
Maths notation is needlessly complex [video] - ricksplat
https://aeon.co/videos/maths-notation-is-needlessly-complex-it-can-and-should-be-better
======
lisper
This presentation misses a more fundamental point. 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. That relationship
can be expressed using a two-dimensional spatial notation, but that doesn't
really help you understand the concept at all because there are a lot of
_different_ relationships that are naturally described by putting three things
in a triangle.

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
    

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.

~~~
nnq
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!

~~~
lisper
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
> 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
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.

------
j2kun
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.

~~~
mnemonicsloth
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.

~~~
j2kun
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.

------
jordigh
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](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....](https://en.wikipedia.org/wiki/Royal_Road#A_metaphorical_.E2.80.9CRoyal_Road.E2.80.9D_in_famous_quotations)

~~~
pacala
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.

~~~
jordigh
> 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.

~~~
pacala
> 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.

~~~
jordigh
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.

------
wfo
"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.

~~~
nemetroid
> 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.

~~~
wfo
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.

~~~
nemetroid
> 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.

~~~
wfo
>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.

------
davidivadavid
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.

~~~
sixo
Yeah. The notation should ideally have some obvious spatial
symmetries/structures that correspond with the symmetries of the operation
(sticking a smaller triangle in the corner doesn't LOOK like it cancels out,
and it should.)

------
overlordalex
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).

------
Animats
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?

------
svckr
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:

    
    
         /U\
        /R*I\
    

(Where U = voltage, I = current, R = resistance.)

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?

~~~
baby
I always pick one at random, U=R/I

You really only have 1/2 chance to make a mistake.

------
mannykannot
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.)

------
jerf
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.

~~~
Double_Cast
> _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...](https://betterexplained.com/articles/how-to-analyze-data-using-the-
average/) (harmonic averages section)

The concept describes various processes which contribute towards identical
workloads at different speeds (or rates).

~~~
jerf
I'm not sure how to relate what you said to my post? Wrong reply button?

~~~
Double_Cast
> _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.

~~~
jerf
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.

~~~
Double_Cast
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.

------
paulmd
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.

~~~
Terr_
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

------
iaw
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.

~~~
mcphage
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.

~~~
iaw
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.

------
andreyk
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.

------
_nalply
This is futile. It's just another standard. xkcd said it best:
[https://xkcd.com/927](https://xkcd.com/927)

------
Double_Cast
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."

    
    
      d = m / v
      v = m / d
      m = v * d 
          .
         /m\
        /---\
       /v | d\
      .-------.

------
mrob
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.

~~~
Mickydtron
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.

~~~
Retra
Just use the connecting edge: 2(/)3 = 8, 2(-)8 = 3, 3(\\)8 = 2.

------
dzdt
And we should all learn esperanto because it is so much more rational and
regular than English. Somehow it doesn't work that way.

------
chriswarbo
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.

------
ittekimasu
\- 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...

~~~
agumonkey
I strongly need to reread that book.

------
ricksplat
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.

------
vinchuco
It's a work in progress
[https://en.m.wikipedia.org/wiki/Zenzizenzizenzic](https://en.m.wikipedia.org/wiki/Zenzizenzizenzic)

------
nikdaheratik
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".

------
mankash666
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](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.

~~~
GFK_of_xmaspast
That seems just fine and should be understandable by a reasonably sharp
undergraduate.

~~~
mankash666
Would you care to elucidate the exact math that makes homomorphic search
possible

------
ianai
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.

------
everyone
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

------
vlasev
Let's see how this notation holds up to nesting...

------
chriswarbo
This also brings to mind other attempts to clarify notation, such as
[https://mitpress.mit.edu/sites/default/files/titles/content/...](https://mitpress.mit.edu/sites/default/files/titles/content/sicm_edition_2/chapter009.html)
and
[https://en.wikipedia.org/wiki/Geometric_algebra#Relationship...](https://en.wikipedia.org/wiki/Geometric_algebra#Relationship_with_other_formalisms)

------
serge2k
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.

~~~
jandrese
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?

~~~
htns
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.

~~~
imtringued
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.

------
catpolice
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](https://en.wikipedia.org/wiki/Hyperoperation)
) after multiplication.

Introducing this weird three place operator actually masks the underlying
simplicity of exponentiation and its inverse.

~~~
Chinjut
"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.

~~~
catpolice
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.

~~~
Chinjut
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...)

~~~
catpolice
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 ;)

------
damptowel
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.

------
cttet
I thought it would talk about lack of namespace, implicit overloading of
symbols etc...

~~~
Pxtl
Lack of support from normal unicode text files, particularly with the frequent
use of subscript and superscript.

------
ulkram
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?

------
Kinnard
:(
[https://news.ycombinator.com/item?id=12125756](https://news.ycombinator.com/item?id=12125756)

------
antoineMoPa
Is there a quick way to use this triangle in LaTeX?

~~~
vlasev
A variation of this should do the trick:

\newcommand[3]{\triangle}{{}_{#1}\overset{#3}{\Delta}_{#2}}

------
Practicality
I can't help but wonder if the triangle of power is a Zelda triforce
reference.

~~~
damptowel
I'm sure if would capture imaginations if it were called the "triforce" :)

