4 ms·

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 promote

by nnq 10y ago

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 EX(b, e, x) := { pow(b, e) == x } And then the notational rule saying that "substitution a parameter for a star/asterix in an expression is defined as the value obtained by solving that expression for the substituted parameter", so now we have: 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 here you go: "triangles notation" in "words" :) 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!

Chinjut 10y ago
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.
- posterboy 10y ago
  > 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.
  - Chinjut 10y ago
    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).
    - posterboy 10y ago
      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_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.
drostie 10y ago
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.
lisper 10y ago
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.
- nnq 10y ago
  > 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 :)
  - lisper 10y ago
    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.
- nnq 10y ago
  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...)
adrianratnapala 10y ago
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.

by nnq 10y ago

Chinjut 10y ago

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.

posterboy 10y ago
> 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.
- Chinjut 10y ago
  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).
  - posterboy 10y ago
    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_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.

drostie 10y ago

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.

lisper 10y ago

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.

nnq 10y ago
> 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 :)
- lisper 10y ago
  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.
nnq 10y ago
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...)

adrianratnapala 10y ago

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.