By "Sylvester's Law of Inertia," I mean:
http://en.wikipedia.org/wiki/Sylvester%27s_law_of_inertia
How does the name "Law of Inertia" fit with the statement of the theorem? I guess it's from physics, but… I just don't see the connection.
Why It’s Called Sylvester’s Law of Inertia – Linear Algebra History
math-historysoft-question
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Vectors should be thought of, at a first approximation, as "numbers with direction". For physical phenomena which carries a direction, such as velocity and displacement, vectors are immensely useful.
The concept of a number with direction most likely dates to antiquity, as the making of maps and sign-posts already implicitly incorporates the notion. The modern representation of a vector/point in space with an ordered triplet of numbers is often attributed to the advent of analytical geometry due to the philosopher Rene Descartes.
A different notion of vectors also arose with the "discovery" of the complex numbers by Jerome Cardan: the imaginary numbers can be thought of as living on a different direction as the real scalars (so the complex numbers form a real vector space).
Over the past 400 years or so the notion of vector gradually evolved to become what we know today, with contributions from branches of mathematics that developed into modern analysis and algebra. A nice summary of that period of development is available here. See Michael Crowe's book for a fuller description of also the Greek contributions and the influences from the 16th century in this matter.
In short, vectors shouldn't be thought of as being "invented", nor should it be attributed to one person alone.
About question n°1 :
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
- Florian Cajori, Origin of the Name "Mathematical Induction" (1918) :
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.
Best Answer
The quote in Mariano's answer is from the introduction to Sylvester's paper. Typical of Sylvester's mathematical papers, he used so many nonstandard terms in that paper that he appended a five-page "Glossary of new or unusual Terms, or of Terms used in a new or unusual sense in the preceding Memoir". There he lists:
Sylvester did similarly for many mathematical terms, i.e. coined them or used them in a "new or unusual ways" mathematically. You can find many such examples in Jeff Miller's Earliest Known Uses of Some of the Words of mathematics, including: allotrious factor, anallagmatic, Bezoutiant, catalecticant, combinant covariant cumulant cyclotomy, cyclotomic, dialytic, discriminant, Hessian, invariant, isomorphic, Jacobian, latent, law of intertia of quadratic forms, matrix, minor, nullity, plagiograph, quintic, Schur complement, sequence, syzygy, totient, tree, umbral calculus, umbral notation, universal algebra, x/y/z-coordinate, zero matrix, zetaic multiplication. Please see each entry for Sylvester's role - some are major, others are minor.
Apparently Sylvester's penchant for colorfully naming mathematical objects arose from his love of language and poetry. Indeed, Karen Parshall wrote:
Sylvester wrote about such:
You can find a short Sylvester biography here.