Necessary and Sufficient

Problem: On a weekend 2 young mountain climbers decided to climb and then descend a local mountain peak. Together they started off on Saturday morning and reached the peak Saturday night. They camped overnight on the summit and promptly started down the mountain on Sunday morning reaching their starting point Sunday afternoon. Prove that at some time of day they were at the exact same altitude on Saturday and Sunday. The two climbers always stay together.

Solutiuon: This isn't too difficult when you think it through. For any time t, let f(t) be the altitude of the climbers at time t on Sunday minus their altitude at the same time on Saturday. Now, as they climbers start their ascent on Saturday morning f(t) must be positive however, on Saturday night f(t) must be negative. Thus, by virtue of the intermediate value theorem, f(t) must be 0 at some point meaning that the climbers were at the same altitude on both Saturday and Sunday at some point in time. QED

Finance, business, and indeed many fields share commonality with mathematics in that there is a need to separate fact from fiction. However, there is much formal terminology that is subtly but significantly different in meaning. What follows is a small sample of the formal terms often used by academics and well-educated practitioners:

Axiom - an axiom or postulate is a proposition that is not proved or demonstrated but taken for granted. Therefore, axioms often serve as a starting point for deducing and inferring other truths.

Theorem - In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms.

Conjecture - In mathematics, a conjecture is a proposition that is unproven but appears correct and has not been disproved.

Lemma - In mathematics, a lemma is a proven proposition which is used as a stepping stone to a larger result rather than as a statement in-and-of itself. There is no formal distinction between a lemma and a theorem, only one of usage and convention.

Corollary - A corollary is a statement of fact that follows readily from a previous statement, usually a theorem. The use of the term corollary, rather than proposition or theorem, is intrinsically subjective.

Axiomatization - the act or process of reducing something to a system of axioms. This happens a lot in quantitative finance but not everyone approves of this.

Hypothesis - A hypothesis is a proposed explanation, that can be tested, for an observable phenomenon. More specifically, in the mathematical sense statistical hypothesis testing is used to evaluate the hypothesis. Thus, the test is a method of making decisions about the certainty of the stated hypotheses using experimental data. In statistical hypothesis testing a result is called statistically significant if it is unlikely to have occurred by chance.

Epistemic Circularity - An epistemically circular argument defends the reliability of a source of belief by relying on premises that are themselves based on the source. You wont sound convincing if you put forward epistemically circular arguments! NB: this is used in philosophy more so than mathematics (hence the photo of Rene Descartes) but non-the-less a useful term to know.

Legendary investment sage, Warren Buffer, was speaking to a journalist about the financial crisis of 2008 when he came out with a fairly accurate summary of how good ideas go bad. Buffet said... there's a "natural progression" to how good new ideas go wrong. He called this progression the "three I's." First come the innovators, who see opportunities that others don't. Then come the imitators, who copy what the innovators have done. And then come the idiots, whose avarice undoes the very innovations they are trying to use to get rich.

I reckon that cycle applies to many new ideas not just those dreamed up by traders at investment banks.

You can read more on the Harvard Business Review website.