Though it’s the summer months I’m happy to say the Carnot Cycle thermodynamics blog is still posting. He had been writing about Jacobus Henricus van ‘t Hoff, first recipient of the Nobel Prize in Chemistry. In the 1880s van ‘t Hoff was studying the osmosis. In April’s essay Carnot Cycle described the problem, and how van ‘t Hoff passed up a correct formula describing osmotic pressure in favor of an attractive but wrong alternative.

In this month’s essay Carnot Cycle continues the topic. It particularly goes over just how van ‘t Hoff got to his mistaken idea. It’s not that he started out wrong. He began from a good start and derived a mistaken formula. The derivation involved a string of assumptions and simplifications and approximations, of the kind that must be made to go from starting principles to a specific problem. He was guided by an idea of what the answer ought to look like, though, and that led him astray. The blog describes what he did and why it would look reasonable in the circumstance. It’s worth reading to see what actual mathematics, the kind that doesn’t have known answers, is like.

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## Author: Joseph Nebus

I was born 198 years to the day after Johnny Appleseed. The differences between us do not end there.
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So mathematics is essentially exploration of the unknown?

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I’m not sure exactly how I would describe mathematics. Part of it does feel like an exploration of the unknown: we set out basic rules and find implications that aren’t obvious. A lot of work does feel like experimentation and discovery, just as one might do in a science. But it does seem bizarre to imagine that the logical consequences of our chosen premises are unknown; it seems like saying that a chess move might need discovery. I’m not sure how to represent it all. Possibly there’s no representing it all as one thing; there are several strands of thought that run through mathematics, I believe.

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Thank you for this honest and lucid response. Strikes me it’s a language which avoids the pitfalls of imprecision and emotionality.

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I think avoiding imprecision and emotionality are considered ideals, yes. And a fully mature, cleaned-up mathematical field has got its important work set up and defined in ways that are precise and avoid emotional appeals. But when working out a problem, especially a new and exciting one, there are many provisional definitions and ambiguities discovered late in the paper and all that. Mathematicians are humans and their lives are all over their work, necessarily. We try to look good when strangers peek in, which is again a most human thing to do.

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Thanks for humanising the world of mathematics for me. You have the skills of a natural teacher.

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