This is a good day to read John Armstrong’s post on Lincoln and proof.
Send to KindleLincoln and proof
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Send to KindleHolding on to the old ways
Many middle class Americans over seventy grew up hearing and singing to a pipe organ on Sundays. Far, far more young people have heard electronic keyboards than have heard pipe organs. I have an acquaintance who trained as a pipe organist who now plays for two churches on Sunday. In at least one of them he plays a keyboard (I don’t know about the other). He hates it when people tell him how wonderful his keyboard playing is. This is not the real thing, and you have never heard a real organ and you have no conception of how tacky a keyboard sounds compared to the real thing! (He also would not like it that I wrote “pipe organ”. “Organ” ought to mean what it always meant!) (See Note). He is witnessing a change in the world and he doesn’t like it.
I once attended a discussion in an Episcopal church that featured a progressive churchman and a traditional churchman in a discussion. Being one of the usual Anglican it’s-all-wonderful but-not necessarily-literally-true people, I was really wanting to hear what the traditionalist would say. Well, his argument was just like my friend the pipe (sorry) organist. For one thing, he made a vehement defense of King James English. In particular that the distinction between “thou” and “you” was important: “Thou” was the form you used with family and close friends and — God! That meant something and modern translations lose the distinction.
Well, I was disappointed. All his arguments were railing against the world changing, which means they weren’t arguments at all but merely venting at the world going to hell.
I’ll conjecture that not one American in 500 knows about the familiar-formal distinction between thou and you in 16th century English, and those who are familiar with “thou” in religious usage think it is something highly formal because after all it is used for God! (See note 2.) This completely reverses the point of the traditionalist who was saying that the use of “thou” means we are addressing God like someone close, like a friend or a member of the family. Must have been as irritating as hearing someone praise your playing an electronic keyboard when they have never heard an organ.
Now, I love language and I love King James English, not least because I grew up with it. But it is dead. I love singing in Latin, too. And anyway, if you want something like King James English that is a living language, try German. It has lots of those old distinctions and more that King James never heard of. KJ me no KJ, try Luther’s translation if you want old style language!
Note 1: Years ago my father got upset when I referred to a car with a bench seat. When he found out what I meant he was annoyed. That’s what you call a seat. The rise of bucket seats had left him behind. I knew someone else who dislikes “acoustic guitar” for similar reasons. And wait till the greenies get told that sea salt is inorganic.
Note 2. There are churches here and there with names like “Thee Greater Blessing Church of God.” (An example.) That’s taking confusion to a new level.
Send to KindleTypical examples
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”.
Send to KindleMathematical definitions
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.
Send to KindleMath and the Modules of the Mind
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
Send to KindleBluetooth
A friend of mine said she didn’t know what Bluetooth is. This is a brief explanation. Bluetooth has aspects that might interest liberal-artsy people.
Bluetooth is a method for transmitting data over short distances, so you don’t need cables. It is transmitted over a frequency that changes rapidly within a narrow band so it doesn’t interfere with other devices using the same method. This method is called frequency hopping.
One method to implement frequency hopping was patented in 1942 by Hedy Lamarr (the actress) and George Antheil (the composer), who wanted to use it to send radio guidance to torpedoes that could not be interfered with. The idea was to use a piano roll to change the frequency rapidly. As you might expect, their method used 88 different frequencies. They never benefited from their patent because clocks in those days weren’t accurate enough.
Hedy Lamarr lived in prewar Germany and was mathematically talented. She married an arms manufacturer who took her to meetings where she learned about military technology. She was disgusted with his fascist tendencies (they were both Jewish) and one day in 1937 arranged to go to a party wearing all her expensive jewelry. She and her maid drugged the husband and she escaped to England with the jewelry. She met Louis B. Mayer and wound up starring in a number of films.
George Antheil grew up in Trenton, New Jersey, and acquired the founder of the Curtis Institute of Music as a patron. His most famous work is Ballet Mécanique, which featured (among other things) several player pianos and three airplane propellers with leather strips flapping in them. At an early concert they blew off toupees and hats from members of the audience. Later concerts had them pointing at the ceiling. I heard a recording of it many years ago and was fascinated by it. In spite of what the description suggests, it is real music.
Bluetooth is named after Haraldr Blátönn, who was King of Denmark in the tent
h century. Runestones he erected still exist, in Jelling in Denmark. They have been standing outside for a thousand years but soon they will be moved indoors. If your computer or music player has Bluetooth, you will see its logo on the instrument. The logo is a “bind-rune” composed of the runes for H and B.
Send to KindleAbstractmath.org after four years
I have been working on the abstractmath website for about four years now (with time off for three major operations). Much has been written, but there are still lots of stubs that need to be filled in. Also much of it needs editing for stylistic uniformity, and for filling in details and providing more examples in some hastily written sections that read like outlines. Not to mention correcting errors, which seem to multiply when I am not looking. The website consists of four main parts and some ancillary chapters. I will go into more detail about some of the parts in later articles.
The languages of math. This is a description of mathematical English and the symbolic language of math (which are two different languages!) with an emphasis on the problems they cause people new to abstract math (roughly, math after calculus). At this point, I have completed a fairly thorough edit of the whole chapter that makes it almost presentable. Start with the Introduction.
Proofs. Mathematical proofs are a central problem for abstract math newbies. People interested in abstract math must learn to read and understand proofs. A proof is narrated in mathematical English. A proof has a logical structure. The reader must extract the logical structure from the narrative form. The chapter on proofs gives examples of proofs and discusses the logical structure and its relationship with the narration. The introduction to the chapter on proofs tells more about it.
Understanding math. There are certain barriers to understanding math that are difficult to get over. Mathematicians, math educators and philosophers work on various aspects of these problems and this chapter draws on their work and my own observations as a mathematician and a teacher.
All true statements about a math object must follow from the definition. That sounds clear enough. But in fact there are subtleties about definitions teachers may not tell students about because they are not aware of them themselves. For example, a definition can really mislead you about how to think about a math object.
The section on math objects breaks new ground (in my opinion) about how to think about them. I also discuss representations and models and images and metaphors (which I think is especially important), and in shorter articles about other topics such as abstraction and pattern recognition.
Doing math. This chapter points out useful behaviors and dysfunctional behaviors in doing math, with concrete examples. Beginners need to be told that when proving an elementary theorem they need to rewrite what is to be proved according to the definitions. Were you ever told that? (If you went to a Jesuit high school, you probably were.) Beginners need to be told that they should not try the same computational trick over and over even though it doesn’t work. That they need to look at examples. That they need to zoom in and out, looking at a detail and then the big picture. We need someone to make movies illustrating these things.
These other articles are outside the main organization:
Topic articles. Sets, real numbers, functions, and so on. In each case I talk just a bit about the topic to get the newbie over the initial hump.
Diagnostic examples. Examples chosen to evoke a misunderstanding, with a link to where it is explained. This needs to be greatly expanded.
Attitudes. This explains my point of view in doing abstractmath.org. I expect to rewrite it.
Send to KindleTwin Stories I: Political Correctness
1. Cannon to the left of us. A few years ago, you could be rebuked in Glasgow if you asked for black coffee. You had to ask for “coffee without cream”. This reportedly happened in Glasgow according to the Global Language Monitor.
Apparently the Enforcers were sometimes a bit confused. The report said that when one person in Glasgow asked for coffee without milk, the server said: “We don’t have any milk, you’ll have to have coffee without cream.”
I remember reading about an edict from the London County Council saying the same thing, but haven’t found a reference to it on the web. As I remember it, Londoners made a big joke out of it. When they would go outside on a moonless night, they would say, “It’s a night without cream.”
By the way, there is a problem with tea. “Black tea” refers to the color of the leaves, not to any milk withheld. (*Sigh*) The English language has as many traps for us as, well, math.
2. Cannon to the right of us. The American Life League has objected to the following press release from Krispy Kreme Doughnuts, Inc:
Krispy Kreme Doughnuts, Inc. (NYSE: KKD) is honoring American’s sense of pride and freedom of choice on Inauguration Day, by offering a free doughnut of choice to every customer on this historic day, Jan. 20. By doing so, participating Krispy Kreme stores nationwide are making an oath to tasty goodies — just another reminder of how oh-so-sweet “free” can be.
The League says: “Celebrating his inauguration with “Freedom of Choice” doughnuts – only two days before the anniversary of the Supreme Court decision to decriminalize abortion – is not only extremely tacky, it’s disrespectful and insensitive and makes a mockery of a national tragedy.”
Mark Liberman of The Language Log has something to say about this. Wait till he hears that Krispy Kreme backed down and stopped using the word “choice”.
Send to KindleConstructivism and Platonism — a Third Way
This is my response to Phil Wilson's article on Constructivism in a recent Plus Magazine, in particular to the following paragraph (but please read the whole article!) where he talks about
'…how firmly entrenched is the realist view that mathematical objects exist independently of the human mind, "out there" somewhere just waiting to be discovered, even in our every day conception of objects as fundamental as the real numbers. Intuitionism is radically antirealist: antirealist in that it claims mathematical objects only come into existence once they are constructed by a human mind (a sad quirk of language that this is called "anti"realist), and radical since it seeks to recast all of mathematics in this light.'
We don't have to choose between the view that mathematical objects exist independently of the human mind and the view that they only come into existence once they are constructed by a human mind. There is a third approach: We think about mathematical objects as if they exist independently of the human mind. In particular, mathematicians have gotten away with pretending that all the digits of a real number exist all at once and proving theorems such as trichotomy based on that view, without running into contradictions. The justification is just that: it works.
This approach has the advantage that our brain has a whole system of thinking about physical objects. We use this system to think about other things such as Sherlock Holmes and pi and appointment schedules and it works quite well. It doesn't work perfectly: for example, physical objects change over time and affect each other, whereas we must think of mathematical objects as eternal and inert if our proof techniques are going to work properly. Indeed, it is thinking of the decimal digits of pi (for example) as "going toward infinity" that gets students into trouble with limits.
Even so, objectification, if that is the right word, has worked very well for mathematicians and we don't need to give it up, nor do we need to be Platonists — we need only act as if we are Platonists.
I wrote about this in several places:
A scientific view of mathematics
Send to KindleParameters
I have been putting off writing the section on parameters in abstractmath.org because I don’t very well understand the connections between the different ways the word is used.
A parametrized family of functions, say f_a(x) = x^2+a, is easily handled by using the adjunction that makes the category of sets cartesian closed. That is, the three ways of looking at it, function of x parametrized by a, a function of a parametrized by x, and a function of two variables, are all naturally equivalent in a completely satisfactory way.
When you talk about taking a curve defined by an equation, e.g. x^2 + y^2 =1, and parametrizing it, that is a different situation; I have never thought about it from a categorical point of view but it is not hard to explain at the level of abstractmath.org.
What I don’t understand is if there is any real connection between these two meanings. I would welcome comments!
There are other meanings of “parameter”. One is the usage, “An important parameter of a finite group is its order.” I am not sure how often this occurs, and I look at the usage with some disapproval. The order of a finite group is more like an invariant (e.g. under isomorphism) than a parameter. Now do I have to write about invariants, too? Ye gods, do I have to do everything myself?
Another usage is in programming languages, where it can be analyzed simply as a variable. But they have different kinds of parameters that are implemented differently.
What I really need to do is spend a day searching for the word in JStor. I swore when I started doing abstractmath.org I wasn’t going to do any more lexicographical research (after the Handbook.). But maybe looking around for a few hours won’t hurt.
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