Philosophy Dictionary of ArgumentsHome
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| Fixed point: a point that satisfies the equation f (x) = x is a fixed point i.e. x is mapped to itself. S.A. Kripke based his alternative theory of truth from 1975 on fixed points in order to resolve the problem of paradoxes when dealing with self-reference. (Kripke, S., 1975. Outline of a Theory of Truth, The Journal of Philosophy, 72 690-716.). See also self-reference, paradoxes, liar paradox, truth theory._____________Annotation: The above characterizations of concepts are neither definitions nor exhausting presentations of problems related to them. Instead, they are intended to give a short introduction to the contributions below. – Lexicon of Arguments. | |||
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Peter Gärdenfors on Fixed Points - Dictionary of Arguments
I 97 Fixed point/communication/Gärdenfors: a fixed point in a communication is reached when, for example, a person sees that the other person looks in the direction of the object they are referring to. --- I 99 There should also be a consistency between the mental representations for the consistency of word meanings. Communication is also possible without this: e.g. children often have fewer domains in the representation of their terms or the domains are differently weighted. Equilibrium: Communication can work restrictedly before the equilibrium of the partners (the same level of information) is reached. --- I 100 Signal game/Jäger/Rooij/Gärdenfors: (Jäger & van Rooij, 2007)(1): randomly selected color samples are ordered by a second person. The goal of the game is to achieve an equal division of the color space in regions. (Nash-equilibrium or fixed point). Gärdenfors: thesis: this is achieved if the conceptual spaces are convex and compact. --- I 101 Equilibrium/Fixed point/Gärdenfors: further experiments have shown that repeated interactions lead to a stable communication system. (E.g. Hurford, 1999,(2) Kirby, 1999,(3) Steels, 1999,(4) Kaplan, 2000,(5) Steels & Belpaeme, 2005(6)). --- I 102 Meanings: do not necessarily have to change when the composition of the communicators involved changes or new parties join or disappear. Fixed point/Dewey/Gärdenfors: (Dewey 1929, p. 178) (7): in order for V to understand A's moves, he must react to the thing from A's standpoint of view. So not I-centered and vice versa. Thus, something is literally made into a common. --- I 104 Fixed point theorem/Gärdenfors: in order to achieve fixed points, it is not necessary for the conceptual spaces of the participants to be identical, nor that they divide the spaces equally. ---- I 105 We assume that the rooms are convex and compact. The following theorem from Warglien & Gärdenfors (2013)(8) is a consequence of Brouwer's fixed point theorem (Brouwer 1910)(9): Theorem: every semantic reaction function, which is a continuous mapping of a convex compact set on itself, has at least one fixed point. That is, there will always be a fixed point representing a Meeting of Minds. Conceptual spaces: that they are assumed to be convex makes the communication flowing and memory performance efficient. --- I 106 Gärdenfors: I do not mean that convex spaces are a reliable representation of our world, but that, because they are effective, they will be widespread. Fixed points: the fixed point approach allows to consider a variety of types of communication such as color determinations and negotiations. The fixed-point theorem guarantees that the consciousness of the participants together (> Meeting of Minds) but it does not show how the semantic reaction function emerges from the communicative interaction. --- I 109 Fixed Point/Communication/Gärdenfors: how do we know if a fixed point (balance, agreement) has been reached? --- I 110 If the listener believes to understand, this is not a guarantee for a meeting of minds. 1. Jäger, G., & van Rooij, R. (2007). Language structure: Psychological and social constraints. Synthese, 159, 99–130. 2. Hurford, J. (1999). The evolution of language and languages. In R. Dunbar, C. Knight, & C. Power (Eds.), The evolution of culture (pp. 173–193). Edinburgh: Edinburgh University Press. 3. Kirby, S. (1999). Function, selection, and innateness: The emergence of language universals. Oxford: Oxford University Press. 4. Steels, L. (1999). The talking heads experiment. Antwerp: Laboratorium. 5. Kaplan, F. (2000). L’émergence d’un lexique dans une population d’agents autonomes. Paris: Laboratoire d’Informatique de Paris 6. 6. Steels, L., & Belpaeme, T. (2005). Coordinating perceptually grounded categories through language: A case study for colour. Behavioral and Brain Sciences, 28, 469–489. 7. Dewey, J. (1929). Experience and nature. New York: Dover. 8. Gärdenfors, P., & Warglien, M. (2013). The development of semantic space for pointing and verbal communication. In J. Hudson, U. Magnusson, & C. Paradis (Eds.), Conceptual spaces and the construal of spatial meaning: Empirical evidence from human communication (pp. 29–42). Cambridge: Cambridge University Press. 9. Brouwer, L. E. J. (1910). Über ein eindeutige, stetige Transformation von Flächen in sich. Mathematische Annalen, 69, 176–180._____________Explanation of symbols: Roman numerals indicate the source, arabic numerals indicate the page number. The corresponding books are indicated on the right hand side. ((s)…): Comment by the sender of the contribution. Translations: Dictionary of Arguments The note [Concept/Author], [Author1]Vs[Author2] or [Author]Vs[term] resp. "problem:"/"solution:", "old:"/"new:" and "thesis:" is an addition from the Dictionary of Arguments. If a German edition is specified, the page numbers refer to this edition. |
Gä I P. Gärdenfors The Geometry of Meaning Cambridge 2014 |
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> Counter arguments against Gärdenfors
> Counter arguments in relation to Fixed Points
Ed. Martin Schulz, access date 2024-04-19