Formulation of Symbol Grounding Problem
To answer this question we have to formulate the symbol grounding problem itself (Harnad 1990):
First we have to define "symbol": A symbol is any object that is part of a symbol system. (The notion of single symbol in isolation is not a useful one.) Symbols are arbitrary in their shape. A symbol system is a set of symbols and syntactic rules for manipulating them on the basis of their shapes (not their meanings). The symbols are systematically interpretable as having meanings and referents, but their shape is arbitrary in relation to their meanings and the shape of their referents.
A numeral is as good an example as any: Numerals (e.g., "1," "2," "3,") are part of a symbol system (arithmetic) consisting of shape-based rules for combining the symbols into ruleful strings. "2" means what we mean by "two", but its shape in no way resembles, nor is it connected to, "two-ness." Yet the symbol system is systematically interpretable as making true statements about numbers (e.g. "1 + 1 = 2").
It is critical to understand the property that the symbol-manipulation rules are based on shape rather than meaning (the symbols are treated as primitive and undefined, insofar as the rules are concerned), yet the symbols and their ruleful combinations are all meaningfully interpretable. It should be evident in the case of formal arithmetic, that although the symbols make sense, that sense is in our heads and not in the symbol system. The numerals in a running desk calculator are as meaningless as the numerals on a page of hand-calculations. Only in our minds do they take on meaning (Harnad 1994).
This is not to deprecate the property of systematic interpretability: We select and design formal symbol systems (algorithms) precisely because we want to know and use their systematic properties; the systematic correspondence between scratches on paper and quantities in the universe is a remarkable and extremely powerful property. But it is not the same thing as meaning, which is a property of certain things going on in our heads.
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