3. Mathematics: different sizes of infinity and Hilbert's Hotel

Mathematics: different sizes of infinity and Hilbert's Hotel

Cool. None of that changes the fact that he's historically been known for his contributions to the field of mathematics. None of that diminishes the fact that his work influenced other mathematicians and shaped the discourse of mathematics. Sorry if I didn't hedge enough or research Bolzano's life story enough before making my initial statement. But I don't see how it matters in the end. I'm not a Bolzano scholar. I've read a few hundred pages of his work, and all of his work that I am aware of is focused on Mathematics. When he is mentioned in academic discourse, it is for his contributions to mathematics. Dedkind's work on number theory contains a reference to the ego in his proofs. That wouldn't fly today, but that doesn't make the work about psychology instead of mathematics. It just means his metaphysical position on numbers lead him to believe that was what the proof needed to succeed. To say that you can find stuff in the work that you wouldn't accept as strictly mathematics doesn't change the fact that is a contribution to the field of mathematics. You'll find plenty of things from Newton which we wouldn't take to be good math or physics, but as just plain theology. Does that diminish his contribution as being mainly toward physics and mathematics? If not, then pointing out something from Bolzano's writings that is more religious than mathematical doesn't either. You'll find that mathematical work before the twentieth century often had a much different flair. That doesn't change the fact that the works were considered contributions to the field of mathematics. The fact that Descartes wrote some books on philosophy doesn't change the work he did on mathematics into philosophy. Bolzano is known for his work on Mathematics. Does that resemble mathematical work today? Some of it does, and some of it doesn't. Further, you're not going to get a good feel of Bolzano's contributions by pointing out a paragraph that you disagree with. Bolzano's contributions to analysis, geometry, and thinking about infinity don't just go away because they don't read like a mathematics paper published in 2015. So read all of the wikipedia articles you'd like. Nobody in the academic world gives a shit what wikipedia says. When philosophers study Bolzano, it's in relation to the foundations of mathematics and the analysis of the role of proof in mathematical practice. When historians read Bolzano it's in relation to the history of mathematics. When mathematicians read him, it's to see the original version of the mathematical proof that was given. So it's great that you've found a reason to win your personal battle against my careless use of language earlier, but it doesn't change the fact that Bolzano is known more as a mathematician than anything else.

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