“Modern positivism…expresses criticism against the naïve use of certain terms… by the general postulate that the question whether a given sentence has any meaning… should always be thoroughly and critically examined. This… is derived from mathematical logic. The procedure of natural science is pictured as an attachment of symbols to the phenomena. The symbols can, as in mathematics, be combined according to certain rules… However, a combination of symbols that does not comply with the rules is not wrong but conveys no meaning.
The obvious difficulty in this argument is the lack of any general criterion as to when a sentence should be considered meaningless. A definite decision is possible only when the sentence belongs to a closed system of concepts and axioms, which in the development of natural science will be rather the exception than the rule. In some case the conjecture that a certain sentence is meaningless has historically led to important progress… new connections which would have been impossible if the sentence had a meaning. An example… sentence: "In which orbit does the electron move around the nucleus?"”

— Werner Heisenberg

But generally the positivistic scheme taken from mathematical logic is too narrow in a description of nature which necessarily uses words and concepts that are only vaguely defined.
Physics and Philosophy (1958)

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Werner Heisenberg42

German theoretical physicist 1901–1976

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“We can now return to the distinction between language and symbolism. A symbol is language and yet not language. A mathematical or logical or any other kind of symbol is invented to serve a purpose purely scientific; it is supposed to have no emotional expressiveness whatever. But when once a particular symbolism has been taken into use and mastered, it reacquires the emotional expressiveness of language proper. Every mathematician knows this. At the same time, the emotions which mathematicians find expressed in their symbols are not emotions in general, they are the peculiar emotions belonging to mathematical thinking.”

R. G. Collingwood (1889–1943) British historian and philosopher

Source: The Principles of Art (1938), p. 268

“…Once you entomb mathematics in an artificial language à la Hilbert, once you set up a completely formal axiomatic system, then you can forget that it has any meaning and just look at it as a game that you play with marks on paper that enable you to deduce theorems from axioms. You can forget about the meaning of the game, the game of mathematical reasoning, it's just combinatorial play with symbols! There are certain rules, and you can study these rules and forget that they have any meaning!”

Gregory Chaitin (1947) Argentinian mathematician and computer scientist

1999 Lecture—"A Century of Controversy over the Foundations of Mathematics" at U. Massachusetts at Lowell, quoted in [2012, Conversations with a Mathematician: Math, Art, Science and the Limits of Reason, Springer, https://books.google.com/books?id=DczTBwAAQBAJ&pg=PA15] p. 15

“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)

“By the logical syntax of a language, we mean the formal theory of the linguistic forms of that language -- the systematic statement of the formal rules which govern it together with the development of the consequences which follow from these rules. A theory, a rule, a definition, or the like is to be called formal when no reference is made in it either to the meaning of the symbols (for examples, the words) or to the sense of the expressions (e. g. the sentences), but simply and solely to the kinds and order of the symbols from which the expressions are constructed.”

Rudolf Carnap (1891–1970) German philosopher

Source: Logical Syntax of Language, 1934/1937, p. 1

“If to-day you ask a physicist what he has finally made out the æther or the electron to be, the answer will not be a description in terms of billiard balls or fly-wheels or anything concrete; he will point instead to a number of symbols and a set of mathematical equations which they satisfy. What do the symbols stand for? The mysterious reply is given that physics is indifferent to that; it has no means of probing beneath the symbolism. To understand the phenomena of the physical world it is necessary to know the equations which the symbols obey but not the nature of that which is being symbolised. … this newer outlook has modified the challenge from the material to the spiritual world.”

Arthur Stanley Eddington (1882–1944) British astrophysicist

Science and the Unseen World (1929)

“In a strict usage the same symbol should never represent the act of sincerely asserting something and the content of what is asserted. For the symbolic distinction between the two, Frege has introduced the 'signpost' symbol. … \vdash p is to signify the actual assertion of p, while the bare symbol p must henceforth be used only as part of a sentence. … It should be clear from the modality of a sentence whether it is a question, a command, an invective, a complaint or an allegation of fact.”

Michael Polanyi (1891–1976) Hungarian-British polymath

Source: Personal Knowledge (1958), p. 27

“It remains a real world if there is a background to the symbols — an unknown quantity which the mathematical symbol x stands for. We think we are not wholly cut off from this background. It is to this background that our own personality and consciousness belong, and those spiritual aspects of our nature not to be described by any symbolism … to which mathematical physics has hitherto restricted itself.”

Arthur Stanley Eddington (1882–1944) British astrophysicist

Science and the Unseen World (1929)

“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.

“The fact that all Mathematics is Symbolic Logic is one of the greatest discoveries of our age”

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

Principles of Mathematics (1903), Ch. I: Definition of Pure Mathematics, p. 5
1900s
Context: The fact that all Mathematics is Symbolic Logic is one of the greatest discoveries of our age; and when this fact has been established, the remainder of the principles of mathematics consists in the analysis of Symbolic Logic itself.

“Man's achievements rest upon the use of symbols.... we must consider ourselves as a symbolic, semantic class of life, and those who rule the symbols, rule us.”

Alfred Korzybski (1879–1950) Polish scientist and philosopher

Source: Science and Sanity (1933), p. 76.

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