"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."

Mathematical 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.

Daniel the mathematician

My favorite current mystery novels are the Brodie Farrell mysteries by Jo Bannister. One character is a high school math teacher (as Americans would call him) named Daniel Hood. Here are some things he says about math in Breaking Faith:

"Numbers are different. Numbers I can do. There are no nuances, no room for debate. They always and only mean one thing: you can't twist them to mean something else... (page 63).

"Look, I'm a mathematician. I've never had an original thought in my life. I can't imagine how you create something entirely new..." (page 116).

Daniel is expressing a widely-held attitude, but he is wrong. Mathematical research does have a place for applying a known method to a problem that obviously can be solved in that way. But math usually requires more creativity than that.

1. To solve a hard problem, you may have to recognize a non-obvious pattern in the problem that reveals its amenability to a known method...
2. ...or you may have to reformulate a known method in a non-obvious way to solve the problem.
3. Sometimes you have to invent a genuinely new method to solve a problem.
4. Sometimes you accomplishment is coming up with a new kind of problem and solving it, or part of it.
5. Sometimes in trying to understand a type of mathematical object you have a creative insight that seems to come out of nowhere, an "entirely new" way to think of or describe a mathematical phenomenon.

Most research papers fit (1) or (2). They both involve what I would call minor-league creativity and can get your paper published. (3), (4) and (5) can get you wide recognition in your field. In exceptional cases you may receive one of the several prizes awarded to mathematicians (Fields medal, Wolf prize). Then many mathematicians outside your field will have heard of you.

When you do serious math research you always find yourself thinking of the objects involved in many ways. (For example, if you "twist" the meaning of complex numbers to be vectors in 2-space then you get the Argand representation, which makes some baffling things about complex numbers obvious.) You cannot do serious math without having several metaphors for each object in your head at once.

Jo Bannister may have been putting her understanding of math in Daniel's mouth. Or she might think that many high-school math teachers think that way even though she knows better. For all I know, many high-school math teachers do think that way.

Default meanings

Concerning Michael Barr's comments on my post on terminology, I remember going to a meeting of topologists in 1965 or 1966 in which people kept spouting nonsense about free groups. The reason it was nonsense was that they were talking about free abelian groups without saying so. That may have been the first time I became aware of default meaning in different groups of mathematicians.

I became aware of default meanings in ethnic and regional groups long before that, when I joined the Air Force after never having been outside the deep south and discovered that other people thought "sweet milk" and "ink pen" were weird things to say.

Representations IV

Mark Meckes recently commented on my post on writing Astounding Math Stories that students "usually think of decimal expansions as formal expressions". His point is well taken, but I would go further and say that they think of real numbers as decimal expansions, hence as formal expressions.

For example, 1/3 is approximately 0.333... However, 1/3 is exactly 1/3. The expression "1/3" is more exact than the decimal expansion. Similarly \sqrt 2 is defined exactly as the positive real number whose square is 2. And of course there are still other representations of some or all real numbers, for example binary notation, representations as limits, as solutions of equations, and so on.

The main thing to understand is that every interesting mathematical object has several representations, each representation coming from a different system of metaphors. And if you are going to understand math you have to be aware of various representations of the same object and hold (some of) the details of several of them in your head at once. Even "2 + 3 = 5" is talking about two different representations of a number simultaneously. William Thurston once said that it was a revelation to him as a child that when you divide 127 by 23 you get 127/23. That notation"127/23" tells you two things: exactly what the number is and one way it is related to other numbers. That kind of phenomenon is what makes math work.

I went on and on about this stuff in abstractmath.org here and here.

Writing Astounding Math Stories

I have written a second Astounding Math Story, this one about factoring integers and primality testing, published here. (I announced ASM here.)

These stories are aimed at people interested in math but not very far along in studying abstract math, the same audience that my abstractmath website is concerned with. That makes the stories hard to write.

For one thing, they need to be streamlined as much as possible. Each story should explain one astounding fact clearly, making as few mathematical demands on the reader as possible. I have put in links, mostly to Wikipedia, to explain concepts used in the story. This story has lots of footnotes telling about further developments and fine points.

Another point is that it is sometimes hard to convince students that they need to be astounded! I mentioned the phenomenon that primality testing is faster than factorization many times in my teaching (mostly to computing science students). Often I had to work hard to get them to realize that there was something shocking about this: Being a composite means having proper factors, but the fast ways of discovering compositeness tell you it is composite without giving any clue as to what the proper factors are. Research mathematicians are familiar with the idea of proving something exists without being able to say what it is, but students often have to be led by the nose to grasp this idea.

With this story I have experimented with making it a dialog. That makes it easier to write about a conflict between new ideas and old presuppositions. I would love to get comments about how these stories are written as well as the math involved.

Experienced research mathematicians will probably not be Astounded by these stories. But the ones I am writing about have often given mathematicians in previous centuries a lot of trouble -- they seemed unbelievable or contradictory. And modern students have trouble with these ideas, too. In the current ASM I mention Kronecker's problem with nonconstructive existence proofs.

One of the worst problems comes with infinite decimal expansions of real numbers (which I intend to write about). You can prove that 1.000... - .999... but the students don't really believe it. That may be because they don't really believe that all the decimal digits are really there. (A lot of philosophers don't believe this either, but almost all research mathematicians talk and act as if they do.)

By the way, it is amazing how often there is an article in Wikipedia that says just what you want the reader to know (and of course usually a lot more) and does it pretty well. Lately I have run across just one exception, the articles on context. I have been writing about context in mathematical writing for abmath (don't look, it isn't there yet) and have wound up going into more detail than I wanted because I could not refer to Wikipedia.

What I learned in school is the Only Truth

Michael Barr recently commented on another post about a Dutch student who insisted that the words "long" and "short" in English referred to qualitative differences such as that between "ride" and "rid", whereas linguists use the words to refer to temporal length, such as the different in the vowels between "hid" and "hit".

I assume the student acquired the qualitative meaning from English courses in school; that meaning is very still widely used in English classes in the USA and Britain, so that (I'll bet) nearly any person on the street in the USA would expect the meaning of "long" and "short" to be the difference between "ride" and "rid".

This is an example of a phenomenon mathematicians have to put up with too. We know that the same word or symbol can have many different meanings in math, but people who know a little math assume that all meanings that they learned in whatever math courses they took are universal and set in granite. They are startled that "pi" can mean anything other that what they think it means. Someone recently started talking to me about "phi" as if I should know what it means, but I recovered fairly quickly, since I had become vaguely aware that it means the so-called golden ratio to laymen. In my experience mathematicians mostly use phi to denote some function.

When I taught, I was constantly in trouble with students who told me that 0 was not a natural number if the textbook said it was, or was a natural number if the textbook said it wasn't, because that was the definition in some previous course they had taken.

I have written about this phenomenon here and here.

Unvoicing final consonants in English

Younger people frequently devoice final d while keeping the length of the vowel, so for example "road" ends with a t sound. The word does not become homophonic with "rote" because the latter has a shorter vowel, shorter in the time it takes to pronounce it ("long" and "short" are often used to denote vowel quality but I am not talking about that here.)

I became especially aware of this when I started singing in choirs twenty years ago. A person standing next to me would sing "Lord" ending in a distinct "t" sound, often released. I noticed younger people doing it more than older people. I have mentioned this occasionally in choir practice, and some leaders really don't want me to end "Lord" with a voiced d, at least not a released voiced d.

Linguists have noticed this, but the articles I have found mostly discuss it happening in African American speech. But I swear I have heard it many times from white native English speakers.

Last week for the umpteenth time I was in the hospital. I told the nurse I would like to walk around the halls. She asked me, "Would you like a rope?" I asked her to repeat herself and finally decided she was saying "robe". She released the p, too. She sounded like a native English speaker from Minnesota. I couldn't tell if she was lengthening the vowel. I suppose I could have tried to elicit a minimal pair by asking her something like, "What do you hang people with in this hospital?" but, no...

Cribles

Mark Meckes' comments on technical words in English reminds me of an incident from the Ancient Days, namely 1966. Well, 1966-ish. A visiting Belgian mathematician in my department talked about category theory. One concept that came up was that of "crible", which was its name in French. It is a family of arrows with a common target (the sources can vary). He couldn't think of the English translation of "crible" so he said something like this: "The best way I can describe this is to think of a soldier in the trenches in World War I who suddenly stands up and is shot full of holes by many machine gun bullets."

We were completely baffled by this explanation. Is there an English word that describes a person with lots of bullet holes? You can see where the picture comes from by thinking of the target as a person with lots of arrows stuck in him, like Saint Sebastian, or Hagar the Horrible on a bad day.

The English word he wanted is "sieve" and that is the usual name of the concept today. In the sixties, many English speaking mathematicians called it "crible" but that usage died out as far as I know. A few tried to pronounce it the French way, but no one understood them, so they spelled it, and then most people said "cribble".

A Scientific View of Mathematics

Over most of the history of human thinking, both philosophers and theologians have come up with explanations of some natural phenomenon, only to be faced with scientific investigations that give a successful evidence-based explanation of the phenomenon they have written about.

The theologians and philosophers eventually lose the argument. The word “eventually” means that (1) the scientific investigation has to produce a pretty solid theory that explains a lot of the evidence and (2) the older theologians and philosophers have to die, since they rarely change their minds about stuff after the age of 50. (Some theologians and philosophers still continue to argue for things such as flat earth, intelligent design and dualism, but they are not really engaging in the intellectual world’s ongoing conversation.)

It is now possible to investigate the theory and practice of mathematics using evidence-based scientific reasoning. In particular, recent findings in neuroscience and cognitive theory make it plausible to provide a description of mathematics that is based on the interaction between brain, body and culture. By continuing to study mathematics and its practice scientifically we can hope to come up with a theory of mathematics that will be a part of cognitive theory.

I have been writing about bits and pieces of this idea for a long time, and so have many others. What I am going to give here is a lacunary sketch of my current thoughts with references. Most of these ideas originated with other people!

Math is an activity of our brains.
Our brains contain ideas. These ideas are real physical structures, organizations of neurons or something. [MO], [TaBa2002] I will call them PSB’s (physical structures in the brain). This is early days in neuroscience and exactly how the ideas exist physically in the brain is still controversial. I assert only that ideas are physical, nature in part yet to be determined.
Among our ideas are representations of objects and lists of rules.

Objects are represented in our brains.
The objects represented in our brains may be physical objects, fictional objects or abstract objects, including mathematical objects [AbMO], [Her97].

There is presumably a PSB that is triggered when you think about any kind of object. It is triggered if you think about the Parthenon, Sherlock Holmes, or the function f that takes a real number x to x^2. (If you are not an experienced mathematician, you might in fact not think about f as an object, but rather as rule or procedure. This can cause serious difficulties for students in calculus classes who are faced with such concepts as “the derivative of f”.)

There is no doubt another PSB’s that recognize that the Parthenon is a physical object in contrast to the squaring function, which is an abstract object. But our brain clearly recognizes both as an object because we talk about physical, fictional and abstract objects using the same grammatical structures and we think about them using similar mental operations. [AbLM]

The fact that we think and talk about the set of all real numbers (for example) as an object is explained by this PSB. It does not imply that the set of all real numbers exists anywhere, physically or ideally.

Our brains are organized to follow rules.
We apparently have a PSB that implements a rule-following mode. (This has been studied but I don’t have a good reference for it.) We learn and follow rules very easily when we play any game, baseball, chess or whatever. We also learn rules for algebraic manipulation and for mathematical reasoning using the rule-following mode. People who are good at math seem to engage the rule-following mode easily in these situations.

Math is communicated among people using the languages of math.
Math has several languages. Mathematical English is a special dialect of English with some disconcertingly different rules. Other major languages have a similar special dialect. The symbolic language of math is a special purpose system that is largely independent of any particular natural language. Graphs, geometric drawings and diagrams form a system for communication as well. The various systems are intertwined with each other in conversation, in lectures and in written math. [AbLM], [O’H], [Wel2003].

When we do math we think about math objects as if they were things.
Conceiving of math as talking about (abstract) objects enables us to think about it using the machinery in our brain we use to think about physical objects. This machinery is highly developed and uses metaphors and physical reasoning (maybe using mirror neurons). We could not do math without it. [LakNun], [WMN], [AbImMet].

Useful mathematical ideas tend to come from our physical experiences with our body and the world.
This is a thesis of [LakNun]. This understanding of the origins of mathematical objects might be developable into an explanation of the “unreasonable effectiveness of mathematics”. Note that you have to explain the effectiveness of mathematical reasoning as well as the usefulness of the objects we talk about.

Mathematical computation and reasoning lead to consistent results.
When we find inconsistent results using math we expect to find a mistake somewhere, and we usually do. This claim is about both numerical and algebraic computation and also formal mathematical reasoning. This phenomenon gives us confidence that mathematical processing is dependable.

This is what was behind my point about actual infinity in [AI]. When we envision the real numbers (for example) as an infinite set that exists all at once, and follow the correct rules of mathematical reasoning, it all works.

It’s also true that we now know of genuine limitations to what we can know, because of incompleteness results as well as cardinality results that say, for example, that there is an uncountably infinite number of real numbers that we cannot refer to individually.

Since in particular we can’t prove the consistency of a system within the system, our experience of consistent results is the only evidence we have that mathematics really works and can be applied to the world.

References

[AbImMet] Images and metaphors
[AbLM] The languages of mathematics
[AbMO] Mathematical objects
[AI] Actual infinity
[MO] Gyre&Gimble (2007), Mathematical objects are “out there” ?
[LM] Gyre&Gimble (2007), The languages of mathematics
[NM] Gyre&Gimble (2007), More about neurons and math
[RIII] Gyre&Gimble (2008), Representations III: Rigor and rigor mortis.
[Her97] Hersh, R. What is Mathematics, Really? Oxford University Press, 1997. ISBN 978-0195113686
[LakNun] Lakoff, G. and R. E. Nüñez (2000). Where Mathematics Comes From. Basic Books. ISBN 978-0465037711.
[O’H] O'Halloran, K. L. (2005), Mathematical Discourse: Language, Symbolism And Visual Images. Continuum International Publishing Group. ISBN 978-0826468574
[TaBa2002] Tall, David and Barnard, Tony (2002). Cognitive units, connections and compression in mathematical thinking
[Ta2001] Tall, David (2001). Natural and formal infinities.
[Wel2003] Wells, C. (2003). The Handbook of Mathematical Discourse.
[WMN] Wikipedia on mirror neurons.