Category Archives: math

astounding math, math

Astounding Math Stories

2 Comments

Astounding Math Stories

I am in the process of selling my old collection of Astounding Science Fiction on Ebay. It occurred to me that math needs an Astounding Math magazine. It could contain short descriptions of some mathematical facts that are really weird that could awaken people’s interest in math. Well, geeky people anyway.

These astounding facts also illustrate some important ideas in math. For one thing, some of them are not astounding when you get the right representation or proof (for example e^{iπ} = –1). And some are simply frauds (infinite cardinals).

Each of the examples below will be fleshed out and given references. The first one, the Perrin function, has already been posted on the Astounding Math Stories website. Comments and suggestions are earnestly desired.

Perrin pseudoprimes

The Perrin function P(n) is a certain easily-defined function on the natural numbers with this property:

For all integers n e^{iπ} = –1

This is Astounding. Who would have thought that the numbers e, i and π would be related in this way? Well, actually, it is not hard to understand why it is true if you use Euler’s formula in the context of the Argand representation. And the fact that Euler’s formula works is not very difficult to understand. (Perhaps Euler’s formula is nevertheless Astounding. Feynman thought it was.) This is an excellent example of the ratchet effect: An amazing or incomprehensible statement about math suddenly becomes totally obvious and you can’t understand why you didn’t understand it before!

Infinite cardinals

There are “as many” integers as there are rational numbers. This is Astounding!

Really? In fact, this statement is a fraud. It depends on defining the cardinality of a set in terms of bijections (not a fraud) and then referring to the cardinality of the integers or rationals in terms of words like “many”.

What is happening is that, for finite sets, the cardinality function on sets Card(S) means the same as #(S). the number of elements of S. On infinite sets, the cardinality function does not have some of the familiar properties of the number of elements of a finite set; in particular, it can happen that S is a proper subset of T but Card(S) = Card(T). Unless you specifically state, “When I say S has as many elements as T, I mean Card(S) = Card(T)”, then you are deliberately using the word “many” with a nonstandard meaning that the listener may not know. This is like the politician who told his audience that his opponent was a “sexagenerian”. Playing on someone’s ignorance is FRAUD.

Composite numbers
It is possible to prove that an integer n is composite without knowing any nontrivial factors of n. This is Astounding! How could you show it is composite without finding a nontrivial factor? This is a consciousness-raising example that shows that just because you can’t think of how to do something doesn’t mean it is impossible.

To show that n is not a prime, all you have to do is find an integer a for which a^n – a is not divisible by n (Fermat’s Little Theorem). That is not even very hard to prove.

Humongous numbers

As far as we know, π(x) − li(x) changes sign for the first time at some humongous number. 1.397×10^316 is a LARGE number. Other even huger numbers are Moser’s number and Graham’s number. One could also refer to Ackerman’s function. All these are consciousness raising examples that show how big numbers can get.

Many years ago the Little Girl Next Door asked me what the next number after a trillion is. I said, a trillion and one. She was Disgusted.

Function Spaces

The Astounding thing about function spaces is that each point in a function space is a function. A whole function, like sin x or the Riemann Zeta function. Not its values, not its formula, but the whole thing. This story is going to be difficult to write, but if I can carry it off it may help the student along the way to understanding the concept of an encapsulated mathematical object.

Other topics:

The monster group

1^2+2^2+3^2+…+24^2 = 702 and the Leech Lattice

41 and 163

Send to Kindle
language, language of math, math, representations

Set notation

1 Comment

Students commonly think that the notation “{Ø}” denotes the empty set. Many secondary school teachers think this, too.

Mistakes in reading math notation occur because the reader’s understanding of the notation system is different from the author’s. The most common bits of the symbolic language of math have fairly standard interpretations that most mathematicians agree on most of the time. Students develop their own non-standard interpretation for many reasons, including especially cognitive dissonance from ordinary usage and ambiguous statements by teachers.

I believe (from teaching experience) that when a student sees “{1, 2, 3, 5}” they think, “That is the set 1, 2, 3 and 5”. The (incorrect) rule they follow is that the curly braces mean that what is inside them is a set. So clearly “{Ø}” is the empty set because the symbol for the empty set is inside the braces.

However, “1, 2, 3 and 5” is not a set, it is the names of four integers. A set is not its elements. It is a single mathematical object that is different from its elements but determined exactly by what its elements are. The correct understanding of set notation is that what is inside the braces is an expression that tells you what the elements of the set are. This expression may be a list, as in “{1, 2, 3, 5}”, or it may be a statement in setbuilder format, as in “{x x > 1}”. According to this rule, “{Ø}” denotes the singleton set whose only element is the empty set.

This posting is based on the belief that that mathematical notation has a standard, (mostly) agreed-on interpretation. I made this attitude explicit in the second paragraph. Teachers rarely make it explicit; they merely assume it if they think about it at all.

The student’s interpretation is a natural one. (Proof: So many of them make that interpretation!) Did the teacher tell the student that math notation has a standard interpretation and that this is not always what an otherwise literate person would expect? Did the teacher explain the specific and rather subtle rule about set notation that I described two paragraphs above? If not, the student does not deserve to be ridiculed for making this mistake.

Many people who get advanced degrees in math understood the correct rule for set notation when they first learned it, without having to be told. Being good at abstract math requires that kind of talent, which is linguistic as well as mathematical. Most students in abstract math classes are not going to get an advanced degree in math and don’t have that talent. They need to be taught things explicitly that the hotshots knew without being told. If all math teachers had this attitude there would be fewer people who hate math.

PS: My claim about how students think that leads them to believe that “{Ø}” denotes the empty set is a testable claim. There are many reports in the math ed literature from investigators who have been able to get students to talk about what they understand, for example, while working a word problem, but I don’t know of any reports about my assertion about “{Ø}” .

Send to Kindle
math, representations, understanding math

More about neurons and math

3 Comments

In the last post I talked about a neuron assembly in the brain that when it fires makes you feel you have been in the current situation before, and another neuron assembly that makes you feel that you are dealing with a persistent object with consistent behavior. I want to make it clear that I don’t know precisely how these brain functions are implemented, and I know of no research literature on these topics.

Brain research has shown that many different kinds of behavior, including thinking about different real and unreal things, causes activity in specific parts of the brain. I claim that the idea that there is a déjà-vu site and a persistent-thing-recognizing site is plausible and consistent with what we know about the brain. And they are far more plausible than any explanation of déjà-vu as coming from past lives or any explanation that mathematical objects are real and live in some ideal non-physical realm that we have no evidence for at all.

Another point: If our perception that when we think about and calculate with math objects we are dealing with things that are “out there” comes from the way our brains are organized, then we mathematicians should feel free to think about them and talk about them that way. We are making use of a brain mechanism that presumably evolved to cope with physical reality, as well as a general metaphor-mechanism that everyone makes use of to think about both physical and non-physical situations in a productive and creative way.

This point of view about metaphors has a lot of literature: see the section “about metaphors” here.

Again, it is a reasonable hypothesis that the metaphor-mechanism is implemented in some physical way in the brain that involves neurons and their connections.

To sum up, when we mathematicians think and act like Platonists we are using some of the main mechanisms of our brain for learning and creativity, and we should go ahead and be Platonists in action, without feeling embarrassed about it and without subscribing to any idealistic airy-fairyness.

Send to Kindle
math, representations, understanding math

Mathematical Objects are "out there"?

5 Comments

(This article is continued in More about math and neurons).

Sometimes we have a feeling of déjà vu in a situation where we know we have never been before. I have had two very strong occurrences of that in my life. One was when I saw St Cuthbert’s Church in Wells in England, and the other was the first time I saw St Martin’s in the Fields in London. Now my ancestors are mostly from England, some even from the south of England (the English ancestors of most white southern Americans are from the north of England, as are some of mine). Was this ancestral memory? Was it memories from a Previous Life? Well, I didn’t believe that, but the feeling was remarkably strong.

Many years later I discovered the reasons for the feelings in both cases. Adelbert Stone Chapel on the Case Western Reserve University campus in Cleveland (where I taught for 35 years) is an exact copy (on the outside) of St Cuthbert’s Church. Independent Presbyterian Church in Savannah is a three quarters size copy of St Martin’s in the Fields, and when I lived in Savannah as a teenager I frequently rode past that church on the bus.

There is presumably a neuron assembly (or something like that) in the brain devoted to recognizing things “I have seen before”. No doubt this can be triggered in the brain by mistake. Being triggered does not have to mean you had a previous life, it may mean a mistake in the recognition devices in your brain. The fact that I eventually understood my two experiences is in fact irrelevant. If you have the feeling of déjà vu and know you haven’t been there before and you never are able to explain it it still doesn’t prove you had a previous life or anything else supernatural. The feeling means only that a certain part of your brain was triggered and you don't know why.

When I deal with mathematical objects such as numbers, spaces, or groups I tend to think of them as “things” that are “out there”. Every time I investigate the number 42, it is even. Every time I investigate the alternating group on 6 letters it is simple. If I prove a new theorem it feels as if I have discovered the theorem.

There is also presumably another neuron assembly that recognizes that something is “out there” when I have repeatable and consistent experiences with it. Every time I push the button on my car door the door will open, except sometimes and then I consistently discover that it is locked and can be unlocked with my key. Every time I experiment with the number 111 it turns out to be 3 times 37. If some math calculation does not give the same answer the second time I frequently find that I made a mistake. I know this feeling of consistent “out there” behavior does not prove that numbers and other math objects are physical objects. The feeling originates in a brain arranged to detect consistent behavior. The feeling is not evidence that math objects exist in some ideal space.

Send to Kindle
language, language of math, math

The languages of mathematics

2 Comments

Conjecture: Mathematical English (ME) and the symbolic language of math (SL) are two distinct languages, not dialects of the same language.

I have asserted this in several places (Handbook, abstractmath.org) but I am not a linguist and it could be that linguists would disagree with this conjecture, or that the study of a mathematical corpus would reveal that another theoretical take on the situation would be more appropriate.

Some relevant points are listed below. I intend to expand on them in later posts.

1) Is ME a dialect of English or a register of English? Or does it have some other relationship to English?

2) ME appears to have several dialects or registers. One register is that used for what mathematicians call “formal proofs”. These are not formal in the sense of first order predicate logic, but their language is constrained, with the intent of making it easier to see the logical structure of the argument. Another register is that of “intuitive [or informal] explanations”. This is more like standard English.

3) The SL is clearly not a spoken language. It is a two-dimensional written language using symbols from English and other languages and some symbols native only to math. People do try to speak formulas aloud occasionally but this is well known to be difficult and can be done successfully only for fairly simple expressions.

4) There are other non-spoken languages such as ASL for example. I don’t know whether there are other non-spoken languages that are written. I don’t think dead languages count.

Send to Kindle
language of math, math, misc

Mass nouns in math writing

1 Comment

Mass nouns seem to be rare in math writing. I have done a little poking around the math journals in JStor and have several observations to make. All are tentative observations based on a small amount of evidence (and thinking).

“Space” as a mass noun

“Space” as a mass noun was common before WWII but is rare now. A search for “in space” (in quotes to make it the phrase that is searched for) gives mostly references to outer space and to very old papers, mostly before 1930. Conjecture: The disappearance of mass nouns in math writing is a consequence of the rise of structural thinking in math.

One recent paper where “space” occurs seemingly as a mass noun, in the title no less, is: F. W. Lawvere, Categories of Space and Quantity, in J. Echeverria et al. eds. The Space of Mathematics: Philosophical, Epistemological and Historical Explorations, DeGruyter, Berlin (1992), 14-30. However, the word “space” appears as a mass noun only once in the body of the paper (according to my hasty scan) and many times as a count noun. Anyway I am not sure it is being used as a count noun in the title. It is paired with “quantity”, which is surely an abstract noun, not a count noun.

Areas of math as mass nouns

Areas of math are commonly used as mass nouns, for example, “Using calculus, we see that the function has one maximum”, or “the result follows by straightforward algebra”. The language of math contains several sublanguages with different uses (symbolic language, rigorous language, “rich” language) and one of them is the metalanguage used for talking about doing math, as those examples surely are.

Mass nouns and plurals

In the paper La Palme Reyes M., Macnamara J. and G. E. Reyes (1999). Count nouns, mass nouns and their transformations: a unified category-theoretic semantics, in Language, logic and concepts: Bradford Book, MIT Press, Cambridge, Ma, 1999, pp 427-452, the authors say that plural nouns are mass nouns, in fact they are the free mass nouns corresponding to count nouns under the adjunction developed in that paper. (The Wikipedia article on mass nouns doesn’t seem to regard plurals of count nouns as mass nouns.) Now plurals are mass nouns with atoms (like “furniture” rather than like “water”). Of course, plurals occur all over the place in math writing. Conjecture: In rigorous math prose the only mass nouns that occur are plurals, or at least are mass nouns with atoms.

I am suspicious of the way Reyes, Macnamara and Reyes smush together mass nouns with atoms (furniture) and mass nouns without atoms (water). (“Atom” means in the lattice of parts. “Some of my furniture” can include a bed and two tables, but not the leg of a table. “Water” is treated in language as if it were infinitely divisible. Of course it really does have atoms in the physical sense.)

These two kinds of mass nouns behave differently in many ways. The most important is that plural nouns can refer to either distributive plurals or collective plurals. (“All groups have identities” is distributive, “the voters were in favor of the proposition” is collective.) I doubt that these different kinds of mass nouns constitute a natural grammatical class.

Send to Kindle