Varieties of Pluralism and Objectivity in Mathematics
The phrase ‘mathematical foundation’ has shifted in meaning since the end of the nineteenth century. It used to mean a consistent general theory in mathematics, based on basic principles and ideas (later axioms) to which the rest of mathematics could be reduced. There was supposed to be only one foundational theory and it was to carry the philosophical weight of giving the ultimate ontology and truth of mathematics. Under this conception of ‘foundation’ pluralism in foundations of mathematics is a contradiction.
More recently, the phrase has come to mean a perspective from which we can see, or in which we can interpret, much of mathematics; it has lost the realist-type metaphysical, essentialist importance. The latter has been replaced with an emphasis on epistemology. The more recent use of the phrase shows a lack of concern for absolute ontology, truth, uniqueness and sometimes even consistency. It is only under the more modern conception of ‘foundation’ that pluralism in mathematical foundations is conceptually possible.
Several problems beset the pluralist in mathematical foundations. The problems include, at least: paradox, rampant relativism, loss of meaning, insurmountable complexity and a rising suspicion that we can say anything meaningful about truth and objectivity in mathematics. Many of these are related to each other, and many can be overcome, explained, accounted for and dissolved by concentrating on crosschecking, fixtures and rigour of proof. Moreover, apart from being a defensible position, there are a lot of advantages to pluralism in foundations. These include: a sensitivity to the practice of mathematics, a more faithful account of the objectivity of mathematics and a deeper understanding of mathematics.
The claim I defend in the paper is that we stand to learn more, not less, by adopting a pluralist attitude. I defend the claim by looking at the examples of set theory and homotopy type theory, as alternative viewpoints from which we can learn about mathematics. As the claim is defended, it will become apparent that ‘pluralism in mathematical foundations’ is neither an oxymoron, nor a contradiction, at least not in any threatening sense. On the contrary, it is the tension between different foundations that spurs new developments in mathematics. The tension might be called ‘a fruitful meta-contradiction’.
I take my prompts from Kauffman’s idea of eigenform, Hersh’s idea of thinking of mathematical theories as models and from my own philosophical position: pluralism in mathematics. I also take some hints from the literature on philosophy of chemistry, especially the pluralism of Chang and Schummer.
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