On Semiotics: On Signs (1900-1949)

"All our thinking is performed upon signs of some kind or other, either imagined or actually perceived. The best thinking, especially on mathematical subjects, is done by experimenting in the imagination upon a diagram or other scheme, and it facilitates the thought to have it before one’s eyes." (Charles S Peirce, “The Principles of Mathematics”, cca. 1902) 

"A sign is a thing which is the representative, or deputy, of another thing for the purpose of affecting a mind. […] The utility of icons is evidenced by the diagrams of the mathematician, whether they involve continuity, like geometrical figures, or are arrays of discrete objects like a body of algebraical formulae, all of which are icons. Icons have to be used in all thinking." (Charles S Peirce, [manuscript] 1903)

"Many diagrams resemble their objects not at all in looks; it is only in respect to the relations of their parts that their likeliness consists. […] When, in algebra, we write equations under one another in a regular array, especially when we put resembling letters for corresponding coefficients, the array is an icon. […] In fact, every algebraic equation is an icon, in so far as it exhibits, by means of the algebraic signs (which are not themselves icons), the relations of the quantities concerned." (Charles S Peirce, "New Elements", 1904)

"The edifice of science is not raised like a dwelling, in which the foundations are first firmly laid and only then one proceeds to construct and to enlarge the rooms. Science prefers to secure as soon as possible comfortable spaces to wander around and only subsequently, when signs appear here and there that the loose foundations are not able to sustain the expansion of the rooms, it sets about supporting and fortifying them. This is not a weakness, but rather the right and healthy path of development." (David Hilbert, "The Logical Principles of Mathematical Thinking", 1905)

"I call the combination of a concept and a sound-image a sign, but in current usage the term generally designates only a sound-image, a word, for example" (arbor, etc.). One tends to forget that arbor is called a sign only because it carries the concept ‘tree’, with the result that the idea of the sensory part implies the idea of the whole." (Ferdinand de Saussure, "Course in General Linguistics", 1915

"The logic of things, i.e., of the material concepts and relations on which the structure of a science rests, cannot be separated by the logic of signs. For the sign is no mere accidental cloak of the idea, but its necessary and essential organ. It serves not merely to communicate a complete and given thought content, but is an instrument, by means of which this content develops and fully defines itself. […] Consequently, all truly strict and exact thought is sustained by the symbolic and semiotics on which it is based." (Ernst Cassirer, "The Philosophy of Symbolic Forms", 1923)

"The words in a poem, (or more exactly, syllables) are vocal signs that convey an intangible essence (the pattern of feeling) that vanishes the moment we approach it with an analytical intelligence." (Herbert Read, "What is a Poem", 1926)

"The words of the language, as they are written or spoken, do not seem to play any role in any mechanism of thought. The physical entities which seem to serve as elements in thought are certain signs and more or less clear images which can be 'voluntarily' reproduced or combined. […] But taken from a psychological viewpoint, this combinatory play seems to be the essential feature in productive thought - before there is any connection with logical construction in words or other kinds of signs which can be communicated to others. The above-mentioned elements are, in my case, of visual and some of muscular type. Conventional words or other signs have to be sought for laboriously only in a secondary stage, when the mentioned associative play is sufficiently established and can be reproduced at will." (Albert Einstein, [letter to Hadamard, in" (Jacques Hadamard, "The Psychology of Invention in the Mathematical Field,1945)])