Abstract
In this chapter I take the reader on a journey from a naïve realist position through to the beginnings of pluralism. Some simplifying assumptions are made, but this is done in order to introduce some of the concepts we find in pluralism, not to defeat all realist positions. In particular, in order to set the stage, the naïve realist will take Zermelo Fraenkel set theory to be the foundation for mathematics in a philosophically robust sense of capturing the essence, ontology and absolute truth of mathematics. The reader is given several reasons to abandon the naïve realist conception and to consider a more pluralist conception. The main aspect of pluralism discussed here is pluralism in foundations. ‘Pluralism in foundations’ is an oxymoron, and therefore, is unstable. Some other aspects of pluralism are then introduced: pluralism in perspective, pluralism in methodology and pluralism in measure of success.
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Notes
- 1.
As we shall see in the subsequent chapters, I shall take the reader through journeys with other starting points: naturalism, structuralism and formalism. Balaguer (1998) works through different versions of realism, and teaches us to use the word carefully. It is quite possible that no one presently holds the position I give here. It is, admittedly, a caricature. That does not matter for present purposes, since (1) the point is to start from a familiar position, not an occupied and carefully defended position, and (2) this chapter is not meant as a knock-down argument against realism in all its forms. Rather, we begin with a naïve and familiar view in order to introduce pluralism.
- 2.
This is quite different from a ‘full-blooded realism’ (Balaguer 1998, 5).
- 3.
The definitions in this chapter are to be read as working definitions. As such, in a more exacting context, they might require further refinement. Definitions are repeated in the glossary.
- 4.
Obviously, being a professional mathematician does not preclude one from having philosophical thoughts or from writing quite philosophically about mathematics. The distinction here is not professional but conceptual, in the sense of philosophical and mathematical problems or puzzles requiring different sorts of solution.
- 5.
By ‘faithfully’ I mean that the language being translated into, here the language of ZF, has the expressive power to capture the nuances of the original concepts as expressed in the original language. A test for loyalty of a definition, say, would be that analogues of all of the same theorems can be derived when the definition is expressed in ZF as can be derived using the definition of the original language, all other definitions, theorems, lemmas and proof techniques remaining equal. In contrast, a reduction would be unfaithful if fewer or more (non-equivalent) theorems could be derived. Ancient Latin can only make an unfaithful translation of a modern computer manual.
- 6.
This point was made in the oral presentation of the material, (Computability in Europe 2010) but is not obvious in the written version.
- 7.
Ontology is usually presupposed to be consistent. There are no impossible objects, there are no pairs of objects whose existence precludes each other. Of course, paraconsistent ontologies are a different matter (no pun intended). For our gross sketch, we need not consider this added complexity.
- 8.
In the last chapter of the book, I am more explicit and subtle than this. It will turn out that the pluralist is, in some respects, a type of sceptic, and he is neutral on the realist, idealist axis of debate; but this added subtlety will be introduced in due course.
- 9.
The idea of The Book of Proofs has a history. In the original conception, all perfect proofs were entered. There was no guarantee that there would only be one proof for each theorem, since there was no presupposition that there was only one founding theory for mathematics. However, if we assume monism in foundations, then The Book of Proofs will only have one proof per theorem.
- 10.
If our second-order quantifiers are Henkin quantifiers, then we only appear to have greater expressive power than in first-order ZF. See for example Väänänen (2001, 504–505).
- 11.
For a discussion of the difficulty in interpreting Frege on this issue see Boolos (1998, 225).
- 12.
Under a substitutional reading, V = L. Definitions are in the next section. For those in the know: under an objectual reading, V = L is independent. So the size of the set theoretic universe is decided if we insist that ZF2 includes a substitutional reading of the quantifiers.
- 13.
In systems which allow empty domains, universal formulas (ones with the universal quantifier as the main operator) do not entail the existential counter-part. So ∀x(Fx)\( \vdash \) ∃x (Fx) is not valid in such systems. Some of these constructive systems have a separate second-order predicate E, which is a sort of metaphysical constant that does indicate ontological commitment. The existential quantifier is read strictly as ‘some’, never as ‘there exists’.
- 14.
For the realist, we would prefer the word ‘absolute’. If the anti-realist is of the cloth marked: ‘truth is epistemically constrained’, then he could still very well be convinced that there is a unique such truth, but it is determined by our epistemology.
- 15.
There are different versions of the axiom, or family of axioms of choice. We shall only be as specific as we think it necessary for the purposes of the considerations being made.
- 16.
An extension of a theory is conservative if no new theorems can be proved, so really the ‘extension’ is in redundant shortcuts. A theory is non-conservatively extended if new theorems can be proven.
- 17.
A trivial theory in mathematics is one where every well-formed formula written in the language of the theory is true, so in particular, the negation of every formula is also true. This make the theory quite useless. To make such a theory, one would have to consider ex contradictione quodlibet inferences to be valid, and there would have to be a contradiction derivable in the theory from the axioms. We could then prove any formula using ex contradictione quodlibet arguments. We shall visit trivialism several times in this book.
- 18.
It is possible that no extension is correct.
- 19.
The rub lies in what is to count as a solution. I think it is safe to say that a naïve view of solution was assumed in these writings. That is, a solution is a definite and unique answer. With the Gödel archives being made increasingly available, this view of Gödel might be revised.
- 20.
Gödel was not using ZF as a foundation. He, with Bernays had developed their own set theory.
- 21.
Gödel is usually interpreted as having hoped and believed that we would eventually find some very powerful axioms that would determine the correct and unique truth for us. This interpretation of Gödel might soon be revised.
- 22.
Of course, if the foundational monist is correct, then he is right to try to force mathematicians to stay on the straight and narrow path. It is the same with religious fundamentalists. If they are correct in their beliefs, then they are doing exactly the right thing to try to force others to adhere to the correct faith. Until more evidence is in, however, such attempts at forcing are, at best, patronising.
- 23.
This point was very nicely made by Sebastik in the Logic Colloquium 2010 presentation: On Bolzano’s Beyträge, Paris, 31 July 2010. There he points out that he will insist on the use of the word ‘set’ when translating Bolzano, but to remember that the word ‘set’ is ambiguous, especially in Bolzano. This is nothing new, and it is not outrageous to think of set this way. As a point of comparison, Sebastik points out that he word ‘atom’ too meant something very different to Democritus than it did to Bohr. So while many modern readers might see the word ‘set’ and think immediately of ZF sets, and therefore be confused or feel deceived when reading Bolzano, they should instead realise that ZF has no monopoly over the use of the term.
- 24.
See the discussion of the proposed axiom V = L above.
- 25.
There is a heated debate between the category theorists and the set theorists on the ‘Foundations of Mathematics’ website. The argument is over whether set theory can say everything category theory can, so is the more fundamental theory, or whether category theory can say everything set theory can, so is the more fundamental. It seems, from the outside, that at this point in the debate, the two theories, or programmes, climb against each other. At present there is no obvious end to the debate.
- 26.
Shapiro does make some conciliatory remarks about being more general, and adopting alternative perspectives. Nevertheless, while he acknowledges that this is a possibility, he proceeds as though model theory is the only perspective. There is some tension in his writing. Whether the model theory perspective is an end point or a starting point depends on one’s reading of Shapiro. I invite him to join me in becoming a pluralist in perspective, if he is not so already.
- 27.
Epistemia should not be confused with heuristics, how we learn, our private experience of knowledge, how we come to form beliefs and so on. Epistemia is an idealised notion of knowledge tout court.
- 28.
The term “maverick philosophers of mathematics” appears in a conference announcement for a conference in June 2009 held at the university of Hertfordshire. Originally, it comes from Asprey and Kitcher (1988, 17).
- 29.
Some philosophers would not count Hilbert as a philosopher. For them, he was a mathematician, whose mathematics and suggested programme had philosophical implications. I prefer to err on the side of generosity, and allow him into the philosophical fold.
- 30.
There are different ways of stating the thesis. The point is that in most versions there will be an irreducibly vague or ambiguous philosophical term, otherwise we do not have an interesting thesis, but instead we have a tautology, or stipulative definition (Folina 1998).
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Friend, M. (2014). The Journey from Realism to Pluralism. In: Pluralism in Mathematics: A New Position in Philosophy of Mathematics. Logic, Epistemology, and the Unity of Science, vol 32. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-7058-4_2
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