ON THE STATUS OF FORMAL SYSTEMS AND THE PRODUCTION OF TEXT




"The design of the following treatise is to investigate the fundamental laws of those operations of the mind by which reasoning is performed; to give expression to them in the symbolical language of a Calculus, and upon this foundation to establish the science of Logic and construct its method…"

(George Boole, 1854, 1958, p. 1)

 

At the turn of the twentieth century a paradigmatic revolution occurred in mathematical philosophy. The status given to mathematical systems and logical systems was radically altered; and, an intensive examination of the relations between these two systems was begun.

One of the central characteristics of this revolution may be grasped in relation to the work of Boole just prior to the turn of the century. In a work aimed at representing the "laws of thought," Boole developed powerful logical calculus which helped lay the foundations of modern symbolic logic. An while the Boolean algebra remains a central contribution, the paradigmatic revolution required a clear partitioning of formal systems from anything having to do with mind or thought.

As Frege (1959) stated it: "Mathematics is not concerned with the nature of our mind, and the answer to any question whatsoever in psychology must be for mathematics a matter of complete indifference (p. 105e)." A radical undoing of the psychologization of mathematics and logic was begun. Logic, which from the time of Aristotle to that of Boole explicitly considered itself directed to the analysis of thought and reasoning, was now to be purged of all psychologizing ideologies. Logics were now viewed as formal systems.

A process central to this accomplishment was begun in the mid-nineteenth century when mathematicians for the first time took up the investigation of logic. They soon were able to demonstrate two important characteristics of logical systems. First, that in addition to syllogistic logic, there was an unlimited number of logical systems (inferential forms of increasingly complex character). And second, in consequence of the unlimited producibility of logical systems, no complete list of valid forms could possibly be drawn up.

In consequence, logics had to be given the status of formal systems. No longer were they to be viewed as being "about" logic, or "about" thinking. And further, if no complete list of valid forms could be produced, there would be no possible way to argue that one particular formal system could be the uniquely true representation of the laws of production (i.e. "thought") of all formal and informal sytems. A formalism was a formalism, and no ontological status could be reasonably attributed to it.

The formalism is to be seen as a production. It is not to be uncritically identified with the conditions of its production. Certainly, there is no reason to consider that it formally represents the conditions of its production. Once produced, it is to be dealt with and manipulated according to its own characteristics. The formal system is completely indifferent to its conditions of production and to the lives of its manipulators.

This "indifference" represents a radical change in status, and as noted is critical to modern mathematical logic. However, while formal systems cannot provide representations of the conditions of their production, can we still investigate this issue without falling into the fallacy of psychologizing reification?

To do so, not only must we always be clear about the status of formal sytems, but we must also do away with any concern with the laws of thought. Not only can logic not be identified with thought, but thought does not give rise to logic. The concern is not with an abstract and hypothesized system of thought, but with the actual conditions of the production of formal systems.

While formal systems are abstract and formal, their production and continued availability are strictly dependent upon palpable conditions and spatio-temporal actions. Formal systems cannot be reduced to the conditions of their production, nor is their analytic character in any way affected by any understanding of their conditions of production.

An investigation of the conditions of production is an empirical science, and is thus radically different in status from any formal system. As an empirical science, it does not aim at any truthful description. It aims at the production of a heuristically valuable account. And accounts and formal systems should never be confused (though an interpretative model of the latter may function as a component of the former).

The investigation begins with the notion that the production and continued availability of formal systems is an empirical issue. Formal systems neither come from the air or the mind, nor do they live in the air of the mind.

The logician Henkin in an article on formal sytems (1967) makes the following revealing points. They are entered in Henkin's account as bracketed qualifications to his construction of a model of formal systems.

[The particular shapes of the several symbols employed in this or any other formal system are of no interest, so we never bother to describe or exhibit these symbols. The symbol "=" is our name for the identity symbol of System 1, but the identity symbol itself may have any shape whatsoever. Similarly, we employ "C1", "C2", and "C3" as names for three other symbols with whose forms we have no concern, except that we shall understand that they are distinct from one another and from the symbol =. A similar understanding will be implicit in our description of other formal systems.]…[When we write "Ci = Cj", as above, we are referring to the string of three symbols consisting of Ci followed by Cj. (In some works this string is denoted by "Ci Ç » Ç Cj", where the symbol " Ç " denotes the operation of concatenation.) A similar understanding holds hereafter whenever a string of symbol-names is used.]… [We use the symbol "=", as is usual throughout mathematics, to indicate that the two expressions between which it is placed denote the same thing. Similarly, " ¹ " is used to indicate that the expression on both sides of it denote two different things.] (p. 62-63, emphasis added).

And indeed, the investigation of formal sytstems qua formal systems requires this "similar understanding." Shapes and forms are of no concern or interest, save that they be "recognized" as distinct. The formal sytem is rightfully separated from its conditions of production and use. However, for those concerned with the conditions their traces are insistently present, though glossed over by the use of extra-formal terms and operations. The most obvious glosses are the "we shall understand" distinctness and concatenation rules.

There can be no formal system for us without the use of shaped markers; nor can there be a formal sytem without activities which give rise to distinctness. Activities of placing, separating, joining and moving are essential to the possibility of producing a formal system. From the empirical point of view, a formal system is a type of text. It requires the use of graphic marks. And it requires the human activities of operating on text. Distinctness is a textual accomplishment. Placing, separating, joining and moving are activities and products of textual manipulation. From an empirical point of view, they are not to be identified with extra-formal abstract locutions such as "operations of concatenation." They depend upon actual concrete operations of textual concatenation. The making of the markers, the moving of the marks and the reading of the marks as distinct as separate or as joined are all to be seen as textual activities and accomplishments.

Thus, an understanding of the production and use of formal systems requires an empirical account of the production and use of text.

The necessarily textual character of other formal systems, such as geometries, should be immediately apparent. For example, Frege (1959) points out that Euclidean proofs require the cutting off of line segments, the location of points, the joining of points, and so on. From an empirical point of view, these geometrical operations are to be seen as dependent upon producing and reading marked texts.

In sum, an empirical understanding of the production and use of formal systems requires that they be approached as a class of textual accounts. This may be contrasted with previous accounts which urge that they be aproached as a class of abstract entities or ideas.





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E-Mail: Laurence E. Heglar, PhD