## Anatomizing An Error

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.

• #### davekingsbury 11:20 pm on Sunday, 24 July, 2016 Permalink | Reply

So mathematics is essentially exploration of the unknown?

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• #### Joseph Nebus 4:36 am on Tuesday, 26 July, 2016 Permalink | Reply

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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• #### davekingsbury 7:24 am on Tuesday, 26 July, 2016 Permalink | Reply

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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• #### Joseph Nebus 5:54 am on Wednesday, 27 July, 2016 Permalink | Reply

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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• #### davekingsbury 8:15 am on Wednesday, 27 July, 2016 Permalink | Reply

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

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## JH van ‘t Hoff and the Gaseous Theory of Solutions; also, Pricing Games

Do you ever think about why stuff dissolves? Like, why a spoon of sugar in a glass of water should seem to disappear instead of turning into a slight change in the water’s clarity? Well, sure, in those moods when you look at the world as a child does, not accepting that life is just like that and instead can imagine it being otherwise. Take that sort of question and put it to adult inquiry and you get great science.

Peter Mander of the Carnot Cycle blog this month writes a tale about Jacobus Henricus van ‘t Hoff, the first winner of a Nobel Prize for Chemistry. In 1883, on hearing of an interesting experiment with semipermeable membranes, van ‘t Hoff had a brilliant insight about why things go into solution, and how. The insight had only one little problem. It makes for fine reading about the history of chemistry and of its mathematical study.

In other, television-related news, the United States edition of The Price Is Right included a mention of “square root day” yesterday, 4/4/16. It was in the game “Cover-Up”, in which the contestant tries making successively better guesses at the price of a car. This they do by covering up wrong digits with new guesses. For the start of the game, before the contestant’s made any guesses, they need something irrelevant to the game to be on the board. So, they put up mock calendar pages for 1/1/2001, 2/2/2004, 3/3/2009, 4/4/2016, and finally a card reading $\sqrt{DAY}$. The game show also had a round devoted to Pi Day a few weeks back. So I suppose they’re trying to reach out to people into pop mathematics. It’s cute.

• #### Marta Frant 5:27 am on Thursday, 7 April, 2016 Permalink | Reply

Questions, questions, questions… The constant ‘why’ is what makes the world go around.

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• #### Joseph Nebus 2:07 am on Saturday, 9 April, 2016 Permalink | Reply

‘Why’ is indeed one of the big questions. ‘What’ and ‘The Heck?’ are also pretty important.

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