Abstract
The reader is introduced to pluralism from the starting point of structuralism. The starting position is Shapiro’s structuralist position. The pluralist is inspired by Shapiro’s position, especially by his anti-foundationalism and by his self-avowed pluralism. The pluralist joins Shapiro in argument against a naïve realist position. Thus, in this chapter, the arguments of Chap. 2 are revisited and made stronger and more precise. Similarly, the pluralist joins Shapiro in endorsing the idea of there being several mathematical structures or theories each only compared to others from a meta-perspective/theory/structure, which, in turn, can only be judged, or compared, from a further meta-perspective/theory/structure. However, the pluralist pushes Shapiro’s pluralism further. In particular, the pluralist will not be confined to the perspective guided and blinkered by classical second-order logic and model theory. What is in dispute, is both the classical conception of logic, especially the idea that inconsistency necessarily leads to disaster in the form of triviality, and the notion of success in mathematics. To make the last point, a distinction is drawn between the optimal and the maximal pluralist. The pluralism advocated in this book is a maximal pluralism.
The end of this chapter is written in a more aggressive style than the previous ones. This is simply for reasons of alleviating boredom.
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Notes
- 1.
Taking this attitude is pluralistic. As we shall see from the temper of the writing, the pluralist does not lack in combative spirit, rather, at the end of the day, what the pluralist hopes for is a clarification of positions and a deepening of understanding of positions: a sense of why one holds a particular position, and what some opponents have to say.
- 2.
Lobachevsky developed hyperbolic geometry. It was one of the first non-Euclidean geometries.
- 3.
Subtraction is a binary operator. Negation is an unary operator, or connective.
- 4.
Some readers might think that such revisionist arguments are absurd. In particular, they will think so if they are not seeped in the tradition of the philosophy of mathematics. If the reader does think that the arguments are absurd, then so much the better for the pluralist. This just makes his job easier, and his position more easily accepted. I applaud the anti-revisionist attitude, and I think that it is pretty plain and obvious in the contemporary climate. Pluralism is a thoroughly modern position, in this respect. Note the reverse: pluralism is a radical position. It does break from past philosophies. It would be irresponsible of me to not warn a reader of this if she has not been seeped in the tradition.
- 5.
In the first chapter I use ZF as The Foundation. This is because there, I wanted to discuss certain points about the axiom of choice. ZFC is a much more plausible foundation, since the axiom of choice is rife in mathematical practice. The switch also illustrates the flexibility of the pluralist. I could have also chosen category theory, or paraconsistent set theory, as The Foundation. I chose to stay on relatively familiar territory.
- 6.
Many philosophers are leery of using the term ‘essence’, so euphemisms are used instead. Feel free to substitute in your favourite circumlocution.
- 7.
Some philosophers would call this move ‘the naturalistic fallacy’: the fallacy of inferring a norm from a description.
- 8.
I am ignoring Frege’s later work of 1914 where he tries to found mathematics on geometry. In the above, read ‘Frege’ to mean the Frege of Begriffsschrift, Grundlagen and Grundgesetze.
- 9.
Vopĕnka is highly revisionary of mathematics too. But he proposes a different founding theory. This does not interest me here. What is important is that we should realise that the Platonist or realist proposal to found mathematics in set theory is sometimes taken to be normative of mathematics.
- 10.
Already this is interesting for the pluralist, since it instanciates the claim that what the mathematical community “takes to be The Foundation” changes over time.
- 11.
Model theory is used also. But my point remains. To illustrate: model theorists sometimes complain to Harizanov that she is drawing distinctions not recognised by model theory. For example, she sometimes insists on more than ‘uniqueness up to isomorphism’. She insists on including certain properties used to measure complexity when she is identifying structures or patterns. Model theorists cannot recognise these properties. This illustrates that there are norms which are not recognised by model theory, but are used in complexity theory. To illustrate the second point, Harizanov’s reaction is not to stop doing her mathematical work, or to consider what she is doing is not mathematics. Rather, she suggests to the model theorists that they should pay attention to more than what they can ‘see’ from a model theorist’s perspective. The illustration comes from conversation. There is no written reference. However, Rodin (2010, 25) gives a further example.
- 12.
The distinction is, of course, somewhat artificial, and if we do not accept it, then we re-phrase the structure of the foundationalist philosophy appropriately. Many mathematicians are also philosophers, and the same person can play both roles. I follow Colyvan in not recognising a clear distinction between philosophy and mathematics, either in terms of persons or in terms of roles. Despite my agreement with Colyvan, it will be useful for the arguments here to adopt this artificial distinction.
- 13.
It is exactly on these sorts of grounds that Maddy, in her earlier work forges a realist naturalist philosophy of mathematics.
- 14.
Oddly, Maddy was reluctant to make this observation, or take it seriously. This is one of the contentions between Maddy and the pluralist, which we saw in Chap. 3.
- 15.
‘Bad’, of course, is an over-simplification, especially in light of Hilbert’s famously stating that he was not willing to be expelled from the paradise Cantor had introduced to mathematics. Nevertheless, there is a tension in Hilbert’s attitudes towards the finitistic and the ideal.
- 16.
Frege’s logicism (paradoxes aside) is a little more subtle, since we can take logicism in a fairly neutral way. I don’t think that this is loyal to Frege, but I do think it is an interesting position. Under the neutral reading, the analytic part of mathematics is not so much ‘best’, but just analytic. We have a description, and no norm. Then it is just philosophically interesting to know what the scope is of the analytic part. The pluralist argument against the neutral reading of Frege, concerns much more the further part of this chapter, where we consider ‘bad’ mathematics.
- 17.
Brouwer agrees with this, so in this respect, he too, parts company with the monist. The issue about where Brouwer fits in my account is quite subtle. Where Brouwer and I part company is in his emphasis on intuition. I think that mathematical intuition is interesting, but I disagree with Brouwer that “mathematics takes place in the mind”, if we interpret the omitted quantifier at the beginning of the quotation as ‘all’. I save this issue for a paper.
- 18.
Hrbacek et al. (2009), are all working on a way of doing calculus using ZF set theory, but they add a notion of small and large relative to a frame of reference. There are many layers of ZF sitting on top of each other, so a number is very small relative to where one is sitting. They have tried teaching calculus in the classroom in this way, and found that it is much more intuitive for the students than calculus as it is normally taught! The very fact that there are mathematicians working on this shows us that the relationship between ZF and calculus is strained.
- 19.
We have to be very careful about quantifying over mathematical results, for, adding almost any axiom will add an effectively enumerable number of new theorems, so then we might count only ‘important’ new results, but how these are determined/chosen is again a problem; at least at any given time, since we might later discover that a theorem or result is important only many years later.
- 20.
It might turn out to be a matter of emphasis. Obviously every philosopher of mathematics, has to have been exposed to some existing mathematics, and taken that as a starting point. The difference in emphasis is over the hesitation or reluctance with which a naturalist will think he can tell the mathematician what counts as mathematics, once the technical result on his foundation has been demonstrated.
- 21.
There are even worse cases, from a foundationalist point of view. Kauffman showed me an example of a knot. He then translated from the language of knot theory into the language of set theory. The knot then seems to be an impossible object, since it is a knot where a ∈ b and b ∈ a, and this is set theoretically impossible. Assuming that the translation from knot theory to set theory is best possible, in some sense, then it is surprising that this makes no difference to the practice of knot theory. They do not defer to set theory at all, except to use the language on occasion.
- 22.
I really did choose this arbitrarily. The only constraint was to look for a short proof. I picked up the closest technical mathematical textbook I had to hand, and opened it to a middle page, and looked for a short proof. I do not think that it is important that the proof is of a meta-lemma, rather than simply an object-level lemma. Even the point about meta-language and object language still holds, since this lemma and proof mix meta-meta-level, meta-level and object-level languages.
- 23.
There is a lot more to be said about proof and the nature of proof. For a more thorough discussion, see Chap. 12.
- 24.
“…what the mathematician says [about the philosophy of mathematics] is no more reliable as a guide to the interpretation of their work than what artists say about their work, or musicians [about theirs].” (Potter 2004, 4), Even if we do not quite have such a strong point of view, it remains that mathematicians express very different philosophical attitudes. At the risk of being repetitive, my personal observation is that most mathematicians are pluralists.
- 25.
This is because, for example, 2 + 8 = 10 is false in arithmetic mod 8, where 2 + 8 = 2.
- 26.
In later parts of the book, I shall revise the simplifying assumption. Not only can we revise the assumption, but we must revise it when moving outside the model theoretic perspective. So the assumption does not offend the structuralist, but it will, at some point, be revised by the pluralist.
- 27.
The definition of Dedekind infinite is that a set is Dedekind infinite iff it has a proper sub-set with which it can be placed into one-to-one correspondence. The natural numbers are Dedekind infinite, as are the integers, the rationals, the reals and so on. Finite sets have no proper sub-set which can be placed into one-to-one correspondence with them. To capture the notion of Dedekind infinite, we need the expressive power of second-order logic. See Shapiro (1991, 100). The formula for set X being infinite is: INF(X): ∃f[∀x∀y(fx = fy → x = y) & → ∀x(Xx → Xfx) & ∃y(Xy & ∀x(Xx → fx ≠ y))]. This is read: There is a function which is such that if (two) of its values are identical, then the (two) arguments are equal. Moreover, the function operates on a proper subset of the set X.
- 28.
The title of Shapiro’s first book on structuralism is: Foundations Without Foundationalism The Case for Second-Order Logic. Note the “Without Foundationalism”. Foundationalism is identified with what I have been calling monism and dualism. Shapiro is anti-foundationalist in the sense that all mathematical theories which he recognizes are on a par. The ‘foundation’ is model theory. Model theory allows him to individuate mathematical theories (as structures). The model theory does not favour one structure as against another.
- 29.
This could be turned into a criticism of Shapiro’s structuralism, but it could equally be launched against the pluralist. However, in the next part we shall see the pluralist defuse it. The criticism is made indirectly in Potter and Sullivan (1997). The criticism is that Shapiro makes different ontological and metaphysical claims concerning individual models, on the one hand, and model theory itself, on the other.
- 30.
It depends on how we single these out. If all it takes for a theory to be intensional is that it have an intentional operator, then some intensional theories are extensional, and can be recognised by model theory. An example is a modal logic where the modal operators have terms within their scope. Such a logic will be extensional (models will be unique, and identified, up to isomorphism). See Melia (2003, 2–4). In contrast, a modal logic with whole formulas within their scope will suffer from ‘opacity of translation’, and are, therefore, not extensional theories. Model theory is extensional, even if it has no axiom of extensionality (since model theory has no axioms at all, i.e., it is not presented axiomatically). An extensional theory cannot recognise the differences between wffs with intentional operators because of the opacity of translation.
- 31.
We postpone discussion of these to Chap. 6.
- 32.
Distinguish between not having a known truth value (now), not having a truth-value at all, and having the ‘truth-value’ ‘unknown’, or ‘undetermined’. We sometimes use ‘U’ to indicate ‘unknown’ or ‘undecided’ and treat it as a truth-value, and make three-valued truth-tables with T, F and U, each as admissible ‘truth-values’. Strictly speaking this is sloppy. ‘Unknown’ or ‘undetermined’ are not truth-values. They are indefinite place holders for a truth-value. They are ambiguous between “there has to be a truth value T or F (not neither or both) but we have not worked it out yet” and “we do not even know if there is a truth-value T or F”. Above, I am not counting lack of truth-value as a truth-value. The parameters for what counts as knowable will depend on the resources we think we are allowed and to some extent on our theory of knowledge.
- 33.
A dialetheist is someone who holds that some sentences (or well-formed formulas) are both true and false. In particular they are true. We shall be introduced to the dialetheist more formally later.
- 34.
A Gödelian optimist is someone who has faith that we shall one day, or that it is (a priori) in principle possible for us to find The Foundation, or the absolute truth about an axiom. In particular, the Gödelian optimist thinks that we shall eventually determine, for example, whether or not the continuum hypothesis is true. We shall do this by finding a new very powerful axiom which will help us to derive the solution.
- 35.
For the distinction between an ‘informative’ and a merely ‘technical’ semantics see Priest (2006, 181). The distinction is not always clear, but roughly there are two parts to being informative: the intention behind the development of the semantics, and the ‘sense’ we can make of the semantics post facto. Intentions first: ‘technical’ semantics are developed with the intension of solving a problem, to provide a model for the syntax. In contrast, ‘informative’ semantics are developed in response to intuitions and ideas, which hold the formal theory responsible (we can judge the success of the formal theory by comparing it to the original intentions. For example, if my intention is to develop a temporal logic to reflect norms of reasoning over temporally indexed propositions, then my formal theory is judged with reference to the supposed norms). The post facto sense concerns what we make of the formal theory after is has been developed. For example, we might find that a purely technically developed semantics turns out to have an application, which makes ‘sense’ of the semantics. An example is quantum logic. In contrast, a technical semantics has only the intention, say, of proving consistency: if there is a model for a set of formulas, then that set is consistent. In this case, we just mechanically ‘give a semantics’, but we do not do so as an act of interpretation, which adds dimension to our understanding.
- 36.
There is plenty of sociological evidence for this. Witness publications by ‘major’ publishers, both as books and in journals; numbers, sizes, and sections newly contained in conferences. One telling example is the history of the world congress on paraconsistent logic.
- 37.
Shapiro’s pluralist structuralism cannot recognise paraconsistent logics and mathematics, since they cannot have a structure, since the logic he uses is classical second-order logic, and only consistent theories have a model.
- 38.
I should like to thank Goethe and Sundholm for sustaining some of these criticisms against me in conversation. Note that they were much more delicate and kind in their tone than what is reported in the imagined quotations!
- 39.
The term ‘continental’ was used, and sometimes still is used, by some Anglo-Saxon philosophers as a term of abuse, and it generally refers to the sort of work being done by some present day French and German philosophers, for the most part. The use of the term here is meant as an ironic joke. As we shall see, some of my leanings are distinctly ‘continental’ since having to do with the notion of meaning, the politics of meaning and communication. My reason for not discussing the ‘continental’ theories in more detail is that I am not sufficiently familiar with them to feel ready to discuss them in writing. While no names are mentioned, see the discussion of meaning earlier in this chapter, for an instance of these sympathies.
- 40.
I can only work towards attaining my posted ambitions.
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Friend, M. (2014). From Structuralism 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_4
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