The joke, the Response

Two plus two equals WHAT?

The Joke:
A mathematician, an accountant and an economist apply for the same job. The interviewer calls in the mathematician and asks "What do two plus two equal?" The mathematician replies "Four." The interviewer asks "Four, exactly?" The mathematician looks at the interviewer incredulously and says "Yes, four, exactly."

Then the interviewer calls in the accountant and asks the same question: "What do two plus two equal?" The accountant says, "On average, four -- give or take 10%, but on average, four."

Then the interviewer calls in the economist and poses the same question: "What do two plus two equal?" The economist gets up, locks the door, closes the shade, sits down next to the interviewer, and says,"What do you want it to equal?"


The Response:
I enjoyed the joke, but I couldn't help making a few minor observations.

1) Isn't it usually the ACCOUNTANT who asks, "What do you want it to be?" since they are best known for their "creative" arithmetic?

2) No mathematician I know would continue to give the answer "four" when pushed. Engineers are the same. Of course, as you are well aware, two plus two equals five (for large values of two) or even three (for small values of two). "Two," given as it is to just one significant figure (it was described as neither exact nor integral), can take any value in the range [1.5, 2.5] where the right hand limit is open. The mathematician in the joke should have picked this up when challenged with the word "Exactly?" after the initial response.

3) The tolerances quoted by the accountant in the story are now transparently incorrect. Given the error bounds of the original values, two plus two can therefore lie anywhere in the range[3.0, 5], which for the median value of "4" actually represents error tolerances of ±25% not ±10%.The use of the word "on average," however, is justified by reference to the Central Limit Theorem, which states that the mean/median of all the sample means/medians drawn from a given population interval is equivalent to the mean/median of that population interval, given the limiting case of all possible samples. The population has not been clearly defined but can be inferred from the given data without loss of generality. Perhaps if the accountant had paid a little more attention in his Statistics and Real Analysis classes at College, he wouldn't have been quite so naive in his unqualified response.

4) The economist had presumably read "Principia Mathematica" by Bertrand Russell and Alfred North Whitehead, which proved conclusively on page 362, that given sufficiently rigorous definitions of unity, addition and equality, 1 + 1 = 2. I presume, therefore, that the economist was clarifying the question with the interviewer so that the repeated "two" in the question could be identified with the above - citedRussell/Whitehead definition of "2," and hence carry with it the proven equivalence with "1 + 1. "Given further the identification of "plus" with the Russell/Whitehead "+," it would then be reasonable to equate "two plus two" with "2 + 2" and hence obtain the answer "1 + 1 + 1 + 1." Since this construct has yet to be formally defined as "4", unless implicitly assuming John Conway's Construct of the Natural Numbers (a set theory argument), it is right to question the interviewer as carefully as he did.

5) I personally made the answer 100. But then I was working in binary.

Warmest regards always,
Stephen Froggatt
Head of Mathematics
Oaks Park High School
Ilford, UK