|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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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.
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.
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.
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)).
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.
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.
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.
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.
Fixed Point/Communication/Gärdenfors: how do we know if a fixed point (balance, agreement) has been reached?
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.
The Geometry of Meaning Cambridge 2014