About Precision

This is a prescriptive handout: for more on precision in general, consult the webpage at http://www.math.usf.edu/~mccolm/pedagogy/PRlong.html.

Precision is difficult, both in understanding situations and in solving problems. It is not difficult to get or communicate the main idea of something, but it is quite a different matter to deal with details. Precision is like clockwork: all the gears must mesh if the clock is to work, much less keep accurate time. Precision is necessary if something is to actually succeed, especially in mathematics, science, and engineering.

If all you want is the main idea, then you can fudge a bit. But if you need to get it right, you need to be careful (as in full of care):

  • Break the problem down into small steps. Smaller steps are easier to deal in isolation. But keep the overall architecture in mind (or write it out as an outline or a diagram).
  • Don't skim past steps: any skipped or skimped step can lead to sunk ships. (Remember that the Titanic sank because of bad rivets.) If you cannot get a step, try to isolate it, and ask about it.
This is the reductionist programme: reduce a problem to its components, deal with the components, and then, having dealt with the components, deal with the problem. This is time-consuming, but it is the most effective method of dealing with problems requiring precision.

Precision is most obvious in mathematics, where each step in a computation requires justification. It can lead to confusion: because we are more comfortable with holistic (and imprecise) notions, precision leads to complexities that we find difficult. In mathematics, science, engineering, law, philosophy, finance, etc., people who are not used to differences between things can find precise terminology (i.e., jargon) confusing: but follow the reductionist programme (look at the words, assemble the words into phrases, phrases into sentences, keeping in mind the architecture of the problem) and, with some elbow grease, one can figure out what is going on.

Precision is often mistaken for obfuscation (and jargon is often used to confuse or deceive). But the real reason for precision is because those who are used to it find it more convenient to communicate in precise terms than to re-explain everything they do every time they do it. So the best thing to do is to master the jargon.

One final point: we often rely on metaphors and analogies for thought. But these illuminations are usually imprecise: when thinking precisely, we have precise examples of more abstract situations. This makes the abstractions more difficult to deal with. It helps if one can think in terms of the examples (e.g., think of a particular triangle when thinking of triangles), but be aware that the example one is thinking of is only an example, and in working out the problem, the steps have to work for any example, not just the one in mind.

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