... when you look at a page full of equations and think "Ahh! That's scary!", all you're really looking at is a bunch of simple ideas in a highly abbreviated form. This is true for every part of mathematics: deconstructing the abbreviations is more than half the battle.

A sizable proportion of textbooks and lectures just say that a function is "a rule that assigns one number to another number," ... For some (including myself when I was first exposed to the idea), this is a rather confusing conceptual leap.

The first letter of the word "something" in Arabic makes a sound similar to the sound "sh" in English. It turns out that the Spanish language has no "sh" sound, so when all of this Arabic mathematics was translated into Spanish, the Spanish translators chose the closest thing they could think of. This was the Greek letter "chi" ... The letter chi looks like this: χ ... Later as you might expect this χ turned into the familiar letter x from the Latin alphabet.

... when I'm explaining some piece of mathematics to someone, and I change abbreviations so that we can remember what we're talking about, one of the most common things I hear is "Oh! I didn't know we could do that!" It's important that we practice changing abbreviations, because a lot of ideas in mathematics look scary and complicated when we use one set of abbreviations, but suddenly seem obvious when we use another.

We will use ☰ in any equality that is true simply because of some abbreviation that we're using, and not because of any mathematics that you missed.

Another way of using equals shows up whenever we're forcing something to be true, and seeing what happens as a result.

What I cannot create, I do not understand.
- Richard Feynman

The process of inventing a mathematical concept consists of attempting to translate [a] vague qualitative idea into a precise quantitative one ... Creation is simply translation from qualitative to quantitative.

The goal should be, not to implant in the student's mind every fact that the teacher knows now; but rather to implant a way of thinking that will enable the student, in the future, to learn in one year what the teacher learned in two years. ... As I came to realize this, my style in teaching changed from giving a smattering of dozens of isolated details, to analyzing only a few problems, but in some real depth.
- E. T. Jaynes

Mathematics is an entire world where nothing is accidental, and where the mind can train itself with an intensity and precision unmatched by any other subject.

Mathematics is about sentences that look like: "If this is true, then that is true."

... you should not try to memorize the steps [of an] argument, but rather to understand the argument well enough that if you ever forget [the resulting formulas] (which you should), then you can reinvent them for yourself on the spot in a few seconds.

New contexts act like a sieve that filters out the possibility of memorization.

I know it might seem weird to solve a harder problem by using a simpler method, but this strategy turns out to work all over the place in mathematics. It's really fortunate that things work out that way!

... the fact that multiplication works the same both ways (i.e. a*b = b*a) just says that the area of a rectangle doesn't change when we turn it on its side.

... our choice of a formal definition often depends (more than anyone wants to admit) on subjective aesthetic preferences, i.e., what we think is pretty.

A steep thing is equally steep whether we climb it when we're underground or when we're inside an airplane.

The essence of mathematics lies entirely in its freedom.
- Georg Cantor

As is always the case when inventing a mathematical concept, the definition we finally arrived at was built from a strange blend of translation and aesthetics: some of our definition's behaviors came from our desire to make it behave like our everyday concept, while other came from a desire to make the resulting definition as elegant and simple to deal with as possible...

Textbooks call abbreviations "symbols," and they call sentences "equations" or "formulas."

... for decades we've been teaching students everything they need to understand [time dilation], and then simply not bothering to show it to them. Why? Because special relativity is an "advanced" topic that doesn't quite "belong" in an introductory physics class, nor does it seem to belong in the courses on Euclidean geometry in which students first learn the mathematics needed to understand the argument itself. The argument is homeless.

... a surprising amount of knowledge can be squeezed out of the simple fact that drawing a picture on something doesn't change its area.

If two things really are equal (i.e. identical), and we modify both in exactly the same way, then (although the two things will both change individually) they will still be identical to each other after the modifications ... identical modifications to identical objects must lead to identical results.

Here's the seemingly impossible fact about light: if you "throw" some light ... and then you immediately run up behind it at 99% of the speed of light, then it won't look like the light is moving away from you at 1% of the speed of light! Rather, it will still look like the light is moving away from you at the full speed of light...

... people usually use the letter c to stand for the speed of light. Basically, c is the first letter of the Latin word for "swiftness," and the speed of light is quite literally the fastest anything in our universe is able to go...

Calculus is the most powerful weapon of thought yet devised by the wit of man.
- W. B. Smith

By stumbling upon a familiar location, you've managed to reduce the problem to one you've already solved in the past, and the rest is easy. That's all of mathematics!

All of Calculus: If we zoom in on curvy stuff, it starts to look more and more straight ... What's more, if we were to zoom in "infinitely far" (whatever that means), then any curvy thing would look exactly straight.

... if we can invent an "infinite magnifying glass" -- then we'll be able to turn any problem involving curvy stuff (a hard problem) into a problem involving straight stuff (an easy problem).

If we forget about mathematics and just stare at a curvy thing ... it's very clear that some places are more steep than others ... Is there any way to make sense of the idea of steepness at a single point of some general curvy thing?

What does it mean for two points to be infinitely close to each other?

To this day calculus all over the world is being taught as a study of limit processes instead of what it really is: infinitesimal analysis.
- Rudy Rucker

If the idea of infinitely small numbers scares you a little, you're not alone! ... How can we act like it's not zero, and then a few lines later act like it is zero?

One of the contraptions that can be used to formalize the idea of infinitely small numbers is called an "ultrafilter." Ultrafilters are pretty complicated and never mentioned in the introductory textbooks.

... the contraption you'll see in all the standard intro textbooks is called a "limit," ... it lets mathematicians get all the benefits of using infinitely small numbers, without giving the idea of infinitely small numbers any of the credit.

It's less confusing if we realize that limits are just one of several (optional!) contraptions that let us avoid worrying about the meaning of infinitely small numbers if we want to.

... to change an expression involving regular numbers to an expression involving infinitely small ones, just change the Greek alphabet to the Latin alphabet (i.e., change Δ to d) ... Similarly, we'll see later that when the textbooks pass from the letter Σ (the Greek S, which stands for the word "sum") to its corresponding Latin letter S (they actually write ∫, which kind of looks like an S), they're doing a similar trick. ... we'll soon see how switching back and forth between different abbreviations has a surprising ability to make complicated things seem simple ...

Since a machine's derivative tells us the slope of the machine at that point, we can make use of the following convenient fact: all flat points of a machine are places where that machine's derivate is zero. As such, we can find the flat points of a machine m by forcing its derivative to be zero ... and then trying to figure out which numbers x make that sentence true.

Not all the flat points are extremes, but both of the extremes are flat points.

If we're at a max or min, then the derivative is zero

... the basic principle that "extremes are usually places where the derivative equals zero" will continue to apply no matter how far we go, and no matter how strange our mathematical universe becomes.

... the ultra-cautious "how do I know if I'm right?" feeling we all develop in mathematics classes is a feeling that we should, at least partially, learn to abandon. When we're inventing mathematics (or anything) for ourselves, we don't know if we're right. No one ever does.

The old saying among physicists appears to be true: too much rigor can, and does, lead to rigor mortis.

... the notation dM/dx, which emphasizes that the derivative of M can be thought of as the "rise over run" between two points that are infinitely close to each other.

... you'll occasionally hear people talking about strange things like "negative powers" or "fractional powers" or "zeroth powers" ...

.... how should we chooe to generalize the concept of powers? Obviously, any generalization is useless unless it is useful.

I have no idea what (stuff)# means when # isn't a whole number... But I really want to hold on to the sentence:

(stuff)n+m = (stuff)n(stuff)m

So I'll force (stuff)# to mean: whatever it has to mean in order to make that sentence keep being true.

This style of thought lies at the core of mathematical invention, and a surprising number of mathematical concepts are invented in exactly this way.

... (stuff)0 has to be whatever number doesn't change things when you multiply by it. So I guess (stuff)0 has to be 1.

1 = (stuff)0 = (stuff)#(stuff)-#

(stuff)-# = 1/(stuff)#

(stuff)1/n is any number that can say the following sentence without lying: "Multiply me by myself n times and you get (stuff)."

We have a habit in writing articles published in scientific journals to make the work as finished as possible, to cover all the tracks, to not worry about the blind alleys or to describe how you had the wrong idea first, and so on. So there isn't any place to publish, in a dignified manner, what you actually did in order to get to do the work...
- Richard Feynman

If M(x) ☰ f(x)g(x)

then M'(x) = f'(x)g(x) + f(x)g'(x)

What if we made an argument, in abbreviated form, that whenever we know the pattern is true for one number of machines multiplied together, then it has to be true for the next number? Then we would automatically know that the pattern was true for every number. ... Textbooks call this style of reasoning "mathematical induction." ... Despite all [the] unrelated meanings of induction, if you hear it used after the word "mathematical," then they're talking about this ladder-like process of reasoning.