I’m currently reading The Undercover Economist by Tim Harford, presenter of Radio 4 maths show More or Less. It’s very good, but one thing is stopping me from giving it an unqualified recommendation: it’s full of passages like this:
[T]he government spends three hundred dollars per person (five times less than the British government and seven times less than the American government)
Because of its lousy education system, Cameroon is perhaps twice as poor as it could be.
The poorest tenth of the population spends almost seven times less on fuel than the richest tenth, as a percentage of their much smaller income.
In case you don’t see what I’m getting at, my problem is with the construction “$n$ times smaller/less/poorer”. It is, I think, in relatively common usage, but its multiple appearances in this particular book are what annoyed me into writing this post. In a moment I’m going to attempt to justify my dislike of it, but my objection isn’t directly a consequence of this justification: the phrasing instinctively sounds jarring and awful to me. I’d be interested to know whether other people feel the same way or if those passages of text were completely unremarkable for everyone else, and whether people’s reaction correlates in any way with their level of maths proficiency.
Now, on to some post hoc rationalisation of my strange linguistic prejudices. Suppose I have £2000 and you have £6000. I could say “You are three times richer than me” and I’m sure nobody would argue with me (least of all you). Tim Harford might say “Paul is three times poorer than you”. Now, my statement is simple. Your money is literally my money, three times. If you had another £2000 (my riches another time), you would be four times richer than me. Another £2000, another time richer. Tim’s statement about poorness is not so clear cut. There’s a cosmetic logic to it: poorness is the opposite of richness so if you’re three times richer than me then it stands to reason I should be three times poorer than you. But let’s look a little more closely. What if I wish to become another time poorer than you? I have to lose £500, so that I have £1500: one quarter of your fortune; I am four times poorer than you. However, if I wish now to be five times poorer than you I have to lose just £300 to get me down to £1200, a fifth of your money. The concept of a ‘time’ here has no meaning. The problem is of course that “times” means multiplication but Tim is talking about division.
Of course, the pedantic mathematician will interject here, division is just multiplying by an inverse. Quite right. Everything is solved if we discuss poorness in inverse pounds. So I am poor to the tune of £⁻¹(1/2000) = £⁻¹0.0005 and you are considerably less poor, with a mere £⁻¹0.00016667. The number representing my poorness is now three times the number representing your poorness, and every time I wish to become another time poorer than you I just need to add your poorness on to my poorness: £⁻¹0.00016667 + £⁻¹0.0005 = £⁻¹0.0006667, which in richness is indeed £1500. This way of thinking about things is the only way to deal with poorness in a manner consistent with Tim’s hypothetical statement “Paul is three times poorer than you”, or indeed with his actual statements in the book.
Plainly this is a ridiculous state of affairs. Nobody, not even Tim Harford, thinks in inverse pounds, and this I think has to do with why the passages quoted above sound so strange to me.
Incidentally, I wondered while writing this whether anyone else has expressed similar annoyance about this topic. It’s obviously a tricky thing to search for but I decided that Googling the phrase “three times smaller” might help. The first result was a post complaining about the same thing but for slightly different reasons at the rather good blog TYWKIWDBI. The only comment was from sometime Aperiodical blogger and all-the-time me brother Andrew Taylor saying many of the things I am saying here (including the good point that there are some inverse pairs in science that do make perfectly good sense such as resistance and conductance). This could of course be a massive coincidence, or Google could have served me that page first because its sinister Google-brain knows Andrew is my brother (though the same search in incognito mode delivered the same result). But very possibly I read that post and comment two years ago (I have no recollection of doing but that proves little) in which case I must thank the author and Andrew for alerting me to this and thereby spoiling for me a perfectly good book.
In a further coincidence, just before this post went to press the excellent Guardian Style Guide has waded into the debate on Twitter, in response to a query on the same subject. Thankfully for my self-respect, they come down on my side:
distinctly queasy. RT @k_w_a_k: @guardianstyle Ok, how do you feel about “three times less” as a comparison?
— Guardian style guide (@guardianstyle) July 11, 2012