Category Archives: understanding math

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Variables

2009/05/15 Charles Wells 4 Comments

One of the themes of abstractmath.org is that we should pay attention to how we think about mathematical objects. This is not the same questions as “What are mathematical objects?”. This post addresses the question: How do we think about variables? What follows are extracts from newly rewrittens sections from Variables and Substitution and Mathematical Objects.

Role playing

If the author says “x is a real variable” then x plays the role of a real number in whatever expression it occurs in. It is like an actor in a play. If the producer says Dwayne will play Polonius you know that Dwayne will hide behind a curtain at a certain point in the play. When x occurs in the expression $x^3-1$ you know that if a number is substituted for x in the expression, the expression will then denote the result of cubing the number and subtracting 1 from it.

Slot or cell

The variable x is a slot into which you can put any real number. If you plug 3 into x in the expression $x^3-1$ you will get 26.

This is like a blank cell in a spreadsheet. If you define another cell with the formula “ $=x^3-1$ ” and put 3 in the cell representing x, the other cell will contain 26.

What’s wrong with this metaphor: In Excel, a blank cell is automatically set to 0. To be a better metaphor the cell shouldn’t have a value until it is given one, and the cell with the formula “ $= x^3-1$ ” should say “undefined!”. (I am not saying this would make Excel a better spreadsheet. Excel was not invented so that I could make a point about variables.)

Variable mathematical object

The two metaphors above refer to the name x. You can instead think of x as a variable mathematical object, meaning x is a genuine mathematical object, but with limitations about what you can say or think about it. This sort of thinking works for both the symbolic language and mathematical English, and it works for any kind of mathematical structure (“Let G be an Abelian group…”), not just numbers in a symbolic expression. There are two related points of view:

1. Some statements about the object are neither true nor false.

This means x is a genuine mathematical object and you can make assertions about it, but some of the assertions might have no truth value. From “Let x be a real number” you know these things:

The assertion “Either x > 0 or $x \leq 0$ ” is true.
The assertion “ $x^2 = -1$ ” is false.
The assertion “x > 0” is neither true nor false.

The assertion “x is a real number” is in a certain sense the most general true statement you can make about x. In other words, x is a mathematical object given by an incomplete specification, so you are limited in what you can say about it or in what conclusions you can draw about it.

If you say, “Let n be an integer divisible by 4, you cannot assume it is 8 or 12, for example. In other words, the statement “n is divisible by 4” is true, and “n = 3” is false, but the statement “n = 8” is neither true nor false, and you can’t derive any conclusions from n being 8.

2. The object is fixed but some things are not known about it.

If you say x is a real number, you know x is a real number (duh) and:

You know x is either positive or nonnegative.

You know $x^2$ is not equal to any negative number.

You don’t know whether x is positive or not.

This way of looking at it involves thinking of x as a particular real number. During the process of solving the equation $x^2-5x=-6$ you are thinking of x as a specific real number, but you don’t know which one.

These points of view (1) and (2) provide genuinely different metaphors for variables. In (1) I say certain statements are neither true nor false, but (2) suggests that all statements about the object are either true or false but you don’t know which. However, note when solving the equation
$x^2-5x=-6$ that, when you are finished, you still don’t know whether x = 2 or x = 3. This factcauses me cognitive dissonance, but the point of view that some statements are neither true nor false upsets other people. I prefer (1) over (2) but I have to admit that (1) is much less familiar to most mathematicians.

View (1) is advocated by category theorists because it allows you to think of a quantity holistically as a single thing rather than as a table of values. The height of a cannonball is different at different times but the “height” is nevertheless one continuous mathematical quantity. People who know more about history than I do believe that that is the simple and uncomplicated way nineteenth-century mathematicians thought about variable quantities.

We need good tools to do math. This means good images and metaphors as well as good tools for reasoning. Having simple and uncomplicated ways to think about math objects (along with guidelines for the way you think about them, such as dropping the law of the excluded middle in some cases!) is every bit as important as making sure our reasoning follows carefully thought out rules that lead from truth only to truth.

Note: Heyting valued logic actually provides sound but non-classical reasoning for thinking about variable objects, but most mathematicians with sound intuitions nevertheless use classical reasoning and come up with correct conclusions. Some of us are now in the practice of using non-classical logic to study differentials and other things, and that is a Good Thing, but it would be a complete misunderstanding if you read this post as advocating that mathematicians change over to that way of doing things. This post is about how we think about variability.

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Sets don't have to be homogeneous?

2009/04/24 Charles Wells 4 Comments

Colm Bhandal commented on my article on sets in abstractmath.org.

Let me first of all say that I am impressed with your website. It gave
me a few very good insights into set notation. Now, I’ll get straight
to the point. While reading your page, I came across a section
claiming that:

“Sets do not have to be homogeneous in any sense”

This confused me for a while, as I was of the opinion that all objects in a set were of the same type. After thinking about it for a while, I came to a conclusion:

A set defines a level of abstraction at which all objects are homogeneous, though they may not be so at other levels of abstraction.

Taking the example on your page, the set {PI^2, M, f, 42, -1/e^2} contains two irrational numbers, a matrix, a function, and a whole number. Thus, the elements are not homogeneous from one perspective (level of abstraction as I call it) in that they are spread across four known sets. However, in another sense they are homogeneous, in that they are all mathematical objects. Sure, this is a very high level of abstraction: A mathematical object could be a lot of things,
but it still allows every object in the set to be treated homogeneously i.e. as mathematical objects.

You are right. I think I had better say “The elements of a set do not have to be ‘all of the same kind’ in the sense of that phrase in everyday speech.” Of course, a mathematician would say the elements of a set S are “all of the same kind”, the “kind” being elements of S.

Apparently, according to the way our brains work, there are natural kinds and artificial kinds. There is something going on in my students’ minds that cause them to be bothered by sets like that given about or even sets such as {1,3,5,6,7,9,11} (see the Handbook, page 279). Philosophers talk about “natural kinds” but they seem to be referring to whether they exist in the world. What I am talking about is a construct in our brain that makes “cat” a natural kind and “blue-eyed OR calico cat” an artificial kind. Any teacher of abstract math knows that this construct exists and has to be overcome by talking about how sets can be arbitrary, functions can be arbitrary, and so on, and that’s OK.

This distinction seems to be built into our brains. A large part of abstractmath.org is devoted to pointing out the clashes between mathematical thinking and everyday thinking.

Disclaimer: When I say the distinction is “built into our brains” I am not claiming that it is or is not inborn; it may be a result of cultural conditioning. What seems most likely to me is that our brains are wired to think in terms of natural kinds, but culture may affect which kinds they learn. Congnitive theorists have studied this; they call them “natural categories” and the study is part of prototype theory. I seem to remember reading that they have some evidence that babies are born with the tendency to learn natural categories, but I don’t have a reference.

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How "math is logic" ruined math for a generation

2009/04/03 Charles Wells 3 Comments

Mark Meckes responded to my statement

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.

with this comment:

I may be misreading your analysis of the second straw man, but you seem to imply that “people thinking philosophy of math is merely a matter of logic and set theory” has done great damage to mathematics. I think that’s quite an overstatement. It means that in practice, mathematicians find philosophy of mathematics to be irrelevant and useless. Perhaps philosophers of mathematics could in principle have something to say that mathematicians would find helpful but in practice they don’t; however, we’re getting along quite well without their help.

On the other hand, maybe you only meant that people who think “philosophy of math is merely a matter of logic and set theory” are handicapped in their own ability to do mathematics. Again, I think most mathematicians get along fine just not thinking about philosophy.

Mark is right that at least this aspect of philosophy of math is irrelevant and useless to mathematicians. But my remark that the attitude that “philosophy of math is merely a matter of logic and set theory” is ruinous to math was sloppy, it was not what I should have said. I was thinking of a related phenomenon which was ruinous to math communication and teaching.

By the 1950’s many mathematicians adopted the attitude that all math is is theorem and proof. Images, metaphors and the like were regarded as misleading and resulting in incorrect proofs. (I am not going to get into how this attitude came about). Teachers and colloquium lecturers suppressed intuitive insights and motivations in their talks and just stated the theorem and went through the proof.

I believe both expository and research papers were affected by this as well, but I would not be able to defend that with citations.

I was a math student 1959 through 1965. My undergraduate calculus (and advanced calculus) teacher was a very good teacher but he was affected by this tendency. He knew he had to give us intuitive insights but he would say things like “close the door” and “don’t tell anyone I said this” before he did. His attitude seemed to be that that was not real math and was slightly shameful to talk about. Most of my other undergrad teachers simply did not give us insights.

In graduate school I had courses in Lie Algebra and Mathematical Logic from the same teacher. He was excellent at giving us theorem-proof lectures, much better than most teachers, but he never gave us any geometric insights into Lie Algebra (I never heard him say anything about differential equations!) or any idea of the significance of mathematical logic. We went through Killing’s classification theorem and Gödel’s incompleteness theorem in a very thorough way and I came out of his courses pleased with my understanding of the subject matter. But I had no idea what either one of them had to do with any other part of math.

I had another teacher for several courses in algebra and various levels of number theory. He was not much for insights, metaphors, etc, but he did do well in explaining how you come up with a proof. My teacher in point set topology was absolutely awful and turned me off the Moore Method forever. The Moore method seems to be based on: don’t give the student any insights whatever. I have to say that one of my fellow students thought the Moore method was the best thing since sliced bread and went on to get a degree from this teacher.

These dismal years in math teaching lasted through the seventies and perhaps into the eighties. Apparently now younger professors are much more into insights, images and metaphors and to some extent into pointing out connections with the rest of math and science. Since I have been retired since 1999 I don’t have much exposure to the newer generation and I am not sure how thoroughly things have changed.

One noticeable phenomenon was that category theorists (I got into category theory in the mid seventies) were very assiduous in lectures and to some extent in papers in giving motivation and insight. It may be that attitudes varied a lot between different disciplines.

This Dark Ages of math teaching was one of the motivations for abstractmath.org. My belief is that not only should we give the students insights, images and metaphors to think about objects, and so on, but that we should be upfront about it: Tell them what we are doing (don’t just mutter the word “intuitive”) and point out that these insights are necessary for understanding but are dangerous when used in proofs. Tell them these things with examples. In every class.

My other main motivation for abstractmath.org was the way math language causes difficulties. But that is another story.

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Mental Representations in Math

2009/04/03 Charles Wells 3 Comments

This post is part of the abstractmath article on images and metaphors. I have had some new insights into the subject of mental representations and have incorporated them in this rewritten version (which omits some examples). I would welcome comments.

Mathematicians who work with a particular kind of mathematical object have mental representations of that type of object that help them understand it. These mental representations come in various forms:

Visual images, for example of what a right triangle looks like.
Notation, for example visualizing the square root of 2 by the symbol “ $\sqrt{2}$ “. Of course, in a sense notation is also a physical representation of the number. An important fact: A mathematical object may be referred to by many different notations. There are examples here and here. (If you think deeply about the role notation plays in your head and on paper you can easily get a headache.)
Kinetic understanding, for example thinking of the value of a function as approaching a limit by imagining yourself moving along the graph of the function.
Metaphorical understanding, for example thinking of a function such as as a machine that turns one number into another: for example, when you put in 3 out comes 9. See also literalism and this post on Gyre&Gimble.

Example

Consider the function $h(t)=25-(t-5)^2$ . The chapter on images and metaphors for functions describes many ways to think about functions. A few of them are considered here.

Visual images You can picture this function in terms of its graph, which is a parabola. You can think of it more physically, as like the Gateway Arch. The graph visualization suggests that the function has a single maximum point that appears to occur at t = 5.

I personally use visual placement to remember relationships between abstract objects, as well. For example, if I think of three groups, two of which are isomorphic (for example $C_2$ and $\text{Alt}_3$ .), I picture them as in different places with a connection between the two isomorphic ones. I know of no research on this.

Notation You can think of the function as its formula . The formula tells you that its graph will be a parabola (if you know that quadratics give parabolas) and it tells you instantly without calculus that its maximum will be at (see ratchet effect).

Another formula for the same function is $-t^2+10t$ . The formula is only a representation of the function. It is not the same thing as the function. The functions h(t) and k(t) defined on the real numbers by $h(t)=25-(t-5)^2$ and $k(t)=-t^2+10t$ are the same function; in other words, h = k.

Kinetic The function h(t) could model the height over time of a physical object, perhaps a ball shot vertically upwards on a planet with no atmosphere. You could think of the ball starting at time t = 0 at elevation 0, reaching an elevation of (for example) 16 units at time t = 2, and landing at t = 10. You are imagining a physical event continuing over time, not just as a picture but as a feeling of going up and down (see mirror neuron). This feeling of the ball going up and down is attached in your brain to your understanding of the function h(t).

Although h(t) models the height of the ball, it is not the same thing as the height of the ball. A mathematical object may have a relationship in our mind to physical processes or situations but is distinct from them.

According to this report, kinetic understanding can also help with learning math that does not involve pictures. I know that when I think of evaluating the function at 3, I visualize 3 moving into the x slot and then the formula transforming itself into 10. I remember doing this even before I had ever heard of the Transformers.

Metaphor One metaphor for functions is that it is a machine that turns one number into another. For example, the function h(t) turns 0 into 0 (which is therefore a fixed point) and 5 into 25. It also turns 10 into 0, so once you have the output you can’t necessarily tell what the input was (the function is not injective).

More examples

¨ “Continuous functions don’t have gaps in the graph“. This is a visual image.
¨ You may think of the real numbers as lying along a straight line (the real line) that extends infinitely far in both directions. This is both visual and a metaphor (a real number “is” a place on the real line).
¨ You can think of the set containing 1, 3 and 5 and nothing else in terms of its list notation {1, 3, 5}. But remember that {5, 1,3} is the same set. In other words the list notation has irrelevant features – the order in which the elements are listed in this case.
The interior of a closed curve or a sphere is called that because it is like the interior in the everyday sense of a bucket or a house. Note The boundary of a real-life container such as a bucket has thickness, in contrast to the boundary of a closed curve or a sphere. This observation illustrates my description of a metaphor as identifying part of one situation with part of another. One aspect is emphasized; another aspect, where they may differ, is ignored.

Uses of mental representations

Integers and metaphors make up what is arguably the most important part of the mathematician’s understanding of the concept.

Mental representations of mathematical objects using metaphors and images are necessary for understanding and communicating about them (especially with types of objects that are new to us).
They are necessary for seeing how the theory can be applied.
They are useful for coming up with proofs.

Many representations

Different mental representations of the same kind of object help you understand different aspects of the object.

Every important mathematical object has many representations and skilled mathematicians generally have several of them in mind at once.

New concepts and old ones

We especially depend on metaphors and images to understand a math concept that is new to us. But if we work with it for awhile, finding lots of examples, and eventually proving theorems and providing counterexamples to conjectures, we begin to understand the concept in its own terms and the images and metaphors tend to fade away from our awareness…

Then, when someone asks us about this concept that we are now experts with, we trundle out our old images and metaphors – and are often surprised at how difficult and misleading our listener finds them!

Some mathematicians retreat from images and metaphors because of this and refuse to do more than state the definition and some theorems about the concept. They are wrong to do this. That behavior encourages the attitude of many people that

mathematicians can’t explain things
math concepts are incomprehensible or bizarre
you have to have a mathematical mind to understand math

All three of these statements are half-truths. There is no doubt that a lot of abstract math is hard to understand, but understanding is certainly made easier with the use of images and metaphors

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Stances

2009/04/03 Charles Wells 4 Comments

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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Proofs without dry bones

2009/03/24 Charles Wells 3 Comments

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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Constraints on the Philosophy of Mathematics

2009/03/18 Charles Wells Leave a comment

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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Typical examples

2009/02/11 Charles Wells 2 Comments

There is a sense in which a robin is a typical bird and a penguin is not a typical bird. (See Reference 1.) Mathematical objects are like this and not like this.

A mathematical definition is simply a list of properties. If an object has all the properties, it is an example of the definition. If it doesn’t, it is not an example. So in some basic sense no example is any more typical than any other. This is in the sense of rigorous thinking described in the abstractmath chapter on images and metaphors.

In fact mathematicians are often strongly opinionated about examples: Some are typical, some are trivial, some are monstrous, some are surprising. A dihedral group is more nearly a typical finite group than a cyclic group. The real numbers on addition is not typical; it has very special properties. The monster group is the largest finite simple group; hardly typical. Its smallest faithful representation as complex matrices involves matrices that are of size 196,883. The monster group deserves its name. Nevertheless, the monster group is no more or less a group than the trivial group.

People communicating abstract math need to keep these two ideas in mind. In the rigorous sense, any group is just as much a group as any other. But when you think of an example of a group, you need to find one that is likely to provide some information. This depends on the problem. For some problems you might want to think of dihedral groups. For others, large abelian p-groups. The trivial group and the monster group are not usually good examples. This way of thinking is not merely a matter of psychology, but it probably can’t be made rigorous either. It is part of the way a mathematician thinks, and this aspect of doing math needs to be taught explicitly.

Reference 1. Lakoff, G. (1986), Women, Fire, and Dangerous Things. The University of Chicago Press. He calls typical examples “prototypical”.

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Mathematical definitions

2009/02/11 Charles Wells 1 Comment

The definition of a concept in math has properties that are different from definitions in other subjects:

• Every correct statement about the concept follows logically from its definition.
• An example of the concept fits all the requirements of the definition (not just most of them).
• Every math object that fits all the requirements of the definition is an example of the concept.
• Mathematical definitions are crisp, not fuzzy.
• The definition gives a small amount of structural information and properties that are enough to determine the concept.
• Usually, much else is known about the concept besides what is in the definition.
• The info in the definition may not be the most important things to know about the concept.
• The same concept can have very different-looking definitions. It may be difficult to prove they give the same concept.
• Math texts use special wording to give definitions. Newcomers may not understand that the intent of an assertion is that it is a definition.

How many college math teachers ever explain these things?

I will expand on some of these concepts in future posts.

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math, representations, understanding math

Math and the Modules of the Mind

2009/01/30 Charles Wells 6 Comments

I have written (references below) about the way we seem to think about math objects using our mind’s mechanisms for thinking about physical objects. What I want to do in this post is to establish a vocabulary for talking about these ideas that is carefully enough defined that what I say presupposes as little as possible about how our mind behaves. (But it does presuppose some things.) This is roughly like Gregor Mendel’s formulation of the laws of inheritance, which gave precise descriptions of how characteristics were inherited while saying nothing at all about the mechanism.

I will use module as a name for the systems in the mind that perform various tasks.

Examples of modules

a) We have an “I’ve seen this before module” that I talked about here.

b) When we see a table, our mind has a module that recognizes it as a table, a module that notes that it is nearby, and in particular a module that notes that it is a physical object. The physical-object module is connected to many other modules, including for example expectations of what we would feel if we touched it, and in particular connections to our language-producing module that has us talk about it in a certain way (a table, the table, my table, and so on.)

c) We also have a module for abstract objects. Abstract objects are discussed in detail in the math objects chapter of abstractmath.org. A schedule is an abstract object, and so is the month of November. They are not mathematical objects because they affect people and change over time. (More about this here.) For example, the statement “it is now November” is true sometimes and false sometimes. Abstract objects are also not abstractions, like “beauty” and “love” which are not thought of as objects.

d) We talk about numbers in some ways like we talk about physical objects. We say “3 is a number”. We say “I am thinking of the only even prime”. But if we point and say, “Look, there is a 3”, we know that we have shifted ground and are talking about, not the number 3, but about a physical representation of the number 3. That’s because numbers trigger our abstract object module and our math object module, but not our physical object module. (Back and fill time: if you are not a mathematician, your mind may not have a math object module. People are not all the same.)

More about modules

My first choice for a name for these systems would have been object, as in object-oriented programming, but this discussion has too many things called objects already. Now let’s clear up some possible misconceptions:

e) I am talking about a module of the mind. My best guess would be that the mind is a function of the brain and its relationship with the world, but I am not presuppposing that. Whatever the mind is, it obviously has a system for recognizing that something is a physical object or a color or a thought or whatever. (Not all the modules are recognizers; some of them initiate actions or feelings.)

f) It seems likely that each module is a neuron together with its connections to other neurons, with some connections stronger than others (our concepts are fuzzy, not Boolean). But maybe a module is many neurons working together. Or maybe it is like a module in a computer program, that is instantiated anew each time it is called, so that a module does not have a fixed place in the brain. But it doesn’t matter. A module is whatever it is that carries out a particular function. Something has to carry out such functions.

Math objects

The modules in a mathematician’s mind that deal with math objects use some of the same machinery that the mind uses for physical objects.

g) You can do things to them. You can add two numbers. You can evaluate a function at an input. You can take the derivative of some functions.

h) You can discover properties of some kinds of math objects. (Every differentiable function is continuous.)

i) Names of some math objects are treated as proper nouns (such as “42”) and others as common nouns (such as “a prime”.)

I maintain that these phenomena are evidence that the systems in your mind for thinking about physical objects are sometimes useful for thinking about math objects.

Different ways of thinking about math objects.

j) You can construct a mathematical object that is new to you. You may feel that you invented it, that it didn’t exist before you created it. That’s your I just created this module acting. If you feel this way, you may think math is constantly evolving.

k) Many mathematicians feel that math objects are all already there. That’s a module that recognizes that math objects don't come into or go out of existence.

l) When you are trying to understand math objects you use all sorts of physical representations (graphs, diagrams) and mental representations (metaphors, images). You say things like, “This cubic curve goes up to positive infinity in the negative direction” and “This function vanishes at 2” and “Think of a Möbius strip as the unit square with two parallel sides identified in the reverse direction.”

m) When you are trying to prove something about math objects mathematicians generally think of math objects as eternal and inert (not affecting anything else). For example, you replace “the slope of the secant gets closer and closer to the slope of the tangent” by an epsilon-delta argument in which everything you talk about is treated as if it is unchanging and permanent. (See my discussion of the rigorous view.)

Consequences

When you have a feeling of déjà vu, it is because something has triggered your “I have seen this before” module (see (a)). It does not mean you have seen it before.

When you say “the number 3” is odd, that is a convenient way of talking about it (see (d) above), but it doesn’t mean that there is really only one number three.

If you say the function x^2 takes 3 to 9 it doesn’t have physical consequences like “Take me to the bank” might have. You are using your transport module but in a pretend way (you are using the pretend module!).

When you think you have constructed a new math object (see (j)), your mental modules leave you feeling that the object didn’t exist before. When you think you have discovered a new math object (see (k)), your modules leave you feeling that it did exist before. Neither of those feelings say anything about reality, and you can even have both feelings at the same time.

When you think about math objects as eternal and inert (see (m)) you are using your eternal and inert modules in a pretend way. This does not constitute an assertion that they are eternal and inert.

Is this philosophy?

My descriptions of how we think about math are testable claims about the behavior of our mind, expressed in terms of modules whose behavior I (partially) specify but whose nature I don’t specify. Just as Mendel’s Laws turned out to be explained by the real behavior of chromosomes under meiosis, the phenomena I describe may someday turn out to be explained by whatever instantiation the modules actually have – except for those phenomena that I have described wrongly, of course – that is what “testable” means!

So what I am doing is science, not philosophy, right?

Now my metaphor-producing module presents the familiar picture of philosophy and science as being adjacent countries, with science intermittently taking over pieces of philosophy’s territory…

Links to my other articles in this thread

Math objects in abstractmath.org
Mathematical objects are “out there”?
Neurons and math
A scientific view of mathematics (has many references to what other people have said about math objects)
Constructivism and Platonism

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Role playing

Slot or cell

Variable mathematical object

1. Some statements about the object are neither true nor false.

2. The object is fixed but some things are not known about it.

Example

More examples

Uses of mental representations

Many representations

New concepts and old ones

math, language and other things that may show up in the wabe