the interesting work of John D. Godino " theory of the semiotic functions ontological-semiotic approach of mathematical cognition and instruction " draw the following comments:
"The application of ontological semantics of realistic mathematics corresponds to a Platonic view of mathematical objects (concepts, propositions, theories, contexts, ...). According to this philosophical position, the concepts and mathematical structures have a real existence independent of humanity, in some ideal domain. Mathematical knowledge is to discover the existing relationships that connect these objects. "
" This concept also implies an absolutist view of mathematical knowledge in the sense that it is regarded as a safe and immutable truths .
"Despite the distinguished representatives of this movement, among which include Frege, Russell, Cantor, Bernays, Hardy, Gödel, ..., there are new trends in philosophy of mathematics bring severe criticism absolutist perspective Platonic mathematics. A summary of these reviews and a view of mathematics from a fallible, based on the conventions of Wittgenstein and Lakatos' quasi-empiricism, can be found in Ernest (1991). "
" From the point of epistemological, pragmatic definition of meaning "Is much more satisfying than figurative realist theory: the disappearance of the concepts and propositions as independent data language, also dispels the problem of how these entities can be known, and we approach the phenomena that justify the dependence of thought and the experience with the language "(Kutschera, 1979, p.148)."
"From our point of view, the ontological assumptions of social constructivism as a philosophy of mathematics (Ernest, 1998) are also the adoption of the pragmatic theories of meaning. Mathematical objects should be considered as symbols cultural units, emerging from a set of applications related to problem-solving activities that make certain groups of people and evolve over time. "
" As explained so far, we find a dilemma between realistic and pragmatic theories seems difficult to overcome. However, Ullmann (1962) presents the type of pragmatic theories (he calls operational or contextual) as a valid complement realistic type theories (he calls referential). "
" This observation seems essential Ullmann and serves as a support for the model of meaning we propose in Chapter 3. For us the meaning begins to be pragmatic on the context, but there are types of uses that help to guide the teaching and learning mathematics. These types of applications are objectified through language and are the leaders of the vocabulary building. "
Source: Symbiotic's Blog
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