Bernays, Dooyeweerd and Gödel – the remarkable convergence in their reflections on the foundations of mathematics

Abstract

In spite of differences the thought of Bernays, Dooyeweerd and Gödel evinces a remarkable convergence. This is particularly the case in respect of the acknowledgement of the difference between the discrete and the continuous, the foundational position of number and the fact that the idea of continuity is derived from space (geometry – Bernays). What is furthermore similar is the recognition of what is primitive (and indefinable) as well as the account of the coherence of what is unique, such as when Gödel observes something quasi-spatial character of sets. It is shown that Dooyeweerd’s theory of modal aspects provides a philosophical framework that exceeds his own restrictive understanding of infinity )to the potential infinite) and at the same time makes it possible to account for key insights found in the thought of Bernays and Gödel. When Laugwitz says that discreteness rules within the sphere of the numerical, he says nothing more than what Dooyeweerd had in mind with his idea that discrete quantity, as the meaning-nucleus of the arithmetical aspect, qualifies every element within the structure of the quantitative aspect. And when Bernays says that analysis expresses the idea of the continuum in arithmetical language his mode of speech is equivalent to saying that mathematical analysis could seen as being founded upon the spatial anticipation within the modal structure of the arithmetical aspect. The view of the actual infinite (the at once infinite) in terms of an “as if” approach (Bernays), that is, as appreciated as a regulative hypothesis through which
every successively infinite multiplicity of numbers could be envisaged as being giving all at once, as an infinite totality, provides a sound understanding of the at once infinite and makes it plain why every form of arithmeticism fails. Such attempts have to call upon Cantor’s proof on the non-denumerability of the real numbers – and this proof pre-supposes the use of the at once infinite which, in turn, pre-supposes the (irreducibile) spatial order of simultaneity and the patial whole-parts relation.