Effect Measure

The mathematics of the boiled frog

An interesting blog post came my way yesterday about The Legend of the Boiling Frog. The gist of the post was that the legend was just that: urban (or science) legend. The post apparently started out with a query to a noted biologist who studies amphibians (I originally wrote, “a noted amphibian biologist” until I realized I didn’t know if were any biologists who were amphibians):

“I am writing a weekly column for Die Zeit, Germany’s major weekly paper, on scientific urban legends that my readers ask me about. Now you surely have heard the story of the boiling frog that is often told by consultants or activists: If you put a frog in boiling water, he will try to escape. If you put him in cold water and heat it gradually, the frog will remain in place until he’s boiled, because that’s the lesson, to him (and consequently to us) gradual change is not perceivable. Frankly, I don’t buy this. But I am looking for professional advice (and I don’t want to boil frogs). Can you help me with that question? Thanks! Christoph Droesser, Hamburg, Germany”
(Whit Gibbons, University of Georgia)

The original addressee (Joe Pechmann at the University of New Orleans) didn’t know and neither did blogger Wit Gibbons who then asked Victor Hutchison at the University of Oklahoma, who replied:

“The legend is entirely incorrect! The ‘critical thermal maxima’ of many species of frogs have been determined by several investigators. In this procedure, the water in which a frog is submerged is heated gradually at about 2 degrees Fahrenheit per minute. As the temperature of the water is gradually increased, the frog will eventually become more and more active in attempts to escape the heated water. If the container size and opening allow the frog to jump out, it will do so.”

OK. So it’s not literally correct for frogs. But it’s a metaphor and metaphors are used for many purposes, including rhetorical ones, whether you are a Baptist preacher or an environmental activist or a politician. Now that that’s out of the way, let’s talk more about the science content. Because there is some interesting science and mathematics to discuss with examples like this. Here’s one I remember hearing many years ago from mathematician Fred Roberts (at Rutgers) whose book (now back in print), Measurement Theory, I highly recommend.

A person takes her coffee black. Not only doesn’t she put sugar or milk in it, she doesn’t put salt in it, either. Salty coffee? Yuck. But if you take a grain of salt — just one little grain — and put it in her coffee, she won’t be able to taste the difference. So put another grain of salt in it. She still probably wouldn’t be able to tell the difference. So put another one in. By the time you’ve put a teaspoon’s worth in, she’ll clearly be able to tell there’s something wrong. Yet at each step she couldn’t distinguish the difference from the previous one. A grain of salt was below the “just noticeable difference” of her salt tasting apparatus.

There is some interesting mathematics behind this. What we have here is a pairwise comparison or relation at each step. If we just fasten on comparisons that are below a noticeable difference (like two cups of coffee whose salt content differs by only a single grain of salt, then we say the two cups of coffee are in a relation of indifference and denote it cup1 I cup2. This is called a binary relation where the two cups of coffee are related by virtue of the taster not being able to distinguish between them. This kind of relation is really an ordered pair of coffee cups. If you have, say a thousand cups of coffee of different saltiness, you can put them into pairs if a taster can’t tell the difference but don’t include them in the same pair if they can tell the difference. Thus cup1 and cup2 are in the same pair (and probably cup3 and cup11 and those in between are all in pairs if the number is the grains of salt in them; after all, I doubt I could tell the difference between a cup with 1 or 3 or 5 grains of salt and one with 11 grains of salt). But cup 1 and cup20000 are not in the same pair. There’s enough salt in cup20000 (20,000 grains of salt) so that I could tell the difference between it and one with 1 grain of salt.

So this relation of Indifference (not being able to taste the difference) is a real and determinate relationship between any pair of coffee cups. Either they are in the same pair or they aren’t (e.g., cup20000 an cup20001 are also probably in the same pair).

Now there is a branch of discrete mathematics called order theory that involves properties of binary relations and there are all sorts of binary relations with interesting properties. Equality is a binary relation with the properties of symmetry (x=y implies y=x), reflexivity (x=x) and transitivity (x=y and y=z implies x=z). Here’s another common order, called a partial order. A good example is “less than or equal to.” It is also reflexive (x?x), and transitive (x?y and y?z implies x?z) but not symmetric (x?y doesn’t necessarily mean y?x; if both are true then x = y; this is a property called anti-symmetry). Binary relations that are reflexive, transitive and anti-symmetric like ? are partial orders.

Both “equals” and ? (one an equivalence relation, the other a partial order) are transitive. Transitivity is so common we take it for granted, but lots of common relations aren’t transitive. Consider the kinship relation, “parent of”. If Bob is the parent of Sue, and Sue is the parent of Joe, Bob isn’t the parent of Joe. On the other hand, being a sib is transitive. If Bob is Sue’s sib and Sue is Joe’s sib, then Bob is also Joe’s sib. So now what about our salty coffee Indifference relation?

It’s reflexive and symmetric, but what about transitive? If cup1 I cup2 and cup2 I cup3, is cup1 I cup3? Yes, probably. Who can tell the difference between 1 grain of salt and 3 grains of salt? But the thing about transitivity is that it has to hold for all pairs in the binary relation. And Indifference clearly does not. It only holds up to the point in the chain cup1 I cup2 I. . .I cup(j-1) when we reach a noticeable difference to the taster between cup1 and cupj even though there is no noticeable difference between cupj and cup(j-1). Psychologists, who study pairwise comparisons like this, call this first detectable difference between cup1 and cupj the just noticeable difference or jnd. At that point cup1 Indifferent cupj. Indifference is not fully transitive.

There is a lot of fascinating mathematics in order theory and this is just a rudimentary illustration of a type of order. The fallacy in the frog legend is that the behavior of the frog as the water is heated is strongly transitive. At some point the frog notices the difference and tries to jump out of the pot. You can’t line up a chain of indifferent pairwise comparisons and expect the two ends to also be indifferent.

This is the kind of post you get when I don’t feel like writing about flu. Or maybe you didn’t notice the difference.


  1. #1 Bob O'H
    December 8, 2009

    If Bob is the parent of Sue, and Sue is the parent of Joe, Bob isn’t the parent of Joe.

    Um, well, not necessarily…

  2. #2 The Science Pundit
    December 8, 2009

    The salty coffee problem is a variation of the Sorites Paradox. These paradoxes can get quite sticky if you approach them with preconceptions of bivalence, such as when creationists say that evolution is false because “you never see one species give birth to another species.”

  3. #3 revere
    December 8, 2009

    BobOH: LOL. Yes, you are right. I guess my world is tamer than the real world.

    The Science Pundit: Interesting point. Here’s a question. Does an order relation that is reflexive and symmetric but not transitive have a name? Weak orders are strongly transitive. This seems like a weak preference but what kind of binary relation is that?

  4. #4 mc
    December 8, 2009

    I think the sorites paradox is the right term, but that the notion of transitivity causing confusion. I would explain it in epidemiological-like terms: you’re making a binary variable (salty vs. not salty) out of a continuous variable (number of grains of salt) and you’re losing ‘information’ by dichotomizing.

  5. #5 IanW
    December 8, 2009

    I thnk you should blog about bird flu with specific reference to Coccyzus americanus and title it “One Flu Over the Cuckoo’s Nest”….

  6. #6 floormaster squeeze
    December 8, 2009

    I am a bit removed from logic, mathematics and/or set theory these days but in math at least relations that are reflexive and symmetric but not transitive are considered “dependency relations” (but I had to look that up so I am probably missing some important nuance).

    My own recollection of set orderings had me thinking that that reflexive and symmetric ordering that was not transitive was called “acyclic”. Looking “acyclic” up seems to indicate some similar specific uses but I cannot tell if this is the right term for a general case.

  7. #7 mark
    December 8, 2009

    I’m a philosopher. The Sorites Paradox has nothing in particular to do with phenomenal continua — it is a more general problem. The original version is about heaps. (Soros is Greek for “heap”.) One grain of sand does not make a heap. Plausibly, if n grains of sand do not make a heap then n+1 grains of sand don’t either. But for a suitably large n, n grains of sand do make a heap. This paradox is of central importance in the massive literature on what philosophers call “vagueness”. Although the Sorites Paradox may seem silly when one first encounters it, in fact it has no easy solution and has important ramifications in other areas of philosophy. Some theories of vagueness involve giving up bivalence, while others do not.

    A good (but challenging) paper on phenomenal continua is Delia Graff Fara’s “Phenomenal Continua and the Sorites”. A free preprint is here, and the published version is here. Graff Fara argues for the unintuitive thesis that phenomenal indiscriminability is transitive.

  8. #8 Cate
    December 8, 2009

    JNDs! Yes! A lost art of psychology — but very familiar to good athletes who tacitly hone their JND detections to great advantage. The great basketball players recalibrate for JND floor spring deviations – the rest of us are oblivious.

    The boiled frog story is alive and well in lobster cooking. I believed it until, while crewing on a sailboat in college, I caught a 22 pound lobster and tried to use a gradual temperature increase to put it to sleep before cooking. Pacifist that I am, I had nightmares for years afterward. At what point can you contritely decide to stop cooking and return a mad thrashing life-loving monster lobster to the ocean with any assurance that it could survive? We took a vote. Our prisoner seemed past survival stage. But that did not mean it had given up. We — three strong sailors — kicked our galley to smithereens while wrestling to keep a mere bug in a pot. The others said it was yummy.

    Sorry, not elegant like a Sorites Paradox.

  9. #9 Bryan Watt
    December 8, 2009

    A tiny pinch of salt improves the taste of coffee. Try it.

  10. #10 Susan Och
    December 8, 2009

    But there’s not much worse than coffee made with overly-softened water. Tried it.

  11. #11 ECaruthers
    December 9, 2009

    When testing for detection thresholds of odors, smell scientists use a dilution series with moderately large steps – usually 1, 1/2, 1/4, 1/8, …. Here the problem is to use binary tests to find about a continuous variable.

  12. #12 Paula
    December 9, 2009

    IanW–nice but did this already. When H5N1 was big news here a few years ago, if a flock of crows went overhead one might chant aloud, “One flu east and one flu west and one flu over—now-go-somewhere-else.” Cuckoo, yes?

  13. #13 Anonymous
    December 9, 2009

    (I originally wrote, “a noted amphibian biologist” until I realized I didn’t know if were any biologists who were amphibians)

    There was one in the Biology Department of Miskatonic University, but he went missing after doing some genealogical research while on sabbatical.

    And of course the frogs will jump out of the heated water, you’re supposed to use milk!

  14. #14 Alex
    December 9, 2009

    Here’s the full story on the boiling frogs thing:


    James Fallows has dozens of posts on correcting the boiling frogs meme:


  15. #15 ginger
    December 10, 2009

    This was a really fun post. I thought of it again today, when I was talking to my husband about the possibility that Australia is going to eliminate the five-cent piece. Australia eliminated its penny and twopenny some time ago, so we already round to the nearest nickel (I’m American and can’t seem to stop using that word, and the Aussies have NO idea what I’m talking about). Even though the dollar, as currency, is built around the idea of 100 pennies to the dollar, and even though prices and card transactions operate at the penny-level JND, cash transactions have a nickel-level JND.

  16. #16 mistah charley, ph.d.
    December 11, 2009

    The JND example Weinberg uses in Secrets of Consulting has to do with gradually decreasing the number of sesame seeds on a hamburger bun. The moral:”No difference and no difference and no difference…makes a difference.”

  17. #17 anon
    December 13, 2009

    I first heard about it as the infinite-students-paradox
    which they used to tell math-students when teaching
    natural induction.
    1) 1 student does easily fit into the lecture-room
    2) if n students do fit into the room, they can always
    crowd together and make place for one new student.

    the problem is that “always” really is only “almost always”

  18. #18 Alex
    February 10, 2010

    “This is the kind of post you get when I don’t feel like writing about flu. Or maybe you didn’t notice the difference.”

    And not a bad post at all. I’m in Math myself and if you like posting stuff about Math, go ahead, I like reading about it. You seem interested in Order Theory. A nice result which originated from Order Theory and Set Theory is called Zorn’s Lemma. The Axiom of Choice is another. Check them out if you’re interested.

  19. #19 revere
    February 10, 2010

    Alex: I do mathematics as part of my research and my son is a mathematician (actually now he’s a biostatistician but his PhD was in low dimensional topology). I’m not sure I’d call the Axiom of Choice a “result” (isn’t that why it’s called an Axiom?) since you have to assume it to get a lot of the results we take for granted, but all my partial orders are finite so I don’t have to worry about it (there may be a lot of people on earth but they are still a finite number). I’ve done a fair amount of math here and if you go to the sidebar and click on “antiviral models” you’ll find a 17 part series taking apart a single paper on mathematical modeling, paragraph by paragraph and equation by equation. But most of the math I do wouldn’t be of great interest here (orthomodular lattices and quantum logic and applications to epidemiology). What kind of math do you do?

  20. #20 sdf2
    February 21, 2010

    The boiling frog story actually is literally true. Check out the current wikipedia article.

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