Thursday, May 29, 2008

Why we should love logarithms
[More Musement from Nature News.]

The tendency of 'uneducated' people to compress the number scale for big numbers is actually an admirable way of measuring the world.


I'd never have guessed, in the days when I used to paw through my grubby book of logarithms in maths classes, that I'd come to look back with fondness on these tables of cryptic decimals. In those days the most basic of electronic calculators was the size of a laptop and about as expensive in real terms, so books of logarithms were the quickest way to multiply large numbers (see 'What are logarithms'.

Of course, logarithms remain central to any advanced study of mathematics. But as they are no longer a practical arithmetic tool, one can’t now assume general familiarity with them. And so, countless popular science books contain potted guides to using exponential notation and interpreting logarithmic axes on graphs. Why do they need to do this? Because logarithmic scaling is the natural system for magnitudes of quantities in the sciences.

That's why a new claim that logarithmic mapping of numbers is the natural, intuitive scheme for humans rings true. Stanislas Dehaene of the Federative Institute of Research in Gif-sur-Yvette, France, and his co-workers report in Science [1] that both adults and children of an Amazonian tribe called the Mundurucu, who have had almost no exposure to the linear counting scale of the industrialized world, judge magnitudes on a logarithmic basis.

Down the line


The researchers presented their subjects with a computerized task in which they were asked to locate on a line the points that best signified the number of various stimuli (dots, sequences of tones or spoken words) in the ranges from 1 to 10 and from 10 to 100. One end of the line corresponded to 1, say, and the other to 10; where on this line should 6 sit? The results showed that the Amazonians had a clear tendency to apportion the divisions logarithmically, which means that successive numbers get progressively closer together as they get bigger.

The same behaviour has previously been seen in young children from the West [2]. But adults instead use a linear scaling, in which the distance between each number is the same irrespective of their magnitude. This could be because adults are taught that is how numbers are 'really' distributed, or it could be that some intrinsic aspect of brain development creates a greater predisposition to linear scaling as we mature. To distinguish between these possibilities, Dehaene and his colleagues tested an adult population that was 'uncontaminated' by schooling.

The implication of their finding, they say, is that "the concept of a linear number line seems to be a cultural invention that fails to develop in the absence of formal education". If this study were done in the nineteenth century (and aside from the computerized methodology, it could just as easily have been), we can feel pretty sure that it would have been accompanied by some patronizing comment about how 'primitive' people have failed to acquire the requisite mathematical sophistication.

Today's anthropology is more enlightened, and indeed Dehaene and his team have previously revealed the impressive subtlety of Mundurucu concepts of number and space, despite the culture having no words for numbers greater than five [3,4].

Everything in perspective


But in any event, the proper conclusion is surely that it is our own intuitive sense of number that is somehow awry. The notion of a decreasing distance between numbers makes perfect sense once we think about that difference in proportionate terms: 1,001 is clearly more akin to 1,000 than 2 is to 1. We can even quantify those degrees of likeness. If we space numbers along a scale such that the distances between them reflect the proportion by which they increment the previous number, then the distance of a number n from 1 is given by the harmonic series, the sum of 1 + 1/2 + 1/3 + 1/4 and so on up to 1/n. This distance is roughly proportional to the logarithm of n.

This, it is often said, is why life seems to speed up as we get older: each passing year is a smaller proportion of our whole life. In perceptual terms, the clock ticks with an ever faster beat.

But wait, you might say – surely 'real' quantities are linear? A kilometre is a kilometre whether we have travelled 1 or 100 already, and it takes us the same time to traverse at constant speed. Well, yes and no. Many creatures, execute random walks or the curious punctuated random walks called Lévy flights [watch out for next week's issue of Nature on this...], in which migrations over a fixed increment in distance takes an ever longer time. Besides, we can usually assume that an animal capable of covering 100 kilometres could manage 101, but not necessarily that one capable of 1 kilometre could manage 2 kilometres (try it with a young child).

Yet the logarithmic character of nature goes deeper than that. For scientists, just about all magnitude scales are most meaningful when expressed logarithmically, a fact memorably demonstrated in the vision of the Universe depicted in the celebrated 1977 film Powers of Ten The femtometre (10**-15 metres) is the scale of the atomic nucleus, the nanometre (10**-9 metres) that of molecular systems, the micrometre (10**-6 metres) the scale of the living cell, and so on. Cosmological eras demand logarithmically fine time divisions as we move closer back towards the Big Bang. The immense variation in the size of earthquakes is tamed by the logarithmic magnitude scale, in which (roughly speaking) an increase of one degree of magnitude corresponds to a tenfold increase in energy. The same is true of the decibel scale for sound intensity, and the pH scale of acidity.

Law of the land

Indeed, the relationship between earthquake magnitude and frequency is one of the best known of the ubiquitous natural power laws, in which some quantity is proportional to the n th power of another. These relationships are best depicted with logarithmic scaling: on logarithmic axes, they look linear. Power laws have been discovered not only for landslides and solar flares but for many aspects of human culture: word-use frequency, say, or size-frequency relationships of wars, towns and website connections.

All these things could be understood much more readily if we could continue to use the logarithmic number scaling with which we are apparently endowed intuitively. So why do we devote so much energy to replacing it with linear scaling?

Linearity betrays an obsession with precision. That might incline us to expect an origin in engineering or surveying, but actually it isn't clear that this is true. The greater the number of units in a structure's dimension, the less that small errors matter: a temple intended to be 100 cubits long could probably accommodate 101 cubits, and in fact often did, because early surveying methods were far from perfect. And in any event, such dimensions were often determined by relative proportions rather than by absolute numbers. It seems more conceivable that a linear mentality stemmed from trade: if you're paying for 100 sheep, you don't want to be given 99, and the seller wants to make sure he doesn't give you 101. And if traders want to balance their books, these exact numbers matter.

Yet logarithmic thinking doesn't go away entirely. Dehaene and his colleagues show that it remains even in Westerners for very large numbers, and it is implicit in the skill of numerical approximation. Counting that uses a base system, such as our base 10, also demands a kind of logarithmic terminology: you need a new word or symbol only for successive powers of ten (as found both in ancient Egypt and China).

All in all, there are good arguments why an ability to think logarithmically is valuable. Does a conventional education perhaps suppress it more than it should?


References

1. Dehaene, S. , Izard, V. , Spelke, E. & Pica, P. Science 320, 1217–1220 (2008).
2. Booth, J. L. & Siegler, R. S. Dev. Psychol. 42, 189–201 (2006).
3. Pica, P. , Lemer, C. , Izard, V. & Dehaene, S. Science 306, 499–503 (2004).
4. Dehaene, S. , Izard, V. , Pica, P. & Spelke, E. Science 311, 381–384 (2006).

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