I have just posted a new Astounding Math Story. I have also rewritten the other two stories.
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Send to KindleNaming real numbers
I am considering an Astounding Math story about how you can’t name an arbitrary real number. In this blog I will describe some of the math technicalities, teaching problems and writing problems that arise in writing the story. It would be great if other math popularizers would blog about the problems they faced in their writing and what decisions they made about them.
Consider first that we can name every rational number. You can describe rational numbers in terms of equivalence classes of expressions of the form “m/n” where m and n are names of integers, n not zero. Then any expression of the form “m/n” names a specific rational number and every rational number can be named in this way. Yes, the naming can be made unique by using expressions in lowest terms, but that is not the point here, which is that in theory you really can name every rational number.
Every real number has a decimal expansion. The expansion is nearly unique: some rational numbers have two different decimal expansions, but that is all (see (2) below). We may define a decimal expansion precisely using the “regular” expression [-]?[0..9]*[.][0..9]^(infinity). This means:
One minus sign or nothing
followed by
A string of any finite length of decimal digits
followed by
A decimal point
followed by
An infinite string of decimal digits.
This is not really a regular expression since regex’s don’t allow you to specify an infinite string. But it is a precise definition of decimal expansion, and every decimal expansion refers to a specific real number. You can then define the real numbers as equivalence classes of regular expressions of decimal digits, with each equivalence class containing one or two members.
The Astounding thing is that as a result of this construction
a) You can describe precisely the set of real numbers.
b) Each real number has a description as an infinitely long decimal expansion.
c) You cannot give a name to every real number. That’s because the description is an infinite sequence and you cannot give every infinite sequence even in theory (see (3).)
d) So when mathematicians deal with real numbers, they are dealing with things that in most cases they cannot refer to.
Complications
My purpose in writing Astounding Math Stories is to get people who are already somewhat familiar with math to have their consciousness raised about all the fascinating things that go on in math. This requires a delicate balancing act when I write them.
1) I get comments from readers like this one: “That is not astounding. I already knew about it.” This is probably inevitable. In the case of the names of real numbers, however, I’ll bet there are practicing mathematicians who understand item (d) implicitly but have never heard it said out loud.
2) You have to say precisely which two infinite sequences are in the same class. When I was teaching discrete math in the eighties and nineties, I realized that I had never seen this written out explicitly. Every description depended heavily on pattern recognition, as in the description “I am referring to the phenomenon that for example 0.9999… denotes the same number as 1.0000…” (See remark (6).) I included a nearly explicit description in my discrete math class notes (page 12).
Perhaps this problem should be slurred over. Really every real number has one decimal expansion. That thing with the 9’s is just a technicality. (This makes me a heretic. Mathematicians don’t usually say things like that.)
3) You cannot give the name of every real number because the set of linguistic expressions is countable and the set of infinitely long decimal expansions is uncountable. Do I just quote this fact? Do I write another Astounding Math Story about infinite cardinality? Probably.
Some would object that you can’t give the name of every rational number either. But there is a name (a finitely long linguistic expression) for every rational number. You can’t in physical fact “give” the humongous ones but that is a practical problem. In contrast, most real numbers have no linguistic expression naming them.
4) I need to keep the demands on the reader as low as is reasonable, but not lower. A minor example in this case is that I express everything in terms of decimal expansion instead of binary expansion or Cauchy sequences (more abstract) or Dedekind cuts (even more abstract). In theory, binary expansions are not any more abstract than decimal expansions and require less data, but in fact most of the people I am trying to attract are less familiar with binary than decimal, and that drags on their understanding.
5) When I say “You can describe a real number as an infinitely long decimal expansion” you run into the ubiquitous difficulties math-newbies have with infinite sequences. Namely, they think of then as progressing through time, so you never get to “the end”. In fact, experienced mathematicians think of an infinite sequence as existing all at once: every entry is there now.
Students complain that they can’t “visualize” the entries all at once, but that is not the point: You are not suppose to visualize the whole sequence at once, you are suppose to think and talk about the entries as if they are all there. That is, assent to the concept that the whole thing is “there”, which is not the same thing as visualizing it. (I also wrote about this phenomenon in abstractmath and in a previous blog.)
So when I write about the infinitely long decimal expansions I know many readers’ understanding will falter right there; they will not be able to take in the rest of what I write. What do I do about this? Well, I suppose I could include the last paragraph!
Note: This discussion is not about what infinite sequences “really are”, but about how you think about them. This way of thinking about them has been around for a couple of centuries and have produced many useful theorems and no known contradictions. Philosophers may have a problem with this point of view, but mathematicians don’t.
6) Studies show that most math students do not believe that 0.999… is the same number as 1.000… Some mathematicians I know say things like: “Why are you writing for people like that? They are too stupid to understand anything about abstract math.” But it is time mathematicians stopped insisting that there is no point in getting people who are not especially talented in math interested in math, or trying to explain anything to them. In fact, it appears to me that this elitist attitude is in the process of dying out. It had better be dying out.
Send to KindleAutomatic spelling using different systems
In a previous post, I discussed how we should read in the near future, with an electronic Reader that allowed searching and access to all texts. One particular service by the Reader that it is reasonable to expect is to provide the text in the spelling system you want to use. Of course, each spelling system would have to be implemented by someone or some organization.
This is already known to be possible, since there is software that will read printed text to you. Software that vocalizes has to identify how each word is pronounced. If that can be done it should be reasonable to hope to have software that implements some alternative spelling system.
As I mentioned in my post on automatic spelling reform, this vocalizing software is pretty good: For example, the English reader in Excel reads “I have read [pronounced “red”] the book” and “I will read [“reed”] the book” correctly. The software can never be perfect (people can’t read every text completely correctly), but it is already good enough.
Imagine you could push a button and get:
a. The text, in any language, printed in the International Phonetic Alphabet.
b. Russian text printed with accents added.
c. Latin text with long marks added.
d. French text printed with dots under silent letters.
e. English text printed in American spelling, or in British spelling.
f. English text printed using your favorite spelling reform.
g. Mandarin in either standard characters or simplified characters.
h. Mandarin in some phonetic transcription that shows tones. (This is likely to be much harder to carry out than the previous tasks. Maybe it can’t be done satisfactorily.)
One type of reform for English would be text printed with the standard spelling but with diacritic marks added to indicate the pronunciation. This would keep the standard spelling in front of someone learning English but would clue her in on how to pronounce it as well. Here is an example I cooked up:
This is just a demonstration of what ought to be possible. Some features of this particular system:
a. Dot under a letter indicates it is silent.
b. Dot over a vowel indicates it is schwa.
c. Long and short vowels indicated by usual American dictionary symbols.
d. Other common vowel pronunciations indicated by diacritics.
e. Rare pronunciations, as in “to” and “of”, indicated by a small letter written above.
f. Each consonant has zero or one pronunciation that is unmarked.
g. All pronunciations are marked for vowels.
None of these choices may be ideal. I made choices that I hope require as little memorization as possible. For example, the reader need not memorize the rules for the two pronunciations of the past ending –ed or the plural ending –s. The example above does not indicate stress, but that should be done too.
Note that this system is not meant to be written or typed by anyone. It will be created by a computer program running on text in ordinary English spelling.
To provide all this would provide lots of coding and each specific ability might have to be purchased separately, although it seems to me that it would be in the interest of a country trying to promote its language to provide appropriate software free.
Send to KindleReading in the near future
I find reading, especially reading technical exposition, is frustrating because there are so many ways it could be made better in this age of computers. This post will suggest some of the improvements that could be made. I hope dozens of people are already working hard on these ideas. If not, then get busy! These ideas are based on the following assumptions, which I think could reasonably be realized in the next few years. Indeed, most of them already exist in some form. Amazon’s Kindle has some of these properties. The idea is to be able to read text in the same comfortable way we read a book, but with the ability to do other things, some of them problematical, that I will talk about here. Links and searching Searching and links are widely used now and I think they are the best thing since sliced bread. One of the sites that makes the best use of links is Wikipedia, and some of the news blogs do a pretty good job too. In these Modern Times it is annoying to have to read a dead old pack of paper (dopop, pronounce “dope-op”) instead of an interactive book that better fits the habits of us grasshoppers. Technical books are the most annoying, but even novels sometime make a reference to a character not mentioned for awhile or to some obscure rock group, and it would be great to stop right there and find out more about them. (By the way, in Ancient Times – 1980 – mystery novelists had a Rule that no two main characters in a novel would have names starting with the same letter. This is no longer true. Why?) It would be straightforward to provide the ability to search the book you are reading, as well as to mark some phrase and search the internet for it. On the other hand, links in general have to be supplied by the author or at least by someone familiar with the ideas in the text. That is labor intensive. A Wikipedia-like solution to this problem would be to allow other people to add links either freely or under editorial control. (“Read Hamlet, with links provided by the Helsingborg University History Department” – or “with links provided by Linkopedia”.) Wikipedia itself seems to have people who spend some time adding links to its own articles, and I think in some cases they even have robots to do it. For example, most years in recorded history have an article devoted to it, and a mention of a year in some other article seems to quickly acquire a link to the article about that year. Every piece of text should be obtainable and most of them should be free One of my practices in writing abstractmath.org is to provide links that are mostly to electronic texts. (There are a couple of lists of dopops here and there in it.) That is because most younger people and many older ones have the following characteristics: I have been frustrated many times by clicking on a link to a math or linguistics article only to discover that I have to Pay Money to read it. This happens even though Case Western Reserve University (where I am professor emeritus) makes many journals free for downloading. I have notices that this situation is worse in math education and in philosophy than it is in math; indeed many researchers in those fields don’t post their papers on their own websites (many others do). This is not the way Modern Scientific Research should be run! This problem does not have an easy solution and I am not clever enough to propose one. I am aware that some universities as well as Google are working on this, but it will be a long time coming. People who do research or expository writing should be paid for what they do, but I don’t know how to arrange it so that some poor young person named Ramanujan can look up the latest research on Eisenstein series and pay only an anna for it. I have more to say, but it must wait until a later post.
Send to KindleRenaming technical concepts
Here are some thoughts about the names of mathematical objects. I don’t make recommendations about how to name things; I am just analyzing some aspects of how names are given and used. I have written about some of these ideas in abstractmath.org, under Names and Semantic Contamination.
Some objects have names from Latin or Greek, such as “matrix” or “homomorphism”, that don’t give the reader a clue as to what they mean, unless the reader has a substantial vocabulary of Latin and Greek roots.
Some are named after people, such as “Riemann sum” and “Hausdorff space”. They don’t have suggestive names either. Well, they suggest that the person they are named after discovered it, but that is not always true; for example L’Hôpital’s rule was discovered by some Bernoulli or other.
You could call both types of names learnèd names.
Others concepts have names that are English words, such as “slope” or “group”. I will call them commonword names. Some of these suggest some aspect of their meaning; “slope” certainly does and so do “truth set” and “variable”. But “group” only suggests that it is a bunch of things; it does not suggest the primary group datum, namely the binary operation. Not only that, but too many commonword names suggest the wrong ideas, for example “real” and “imaginary”.
In contrast, learnèd names don’t usually suggest the wrong things, but they can and do intimidate people.
One upon a time, Roger Godement and Peter J. Huber came up with an important construction for adjunctions in category theory. They called it the standard construction. That commonword name communicates very little. They named it that because it kept coming up in their work. Well, derivatives and integrals are each more deserving of the name. Eilenberg and Moore renamed them triples, which suggests nothing useful except that the concept is given by three data. Well, so are rings. Saunders Mac Lane renamed them again, calling them monads, a learnèd name that suggests nothing except possibly an illusory connection with a certain philosophical concept.
Perhaps learnèd names are better, since they don’t suggest the wrong things. In that case “monad” is better than the other names, but I have a personal prejudice since I have co-authored two books that called them “triples”.
Some writers of popularizations of math and science avoid using the names of certain concepts that suggest the wrong things. In Symmetry and the Monster, by Mark Ronan, the author talks about “atoms of symmetry” instead of “simple groups”, on the grounds that “simple group” is misleading (the Monster Group is simple!) and doesn’t suggest the important property they have. He called involutions “mirror symmetries”, which is appopriately suggestive. Centralizers of involutions became “cross-sections”, which I don’t understand; it must be based on a way of thinking about them that I am not aware of. He doesn’t change the name of the Monster Group, though; that is a terrific name.
Frank Wilczek, in The Lightness of Being, used “core theory” for the theory in particle physics that is commonly called the “standard model”. I suppose that really is more suggestive of its current place in physics, since as far as I know all modern theories build on it.
Marcus du Sautoy, in The Music of the Primes (HarperCollins, 2003), also introduces new names for concepts. His description of the meanings of the many concepts he discusses uses some great metaphors that clearly communicate the ideas. He talks about the “landscape” of the zeta function, how Riemann “extended the landscape to the west”, and refers to its zeroes as its places “at sea level”. But he also calls them by their normal mathematical name “zeroes”. (I could have done without his reference to the “ley line of zeroes”.) He refers to modular arithmetic as “clock calculators” and in one parenthetical remark explains that modular arithmetic is what he means.
Summary
The problem with learnèd names is that they don’t give you a clue about the meaning, and for some students (co-intimidators) they induce anxiety.
The problems with commonword naming are that what a commonword name suggests can give you only one connotation and it is hard to find the best one, and almost any choice produces a metaphor that suggests some incorrect ideas. Furthermore, beginning abstract math students are way too likely to be stuck on one metaphor per mathematical object and commonword names only encourage this behavior. I have written about that here and here.
One problem with popular renaming is that the interested reader has a hard time searching the internet for more information about it, unless she noticed that one place in the book where the fact that it was not the standard name was mentioned.
Send to KindleTwo science books in the modern expository style
Symmetry and the Monster: The Story of One of the Greatest Quests of Mathematics, By Mark Ronan. Oxford University Press, 2006.
This is an excellent way for the non-mathematician to learn about what is going on in the attempt to classify symmetries by discovering all the finite “simple groups”. The last one found was the Monster Group and the classification was completed in 1982. This book is full of fascinating information about how this came about and the tantalizing connections between physics and the Monster that have been discovered since.
The Lightness of Being, by Frank Wilczek. Basic Books, 2008
This book is an exposition for the layman of the modern theory of particle physics – the Standard Model, Supersymmetry and other possible extensions. I recommend it for anyone interested in the subject.
These two books are examples of the modern trend in science expository writing, using metaphors, anecdotes, graphs and speculation to try to communicate an understanding of how the scientists involved think about the subject and what their motivations are. Ronan and Wilczek use much the same approach that I have been using in abstractmath.org and it has made me think about what works and what doesn’t.
I will be writing about my reactions to the writing in such books in future posts.
Send to KindleEnglish pronunciation heads for catastrophe
The Yiddish word for “beautiful” sounds like shay-na (that is the feminine form). The first syllable rhymes with “say”. It was once natural, if you saw the spelling “shana”, to pronounce it “shay-na” and so people gave their daughters the name “Shana”. Younger, educated people now expect that a word that looks foreign should be pronounced (more or less) with European vowels (essentially those of Spanish), so that they would say “shah-na”. The first syllable rhymes with “pa”. So some people started naming their daughters “Shaina” to avoid this, and others named their daughters “Shana” and pronounced it “shah-na”.
In many European languages (not French) “Shaina” would be pronounce “shy-na”. So how do you spell the name so that it is pronounced “shay-na” by every native English speaker? Probably Shayna is the best choice, and there are people named Shayna. There are also people named Shawna because their parents heard the name “Shana” pronounced that way and thought it was spelled “Shawna”. (You can check these claims on Google.)
Once upon a time almost every American would say “shay-na” when seeing the spelling “Shana”. For Bernstein they would say “burn-steen” not “bairn-shtine” (the conductor said “burn-steen”). Churchill said “nazzi” for Nazi instead of “nah-tsee”. People rhymed “Lagrange” with “range” and accented “Berlin” and “Madrid” on the first syllable.
People without college education still do these things. But the educated classes have come to believe that they must pronounced foreign words as if they were speaking that language. They are inconsistent about it, they still say “Paris” to rhyme with “Harris”, but they pronounce less common foreign words in the way they think they are pronounced in the original language.
This comes out of snobbery (I am educated and you are not) but also out of genuine enjoyment at saying things in foreign languages. It is not all snobbery.
But whatever the motive, it is causing a disaster. English has always had irregular spelling, but in fact people could make a stab at saying a word they say in print and often get it right. The situation is much worse now. We have gotten to the point where we simply have to know how lots of words are pronounced. The spelling gives no clue.
This is a catastrophe for English spelling.
Send to KindleUnconstructed existence
Astounding Math Stories is intended to attract the interest of bright, technically oriented non-mathematicians or beginning math students by telling them about some of the phenomena in math that are more surprising than, say, Venn Diagrams. This is turning out to be hard to do. You get responses from sophisticated people who say, in effect, “Well I knew that!” or “That’s not surprising!”. But sometimes they uncover real difficulties. Example:
Last August, Pubkeybreaker posted this message on sci.math about Finding Factors or Not:
I’m not sure why being able to show that an integer is composite without finding a factor is “astounding”.
Even at a beginning, first year algebra level we know that linear equations (with real coefficients) have a real solution, without knowing what the solution is. Similarly for any odd degree polynomial. We know that positive real numbers have a positive square root without knowing its value…. etc. etc.
Why should it be astounding that we know something
exists yet not know its value?
This brings up a sticky point. I was trying to find a simpler example of the sort of nonconstructive proof of existence that Hilbert used for the finite basis theorem. In that proof there is an inductive construction where at each step you find a polynomial of minimal degree in a certain ideal. But the ideal is infinite, so how do you know when you have found a minimal degree member? That is why it is nonconstructive.
Now Fermat’s Little Theorem gives you a way (if it works — it doesn’t always) of finding a proper factor of a composite number without showing how to calculate it. But finding a proper factor of a composite number has an obvious algorithm: try every number up to its square root. This is a finite but slow process; it is not nonconstructive.
What FLT does is give you a fast way of showing the existence of the proper factor without give you a fast way to find it. (See the appendix below.) That is simply not the same situation as in the finite basis theorem.
Clearly I need to rewrite the FLT story in such a way as to emphasize speed-of-algorithm difference instead of the nonconstructive part. Then I need to write another Astounding Math Story that describes a truly nonconstructive proof directly. The Hilbert Basis Theorem itself might be a candidate; the ideas involved are not too hard, although they are more advanced than I was hoping for for Astounding Math Stories.
Appendix
Here are some more detailed comments on Pubkeybreaker’s specific examples:
The proof that odd degree real polynomials have a real root is that such a function goes to positive, resp. negative infinity in the positive, resp. negative direction, so must cross the x axis. This immediately suggests an algorithm for finding the solution: choose larger and larger inputs in both directions until you find one that gives positive output and one that gives negative output. Now you have trapped the solution between two real numbers. Now use bisection. The square root can be found in a similar way, although of course there are faster algorithms.
The proof that a number is composite using the contrapositive of Fermat’s Little Theorem does not obviously (or even non-obviously) suggest a way to find a proper factor of the number. Not only that, but people have fancier ways (only partly based on Fermat’s Little Theorem) to test for primality that always work, and work faster than any known method of producing an explicit proper factor. This is a real contrast to the situation with cubic polynomials.
Send to Kindle"Mathematical objects" rewritten
Vaughan Pratt has rewritten the Wikipedia article on mathematical object, and the result is a great improvement.
I still think that a suitable approach would be to pick out mathematical objects among all abstract objects by specific properties they have.
- A mathematical object is always an abstract object that satisfies certain axioms specific to the object.
- A mathematical object is inert and unchanging.
- A mathematical object is defined crisply, no fuzzy allowed.
- And so on.
I wrote about some of those points here, but not the part about axioms. Watch this space!
These ideas are not settled. As one commenter said, Wikipedia articles should not be the product of current research. What Vaughan has written is about right for now.
Specific comments:
1) Above, I reworded my comment about mathematical objects satisfying axioms in response to an objection by a reader. For example, a model of untyped lambda calculus is an object S for which the function space S -> S is isomorphic to S. Such a thing does not exist in the category of sets, but it does in the category of topological semigroups and also in the realizability topos.
2) In a note to the category mailing list, Vaughan also said:
“It might also be worth mentioning coalgebras, and perhaps more importantly dually defined structures such as locales which are understood better in terms of the morphisms from them to a cogenerator rather than those to them from a generator, i.e. dual elements rather than elements. Also toposes as a more general codomain of the forgetful functor than the particular topos Set.”
Send to KindleMathematical objects in Wikipedia
The Wikipedia article on “mathematical object” needs to be rewritten. It begins: “In mathematics any subject of mathematical research that can be expressed in terms of set theory is a mathematical object”. It also says, “Outside mathematics, a mathematical object is an abstract object that is referred to or occurs in mathematics”.
That is defining mathematical object in terms of a historical detail. Yes, it has been shown that most mathematical objects can be constructed from sets in one way or another, often in strange or unintuitive ways. However, that is a theorem, not a main property of mathematical objects. Besides, there are objects that exist in other categories but not in sets, such as models of untyped lambda calculus. Those categories involve proper classes of objects, not sets of them.
I would prefer some definition such as this: “A mathematical object is an abstract object defined by axioms”, together with explanations of abstract object and axiom. Mathematical objects should be distinguished from abstract objects such as “schedule” that change over time and also from objects in narrative fiction.
The article should describe the different points of view taken by philosophers and mathematicians who have written about the idea. It should refer to some of these articles and books:
Davis and Hersh, The Mathematical Experience (Mariner Books, 1999), sections on Mathematical Objects and Structures: Existence and on True Facts about Imaginary Objects.
Goodman’s article in New Directions in the Philosophy of Science, Princeton, 1998.
Hersh, What is Mathematics, Really?, Oxford University Press, 1997.
Stanford Encyclopedia of philosophy article on abstract object.
I have written about mathematical objects at length on abstractmath.org. It pulls together many of the ideas in the articles listed above.
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