Philosophy of Mathematics: A Contemporary Introduction to the World of Proofs and Pictures by James Robert Brown (1999)

This book looks pretty nice – current, accessible and far-reaching. I’m interested in the chapter about knots and notation. That’s what I’m thinking of focusing on in my own project.

Brown begins on p. 84 comparing maths’ tying of notation to innovation with that seen in poetry. He refers to the advantages of the Arabic numerals. So then we go on to Dowker notation, which involves giving each crossing two numbers, Conway notation, which deals with tangles, and neatly expresses each turn as a positive or negative number and turns them as fractions. Interestingly, two tangles are equivalent if their associated continued fractions are equal. Like these guys.

2016-01-24 14.27.19

Knots can also be expressed as polynomials; Alexander, Jones, HOMFLY. Brown notes that these proofs involve little pictures, hieroglyphs. They’re picture-proofs.

This project is listed in the bibliography: http://grantome.com/grant/NSF/SES-9412895

He talks about grain size in different modes. I need to find some relevant papers of his.

Brown asks

Does a particular notation create or merely reveal the properties of objects? p. 94

That’s a very interesting question indeed. He considers formalist and Platonist views of whether a parabola has a degree, and notes that the identification of that feature depends upon the Cartesian notation. Brown then goes on to note Frege’s idea of sense and reference (interesting, I’ll have to look into how this relates to other ideas in linguistics) and adds his own: computational role. This bit is… interesting. I’m not sure. Is he just talking about affordances here?

He rounds off the chapter saying that there’s barely anything written on this, bar Cajori’s great tome on notation (I have it. Good for reference, utterly unreadable). I wonder if that’s true? This was 1999… I hope so. I’m checking out Kenneth Manders.

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