mathematical discussion of neural networks,
http://neuralnetworksanddeeplearning.com/chap1.html#eqtn7
you're baffled because you don't know what those symbols mean, there's likely far deeper things that are unfamiliar, such as differentials, rate of change, derivatives, and the multivariable chain rule. A couple of days ago we had someone come to ##math in Freenode asking for help with this, and I tried, but the guy had never had any calculus training whatsoever. Normally going from no calculus to the multivariable chain rule as applied to differentials or as a best linear approximation takes at least three semesters in university, and I don't think this path to enlightenment could be shortened much more.
I guess I am being very old school and reiterating that the royal road everyone's been looking for for the past couple millenia just doesn't exist.
https://en.wikipedia.org/wiki/Royal_Road#A_metaphorical_.E2....
A lot of math is simple and logical. Notation is a major obstacle, because of a. the sheer amount of various notations which are rarely used by a non-expert in a particular subfield, thus easily forgotten and b. un-searchability. Reading a math text becomes an exercise in jumping around figuring out what the author meant with a particular notation, which usually leads to more notations, etc.
Heck, coding CRUD apps I have access to better notation exploration tools than a mathematician. I can navigate to a definition or check the unit tests in seconds. I don't have to instantly remember the purpose of every function, I can simply look up the precise details on the fly. I can rely that the authors have not overused symbolic operators, so I can easily search StackOverflow for tips on the framework design. No more rote memorization of hundreds of tiny factoids that may or may not be germane to the problem at hand.
Heck, coding CRUD apps I have access to better notation exploration tools than a mathematician. I can navigate to a definition or check the unit tests in seconds. I don't have to instantly remember the purpose of every function, I can simply look up the precise details on the fly. I can rely that the authors have not overused symbolic operators, so I can easily search StackOverflow for tips on the framework design. No more rote memorization of hundreds of tiny factoids that may or may not be germane to the problem at hand.
> A lot of math is simple and logical
And I suppose you are implying a lot of math is illogical. :-)
> Heck, coding CRUD apps
Yeah, I hear the "bad notation" argument frequently from programmers, precisely for the reasons you describe: because you can't read a mathematical textbook with a programmer's IDE.
Some mathematical texts will do you the favour of having a list of some notation that they consider to be idiosyncratic to their own text, but few would go as far as to letting you look up the definition of plus and minus. In general, though, mathematics isn't programming, despite some similarities and analogies between the two. The symbol for the partial derivative is used nearly universally to mean partial derivatives of some kind, so there's a lot of tradition that writers of mathematical texts expect you to know.
Programming has similar traditions that are baffling to outsiders but as invisible to the programmers as water is to fish. For example, in programming it is understood that everyone is capable of easily handling plain text files. Have you ever seen a newcomer try to write source code in Microsoft Word? I have.
Wikipedia and other online mathematical texts can help a little by hyperlinking the text to explain new notation or terminology, and sometimes texts are just plain bad in that they use idiosyncratic notation without explanation. In the last case, it takes effort to work out from context the likely meaning of a symbol.
In general, though, I stand by my thesis that notation is not the biggest obstacle to mathematics just like learning to use a text editor or IDE is not the biggest obstacle to programming. The fundamental ideas behind the practices of each are much deeper than the superficial aspects of text and notation.
And I suppose you are implying a lot of math is illogical. :-)
> Heck, coding CRUD apps
Yeah, I hear the "bad notation" argument frequently from programmers, precisely for the reasons you describe: because you can't read a mathematical textbook with a programmer's IDE.
Some mathematical texts will do you the favour of having a list of some notation that they consider to be idiosyncratic to their own text, but few would go as far as to letting you look up the definition of plus and minus. In general, though, mathematics isn't programming, despite some similarities and analogies between the two. The symbol for the partial derivative is used nearly universally to mean partial derivatives of some kind, so there's a lot of tradition that writers of mathematical texts expect you to know.
Programming has similar traditions that are baffling to outsiders but as invisible to the programmers as water is to fish. For example, in programming it is understood that everyone is capable of easily handling plain text files. Have you ever seen a newcomer try to write source code in Microsoft Word? I have.
Wikipedia and other online mathematical texts can help a little by hyperlinking the text to explain new notation or terminology, and sometimes texts are just plain bad in that they use idiosyncratic notation without explanation. In the last case, it takes effort to work out from context the likely meaning of a symbol.
In general, though, I stand by my thesis that notation is not the biggest obstacle to mathematics just like learning to use a text editor or IDE is not the biggest obstacle to programming. The fundamental ideas behind the practices of each are much deeper than the superficial aspects of text and notation.
> Normally going from no calculus to the multivariable chain rule as applied to differentials or as a best linear approximation takes at least three semesters in university.
Multivariable chain rule is much simpler than the 18 months learning curve would imply. It boils down to figuring out what a function is, what a derivative is, what the chain rule is and generalizing to multiple dimensions. We could probably teach it to a sufficiently logically apt high-schooler within a week, provided we could find a sufficiently motivating use-case.
The lack of compelling use-cases being the other major obstacle in learning math. Why bother rote memorizing tens of concepts and hundreds of factoids, when a lot of math texts pride themselves of building the perfect theory in abstract, decoupled from the original motivations.
Multivariable chain rule is much simpler than the 18 months learning curve would imply. It boils down to figuring out what a function is, what a derivative is, what the chain rule is and generalizing to multiple dimensions. We could probably teach it to a sufficiently logically apt high-schooler within a week, provided we could find a sufficiently motivating use-case.
The lack of compelling use-cases being the other major obstacle in learning math. Why bother rote memorizing tens of concepts and hundreds of factoids, when a lot of math texts pride themselves of building the perfect theory in abstract, decoupled from the original motivations.
I personally don't find a lot of motivation to learn calculus because engineers in the 19th century needed to figure out how to make better steam engines or cannons or because physicists wanted to understand electromagnetism. My own motivation was: okay, neat, derivatives. This says a lot about a function! What happens now if we try to do this with more variables? Oh, wow, look, the chain rule gets all weird now and grows additions it didn't have before!
For other people, I guess you need to find a different motivation. Maybe neural networks will do it for some. I must admit that I picked up a neural networks text in 1995 because I wanted to build robots, didn't understand a word of it, and ten years later I got a degree in mathematics having long ago forgotten about the neural networks book which I only recently picked up again. But regardless, ever since I was a little kid, mathematics is just something that naturally attracted me.
We should not minimise the intrinsic interest of the subject itself either. There is an artistic side to mathematics, where we do it because it's beautiful for its own sake. Not all mathematics needs a purely practical reason to justify its study.
For other people, I guess you need to find a different motivation. Maybe neural networks will do it for some. I must admit that I picked up a neural networks text in 1995 because I wanted to build robots, didn't understand a word of it, and ten years later I got a degree in mathematics having long ago forgotten about the neural networks book which I only recently picked up again. But regardless, ever since I was a little kid, mathematics is just something that naturally attracted me.
We should not minimise the intrinsic interest of the subject itself either. There is an artistic side to mathematics, where we do it because it's beautiful for its own sake. Not all mathematics needs a purely practical reason to justify its study.
> Multivariable chain rule is much simpler than the 18 months learning curve would imply. .... We could probably teach it to a sufficiently logically apt high-schooler within a week, provided we could find a sufficiently motivating use-case.
Supposing this were the case, given that there's no shortage of homeschooling and "alternative" high schools out there, can you find a case where this has actually happened.
Supposing this were the case, given that there's no shortage of homeschooling and "alternative" high schools out there, can you find a case where this has actually happened.
According to the trivium [0], the path to learning consists of a sequence: grammar; dialectic; and rhetoric. In my head I think of it in terms of vocab, logic, and application (respectively).
Notation may not be the hardest part. It is however (as you say) the first part. If our goal is to make math more readily accessable to the masses, perhaps the greatest immediate gains entail improving the notation to better reflect the concepts -- exactly because it is the first barrier.
[0] https://en.wikipedia.org/wiki/Trivium
Notation may not be the hardest part. It is however (as you say) the first part. If our goal is to make math more readily accessable to the masses, perhaps the greatest immediate gains entail improving the notation to better reflect the concepts -- exactly because it is the first barrier.
[0] https://en.wikipedia.org/wiki/Trivium
> Sure, notation can be better, and maybe this triangle of power is a cute way to make it better. Notation changes and improves all the time, btw. Well, all the time in the mathematical scale of time, which is two or three millenia. In this scale, things like the Greek letter for the ratio of circumference to diameter are remarkably modern, merely 300 years old. Notation for linear algebra is even newer, all from the 20th century.
Forget linear algebra, modern notation for regular algebra is from the 19th century. And the form of the notation absolutely does matter to learning and comprehension.
Ancient Babylonians knew how to solve quadratic equations, but you basically had to be a professional mathematician to understand the way they phrased problems and work with their notation.
More well known is the notoriety of Roman numerals for doing basic arithmetic. It takes more than 20 pages to explain fractions with Roman numerals: http://dmaher.org/Publications/romanarithmetic.pdf
Forget linear algebra, modern notation for regular algebra is from the 19th century. And the form of the notation absolutely does matter to learning and comprehension.
Ancient Babylonians knew how to solve quadratic equations, but you basically had to be a professional mathematician to understand the way they phrased problems and work with their notation.
More well known is the notoriety of Roman numerals for doing basic arithmetic. It takes more than 20 pages to explain fractions with Roman numerals: http://dmaher.org/Publications/romanarithmetic.pdf
It sure doesn't help that often it's forgotten to teach notation in the beginning. Not in high school but at university level this is my experience. And it's incredibly hard to google on your own.
It's like that for other skills too, like how to write a proof. I guess it is assumed that you will simply assimilate it with time, but I have met students in their third year who still struggled with e.g. properly introducing variables.
I honestly find the concepts far easier than the syntax, and I am used to weird syntax.
It's not just the notation though. It's using crappy notation in place of explanation or as a short-cut.
While some notation is "modern" in absolute linear terms, it's really not in any practical sense. The entire premise of most mathematical writing, exposition, and notation, is still that of thousands of years ago: writing takes significant time, writing surfaces are expensive, communication lag is long, and cultural/linguistic barriers between people are significant. That is simply not very true any more.
There is no reason we should be writing, teaching, and discussing math as though it was still a thousand years ago. We have computers, we have amazing printing presses, we have relatively cheap material. There should be no reason we're still pulling this "as the reader will clearly see" crap with this massively dense short-cut notation.
It's not just the notation though. It's using crappy notation in place of explanation or as a short-cut.
While some notation is "modern" in absolute linear terms, it's really not in any practical sense. The entire premise of most mathematical writing, exposition, and notation, is still that of thousands of years ago: writing takes significant time, writing surfaces are expensive, communication lag is long, and cultural/linguistic barriers between people are significant. That is simply not very true any more.
There is no reason we should be writing, teaching, and discussing math as though it was still a thousand years ago. We have computers, we have amazing printing presses, we have relatively cheap material. There should be no reason we're still pulling this "as the reader will clearly see" crap with this massively dense short-cut notation.
Can you recommend an approach to learn high level math on one's own? Any particular books that are good for that purpose?
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Beautiful, elegant, and I disagree it should be taught.
∆ Is used for delta's so frequently that I could see some nasty issues arising in notes for higher level maths.
I think this could be an excellent teaching tool, to be honest I sometimes pull log notation back into exponential with a variable if I can't remember it immediately, but I think this guy didn't read Feynman's biography where he discusses the problems with creating notation.
∆ Is used for delta's so frequently that I could see some nasty issues arising in notes for higher level maths.
I think this could be an excellent teaching tool, to be honest I sometimes pull log notation back into exponential with a variable if I can't remember it immediately, but I think this guy didn't read Feynman's biography where he discusses the problems with creating notation.
I don't think this notation would get confused with deltas; how it's used is pretty different, and overloading symbols (especially once you're in higher levels) is common enough.
Overloaded symbols became a nightmare in some of my later undergraduate work.
It is not the math classes where I'm worried that this would be a problem, it's in the engineering courses. There's a few situations in mechanics/structures/materials that I can think of where using ∆ instead of the existing 3 notations would become nearly untenable. In these circumstances each "line" of a solution/step usually took the form of more than one page.
It is not the math classes where I'm worried that this would be a problem, it's in the engineering courses. There's a few situations in mechanics/structures/materials that I can think of where using ∆ instead of the existing 3 notations would become nearly untenable. In these circumstances each "line" of a solution/step usually took the form of more than one page.
δ could be used instead (lowercase delta). Root/exponent/log is so common that it should get priority for the best notation.
I agree in theory, but in practicality I don't.
∆ and δ have distinct and separate meanings in some of my coursework that can't be exchanged. The nomenclature/symbolism is so screwed up in some engineering fields that it would either require all students to revert to the existing symbolism or a level of investment in rewriting literature from research academics that created this screwed system in the first place.
I think Feynman had it right, he could invent the best symbols in the world for every operation but no one will use them.
∆ and δ have distinct and separate meanings in some of my coursework that can't be exchanged. The nomenclature/symbolism is so screwed up in some engineering fields that it would either require all students to revert to the existing symbolism or a level of investment in rewriting literature from research academics that created this screwed system in the first place.
I think Feynman had it right, he could invent the best symbols in the world for every operation but no one will use them.
I like the problem statement, I like the use of a 2D representation, but the triangle symbol as drawn in that presentation is gargantuan. I think the symbol needs more work.
One thought that comes to mind is that I'm not convinced the bottom part of the triangle should be there. It implies a direction connection that I'm not sure exists. Removing that overlaps with ∧, logical and, though.
Another that comes to mind is that I can't think of another mathematical symbol that has such divergence in meaning depending on what is left blank like that. The closest I know of is integral, where you can leave the from and to parts blank for a symbolic integral, but that's still not like leaving those blank turns the integral into a differentiation (the opposite), depending. I'm sure there's something else somewhere up in math, too, but nothing your average student will hit.
Similarly, note that filling in all three corners of that symbol is actually an equation. I'm also not aware of any other symbols that constitute entire equations on their own. In fact hiding away an = symbol is probably a big strike against the idea as if anything standard math education underplays and abuses that most fundamental of symbols; let's not add to that. Again, somewhere up in higher maths than I've gotten to there may be symbols that constitute entire equations, but it won't be something most students see.
Also I think once that symbol is being shown with full expressions rather than cute little single-digit numbers or single-letter variables, it's going become very difficult to deal with.
I think there's something to this, though. I'm criticizing in the spirit of continuing to move forward. (I'm aggressively hostile to the idea that math notation is perfected and debating better alternatives is some sort of betrayal or something.) Personally I'd seek out a smaller, inline symbol that may visually reference a richer presentation (which may not be this literal triangle) for a nicer didactic experience, but doesn't literally draw it out in the formula.
One thought that comes to mind is that I'm not convinced the bottom part of the triangle should be there. It implies a direction connection that I'm not sure exists. Removing that overlaps with ∧, logical and, though.
Another that comes to mind is that I can't think of another mathematical symbol that has such divergence in meaning depending on what is left blank like that. The closest I know of is integral, where you can leave the from and to parts blank for a symbolic integral, but that's still not like leaving those blank turns the integral into a differentiation (the opposite), depending. I'm sure there's something else somewhere up in math, too, but nothing your average student will hit.
Similarly, note that filling in all three corners of that symbol is actually an equation. I'm also not aware of any other symbols that constitute entire equations on their own. In fact hiding away an = symbol is probably a big strike against the idea as if anything standard math education underplays and abuses that most fundamental of symbols; let's not add to that. Again, somewhere up in higher maths than I've gotten to there may be symbols that constitute entire equations, but it won't be something most students see.
Also I think once that symbol is being shown with full expressions rather than cute little single-digit numbers or single-letter variables, it's going become very difficult to deal with.
I think there's something to this, though. I'm criticizing in the spirit of continuing to move forward. (I'm aggressively hostile to the idea that math notation is perfected and debating better alternatives is some sort of betrayal or something.) Personally I'd seek out a smaller, inline symbol that may visually reference a richer presentation (which may not be this literal triangle) for a nicer didactic experience, but doesn't literally draw it out in the formula.
> Similarly, note that filling in all three corners of that symbol is actually an equation. I'm also not aware of any other symbols that constitute entire equations on their own.
Multiplication and division can also be put into a triangle, such that filling in each corner constitutes an equation. See my comment [0] about the density equation. I suspect such triangles reflect some deeper relation of abstract algebra.
[0] https://news.ycombinator.com/reply?id=12138817&goto=threads%...
Multiplication and division can also be put into a triangle, such that filling in each corner constitutes an equation. See my comment [0] about the density equation. I suspect such triangles reflect some deeper relation of abstract algebra.
[0] https://news.ycombinator.com/reply?id=12138817&goto=threads%...
Well, sure, we can put whatever we like in explanatory diagrams. I'm not aware of recognized common mathematical symbols that constitute equations on their own, though. This video did not seem to be proposing a diagramming method, because if that's all it was there wouldn't be any controversy to speak of; it appeared to be proposing a new family of operators. That's what I was criticizing.
The triangle being an equation by itself is a bit of a bummer. The obvious followup to yours removing a line would be to remove two. Exponentiation would be:
But what I like best with this idea is the possibility of replacing the line by some arbitrary symbol (including no symbol at all). That way, other triangular relations could be expressed without stepping on each other's namespace.
By the way, there is an obvious direction we haven't used yet:
---
Now that I think of it, hiding the `=` sign may not be so bad in some cases. Relations don't have to be limited with the identity and orders (<, >, ≤, ≥). They don't have to be limited to relations with 2 variables either. Other relations would have to justify their usefulness to warrant a special symbol of their own however.
e
/
x
e
\
b
x—b
But what I like best with this idea is the possibility of replacing the line by some arbitrary symbol (including no symbol at all). That way, other triangular relations could be expressed without stepping on each other's namespace.
By the way, there is an obvious direction we haven't used yet:
a
|
b
---
Now that I think of it, hiding the `=` sign may not be so bad in some cases. Relations don't have to be limited with the identity and orders (<, >, ≤, ≥). They don't have to be limited to relations with 2 variables either. Other relations would have to justify their usefulness to warrant a special symbol of their own however.
> One thought that comes to mind is that I'm not convinced the bottom part of the triangle should be there. It implies a direction connection that I'm not sure exists.
The operator in question is actually quite common.
> Buying stocks. Suppose you buy $1000 worth of stocks each month, no matter the price (dollar cost averaging). You pay $25/share in Jan, $30/share in Feb, and $35/share in March. What was the average price paid? It is 3 / (1/25 + 1/30 + 1/35) = $29.43 (since you bought more at the lower price, and less at the more expensive one). And you have $3000 / 29.43 = 101.94 shares. The “workload” is a bit abstract — it’s turning dollars into shares. Some months use more dollars to buy a share than others, and in this case a high rate is bad.
https://betterexplained.com/articles/how-to-analyze-data-usi... (harmonic averages section)
The concept describes various processes which contribute towards identical workloads at different speeds (or rates).
The operator in question is actually quite common.
> Buying stocks. Suppose you buy $1000 worth of stocks each month, no matter the price (dollar cost averaging). You pay $25/share in Jan, $30/share in Feb, and $35/share in March. What was the average price paid? It is 3 / (1/25 + 1/30 + 1/35) = $29.43 (since you bought more at the lower price, and less at the more expensive one). And you have $3000 / 29.43 = 101.94 shares. The “workload” is a bit abstract — it’s turning dollars into shares. Some months use more dollars to buy a share than others, and in this case a high rate is bad.
https://betterexplained.com/articles/how-to-analyze-data-usi... (harmonic averages section)
The concept describes various processes which contribute towards identical workloads at different speeds (or rates).
[deleted]
I'm not sure how to relate what you said to my post? Wrong reply button?
> One thought that comes to mind is that I'm not convinced the bottom part of the triangle should be there. It implies a direction connection that I'm not sure exists.
I should have included this quote to begin with. I'll put it in now.
In any case. A direct connection does exist. If you follow the link, there are other examples where the (+) operator would be useful besides parallel resistance.
I should have included this quote to begin with. I'll put it in now.
In any case. A direct connection does exist. If you follow the link, there are other examples where the (+) operator would be useful besides parallel resistance.
Now I see. Thanks.
I still think it implies a degree of symmetry that will confuse students, though, and that the notation can be further improved from what was proposed there. The operation in question is not trilaterally symmetric, and using a trilaterally symmetric symbol is probably misleading. The root symbol is arbitrary, but at least it doesn't promise nonexistent symmetries.
I still think it implies a degree of symmetry that will confuse students, though, and that the notation can be further improved from what was proposed there. The operation in question is not trilaterally symmetric, and using a trilaterally symmetric symbol is probably misleading. The root symbol is arbitrary, but at least it doesn't promise nonexistent symmetries.
The root symbol (as a multi-valued function) returns the roots of unity (multiplied by a scalar). Which is like, the posterchild of imaginary symmetries. </pedantry>
More seriously. The selling point of the triangle operator is that it highlights isomorphisms between the three more traditional operators as a cyclic group. Which is a symmetry, just not the commutativity that students might naively expect. So I suppose it's a double-edged sword. I agree that it sucks as an operator. Nonetheless, the video could help a lot of students.
More seriously. The selling point of the triangle operator is that it highlights isomorphisms between the three more traditional operators as a cyclic group. Which is a symmetry, just not the commutativity that students might naively expect. So I suppose it's a double-edged sword. I agree that it sucks as an operator. Nonetheless, the video could help a lot of students.
I am sympathetic to the aims here. I think it makes a lot of sense to represent the relation (a,b,c) where a^b = c in a unified way. Of course, it may also make sense to represent other ternary relations in similar ways, too. You could put something inside the triangle to represent which relation is intended.
It's even worse than shown in the video. Consider:
sin^2(x) = (sin(x))^2
sin^-1(x) = arcsin(x)
Switching to triangle notation would help remove this confusing overloading of the superscript operator. The only drawback I see with triangle notation is there's no obvious way to type it on a single line.
sin^2(x) = (sin(x))^2
sin^-1(x) = arcsin(x)
Switching to triangle notation would help remove this confusing overloading of the superscript operator. The only drawback I see with triangle notation is there's no obvious way to type it on a single line.
You might be able to approximate the triangle in line by using slash and backslash. 2/3\ = 8. 2/\8 = 3. /3\8 = 2. Not nearly as pretty, but if you're familiar with the triangle in a hand written context, it might get the point across.
Just use the connecting edge: 2(/)3 = 8, 2(-)8 = 3, 3(\)8 = 2.
There are just overloaded notations from two different contexts. sin² can mean either:
* x → sin(x)² (in multiplicative group)
* x → sin(sin(x))² (in composition group)
Similarly, sin⁻¹ can mean either:
* x → 1 / sin(x) (in multiplicative group)
* x → arcsin(x) (in composition group)
* x → sin(x)² (in multiplicative group)
* x → sin(sin(x))² (in composition group)
Similarly, sin⁻¹ can mean either:
* x → 1 / sin(x) (in multiplicative group)
* x → arcsin(x) (in composition group)
> The only drawback I see with triangle notation is there's no obvious way to type it on a single line.
Definitely a valid complaint, although nth-root and log-base-b are already poor for typing on a single line.
Definitely a valid complaint, although nth-root and log-base-b are already poor for typing on a single line.
"It's like it's a different language!" (incredulously)
Yes, it is a different language. It is built for expressing ideas about deduction and quantity clearly, flexibly, creatively, and it works beautifully. Saying 2^3 = 8 and log_2 8 = 3 sort of get at the same fact but not really, that's misunderstanding their purpose. They express that we are evaluating functions. You can express 8 - 2 = 6 or 6 + 2 = 8 "in the same way" (Crazy! How can we have two notations +, - when they are just inverses of each other?! We shouldn't give ourselves language to express both "the difference between 8 and 2 is 6", and "2 more than 6 is 8" because they happen to be rearrangements of the same equation!) but the equations are used to convey meaning in different ways in different contexts.
The second example, the 8^(1/3) is not even equal to 2, it's equal to three values, two of them are imaginary. It's important to have notation for "the thing that when you cube it is equal to 8" distinct and understood so that when you begin understanding imaginary numbers (high school iirc) you have context. Then you can explain the definite article in that quoted sentence is actually inaccurate. What if he had selected an example with two real roots, like sqrt(4)? What should we put in the the "triangle of power"? The positive branch cut? Okay, now you have to explain that you're really doing a different operation now, that has multiple answers, but we are going to pick one of them and put it there, but we have to remember that it could be either. Which is best expressed using separate notation to explain the operation you are doing that isn't even a function.
The fact of the matter is that these ideas are distinct and relating them is a separate, worthwhile exercise that helps understand the structure of exponentiation and its inverses.
And in doing mathematics you will find that switching to equivalent but more informative or clean or applicable notation is one of the most valuable workhorses we have for solving simple problems.
Yes, it is a different language. It is built for expressing ideas about deduction and quantity clearly, flexibly, creatively, and it works beautifully. Saying 2^3 = 8 and log_2 8 = 3 sort of get at the same fact but not really, that's misunderstanding their purpose. They express that we are evaluating functions. You can express 8 - 2 = 6 or 6 + 2 = 8 "in the same way" (Crazy! How can we have two notations +, - when they are just inverses of each other?! We shouldn't give ourselves language to express both "the difference between 8 and 2 is 6", and "2 more than 6 is 8" because they happen to be rearrangements of the same equation!) but the equations are used to convey meaning in different ways in different contexts.
The second example, the 8^(1/3) is not even equal to 2, it's equal to three values, two of them are imaginary. It's important to have notation for "the thing that when you cube it is equal to 8" distinct and understood so that when you begin understanding imaginary numbers (high school iirc) you have context. Then you can explain the definite article in that quoted sentence is actually inaccurate. What if he had selected an example with two real roots, like sqrt(4)? What should we put in the the "triangle of power"? The positive branch cut? Okay, now you have to explain that you're really doing a different operation now, that has multiple answers, but we are going to pick one of them and put it there, but we have to remember that it could be either. Which is best expressed using separate notation to explain the operation you are doing that isn't even a function.
The fact of the matter is that these ideas are distinct and relating them is a separate, worthwhile exercise that helps understand the structure of exponentiation and its inverses.
And in doing mathematics you will find that switching to equivalent but more informative or clean or applicable notation is one of the most valuable workhorses we have for solving simple problems.
> The second example, the 8^(1/3) is not even equal to 2, it's equal to three values
No, the expression "8^(1/3)" is equal to 2 and only 2, even though there are three values of x for which it holds that x^3 = 8.
> What if he had selected an example with two real roots, like sqrt(4)?
The notation √a refers to the principal square root of a, which is defined to be positive. You might recall expressions such as "x = +/- sqrt(a)", which would be redundant if sqrt itself was a multivalued function. a^b, for rational b, is defined in a similar manner.
No, the expression "8^(1/3)" is equal to 2 and only 2, even though there are three values of x for which it holds that x^3 = 8.
> What if he had selected an example with two real roots, like sqrt(4)?
The notation √a refers to the principal square root of a, which is defined to be positive. You might recall expressions such as "x = +/- sqrt(a)", which would be redundant if sqrt itself was a multivalued function. a^b, for rational b, is defined in a similar manner.
It depends entirely on the context -- mathematical notation, as always, depends upon the level of instruction, textbook, context, mathematician etc. In algebraic geometry we take the radical ideal sqrt(I) for an ideal I which is most certainly a set of values. When you're teaching it for the first time (as is the context we are discussing right now) it's quite important to reinforce how many square roots there are. Context generally makes it clear (among people who already understand what is happening) which version you mean as is usually the case. Similarly with logs, though we have the nice log/Log distinction which I suppose has become standard-ish. There is literally no reason we select the positive branch other than notational convenience and one of the biggest mistakes I've seen with freshmen students trying to learn calculus is algebra mistakes like this -- taking a square root and only considering the positive branch. Because they were taught square root of 4 is 2. And that's true sometimes (when you are computing sqrt(x)), and not others (when you take the square root of something for the purpose of solving an equation). I've noticed some online resources recently are very careful about referring to sqrt(x) as the "principal square root function", something which is very good but that I have never once heard anyone say in real life, teacher or mathematician.
Obnoxious irrelevant pedantry aside, context matters. You understand the concern here: simplifying notation masks the actual mathematics. These operations aren't identical or as simple as the video would have us believe, distinct notation exists for a reason -- to separate separated concepts.
Obnoxious irrelevant pedantry aside, context matters. You understand the concern here: simplifying notation masks the actual mathematics. These operations aren't identical or as simple as the video would have us believe, distinct notation exists for a reason -- to separate separated concepts.
> In algebraic geometry we take the radical ideal sqrt(I) for an ideal I which is most certainly a set of values.
But that's just overloaded syntax, isn't it? We're concerned with real numbers here.
> There is literally no reason we select the positive branch other than notational convenience
Agreed, it's just a convention, though a very helpful one.
> I've noticed some online resources recently are very careful about referring to sqrt(x) as the "principal square root function", something which is very good but that I have never once heard anyone say in real life, teacher or mathematician.
I had to look up that one, since my maths education wasn't in English. I agree that it's nicer than what appears e.g. in my textbook: "the square root is always a positive number or zero", which conflicts wiwitthe definition that's on Wikipedia. It's probably the conflicting uses of "square root" to refer to two different things (all solutions, which is more relevant analytically, or the unique positive solution, which is more helpful for notation) that causes the kinds of freshman errors you mention.
> You understand the concern here: simplifying notation masks the actual mathematics.
My point is that it doesn't mask the actual mathematics any more than the regular notation already does. Which also is a problem, but not the one at hand.
But that's just overloaded syntax, isn't it? We're concerned with real numbers here.
> There is literally no reason we select the positive branch other than notational convenience
Agreed, it's just a convention, though a very helpful one.
> I've noticed some online resources recently are very careful about referring to sqrt(x) as the "principal square root function", something which is very good but that I have never once heard anyone say in real life, teacher or mathematician.
I had to look up that one, since my maths education wasn't in English. I agree that it's nicer than what appears e.g. in my textbook: "the square root is always a positive number or zero", which conflicts wiwitthe definition that's on Wikipedia. It's probably the conflicting uses of "square root" to refer to two different things (all solutions, which is more relevant analytically, or the unique positive solution, which is more helpful for notation) that causes the kinds of freshman errors you mention.
> You understand the concern here: simplifying notation masks the actual mathematics.
My point is that it doesn't mask the actual mathematics any more than the regular notation already does. Which also is a problem, but not the one at hand.
>But that's just overloaded syntax, isn't it? We're concerned with real numbers here.
Yes, you're 100% correct, I was just giving an example for why it's natural for some (me included) to think of sqrt(x) as a solution set, I suppose.
>My point is that it doesn't mask the actual mathematics any more than the regular notation already does. Which also is a problem, but not the one at hand.
Well, the normal notation is a little confusing yes, but the new notation makes it worse -- they propose marrying the principal sqrt function, which makes an arbitrary choice and drops information, to the log and exponential function, which both do not over R, and for which the exponential function does not over C.
At least we teach three separate concepts and then unify them later as best as we can, as opposed to trying to pretend they are all the same.
Yes, you're 100% correct, I was just giving an example for why it's natural for some (me included) to think of sqrt(x) as a solution set, I suppose.
>My point is that it doesn't mask the actual mathematics any more than the regular notation already does. Which also is a problem, but not the one at hand.
Well, the normal notation is a little confusing yes, but the new notation makes it worse -- they propose marrying the principal sqrt function, which makes an arbitrary choice and drops information, to the log and exponential function, which both do not over R, and for which the exponential function does not over C.
At least we teach three separate concepts and then unify them later as best as we can, as opposed to trying to pretend they are all the same.
cant upvote enough... the triangle notation is actually "wrong"
Let's see how this notation holds up to nesting...
While this post limits itself to basic mathematics, the notations in group theory and advanced linear algebra JUST DON'T MAKE SENSE. Please read the "Brakerski’s Homomorphic Cryptosystem", section 3.3, of this ( https://eprint.iacr.org/2015/137.pdf ) paper.
I had to read it 10 times to truly understand what the notation was trying to say! Fail.
I had to read it 10 times to truly understand what the notation was trying to say! Fail.
That seems just fine and should be understandable by a reasonably sharp undergraduate.
Would you care to elucidate the exact math that makes homomorphic search possible
Math notation is an endlessly interesting subject, but I must say I wasn't very impressed with that idea.
If we're going to give up some notation to adopt another, it would need to have some serious and obvious advantages.
I waited for that throughout the video. The author consistently seems to assume that what he's saying is "intuitive". It isn't.
Why should I put a particular number in a particular corner of the triangle? How does it help computation? I see triangles being nested within triangles and fusing together according to rules that seem completely arbitrary.
Certainly, using our visual apparatus to help us complete computations without having to think about it can be appealing, but that's probably not the most convincing example.
If we're going to give up some notation to adopt another, it would need to have some serious and obvious advantages.
I waited for that throughout the video. The author consistently seems to assume that what he's saying is "intuitive". It isn't.
Why should I put a particular number in a particular corner of the triangle? How does it help computation? I see triangles being nested within triangles and fusing together according to rules that seem completely arbitrary.
Certainly, using our visual apparatus to help us complete computations without having to think about it can be appealing, but that's probably not the most convincing example.
In one of my (middleschool?) science classes, my teacher explained the density equation in terms of a triangle. Of course, we didn't adopt it into our notation. But I think that just having shown the class the diagram helped a lot of my peers.
Later in high school (in a different state), a mathematically-challenged friend was studying logarithms during study hall. Our curriculum used this bizarre, three-step arrow rule to transform logs into the familiar (y = b^e). Having remembered the density triangle, I showed my friend a similar diagram for logs. He said "Thanks. I was probably going to get a zero on the next quiz. I might actually pass now."
Later in high school (in a different state), a mathematically-challenged friend was studying logarithms during study hall. Our curriculum used this bizarre, three-step arrow rule to transform logs into the familiar (y = b^e). Having remembered the density triangle, I showed my friend a similar diagram for logs. He said "Thanks. I was probably going to get a zero on the next quiz. I might actually pass now."
d = m / v
v = m / d
m = v * d
.
/m\
/---\
/v | d\
.-------.
What you really want students to understand is that expt, log, and nth-root are functions that are related in the following way:
It's really that simple. No fancy notation needed. In fact, fancy notation always gets in the way of understanding because people naturally think in words, not in spatial relationships. Mathematical notation was invented not because it aids understanding, but because when you're writing math with pen and ink it's faster and uses less paper to use Greek letters and spatial relationships than full words. But when you're on a computer, it's easier to write out the names of functions, and that is actually a better impedance match to people's natural mental processes, which involve language.
[UPDATE] I would like to revise this: not everyone thinks in language. But everyone communicates in language. For communicating mathematical concepts, language is the best tool we have. There's a reason that the symbology in math papers is invariably wrapped in natural language. It's the same reason that the video has a narration. It wouldn't make any sense otherwise.