foundations of mathematics: Peter Aczel argues for a plurality of conceptual frameworks, coherence

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Peter Aczel argues for a plurality of conceptual frameworks, coherence
Does he mean multiple foundations?

Peter Aczel, Conceptual frameworks for foundations

Abstract

I will claim that, in view of the lack of resolution between the competing conceptual frameworks of the classical period and after and in view of Gödel's incompleteness theorems and later work in proof theory, there can be no absolute conceptual framework for mathematics.

Instead, in pursuing the philosophy of mathematics one should allow for a plurality of conceptual frameworks that will vary in the kind of mathematical practise and its extent that the framework covers and the degree of coherence of the ideas involved in the framework.

I will attempt to make clear an objective notion of conceptual framework that can combine a kind of mathematical practise with a belief system and with formal systems that can represent the practise or encapsulate the beliefs. Among the key examples of conceptual frameworks will be those from the classical period.