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Note: You are looking at a static copy of the former PineWiki site, used for class notes by James Aspnes from 2003 to 2012. Many mathematical formulas are broken, and there are likely to be other bugs as well. These will most likely not be fixed. You may be able to find more up-to-date versions of some of these notes at http://www.cs.yale.edu/homes/aspnes/#classes.

1. So why do I need to learn all this nasty mathematics?

Why you should know about mathematics, if you are interested in ComputerScience: or, more specifically, why you should take CS202 or a comparable course:

So in order to understand computations, we need a language that allows us to reason about things we can't see and can't touch, that are too big for us to understand, but that nonetheless follow strict, simple, well-defined rules. We'd like our reasoning to be consistent: any two people using the language should (barring errors) obtain the same conclusions from the same information. Computer scientists are good at inventing languages, so we could invent a new one for this particular purpose, but we don't have to: the exact same problem has been vexing philosophers, theologians, and mathematicians for much longer than computers have been around, and they've had a lot of time to think about how to make such a language work. Philosophers and theologians are still working on the consistency part, but mathematicians (mostly) got it in the early 20th-century. Because the first virtue of a computer scientist is laziness, we are going to steal their code.

2. But isn't math hard?

Yes and no. The human brain is not really designed to do formal mathematical reasoning, which is why most mathematics was invented in the last few centuries and why even apparently simple things like learning how to count or add require years of training, usually done at an early age so the pain will be forgotten later. But mathematical reasoning is very close to legal reasoning, which we do seem to be unusually good at. There is very little structural difference between the sentences


but because the first is about boring numbers and the second is about fascinating social relationships and rules, most people have a much easier time deducing that to show somebody is royal we need to start with some known royal and follow a chain of descendants than they have deducing that to show that some number is in the set S. we need to start with some known element of S and show that repeatedly adding 1 gets us to the number we want. And yet to a logician these are the same processes of reasoning.

So why is statement (1) trickier to think about than statement (2)? Part of the difference is familiarity—we are all taught from an early age what it means to be somebody's child, to take on a particular social role, etc. For mathematical concepts, this familiarity comes with exposure and practice, just as with learning any other language. But part of the difference is that we humans are wired to understand and appreciate social and legal rules: we are very good at figuring out the implications of a (hypothetical) rule that says that any contract to sell a good to a consumer for $100 or more can be cancelled by the consumer within 72 hours of signing it provided the good has not yet been delivered, but we are not so good at figuring out the implications of a rule that says that a number is composite if and only if it is the product of two integer factors neither of which is 1. It's a lot easier to imagine having to cancel a contract to buy swampland in Florida that you signed last night while drunk than having to prove that 82 is composite. But again: there is nothing more natural about contracts than about numbers, and if anything the conditions for our contract to be breakable are more complicated than the conditions for a number to be composite.

3. Thinking about math with your heart

There are two things you need to be able to do to get good at mathematics (the creative kind that involves writing proofs, not the mechanical kind that involves grinding out answers according to formulas). One of them is to learn the language: to attain what mathematicians call mathematical maturity. You'll do that in CS202, if you pay attention. But the other is to learn how to activate the parts of your brain that are good at mathematical-style reasoning when you do math—the parts evolved to detect when the other primates in your primitive band of hunter-gatherers are cheating.

To do this it helps to get a little angry, and imagine that finishing a proof or unraveling a definition is the only thing that will stop your worst enemy from taking some valuable prize that you deserve. (If you don't have a worst enemy, there is always the UniversalQuantifier.) But whatever motivation you choose, you need to be fully engaged in what you are doing. Your brain is smart enough to know when you don't care about something, and if you don't believe that thinking about math is important, it will think about something else.


2014-06-17 11:58