A plea for notational common sense
Tradition can often be a retarding force in mathematics. Advancements in logic were held back by centuries due to a collective belief that Aristotle had it all figured out and there was nothing left to discover. Non-Euclidean geometry wasn’t seriously studied until the 1800s because no one could think of the parallel postulate being false.
We nowadays tend to recognize the dangers of blindly following tradition, yet in virtually all areas of mathematics we continue to adhere to notational lunacy, and pass it on to future generations starting as early as in junior high school.
This lunacy is so deeply ingrained in our minds that we see nothing wrong with the following expression:
Let’s decipher the above. We start with a number x, apply f, square the result, and finally apply sin. Yet how do we recover this order of operations from the symbols above? Starting at x in the middle we move left to f, right to the 
, and far left to the sin.
Although we read and write left-to-right, and so naturally imagine time evolving in this direction, the notation f(x) (due to Euler) means the opposite: start with x, then apply f. Even worse is when we have an expression like f(g(h(x))). The very first operation is written on the very right, and the final operation is written on the very left, in complete contrast to how we normally think.
Worse still is that we do read some operations left-to-right, like raising to a power, and so we end up mentally zig-zagging across mathematical writing to even decipher mathematical expressions.
But worst of all is the following: when we write a function 
, A denotes the input domain, and B denotes the output domain: input on the left, output on the right. As we’d expect in a sane world. If we also have a function 
, this left-to-right notation suggests we should be able to follow the arrows (A to B to C) to compose f and g. Yet, of course, the composite is called gf rather than fg. When reading gf we are meant to mentally reverse the direction of the arrows, so that 
and 
, although we never write this way!
This is psychotic behaviour, and we should put a stop to it. I suggest we forget tradition, and instead succumb to notational common sense to write expressions like x.f.g rather than g(f(x)).
This may seem bizarre at first, but at least one group of scientists adopted this notation years ago. In object-oriented programming languages a mental shift is required, where one stops thinking that functions have certain inputs, and instead starts to see that inputs have certain functions. In doing so, the left-to-right notation becomes completely natural, and we would write deck.sort.pop to sort a deck of cards and remove the last one, rather than pop(sort(deck)) as we’d expect in the mathematicians’ ridiculous notation.
I believe that a notational shift would make all of our lives easier in the long run, but of course it will likely never happen. Just as negative charge will always correspond to an excess of electrons, Euler’s notation is an unfortunate convention that is simply known by too many people. As a result, already-difficult subjects like category theory are made even more maddening for silly reasons, and high school students have far more difficulty learning precalculus than they should.
Help us, Barack Obama, we need Change.
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does not exist (at least not in in the traditional sense) and is analogous to the alternating sequence { 1, -1, 1, -1, … }. But what of squaring the x? The function 
alternates between positive and negative as well, so one might be tempted to say that its integral also diverges.
, the kth 
(where C is for Christmas.)
, so (assuming n is even)
.
as we hoped!
to any real (or complex!) number, i.e.
?
and 
, it is a reasonable guess that 
. This indeed turns out to be the case, and so we may naturally define a generalized Christmas function
