“A successful attempt to express logical propositions by symbols, the laws of whose combinations should be founded upon the laws of the mental processes which they represent, would, so far, be a step towards a philosophical language.”

Source: 1840s, The Mathematical Analysis of Logic, 1847, p. 5

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George Boole39

English mathematician, philosopher and logician 1815–1864

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“The viewpoint of the formalist must lead to the conviction that if other symbolic formulas should be substituted for the ones that now represent the fundamental mathematical relations and the mathematical-logical laws, the absence of the sensation of delight, called "consciousness of legitimacy," which might be the result of such substitution would not in the least invalidate its mathematical exactness. To the philosopher or to the anthropologist, but not to the mathematician, belongs the task of investigating why certain systems of symbolic logic rather than others may be effectively projected upon nature. Not to the mathematician, but to the psychologist, belongs the task of explaining why we believe in certain systems of symbolic logic and not in others, in particular why we are averse to the so-called contradictory systems in which the negative as well as the positive of certain propositions are valid.”

L. E. J. Brouwer (1881–1966) Dutch mathematician and logician

as translated by Arnold Dresden from: Brouwer, L. E. J. (1913). Intuitionism and formalism. Bulletin of the American Mathematical Society, 20(2), 81–96. (quote on p. 84)

“There are a number of theorems in ordinary algebra, which, though apparently proved to be true only for symbols representing numbers, admit of a much more extended application. Such theorems depend only on the laws of combination to which the symbols are subject, and are therefore true for all symbols, whatever their nature may be, which are subject to the same laws of combination. The laws with which we have here concern are few in number, and may be stated in the following manner. Let a, b represent two operations, u, v two subjects on which they operate, then the laws are
(1) ab(u) = ba (u),
(2) a(u + v) = a (u) + a (v),
(3) am. an. u = am + n. u.
The first of these laws is called the commutative law, and symbols which are subject to it are called commutative symbols. The second law is called distributive, and the symbols subject to it distributive symbols. The third law is not so much a law of combination of the operation denoted by a, but rather of the operation performed on a, which is indicated by the index affixed to a. It may be conveniently called the law of repetition, since the most obvious and important case of it is that in which m and n are integers, and am therefore indicates the repetition m times of the operation a.”

Duncan Gregory (1813–1844) British mathematician

That these are the laws employed in the demonstration of the principal theorems in Algebra, a slight examination of the processes will easily shew ; but they are not confined to symbols of numbers ; they apply also to the symbol used to denote differentiation. <br class="br"> p. 237 http://books.google.com/books?id=8lQ7AQAAIAAJ&pg=PA237; Highlighted section cited in: George Boole " Mr Boole on a General Method in Analysis http://books.google.com/books?pg=PA225-IA15&id=aGwOAAAAIAAJ&hl," Philosophical Transactions, Vol. 134 (1844), p. 225; Other section (partly) cited in: James Gasser (2000) A Boole Anthology: Recent and Classical Studies in the Logic of George Boole,, p. 52 <br class="br">Examples of the processes of the differential and integral calculus, (1841)

“To deduce the laws of the symbols of Logic from a consideration of those operations of the mind which are implied in the strict use of language as an instrument of reasoning.”

George Boole (1815–1864) English mathematician, philosopher and logician

Source: 1850s, An Investigation of the Laws of Thought (1854), p. 42

“All language is symbolic, so far as it is applied to mental and spiritual phenomena and action.”

Albert Pike book Morals and Dogma of the Ancient and Accepted Scottish Rite of Freemasonry

Source: Morals and Dogma of the Ancient and Accepted Scottish Rite of Freemasonry (1871), Ch. III : The Master, p. 62
Context: All religious expression is symbolism; since we can describe only what we see, and the true objects of religion are The Seen. The earliest instruments of education were symbols; and they and all other religious forms differed and still differ according to external circumstances and imagery, and according to differences of knowledge and mental cultivation. All language is symbolic, so far as it is applied to mental and spiritual phenomena and action. All words have, primarily, a material sense, howsoever they may afterward get, for the ignorant, a spiritual non-sense. To "retract," for example, is to draw back, and when applied to a statement, is symbolic, as much so as a picture of an arm drawn back, to express the same thing, would he. The very word " spirit" means " breath," from the Latin verb spiro, breathe.

“In treating of the practical application of scientific principles, an algebraical formula should only be employed when its shortness and simplicity are such as to render it a clearer expression of a proposition or rule than common language would be, and when there is no difficulty in keeping the thing represented by each symbol constantly before the mind.”

William John Macquorn Rankine (1820–1872) civil engineer

"On the Harmony of Theory and Practice in Mechanics" (Jan. 3, 1856)
Context: In treating of the practical application of scientific principles, an algebraical formula should only be employed when its shortness and simplicity are such as to render it a clearer expression of a proposition or rule than common language would be, and when there is no difficulty in keeping the thing represented by each symbol constantly before the mind.<!--p. 177

“Let us conceive, then, of an algebra in which the symbols x, y z etc. admit indifferently of the values 0 and 1, and of these values alone The laws, the axioms, and the processes, of such an Algebra will be identical in their whole extend with the laws, the axioms, and the processes of an Algebra of Logic. Difference of interpretation will alone divide them. Upon this principle the method of the following work is established.”

George Boole (1815–1864) English mathematician, philosopher and logician

Source: 1850s, An Investigation of the Laws of Thought (1854), p. 37; Cited in: William Torrey Harris (1879) The Journal of Speculative Philosophy, p. 109

“In [Aristotle’s] formal logic, thought is organized in a manner very different from that of the Platonic dialogue. In this formal logic, thought is indifferent toward its objects. Whether they are mental or physical, whether they pertain to society or to nature, they become subject to the same general laws of organization, calculation, and conclusion — but they do so as fungible signs or symbols, in abstraction from their particular “substance.” This general quality (quantitative quality) is the precondition of law and order — in logic as well as in society — the price of universal control.”

Herbert Marcuse book One-Dimensional Man

Source: One-Dimensional Man (1964), p. 136

“In this chapter I shall collect those Theorems in the Differential Calculus which, depending only on the laws of combination of the symbols of differentiation, and not on the functions which are operated on by these symbols, may be proved by the method of the separation of the symbols : but as the principles of this method have not as yet found a place in the elementary works on the Calculus, I shall first state? briefly the theory on which it is founded.”

Duncan Gregory (1813–1844) British mathematician

Source: Examples of the processes of the differential and integral calculus, (1841), p. 237; Lead paragraph of Ch. XV, On General Theorems in the Differential Calculus,; Cited in: James Gasser (2000) A Boole Anthology: Recent and Classical Studies in the Logic of George Boole,, p. 52

“Let x represent an act of the mind by which we fix our regard upon that portion of time for which the proposition X is true; and let this meaning be understood when it is asserted that x denote the time for which the proposition X is true. (...) We shall term x the representative symbol of the proposition X.”

George Boole (1815–1864) English mathematician, philosopher and logician

Source: 1850s, An Investigation of the Laws of Thought (1854), p. 165; As cited in: James Joseph Sylvester, ‎James Whitbread Lee Glaisher (1910) The Quarterly Journal of Pure and Applied Mathematics. p. 350

“… The answer to this problem is: as implied by Hume, we certainly are not justified in reasoning from an instance to the truth of the corresponding law. But to this negative result a second result, equally negative, may be added: we are justified in reasoning from a counterinstance to the falsity of the corresponding universal law (that is, of any law of which it is a counterinstance). Or in other words, from a purely logical point of view, the acceptance of one counterinstance to 'All swans are white' implies the falsity of the law 'All swans are white' - that law, that is, whose counterinstance we accepted. Induction is logically invalid; but refutation or falsification is a logically valid way of arguing from a single counterinstance to - or, rather, against - the corresponding law. This shows that I continue to agree with Hume's negative logical result; but I extend it. This logical situation is completely independent of any question of whether we would, in practice, accept a single counterinstance - for example, a solitary black swan - in refutation of a so far highly successful law. I do not suggest that we would necessarily be so easily satisfied; we might well suspect that the black specimen before us was not a swan.”

Karl Popper book The Logic of Scientific Discovery

Source: The Logic of Scientific Discovery (1934), Ch. 1 "A Survey of Some Fundamental Problems", Section I: The Problem of Induction http://dieoff.org/page126.htm p. 27

“A successful attempt to express logical propositions by symbols, the laws of whose combinations should be founded upon the laws of the mental processes which they represent, would, so far, be a step towards a philosophical language.”

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George Boole39

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