James J Sylvester - Collected Quotes

“Arithmetic has for its object the properties of number in the abstract. In algebra, viewed as a science of operations, order is the predominating idea. The business of geometry is with the evolution of the properties of space, or of bodies viewed as existing in space.” (James J Sylvester, “A Probationary Lecture on Geometry”, 1844)

"Partitions constitute the sphere in which analysis lives, moves, and has its being; and no power of language can exaggerate or paint too forcibly the importance of this till recently almost neglected, but vast, subtle, and universally permeating, element of algebraical thought and expression." (James J Sylvester, "On the Partition of Numbers", 1857)

“The world of ideas which it discloses or illuminates, the contemplation of divine beauty and order which it induces, the harmonious connexion of its parts, the infinite hierarchy and absolute evidence of the truths with which it is concerned, these, and such like, are the surest grounds of the title of mathematics to human regard, and would remain unimpeached and unimpaired were the plan of the universe unrolled like a map at our feet, and the mind of man qualified to take in the whole scheme of creation at a glance.” (James J Sylvester, “The Study That Knows Nothing of Observation”, 1869) 

"I think it would be desirable that this form of word [mathematics] should be reserved for the applications of the science, and that we should use mathematic in the singular to denote the science itself, in the same way as we speak of logic, rhetoric, or (own sister to algebra) music.” (James J Sylvester, Collected Mathematical Papers, 1869)

“It is the constant aim of the mathematician to reduce all his expressions to their lowest terms, to retrench every superfluous word and phrase, and to condense the Maximum of meaning into the Minimum of language.” (James J Sylvester, 1877)

“’Divide et impera’ is as true in algebra as in statecraft; but no less true and even more fertile is the maxim ‘auge et impera’. The more to do or to prove, the easier the doing or the proof.” (James J Sylvester, “Proof of the Fundamental Theorem of Invariants”, Philosophic Magazine, 1878)

“It always seems to me absurd to speak of a complete proof, or of a theorem being rigorously demonstrated. An incomplete proof is no proof, and a mathematical truth not rigorously demonstrated is not demonstrated at all.” (James J Sylvester, "On certain inequalities related to prime numbers", Nature Vol. 38, 1888)

"The world of ideas which it [mathematics] discloses or illuminates, the contemplation of divine beauty and order which it induces, the harmonious connection of its parts, the infinite hierarchy and absolute evidence of the truths with which mathematical science is concerned, these, and such like, are the surest groimds of its title of human regard, and would remain unimpaired were the plan of the universe unrolled like a map at our feet, and the mind of man qualified to take in the whole scheme of creation at a glance.” (James J Sylvester, "A Plea for the Mathematician", Nature, 1908)

"May not Music be described as the Mathematics of sense, and Mathematics as the Music of reason?” (James J Sylvester)

“No mathematician nowadays sets any store on the discovery of isolated theorems, except as affording hints of an unsuspected new sphere of thought, like meteorites detached from some undiscovered planetary orb of speculation.” (James J Sylvester)

“The existence of an odd perfect number – its escape, so to say, from the complex web of conditions which hem it in on all sides – would be little short of a miracle.” (James J Sylvester)