Symbols as Physical Objects
As mentioned above, it is well-known that symbols can be used as semantic and syntactic objects. I would, however, like to point out that symbols can also play a third and qualitatively different role in mathematical cognition, and that is the role as plain physical objects. This use of symbols is often manifested in pen-and-paper calculations. When for instance we multiply two numbers using pen and paper, the results of the sub-calculations are carefully arranged in columns and rows, and we use the previously written results as visual cues as to where we should write the next sub-result. In other words, the physical layout of the symbols is used as a way to guide the (epistemic) actions we perform on them (see [18, p. 125] for more elaborate examples).
The use of symbols as objects is also clear in the case of matrix multiplication. Here, the usual arrangement of the elements of matrices in columns and rows is a considerable help when we have to locate the elements we are about to operate on in a particular step in the process (cf. [11, p. 242]) . Notice, that the algebraic structure of matrix multiplication is completely independent of the usual physical layout of the symbols; the product of an m x n matrix A ¼ ½aij] with an n x p matrix B ¼ ½bjk ] can simply be defined as the m x p matrix, whose ik-entry is the sum:
(see e.g. [29, p. 178]). So in theory, it would be possible to perform matrix multiplication on two unsorted lists A and B of indexed elements. In that case, it would however pose a considerable task to find the right elements to operate on. By arranging the elements of the matrices in columns and rows in the usual way, the cost of this task is markedly reduced: The sum given above is simply the dot product of the ith row of A and the kth column of B, and if you know that, it is easy to find the elements you need. Thus, the multiplication process is clearly guided by the physical layout of the matrices.
It has also been suggested that the actual physical, or rather: typographical layout of symbolic representation of mathematical content has in some cases inspired new theorems and theoretical developments. Leibniz' derivation of the general product formula for differentiation is a case in point. Using the standard symbolism, the formula can be stated as:
It has been suggested that Leibniz' derived the formula by making a few, inspired substitutions in Newton's binominal formula:
(see [18, 24, p. 155] for further elaboration and more examples).
Words as Abstract Symbols
Finally, we might compare the use of abstract mathematical symbols with the use of written words. It should be noted that words are also abstract symbols (at least in alphabetic systems). In general, the physical appearance of a word has no likeness with the object, the word is supposed to represent; The word-picture ''point'', say, does not look like a point, and the word-picture ''eight'' does not have any more likeness with eight units than the abstract number-symbol ''8''.
Furthermore, words can carry mathematical content just as well as mathematical symbols. Of course, in general symbols allow a much shorter and more compact representation of a given content, but that is, in my view, only a superficial difference between the two representational forms. The important difference between written words and symbols is the fact that mathematical symbols, besides their role as bearers of content, can also be treated as syntactic and physical objects. With a few rare exceptions (such as avant-garde poetry), written words are never used as more than semantic objects; they cannot be used for purely syntactic transformations or as purely physical objects. For this reason, there are qualitative differences between written words and written mathematical symbols. We can simply do more with symbols than we can do with written words (cf. [18, p. 136]) .