Maths notation is needlessly complex [video]
aeon.co
aeon.co
What you really want students to understand is that expt, log, and nth-root are functions that are related in the following way:
expt(b, n) = x
log-x-base-b(x, b) = n
nth-root(x, n) = b
[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.
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.
When you are first exposed to the First Isomorphism Theorem for Rings, a thousand words can't do what that simple triangle diagram can.
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).
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 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 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.
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.
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.”
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.
Yes, you're right. But everyone communicates in language. I've updated the original comment.
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 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 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.
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.
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.
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.
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.
> 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.
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 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 ;-)
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 :)
I bet if Professor Penrose rewrote his book in code-language it would be much easier to learn, at least for software engineers!
-- 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
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.
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!
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.
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.
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).
> 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.
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 am certainly not arguing against teaching higher level concepts.
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 :)
> 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...)
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.
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.
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.
(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.")
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.
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.
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.
> 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.
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.
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....
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.
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.
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.
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.
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.
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.
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.
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
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.
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.
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.
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.
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.
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.
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).
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?
/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?
You really only have 1/2 chance to make a mistake.
The video is also plagued with distracting visual effects (there are some that are worthwhile, but most are not.)
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.
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).
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 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.
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.
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%...
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.
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.
There may be a benefit to capturing that sqrt(square(x)) != x
∆ 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.
∆ 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.
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.
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.
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\
.-------.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.
Definitely a valid complaint, although nth-root and log-base-b are already poor for typing on a single line.
* 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)
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.
- 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...
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".
I had to read it 10 times to truly understand what the notation was trying to say! Fail.
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.
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?
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.
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.
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].
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]"
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.
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.
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.
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...)
I really like the point he's making, just not sure if this triangle notation is the most practical.
\newcommand[3]{\triangle}{{}_{#1}\overset{#3}{\Delta}_{#2}}