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.

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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.

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The infinite sequence 1, 1/2, 1/3, 1/4, … is commonly described as “getting closer and closer to 0”. This fits with the mental representation (metaphor) students have of the sequence as something whose successive entries are revealed or created over time.

People who are used to abstract math have another handle on that sequence: It is the function whose value at each positive integer n is 1/n. Stated that way, the sequence is being pictured as existing all at once (another metaphor). It is a completed infinity or actual infinity. Completed infinities occur all over modern math and mathematicians rarely remark on them. But some people outside math, in particular philosophers, get all hot and bothered about it.

I have not found a good reference on the internet to the idea of completed infinity. The article on actual infinity in Wikipedia gives lots of reasons against the idea but few reasons in favor of it. Many other articles on the internet also mostly discuss the opposition to the idea.

The phrase “the function whose value at each positive integer n is 1/n” is a definite and clear description of the sequence. Mathematicians have proved all sorts of statement about the sequence using classical logic; it is a Cauchy sequence converging to 0, for example. These statements all seem to be consistent with each other and with other parts of math. To put this in another way, we have a clear syntax for talking about this sequence and others, and we have built up a fund of statements about it that all hang together.

As always when talking about anything, we use metaphors and images when talking about this sequence. The image of the sequence as getting closer and closer to zero is one, which we could call the Xeno image. The image of the sequence as existing all at once is another, let’s say the actual-infinity image.

Neither image says anything about physical reality. The sequence is of course represented physically in our brains by arrangements of neurons or some such thing, but in no way does that imply that every entry is represented in our brain. What is represented in the brain are properties of the sequence, the metaphors and images we have mentioned (and others), relationships with similar sequences, and so on, what the math ed people call our schema of the concept. (See [1] and [2].)

I propose that the trouble philosophers and students have with the actual-infinity image occurs because there is a physical arrangement in a brain that serves to recognize big bunches of individual things. (Compare this post.) It may be triggered, for example, if you look out of a high window and see a big crowd of people standing in the street. You have a person-recognizer in your brain and you also have a big-bunch recognizer in your brain, which works with the person-recognizer to identify a crowd of people. (I talked about a similar idea, the I-saw-this-before recognizer, here).

A smart person soon realizes that there is a difference between a crowd of people and an infinite sequence, namely that the crowd is large but finite whereas the sequence is infinite. Well, that is true. So what? The crowd may be finite but you still can’t hold an image of each individual in the crowd in your head. The finite-infinite difference is indeed a difference, just as the idea that one is composed of physical objects (people) and the other is composed of mathematical objects (numbers). You can reason about both the crowd and the sequence, make discoveries about them, and so on.

Why is there a difficulty? Perhaps it is because a device in your brain that is usually used for bunches of physical objects is now being triggered by a bunch of abstract objects. That can be disconcerting.

But: You don’t have to get all upset about something just because it disconcerts you.

Now, then, no one ever needs to worry abou
t actual infinity again. (Fat chance).

REFERENCES

[1] Ed Dubinsky and Michael A. McDonald, APOS: A Constructivist Theory of Learning

in Undergraduate Mathematics Education Research

[2] David Tall, Reflections on APOS theory in Elementary and Advanced Mathematical Thinking.

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