One of the things I don’t think I did well in my post on A Mathematician’s Lament is to properly state that I disagree with Paul Lockhart that Mathematics is inherently useless, but I understand why he takes that position. Mathematics is one of the most useful tools that mankind has ever invented. But to make that statement to someone who doesn’t know mathematics (such as, for example, a grade student), will be met with the question “But what is it good for?”

Note that this question gets raised about computer science as well- take, for example, some of the responses I’ve gotten from this post. So I’d like to tackle this question head on, and hopefully provide my answer to this eternal question.

The problem with this question is that it is making false assumptions- not unlike the classic “have you stopped beating your wife yet?” These false assumptions are:

1. There are a finite (small) number of uses of mathematics, and that the knowledge is only usefully in these cases,


2. that these uses are easily enumerable and understandable to the non-knowledgeable, both as to the utility of the uses and the applicability of the mathematics to the problem, and


3. The problem can not be easily avoided.

All three of these assumptions are false. The limits of the applicability of math (and computer science) are mainly the limitations of our imagination. All the best mathematicians and programmers I know are constantly playing a little game in their heads- asking themselves how <insert random bit of math or CS here> might apply to <insert problem they’re working on here>. Many times the answer, after a few seconds or a minute or two considering the problem, is that it’s not really applicable. Sometimes, however, the game provides a crucial insight.

Nor can the utility of a piece of mathematical or computer science knowledge to a given problem be judged before it is known, for obvious reasons. The utility of the knowledge is in the application. The details can be vague- but the general idea has to be grasped before applicability can be judged in even the crudest sense. Applying mathematics or computer science to a problem is often an indirect process, where the generic idea needs to be mapped to a specific problem instance. This process is not unlike an analogy, or poetry.

If the idea has limited applicability, than the understanding of the idea is not necessary if you avoid those situations. If the utility of the idea is limited simply by your imagination, then sooner or later you’re going to be facing a problem that you can apply the math or CS to.

I considered supplying examples of this process- which is a popular response by teachers when asked this question. But upon reflection, I realized this defeats the purpose. Supply your own example! Pick a random piece of math or computer science you know, and as your day goes by, look for things you might apply it to. Ask yourself how you might apply that skill to the problem currently facing you- and what it might tell you about the problem. The examples you will find will be more relevant to you than any I might apply.

All the math and CS do is supply the solutions- you supply the problems.

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