CEPA eprint 1463 (EVG-175)
A rejoinder to Otte
Glasersfeld E. von (1995) A rejoinder to Otte. Philosophy of Mathematics Newsletter 8: 18–19. Available at http://cepa.info/1463
From the perspective of the oldthe new is always wrong.E. Weichselbaum
Reading Michael Otte’s contribution to Paul Ernest’s recent book on Constructing Mathematical Knowledge (Falmer Press, 1994), I realized once again how difficult it is to prevent philosophically trained readers from grafting their interpretation of constructivism upon the prevailing philosophical presuppositions. Otte’s title “Is Radical Constructivism Coherent?” heralds an examination of the internal consistency or non-contradictoriness of the constructivist proposal. However, Otte proceeds to criticize it for not being consistent with conventional epistemology. He brings to bear a number of erudite arguments that would clinch his dismissal of radical constructivism if – and only if – the constructivist orientation were based on and required the metaphysical assumptions of traditional theories of knowledge.
As Otte refers to me several times in his article, I feel justified in trying once more to clear up the misunderstanding which, by the way, has led other authors to make similar but sometimes far less civilized comments. In fact, it was a pleasure to read Otte’s critique, because he approaches the subject from a fund of knowledge and with a clarity that no other critic I have so far come across could muster. Nevertheless I contend that his analysis misses the fundamental element that sets radical constructivism apart from the epistemological tradition. This becomes patently obvious in the concluding paragraph of his paper.
Epistemology is not independent of metaphysics, because if we insist on identifying the object with the definition theory gives of it, we also. perhaps unwittingly, tend to define the human subject. This is in opposition to the idea that the essence of man is existential freedom. Mathematics may in part construct its own reality but always in face of the continuum of yet undefined real possibility. Otherwise such a construction loses its subject becoming instead a quasi-mechanistic process, as in the case of radical constructivism. (p.61)
The initial statement in this quotation formulates the very dogma that, as I have reiterated many times, prompted constructivism to break with tradition and to devise a theory of cognition that does without the metaphysical assumptions inherent in any ontology. Constructivism is intended to provide a model of how we may come to know the things we call ‘objects’ and is in no way concerned with what their ontological source might be. At best it may define objects we know. It has no ambitions concerning objects as they might be independent of a knowing subject. Nor does it want to define the knowing subject as an ontological entity, The subject enters merely as the active agent of construction, characterized by the repeatable operations it seems to be carrying out. As Otte says of mathematics, the subject “constructs its own reality,” and its construction, too, faces a continuum of undefined possibilities (and impossibilities) that arise from the constructs with which it started. These possibilities and impossibilities are ‘real’ in the world of knowing, not ontologically real in the world of being.
In Otte’s piece, the misunderstanding is launched at the beginning by his reference to Parmenides, who first spoke of an “identity between thinking and being” (p.50). I would suggest that the Parmenidean dictum is ambiguous: if “being” is taken in the ontological sense, the statement is that of a mystic who believes that our thinking and knowing is divinely preordained to coincide with the world as it is; but I prefer to take it in Berkeley’s sense, namely that all that ‘exists’ for us is what we perceive and conceive. If one accepts this second interpretation, one is free to adopt any metaphysical belief one chooses, including Berkeley’s that the existence of the real world is vouchsafed by the fact that God perceives it. Hence I do not think that constructivism demeans “the dignity of the individual human subject” (p.51). On the contrary, it enhances it by strengthening an attitude of tolerance and freedom, since it never resorts to claims of ontological truth as a weapon against others.
The confusion surfaces at other points. When Otte argues against Luhmann and says “We talk with other subjects about objects…” (p.52), he implies that these objects are ‘things in themselves’ and not, as Luhmann and I would say, constructs arising from a subject’s distinctions in his or her experiential world.
Being a meticulous thinker, Otte raises a point that goes to the heart of the matter. “If we represent an idea is there something beyond the individual symbolic representation? Are numbers different from numerals?” (p.55). In Children’s Counting Types (Steffe et al., 1983), we presented the constructivist answer to that question. Numerals “represent” numbers to any subject who has acquired the mental operations necessary to construct numbers and is able to re-present them to him or herself. For a constructivist, this is analogous to the functioning of non-mathematical language. Words do not ‘refer’ to things in themselves, but they have the power to call forth re-presentations of experiences the subject has associated with them. Constructivism is simply a model that suggests how what we know and re-present to ourselves might be constructed in the first place. But for metaphysical realists it is as difficult to give up the traditional theory of objective reference as to relinquish the age old belief that knowledge somehow must reflect ontology.
Otte, M. (1994) Is radical constructivism coherent?. In P.Ernest (Ed.) Constructing mathematical knowledge: Epistemology and mathematical education (50-61). London: The Falmer Press. http://cepa.info/3654
Steffe, L.P., Glasersfeld, E.von, Richards, J., & Cobb, P. (1983) Children’s counting types: Philosophy, theory, and application. New York: Praeger.
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