Abstract
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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Notes
- 1.
There are other aspects in which one can be a pluralist. Since I shall be focusing on the question of objectivity, these aspects are sufficient to draw out the problem.
- 2.
I think that no one would claim a complete knowledge of mathematics, but some people might well claim a complete knowledge of some parts of mathematics, say, the theorem ‘2 + 2 = 4’ in Peano arithmetic.
- 3.
Arguably, there is always some ambiguity in any concept. For example, there is always ambiguity in context. However, the ambiguity might not be important for our concerns or purposes, or it might not be recognised yet, since we do not yet make the subtle distinctions necessary to notice the ambiguity.
- 4.
The exceptions are, for example, when a new way of teaching mathematics is adopted nation-wide. Experiments of this sort were introduced in France under the ‘Code Napoleon’, in the USA as the ‘New Math’ (Sic!) and might very well be the case in some countries with a language not known to many people outside the country – for lack of textbooks, schools of mathematical learning etc. All of these are geographical, historical and politically situated. The subject is not without interest.
- 5.
‘Immediate’ is not meant in the temporal sense of ‘right away’, or in the sense of ‘obvious’. Some constructively ‘immediate truths’ might take a long time to appreciate. It remains, however, that there is no further direct justification we can add to bring about appreciation. At best, we can give a number of examples, and expose someone to them, hoping they will ‘sink in’.
- 6.
This is a sort-of ‘clock arithmetic’, where the numbers go up to 8 and then start again at 1, going in a circle.
- 7.
I’ll explain the terms. Shapiro’s structuralism is a philosophy of mathematics characterised roughly as follows: mathematical theories can all be thought of as structures. A structure consists in a domain of objects together with some predicates, relations and functions that bear between the objects, and give structure to the objects, together with operations we can perform on formulas in a second-order set theoretic language. We use standard classical model theory to compare structures to each other. Group theory studies objects, such as the positive integers, which must include an identity element and operations that correspond closely to addition and multiplication. Group theorists impose their idea of groups, and findings about groups on other areas of mathematics, bringing new insights to those areas. Algebra is also a very basic way of looking at other areas of mathematics. Algebra is about working out which formulas are equivalent to which other formulas. Category theory was proposed as an alternative to set theory. In category theory, we have categories. They are made up of objects and ‘arrows’. The claim of category theorists is that (almost) any part of mathematics can be thought of as a category. Moreover, category theorists compare categories and work out meta-functions (called ‘functors’ in category theory) which take us from one category to another. This, too, tells us about connections between different areas of mathematics. Set theory was proposed as a real foundation for mathematics at the beginning of the twentieth century. It was so proposed because so much of mathematics was found to be reducible to set theory. That is, it was a mathematically comprehensive theory. Alternative ‘foundations’, such as Whitehead and Russell’s type theory, were proposed at the time or since the development of set theory. Homotopy type theory is more recent. Some mathematicians claim that it, too, can be thought of as a foundation for mathematics. From what I understand, it is based on the constructive dependent type-theory of Martin-Löf. What the homotopy type theorists bring to the table is not only the very sophisticated individuation of statements in mathematics in terms of type, but also the homomorphisms, that is, roughly, the relations and functions, between the types that are normally glossed over as equality. Each of the above areas of mathematics is foundational in the sense that they can make the claim that they can ‘see’ or ‘interpret’ much of what is counted as ‘mathematics’ today. Moreover, each theory brings its own way of looking and of seeing mathematics. We learn something from each, and we would deprive ourselves of understanding and insight if we were to ignore some.
- 8.
James Weatherall “Understanding gauge”. Paper presented at the conference: Logic, Relativity and Beyond 2015, Budapest, 9–13 August 2015.
- 9.
For the paraconsistent logicians: substitute ‘coherent’ or ‘non-trivial’ for ‘consistent’.
- 10.
I deliberately use the qualifier ‘mere’ to distance this attitude of giving up on the objectivity of mathematics from the Field’s factionalism which is a positive account of mathematics as fiction.
- 11.
A theory is consistent if and only if it has a consistent model, or less impredicatively, a theory is consistent just in case the underlying logic, or formal representations of reasoning is/are not paraconsistent or relevant, and there are no contradictions. A theory is trivial if and only if every well-formed formula of in the language of the theory can be derived in the theory. A theory is paraconsistent, or relevant, if and only if inconsistency in the theory does not engender triviality.
- 12.
The question whether intentionally working within a trivial theory should be acceptable, or under what circumstances it should be acceptable is under investigation by Luis Estrada Gonzalez.
- 13.
This has not been proved.
- 14.
It is an interesting and open question whether any correct proof can be arranged in a chunk-and-permeate fashion. It is an interesting and open problem whether any incorrect proof can also be so arranged, and if so what this means. These questions are under investigation by the author and others, especially my colleagues in Mexico. The questions are quite delicate especially if we are allowed to use paraconsistent reasoning or even trivial representations of reasoning within a chunk, but then if we have to use such representations then this might reveal too high a price for acceptance or objectivity of the conclusion. These are all open questions.
- 15.
The word ‘fixtures’ is supposed to be suggestive of the notion of a fixed point in mathematics, but it is a little looser than that of fixed point.
- 16.
This was less the case in the past, especially when geometry and arithmetic were kept quite separate. The later interaction between the two was also important. Today, we do not so much use arithmetic to check geometry and geometry to check arithmetic, but rather, we use set theory, and the ultimate tool: model theory, to do this. Or we use a combination of the two.
- 17.
Explaining the vocabulary: an embedding of one structure into another is a demonstration that the embedded structure is a part of a greater structure. A theory is complete if and only if the semantics and the syntax match in what they consider to be theorems (on the syntactic side) or be true (on the semantic side). That is, the syntax will prove all and only the truths of the theory. A theory is compact if and only if when an infinite set of theorems of the theory has a model, every finite set of theorems also has a model. The upward Lôwenheim-Skolem property states that if a countable theory has a countable model then it has a model for ‘every’ cardinal that is greater than countable. The downward Löwenheim-Skolem theorem states that if a countable theory has a model of countable size, then it has a model of every cardinality less than countable (i.e., of finite size). A theory is categorical if and only if all of its models have the same cardinality. A theory is decidable if and only if the syntax of the theory can decide in every case whether a conclusion follows from a set of premises or not (Shoenfield 1967).
- 18.
Equi-consistency is the proved result that if one theory is consistent then so is the other. That is, they are either both consistent, or both inconsistent.
- 19.
This is in a loose sense of ‘set’, i.e., not tied to a particular set theory.
- 20.
Both are widely held in the literature in philosophy of mathematics. And both are very suspect claims in light of pluralism.
- 21.
If we think that mathematics is more certain than observation, then none of it is ‘checked’ by observation.
- 22.
Renormalisation is a method of eliminating infinite quantities in certain calculations in electrodynamics. In electrodynamics, calculations involve finite quantities, this is plain from our physical conceptions. But the mathematical theory used includes infinite quantities, and they are the result of some straight experimental calculations. This is embarrassing, since it does not fit our physical conception. We therefore systematically (have a method to) eliminate such embarrassing quantities built into the use of the mathematical theory.
References
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Friend, M. (2019). Varieties of Pluralism and Objectivity in Mathematics. In: Centrone, S., Kant, D., Sarikaya, D. (eds) Reflections on the Foundations of Mathematics. Synthese Library, vol 407. Springer, Cham. https://doi.org/10.1007/978-3-030-15655-8_15
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