Home > Uncategorized > Is 0.999 … = 1? (wonkish)

Is 0.999 … = 1? (wonkish)

from Lars Syll

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 too 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 …

1. March 3, 2016 at 8:14 pm

Numbers are a human invention. When you say 1 and 0 existe, it is as usefull numbers, but you d’ont say any thing else that you limit the space in two areas, or more. But space is continuous. If you want to separate it, you will find that is impossible a the Plank analysis.

2. March 3, 2016 at 8:30 pm

You are right. That is to say that real world with mathematics language cannot be used for economics as Keynes said. Only probabilities are usefull. That is why in my book I warned people that the economics models that I have found are not useful for predictive solutions. One and zero, yes, but you have to be carefull for descriptive economics.

3. March 3, 2016 at 10:02 pm

A bit more recently than Cauchy (in fact, around 1966, in the early days of trying to teach mathematics to stupid computers which hadn’t learnt to jump from premises to conclusions but required every step to be spelled out), it was realised

a. What we read as numbers are numerical notations of different types and forms, for which different forms of processing methods are required. We use not only numerals but count our fingers (digits), use positional notation on an abacus, power notations and logarithms to shorten the large numbers encountered in science etc; we use decimal notation in human contexts and binary notation for computers etc. Mathematicians [if not all users of maths mathematics, remembering geometry preceded continuous measures] distinguish natural counting), integer (counting including negatives), rational (integers dividing without remainder), real (continuous) and complex (two-dimensional) number types, along with circular and exponential as well as unitary bases and randomised code numbers. So-called reals represent measures of time or spatial distance and, like the simpler forms, are special cases of complex number form, from which a four-dimensional (quaternion) representation of the spact-time continuum can be constructed

b. What we read as numbers in mathematical models, in reality refer to some aspect of an object or process. Thus there is no such thing as a price, it is always a price ‘of’ something. Abstracting the reality in a monetary ‘bottom line’ is like taking logarithms without reversing the process, i.e. presenting a logarithm rather than its anti-logarithm as the result being sought.

c. What we read as numbers in algorithmic computations are very often index numbers, referring not just to number form or the numeric form of a variable but to specific variables in a list of variables of the same type or (in list processing terms) the address of the next variable in an ordered list. In other words, there are four possible levels of reference, not just references to variables and their values but also references to processing and memory.

4. March 4, 2016 at 8:46 am

First of all, 0.999… is just another expression for 1. Just calculate 1 – 0.999…, what do you get? But that has nothing to do with economics.

The problem of Neoclassical economics is not mathematics. Neoclassical economics has bad, really really bad assumptions. Bad means those assumptions are clearly not given in the real world. The assumptions are no generalizations or approximations but are absurd fictions. They are anti-axioms. That is it already. Bad assumptions means bad models means irrelevant at best.

Not that it matters but additional to that neoclassical “models” are inconsistent disparate scraps. They do not work together. They do not form a complete theory that could represent an economy. They can be best described as “not even wrong” (https://en.wikipedia.org/wiki/Not_even_wrong).

5. March 4, 2016 at 9:01 am

Louis said: “the economics models that I have found are not useful for predictive solutions”.

Having slept on this and Pavlov’s comment on non-exact settlement of accounts, I woke up dreaming of railway time-tables! I suspect Louis and most economists would not think of these as in mathematical language, yet dates and times are a numerical form with their own complex processing. Such time-tables, however, are not descriptive but predictive. They make possible the synchronisation of events, prioritisation in the event of conflict and give and take in compensating for errors and failures which are of the essence of economic efficiency and reliability, i.e. our making predictions as nearly as possible true.

• March 4, 2016 at 3:37 pm

In the 1980s, I consulted a Brit Rail time table, for the last evening train from Claygate to Waterloo, London. I waited for a long time, but it never came. I had to take a taxi to London because BR had cancelled the service without notifying he public. So much for predictability.

• March 4, 2016 at 11:38 pm

LOL ! I hope you persuaded BR to pay for your taxi! Seriously, I was trying to say preductability doesn’t just happen, it has to made to happen. In a network of events all as liable to such failings as the one you experienced, the big issues are not knowing one is failing and the need to avoid knock-on effects. hat’s where time-tabling and mathematical techniques like critical path analysis come into their own.

6. March 4, 2016 at 2:26 pm

“Davetaylor” said : the essence of economic efficiency and reliability, i.e. our making predictions as nearly as possible true”. Yes you are right but only for the short time. That is why economics are always working on cycles, bringing errors and fixing them (we call them crisises), floating as a cork in the sea. That why I said that predictictions are not easing. That is why I prefer 3 predictive models, low, middle or high. Nobody can say which is the good one.