by ‘Cauchy logic’ in economics What is 0.999 …, really? Is it 1? Or is it some number infinitesimally less than 1? The right answer is to unmask the question. 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.
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‘Cauchy logic’ in economics
What is 0.999 …, really? Is it 1? Or is it some number infinitesimally less than 1?
The right answer is to unmask the question. 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 ponder even for economists all to fond of uncritically following the mathematical way when applying their mathematical 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 …
Accepting Haavelmo’s domain of probability theory and sample space of infinite populations – just as Fisher’s “hypothetical infinite population, of which the actual data are regarded as constituting a random sample”, von Mises’s “collective” or Gibbs’s ”ensemble” – also implies that judgments are made on the basis of observations that are actually never made!
Infinitely repeated trials or samplings never take place in the real world. So that cannot be a sound inductive basis for a science with aspirations of explaining real-world socio-economic processes, structures or events. It’s — just as the Cauchy mathematical logic of ‘defining’ away problems — not tenable.
In social sciences — including economics — it’s always wise to ponder C. S. Peirce’s remark that universes are not as common as peanuts …