Definitions and relations system in the EOS below is included the contribution made by James H. Romero C. and Pedro Javier Rojas G. (Universidad Distrital, Bogotá, Colombia) forum, conference-idm-cos, on 21/2/2008.
When a theory is complex, made vague descriptions is necessary, its contextual nature and the ability to build relationships are essential, but since our formation "traditional" discipline like mathematics, we get used to a way of acting from "the" theory of sets, at least in the university context, "with a bivalent logic, which has led to consider as a necessity that any definition that is made is completely" bordeable "-as, given an object, we decide whether or not it meets As required in the definition.
Now, if we are talking about the pragmatic (linked with space and time), it can not be approached from the theory of sets, which required just rid of space and time. Although set theory arises from specific types of practice, was not intended to capture those systems practice. If we review a number of interventions made in the context of this seminar, we may be able to recognize that in some ways, we are attempting to address the complexity of the issues raised
from the EOS using forms or procedures logical and useful language for set theory to study the "classical" (compatible with bivalent logic) perspective, for example, strengthens in us the idea that if a proposition has a truth value, its negation is a truth value otherwise, which means acceptance of the principle of the excluded; principle from other successful mathematical theories can not be assumed, as in category theory, where such failure is a requirement.
When it comes to successful not only in terms of mathematical development but also in what he has to do with their technological applications (in communications, artificial intelligence and computer among others. In particular, we recognize that the use of fuzzy logic in the design of software to do things
with bivalent logic is impossible, as capturing basic archetypes in the scanner as a tool to convert image to text-). It must be remembered that in this community of practice there are many mathematicians who do not accept the principle of the excluded, among them are
authorities at the global and Pearce, Brouwer, Grothendieck and Thom [in Colombia we can cite two well-known mathematicians: Xavier Caicedo Fernando Zalamea and Arnold Oostra].
From what discussed above, we emphasize that, even in mathematics, there are ways different reasoning, they assume different bivalent logic and the theory of the EOS is participating in a more compatible with these different ways (not exactly new) to argue, reason, and do math.
Moreover, turning to one of the documents submitted to the Seminar (Godino, 2003), we emphasize two facts related to our way of "seeing the world '
(1) The intention of" edge "with a definition. If the production of thought and instruments are inseparable, the instrument is thought-it is impossible to separate language and thought (and also sign language). Thus, in the typology proposed mathematical objects, can recognize objects as language, situations and procedures, which are distinguishable, recognizable, characterizable, but not "strongly" separable. Speaking of clear and distinct ideas does not always mean separation, while
is impossible to eliminate certain "close" and / or overlapping. As an example of a recognition of this impossibility, we return the following quote by Godino (p. 15-16):
Sfard rejects the conception proposed by the signs and meanings as independent entities and adopts the view of psychologists like Vygotsky and semiotics as Peirce, that the signs (language in general) has a constitutive role objects of thought and not merely representational. Basically agrees with Wittgenstein postulated that the "meaning of a word is in use in the language, but also has the conviction that from the psychological point of view, the problem of meaning can not be reduced only to the analysis language.
The central thesis Sfard argues in this paper is that "the mathematical discourse and its objects are mutually constitutive discursive activity, including the continued production of symbols, is what creates the need for mathematical objects, and are mathematical objects (or rather the use of symbols mediated by objects) the which, in turn, influence the discourse and brings in new directions "
[From Godino (2003). theoretical frameworks on mathematical cognition, accessed February 14, 2008 at: http://www . ugr.es / ~ jgodino/fundamentos-teoricos/02_MarcosCM.pdf ]
(2) The pragmatic nature of the EOS. Since the proposed Godino, explicitly recognizes this fact, it is precisely from there that discusses the failure As proposed by Wittgenstein (p. 17):
[...] Wittgenstein's philosophy seems insufficient to base her analysis of the processes of mathematical study.
Designing mathematics and grammar the use of symbols and expressions solve the problem of explaining the necessity of the propositions, but not to explain the effectiveness of its application, or the motivation for its adoption. How do you create the rules? Not enough to know to follow the rules, we must understand their motivation, their application, and all knowledge derive new rules useful to organize our worlds.
Pragmatics requires the use of non-bivalent logic, as all human activity is multirelacional, is managed as multiple "planes" of experience of the subject / / society. Bulhões In terms of cognitive, expressive and appealing; from Grize: cognitive, expressive and regulatory, from Habermas is added to those proposed by Grize mandatory-level expression of power. One author, probably best known for the participants of this seminar is Duval, it captures the mathematical discourse this interrelatedness of the different levels present in a discursive experience in mathematics proposing three components of the sense of propositions contained in a speech.
insist, when it comes to human activity requires taking into account that this activity exists and runs at various levels, which interact and are structured in ways that respond temporarily to the structures of the action. Other examples
theories built from a complex environment with different ways of looking at the world, Beam Theory, Topos, Motives (mathematical theory), field theory of Bourdieu (Sociological Theory), ...
Finally, do not pretend here to exhaust rich and fruitful discussion, as it can be assumed, however, will try soon develop and complement some of the ideas briefly exposed. Jaime H.
Romero C. and Pedro Javier Rojas G
(Contribution to the forum, seminar-idm-eos, held on 21.2.2008)
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