“The 'language theory' is inadequate as a description of the nature of mathematics.”

100 Years of Mathematics: a Personal Viewpoint (1981)

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George Frederick James Temple 21
British mathematician 1901–1992

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“The theory of the nature of mathematics is extremely reactionary. We do not subscribe to the fairly recent notion that mathematics is an abstract language based, say, on set theory. In many ways, it is unfortunate that philosophers and mathematicians like Russell and Hilbert were able to tell such a convincing story about the meaning-free formalism of mathematics. In Greek, mathematics simply meant learning, and we have adapted this... to define the term as "learing to decide."”

C. West Churchman (1913–2004) American philosopher and systems scientist

Mathematics is a way of preparing for decisions through thinking. Sets and classes provide one way to subdivide a problem for decision preparation; a set derives its meaning from decision making, and not vice versa.

C. West Churchman, Leonard Auerbach, Simcha Sadan, Thinking for Decisions: Deductive Quantitative Methods (1975) Preface.
1960s - 1970s

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“Mathematics is not just a language. Mathematics is a language plus reasoning.”

Source: The Character of Physical Law (1965), chapter 2, “The Relation of Mathematics to Physics”
Context: Mathematics is not just a language. Mathematics is a language plus reasoning. It's like a language plus logic. Mathematics is a tool for reasoning. It's, in fact, a big collection of the results of some person's careful thought and reasoning. By mathematics, it is possible to connect one statement to another.

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“There is always the danger that the use of traditional grammatical terms with reference to a wide variety of languages may be taken to imply a secret belief in universal grammar. Every analysis of a particular ‘language’ must of necessity determine the values of the ad hoc categories to which traditional names are given. What is here being sketched is a general linguistic theory applicable to particular linguistic descriptions, not a theory of universals for general linguistic description.”

John Rupert Firth (1890–1960) English linguist

Source: "A synopsis of linguistic theory 1930-1955." 1957, p. 21; as cited in: Olivares, Beatriz Enriqueta Quiroz. The interpersonal and experiential grammar of Chilean Spanish: Towards a principled Systemic-Functional description based on axial argumentation. Diss. University of Sydney, 2013.

“Science does not speak of the world in the language of words alone, and in many cases it simply cannot do so. The natural language of science is a synergistic integration of words, diagrams, pictures, graphs, maps, equations, tables, charts, and other forms of visual and mathematical expression… [Science thus consists of] the languages of visual representation, the languages of mathematical symbolism, and the languages of experimental operations.”

Jay Lemke (1946) American academic

Jay Lemke (2003), "Teaching all the languages of science: Words , symbols, images and actions," p. 3; as cited in: Scott, Phil, Hilary Asoko, and John Leach. "Student conceptions and conceptual learning in science." Handbook of research on science education (2007): 31-56.

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