Short Answer
By logical standards, mathematical induction is a form of deduction. From WP:
Although its name may suggest otherwise, mathematical induction should not be confused with inductive reasoning as used in philosophy (see Problem of induction). The mathematical method examines infinitely many cases to prove a general statement, but does so by a finite chain of deductive reasoning involving the variable n, which can take infinitely many values.
The reason the name of the mathematical proof procedure is confusing is that there is more than one definition of induction just as there is more than one definition of deduction. There are two competing uses of induction, the non-technical definition which means an inference that moves from specific premises to a general, and a more technical one where an inference doesn't necessarily, but only probabilistically arrives at a conclusion. Mathematical induction simply uses the first definition, the one NOT used by logicians and philosophers contemporaneously.
Mathematicians use the general definition of induction: moving from specific instances to general rules.
Philosophers use the technical definition of induction: using strong premises in a cogent argument to arrive at a probabilistic conclusion.
At this point, the terminology is entrenched, so it is what it is.
Long Answer
Before commenting on mathematical induction, it's first to know something from the philosophy of language. Attitudes on language use and definitions can fall under two general categories, language prescriptivism, which is the belief that words have a "correct" way of being used, and language descriptivism, which is the belief that there isn't. It's very hard to "enforce" the specific use of words, especially over a great number, and so often, words are used by convention.
Now, this explains the fact there are two definitions of deduction, and two definitions of induction. Let's examine deduction from MW's entry 'deduction', emphasis mine:
2 a: the deriving of a conclusion by reasoning based on reasoning
b : a conclusion reached by logical deduction
So, deduction is either a synonym for inference in (a) which is a lay usage or a specific method where the conclusion follows from the premises (b) which is the philosophical technical definition of deduction.
Likewise, induction has two definitions. From WP's 'induction', emphasis mine:
2 a (1): inference of a generalized conclusion from particular instances
b : mathematical demonstration of the validity of a law concerning all the positive integers by proving that it holds for the integer 1 and that if it holds for an arbitrarily chosen positive integer k, it must hold for the integer k + 1
Note that philosophers take induction a step further than 2a by essentially exending induction technically. This is called a precising definition.
So, we've suggested that language can't really be prescribed in the long run, and that we have two or more definitions for induction. Let's follow this last observation to its conclusion.
Language use is shaped by the norms of a language community; this is a tremendously influential idea that is often described in terms of the notion of language-game, and idea advocated by later Ludwig Wittgentstein. For our purposes, let's just say that there are mathematical language communities and philosophical ones.
Mathematicians are not usually philosophers, and they simply don't use words in the same way as philosophers as a general rule (though there are philosophers of math who use both mathematical and philosophical jargon). Therefore, in accordance with the general use of induction, that is moving from specific instances to a general rule, mathematical induction IS induction. The base case, the induction step, and the induction hypothesis are premises that lead to the conclusion which is a general rule that holds for all statements of the type instantiated by the base case. Since the three premises guarantee the conclusion, it is logical deduction, and not logical induction. Deduction deals with soundness and validity, but induction deals with strength and cogency.
Now, you might protest, that this is confusing, and certainly you'd be right to. But sometimes language usage is what it is. There is no academy that votes on mathematical terminology. There is no academy that votes on logical terminology. And if you have a few years under your belt, you'd know that people don't follow the rules anyway. If they did, we English speakers would be speaking Old English still. All languages change, and language use reflects the people who use it; this is true especially in the Anglo-American tradition, definitions are built on utility and consensus, not by the language police as in other cultures.
