One reason why people are scared of/hate mathematics is that once people see an expression like
they run and hide. Let us step back for a moment and try to see what’s really going on here, so that we can better understand this fear, and possibly do away with some of it.
The above expression is scary not because the ideas inherently hidden within it are bad or ugly, but rather only because we do not understand the notation of the expression. In other words, we don’t understand the chosen symbols used to represent the ideas. It might as well be Egyptian hieroglyphics for all we know. In fact, math could have been written down in Egyptian hieroglyphics and its logical content would still be exactly the same. Why is this? To answer this question, we need to better understand the role that notation plays for us in mathematics. In doing so, we’ll understand one of the many reasons why math is in fact not so scary.
I’ll begin with a question:
What do I mean when I write down the number 3?
Note that I’m not asking “what is the number 3”, but rather I am asking what the meaning of the symbol “3” is. This is a subtle but extremely important issue. The symbol 3 is shorthand notation for the abstract concept of “three” which we keep in our heads. Thus, when I write down 3 and you read 3, I can know that we’re keeping the same abstract idea in our minds. In other words, the symbol 3 means nothing without the abstract meaning that we give it. Moreover, it is simply a shorter, more concise way of keeping track of the abstract notion of “three-ness”. We could have written any other symbol in its place, and its abstract meaning would be the same.
Let us take an example that is even closer to home. Consider, for example, the words on this page. You’re looking at this strange concoction of squiggly and straight lines, interspersed with blank white space, and you’re deriving from it certain meaning. Simply put, you’re reading. You’re extracting meaning from the abstract ideas that are represented by a relatively arbitrary mixture of symbols. Moreover, the meaning derived from these symbols is independent of the symbols themselves, i.e., we could have all agreed to use other symbols and the abstract ideas would be left intact, so long as we agreed on what the symbols meant. Therein lies the creation of new/different languages!
In addition to representing words (abstract ideas) by symbols on a page, we can define new words based on old words. In doing so, we can have shorter ways of expressing more and more meaning. For example, we defined the word “bus” so that we could encapsulate all of the meaning of “bigger version of a car, designed for transporting large amounts of people” without having to always write the whole thing down. Thus, when you read the word “bus” you automatically associate it with this larger meaning, and the word “bus” is just a convenient abbreviation. We then can use this word to define a new word, namely “school bus”. This way we can simply say “school bus” instead of always having to say “bigger version of a car, designed for transporting large amounts of people to and from school”.
Math is no different. We use symbols to wrap up more and more meaning into less and less writing. This has no effect on the logical content of a given mathematical statement or equation—notation is simply used so that we humans can communicate mathematics to each other. Thus, if you can read (which I’m assuming you can since if you can’t you’ll never know that I’m assuming you can) then you are already doing exactly what you were scared to do by looking at the above equation—namely, deriving abstract information from a language designed to capture and emphasize certain ideas.
Here’s a quick way to become less scared of math. Consider the term “symplectic manifold”. There is a decent chance the reader has no idea what a symplectic manifold is, and therefore is confused, and therefore is worried that he’ll never understand math, and therefore is scared of math forever, and therefore tries to learn guitar instead, and therefore becomes that guy at parties always playing the guitar and pissing people off. I’m here to argue that there’s a better way.
As mentioned before, the words “symplectic manifold” are just abbreviations for other constructions, which are in turn abbreviations for other constructions—just like “school bus” (and just like the symbols in the expression at the top of the page—each standing for some larger, well-constructed idea). In fact, a symplectic manifold is nothing but a differentiable manifold with some additional properties, and a differentiable manifold is nothing but a manifold with some additional properties, and a manifold is nothing but a topological space with some additional properties, and a topological space is nothing but a set with some additional properties, and we study sets in the second lesson of this site, requiring no further background! Of course I’m not trying to say that a symplectic manifold is a particularly easy thing to study, but I am saying that there’s nothing to be inherently scared of. If you can make it through lessons 1 and 2 (and lesson 1 is hardly even a lesson), then you’re well on your way to learning what a symplectic manifold is. BOOM. All of mathematics is now at your fingertips…
“What we need are *notions*, not notations.” — Carl Friedrich Gauss
But we need the latter to express the former (expanding the scope of notation to include gestures and speech).† This begs the question of how those initial frameworks were developed: while I understand that an infant could understand that a ‘tree’ refers to those tall, green-looking things; how do kids actually learn enough to be able to understand abstract concepts like ‘liberty’? I must spend some time with infants (I’m a child myself, BTW).
† Or do we? I for one have always fantasized about a device that could transfer thoughts or mental images that we weren’t able to articulate with words to recreate with a paintbrush (physical, or virtual), perhaps by directly interfacing with brainwaves. Since human brains evolved as independent learning systems, I don’t think there’d be a common protocol for such a transfer of intention; are those thoughts consigned to isolation?
For that matter, is it even possible to have such thoughts? Obviously, we could dream up the definition of a bus; but as the book 1984 demonstrated as well as an experiment about recalling memory at different ages demonstrated, not having the words for something can prevent you yourself from being able to think properly about it.
‡ (Don’t go hunting for a previous occurrence of this footnote; there isn’t any.) I couldn’t submit this comment the first time I’d tried (and unfortunately WordPress breaks the normal behaviour of browsers filling in previous form values on hitting the back button); but remarkably, the second time around I could make everything much more concise (which is always a good thing, I believe) and clear; a benefit of revision that today’s teens — teens that have shunned the thoughtful framework of (e)mail for the ephemeralness of IM and Snapchat and who are stuck in English classes and standardized tests (I’m looking at you, SAT I) that don’t leave room for editing — don’t get to experience.
These are all very interesting ideas, especially as related to math. In particular, in math we’re always finding (inventing?) ideas that previously had no existence in language, and therefore giving them new names. An example that you’ll see in the lessons is the idea of a group. A group is an abstract idea (much more abstract even than “liberty”) that needs its own word, for no other word in English (or any other language, I would bet) represents an idea that is in any way similar to the idea of a group. Namely, one cannot say that a group is like a flower, for example, because it simply it isn’t—not even close. Thus, when a child learns that a tree is one of those big green/brown things, they might relate it to a larger version of a flower or bush, but with some differences, or vice-versa (thinking of a flower as a smaller version of a tree). This isn’t possible with ideas like groups (or almost all mathematical structures, for example), at least not with words that relate to ideas that have PHYSICAL manifestations. This is one of the beautiful things about mathematics: the ideas that we relate mathematical ideas to are OTHER MATHEMATICAL STRUCTURES. So, even though no one has ever “seen” a group, and no one has ever “seen” a category (which is just another mathematical structure that we haven’t gotten to yet in the site), we (i.e., humans, and/or mathematicians) can still relate them to each other. This is remarkable when you think about it—it gives credence not only to the idea that mathematical structures “live” in some non-physical realm, but also to the (remarkable) idea that we can still access them and study them! This is very much in line with the ideas I discussed in the post The Three Worlds. There is MUCH to discuss on these issues but I have to skype with someone soon! So I’ll leave this for now and I’ll get to your other (very insightful) comments soon. Thanks for reading!
> it gives credence not only to the idea that mathematical structures “live” in some non-physical realm
I disagree with the realist/Platonist view, as I’ve commented on your Three Worlds post.
> I’ll leave this for now
Do get back to this later, as I’d be interested in seeing my ideas expanded and in what you have to say.