Abstract
A basic question in the philosophy of mathematics concerns the analytic or the synthetic character of mathematics (cf. Kant). Since the mathematics of Greece, a relevant mapping of geometry to phenomena of space (architecture, astronomy, etc.) and musical scales had been accepted. In his philosophy of symbolic forms, Ernst Cassirer considered a gradient that links mathematics with language and myth. They have in common the symbolic mapping of phenomena in human apperception to systems of signs. This chapter concerns the relation between mathematics and linguistic, visual, and musical signs. In a first step, we reconsider Plato’s geometrical explanation of the universe and the human mind in his dialogue “Timaeus” and the use of a similar strategy in René Thom’s topological semantics that applies major results of catastrophe theory. Thom’s derivation of a list of semantic archetypes and their linguistic applications are summarized regarding results in Wildgen (1982). Beyond the semiotics of language, a further application shows the relevance of geometrical archetypes in classical paintings (e.g. Leonardo da Vinci) and tonal music (e.g. the blues schema). The final remarks consider evolutionary aspects of archetypal structures in semiosis.
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Notes
- 1.
With the increase of distance, new and important values are added. These are mainly: the greater independence from actual use and its contexts, the capacity of generalization, and thus of further insights useful in later uses, and possibly a global understanding.
- 2.
In the fourth chapter of his “Phänomenologie der Erkenntnis”, Cassirer says (with Leibniz) that mathematics should have a proper “fundamentum in re”, i.e. only in the context of scientific theories do they have the status of a symbolic form, become operative as conceptual tools for the enhancement of scientific knowledge. Cf. Cassirer, [1929] 1982: 418f. I thank the reviewer for his question on this behalf.
- 3.
In the epistemology of Analytic Philosophy (e.g. in the work of Wolfgang Stegmüller) one tries to reorganize different empirical theories based on set theory and logic (which are thus assumed to be universal tools for the organization of empirical knowledge). In the continuity of this program, Mormann proposed to use category theory for the same aim (cf. Balzer and Moulines 1996: Chap. 13). Applications to linguistics and more specifically to semantics are rare if not inexistent. Insofar as all other formalizations can be reanalyzed using category theory, most algebraic grammars and logical semantics should allow for a reformulation in the context of category theory. This will however not bring new insights for linguists.
- 4.
The critique of George Lakoff is directed against Chomskyan and logical models in linguistics. Lakoff (1987): Part II is a critique if the Objectivist Paradigm “reality is structured in a way that can be modeled by set-theoretical models” (ibid.: 159) and of the Formalist Enterprise “natural language syntax and semantics is just a special case of formal syntax and semantics” (ibid.: 219).
- 5.
The family Ak of cuspoïds is the real resolution of the (complex) cyclic group already described by Felix Klein in 1874; see for more details Slodowy (1988: 77–80).
- 6.
- 7.
Cf. Verstegen (2005) for an overview of Arnheim’s contributions to the psychology of vision and visual art.
- 8.
This short analysis is my response to a suggestion by the reviewer. The source of the message, God, can be associated with the white surface between two Italian cypresses. As in other paintings by Leonardo, e.g. “The virgin of the rocks”, 1483–1485, the theological content may be heretical.
- 9.
A specific consequence of the mathematical conceptualization of dynamics is the calculus of vectors introduced by William Rowan Hamilton (1805–1865). In Fig. 10 three force-vectors and four sight vectors are given. In a catastrophe theoretical frame, one would consider two vector-fields, one sight field and one grasp-field and their attractors (minima) and repellors (maxima).
- 10.
The area between the archetypal level and arbitrary rules in semiotic systems may be called semiotic stylistics. In the act of creation or creative use, highly motivated sign gestures and content structures beyond conventionalized patterns may show up. In relation to saliency (“saillance”), these may be highly coordinated simultaneous features. Thus, in an orchestral work, musical gestures are supported by different instrumental groups and cooperate to create a rich fabric of sounds. In the realm of content (“prégnance”), utterances may have a multiplicity of relevant readings, i.e. be enriched, or compress a field of meanings into a very short, allusive expression, e.g. in poetry. This intermediate area asks for a more specific treatment, which lies beyond the scope of the present contribution.
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Wildgen, W. (2020). Structures, Archetypes, and Symbolic Forms. Applied Mathematics in Linguistics and Semiotics. In: Peruzzi, A., Zipoli Caiani, S. (eds) Structures Mères: Semantics, Mathematics, and Cognitive Science. Studies in Applied Philosophy, Epistemology and Rational Ethics, vol 57. Springer, Cham. https://doi.org/10.1007/978-3-030-51821-9_9
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