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Eryk Banatt's Reviews > Proofs and Refutations: The Logic of Mathematical Discovery
Proofs and Refutations: The Logic of Mathematical Discovery
by
by
I picked this up seeing it on a list of Robb Seaton's favorite books". I think I can describe it as "Plato's The Republic meets Philosophy meets History of Mathematics" and that sentence can more or less describe the entirety of the book.
I will admit that the book was a bit challenging for me, and I suspect I will revisit this book when I get a bit better at math, but for what it was I think it was quite readable and I enjoyed it. It was a little dry at times but the dialogue was very interesting and posed some very interesting questions about the way people have approached solving problems throughout history.
A line I thought was pretty interesting is the following:
Of course I would. I certainly wouldn’t call a whale a fish, a radio a noisy box (as aborigines may do), and I am not upset when a physicist refers to glass as a liquid. Progress indeed replaces naive classification by theoretical classification, that is, by theory-generated (proof-generated, or if you like, explanation-generated) classification. Conjectures and concepts both have to pass through the purgatory of proofs and refutations. Naive conjectures and naive concepts are superseded by improved conjectures (theorems) and concepts (proof-generated or theoretical concepts) growing out of the method of proofs and refutations. And as theoretical ideas and concepts supersede naive ideas and concepts, theoretical language supersedes naive language.
This quote reminded me a lot of a great blogpost I read once, The Categories were Made for Man, Not Man for the Categories. The cool part of this part of this passage is the idea that statements have different consistency values depending on the language in which you talk about them - you have certain things that might be true in a naive language (i.e. finned creatures in the ocean are called fish and a whale is a fish) that may be untrue when you drill down into a different language (i.e. whales and tunas are not in the same taxonomic classification and therefore only one can be a fish).
Overall pretty readable for what it is - will revisit again someday.
I will admit that the book was a bit challenging for me, and I suspect I will revisit this book when I get a bit better at math, but for what it was I think it was quite readable and I enjoyed it. It was a little dry at times but the dialogue was very interesting and posed some very interesting questions about the way people have approached solving problems throughout history.
A line I thought was pretty interesting is the following:
Of course I would. I certainly wouldn’t call a whale a fish, a radio a noisy box (as aborigines may do), and I am not upset when a physicist refers to glass as a liquid. Progress indeed replaces naive classification by theoretical classification, that is, by theory-generated (proof-generated, or if you like, explanation-generated) classification. Conjectures and concepts both have to pass through the purgatory of proofs and refutations. Naive conjectures and naive concepts are superseded by improved conjectures (theorems) and concepts (proof-generated or theoretical concepts) growing out of the method of proofs and refutations. And as theoretical ideas and concepts supersede naive ideas and concepts, theoretical language supersedes naive language.
This quote reminded me a lot of a great blogpost I read once, The Categories were Made for Man, Not Man for the Categories. The cool part of this part of this passage is the idea that statements have different consistency values depending on the language in which you talk about them - you have certain things that might be true in a naive language (i.e. finned creatures in the ocean are called fish and a whale is a fish) that may be untrue when you drill down into a different language (i.e. whales and tunas are not in the same taxonomic classification and therefore only one can be a fish).
Overall pretty readable for what it is - will revisit again someday.
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