„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, 1850s, An Investigation of the Laws of Thought (1854), p. 42
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George Boole
matematico e logico britannico 1815 - 1864
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George Boole photo

„There is not only a close analogy between the operations of the mind in general reasoning and its operations in the particular science of Algebra, but there is to a considerable extent an exact agreement in the laws by which the two classes of operations are conducted.“

—  George Boole English mathematician, philosopher and logician 1815 - 1864
1850s, An Investigation of the Laws of Thought (1854), p. 6; As cited in: Leandro N. De Castro, Fernando J. Von Zuben, Recent Developments in Biologically Inspired Computing, Idea Group Inc (IGI), 2005 p. 236

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Duncan Gregory photo

„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 British mathematician 1813 - 1844
Examples of the processes of the differential and integral calculus, (1841), 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. 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

Karl Popper photo

„… 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 Austrian-British philosopher of science 1902 - 1994
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

Robert Chambers (publisher, born 1802) photo
African Spir photo

„The concept of absolute, hence (or whence) springs, in the moral field, the moral laws or norms, represent, in the field of knowledge, the principle of identity, which is the fundamental law of the thought; norms of logic springs from it, that govern the thought (or mind) in the field of science.“

—  African Spir Russian philosopher 1837 - 1890
"Le concept de l'absolu, d'où découlent, dans le domaine moral, les lois ou normes morales, constitue, le principe d'identité, qui est la loi fondamentale de la pensée; il en découle les normes logiques qui régissent la pensée dans le domaine de la science." p. 59 [Hélène Claparède-Spir had underlined - the translator]

Konrad Lorenz photo

„Our freest will underlies strict moral laws, and one of the reasons for our longing for freedom is to prevent our obeying other laws than these.“

—  Konrad Lorenz Austrian zoologist, winner of the Nobel Prize in Physiology or Medicine in 1973. 1903 - 1989
On Aggression (1963), Context: Nobody can seriously believe that free will means that it is left entirely to the will of the individual, as to an irresponsible tyrant, to do or not do whatever he pleases. Our freest will underlies strict moral laws, and one of the reasons for our longing for freedom is to prevent our obeying other laws than these. It is significant that the anguished feeling of not being free is never evoked by the realisation that our behaviour is just as firmly bound to moral laws as physiological processes are to physical ones. We are all agreed that the greatest and most precious freedom of man is identical with the moral laws within him. Increasing knowledge of the natural causes of his own behaviour can certainly increase a man's faculties and enable him to put his free will into action, but it can never diminish his will. If, in the impossible case of an utopian complete and ultimate success of causal analysis, man should ever achieve complete insight into the causality of earthly phenomena, including the workings of his own organism, he would not cease to have a will but it would be in perfect harmony with the incontrovertible lawfulness of the universe, the Weltvernunft of the Logos. This idea is foreign only to our present-day western thought; it was quite familiar to ancient Indian philosophy and to the mystics of the middle ages. Ch. XII : On the Virtue of Scientific Humility

Albert Einstein photo

„The supreme task of the physicist is to arrive at those universal elementary laws from which the cosmos can be built up by pure deduction. There is no logical path to these laws; only intuition, resting on sympathetic understanding of experience, can reach them.“

—  Albert Einstein German-born physicist and founder of the theory of relativity 1879 - 1955
1910s, Principles of Research (1918), Context: The supreme task of the physicist is to arrive at those universal elementary laws from which the cosmos can be built up by pure deduction. There is no logical path to these laws; only intuition, resting on sympathetic understanding of experience, can reach them. In this methodological uncertainty, one might suppose that there were any number of possible systems of theoretical physics all equally well justified; and this opinion is no doubt correct, theoretically. But the development of physics has shown that at any given moment, out of all conceivable constructions, a single one has always proved itself decidedly superior to all the rest. Variant, from Preface to Max Planck's Where is Science Going? (1933): The supreme task of the physicist is the discovery of the most general elementary laws from which the world-picture can be deduced logically. But there is no logical way to the discovery of these elemental laws. There is only the way of intuition, which is helped by a feeling for the order lying behind the appearance, and this Einfühlung [literally, empathy or 'feeling one's way in']' is developed by experience.

Richard Feynman photo

„We can deduce, often, from one part of physics like the law of gravitation, a principle which turns out to be much more valid than the derivation.“

—  Richard Feynman American theoretical physicist 1918 - 1988
The Character of Physical Law (1965), Context: Now we have a problem. We can deduce, often, from one part of physics like the law of gravitation, a principle which turns out to be much more valid than the derivation. This doesn't happen in mathematics, that the theorems come out in places where they're not supposed to be! chapter 2, “ The Relation of Mathematics to Physics http://www.youtube.com/watch?v=M9ZYEb0Vf8U” referring to the law of conservation of angular momentum

Henry Ward Beecher photo

„The laws that are the most operative are the laws which protect life.“

—  Henry Ward Beecher American clergyman and activist 1813 - 1887
Miscellany, Context: Any law that takes hold of a man’s daily life cannot prevail in a community, unless the vast majority of the community are actively in favor of it. The laws that are the most operative are the laws which protect life. Civil Law and the Sabbath sermon (3 December 1882)

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John Quincy Adams photo

„I told him that I thought it was law logic — an artificial system of reasoning, exclusively used in Courts of justice, but good for nothing anywhere else.“

—  John Quincy Adams American politician, 6th president of the United States (in office from 1825 to 1829) 1767 - 1848
Diary record of a comment made by Adams to John Marshall, Charles Francis Adams, Memoirs of John Quincy Adams : Comprising Portions of His Diary from 1795 to 1848 (1875), p. 372

P. D. Ouspensky photo

„Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Etiam egestas wisi a erat. Morbi imperdiet, mauris ac auctor dictum.“