Constructivism

The emphasis in this thesis on interpretation, together with synthesis as an integral and necessary part of it, betray a constructivist point of view: interpretation and logical form are not out there in the data to be found by the reasoner, but they arise through an active and dynamic process of construction.

Constructivism emerged in mathematics mainly as a reaction to a platonistic conception of mathematical objects, that is, a conception of mathematical objects as existing indepen- dently of the mental acts of the mathematician. This is what Brouwer characteristically calls the ‘observational standpoint’ [4, p. 1]. From such a standpoint, mathematical prac- tice is devoted to discovering and exploring already existing mathematical facts. This view is very much like the attitude at the very opposite pole of the one adopted here, namely taking classical logic (or any ‘Universal Logic’) as the logic to recover the logical form in any given set of data. Taking classical logic to be the paradigmatic conceptual or cogni- tive framework (the universal language, the language of thought) forces one to adopt an observational point of view towards logic (cf. Frege). In contrast to this, the active process of interpretation is taken here to be part of what it means to assign a logical form. It is this point that is highlighted further here by explicitly subscribing to a constructivist point of view. Additionally, constructivism will provide some insight into the justification of logical laws.

Constructivism arose with questions of legitimacy concerning mathematical practice. Con- sider for instance constructivism as embodied in intuitionism, which is based on opposition to abstract, axiomatic mathematics and a simultaneous emphasis on internal, mental evi- dence for one’s mathematical assertions. One can view intuitionism as adopting a partic- ular logical form, namely the Brouwer-Heyting-Kolmogorov interpretation of the logical constants.1 In this interpretation the implication, for instance, is defined as a construction which transforms a proof of the antecedent into a proof of the consequent. However, one might object, would not such a view (i.e. intuitionism as arising from a particular logical form) contradict the spirit of intuitionism, which takes logic to follow from mathemat- ics and not the other way around? The next quote gives an example of this tendency in intuitionism:

Logic is not the ground on which I stand ... [a] logical theorem is but a math- ematical theorem of extreme generality; that is to say, logic is a part of math- ematics, and can by no means serve as a foundation for it (Heyting [20, p. 6]).

One way to understand pronouncements such as Heyting’s is via the earlier introduced dis- tinction of reasoning to, and reasoning from an interpretation (see Chapter 2). Heyting is not arguing here that a logical form (defined as parameter-setting resulting from reasoning to an interpretation) cannot serve as a foundation; indeed, one needs a meaning for the logical operators to get started. Rather, what he objects to is the idea that logical laws, the results of reasoning from the interpretation, can ever exhaust the logical form appropriate to intuitionistic mathematical reasoning. For example, the introduction rule for the impli- cation considers only the special case in which the transformation posited by Heyting con- sists in appending a piece of proof (establishing the consequent assuming the antecedent) to a proof of the antecedent. The introduction rule for the implicationA → B, together with the other rules, basically operate only on the information that can be extracted from the formula A, and the possibility to extract information from the proof of the antecedent is not used. Consequently, it is in principle possible that another introduction rule will be proposed which exploits the meaning of the intuitionistic implication to a greater extent. Thus, if the interpretation of the logical constants is that of Brouwer-Heyting-Kolmogorov, reasoning from that interpretation by means of the introduction rule for the implication does not exhaust the interpretation.

The conclusion of the preceding considerations is that it is not inimical to the spirit of intu- itionism to view it as the adoption of a certain logical form, as long as one does not assume that this logical form can be exhausted by concrete inference patterns. In fact, intuitionistic mathematics is chosen here as an example not because of its normative claims, but because it clearly shows how logic can play a constitutive role in cognition (here mathematical cog- nition), and also that when logic plays this role, it often cannot be conceived of as a set of rules only.

That constructivism can be found in other forms of mathematics beside intuitionism is well-known. Posy [32, p.130] makes a distinction between what he calls ‘constructivity of the right’: formalism in Hilbert’s program, and ‘constructivity of the left’: Brouwer’s intuitionism. The existence status of mathematical objects is a key to understanding Posy’s distinction. Roughly put, for constructivists of the left, only those objects exist that are constructed in a (albeit ideal) mind; a familiar example of what such a conception typ- ically excludes is the proof by contradiction. For constructivists of the right, however, objects that exist can in principle be constructed. Notice that formulated in these terms, a platonic conception of mathematics is only incompatible with ‘constructivity of the left’.