# “Mathematical Induction”

I realize this question borders on not qualifying as answerable or mathematical enough, but I would suspect it relevant somehow. I'll remove it if it's not. If you look at some explanations of mathematical induction you can find authors first choose to point out that mathematical induction isn't inductive, in the sense of inductive reasoning, which indicates the term as initially confusing. Would it work out better to rename "mathematical induction" as "mathematical recursion" or have I missed some subtly of how the term "recursion" usually gets used?

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Because that's the way it is. Just like linear programming isn't programming. And a ring isn't round. And ... – marty cohen Aug 31 '11 at 1:52
@Marty I'm aware of that. – Doug Spoonwood Aug 31 '11 at 2:01
@marty: OT, but since you brought it up: LP is indeed "programming", in the sense of the word used by the military brass who funded RAND (where Dantzig was at the time). See this for instance. It just happens that the sense of the word we have nowadays isn't the same as what those uniformed dudes had... – J. M. Aug 31 '11 at 3:09

## 3 Answers

A distinction is often made between (mathematical) induction and recursion, according to which the former is a proof technique, while the latter is a method of defining and constructing mathematical objects.

But even ignoring that distinction, which not everyone makes, the name mathematical induction is far too well established to be worth trying to change. First, it does no real harm: students should learn early on that the meaning of ordinary words used as technical terms can’t reliably be predicted from the everyday senses of the words. (Normal, anyone?) And any attempt to change it would almost certainly be futile anyway.

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Lol, the word "normal" is such a clever example. Normal objects in mathematics are soooo often not "normal" in the usual sense... – Patrick Da Silva Aug 31 '11 at 14:29
I agree with you here. I have no illusions that all that many people would ever change using the term, and I'm not saying they should. I only really imagine saying "mathematical recursion" when explaining the concept initially, or when trying to clear up that mathematical induction is NOT inductive. The problem here isn't though a difference between say "series" in "world series" and "series" in "infinite series", but rather a difference between "induction" in logic, and "induction" in mathematics... which isn't as easy a distinction to make as the former. – Doug Spoonwood Aug 31 '11 at 23:23
@Doug: I don’t see that as a real problem, though: in my experience very few mathematics students are already familiar with the use of the term in logic when they first encounter mathematical induction. – Brian M. Scott Sep 1 '11 at 5:37

I believe the name comes from the fact that P(n) "induces" the truth of P(n+1) for proposition P.

Dictionary definition of induce (one of several): To bring about, produce, or cause: That medicine will induce sleep.

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There are so few things that induce sleep in math nowadays... :P – Patrick Da Silva Aug 31 '11 at 3:52
ODEs do it for me every time! – Codie CodeMonkey Aug 31 '11 at 4:29
I'd have thought it's called "induction" because it proves something is true of all values of something. I.e. it's analogous in that respect to inferring general laws from particular observed instances in the empirical sciences, which is called induction. – Michael Hardy Aug 31 '11 at 13:33
ODEs do not induce sleep... it induces disgusted faces for those who are not interested and excitement for those who are! =P – Patrick Da Silva Aug 31 '11 at 14:28

Let me know if I'm wrong, but when I make my inductive hypothesis I believe I'm doing induction, making a generalization based on an observation ("assume the base the base case holds for a particular n=k...)

Why would you say it's not inductive? What author are you referring to?

And if you agree with what I said than the induction part is more important than the recursion part, because without the induction, you couldn't perform the recursion.

Edit: Your comment does make sense. In that case, this might answer your question: '"Mathematical induction" is unfortunately named, for it is unambiguously a form of deduction. However, it has certain similarities to induction which very likely inspired its name. It is like induction in that it generalizes to a whole class from a smaller sample. In fact, the sample is usually a sample of one, and the class is usually infinite. Mathematical induction is deductive, however, because the sample plus a rule about the unexamined cases actually gives us information about every member of the class. Hence the conclusion of a mathematical induction does not contain more information than was latent in the premises. Mathematical inductions therefore conclude with deductive certainty.'

In other words, you can't check every value, but you assume it must be true for every value, and that's a generalization. Hope that helps.

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I don't agree that you've made a generalization based upon observation... you would need empirical objects for that. Generally authors indicate, wikipedia is an example en.wikipedia.org/wiki/Mathematical_induction, the proof technique as deductive. The argument works by having a base case, and then showing that if the property holds for n, it'll also hold the successor of n. So, then from the base case it holds for the successor of the base case. Then from the successor of the base case it holds for the successor of the successor of the base case... and so on... – Doug Spoonwood Aug 31 '11 at 2:23
Other than the "and so on" part, that rather clearly, in my opinion, qualifies as deductive reasoning by any suitable definition of deductive reasoning. The last part often gets omitted in explanations also. – Doug Spoonwood Aug 31 '11 at 2:26