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.