Abstract
Realist philosophers of mathematics have accounted for the objectivity and robustness of mathematics by recourse to a foundational theory of mathematics that ultimately determines the ontology and truth of mathematics. The methodology for establishing these truths and discovering the ontology was set by the foundational theory. Other traditional philosophers of mathematics, but this time those who are not realists, account for the objectivity of mathematics by fastening on to: an objective account of: epistemology, ontology, truth, epistemology or methodology. One of these has to stay stable. Otherwise, it is traditionally thought, we have a rampant relativism where ‘anything goes’. Pluralism is a relatively new family of positions. The pluralist in mathematics who is pluralist in: epistemology, foundations, methodology, ontology and truth cannot account for the objectivity of mathematics in either the realist or in the other traditional ways. But such a pluralist is not a rampant relativist. In the paper, I look at what it is to be a pluralist in: epistemology, foundations, methodology, ontology and truth. I then give an account of the objectivity and robustness of mathematics in terms of rigour, borrowings, crosschecking and fixtures—all technical terms defined in the paper. This account is an alternative to the realist and traditional accounts of objectivity in mathematics.
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Notes
I should like to thank Amita Chatterjee for her excellent comments as referee for improving the paper. This paper and Mirna Dzamonja’s paper in this issue are complementary. I should also like to thank the Instituto de Investigationes Filosóficas at the Universidad Autónoma de México for support.
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.
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.
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.
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. But see the other papers in this issue! The subject is not without interest.
‘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’.
This is a sort of ‘clock arithmetic’, where the numbers go up to 8 and then start again at 1, going in a circle.
To 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 into 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 functions (called ‘functors’ in category theory) which take us from one category to another, or form a new category. 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 we can use them to ‘see’ or ‘interpret’ much of mathematics. 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.
James Weatherall ‘Understanding gauge’. Paper presented at the conference: Logic, Relativity and Beyond 2015, Budapest, 9–13 August 2015.
For the paraconsistent logicians: substitute ‘coherent’ or ‘non-trivial’ for ‘consistent’.
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.
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 logics 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.
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.
This has not been proved.
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 logics or even trivial logics within a chunk. If we have to use such logics then this might be judged to be too high a price for acceptance or objectivity of the conclusion.
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.
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.
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 are proved to be theorems (on the syntactic side) and truths (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 Löwenheim–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.
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.
This is in a loose sense of ‘set’, i.e. not tied to a particular set theory.
Both are widely held in the literature in philosophy of mathematics. And both are very suspect claims in light of pluralism.
If we think that mathematics is more certain than observation, then none of it is ‘checked’ by observation.
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 surface as 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.
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Shapiro, S. (1991). Foundations without foundationalism; A case for second-order logic. Oxford: Oxford University Press. (Oxford Logic Guides 17).
Tarski, A. (1986). ‘What are Logical Notions?’. J. Corcoran (ed.). History and Philosophy of Logic, 7, 143–54.
Wright, C. (1992). Truth and objectivity. Cambridge, Massachusetts: Harvard University Press.
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Friend, M.I. Varieties of Pluralism and Objectivity in Mathematics. J. Indian Counc. Philos. Res. 34, 425–442 (2017). https://doi.org/10.1007/s40961-016-0085-3
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DOI: https://doi.org/10.1007/s40961-016-0085-3