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You don’t suck at math (Part 2)

November 26, 2009 Leave a comment

In my last post, I promised to give a thorough explanation of the quadratic formula–where it comes from.

Here is the formula again.

\frac{-b \pm \sqrt{b^2-4ac}}{2a}

Now that you’ve seen it, I encourage you to forget it for a while. As I’ve thought about this more, and talked to a few people, I realize that my promise is itself somewhat against the spirit of my last post. That is, I am not convinced that I will be able to give a short explanation of the quadratic formula, without going on a few big tangents–whenever you start talking about math, you realize that there is a lot more going into what you are saying, a lot more ideas that are interacting with the one you are trying to convey, then you thought at first. I consider these tangents, however, to be of much value in themselves. It is often in fudging these extra parts, in order to press on towards your goal, that you are in danger of losing the audience.  So I can’t promise to even get all the way to the quadratic formula in this post, though I do hope to say something interesting, and I will get to the quadratic formula in a future post.

We’re going to first talk a bit about equations. First of all, what is an equation, and what does it mean to “solve” it? Well, let’s consider an example. Here is an equation:

3x-9=0

Solving this means finding a number that, when substituted for x, makes the equation true. If you don’t know how to solve this, that’s nothing to be embarrassed about. And I’m about to go through it anyway. But I want to stress what it is that we are doing. We want to find a number that, when we multiply it by 3, and then subtract 9 from that, we get zero. I could give real world examples of why you might want to do this, but I’m not going to. Doing so conveys the impression that it is only through such applications that math becomes interesting, whereas nearly the opposite is true: if you think that math is only studied for the sake of a series of contrived applications, you are going to find it very boring indeed.

So, solving this equation is pretty simple–we move the -9 over to the other side, making it positive (“add nine to both sides” if you like to think of it in that way):

3x=9

and then we divide by 3:

x=3

We could solve other such equations in a similar manner:

8x+4=0

Answer: x = -\frac{1}{2}

12x-7 =0

Answer: x = \frac{7}{12}

Notice that all of the equations I have given so far have 0 on the right hand side. There is a reason I did this. Basically, every equation is like this, if you think about it. For example, if I had the equation:

8x+5 = 24

I could turn it into an equation with zero on the right hand side by subtracting 24 from both sides:

8x-19 = 0

I could do the same thing with even much more complicated equations, by moving everything over to one side. Here this would actually be an unnecessary step if you were trying to solve the equation for x, but that’s not the point right now. The point is that, in math, whenever two things are essentially the same, we prefer to consider only one of them, and we like to be fairly consistent in that choice, unless it is convenient to switch (which it often will be, so it is important to know how to go back and forth). So the study of solving equations reduces to the study of how to make any particular expression (which is just one side of an equation) equal to zero. This simplifies things, in a way: instead of continually asking, “how to we make ‘blah’ equal to ‘blech’?” we only have to ask “how do we make ‘blah’ become zero?” This is somewhat analogous to the situation where two people are looking for each other–the task often seems simpler if one of them is asked to stand completely still.

Anyway, for the sorts of equations we’ve looked at so far, we can answer the question of how to solve them in general. By that I mean, we can answer it regardless of the different numbers that are showing up in these different equations I’ve given–or, more precisely, we can answer it in terms of those numbers. Let’s consider the “general form” for the sorts of equations we’ve looked at:

ax+b = 0

Here, a and b are thought of as “constants” and x is thought of as the “variable.” When I was first learning algebra, this distinction, I remember, would often enrage me. I had a valid reason: we don’t know what a, x, or b are, so what is the sense in drawing this distinction between two different types of unknowns? Philosophically, it is hard to say–and I won’t try to justify it completely. But I will say that much of the clever work that can be done in algebra is, in some sense, built on taking this distinction seriously, by continually shifting our perspective on what the constants are, and what the variables are.

OK, back to work. How do we solve for x? Well, in the same way we have been solving. First let’s move the b over:

ax = -b

Next, we divide by a:

x = \frac{-b}{a}

And we are done.

The above was really the simplest sort of equation to be solved. I’m sure you’ve seen more complicated equations; for example, here is one:

x^2 + y^2 -1 = 0

To solve this would be to find values for x and y that make the equation true. So here we’d be looking for a pair of numbers–and there may be many such pairs. One such pair is (x,y) = (1, 0). Another is (x,y) = (1/2, \frac{\sqrt{3},} {2}). In fact, there are infinitely many pairs that work. This might seem like a bad thing: who wants a problem that has infinitely many answers? If this bothers you, you can think of all the solutions together as a “set”–then there is only one set of pairs that precisely includes all of the solutions to this equation, and nothing extra. This is a good way of thinking about these things, because often there is a neat geometric description to the set that you are thinking about, allowing us to make a connection between algebra and geometry. It turns out here that the solutions to this equation describe a circle (maybe I’ll explain why, in a future post).

The question I want to ask now is: what made this equation more complicated than the ones before? Well, in our original equations, there was only one variable, x, and it appeared appeared only once, and there wasn’t any exponent on it. Here, however, we have multiple variables (two), and exponents on each of them. Say, now, we wanted to make the “next most complicated” equation to the ones we looked at before. It appears that there are two directions we could go in: we could include more variables, or we could include exponents on the variables that are already there. Doing the former, subject to some further restrictions, leads to the beautiful subject known as linear algebra. To get to the quadratic equation, however, we do the latter: we allow an exponent of 2 on the variable x.

To do that, let us start out with the simplest case of such an equation. Let consider this equation:

x^2 = 9

Now, I know this isn’t in the form I mentioned before, where zero is on the left. It easily could be (just move the nine over to the right hand side), but, right now, that’s not what I want to focus on. The important point here is that many of us probably already know how to solve this equation: set x equal to 3. Since 3 times 3 is nine, this is a solution. Less obviously, there is another solution, if we are considering negative numbers: -3 times -3 is also 9. This can seem like a pedantic point, but it actually represents an important fact that will also be true of the more difficult quadratic equations: typically, they will have two solutions). So when we allow a power of two on the x, there are (except in some special cases that we will be able to describe) two solutions. Isn’t that nice?

Incidentally, many people are bothered by the fact that -3 times -3 is equal to 9. In my next post, I will give an explanation for this, as I also give a general explanation for how you solve all of the equations of the form I just gave. I mean equations of the form:

x^2 = a

where a can be anything. These are a special case of quadratic equations, and we’ll have to understand them before we can even hope to get to the more general situation!

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You don’t suck at math (Part 1)

November 13, 2009 Leave a comment

I want you to imagine that you’ve decided to play softball for your company’s softball team. Your coach, a guy from the accounting department, is a former math teacher, and has taken it upon himself train all of you. You spend an hour or so each day training. These training sessions take place inside one of those company meeting rooms. For the first 40 minutes, you listen to him talk about the different baseball games that he has been to; he draws various diagrams on the whiteboard, explaining in detail the dimensions of each ballpark. Then, for the final 20 minutes–in order to give you a more hands on experience–you watch some tapes of the most important moments in baseball history: Babe Ruth’s famous “called shot,” Bucky Dent’s home run in Fenway park during the division tie breaker, and the victory celebration of the 2004 world series, won by the Boston Red Sox.

Your teacher, who stresses the importance of some personal softball experience in learning the game, also assigns homework each week. Here is your first assignment: go home and make a video tape of yourself swinging the bat. Make sure the video is in focus, and that your whole body is in the frame. Take ten swings left handed, and ten swings right handed. Similar exercises are assigned throughout following weeks. Then your first game of the year comes along. Your team loses 23 to 0. The rest of the season doesn’t go very well either; you lose the first 10 games of the season. After a while, you manage to win a close one, and then a few more after that. But, for the most part, the season is a disappointment. You walk away from the experience believing that you and your teammates just aren’t the sort of people who can be good at softball. You don’t sign up for the team the next year.

This story, as unrealistic as it is, is not so different than the experience people go through when they are “taught” mathematics in school. In short, they are presented with a series of activities that superficially feature the subject matter, but in reality are absolutely unlike doing activity that generated that subject matter, and also quite unlike any secondary activity that allows one to really appreciate that subject matter. Watching baseball, and even mimicking the basic moves that baseball players do, is nothing like playing baseball–it won’t make you good at what they do. Similarly, watching someone drone on about the quadratic equation, or about the cosine and sine functions, or anti-derivatives, won’t make you good at math. Neither, really, will rote memorization of formulas or repeating procedures for doing problems (this may, however, at least get you to where you can pass the test). To get good at math, actual math, you have to sit and think, and explore, and even see things for yourself. Fortunately, this is easier, and more fun.

Many people tell me that they gave up on math when they could no longer follow what was going on in their math class. Well, I’m going to let you in on a secret: I almost never know what is going on in my math class. Granted, my classes are probably harder than the ones you were taking, but the difference isn’t so great as you might imagine. I don’t think I’ve ever been to a math lecture that I followed the entire way through. Losing track of what is going on in a math class is the norm. A good percentage of the time, it isn’t even your fault that you lose track of what is going on. Often, it is the fault of the lecturer. At some point along the way he may do something that actually makes it impossible for you to follow the rest of the lecture–for example, introducing a notation that you aren’t familiar with, and not explaining it. Or he could be assuming knowledge of a concept that he never taught.  The lecturer isn’t necessarily being negligent in all of this. It’s just that explaining mathematics is actually quite difficult–you have to bring people to where you are, without knowing exactly what is missing. Knowing what information to supply so that your audience can see what you see can be even more difficult than the doing of mathematics itself. The format of a daily lecture, in which the students frantically write down everything that the lecturer is writing, while simultaneously attempting to follow the logic of what he is saying, simply doesn’t lend itself well to actual comprehension.

The sort of brain that is good at following a math lecture, in fact, may not be at all the sort of brain that is good at actually grasping mathematical structure. It is true that some brains are better than others at doing math, but I seriously doubt you’ve ever received reliable information on whether or not you’ve got one of the better ones in this regard. Think about this: there was a time when most people were illiterate, when reading and writing was reserved for the scholars. Now virtually everyone can read. This is because we, as a society, have figured out how to really teach reading, how to facilitate a person’s learning of what is actually quite an involved mental task. But I don’t think we’ve accomplished this with math yet.

A person said to me recently that she isn’t good at math because she just isn’t a right brain sort of person. I’ve heard similar comments in the past. My response is that you don’t do math with the right side of your brain, you do it with your whole brain! I’m sorry, my friend, but  you’ve been duped by a false dichotomy. The logical and analytical are not opposed to the creative and fanciful (let alone separated neatly into distinct compartments in your physical brain. The things we believe these days!)–your mind is an organic whole, and I doubt any part of it is completely inactive in anything you do. If you are an artsy or creative type, that is probably going to help you do math, believe it or not. Does your mind have a tendency to go off on tangents? Can you not stay focused on what the teacher is saying? Good! This means that your brain is actually doing something. Don’t fight it! Quit paying attention, and pay attention to the thoughts that you are having (OK, sorry, now I’m sounding like Morpheus from The Matrix).

The main thing that makes math seem difficult is when someone is seeing something that you aren’t, but this isn’t communicated. They have a picture in your head, that you don’t see. Or there is an idea, or a whole train of thought, that is simply missing, and that makes everything seem arbitrary and complicated. Everything is difficult when you don’t know what it really is. For example, I bet you’ve heard of the quadratic formula. You may even remember part of it! You may even know that it’s used to “solve” quadratic equations. At some point in your math education, I bet you were presented with this monster:

\frac{-b \pm \sqrt{b^2-4ac}}{2a}

What you are seeing is the result of moderately long process of reasoning that is not presented to you when you are given the formula. It’s easy to think that, since the result looks complicated, the process itself must have been so, and that it’s something you could never have come up with. But actually, it is merely the result of many steps that are, in themselves, straightforward. These steps constitute a perfectly logical and natural series of thoughts, a series of thoughts that maybe aren’t so different from the sorts of thoughts that you would start to have in your math class. You (or I) might not be able to come up with the quadratic formula on your own, but you can at least get a good idea of how someone did. In my next post, I’m going to give you a down to earth explanation of where this comes from, and how and why a person would come upon it naturally, and I’m going to try to get you to see it in the way that I see it. This, I hope, will give you an idea of how people think about mathematics. Moreover, I hope I can convey the idea that, when we see where these things come from, and see the kind of thinking that caused them, everything becomes a lot more interesting. The quadratic formula can actually be seen as a simple example of the sort of mathematics I study–Algebraic Geometry. I’m hoping, also, that I can (if it seems interesting enough), build up from there a more general description of what Algebraic Geometry is. We’ll have to see how far I get with that, though.

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More than Math

November 6, 2009 Leave a comment

What I call materialism is the belief that the material world is all that exists, that everything else is somehow derivative of a base level reality that is somehow “material.” The most common sort of materialism, scientific materialism, adds to this the belief that science is the most suitable technique for coming to know the truth about this baseline reality. Some people associate this view with rational thinking itself, with a sort of wise willingness to look beyond mere appearances and see things as they really are.

Against this view, I ask that the reader would try to join me in a sort of shift of perspective. We are all aware of the distinction between perception and reality, between the familiar world (clouds, trees, etc) and the world that we imagine is somehow beneath that, the world that is somehow objective and concrete, and is considered as the more fundamental cause of our perceptions itself. Nowadays we tend to think that science is what is best able to study that “baseline reality”: we tend to identify that baseline reality with something like atoms, particles, waves, quantum states, or whatever. Now, whether such a world–a baseline reality that is the cause of all of our subjective experience–exists in general, I am not quite sure, nor even am I sure exactly if this is a meaningful question to ask. But what I want you to try to do now is try to imagine that the world described by science in particular, whatever it is, is not actually as the world “as it really is,” or even a “best approximation” of that world. Instead, I’d like you to try to imagine that the content of science is even more deeply psychological than the familiar world, that it is even further removed than the familiar world is from whatever baseline reality might actually exist. That is, try to think of science itself as the end result of a very psychological process indeed: one of applying a deeply abstract and mentally generated structure (namely, mathematics) to the world that we find ourselves experiencing.

Now, this point of view is actually quite natural, once a person gets used to it. In some sense at least, this has to be what science really is, though I don’t think this is all that it is.  Now, from this new point of view, it is actually quite a remarkable fact that science works at all. That is, it is remarkable that all of these particular abstract mental structures, things we came up with in our mind, do seem to find a correspondence to the reality that we perceive. Some would say that this fact alone refutes the perspective shift that I have endorsed, and is evidence for the truth of something more like scientific materialism. And I actually think this is somewhat correct; science is obviously much more than just a mental construct. But I think it also indicates that we, at least today, do not think deeply enough about the very question of why science works. I believe that our popular ways of thinking about the world do not actually have a satisfactory answer to that question. Further, I believe that it is mainly in answering this question that materialism has gone horribly awry. Or, from another point of view, materialism is caused by a gravely mistaken answer to this question. What I believe is the mistake, and the whole hubris of materialism, is to suppose that a sufficient explanation for the effectiveness of science is simply that some sort of mathematically describable structure is itself a final explanation for what is really there, is a sufficient explanation for reality itself.

Why does mathematics hold such an extraordinary ontological status? I see no good reason that it should. Thus, I wish to make the following proposition: what if we were to give to “natural language” (a terminology that actually already belies a materialist bias) the same ontological weight as we currently give to mathematics? To put it in less philosophical language, what if popcorn and lakes were as fundamentally real as pi and the integers? After all, undoubtedly they are, to most people.

This would, I believe, mean that a great many things would become real to us again! Suppose–imagine–that we simply took it for granted that thought and language are somehow fundamental, that their very existence and association with the world is basic, axiomatic. I think, then, that we would think of the appearances as being predictable when we describe them in the manner that science does precisely because the “thought patterns” being put to use–the ideas and language of mathematics–have a rigid, precise, and predictable structure. Rather than idolizing it, lifting it up as the proper foundation of all other truth, we could simply say that mathematics is the language that we can use to talk in a very unique and useful and beautiful way about the world. And we could say that art, for example, is another such way. Or poetry. The representations of the poet are no good for making the sorts of predictions that science makes, but that is not the point. It is not their nature to do so. The point is that they are just as real; it is only a philosophical bias to say otherwise. From our new perspective, we can then truly say, with philosophical rigor rather than simply mystical vagueness, that an artist knows something about a cat, or a tree, or a mountain range, that a scientist does not (and vice versa). Even if we were to exhaust what mathematics can say about reality, there would still be something very definite left–indeed a great deal left.

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