A mathematician, a physicist, and an engineer are riding a train through Scotland.

The engineer looks out the window, sees a black sheep, and exclaims, “Hey! The sheep in Scotland are black!”

The physicist looks out the window and corrects the engineer, “Strictly speaking, all we know is that there’s at least one black sheep in Scotland.”

The mathematician looks out the window and corrects the physicist, ” Strictly speaking, all we know is that is that at least one side of one sheep is black in Scotland.”

Why should you care about rigor? Two reasons:

- There are many people who find rigorous arguments more convincing. If you learn to construct rigorous-looking arguments you will be able to more effectively convince such people.
- Groups of people who share conventions of rigor have more effective ways of disagreeing and arriving at truth through discussion.

So, what is rigor? wikipedia:

An attempted short definition of

intellectual rigourmight be that no suspicion of double standard be allowed: uniform principles should be applied. This is a test of consistency, over cases, and to individuals or institutions (including the speaker, the speaker’s country and so on).

This means that arguments should be explicit about when and where they are valid. In the opening, the mathematician, physicist and engineer arrived at different conclusions because they started from different premises. The engineer had assumed that the observed sheep was representative of all sheep in Scotland, and the physicist had assumed that each sheep can have only one color.

Explicit assumptions are useful because they allow the quality of your logic to be evaluated separately from the soundness of your premises / assumptions. It allows people to divide up the task of understanding the world, because then ideas can be connected to each other. For example, If I prove that “A implies B” and you prove that “B implies C”, then if from experiments we find that “A is true”, we can infer that C is true. If, however, we find that the “A implies B” proof is wrong, we can still hang on to the “B implies C” piece until we find some other way of proving or measuring “B is true”.

Despite this great benefit, however, people do not usually make all assumptions explicit – there are simply too many assumptions to exhaust explicitly. This being the case, the communication of rigorous arguments requires an understanding exist between speaker and listener, in the form of a body of common assumptions. In the presence of such a convention, only assumptions new or contrary to the convention need to be stated explicitly.

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A rigorous argument is composed of three types of statements – premises, logic and conclusions. When conclusions are wrong, either the premises or the logic is wrong. If conclusions are observed to be wrong and yet both logic and premises are correct, people get thrown into a panic. This is because it means that one of the unstated, implicit assumptions must be wrong. Erroneous implicit assumptions are troublesome because you don’t know where else they could pop up.

The ideal rigorous argument only has explicit assumptions. This is an unattainable ideal, because in reality it is impossible to state all assumptions. The ways in which this shortcoming is overcome are complex indeed. wikipedia:

A variety of 20th century movements, including liberalism and neo-classicism, traced their intellectual heritage back to the Enlightenment, and away from the purported emotionalism of the 19th century. Geometric order, rigor and reductionism were seen as Enlightenment virtues. The modern movement points to reductionism and rationality as crucial aspects of Enlightenment thinking, of which it is the heir, as opposed to irrationality and emotionalism. In this view, the Enlightenment represents the basis for modern ideas of liberalism against superstition and intolerance.

Reductionism, liberalism and tolerance of diversity all help to manage the body of implicit assumptions. It is ultimately these attitudes that make the ideal of rigor practical as a guiding principle in reality.