# Who introduced the Principle of Mathematical Induction for the first time?

Can you tell me the name of the mathematician, who introduced the Principle of Mathematical Induction for the first time? (with reliable source).

Please don't say De Morgan because I have read the name of the mathematician somewhere, but I have forgotten.

## 5 Answers

The issue is thorny ...

According to Morris Kline, Mathematical Thought from Ancient to Modern Time. Volume I (1972), page 272 [only entry of the Subject Index regarding : mathematical Induction] :

The method was recognized explicitly by Maurolycus in his Arithmetica of 1575 and was used by him to prove, for example, that $1+3+5+ \ldots + (2n+1)=n^2$. Pascal in one of his letters acknowledged Maurolycus's introduction of the method and used it himself in his Traité du triangle arithmétique (1665), wherein he presents what we now call the Pascal triangle.

The modern source is Giovanni Vacca (1872 –1953) Italian mathematician, assistant to Giuseppe Peano and historian of science in his :

with comments in :

Acording to Kline :

the method [of mathematical induction] is implicit even in Euclid's proof of the infinitude of the number of primes [IX, 20].

This point is debatable.

Euclid, The Thirteen Books of the Elements, Vol. 2: Books III - IX (T.L.Heath editor), states IX.20 as follows [page 412] :

Prime numbers are more than any assigned multitude of prime numbers.

The proof is per impossibile. According to Heath's comment [page 413] :

We have here the important proposition that the number of prime numbers is infinite.

Neither in the statement of the proposition nor in its proof Euclid uses the work infinite [apeiros -on : adj, infinite].

In can be helpful to place it in the context of Grrek debate about the infinite : see Aristotle and Mathematics on the actual infinity.

According to the Aristotelian philosophy, we cannot legitimately "conceive" actual infinity; i.e. we have no experience of an infinite "collection" but only of an unlimited iterative process (the potential infinity).

Euclid's statement must be understood in this context : we never have a "complete" infinite set of prime numbers, but we have a "procedure" that, for a finite collection of prime numbers whatever, can "produce" a new prime which is not in the collection.

The only occurrence (not regarding straight lines) I've found of the word infinite (searching into : EUCLID’S ELEMENTS OF GEOMETRY, the Greek text of J.L. Heiberg (1883–1885), edited, and provided with a modern English translation, by Richard Fitzpatrick), is VII,31 [Heath ed, page 332] :

Any composite number is measured by some prime number.

The proof uses the fact that :

For, if it is not found [some prime number which will measure the number before it, which will also measure A], an infinite series of numbers [emphasis added] will measure the number A, each of which is less than the other: which is impossible in numbers.

It seems the only reasonable "candidate" for a precursor of the least number principle [see the comment of Ian Mueller, Philosophy of Mathematics and Deductive Structure in Euclid's Elements (1981, Dover reprint), page 77].

According to :

the Greeks [...] did not have mathematical induction [...].

Regarding Pappus, in

• +1, A very well thought out answer. One subtopic I would like to see added though is how geometric series figures into the story. I can't remember for sure, but I believe geometric sums were a very early result of Eudoxan geometry. – David H Nov 21 '14 at 14:08
• Induction doesn't have to apply to an actually infinite set, it is also needed for finite sets of indefinite length. So another precursor of an inductive argument is Euclid's proof of what is now called Euclidean algorithm aleph0.clarku.edu/~djoyce/java/elements/bookVII./propVII2.html. And that probably goes back to Eudoxus if not early Pythagoreans. – Conifold Nov 21 '14 at 19:57
• Related: Duhem's counter-argument, in his "The Nature of Mathematical Reasoning," to Poincaré's belief that "arithmetic frequently uses a reasoning which is not equivalent to a series of syllogisms of limited number; in reality, it condenses an infinity of successive syllogisms." – Geremia Nov 17 '16 at 2:47

Well, you could go with Plato. From this,

For a start, although the principle itself is not explicitly stated in any ancient Greek text, there are several places that contain precursors of it. Indeed, some historians see the following passage from Plato’s (427-347 BC) dialogue Parmenides (§147a7-c3) as the earliest use of an inductive argument.

The text quotes Parmenides:

Then they must be two, at least, if there is to be contact. - They must. - And if to the two terms a third be added in immediate succession, they will be three, while the contacts [will be] two. - Yes. - And thus, one [term] being continually added, one contact also is added, and it follows that the contacts are one less than the number of terms. For the whole successive number [of terms] exceeds the number of all the contacts as much as the first two exceed the contacts, for being greater in number than the contacts: for afterwards, when an additional term is added, also one contact to the contacts [is added]. - Right. - Then whatever the number of terms, the contacts are always one less. -True.

I attempted to find a second translation, and found this one, but I have been unable to identify this section, even though the section is supposedly listed. I found another transcript here, but it, too does not agree with the given text in the pdf. Still, the dialogue closely resembles the technique of mathematic induction, albeit used in a philosophical context.

The page goes on to add that Euclid and Pappus, too, played a part:

There are, however, several ancient mathematical texts that also contain quasi-inductive arguments. For instance Euclid (~330 - ~ 265 BC) in his Elements employs one to show that every integer is a product of primes. An argument closer to the modern version of induction is in Pappus' (~290-~350 AD) Collectio.

I have been unable to find a similar primary source for Euclid and Pappus. However, this appears to corroborate Euclid's role (though not as detailed as I would have likes) and also says that Wikipedia's claim about Pascal being important (mentioned below from another source) is accurate. I'm working on more primary sources, but at the moment no others are forthcoming.

This says that Pascal played a role in modern times:

Cantor in his Vorlesungen iuber Geschichte der Mathematik' says that Pascal was the originator of the method of complete induction

• I find it highly unsatisfying that you do not reproduce the relevant section of Plato's (and the other ancient Greeks') work in your answer. Please include them in your answer! – Danu Nov 19 '14 at 20:02
• @Danu I appreciate that you didn't merely downvote and not give any feedback (or downvote at all!). Yes, it would be good to get direct quotes from Plato et al.; I hadn't had time to get them earlier. I'll certainly add them it. – HDE 226868 Nov 19 '14 at 21:04

Victor Katz wrote in a discussion at the Historia mathematica discussion list, "as Barnabas Hughes notes, it is difficult to answer the question of the "first instances of proof by induction" unless one carefully defines what one means by "proof by induction." (The same general remark applies to lots of questions about firsts in the history of mathematics.) Noicomachus certainly has an argument which we would find very easy to convert into a modern formal "proof by induction." So do several Islamic authors around the 11th century, including al-Karaji and al-Samaw'al. (See my book, pp. 238-242.) Pascal may well be the first to state the modern principle of mathematical induction explicitly, but even he does not give proofs in the modern style - because he has no notation for a general "n". Thus he generally gives proofs by what I call the "method of generalizable example."

In another post Barnabus Hughes suggests yet an earlier "first use" of induction: If the essence of math induction lies in a process that begins at some small value, which process can be continued to larger values which regardless of their size maintain the pattern one wishes to accept, then I would hazard that Nicomachus of Geresa used the essence of math induction where he discussed figurate numbers (Arithmetica, Bk. 2, CC. 7ff.) In C. 7 he states, "Hence, the triangle is elementary among these

figures; for everything else is resolved into it, but it into nothing else." He then shows how, if the process established for the creation of each figurate number is followed, then that number is always seen. C. 12 shows how the other figures are resolved into triangles, including a table which lists the first five polygonal numbers to the tenth degree of each. I think that Nick established the pattern by induction. Comment?

Barnabas Hughes

According to Wikipedia this was Levi ben Gershon (a.k.a. Gershonides, RALBAG). 1288–1344

The work is notable for its early use of proof by mathematical induction, and pioneering work in combinatorics.

and

Gersonides was also the earliest known mathematician to have used the technique of mathematical induction in a systematic and self-conscious fashion

Remark. The word "induction" is used in a different sense in philosophy. One has to distinguish Mathematical induction from "induction" in philosophy. These are very different things. I am not aware of any ancient Greek use of Mathematical induction.

• About the "modern" sense of induction (as mathematical induction), see André Weil, Number Theory : An approach through history (1984), page 50 for Fermat's critique (1657) of the "inductive method" as an heuristical way to establish a "general" statement in mathematics (call it "incomplete induction") compared to a proof of the said "general" statement, and see page 77 for Fermat's enunciation (1659) of his newly discovered method of proof : the infinite descent. – Mauro ALLEGRANZA Nov 23 '14 at 11:36

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.