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Posts Tagged ‘mathematics’

This is from Bryan Magee’s 15-part series The Great Philosophers, broadcast on BBC2 in 1987. Here he talks with Hilary Putnam about the philosophy of science.

Alan Kay has talked about how most people don’t understand how science worked in the 20th century, much less the present century. Our schools still teach science in a 19th century fashion in terms of how people can approach knowledge. This episode of the show amply demonstrates this disconnect.

As I studied what was said here about scientific thinking up until the late 19th century, it reminded me a lot of how I was taught computer science. It was thought to be a process of adding on to existing knowledge, and sorting out any inconsistencies. With rare exceptions, alternatives to existing models were not discussed, or even made known to students.

I’m going to put the notes I took from this episode below each section of video, because I found it hard to follow the arguments without going through them slowly, and looking at what Magee and Putnam said explicitly.

Magee: The religious world view has been largely replaced by a world view purportedly derived from science, since the 17th century.

Science was seen as a process of “adding” and sorting for 300 years, up until the late 19th century. The view of knowledge was that it grew by accumulation. Science was seen as getting its success from an inductive method. For 300 years educated people thought of the Universe as just matter in motion. They thought if we went on long enough, we’d find out everything about the Universe there was to know. This whole idea has been abandoned by science, but non-scientists think that scientists still think this way.

Kant challenged the “correspondence” view of truth, that theories correspond directly to reality without nuances, that the world makes its truths apparent, and scientists just find out what that is and write it down. Kant said there’s a contribution of the thinking mind. Truth depends on what exists, and on the mind of the observer. Scientists have come to a similar view, that theories are not merely dictated to us by the “facts.”

The categories and interpretations we use, the ideas within which we organize our observations, are our contributions. The world that’s perceived by us is partly contributed to by external effects, and is partly made up of categories and ways of seeing things that come from us. [Mark says: I think Plato's notion of "forms" also applies here, in terms of our theories. We could equate a theory of a phenomenon to its form in our own minds, and we could recognize that if a phenomenon is unfamiliar enough, different people will create different forms of it in their own minds.]

In the late 19th century the views of scientists changed to realize that the old theories could be wrong. The more modern view is that not only is our view of reality partly mind-dependent, but there are alternatives, and the concepts we impose on the world may not be the right ones, we may have to change them, and there is an interaction between what we contribute and what we find out. It was realized that there were alternative descriptions of some things that were equally as valid.

The new conception of science is that theories are constantly being replaced by newer and better ones, which are richer, that explain phenomena more fully, and there’s a mystery to our universe which will never be completely discovered. The view about “truth” that Putnam said was coming into use more and more in science is the idea that there cannot be a separation between what’s considered true and what our standards of assertability are. So the way that mind-dependence comes in is the fact that what’s true and what’s false is partly a function of what our standards of truth and falsity are. That depends on our interests, which change over time. Putnam defined “interests” in a cultural context, such as, in our modern world we’d recognize that there’s a policeman on the corner. Someone from a tribal culture, which may have no formal social services, wouldn’t recognize a policeman, but would instead see “someone in blue” on the corner.

Even facts within theories can have alternatives, as was evidenced by the theory of relativity.

A scientific theory can be useful even if nobody really understands what it means (quantum mechanics). [Mark says: You could also say the same about Newton's theory of gravity.]

Putnam thinks the way in which the old scientific view (pre-19th century) was destructive was that since it saw scientific findings as objective facts which were accumulated over time, then everything else was considered non-knowledge, that couldn’t even be considered true or false.

Real interesting! Putnam and Magee talk about how computer science, through an interaction with computer simulations, has created a growth of knowledge about the human mind. This relates to what I’ve read about Licklider’s use of computers at MIT in the 1950s, in Waldrop’s book, “The Dream Machine.”

Edit 5-4-2011: I had a brief conversation with Alan Kay about the main theme of this program, and I put particular focus on this part, where Magee and Putnam discussed the role that computer science has played in the advancement of knowledge in other areas of research. He zeroed in on this part, saying that it was a misperception of what really happened, at least in relation to Licklider (and Engelbart). He said the advances in knowledge from computer science have been paradigm shifts, not advancements by interaction. He pointed out that the theory of evolution did not begin in the field of biology, as Magee asserts. He also said that computers did not begin as “a self-conscious analogy to the human mind,” but rather as machines to run calculations. I knew that. I thought Magee’s description of this was rather romantic, and other historical accounts have agreed with his assessment, so I let it slide, but I have since had second thoughts about that.

Magee: Most people with college degrees have no idea what Einstein’s theory of relativity is all about, more than 70 years after it was published. It’s done very little to influence their view of the world. Isn’t there a danger that science and mathematics are racing ahead, and the whole range of insight that that’s giving us into the Universe simply isn’t filtering through to the layman?

Putnam: There was a text on Special Relativity called “Space-Time Physics” that was designed for the first month of the first college physics course, and the authors hoped that someday it would be taught in high schools.

The question and answer above really struck a chord with me. First of all, I didn’t encounter Einstein’s theory of relativity in my first semester physics course in college. All we covered, that I remember, was Newtonian mechanics, though at a more detailed and advanced level than what we got in high school.

The discussion that Magee and Putnam had about General Relativity, and the risk we take with science “racing past” the rest of society, I think, makes a good case for Alan Kay’s efforts to teach more advanced math concepts to children, because without that, they’re never going to understand it. To put this in perspective, take a look at General Relativity, and think about the understanding of mathematics that would be required to understand it.

Strangely enough, I got more exposure to Einstein’s theories of relativity when I was in Jr. high school (1982-’85) than I got at any other time. We watched parts of Carl Sagan’s Cosmos series. In it Sagan talked about Einstein’s theories of relativity, and included Einstein’s notion of gravity as warped space-time. Farther back than that, I had been fascinated by the idea of black holes, since I was in 4th grade. I read all I could on them, and I learned some very strange things: that not even light could escape from them, and that matter was destroyed upon entering them, for all intents and purposes, though I think the evidence still supports the idea of conservation of matter. It’s just that the matter gets separated into its component parts, and some of the matter is converted into energy.

I got exposed a bit to a theory of gravity that was different from Einstein’s or Newton’s outside of school, by a toy inventor, of all people. So I knew there were other theories besides those that were commonly accepted. What became clear to me after exposure to these ideas was that gravity is a fascinating mystery. We didn’t know what caused it then, and we don’t know now, not really, except for mass. The question that would not go away for me was what about mass causes it? The idea that mass causes it just because it exists never satisfied me. The other forces were explained through phenomena I could relate to, but gravity was different.

(Update 12-15-2013: I’ve updated the paragraph below with what I think is a more accurate description of the theory, and I’ve added a couple paragraphs to further explain. I was mistaken in saying that astrophysicists think that the distortion in space-time causes us to “slide” in towards the gravity well.)

What Einstein’s theory explains is that gravity is not a force in the way that we think about other forces. His theory said that matter warps space-time, and it is this warping which creates the perception of a force acting on objects and energy. The way Sagan explained it is that we are all “sliding” being pushed inward towards the center of the gravity well, on by this distortion, and what keeps us from going to the center of the well is the outward force exerted by the earth we stand on. Scientists have tried to explain it using an analogy of a large ball setting on a piece of taut fabric, which creates a dip in it. If you toss a marble onto the fabric, it will fall towards the larger ball, due to the anomaly in the fabric. This is not a great analogy, because in it, gravity is pulling the larger ball down creating the dip in the fabric. If you can disregard that fact, what they are trying to get across is its the distortion of the fabric that alters the path of the marble, not any force. Another way this analogy is imperfect is it doesn’t illustrate how space-time is being drawn inward by something that is not yet explained. What Einstein’s theory says is that there is something about matter that causes this distortion in space-time. That could be attributed to a force, but from everything I’ve studied about the theory so far, Einstein didn’t say that. That’s left as “a problem for the reader,” so to speak.

The strange thing about the theory that might seem confusing is it explains that while we are “pushed” inward towards the well, it’s not a push in the Newtonian sense. This “push” is coming from what is called “space-time” itself. It’s a push on everything that constitutes matter and energy, down to our body’s subatomic particles, and photons of light.

Looking at how it is we stand on earth, rather than all matter being drawn into the well, it’s rather like standing on stretchy material that’s constantly being drawn into something, with a “wall” in our way (which does not “catch” the fabric). The closer the material gets to whatever is pulling on it, the more stretched it becomes. As it becomes stretched, so are we, and all the matter around us, because, as we understand, matter is “situated” in space-time. There may be some “friction,” if you will, between us and the material (space-time), that causes us to be drawn in with it, but it’s not total, allowing this “material” to slide past us, while we are repelled outward by the “wall” (forces in the matter we stand on). Hopefully my attempt to explain this isn’t too confusing. I’m groping at understanding it myself.

An example of the way school gets science wrong

High school physics was odd to me, because we talked about stuff like how electrons had both the property of a particle and a wave (as Putnam discussed above), a very interesting idea, but when it came to gravity, all we focused on was Galileo’s and Newton’s notions of it, particularly Newton’s Universal Law of Gravitation. One of the things I remember being emphasized was that “this law is the same throughout the Universe.” For the sake of argument, from our perspective, being on Earth, I was willing to accept that claim, but I remember being a bit skeptical that it was really true. I knew that we hadn’t really explored the whole universe, and that we hadn’t even come close to testing this notion everywhere. I was open to the idea that someday we might find an exception to this notion if we were to theoretically explore the Universe, which is something that may never happen (I didn’t even know about Mercury’s orbit, which doesn’t fit Newton’s theory as well as the orbits of the other planets). So, for practical purposes, we could assume that it’s “universal.”

We talked about Einstein’s theory of Special Relativity, and what was really fascinating about that was E = mc2, that there’s a relationship between matter and energy that’s only “separated” by the square of the speed of light. My memory, though, is we hardly talked about General Relativity at all, except perhaps in a historical context. Looking back on this, it’s rather obvious to see why. In high school science we were expected to get a little more into the details of scientific theories, and work with the math concepts more. The school system hadn’t prepared us to work with the notion of General Relativity at that level. So in effect we skipped it, but the conceit that was presented in class was that Newton’s law of gravity was “the truth.”

I remember being asked a question on a physics test that asked, “If aliens visited Earth, would we find that they have the same knowledge of gravity as we do?” I paused. This was an interesting question to me, because I thought it was asking me to consider what understanding another race of intelligent beings would have about this phenomenon. I asked myself, if space aliens existed that were intelligent enough to build craft for interstellar travel, would they have the same ideas about gravity as we do? I answered, “Maybe.” I added something about how since the aliens had managed to make the journey from whatever star system they came from, that their technology was probably more advanced than ours (I mean, we haven’t tried this yet, so that was a good guess), and maybe they had a better understanding of gravity than we did, particularly what caused it. I hedged a bit, but I guessed that there was probably a link between technological development and greater scientific understanding of our universe. Granted, this was a totally speculative answer, but it was a speculative question, as far as I was concerned.

My physics teacher marked this answer wrong. I was floored! I wondered, “What did she expect?” I asked her about it after class, and she said the answer she expected was something along the lines of, “Yes, because the Law of Gravity is universal.” I was so disappointed (in her). It immediately hit me that, “Oh, yeah. I remember we talked about that.” I could’ve almost kicked myself for thinking that she had asked a thought-provoking question, and falling for it! I was supposed to remember to recall what we had talked about in class. I wasn’t supposed to think on it! Duh! How could I have been so stupid? That’s really how perverse and offensive this was. It brings to mind the fictional short story of “Harrison Bergeron,” now that I think about it… However, trying not to see that she was telling me not to think, I tried to talk her through my reasoning, because I thought I gave a legitimate answer. I told her about the other notions of gravity I knew about, and the questions they raised for me. She wouldn’t hear of it. I think I said in a final protest, “Do you really think we’ve discovered everything there is to know about gravity?!” In any case, she didn’t answer me. I walked out of the classroom exasperated. It was one of the most disillusioning experiences of my life. It made me fume!

Looking back on things like this, she probably didn’t even understand what a good question she had asked. Secondly, there were other instances where this happened in my schooling. Sometimes I wanted to think through things and come up with original answers, not merely regurgitate what I had been fed, and I got penalized for it. She and I had not been getting along for most of the time while I was in her class, and I think it was over issues like this. So this was nothing new, but this incident revealed a disturbing fact to me in a way that was so obvious, I couldn’t just brush it off as a misunderstanding between us: Her approach to science was that we were supposed to accept what she said as truth. We were not supposed to think about it, or question it. The only thinking we were supposed to do was in calculating results from experiments, but a lot of that was applying the “correct” formulas. More memorization. Nevertheless, I got an “A” in her class.

Looking at this from a “mountaintop” view, I think this example shows the split between 20th century scientific thinking, and the 19th century thinking that’s been used to teach science in schools. I saw a discussion recently where Alan Kay talked about this with the No Child Left Behind policy, that for students who were developing an understanding of scientific thinking, they had to, on the one hand, gain real understanding, and on the other, remember to answer “wrongly” on the test. That summed up the experience I describe above! I’ve used my example sometimes when I’ve heard people complain about this, because I can say to them I got the same treatment when I was in my high school science classes, more than 20 years ago. As far as I’m concerned, this policy is just taking that idea of instruction, which has been around for years, to its logical conclusion. It’s now metastasized throughout the public education system, at least in the areas that are tested for proficiency, whereas in my day there were exceptions.

—Mark Miller, http://tekkie.wordpress.com

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I’ve jumped ahead some, since I’m going through the book with a local Lisp users group. They’ve decided to not go through all of the exercises, just some that we select. I may backtrack at some point to cover some of the ones I missed.

I won’t say much about this exercise. If it mystifies you from the start, look up information on “lambda calculus”. I had to review this as well before I started into it, because it had been a couple years since I last looked at stuff like this.

I found that it didn’t help at all to try to execute stuff in a Scheme interpreter. That misses the point. I found it helpful to just write down, or type out in a text editor, a walkthrough of the logic in the “zero” and “add-1″ functions as I used them, to figure out part of the exercise.

Getting to “one” and “two” was pretty easy. What the exercise says to do (it says “use substitution”) is use the functions you are given to figure out what “one” and “two” are. In other words, carry out the logic and see what you get.

The challenging part is figuring out what the “+” operation should be. It helps to look at this at a conceptual level. You don’t have to do a math proof to get this, though understanding how to do an inductive proof helps in giving you ideas for how to proceed with this part (if you need practice with doing an inductive proof, try Exercise 1.13). Keep in mind that this is a computing problem, not a classical mathematics problem (though there is definitely mathematics of a sort going on as you go through the logic).

The idea of the exercise is to get a different idea of what the concept of number can mean via. computing. It helps to be familiar with mathematical logic for this one, to see what you’re really doing, though I think you can get by without it if that’s not one of your competencies. If you’ve ever seen a mathematical explanation for the existence of natural numbers, this is reminiscent of that.

I don’t want to give away the answer, except to mention an article I found, called “The Genius of Alonzo Church (rerun)”, by Mark Chu-Carroll. NOTE: Do not read this article until you’ve done the problem! It will give away the answer. I only mention it because Mark says something quite beautiful about what the answer means. It ties right into a concept in object-oriented programming.

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I came upon a video recently, titled “Dangerous Knowledge,” produced by the BBC. It profiles the life and times of three mathematicians (Georg Cantor, Kurt Gödel, and Alan Turing) and one scientist (Ludwig Boltzmann). For some reason it drew me in. I can’t embed the video here, but you can watch it by following the link.

The show makes allusions to an atheism which I find difficult to relate to this subject of crumbling certainty, the rise of contradictions or imperfections in logical systems, and the decay of what was thought to be permanent or static. Commentators in the show kept making reference to this idea that “God is dead.” It feels irrelevant to me. The important thing going on in the story is the destruction of the concept of the Newtonian mechanical universe. This is not the destruction of Newton’s theories of motion, but rather people’s misperception of his theories. Perhaps this is a European idea. Their concept of God, as it’s portrayed in the show, seemed to be deeply tied to this idea of certainty. Since these four people (not to mention the world around them) were destroying the idea of certainty, their concept of God was being destroyed as well.

Edit 11/17/2012: This was also about the limits of some of our mental perceptual tools, math and science. I think what drew me to this is it felt closer to the truth–that the notion that mathematics provides clarity, truth, and that our world or Universe are permanent was just a prejudice, which doesn’t hold up against scrutiny; that in fact our notions of truth have limits, and that everything in our world is in flux, constantly in a state of change.

I had heard of Cantor when I read “The Art of Mathematics,” by Jerry King. He talked about how Cantor had come up with this very controversial idea that there were infinities of different sizes, using the mathematical concept of sets. He proved the paradoxical idea, for example, that the infinite set of even (or odd) numbers was the same size as the infinite set of natural numbers.

What’s interesting to me is that through this program you can see a line of thought, beginning with Cantor, running through several decades, to some of the first baby steps of computer science with Turing. Turing’s inspiration for his mathematical concept of the computer (the Turing machine) came from the work of Gödel, and another mathematician named Hilbert. The work of Gödel’s that Turing was specifically interested in was inspired by the paradoxes raised by Cantor.

The video below is from the television production of Breaking the Code, which was originally written as a play by the same name. It’s about Turing’s life and work deciphering the Nazis’ Enigma encoding/decoding system, and his ideas about computing. The TV production puts the emphasis on Turing’s homosexuality and how that put him in conflict with his own government. The play got much more into his ideas about computing. There are a couple scenes in the TV version that talk about his ideas. This is one of them. I think it’s a perfect epilogue to “Dangerous Knowledge”:

A couple clips of the play, starring the same actor (Derek Jacobi), are shown in the first episode of The Machine That Changed The World, called “Great Brains.” There’s a whole section in it on Turing starting at 43:45 in the video.

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“So, so you think you can tell
Heaven from Hell,
blue skies from pain.
Can you tell a green field from a cold steel rail?
A smile from a veil?
Do you think you can tell?”
— from “Wish You Were Here” by Pink Floyd

I’ve taken some time to get back into the subject of mathematics, and secondarily math education. This is partly because I’ve had this niggling feeling that math is pertinent to my study of computing. I was always told that math was important to computer science, but in my work I didn’t feel it had relevance beyond algebra and discrete structures. The thing is, with the exception of a couple math classes I had in high school I didn’t find the presentation of math that interesting, nor did I feel like I was really grasping it. Nevertheless there was always some part of me that was interested in math generally for some reason.

Alan Kay helped steer me towards some interesting things about mathematics last year, and I’m grateful for that. This started when I watched a video of a presentation he did called “What is Squeak?” and I became intrigued when he said that students aren’t learning what real mathematics is in most American schools. I went to good public schools, but I figured since he talked about most schools I probably wasn’t taught this either. I asked him about this last year, and I realized quickly that no, I wasn’t taught what real mathematics was.

He believes that students should be taught to think of mathematics as mathematicians do, even if they don’t go into it professionally. He’s seen examples of how this can be done. The reason is that math is not about the numbers, arithmetic, symbols, and algebra that we learned in school. It’s a way of thinking. It’s just a matter of translating this mode of thought into forms that children can understand and work with.

For the rest of this post I’ll be talking about a book called The Art of Mathematics, written by Jerry King, who is a professor of mathematics at Lehigh University. This isn’t a book report. He talks about a bunch of things in greater detail than I’ll cover here. So please do read it. I’ll just be talking about what really grabbed me.

It’s a very interesting book. In it King lays bare what is rarely acknowledged, that typically schools and universities do not teach the full depth, the beauty of mathematics. In a typical scenario a student is not exposed to real mathematics until their junior or senior year in college, if they take math courses for that long. Up until then they are fed a steady diet of procedures and techniques for solving math problems, but they don’t get to the essence of what mathematics is. The end result has been generations of students who for the most part fall into two categories (à la C. P. Snow).

There are those who became skilled in some aspects of mathematics for its utility in science and engineering, but have missed its beauty. Then there are those he calls “humanists” who have felt like math isn’t for them. They’ve felt forced to do it in school. They left it by the side of the road as soon as they had the chance, hoping to never deal with it again. They are educated but they are so distant from what mathematics is they don’t recognize it, much less its beauty. We see them out in society all the time. They say, “I’m not good at math,” and they say it with no stigma at all. Most of the rest of society would say, “Yeah, me neither. What was the point?”

None of us would say, “I can’t read” without feeling a profound sense of shame and deprivation, which goes to show that mathematics is not seen as essential in order for people to function in our society. But is that view correct? I think there’s reason to believe it’s not. There are some common everyday reasons I can imagine where having “a taste for mathematics”, as Alan Kay calls it, would be advantageous to all sorts of people in ordinary life (again, I’m not talking about algebra, but mathematical thinking). But more importantly, Kay has also said that in order for our civilization to realize the potential of the computer as a new medium it is essential that we understand (real) mathematics and science. He’s also said for years that “computers form a new kind of math”, a kind that doesn’t fit into the confines of classical math categories.

Jerry King wishes that mathematics would be upgraded to the status of a subject that students should be literate in to be considered educated. He explains why mathematics education at universities has not been good for decades, and touches on some essential ideas that every mathematician knows, which the typical math student is not shown.

Mathematics at its essence is a way of thinking about relationships of abstractions. It’s also a creative/discovery process, something that most math pedagogies don’t teach.

From this one can see right away that math is valuable to science, since deeply embedded in it is the study of the relationships between things. It also has value in computer programming, since in order to get our programs to work we have to understand the concept of abstraction, and understand something about the relationships between things in our code, and entities that are created in a running program.

Bertrand Russell said that mathematics is “p implies q“, and as King explains in his book this point of view allows us to take assumptions and reason about their implications. For example, “If p is true then q is also true”.

Beyond the reasoning skills it engenders, King says there’s also beauty in it which you can experience once you get to the real essence of mathematics. For mathematicians this is the reason they do it. I would say the same is true of computing. It’s the reason I got into it, and still pursue it, though as a field I would not be so bold as to say we know yet what its true essence is.

What King’s book reveals is that mathematicians do not merely have a more advanced knowledge of math in terms of how most of us understand it. They have a very different perspective on it than we do. Mathematicians believe The conventional wisdom among mathematicians is that this perspective is impossible to teach. You either “get it” innately, or you don’t, no matter how you’ve been taught. King is the exception. He contends that everyone has the potential to understand this perspective, and that it would do us good to do so. If only mathematicians could find it in themselves to have a passion for teaching what they really know.

What follows are summaries from my notes, and quotes from his book. I’ll try to make it clear when I’m talking about my own thoughts.

Aesthetics

In my estimation the most important chapter in his book is called “Aesthetics”. King says that mathematicians do mathematics for aesthetic reasons, but the aesthetics of mathematics are not defined anywhere, either in the field or outside it. For mathematicians it’s a personal experience.

The crux of his book is explaining why most people are never exposed to the beauty of mathematics, and it’s a tough problem to solve:

Deeply embedded in our culture lies the notion that mathematics can be truly comprehended only by a gifted minority. Because so few members of the otherwise educated public possess even the rudiments of mathematical knowledge, mathematics has been assigned a special status. Unique among the collection of disciplines–such as philosophy, history, and literature–which in times past formed the basis of both the concepts of liberal education and the core curriculum, mathematics may be set aside under ordinary conditions without social or intellectual consequence.

Behind all this stands the concept–widely held and deeply believed–that there exists in a certain few members of the human race a type of “mathematical mind” which allows them to understand the logical complexities of mathematics. It is believed that just as there are only a few people capable of running 100 meters in less than ten seconds there are only an analogous few capable of understanding mathematics. And just as the inability to sprint at world-class level carries with it no social stigma, neither does the inability to understand mathematics.

Mathematical talent comes to you exactly as does sprinting talent: God either gives it to you or he does not. Or so it is believed.

Such beliefs provide comfort. Through them, members of the public can justify their often awesome mathematical ignorance. And because of them, mathematicians can rationalize their failure to teach–in a way such that the knowledge sticks–even the basics of the glorious discipline which occupies every moment of their conscious thought and almost every ounce of their energy. You cannot be expected to understand mathematics–so goes the myth–unless nature has provided to you the kind of mind necessary for the subject’s comprehension. Nor can you be expected to teach mathematics to people who lack the basic mental equipment for it, just as Minnesota Fats lacks the physical equipment to run the Olympic Trials.

Comforting they may be, but these beliefs have no more validity than astrology.

He speculates that a sense of aesthetics may really make mathematicians who they are, as opposed to the developed senses of logic, precision, the ability to manipulate symbols, or the ability to deal with layers of abstraction. Jules-Henri Poincaré, a French mathematician, believed…

that an “aesthetic sensibility” for mathematics defined the very soul of the mathematician. It acted as a “delicate sieve” without which no one in mathematics can become a “real creator”.

Poincaré is right as rain about beauty and every mathematician knows he is right. You have inside you an aesthetic sensibility toward mathematics which acts on your intuitive mind as a delicate sieve sorting out the elegant and harmonious ideas from those which are merely useless combinations of other ideas. You have it, that is, or you are not a mathematician.

Poincaré, however, did not extend the notion of the aesthetics of mathematics to teaching. He did not believe that students could understand the beauty of mathematics through the educational process, but King is hopeful.

Perhaps Poincaré was wrong. Maybe the notion extends. Perhaps you can bring students to mathematics early on by emphasizing its aesthetic value rather than its utility or its applicability. … I don’t know. But one thing is certain. We will do no harm by trying. For what we do now has failed. Mathematics as understood by mathematicians remains unknown to everyone else. … To the humanist–and to everyone else in the other culture–the subject is something to be shunned.

I refuse to believe that this is the nature of things, that mathematics must remain forever beyond all but a tiny minority of our citizens. The notion that there exists a large subset of the populace who are capable of appreciating and understanding music, art, and literature but are somehow innate mathematical cripples seems to be simultaneously arrogant, apologetic, and just plain wrong. … We can do ourselves no harm … by presenting to our students early on those characteristics of mathematics which, in Poincaré’s words, contain “this character of beauty and elegance, and which are capable of developing in us a sort of aesthetic emotion.”

Instead of aesthetics, the schools give us drill and tedium–and immediately forgettable techniques aimed at unwanted and unwelcome applications.

The other side of this problem is we need to understand…

why people are driven to do important and difficult things for the sake of beauty. We need to understand what Mr. Keats had in mind when he wrote:

“Beauty is truth, and truth beauty–that is all
Ye know on Earth, and all ye need to know.”

In the book King notes an interesting correlation between the beauty of mathematics and scientific and engineering ideas, derived from mathematics, that have been recognized as being very useful. He suggests that perhaps in the case of mathematics beauty brings more than just pleasure, but also advancements in our ability to make new discoveries and build things that work. That would be pretty cool if it was true. :) King also says that mathematics IS a kind of art.

He says that unfortunately there has been no formal study of aesthetics (he suggested philosophy is the appropriate venue for this) which might help the academic community better understand what mathematics is, because philosophers don’t consider art and aesthetics as serious subjects, since they provide amusement and enjoyment. To them serious subjects like truth, justice, and reality are the things worth spending time on.

[S]erious people–most of them–do not deal professionally with matters of amusement and enjoyment. But they ought. Because there is more at stake than understanding what kind of art appeals to whom and why it does. … We need to understand why people are driven to do important and difficult things for the sake of beauty and for its own sake alone.”

My own sense of the reason King says this is it would help educators understand how to teach mathematics so that people understand it, and convey why it’s important.

Quoting Seymour Papert, King says:

[W]hen mathematics is taught in the schools, students are asked to “forget the natural experience of mathematics in order to learn a new set of rules.” Moreover, as we all know, the existing “rule learning process” does not work, has not worked, and–in my view–cannot work.

What is needed is a real understanding of the mathematician’s “personal experience” with his subject. At the highest levels, there can be no doubt that this experience is largely aesthetic. What we must learn first are the characteristics of mathematical aesthetics so that we can talk about mathematical elegance in more than a descriptive manner.

He says this will require rethinking classical aesthetics to include theories (to be developed) that apply to mathematics. It will require bringing teachers into math classrooms in primary and secondary schools who have been touched deeply by their own personal sense of mathematical beauty, so they can communicate that to their students. The problem here is:

We are not … headed that way. The future seems to hold, for mathematics instruction, an increased dependence on technology in the form of computers, hand-held calculators, and video presentation. Surely, I cannot be the only person to notice the clear correlation between the declining mathematical abilities of American students and the aggressive introduction of these technologies into the mathematics classroom. The decline began as the technology came in.

At the university level it will require bringing research mathematicians back to teaching. The problem here, he says, is that mathematicians think their only job in life that’s worth doing is mathematics. If a mathematician talks about mathematics (rather than doing it) they are considered “on their way out”, a former mathematician. In the profession it’s seen as beneath one’s stature, even shameful, to talk about it in descriptive terms. Thankfully there are a few like King who don’t care about that.

Technology and math education

Alan Bradley: Some programs will be thinking soon.
Dr. Walter Gibbs: Won’t that be grand? All the computers and the programs will start thinking and the people will stop.

— Tron (1982)

I want to take a detour here to address King’s point about technology entering the math curriculum in primary and secondary schools. His aversion to technology in the educational setting is understandable, but I think he’s narrowing the focus of his blame too much. In much the same way that college mathematics is taught as a set of rules and techniques, so it is in primary and secondary school. The quality of instruction would be improved somewhat if the technology were taken away, but not a great deal.

I had a wonderful conversation with an engineering professor from Duke University in March. One of the topics we discussed was how math is taught in secondary schools, from her own experience either observing it, or hearing second-hand accounts. It gave me a sinking feeling in my stomach. She said that students are required to buy a specific TI calculator model, and teachers use it as an integral part of their “math” instruction. She used the example of inverting a matrix. Rather than teaching what one is really doing when one inverts a matrix, students are taught to enter the matrix into the calculator and to hit the “invert” button. That’s it. They are not taught where the principle comes from, why they would want to do this, or what it represents. It’s a continuation of the procedural mindset that King criticizes. From this perspective I can see its folly now.

I said to the professor, “That’s not teaching math. That’s teaching how to use the calculator.” Yet the educational system has deluded itself into thinking they are teaching mathematics by doing this. Obviously one needs to know about arithmetic operations, but that’s not all we need to know.

Later I remembered that some academics used to say, “Someday we won’t have to teach math. We will have calculators and computers that will do the math for us.” First of all, as I’m increasingly becoming aware, this displays a fundamental misunderstanding of what math is, and its importance. It’s also widespread. What it reveals is the mistaken notion that mathematics has nothing to contribute to human reasoning, and is only useful for science and engineering where math is used to express ideas and figure things out. And why would it be useful to the general population to understand mathematics? It’s just symbol manipulation and calculation after all, isn’t it? Something that machines are fully capable of doing by themselves.

We can see how the educational establishment, and the “humanists” as King calls them, view mathematics. I’m not saying that calculation and symbol manipulation are unnecessary skills, but let’s be clear (for once!). Calculation is a product of arithmetic, not mathematics. Arithmetic is an outgrowth of mathematics, but they are not the same thing. Symbol manipulation is a part of mathematics, but it’s only a skill within it. It’s like knowing how to write in that realm. Having it does not mean that one understands mathematics.

I disagree with King that technology is at fault for the decline in math education. What he and the professor from Duke are seeing is an extension of the way math has been taught in our school system and in our universities all along! Now, I’ve heard the claims that the current standard in public education in the U.S. (No Child Left Behind) requires teachers to “teach to the test”, and other things. That’s not my point. Procedural thinking in math education has been around for decades.

The perceived role of technology also has something to do with this. If teachers and those who create curriculum believe that progress means machines do more of the thinking for us, then this is what you get.

There’s nothing wrong with a computer or calculator making some tasks easier. It can enhance our work. What we should not do is use it as a crutch, making it automate as much as possible so we no longer have to think, because then we’re really robbing ourselves. Technology should be brought in consciously with an eye towards where it helps us think.

Theory

King gets into some foundational concepts, theory and forms, which he uses later to illustrate what mathematical aesthetics is like.

He says a theory is a framework or set of rules which prescribes the collection of allowable sentences one can make about whatever objects are under consideration.

A theory, therefore, gives you a framework for talking about the objects of the theory. More precisely, it gives you rules of inference from which you can prove theorems about the objects. Truth about the objects consists of sentences about them which are either assumed in the theory or else are provable within the framework of the theory.

Just an aside, but this almost sounds like a description of a programming language on a computer, minus the stuff about theorems and proofs. Could programming languages represent theories of computation? Hmm…something to consider. Anyway…

Extending this idea of mathematical theory a bit King says:

Plato, you will recall, asserted that what one has knowledge of are abstractions, which he called Forms. Roughly, what Plato did was replace real-world objects by their Forms in his epistemology. Each real-world object–say an apple–is a temporal thing which undergoes constant change. The Form of the apple, however, lives in an ideal realm and is eternal and unchanging. It is the Form of the apple and not the apple, said Plato, about which we can have knowledge. Moreover, because Plato’s truth could be known only about Forms and not about real-world objects the Forms were to him more real than the actual objects. … Plato argued, in fact, that Forms represented the only true reality since it was only Forms about which one could have knowledge.

An example of real mathematics

King takes a stab at defining aesthetics in mathematics, in formal terms. I’m not going to cover that here. Read the book. From my own sense of the aesthetics of mathematics I think he defined it well.

He presents a couple of proofs to demonstrate the aesthetics. I’m going to use one of them as a way to discuss what real mathematics is like. It also demonstrates a foundational principle of mathematics: p implies q (also expressed as p=>q, using “=>” as a double-lined arrow). The symbol p represents a hypothesis, and q represents a conclusion.

He puts forward the problem: Without leaving the room let’s set out to prove that there are at least two trees in the world with the same number of leaves.

To make a proof we start with a hypothesis. “The hypothesis is what we assume, the conclusion is what we deduce.” The hypothesis needs to be minimal. This is part of the aesthetics.

Whatever we mean by “trees” and “leaves” it seems obvious that there are more trees in the world than there are leaves on any single tree (probably many more). … So, for our hypothesis, we will assume that the number of trees exceeds the number of leaves on any single tree by some positive number. It is sufficient to assume that the excess is only 1.

If this were written as a typical math problem I had in school it would have been rendered as: “Given t trees in the world and a maximum of m leaves on any one tree, prove that there are at least two trees with the same number of leaves,” and I would’ve felt stuck, because my math training taught that we should only draw knowledge from prior math material (formulas and proofs), and whatever the problem statement gave us.

What I find interesting is that King approaches the problem differently than I would have, and probably most other math students. He uses the ideas about forms and theory to get an idea about how to approach the problem, and he thinks about the relationships of “trees” to “leaves”. What I find interesting and pleasant about the mathematician’s approach is that our imagination, intuition, common sense, and experience can play a part in solving math problems.

We can imagine forms of different trees. We’ve seen hundreds of them in our life. So we have a fair idea of their structure and what they look like. The theory is that there are t trees in the world and a maximum of m leaves on any tree. Using the forms of “trees” and “leaves”, and this theory, we can explore its ideas.

He starts with an idea that fits easily into our notions of the relationships. When I looked at King’s hypothesis I thought, “Of course. I can imagine that,” but I wouldn’t have come up with it, because I wasn’t taught to think of math in these terms.

If I had understood the notions of theory and forms, and their use in mathematics, I might’ve come up with the same assumption right away. Or I could’ve experimented with other ones, and through a process of discarding the theories that didn’t serve the objective of the proof maybe I would’ve eventually hit upon the one King used. I remember Alan Kay saying that children who are learning to think like mathematicians go through a kind of scientific process, trying out ideas and seeing which ones succeed and which ones don’t. Research mathematicians go through the same process.

I believe it was King who also said something I have heard from those in the math reform movement for years, that it’s the process that matters, not the result, and I think I’m beginning to understand this. Math education has been results-oriented for as long as I can remember, creating a system where the only acceptable goal is to get from Point A to Point B as efficiently as possible. What King indicates is that this mindset is part of the problem.

(Update 7-7-2009: I modified the two paragraphs below because I realized I was making some pedagogical points that were perhaps inappropriate. What I wanted to emphasize in the first paragraph was the importance of criticism in the education of math students. In the second, I did not want to diminish the importance of precision. It’s just a matter of where it’s applied.)

I think, however, there’s been a misunderstanding of “it’s the process that matters” when the math reform agenda has been implemented. Logic matters in mathematics, as does the concept of relationships between things. Depending on where students are pedagogically, one subject or both might be good to emphasize. This implies that there needs to be room for criticism and correction of flaws in thinking, something that reform-minded teachers have discarded in the name of protecting self-esteem. This is not a small omission. I think the idea needs to be that how you get to something that’s logically consistent need not be a straight A-to-B process. “The right answer” is the end result of creating something that’s logically consistent. It’s an afterthought. The goal is to reason about the ideas under consideration, and to wonder about the implications of those ideas (not necessarily in that order).

It’s become apparent to me that in addition there are circumstances in the realm of mathematics where a precise answer is not desirable (though precision within proofs is essential), because it gets in the way of exploring and thinking about ideas and how those explored ideas fit into a logical structure.

This brings to mind a story I heard years ago about a female executive interviewing for a position at Microsoft with Bill Gates. It’s been a while, so I may not have this word for word, but it basically went like this. Gates asked her point blank during the interview, “How many gas stations are there in the United States?” The woman responded, “I don’t know.” He pressed her further about why she didn’t know, and she said, “I would need to collect data to find an answer to that question.” Gates got frustrated and asked her, “Are you stupid?”

I heard commentary about this incident from others who had been associated with Microsoft, and what they said was Gates wasn’t expecting her to recall a fact, an exact figure on the number of gas stations. He was looking for a reasonable estimate, and he would’ve looked for a justification of it–a proof, perhaps–without her leaving the room. What was said about this and other “Microsoft questions” is “they really want to see how you think.” My guess is in this instance he was testing the candidate’s understanding of mathematics. I’m also guessing that she had a math education that was similar to mine. I would’ve given the same response.

I talked to Sanjoy Mahajan, a theoretical physicist at MIT who was a panelist at this year’s Conference on World Affairs at the University of Colorado at Boulder. We were talking about math education and he spoke about the ability to estimate, and that this is an important skill in mathematics. He used an example that a math teacher used with students (I can’t remember the name). He said the teacher would ask, “How many trees could fit in this room?” without bringing a single physical tree into it. He wanted the students to visualize, based on the trees they’d seen, about how many would fit, and they’d discuss it.

The setup for the proof on “two trees with the same number of leaves” is similar. In terms of mathematics we don’t have to go out and look at every tree and count their leaves and come up with a maximum. We just make a reasonable estimate, based on our experience with how many trees are, say, in a single forest, and how many leaves we’ve seen on a single tree, and our knowledge that there are many forests in the world. We estimate that there are more trees than leaves on a tree. It seems to work, and that’s our assumption.

Here’s the theorem statement from the book: Let t denote the number of trees in the world and let m denote the maximum number of leaves on any single tree. If t exceeds m + 1, then there exist at least two trees with the same number of leaves.

Just to make the mapping to “p implies q” more obvious I’ll break it down: p = “t exceeds m + 1″, implies = “then”, q = “there exist at least two trees with the same number of leaves”.

Here’s the proof from the book: Since m denotes the maximum number of leaves on any one tree, each tree will possess either 0, 1, 2, 3, … or m leaves. Imagine m + 1 boxes sitting in a row on your floor, each tagged, in order, with the numbers 0, 1, 2, …, m. Now imagine bringing the world’s trees one by one into your room (suspend disbelief). Place each tree in the box which bears the label equal to that tree’s number of leaves (thus, a bare tree goes into box number 0, a tree with one leaf goes into box number 1, etc.).

We have only m + 1 boxes and, by our hypothesis, we have more than this number of trees. Hence, some box must contain at least two trees when trees have been brought into the room. Thus, at least two trees must have the same number of leaves.

Q.E.D.

The reason we were able to prove this was because of our knowledge of combinatorial math, but it’s so simple it even works intuitively.

Aesthetic distance

He introduced this topic of “aesthetic distance” from a philosopher’s perspective on how audiences appreciate art. He then put mathematics in the situation of being the object of appreciation. He said there are people who are too close to mathematics, and those who are too far away. The ones who are too close are scientists and engineers. They have quite a bit of knowledge around mathematics, but “their nose is pressed up against it”. They work with it all the time, but they don’t have sufficient distance from it to understand it and appreciate its beauty. They understand the rules and formulas they were taught, but they don’t really understand what mathematics is. Most others are so far away from mathematics that they have no comprehension of it at all and therefor cannot appreciate its beauty either.

In “the middle” between these two groups are the mathematicians, who are at a sufficient distance to understand what they are looking at, and to appreciate its beauty, and they are a small group compared to the others. With the way things are set up, only mathematicians can experience its beauty, and they aren’t doing anything to change the situation. They’ve locked themselves into their own “room”, so to speak, and they aren’t letting anyone else in.

King discusses at length the reasons why this has happened. To sum it up, it wasn’t always like this. There was a time when mathematics departments actually taught what they knew about math, rather than just procedures and techniques. King says that after 1957, our “Sputnik moment”, math departments at universities were seen as essential and no longer had to compete for money and students. They were even encouraged to engage in more research, for Cold War purposes, rather than teach. This fit in well with what mathematicians really wanted to do anyway–research. Math education, however, deteriorated. The change made mathematicians feel privileged, even superior. The privilege that mathematicians enjoy is in essence the cause of the problem with math education at universities to this day.

Conclusion

This book was an eye opener for me, mainly because it revealed that what I thought was math was not real mathematics, but rather aspects of math (deduction, reduction, symbol manipulation) and techniques derived from math (arithmetic, procedural tricks for solving problems, pattern matching). My sense is my math education in public school was better in terms of quality than what I got in college, but that isn’t necessarily saying much. College math was like how King described it: mostly techniques taught by rote, conveying no understanding. At least in the public schools I attended I got some sense of the aesthetics of mathematics.

I expected to get a sense of “the art of mathematics” from this book. Instead it talks about the subjects I discuss above, and more. I didn’t get a sense of the beauty of mathematics, but then King did say the purpose of the book was to talk about the subject, not delve into it. Nevertheless it is a good introduction for the uninitiated. He came out with a new book just a few months ago called Mathematics in 10 Lessons: The Grand Tour. I have it on my reading list. My sense of it is he gets into the actual subject of mathematics in that book.

—Mark Miller, http://tekkie.wordpress.com

escher

Hand With Reflecting Sphere, M. C. Escher, 1935

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