How not to be wrong2 December, 2016 at 18:21 | Posted in Statistics & Econometrics | 1 Comment
What is 0.999 …, really?
It appears to refer to a kind of sum:
.9 + + 0.09 + 0.009 + 0.0009 + …
But what does that mean? That pesky ellipsis is the real problem. There can be no controversy about what it means to add up two, or three, or a hundred numbers. But infinitely many? That’s a different story. In the real world, you can never have infinitely many heaps. What’s the numerical value of an infinite sum? It doesn’t have one — until we give it one. That was the great innovation of Augustin-Louis Cauchy, who introduced the notion of limit into calculus in the 1820s.
The British number theorist G. H. Hardy … explains it best: “It is broadly true to say that mathematicians before Cauchy asked not, ‘How shall we define 1 – 1 – 1 + 1 – 1 …’ but ‘What is 1 -1 + 1 – 1 + …?'”
No matter how tight a cordon we draw around the number 1, the sum will eventually, after some finite number of steps, penetrate it, and never leave. Under those circumstances, Cauchy said, we should simply define the value of the infinite sum to be 1.
I have no problem with solving problems in mathematics by defining them away. But how about the real world? Maybe that ought to be a question to consider even for economists all to fond of uncritically following the mathematical way when applying their models to the real world, where indeed ‘you can never have infinitely many heaps.’
In econometrics we often run into the ‘Cauchy logic’ — the data is treated as if it were from a larger population, a ‘superpopulation’ where repeated realizations of the data are imagined. Just imagine there could be more worlds than the one we live in and the problem is fixed …