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The aim is to use phenomenology to justify Brouwer’s choice sequences as mathematical objects

The aim is to use phenomenology to justify Brouwer’s choice sequences as mathematical objects.

In chapter 5, I explained Husserl’s principle that transcendental phenomenology provides the ontology for the a priori sciences. What the basic objects, and the axioms governing them, in an a priori science are, is to be disclosed in phenomenological analysis. Now I want to show this idea in action: the constitution analysis of choice sequences, undertaken in the previous chapter, yields a justification of one of the basic principles for choice sequences.

The purpose of this chapter is to bring out a conflict between Brouwer’s and Husserl’s philosophies of mathematics. I will begin by locating a particular point of disagreement, and pin down what it consists in. Disagreement by itself does not warrant speaking of a conflict. A further condition should be fulfilled, namely, the presence of mutual pressure. Two sources of such pressure are identified. Finally, we have to consider the ways in which the conflict could be resolved, and find one that is consistent with our phenomenological approach.

The purpose of this section is threefold. First, to provide a standard to evaluate Husserl’s and Brouwer’s original positions by. Secondly, to let this standard be the methodological clue to the reconstruction of choice sequences in chapter 6. Thirdly, to justify the revisionism implied in that reconstruction; which is all the more urgent since, as I will argue below, the kind of revisionism needed is not apparent in those of Husserl’s writings on which I rely. As the first and second will have to be addressed in serving the third, this section will take the form of an argument for revisionism.

In section 5.1, it was argued that with regard to pure mathematics, phenomenology is capable of ontological judgements. Generally, complete justification for asserting the existence of a supposed object consists in giving a strict constitution analysis; what is specific to the case of pure mathematics is that the laws governing strict constitution of its objects are precisely the laws of categorial formation. In a slogan, for formal objects, transcendental possibility and existence are equivalent. What has to be shown, then, is that choice sequences can be strictly constituted as formal, or purely categorial, objects. (In Tragesser’s felicitous turn of phrase, ‘something is recognizable as being a mathematical object if it can be recognised that it can be completely thought through mathematically’ [217, p. 293].) This will be attempted in two steps. The first is to show that choice sequences can be constituted as objects at all, the second, that this is constitution of purely categorial objects.

In chapter 5, I explained Husserl’s principle that transcendental phenomenology provides the ontology for the a priori sciences. What the basic objects, and the axioms governing them, in an a priori science are, is to be disclosed in phenomenological analysis. Now I want to show this idea in action: the constitution analysis of choice sequences, undertaken in the previous chapter, yields a justification of one of the basic principles for choice sequences.

One correct, phenomenological argument on the issue whether mathematical objects can be dynamic is not Husserl’s (negative) argument, but a reconstruction of Brouwer’s (positive) one. This I have argued by an attempt to justify, phenomenologically, the existence of one particular kind of dynamic object, the choice sequence. The phenomenological analysis was then applied to yield, in an attempt at informal rigour, a justification of the weak continuity principle for numbers.