Most people are not fascinated by mathematics. It is generally viewed as either a pedantic, mundane business made obsolete by calculators, or an esoteric, inscrutable pursuit relegated to a few hardy souls with a head for that sort of thing. However, there are some surprising, even shocking, results which are not really all that esoteric and which may have a profound impact on the very concept of number itself.
Enter Frank Benford, an industrial physicist for General Electric in the 1930's. Back then they did not have computers or even calculators, and probably had to walk twenty miles to and from work every day, in the snow, uphill both ways. So when somebody like Dr. Benford had to multiply two really big numbers like 143 and 2697 they had two choices: sit down with a pencil and paper for about half an hour, or look up the result in a giant book of logarithmic tables for an answer that somebody else already found by sitting down with a pencil and paper for about half an hour. Being a physicist, Frank Benford had to do this quite often and after a while he began to notice something rather strange. The book was laid out in numeric order, which was not so strange, with the numbers starting with 1 at beginning all the way down to the 9's at the end of the book. What Dr. Benford noticed was that the first parts of the book were dirtier, more creased, and generally more worn out than the last parts of the book. The greater wear and tear on the first half of the book indicated that he was looking up numbers starting with 1, 2, and 3 a lot more than those that start with 7, 8,, or 9. Being a physicist, Dr. Benford did not just shrug and think to himself "oh well, that's an interesting coincidence". Being a physicist, he decided there must be something bigger at work here, some strange, interesting reason why he would be using numbers that start with 1 a lot more than numbers that start with 9. And it is strange if you think about it, there are 9 numbers from 1 to 9 (duh) so you would expect that each number would show up one ninth of the time, or about 11.1%.
So Dr Benford started collecting numbers...lots of numbers. He compiled tens of thousands of statistics and found that no matter what the number described, no matter what units the numbers were in, be it feet, meters, degrees or whatever, they all displayed a certain pattern. You guessed it, the 1's, 2’s and 3's occurred as the first digit a lot more than 7's 8's and 9's. Is your mind blown yet? Why would the lengths of rivers, heights of buildings, populations of towns, numbers on tax returns, or even specific heats of materials or molecular weights of compounds follow this pattern? Guess what? It's not a coincidence. It is now a proven mathematical law.
As sure as 1+1=2, numbers starting with a 1 are a lot more common than numbers starting with 9.
That is a very surprising result indeed! So why is that? Why is it a mathematical truth that collections of “random” numbers begin with 1’s more often than 2’s and 2’s more often than 3’s and so on? Is there some sort of Universal conspiracy?
Not really. Benford's law is a little creepy, but it only works for numbers distributed over multiple orders of magnitude. For example, if "town" is defined as settlements of population greater than 500 and less than 1000, then the list of town populations will not follow Benford's law. This is actually a clue as to why numbers are distributed in such a fashion. It's not some spooky conspiracy, it arises from the fact that logarithms of numbers are distributed evenly. Common sense assumes that the first digit of a number in a collection has an equal chance of being a 1 through 9. What actually happens is that the logarithms of numbers are evenly distributed in such a fashion. That means that a number is just as likely to be between 100 and 1000 (logarithm 2 and 3) as between 10,000 and 100,000 (logarithm 4 and 5).
Perhaps it is strange, foreign even, that a concept as abstract as logarithms has such a profound effect on the numbers we use every day. It just so happens that logarithms are not really so abstract after all and the "numbers we use every day" are in fact much more abstract constructions than most people would ever know.
To investigate this statement a little further we shall turn to the work of Stanislas Dahaene, one of Europe’s leading cognitive neuroscientists. He wrote a book called “The Number Sense” in which he outlined his work on how babies think about numbers. Many people would assume that babies are a blank slate and are taught numbers and counting by relating the word for each number with a quantity. However, babies are actually born with an innate sense of number and quantity, it's just not the sense that we're used to thinking in. Babies, in fact, think logarithmically, not linearly. It is only by thorough teaching of "how to count" that they learn to think numerically in terms that we are all familiar with as adults.
One might ask then, what happens if a baby is never taught "how to count"? To answer this question we turn to the Piraha. They're one of the few indigineous, hunter-gatherer tribes left in the Amazon region of South America and they all think about numbers in the same way as babies. An example of this is as follows; what number lies between 1 and 9? The answer, of course, is five. However, a Piraha would answer 3. That is because linearly, in terms of 1,2,3,4,5,6,7,8,9, the number in the middle is 5. But in logarithmic terms, which is really just in terms of ratios, the answer is 3 because 3 is the triple of 1 and the triple of 3 is 9. So 1 is to 3 as 3 is to 9, that's why 3 is exactly between 1 and 9.
Such reasoning might seem much more complex than simply counting from 1 to 9 but, believe it or not, that is how the human brain is wired. We actually have to rewire our brain to think about numbers in the "more simple" linear way.
Hopefully this titillating tidbit of trivia has sparked some reflection on the nature of number and the hidden consequences of imposing abstract ideas such as "one" and "two" etc. on quantities of objects in the real world.
