“The contemporary mathematical and symbolic logic is certainly very different from its classical predecessor, but they share the radical opposition to dialectical logic. In terms of this opposition, the old and the new formal logic express the same mode of thought. it is purged from that “negative” which loomed so large at the origins of logic and of philosophic thought — the experience of the denying, deceptive, falsifying power of the established reality. And with the elimination of this experience, the conceptual effort to sustain the tension between “is” and “ought”, and to subvert the established universe of discourse in the name of its own truth is likewise eliminated from all thought which is to be objective, exact, and scientific. For the scientific subversion of the immediate experience which establishes the truth of science as against that of immediate experience does not develop the concepts which carry in themselves the protest and the refusal. The new scientific truth which they oppose to the accepted one does not contain in itself the judgment that condemns the established reality. … In contrast, dialectical thought is and remains unscientific to the extent to which it is such judgment.”

— Herbert Marcuse, book One-Dimensional Man

Source: One-Dimensional Man (1964), pp. 139-140

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Herbert Marcuse105

German philosopher, sociologist, and political theorist 1898–1979

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“Dialectical logic undoes the abstractions of formal logic and of transcendental philosophy, but it also denies the concreteness of immediate experience. To the extent to which this experience comes to rest with the things as they appear and happen to be, it is a limited and even false experience. It attains its truth if it has freed itself from the deceptive objectivity which conceals the factors behind the facts — that is, if it understands its world as a historical universe, in which the established facts are the work of the historical practice of man.”

Herbert Marcuse book One-Dimensional Man

Source: One-Dimensional Man (1964), p. 141

“If dialectical logic understands contradiction as “necessity” belonging to the very “nature of thought,” … it does so because contradiction belongs to the very nature of the object of thought, to reality, where Reason is still Unreason, and the irrational still the rational. Conversely, all established reality militates against the logic of contradictions — it favors the modes of thought which sustain the established forms of life and the modes of behavior which reproduce and improve them. The given reality has its own logic and its own truth; the effort to comprehend them as such and to transcend them presupposes a different logic, a contradicting truth. They belong to modes of thought which are non-operational in their very structure; they are alien to scientific as well as common-sense operationalism. … These modes of thought appear to be a relic of the past, like all non-scientific and non-empirical philosophy. They recede before a more effective theory and practice of Reason.”

Herbert Marcuse book One-Dimensional Man

Source: One-Dimensional Man (1964), p. 142

“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

“A frequent source of misunderstanding has to do with the interpretation of fuzzy logic. The problem is that the term fuzzy logic has two different meanings. More specifically, in a narrow sense, fuzzy logic, FLn, is a logical system which may be viewed as an extension and generalization of classical multivalued logics. But in a wider sense, fuzzy logic, FLw is almost synonymous with the theory of fuzzy sets. In this context, what is important to recognize is that: (a) FLw is much broader than FLn and subsumes FLn as one of its branches; (b) the agenda of FLn is very different from the agendas of classical multivalued logics; and (c) at this juncture, the term fuzzy logic is usually used in its wide rather than narrow sense, effectively equating fuzzy logic with FLw</sub”

Lotfi A. Zadeh (1921–2017) Electrical engineer and computer scientist

Zadeh (1995) in Foreword of George J. Klir Fuzzy sets and fuzzy logic: theory and applications.
1990s

“The subject of a science is never well circumscribed and there is little use sharpening its definition. However, nobody will deny that physics deals with nature and sociology with human society in some of their aspects. With logic, it is another matter. Logic is usually understood nowadays as a study of certain formal systems, though in former times there were philosophers who held that the subject matter of logic was the formal rules of human thought. In the latter sense it would be an empirical rather than a formal science, though its empirical subject matter would still be fundamentally different from that of psychology of thinking. One interpretation of logic does not exclude the other. Formal approaches are often easier than empirical ones, and for this reason one can understand why logic as a study of formal systems has till now made more progress than logic as a study of the formal rules of thought, even if restricted to scientific thought.”

Hans Freudenthal (1905–1990) Dutch mathematician

Source: Realistic models in probability (1968), p. 1

“People can be fooled into thinking of mathematics as logical, formal, symbolic reasoning. But this is far from reality...computers are far better at formal computation and formal reasoning, but humans are far better mathematicians.”

William Thurston (1946–2012) mathematician

Foreword to Teichmüller Theory

“There is science, logic, reason; there is thought verified by experience. And then there is California.”

Edward Abbey (1927–1989) American author and essayist

“Mathematics has also been developed as a philosophy, in the sense in which this term is defined by A. N. Whitehead as 'the endeavor to frame a coherent, logical and necessary system of general ideas in terms of which every element of our experience can be interpreted'. Substitute 'mathematics' for 'experience' and we have an admirable description of its speculative and philosophic development…. Philosophy of mathematics… has its paradoxes and antimonies, and also diverse schools of thought…”

George Frederick James Temple (1901–1992) British mathematician

100 Years of Mathematics: a Personal Viewpoint (1981)

“So much of modern mathematical work is obviously on the border-line of logic, so much of modern logic is symbolic and formal, that the very close relationship of logic and mathematics has become obvious to every instructed student. The proof of their identity is, of course, a matter of detail: starting with premisses which would be universally admitted to belong to logic, and arriving by deduction at results which as obviously belong to mathematics, we find that there is no point at which a sharp line can be drawn, with logic to the left and mathematics to the right. If there are still those who do not admit the identity of logic and mathematics, we may challenge them to indicate at what point, in the successive definitions and deductions of Principia Mathematica, they consider that logic ends and mathematics begins. It will then be obvious that any answer must be quite arbitrary.”

Bertrand Russell (1872–1970) logician, one of the first analytic philosophers and political activist

Source: 1910s, Introduction to Mathematical Philosophy (1919), Ch. 18: Mathematics and Logic

“An aphorism can be considered a logically completed thought, accepted from the point of view of its meaning for the expression of new sensations, knowledge and meanings.”

Zafar Mirzo (1972)

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