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I'm just starting to learn about the philosophy of mathematics, and was asked to read this paper for a course: Danielle Macbeth, Seeing How it Goes: Paper-and-Pencil Reasoning in Mathematical Practice, which appears in the Philosophia Mathematica. I've posted the paper here in case you can't access it otherwise.

I'm supposed to respond to the following question:

Is Danielle Macbeth's position in the paper in tension with mathematical platonism or, rather, does her position presume it?

I'm curious to hear some thoughts on this question. I'm kind of struggling with this, because in my view the position of Macbeth is that good mathematical notations embody mathematical arguments and reasoning, while the position of a mathematical platonist seems to be that mathematical objects are abstract and independent of human thought. I'm finding it hard to argue convincingly in response to the question above. Thoughts and ideas welcome!

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  • I find the difficulty here to be the phrase 'independent of human thoughts'. Clearly some mathematical (mathematically-describable) objects meet this specification, but the bigger question is whether they are independent of thought regardless of who is doing the thinking. .
    – user20253
    Feb 12, 2018 at 12:58

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There is a tension.

If we agree to characterize the platonist point of view (in mathematics) as:

the metaphysical view that there are abstract mathematical objects whose existence is independent of us and our language, thought, and practices,

then Macbeth's point of view is summarized in the Conclusion:

a good mathematical notation serves not merely to record something but to embody the reasoning, to put the reasoning itself before our eyes. [...] It is in just this way that in all these cases [Euclid, Frege] the chain of reasoning to some significant result is embodied in the writing, put before our eyes.

[...] the contents of mathematically significant concepts and functions are formulated in the various systems of signs in a mathematically tractable way, in a way that enables the mathematical demonstration of significant results.

Mathematical concepts are clearly abstract, but they are embodied in the (historical) mathematical practices (the notations).

Thus, their embodiement is not "independent of language and practices".

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