The Last Superstition – A Refutation of the New Atheism: Mathematics and Geometry
Part Seven? Seriously? Do not be misled – that I am over seven thousand words into a review of a relatively short book does not indicate that this is a good book. I’ll do a more summative review when I’m done.
In the meantime I am onto another section that is reviewing what isn’t in the book – in this case any critical comparison with Feser’s views on the inescapable logic of his position and mathematics.
Why is mathematics important? Fewer tells us early on:
But it is important to understand that, certain details and rhetorical flourishes aside, the core of Plato’s theory is admitted even by many who are unsympathetic to his overall worldview to be highly plausible and defensible, and has always had powerful advocates down to the present day. The reason is that at least something like Plato’s theory is notoriously very hard to avoid if we are to make sense of mathematics, language, science, and the very structure of the world of our experience.
Feser touches on mathematics at several point. He uses 2+2=4 as a standard example of a necessary truth:
Indeed 2+2=4 would remain true even if the entire universe collapsed in on itself.
That quote is taken from a longer section in which he offers nine arguments for Platonic realism:
- The “one over many” argument. (e.g. any specific triangle is a specific case of triangularity in general)
- The argument from geometry. (in short – geometry involves true stuff about abstract objects so how could abstract objects not be real?)
- The argument from mathematics in general. (Which is the second argument but now applied to maths more generally)
- The argument from the nature of propositions. (Which is the third argument but now applied to logic more generally)
- The Argument from science. (Science involves general principles.)
- The vicious regress problem. (If you try and avoid universals as somehow being real you have to appeal to other things such as resemblance but then ‘resemblance’ is a universal too and you are still stuck with universals.)
- The “words are universals too” problem. (Basically argument 6 again – you can’t avoid universals.)
- The argument from the objectivity of concepts and knowledge. (A more complex argument intended to address conceptualism in the way 6 and 7 address nominalism. Interestingly Feser casts this in terms of a mathematical example – when you and I both think about Pythagoras’s theorem we both are thinking of the same thing. That actually isn’t strictly true but it it is true enough to be an issue.)
- The argument from the possibility of communication. (Really just a variation on 8)
Of those points 2, 3 and 4 are specific to the philosophy of mathematics. Point 1 is more general but is core to question within mathematics. Points 5, 6, 7 and 8 are all applicable to mathematics both philosophically and psychologically.
It shouldn’t be a surprise that Platonism has been the most enduring basis for a philosophy of mathematics. There is a sense that even among people who haven’t seriously considered it there is a kind of naive default Platonism that assumes mathematical objects (numbers, shapes etc) are real in some abstract way.
Additionally mathematics demonstrates that there is a rational alternative to science. That may sound a little odd but only because we tend to lump the two together. Science is an empirical discipline – it involves contingent truths and observation. Mathematics on the other hand, offers a way to find apparently universal truths without any need for messy experiments or double-blind trials. That offers hope to somebody like Feser. Maybe there is a legitimate way to get at truths that are not scientific in nature but which can address what I would call the supernatural.
Scientific argument start from empirical premises and draw merely probabilistic conclusions. Mathematical arguments start from purely conceptual premises and draw necessary conclusions. Metaphysical arguments of the sort Aquinas is interested in combine elements of both these other forms of reasoning; they take obvious, though empirical starting points, and try to show that from these starting points, together with certain conceptual premises, certain metaphysical conclusions follow necessarily. And the empirical starting points are always so general that there is no serious doubt of their truth: for example, they will be premises like “More than one object is red,” or “We observe change going on in the world around us.”
In short: observe some general principle about the universe (it changes, there is stuff etc) and use that as axioms and then deduce everything else. What could possibly go wrong? Furthermore both the Western form of mathematics and Catholicism both have philosophical roots in Plato. It is a match made in heaven!
Feser name checks three notable figures in modern mathematics (two of whom are directly relevant to modern logic as well) Frege, Russell and Riemann. Riemann is mentioned only in passing in note 7 in relation to a point about in the section on Natural Law that Feser is making about triangles.
…it is of the essence, nature, or form of a triangle to have three perfectly straight sides . Notice this remains true even if some particular triangle does not have three perfectly straight sides, and indeed even though (as I’ve repeated ad nauseam) every material instance of a triangle has some defect or other.
Feser’s note 7 is a response to what would be a likely objection from people familiar with more modern (i.e. 19th century or later) mathematics:
7. Relativize to Euclidean space if you’re worried about Riemannian triangles. Romanian triangles will, in any case, have their own fixed natures, and thus could be equally well used to make the point.
This is the full extent that Feser addresses a major issue with all his comparisons with mathematics. In short Feser is appealing to a very old view of mathematics and a view that came under increasing challenge in the past few centuries and not because of wilfulness or evil or a desire to distance oneself from religion but because of deep mathematical problems.
The triangles are just the start of it.
The axiomatic model of reasoning was always exemplified by Euclid’s geometry. A set of apparently self evident axioms about geometry such as a line being the shortest distance between two points. From these and using some basic logic more complex geometric propositions could be demonstrated. It was powerful and convincing and also offered hope that reason alone could establish truth. There were still issues with that view not least of which being that really only geometry worked that way – arithmetic was not based on fundamental axioms for example until more modern times – but it remained an influential view.
Of Euclid’s axioms (http://mathworld.wolfram.com/EuclidsPostulates.html ) the fifth one was a bit of an oddity. Whereas the others were very basic the fifth one was a bit more complicated:
If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.
There are many ways of restating this and the easiest one to make sense of is that parallel lines don’t meet and that for a line and a given point not on that line there is a unique parallel line that goes through that point. What this axiom lacks is any sense of it being obviously true. Eculid’s other four axioms read more like self evident definitions and so the fifth one became a perennial mathematical puzzle. The assumption was that rather than an axiom (or postulate) it was a theorem – something that could be proven with the other four axioms alone. Many tried and all failed.
One method of proving it would be proof by contradiction. Assume that the fifth axioms is false and then show using the other four that a contradiction arises. People tried this and found that they couldn’t derive a contradiction. In the early nineteenth century some mathematicians made the conceptual leap: perhaps the fifth axiom did not have to be true.
This notion was revolutionary. It entailed, in effect, the notion of an optional axiom. In normal geometry (the geometry that Euclid had described) you might take the fifth axiom as true but in some other geometry you could replace it with a different rule about parallel lines. For example it is possible to have a geometry in which parallel lines meet exactly twice. Such geometries became known as Non-Euclidean geometries. Mathematicians such as Bolyai, Lobachevsky and Gauss all helped originate this field but Bernhard Riemann gave one of the most extensive treatments of it https://en.wikipedia.org/wiki/Bernhard_Riemann#Riemannian_geometry – hence Feser’s reference to ‘Riemannian triangles’.
Non-Euclidean geometry in itself does not demonstrate that the Platonic view of the reality of mathematical objects is false. It does add another layer of abstraction in so far as all Euclidean triangles share a quality of triangularity that includes some common properties (e.g. the internal angles having a sum of 180 degrees) but then at another layer of abstraction triangles in other geometries share a quality of triangularity that does not include the same range of properties (across several geometries the sum of the internal angles of a triangle varies). Platonism can still work with non-Euclidean geometries but it requires some work.
More importantly what the development of non-euclidean geometry meant was that the approach Feser outlined above for metaphysics is not reliable. Consider how closely Feser’s scheme for metaphysics matches geometry: people make some general observations about apparently fundamental aspect of the universe (e.g. for Aquinas that change happens, for Euclid that parallel lines don’t meet) and then take those observations as core truths and then deduce other things from them. Non-Euclidean geometry demonstrated that this was not a reliable approach – apparently sound fundamental truths about reality might be contingent truths. As time progressed it also became clear that this was not just some mathematical parlour trick; Henri Poincare, Albert Einstein and Emily Noether explored the implications of this radical geometry in physics. Einstein’s general theory of relativity found that gravity represented a geometrical distortion of space time such that the relevant geometry of the space about us was dependent on the mass of the objects in that space.
In short simply claiming even apparently unquestionable premises as self-evident truths is not defensible. The very notion of an axiomatic truth came under question in mathematics and not for sociological reasons or because of religious reasons but because mathematicians were engaging in useful, interesting mathematics that kept bumping up against these problems. Axioms began to look more like arbitrary rules of a game – starting points of systems and that choosing axioms was a pragmatic or aesthetic choice.
Coincident with this was a renewed interest in both logic and the fundamental foundations of mathematics. Fewer mentions both Frege and Bertrand Russell, figures who bridge the nineteenth century and the twentieth. Both were looking at logic and set theory as a way of identifying the most fundamental and the most abstract basis for mathematics. A full treatment is not possible here but as a philosophical or metaphysical exercise the project failed – the same issues of arbitrary choices of axioms (notably Zermelo’s Axiom of Choice in set theory) and paradoxical results prevented any one scheme being clearly the absolute truth. As a mathematical exercise and as an exercise in developing the field of logic the project was a massive success. As I’ve discussed elsewhere by the 1930s mathematics and logic had developed extraordinary new tools and insights. As a discipline it was enjoying unparalleled success but the break with metaphysics was getting deeper and deeper. Simply put there were multiple philosophical perspectives on mathematics, each with flaws and no way of deciding between them.
The practical outcomes of this inquiry into the fundamentals of mathematics helped usher in the age of electronic computing (it helped inspire both Alan Turing and John Von Neumann). It also helped generate alternate logics and a greater understanding of mathematics. What it didn’t do is make the metaphysics of Plato anymore convincing. The simple distinction of whether mathematics is discovered or invented could not be answered but our capacity to at least apparently invent new and weird mathematics on a whim certainly helped tip the scales.
Platonism did not die as a philosophy of mathematics. It is flexible enough to accommodate even the vast and contrary discipline that mathematics is today. Notable twentieth century figures in mathematics (Kurt Godel, Paul Erdos to name but two) operated within a framework of Platonic truth (i.e. mathematical objects were real things in an abstract realm). However the trite examples that Feser offers (2+2=4 rather than say 0 as it would in modulo 4 arithmetic, or pythagoras theorem which doesn’t hold in all geometries etc) are each true only within a specific framework. A mathematical Platonist can still argue that they are all true and real within an abstract realm (presumably with sturdy fences so that entities don’t wander) but what of Feser’s scheme?
Feser argues that his (or rather his view of Aquinas’s scheme) is like mathematics and in doing so he means the naive view of mathematics that has not been sustainable for nearly two centuries. It is not enough to simply declare a set of truths as being axiomatic. Indeed even in Euclid’s day you would have to do the hard work to demonstrate that the deductions from those axioms did not lead to contradictions AND were sufficient to actually deduce useful theorems (which is why Euclid needed the fifth axiom to start with – without it you can’t deduce even some quite basic properties of triangles such as the internal angle sum of 180 degrees that I mentioned earlier).
In short, mathematics is big and crazy and sophisticated. Feser’s sketch of a methodology for metaphysics is naive and one which we know was not sustainable in the optimum conditions of mathematics. Platonism is not dead in mathematics but it is not pre-eminent and more importantly the break between mathematics and metaphysics was more recent than the equivalent break in science. This more recent history is easier to trace and follow and it can be seen that it really did arise out of the practical considerations of mathematics as a discipline and not because of trendy or sinful wilfulness. Science broke from metaphysics earlier (i.e. metaphysics became irrelevant) but mathematics as a discipline did so as well and for the same reasons.