author : John Allen Paulos

Paulos here investigates the differences, and similarities, between everyday and mathematical reasoning. His main investigation in on statistical arguments, with smaller forays into logic and complexity. He contrasts the anecdotes, stories, myths that we use in everyday life, and the mathematical counterparts that have grown and been refined from them. His main thrust is that these give us two connected but different ways of understanding the world, and that we should be alert to these differences. It is important that we don't go about:

It's not that one way is right, and one is wrong. They offer two possible ways of understanding the world.

Each has its problems, and each its advantages. For example, anecdotal reasoning can lead us to make incorrect generalisations, because the very richness of the story that makes it so compelling gives us so much information that there are accidental correlations.

The mathematical approach can help us see when these errors might occur. He illustrates this point with several standard statistical "paradoxes", none the worse for being repeated. One very good point he makes, in the case of "digging the dirt" on celebrities and others, is that we should weigh the evidence presented against the effort taken to produce that evidence:

He has lots of good little anecdotes (deliberately used to be memorable!), although occasionally they don't give enough background or explanation. (Maybe he didn't want to clutter the main text with too much detail, but that's what appendices are for.) For example, he gives an example of a paradox about the odds of poker hands with wildcards, but doesn't explain the details:

On the face of it, this looks weird. How can changing the rules change the probability of the hands? What I think is going on is that, given a set of rules, you decide which of two (or more) possible hands you have, given wildcards, and choose the "better" hand. If the better hand is three of a kind, you will choose your wildcards to make three of a kind even if they could also make two pairs, and three of a kind becomes more likely under this choice. Conversely, if two of a kind is deemed to be the better hand, then, dealt exactly the same hand, you would choose to use any wildcards to make two of a kind, which then, because of this different choice, becomes the more likely hand. (But I may be wrong, since I don't know poker!) The point, however, is that, given choices (wildcards), things become complicated and difficult to order, and that life is full of such choices, and that the choices we make can affect the value of what we have chosen.

There is another place where he doesn't explain everything that's needed. That's in the section on extensional versus intensional definitions. He explains what an extensional definition is ...

... but not what an intensional definition is. The difference is one is by membership {Alice, Bob, Eve}, and one is by property {everyone over 5'6" tall in this room}. It might so happen that two definitions happen to refer to the same members (Alice, Bob, Eve happen to be the only people of the specified height in the room at the moment), but that could change. One common step in mathematical proof to "substitute equals for equals": if I see "2+2", I can substitute "4". But Paulos points out that we cannot substitute an extensional definition for an intensional one (or vice versa), because, what if Alice leaves the room? The important point is that we use intensional definitions all the time, and can easily find ourselves in difficulties, silliness, or even tragedy, if the "equals for equals" rule is followed blindly. The point is that mathematics is being used to model the real world, and the fit isn't perfect, so we need to think about what we are doing:

Converting well-understood English statements into mathematics is non-trivial for a wide range of reasons.

He then makes a very interesting suggestion. Recently, "situational logic" has tried to take context into account. (I've been reading a bit about this lately. The Liar, for example, uses it to analyse certain logical paradoxes. I've also recently read Vicious Circles, again by Barwise, for reasons that can be found in my review. I read Once Upon a Number in order to have a break, and read something a bit different. So I was surprised partway through when Paulos says that Barwise was his thesis advisor! Fortunately I'm already inured to these kind of coincidences, so I didn't mind that, coincidentally, Paulos also has an explanation of how this sort of thing happens all the time.) Anyway, Paulos suggests that there needs to be a similar situational approach to other areas of mathematical modelling:

There is also a section on complexity. In another intriguing suggestion, Paulos speculates about the underlying mathematical cause of self-organisation in Kauffman's Random Boolean Networks:

He also gives a solution to the "problem of induction" that I hadn't come across before (the usual problem is that justifying "the future will be like the past" by using the fact that, in the past, the future has been like the past, is circular). I rather like it, since it fits in with ideas of complexity and emergence rather well:

This idea that induction works across levels is interesting. Levels are important in complexity and emergence (an emergent property usually being described at a higher level that the properties it emerges from). I'd never thought of stream-of-consciousness literature as an example of emergence:

He then goes on to describe a measure of complexity.

His speculation that follows is fascinating:

The reason it is fascinating (to me, at least!) is because of problems I have with some definitions of emergence, that require the observer to experience "surprise" at the emergent property. So, the second time the same thing happens, it's no longer emergent, then? But taking this definition into account, the second time it happens, the complexity of the observer has changed. So maybe there's something more going on here. Hmm.

He finishes off with a nice little example that not all equally probable things are equally probable (depending on how we look at them).

So, all in all, some great food for thought.