# Why is “mathematical induction” called “mathematical”?

One of my whims is that I never write "mathematical induction" but just "induction". We are doing maths, so what is the point about precising? We don't say "Let $f$ be a mathematical function from the mathematical set of mathematical real numbers into itself" for example.

My main question is: why do people consider necessary to add the adjective "mathematical" to this vocabulary? A related question that focuses on the "induction" in the name says that the mathematical induction isn't actually inductive. But, there are many terms in mathematics that doesn't exactly correspond to their common sense counter-parts: amoeba, worms, dessins d'enfants, etc are not called mathematical amoeba, mathematical worms or mathematical dessins d'enfants, etc. See the What is? column in The notices of AMS for more examples.

To make the question answerable, we can narrow the main question to these two ones (I am more interested in the second one):

1. Who coined the expression "mathematical induction"?
2. Is there any discussion or argument from the persons that coined that term about their use of the adjective "mathematical"?

My intuition is that, because a term from the field of logic, it lies in the intersection of maths and philosophy

Who coined the expression "mathematical induction"?

the qualificative "mathematical" was introduced in order to separate this method of proof from the inductive reasoning used in empirical sciences (the "all ravens are black" example); it is common also to call it complete induction, compared to the "incomplete" one used in empirical science.

The reason is straightforward : the mathematical method of proof establish a "generality" ("all odd numbers are not divisible by two") that holds without exception, while the "inductive generalization" established by observation of empirical facts can be subsequently falsified finding a new counter-example.

Note : induction (the non-mathematical one) was already discussed by Aristotle :

Deductions are one of two species of argument recognized by Aristotle. The other species is induction (epagôgê). He has far less to say about this than deduction, doing little more than characterize it as “argument from the particular to the universal”. However, induction (or something very much like it) plays a crucial role in the theory of scientific knowledge in the Posterior Analytics: it is induction, or at any rate a cognitive process that moves from particulars to their generalizations, that is the basis of knowledge of the indemonstrable first principles of sciences.

For the history of the name "mathematical induction", see

The process of reasoning called "mathematical induction" has had several independent origins. It has been traced back to the Swiss Jakob (James) Bernoulli [Opera, Tomus I, Genevae, MDCCXLIV, p. 282, reprinted from Acta eruditorum, Lips., 1686, p. 360. See also Jakob Bernoulli's Ars conjectandi, 1713, p. 95], the Frenchmen B.Pascal [OEuvres completes de Blaise Pascal, Vol. 3, Paris, 1866, p. 248] and P.Fermat [Charles S Peirce in the Century Dictionary, Art."Induction," and in the Monist, Vol. 2, 1892, pp. 539, 545; Peirce called mathematical induction the "Fermatian inference"], and the Italian F.Maurolycus [G.Vacca, Bulletin Am. Math. Soc., Vol. 16, 1909, pp. 70-73].

The process of Fermat differs somewhat from the ordinary mathematical induction; in it there is a descending order of progression, leaping irregularly over perhaps several integers from $$n$$ to $$n - n_1, n - n_1 - n_2$$, etc. Such a process was used still earlier by J.Campanus in his proof of the irrationality of the golden section, which he published in his edition of Euclid (1260).

John Wallis, in his Arithmetica infinitorum (Oxford, 1656), page 15, [uses] phrases like "fiat investigatio per modum inductionis" [...].He speaks, p. 33, of "rationes inductione repertas" and freely relies upon incomplete "induction" in the manner followed in natural science.

Thus, his method has been criticized by Fermat as being "conjectural", i.e.based on a perceived regularity or repeated schema in a group of formuale.

Wallis states (page 306) that Fermat "blames my demonstration by Induction, and pretends to amend it. . . . I look upon Induction as a very good method of Investigation; as that which doth very often lead us to the easy discovery of a General Rule."

For about 140 years after Jakob Bernoulli, the term "induction" was used by mathematicians in a double sense: (1) "Induction" used in mathematics in the manner in which Wallis used it; (2) "Induction" used to designate the argument from $$n$$ to $$n + 1$$. Neither usage was widespread. The former use of "induction" is encountered, for instance, in the Italian translation (1800) of Bossut and Lalande's dictionary,' article "Induction (term in mathematics)." The binomial formula is taken as an example; its treatment merely by verification, for the exponents $$m = 1, m = 2, m = 3$$, etc., is said to be by "Induction." We read that "it is not desirable to use this method, except for want of a better method." H.Wronski (1836) in a similar manner classed "methodes inductionnelles" among the presumptive methods ("methodes presomptives") which lack absolute rigor.

The second use of the word "induction" (to indicate proofs from $$n$$ to $$n + 1$$) was less frequent than the first. More often the process of mathematical induction was used without the assignment of a name. In Germany A.G.Kastner (1771) uses this new "genus inductionis" in proving Newton's formulas on the sums of weakness of Wallis's Induction, then explains Jakob Bernoulli's proof from $$n$$ to $$n + 1$$, but gives it no name. In England, Thomas Simpson [Treatise of Algebra, London, 1755, p. 205.] uses the $$n$$ to $$n + 1$$ proof without designating it by a name, as does much later also George Boole [Calculus of Finite Differences, ed. J.F.Moulton, London, 1880, p. 12.]

A special name was first given by English writers in the early part of the nineteenth century. George Peacock, in his Treatise on Algebra, Cambridge, 1830, under permutations and combinations, speaks (page 201) of a "law of formation extended by induction to any number," using "induction," as yet in the sense of "divination." Later he explains the argument from $$n$$ to $$n + 1$$ and calls it "demonstrative induction" (page 203).

The next publication is one of vital importance in the fixing of names; it is Augustus De Morgan's article "Induction (Mathematics)" in the Penny Cyclopedia, London, 1838. He suggests a new name, namely "successive induction," but at the end of the article he uses incidentally the term "mathematical induction." This is the earliest use of this name that we have seen.

• Good, I wasn't wrong. :-) – Asaf Karagila Dec 25 '14 at 9:04
• Well-researched. – N. F. Taussig Dec 25 '14 at 11:47
• @Taladris: It was pointed out, correctly, that while I answered the question in the title, I didn't quite answer the questions in the body. So I removed it. – Asaf Karagila Dec 25 '14 at 14:47
• I think part of the reasoning in the question was why have the word mathematical in there at all, because then someone might try and do math using non-mathematical induction. – Neil Mar 20 '15 at 23:02

John Wallis used induction as an "analytical" technique of deducing a general result from the first few cases, and was criticized for this by some contemporaries. The adjective is added to make it clear that we are talking about a mathematical procedure rather than a heuristic device.

There is also the Aristotelian Induction or Inductive Reasoning.

The adjective "mathematical" is primarily used to specify that the we are not referring to a logical or a philosophical induction or inductive reasoning.

Because the term induction is used in many fields, why not make things more precise by explicitly naming the field the induction belongs to.

As an example there is the electromagnetic induction

It's called "mathematical induction" because induction is a more general logic term (cf. Aristotle's Posterior Analytics).

Also (from here):

writes:

The process of reasoning called "mathematical induction" has had several independent origins. It has been traced back to the Swiss Jakob (James) Bernoulli, the Frenchmen B. Pascal and P. Fermat, and the Italian F. Maurolycus.

It seems Fermat (1601-1665) might have been the first.

C. S. Peirce says "mathematical induction" is an improper term for "Fermatian inference" (CP 6.116):

In truth, of infinite collections there are but two grades of magnitude, the endless and the innumerable. Just as a finite collection is distinguished from an infinite one by the applicability to it of a special mode of reasoning, the syllogism of transposed quantity, so, as I showed in the paper last referred to, a numerable collection is distinguished from an innumerable one by the applicability to it of a certain mode of reasoning, the Fermatian inference, or, as it is sometimes improperly termed, “mathematical induction.”

He cites Fermat, Opera Omnia (Leipzig, 1911), vol. 1, §§340-351.

Also,

write, (ed. 2) xvii. 355:

In fact, mathematical induction, or reasoning by recurrence, sometimes is referred to as ‘Fermatian induction’, to distinguish it from scientific, or ‘Baconian,’ induction.