For example, by examing the square number 25 we might observe that it equals the sum of the first five odd integers, i.e., 5^2 = 1+3+5+7+9. We might then check a few more squares and by common induction draw the general conclusion that the square of N is always equal to the sum of the first N odd numbers. In contrast, mathematical induction would proceed by first noting that the proposition is trivially true for the case N=1. Moreover, if it's true for any given integer n it is also true for n+1 because (n+1)^2 - n^2 equals 2n+1, which is the (n+1)th odd number. Thus, by mathematical induction it follows that the proposition is true for all N.
Understandably, many mathematicians take it as an insult to have mathematical induction confused with common induction. The crucial difference is that MI requires a formal implicative relation connecting all possible instances of the proposition, whereas CI leaps to a general conclusion simply from the fact that the proposition is true for a finite number of specific instances. Of course, it's easy to construct examples where CI leads to a wrong conclusion but, significantly, CI often leads to correct conclusions. We could devote an entire discussion to "the unreasonable effectiveness of common thought processes", but suffice it to say that for a system of limited complexity the possibilities can often be "spanned" by a finite number of instances.
In any case, questions about the consistency of arithmetic may cause us to view the distinction between MI and CI in a different light. How do we know that (n+1)^2 - n^2 always equals 2n+1? Of course this is a trivial example; in advanced proofs the formal implicative connection can be much less self-evident. Note that when challenged as to the absolute consistency of formal arithmetic, one response was to speak of "the supreme confidence that a century of working with ZF has given us". This, of course, is nothing but common induction. So too are claims that arithmetic must be absolutely consistent because otherwise bridges couldn't stand up and check books wouldn't balance. (These last two are not only common induction, they are bad common induction.)
Based on these reactions, we may wonder whether, ultimately, the two kinds of induction really are as distinct as is generally suppossed. It would seem more accurate to say that mathematical induction reduces a problem to a piece of common induction in which we have the very highest confidence, because it represents the pure abstracted essence of predictability, order, and reason that we've been able to infer from our existential experience. Nevertheless, this inference is ultimately nothing more (or less) than common induction.
Now, did Marilyn actually say any of this? No. It's entirely possible that she really didn't understand the technical difference between mathematical and common induction, and simply confused the two terms, just as she was evidently confused about elliptic curves and hyperbolic geometry and their uses in number theory. Similarly she didn't actually articulate a well-reasoned challenge to proof-by-contradiction within a formal system whose freedom from intrinsic contradicitons has not been (and cannot be) established. In both cases I was reading into her book much more than is actually there. From the standpoint of trying to figure out how smart Marilyn vos Savant is, such extrapolation is obviously not helpful. However, I'm less interested in what Marilyn vos Savant does or doesn't understand than I am in the actual issues themselves, and I'm interested in dealing with the strongest case that can be made, as opposed to arguing with the village idiot. This is why I wasn't too bothered by people saying "hey!, Marilyn never said that!"
Finally, it's clear from the preceeding pages that many people are highly disdainful of attempts to examine the fundamental basis of knowledge. In particular, some mathematicians evidently take it as an afront to the dignity and value of their profession (not to mention their lives) to have such questions raised. (Refer to the thumbnail review of Morris Kline's "Mathematics and the Loss of Certainty" in Part 3.) In general, I think people have varying thresholds of tolerance for self-doubt. For many people the exploration of philosophical questions reaches its zenith at the point of adolescent sophistry, as in "hey dood, did ya ever think that maybe none of this is real, and some demon is just messin' with our minds?" Never progressing further, for the rest of their lives whenever they encounter an issue of fundamental doubt they project their own adolescent interpretation onto the question and dismiss it accordingly.
In any case, the preceeding pages provide a nice catalog of reactions to such questions, including outrage, condesension, bafflement, fascination, and complete disinterest.