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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