All posts by Charles Wells

math, understanding math

Stances

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Philosophy

With the help of some colleagues, I am beginning to understand why I am bothered by most discussions of the philosophy of math.  Philosophers have a stance. Examples:

  • "Math objects are real but not physical."
  • "Mathematics consists of statements" (deducible from axioms, for example).
  • "Mathematics consists of physical activity in the brain."

And so on.  They defend their stances, and as a result of arguments occasionally refine them.  Or even change them radically.  The second part of this post talks about these three stances in a little more detail.

I have a different stance:  I want to gain a scientific understanding of the craft of doing math.

Given this stance, I don't understand how the example statements above help a scientific understanding.    Why would making a proclamation (taking a stance) whose meaning needs to be endlessly dissected help you know what math really is?

In fact if you think about (and argue with others about) any of the three, you can (and people have) come up with lots of subtle observations.  Now, some of those observations may in fact give you a starting point towards a scientific investigation, so taking stances may have some useful results.  But why not start with the specific observations?

Observe yourself and others doing math, noticing

  • specific behaviors that give you forward progress,
  • specific confusions that inhibit progress,
  • unwritten rules (good and bad) that you follow without noticing them,
  • intricate interactions beneath the surface of discourse about math,

and so on.  This may enable you to come up with scientifically testable claims about what happens when doing math.  A lot of work of this sort has already been done, and it is difficult work since much of doing math goes on in our brains and in our interactions with other mathematicians (among other things) without anyone being aware of it.   But it is well worth doing.

But you may object:  "I don't want to take your stance! I want to know what math really is."  Well, can we reliably find out anything about math in any way other than through scientific investigation?   [The preceding statement is not a stance, it is a rhetorical question.]

Analysis of three straw men

The three stances at the beginning of the post are not the only possible ones, so you may object that I have come up with some straw men that are easy to ridicule.  OK, come up with another stance and I will analyze it as well!

"I think math objects are real but not physical."  There are lots of ways of defining "real", but you have to define it in order to investigate the question scientifically.  My favorite is "they have consistent and repeated behavior" like physical objects, and this behavior causes specific modules in the brain that deal with physical objects to deal with math objects in an efficient way.  If you write two or three paragraphs about consistent and repeated behavior that make testable claims then you have a start towards scientifically understanding something about math.   But why talk about "real"?  Isn't "consistent and repeated behavior" more explicit?  (Making it more explicit it makes it easier to find fault with it and modify it or throw it out.  That's science.)

"Mathematics consists of statements".  Same kind of remark:  Define "statement".  (A recursively defined string of symbols?  An assertion with specific properties?)  Philosophers have thought about this a bunch.  So have logicians and computer scientists.  The concept of statement has really deep issues.  You can't approach the question of whether math "is" a bunch of statements until you get into those issues.  Of course, when you do you may come up with specific testable claims that are worth looking into.   But it seems to me that this sort of thinking has mostly resulted in people thinking philosophy of math is merely a matter of logic and set theory.  That point of view has been ruinous to the practice of math.

"Mathematics consists of physical patterns in the brain."   Well, physical events in the brain are certainly associated with doing math, and they are worth finding out about.  (Some progress has already been made.)  But what good is the proclamation: "Math consists of activity in the brain".   What does that mean?  Math "is" math texts and mathematical conversations as well as activity in the brain.   If you want to claim that the brain activity is somehow primary, that may be defendable, but you have to say how it is primary and what its relations are with written and oral discourse.  If you succeed in doing that, the statement "Math consists of activity in the brain" becomes superfluous.

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Curvature

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This post is the result of my first experiment with the capability of including TeX in WordPress blogs (that capability is the reason I switched from Blogger).  This article will eventually appear as an example in abstractmath.org with lots of links to posts in the website that are germane to reading and understanding the article.

parabolasmall

Measuring bending

The curve y=x^2 (graph above) has a fairly sharp bend near the origin but as you move away from the origin in either direction it looks more and more like a straight line.   To measure this bendiness, each curvey=f(x) in the real plane has an associated curvature function  that measures how bent the curve is at each point.  (For this to work, f must have first and second derivatives.)  By definition, the curvature ofy=f(x) at x is given by:

\kappa_f(x)=\frac{\left|f''(x)\right|}{\left(f'(x)^2+1\right)^{3/2}}

The curvature function has the following three properties:

  • The curvature at any point on a straight line is 0.
  • The curvature at any point on a circle of radius r is\frac{1}{r}.  (Proving that this is true of the formula above is a nice freshman calculus exercise.)
  • The circle that best approximates a curve at a point (x,y) is the circle that is tangent to the curve at  (x,y) and that has radius 1/k, where k is the curvature of the curve at (x,y) .   This circle is called the osculating circle at (x,y) . 

Curvature of the parabola

You can calculate that the curve of the parabola y=x^2 at x is given by

\kappa(x) = \frac{2}{\left(4 x^2+1\right)^{3/2}}

For example, the curvature at (0,1) is 2, at the point (1/2, 1/4)  it is  \scriptstyle 1/\sqrt{2}\:\approx\: 0.71, and at (1,1) it is about 0.18.  The radii of the osculating circles are  1/2,  \sqrt{2}, and 5.59 respectively.  For large numbers the curvature is nearly 0; for example, at (10, 100) the curvature is about .00025.  To the eye the parabola near (10,100)  looks like a straight line.

This graph shows the osculating circles at x = 0, 1/2 and 1:

threecircles

You can see animated osculating circles at the Wolfram Demonstration Project (click on “web preview”).  From that site you may download Mathematica Player for free, which allows you to operate the slidebars yourself. 

This graph shows the parabola and its curvature function.

   curvature21

Turning the wheel

If you think of the graph of the curve as a path and you imagine bicycling along the path, the size of the curvature corresponds to the specific angle to the right or left the front wheel must be turned to stay on the path. 

A circle has constant curvature, so to bike around a circle means keeping the front wheel at a constant angle. 

As you can see the curvature of the parabola goes up gradually as you move from a negative x– value to 0, and after that it goes down gradually.   So biking along that path from left to right means gradually turning your wheel to the left, and then at (0,0) you gradually turn it back closer to straight front. 

Notice that going faster or slower makes no difference to the angle you must turn the wheel (as long as you don’t skid).  The curvature at a point on the path depends on the path (which doesn’t move), not on the speed of your bicycle moving along the path. 

Another curve

You may have a seen a model electric train in action.   What I am about to say applies particular to cheap model trains.  They tend to have two kinds of track pieces, straight segments and segments of circles of fixed radius.  You could make a layout with these pieces that looks like this:

circleline1

When the train starts at the left, it goes along a straight track (curvature 0) until it reaches the point (0, 2), where it enters a stretch of constant curvature 1/2.   At (0, 2) the curvature jumps instantaneously from 0 to 1/2.   Of course, “instantaneous” does not exist in the physical world (at this scale — don’t start carrying on about quantum jumps, please).   Where the track starts to curve, the front wheels of the train are forced by the change in the track to suddenly jump from facing straight front to angling right by a fixed amount.  If you have the track on the floor and stand looking down at it, and the train is going pretty fast, you will notice that the front car jerks to the right as it enters the curve.

You can see this in action in this You-Tube movie at 15 seconds and 1:14 minutes.

Fancier model trains have track pieces with varying curvatures.  Look up “model train” on YouTube and you will see dozens of them.

If a highway were laid out like the graph above, and you were driving pretty fast, then at (0,2) you would have to turn your steering wheel suddenly to the right and you would probably swerve a little.  But you probably can’t find any highways like that.  In the 1960’s a Kentucky highway engineer told me that they knew better; they used French Curves with curvature that increases continuously from 0.  Nowadays highway engineers lay out highways using CAD systems that can calculate the track transition curves directly.

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math, proof, representations, understanding math

Proofs without dry bones

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I have discussed images, metaphors and proofs in math in two ways:

(A) A mathematical proof

A monk starts at dawn at the bottom of a mountain and goes up a path to the top, arriving there at dusk. The next morning at dawn he begins to go down the path, arriving at dusk at the place he started from on the previous day. Prove that there is a time of day at which he is at the same place on the path on both days.

Proof: Envision both events occurring on the same day, with a monk starting at the top and another starting at the bottom at the same time and doing the same thing the monk did on different days. They are on the same path, so they must meet each other. The time at which they meet is the time required.

This example comes from Fauconnier, Mappings in Thought and Language, Cambridge Univ. Press, 1997. I discuss it in the Handbook, pages 46 and 153. See the Wikipedia article on conceptual blending.

(B) Rigor and rigor mortis

The following is quoted from a previous post here. See also the discussion in abstractmath.

When we are trying to understand or explain math, we may use various kinds of images and metaphors about the subject matter to construct a colorful and rich representation of the mathematical objects and processes involved. I described some of these briefly here. They can involve thinking of abstract things moving and changing and affecting each other.

When we set out to prove some math statement, we go into what I have called “rigorous mode”. We feel that we have to forget some of the color and excitement of the rich view. We must think of math objects as inert and static. They don’t move or change over time and they don’t interact with other objects or the real world. In other words, pretend that all math objects are dead.

We don’t always go all the way into this rigorous mode, but if we use an image or metaphor in a proof and someone challenges us about it, we may rewrite that part to get rid of the colorful representation and replace it by a calculation or line of reasoning that refers to the math objects as if they were inert and static – dead.

I didn’t contradict myself.
I want to clear up some tension between these two ideas.

The argument in (A) is a genuine mathematical proof, just as it is written. It contains hidden assumptions (enthymemes), but all math proofs contain hidden assumptions. My remarks in (B) do not mean that a proof is not a proof until everything goes dead, but that when challenged you have to abandon some of the colorful and kinetic reasoning to make sure you have it right. (This is a standard mathematical technique (note 1).)

One of the hidden assumptions in (A) is that two monks walking the opposite way on the path over the same interval of time will meet each other. This is based on our physical experience. If someone questions this we have several ways to get more rigorous. One many mathematicians might think of is to model the path as a curve in space and consider two different parametrizations by the unit interval that go in opposite directions. This model can then appeal to the intermediate value theorem to assert that there is a point where the two parametrizations give the same value.

I suppose that argument goes all the way to the dead. In the original argument the monk is moving. But the parametrized curve just sits there. The parametrizations are sets of ordered pairs in R x (R x R x R). Nothing is moving. All is dry bones. Ezekiel has not done his thing yet.

This technique works, I think, because it allows classical logic to be correct. It is not correct in everyday life when things are moving and changing and time is passing.

Avoid models; axiomatize directly
But it certainly is not necessary to rigorize this argument by using parametrizations involving the real numbers. You could instead look at the situation of the monk and make some axioms the events being described. For example, you could presumably make axioms on locations on the path that treat the locations as intervals rather than as points.

The idea is to make axioms that state properties that intervals have but doesn’t say they are intervals. For example that there is a relation “higher than” between locations that must be reflexive and transitive but not antisymmetric. I have not done this, but I would propose that you could do this without recreating the classical real numbers by the axioms. (You would presumably be creating the intuitionistic real numbers.)

Of course, we commonly fall into using the real numbers because methods of modeling using real numbers have been worked out in great detail. Why start from scratch?

About the heading on this section: There is a sense in which “axiomatizing directly” is a way of creating a model. Nevertheless there is a distinction between these two approaches, but I am to confused to say anything about this right now.

First order logic.
It is commonly held that if you rigorize a proof enough you could get it all the way down to a proof in first order logic. You could do this in the case of the proof in (A) but there is a genuine problem in doing this that people don’t pay enough attention to.

The point is you replace the path and the monks by mathematical models (a curve in space) and their actions by parametrizations. The resulting argument calls on well known theorems in real analysis and I have no doubt can be turned into a strict first order logic argument. But the resulting argument is no longer about the monk on the path.

The argument in (A) involves our understanding of a possibly real physical situation along with a metaphorical transference in time of the two walks (a transference that takes place in our brain using techniques (conceptual blending) the brain uses every minute of every day). Changing over to using a mathematical model might get something wrong. Even if the argument using parametrized curves doesn’t have any important flaws (and I don’t believe it does) it is still transferring the argument from one situation to another.

Conclusion:
Mathematical arguments are still mathematical arguments whether they refer to mathematical objects or not. A mathematical argument can be challenged and tested by uncovering hidden assumptions and making them explicit as well as by transferring the argument to a classical mathematical situation.

Note 1. Did you ever hear anyone talking about rigor requiring making images and metaphors dead? This is indeed a standard mathematical technique but it is almost always suppressed, or more likely unnoticed. But I am not claiming to be the first one to reveal it to the world. Some of the members of Bourbaki talked this way. (I have lost the reference to this.)

They certainly killed more metaphors than most mathematicians.

Note 2. This discussion about rigor and dead things is itself a metaphor, so it involves a metametaphor. Metaphors always have something misleading about them. Metametaphorical statements have the potential of being far worse. For example, the notion that mathematics contains some kind of absolute truth is the result of bad metametaphorical thinking.

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Pronouncing the Indefinite Article

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In his speeches, President Obama commonly pronounces the indefinite article “a” like the “a” in “may”.  You can hear it in this speech.  Most people, most of the time, pronounce it with the schwa.   In that speech, I have also caught him pronounce “to” like “too” before a vowel, but with a schwa before a consonant.

Hilary Clinton pronounces “a” that way in speeches, too.  Example.

They may not use this pronunciation mode in ordinary conversation.   Is this a generational change or do they have the same speech teacher?

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math, understanding math

Constraints on the Philosophy of Mathematics

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In a recent blog post I described a specific way in which neuroscience should constrain the philosophy of math. For example, many mathematicians who produce a new kind of mathematical object feel they have discovered something new, so they may believe that mathematical objects are created rather than eternally existing. But identifying something as newly created is presumably the result of a physical process in the brain. So the feeling that an object is new is only indirectly evidence that the object is new.  (Our pattern recognition devices work pretty well with respect to physical objects so that feeling is indeed indirect evidence.)

This constraint on philosophy is not based on any discovery that there really is a process in the brain devoted to recognizing new things. (Déjà vu is probably the result of the opposite process.) It’s just that neuroscience has uncovered very strong evidence that mental events like that are based on physical processes in the brain. Because of that work on other processes, if someone claims that recognizing newness is not based on a physical process in the brain, the burden of proof is on them.  In particular, they have to provide evidence that recognizing that a mathematical object is newly discovered says something about math other than what happened in your brain.

Of course, it will be worthwhile to investigate how the feeling of finding something new arises in the brain in connection with mathematical objects. Understanding the physical basis for how the brain does math has the potential of improving math education, although that may be years down the road.

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The Linguistic Sherlock Holmes

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Spysweeper is a program on our laptop that scans the disk, finds Bad Files and presents them for your inspection.   The list of Bad Files appears and under it is a button that says Quarantine Selected.  You are supposed to click that button to quarantine them.  When this happens I have been annoyed because until I clicked the button I hadn’t made the choice to quarantine them.

The other day I realized that the phrase on the button was meant to say “Quarantine the selected files”, not “You have selected quarantine”.  So I decided that the author of that program was a native speaker of a Slavic language.  Slavic languages use participles that way.  So do some Western European languages but a native speaker of one of them would probably write “Quarantine the selected”, still not idiomatic but not confusing.

Any time you want a linguistic puzzle detected, just ask me.

Of course, I have no evidence that I was correct…

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An early revolution in notation

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This is a reprint of an article by Jeffrey A. Tucker. Musical notation was invented by Guido six hundred years before Viète and Descartes invented modern algebraic notation. –cw

Guido the Great, by Jeffrey A. Tucker

This essay is adapted from Jeffrey’s new book, Sing Like a Catholic.

The people who make modern inventions are often celebrated for improving our lives. But what about those one thousand years ago who laid the very technological foundation of civilization as we know it? They too served the world, but with the primary purpose of contributing to the faith. I’m thinking here of those who solved the architectural problem to build the great cathedrals of the middle ages, and the scientists of the period who took the first steps toward modern medical knowledge.

Also we don’t often consider the innovations in art that make all music possible. There is one person who stands out here: the late 10th and early 11th century Benedictine monk named Guido d’Arezzo (991/992–after 1033). He is credited with fantastic musical innovations that led to the creation of the modern system of notes and staffs, and also the organization of scales that allowed for teaching and writing music.

His contributions have usually been seen as technical innovations and evaluated as such, though known only inside a small circle of music historians. Without his contribution, the music you hear on your iPod and on the radio would not likely exist.

A new book by Angelo Rusconi, synopsized by Patrick Reynolds from the Italian, and appearing in Goldberg #46 (2007), offers a more complete picture of what drove Guido, and the results will be very exciting for anyone who seeks to understand how any serious innovation upsets the status quo, makes enemies, causes a bit of social upheaval, and ultimately makes the world a better place.

Consider the technical feat that Guido undertook. Imagine a world without printed music. How would you go about conveying a tune in printed form? It’s one thing to render words on paper in a way that others can read them. But what about sound? It floats in the air and resists having a physical presence at all.

How can you share the melody without singing it for them, by just writing things down? People had tried since the ancient world without success. Some attempts in the 8th and 9th centuries came a bit closer (but the results look like chicken scratch to us). It was Guido who made the breakthrough with lines and scales that illustrate for the eye what the voice is to sing, and precisely so. His innovation was a beautiful integration of art and science.

And what a remarkable innovation, if you think about it. From the beginning of time until his time, the teaching of music was done by a tiny and ever-arrogant cartel of masters. You had to sing exactly as they instructed you. If they weren’t around, you were stuck. They held the monopoly. To become a master of music, you had to study under one of the greats, and then receive the blessing to become a teacher yourself, and you know that they wanted to limit their numbers. One can imagine that you had to be sycophantic to even get your foot in the door.

Guido’s innovation busted up the cartel. Rusconi shows that Guido’s primary interest was in notating not just music in general but Gregorian chant in particular. He was frustrated that the chant was passed on by oral tradition only. He worried that melodies would be lost, especially given the then new fashion for multi-part improvisation.

So while writers have usually treated Guido as an innovator, what’s been forgotten is that his innovations were driven by the desire to conserve and preserve for future generations. The desire to maintain the chant and pass it on was the key issue for him; the technical aspects of the music and writing were merely tools and not ends in themselves.

And there was an interesting sociological element here. He had become seriously annoyed at the cartel of chant masters and the power they exercised over the monastic community. He wanted the chant to be freed and put into the hands of everyone both inside and outside the monastery walls.

For this reason, his first great project was a notated Antiphoner, a book of melodies: “For, in such a ways, with the help of God I have determined to notate this antiphoner, so that hereafter through it, any intelligent and diligent person can learn a chant, and after he has learned well part of it through a teacher, he recognizes the rest unhesitatingly by himself without a teacher.”

He goes further. Without a written form of music “wretched singers and pupils of singers, even if they should sing every day for a hundred years, will never sing by themselves without a teacher one antiphon, not even a short one, wasting so much time in singing that they could have spent better learning thoroughly sacred and secularly writing.”

The elite musicians resisted his attempt to democratize the knowledge and conserve time. Guido did whatever great innovator does: he freed up resources for other uses even while improving lives.

But as a result of his innovation, his monastery in Pomposa, Italy, tossed him out into the snow. He then went to the Pope, who was very impressed, and gave him a letter of support. With the letter in hand, he went to the Bishop of Arezzo, who took him in so that he could continue his preaching and his work.

Now, one can’t but think of mistakes that have been made over the years with the Gregorian chant: the attempt to keep it the private preserve of musicologists; the dominance of singers by a single master who believes that he knows the one true way; the perception that chant is only for monasteries but not regular people; and on and on.

Here we see Guido embodying the same principle that drove the Solesmes monastery at the early part of the restoration efforts in the late 19th century: innovation in order to preserve, teach, and distribute this glorious music as widely as possible, in the service of the faith. They had the right ordering of priorities: technical innovation in the service of preserving universal truth.

This story illustrates a general principle in the history of technology. There does seem to be a real pattern here. There are those who believe that innovation is for everyone and ought to be accessible to all – that everyone should be permitted to have access to the forms and structures that make for progress. This side loves technical innovation not for its own sake but in the service of great goals.

Then there is the other side, which is reactionary, hates technical innovation, wants to reserve technical forms to a tiny elite, fears freedom, detests the idea of human choice, and advances a kind of Gnosticism over technical forms – always wants it to remain the private preserve of the elect who appoint each other and operate as a kind of guild. This Gnostic guild wants to guard and exclude and privatize, and the people are ultimately their enemy.

This perspective hearkens back to the ancient world where priests served the philosopher kings, and sparingly hand out religious truth to the masses based on what they believe they should know in the service of their agenda.

One can detect these two tendencies from all ages.

Guido didn’t patent his innovation. He didn’t copyright his music. The legal means weren’t available to him, and he wouldn’t have used them if they were. His whole point was to uplift the whole of the culture. Keeping his innovation to himself would have been contrary to that goal. As a result, his use of the staff spread widely. His innovation was infinitely reproducible, and it changed the world.

In his pedagogy, he adapted an existing song to illustrate the scale: Ut Queant Laxis, a hymn to St. John the Baptist, who was then considered the patron saint of singers. On the first syllable of each ascending note, the words were Ut, Re, Mi, Fa, Sol – the very foundation of music pedagogy to this day: do, re, mi, etc.

A millennium later, Guido’s innovation is still with us!

Here is a model for our Arial and all Arial.

March 4, 2009

Jeffrey Tucker [send him mail] is editorial vice president of www.Mises.org.

Copyright © 2009 by LewRockwell.com. Permission to reprint in whole or in part is gladly granted, provided full credit is given.

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Reed College

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Abouot twenty years ago I went to Reed College in Portland, Oregon to give a talk at a computer science meeting. As I walked onto campus, a gardener looked up and said “Howdy”. I thought that was a stereotype, but I guess they really do say that in the west. (Australians really say G’dye Myte, too.)

Then I saw a sign pasted on a wall:

VISUALIZE INDUSTRIAL COLLAPSE

Message to Reed College: All right, you’ve done that, now visualize World Peace, please.

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language

Space is real

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In the late eighties, I was at a church service on the Sunday when they talk about the budget (usually once a year in mainline Protestant churches). After the talk, ten members of the congregation marched up front each carrying a sign with one letter on it. They arranged themselves to spell

I realized then that the use of computers had changed the way we think about spaces. Now a space is something (a particular character) instead of nothing.

This meant Mathematica could use the space as a symbol for multiplication, and everyone under 50 understands it.

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