by Carl Pierer
In the first part of this essay, the axiom of choice was introduced and a rather counterintuitive consequence was shown: the Banach-Tarski Paradox. To recapitulate: the axiom of choice states that, given any collection of non-empty sets, it is possible to choose exactly one element from each of them. This is uncontroversial in the case where the collection is finite. Simply list all the sets and then pick an element from each. Yet, as soon as we consider infinite collections, matters get more complicated. We cannot explicitly write down which element to pick, so we need to give a principled method of choosing. In some cases, this might be straightforward. For example, take an infinite collection of non-empty subsets of the natural numbers. Any such set will contain a least element. Thus, if we pick the least element from each of these sets, we have given a principled method. However, with an infinite collection of non-empty subsets of the real numbers, this particular method does not work. Moreover, there is no obvious alternative principled method. The axiom of choice then states that nonetheless such a method exists, although we do not know it. The axiom of choice entails the Banach-Tarski Paradox, which states that we can break up a ball into 8 pieces, take 4 of them, rotate them around and put them back together to get back the original ball. We can do the same thing with the remaining 4 pieces and get another ball of exactly the same size. This allows us to duplicate the ball.
The second part of this essay demonstrated a useful consequence (or indeed, an equivalent) of the axiom of choice, known as Zorn’s Lemma and looked at a few applications of this Lemma. Two positions have been mapped out in the course of this essay. On the one hand, the axiom has very counterintuitive consequences, so much so that they’ve received the name of a paradox. On the other hand, the axiom proves to be very useful in deducing mathematical propositions. These considerations lead back to the question that had already been raised at the end of the first part: how are we to decide on the status of an axiom, on whether to accept it or reject it?
In this third and final part of the essay, we will take a more philosophical approach to this problem. In particular, we will look at a possible resolution offered by Penelope Maddy in her Defending the Axioms. The solution offered would lead onto further questions about the nature of mathematics: what is mathematics actually about? At the same time, Maddy’s view is based on a certain conception of proof that does not really reflect mathematical practice. The essay, due to limitations, only hints at a different perspective offered by looking at what mathematicians actually do and what role proofs play for them.
In her recent Defending the Axioms, mathematician and philosopher Penelope Maddy raises the question how set theoretic principles can be defended. More precisely, for Maddy mathematics progresses by proofs and proofs have to start somewhere. This somewhere are usually the fundamental assumptions or axioms of set theory. However, in contrast to the old Euclidean axioms, some of the set theoretic axioms do not convince by mere intuitiveness. The question thus becomes what are the proper methods to defend set theoretic axioms. In her book, Maddy attempts to answer this question by sketching how a special field of mathematics (so-called pure mathematics) came into being and how this coincides with questions asking about the foundations of mathematics becoming more pressing. This leads her on to investigate “how progress of pure maths is guided, of which mathematical entities and proof techniques are legitimate, of what constraints our methods must properly satisfy.” [i]
The brief sketch of the history of mathematics Maddy gives, the relation between mathematics and science undergoes a radical change. She begins with Plato, who considered mathematics the highest form of knowledge – as opposed to science, which he considered as uncertain and changeable. At the time of Galileo and Newton, this already began to change. For them mathematics and science were one and the same: “the great thinkers of that time – from Descartes and Galileo to Huygens and Newton – did mathematics as science and science as mathematics without any effort to separate the two.” [ii] Nowadays, this relation has been all but inverted: science, for the positivists of the Vienna Circle and their heirs, is the measure of all things. Any sentence was meaningful only insofar it expressed some content that could be empirically verified. Such a strong and radical position of course threatens to dispense with mathematics altogether (how could a principle such as the axiom of choice ever be empirical verified?). Thus, “their solution was to view mathematics as purely linguistic, as true by the conventions of language, as telling us nothing contentful about the world.”[iii]
How did we go from a conceived and practiced unity of science and mathematics in the 16th and 17th century CE to the radical theses of the Vienna Circle? For Maddy, this happened when mathematics slowly became detached from its worldly roots. In fact, the whole picture of mathematics and what mathematics is about changed. She proposes three ways in which this happened.
1. The pursuit of various purely mathematical goals gradually led mathematicians to new studies not motivated by their immediate application to the world. In the 19th century, gradually mathematics introduced non-empirical concepts, ideas that are not tied to the world we experience: complex numbers, n-dimensional spaces, and the like. This slowly led to the dominant orthodoxy today: Mathematics progresses by its own lights, independent of ties to the physical world. Legitimate mathematical concepts and theories need not have any direct physical interpretation.
2. Mathematical theories were protected from empirical falsification by positing a special realm of abstracta about which they remain true. This was the case for when Riemannian geometry developed in departure from the classical and much more intuitive Euclidean geometry.
3. Finally, the view developed that our best mathematical accounts of physical phenomena aren’t the literal truths Newton took them for, but free-standing abstract models that resemble the world in ways that are complex and sometimes not fully understood.
With these developments, the mathematician finds herself in an entirely new situation: pure mathematics detached itself from empirical science and needed no longer to be justified by application to observable phenomena. The question then arose how else to ground mathematics? “In this brave new world, where can the pure mathematician turn for guidance? How can we properly determine if a new sort of entity is acceptable or a new method of proof reliable? What constrains our methodological choices?” [iv] In this changed setting, mathematicians newly emphasis rigour: whereas earlier generations of mathematicians were happy with imprecise or even erroneous proofs, as long as it worked, the new, pure branch of mathematics had much stricter requirements. The central tool of this process was axiomatisation: as exemplified in Hilbert’s axioms for geometry and
Dedekind’s for the real numbers. Maddy continues:
But it was widely appreciated that simply laying down a list of axioms wasn’t enough to establish that they succeed in describing a genuine structure. … What emerged gradually – in the theory of functions, in Dedekind’s constructions of the reals, in the foundations of arithmetic and elsewhere – is that set theory provides a natural arena in which to interpret the myriad structural descriptions of mathematics, to settle which are and aren’t coherent. [v]
For Maddy, it is set theory which enables us to answer all sorts of mathematical questions: the existence of mathematical objects or structures is determined by finding an instance or surrogate for it in the set-theoretic hierarchy. The provability of propositions is decided by checking whether it follows from the axioms of set theory. As Maddy writes enthusiastically: “In this way, set theory furnishes us with a single tool that can give explicit meaning to questions of existence and coherence; make previously unclear concepts and structures precise; identify perfectly general fundamental assumptions that play out in many different guises in different fields; facilitate interconnections between disparate branches of mathematic now all uniformly represented; open the door to new strong hypotheses to settle old open questions; and so on.” [vi]
This view of set theory, together with the genesis of pure mathematics, provides Maddy with a twofold way to reject the problem posed by the Banach-Tarski paradox. The first is afforded by the third aspect of how pure mathematics became separated form empirical science: In fact, if pure mathematics does not give literal truths about the physical universe, but rather abstract models resembling the world, there is no reason to suppose that the (mathematical) consequences we might derive in the abstract sphere find exact implementations in the physical world. The drastic step to reject the axiom because it leads to impalpable consequences is not needed: instead of modifying our model, understanding that its application to the physical world is subtle might lead us to apply with due care, “making sure not to rely too heavily on its more esoteric aspects.” [vii] The second is to use the second aspect of the story: if we suppose that indeed the physical world is literally modelled by subsets of R3, going through the Banach-Tarski construction we show an impossible consequence of the axiom and hence empirically falsified it. However, as Maddy rightly points out, there is an issue here: why reject the axiom of choice and not some other premise – for instance the assumption that the world is modelled by subsets of R3? She suggests that in light of the success of the axiom of choice, it might be more reasonable to replace this assumption with the idea that the world is better modelled by all measurable subsets of R3 – which would push the Banach-Tarski paradox into the realm of abstracta, thereby depriving it of its bite. As Maddy writes: “Once we have those well-stocked warehouses, any candidate for an empirical confirmation or disconfirmation of the mathematics is more reasonably viewed as confirming or disconfirming the claim that the best model has been identified.”[viii]
Of course, at this point all the argument has done is to show that an empirically unpalpable consequence of an axiom in itself is not conclusive reason to reject the axiom. Nonetheless, it provides us with a good sketch of a way out of the predicament we found ourselves in at the end of the two previous parts of this essay. It would be interesting to continue from here and examine the question what mathematics is actually about – both in Maddy’s view and in that of other philosophers. Unfortunately, such an endeavour would go beyond the scope of the present essay.
An entirely different perspective is open up by challenging the assumption that Maddy departs from: what if proofs do not play this crucial role in mathematics? What if mathematical practice is guided by a different principle? There is a good argument to be made here, following the wonderful Eugenia Cheng’s presentation of morality in mathematics [ix]. In essence, her argument is that proofs – as understood by philosophers – do not play such an important role in mathematical practice. Instead, her point is that what is much more relevant for mathematicians is understanding and a sense of coherence. Regrettably, a full discussion of this theory cannot be given in the present essay.
In this last part of this essay, we have looked at Penelope Maddy’s approach to settling the question and presented a solution to the problem under discussion that draws on the development of pure mathematics as a discipline in its own right and distinct from any physical applications. This, however, raised further questions as to the subject matter of mathematics. In addition, it was suggested that this solution is based on a certain conception of proofs that does not reflect actual mathematical practice. While the further examination of these issues sadly lies beyond the scope of the essay, the present discussion illustrated a nice mathematical consequence of the axiom of choice which runs counter our physical intuitions. In the second part, we showed that there are some consequences to the axiom that are highly productive in mathematics and hence mounted a kind of defence of the axiom. In the present part, we showed one way in which this particular axiom can be defended.
Cheng, E. (2004, January 23). Mathematics, morally.
Talk at the Cambridge University Society for the Philosophy of
. Cambridge. Retrieved from http://cheng.staff.shef.ac.uk/morality/
Maddy, P. (2013).
Defending the Axioms: On the Philosophical Foundations of Set Theory.
Oxford: Oxford University Press.
[i] (Maddy, 2013)
[ii] (Maddy, 2013)
[iii] (Maddy, 2013)
[iv] (Maddy, 2013)
[v] (Maddy, 2013)
[vi] (Maddy, 2013)
[vii] (Maddy, 2013)