Philosophy of mathematics

 I’ve been watching a number of YouTube videos on this topic, although some of them are just podcasts with a fixed-image screen – usually a blackboard of equations. I’ll provide links to the ones I feel most relevant. I’ve discussed this topic before, but these videos have made me reassess and therefore re-analyse different perspectives. My personal prejudice is mathematical Platonism, so while I’ll discuss other philosophical positions, I won’t make any claim to neutrality.

What I’ve found is that you can divide all the various preferred views into 3 broad categories. Mathematics as abstract ‘objects’, which is effectively Platonism; mathematics as a human construct; and mathematics as a descriptive representation of the physical world. These categories remind one of Penrose’s 3 worlds, which I’ve discussed in detail elsewhere. None of the talks I viewed even mention Penrose, so henceforth, neither will I. I contend that all the various non-Platonic ‘schools’, like formalism, constructivism, logicism, nominalism, Aristotlean realism (not an exhaustive list) fall into either of these 2 camps (mental or physical attribution) or possibly a combination of both. 

 

So where to start? Why not start with numbers, as at least a couple of the videos did. We all learn numbers as children, usually by counting objects. And we quickly learned that it’s a concept independent of the objects being counted. In fact, many of us learned the concept by counting on our fingers, which is probably why base 10 arithmetic is so universal. So, in this most simple of examples, we already have a combination of the mental and the physical. I once made the comment on a previous post that humans invented numbers but we didn’t invent the relationships between them. More significantly, we didn’t specify which numbers are prime and which are non-prime – it’s a property that emerges independently of our counting or even what base arithmetic we use. I highlight primes, in particular, because they are called ‘the atoms of mathematics’, and we can even prove that they go to infinity.

 

But having said that, do numbers exist independently of the Universe? (As someone in one of the videos asked.) Ian Stewart was the first person I came across who defined ‘number’ as a concept, which infers they are mental constructs. But, as pointed out in the same video, we have numbers like pi which we can calculate but which are effectively uncountable. Even the natural numbers themselves are infinite and I believe this is the salient feature of mathematics. Anything that’s infinite transcends the Universe, almost by definition. So there will always be aspects of mathematics that will be unknowable, yet, we can ‘prove’ they exist, therefore they must exist outside of space and time. In a nutshell, that’s my best argument for mathematical Platonism.

 

But the infinite nature of mathematics means that even computers can’t deal with a completely accurate version of pi – they can only work with an approximation (as pointed out in the same video). This has led some mathematicians to argue that only computable numbers can be considered part of mathematics. Sydney based mathematician, Norman Wildberger, provides the best arguments I’ve come across for this rather unorthodox view. He claims that the Real numbers don’t exist, and is effectively a crusader for a new mathematical foundation that he believes will reinvent the entire field.

Probably the best talk I heard was a podcast from The Philosopher’s Zone, which is a regular programme on ABC Radio National, where presenter, Alan Saunders, interviewed James Franklin, Professor in the School of Mathematics and Statistics at UNSW (University of New South Wales). I would contend there is a certitude in mathematics we don't find in other fields of human endeavour. Freeman Dyson once argued that a mathematical truth is for all time – it doesn’t get overturned by subsequent discoveries.

 

And one can’t talk about mathematical ‘truth’ without talking about Godel’s Incompleteness Theorem. Godel created a self-referencing system of logic, whereby he created the mathematical equivalent of the ‘liar paradox’ – ‘this statement is false’. He effectively demonstrated that within any ‘formal’ system of mathematics you can’t prove ‘consistency’. This video by Mark Colyvan (Professor of Philosophy and Director of the Sydney Centre for the Foundations of Science), explains it better than I can. I’m not a logician, so I’m not going to expound on something I don’t fully understand, but the message I take from Godel is that he categorically showed there is a fundamental difference between ‘truth’ and ‘proof’ in mathematics. Basically, in any axiom-based mathematical system (that is consistent), there exist mathematical ‘truths’ that can’t be proved. It’s the word axiom that is the key, because, in principle, if one extends the axioms one can possibly find a proof.

 

Extending axioms extends mathematics, which is what we’ve done historically since the Ancient Greeks. I referenced Norman Wildberger earlier, and what I believe he’s attempting with his ‘crusade’, is to limit the axioms we’ve adopted, although he doesn’t specifically say that.

 

Someone on Quora recently claimed that we can have ‘contradictory axioms’, and gave Euclidian and subsequent geometries as an example. However, I would argue that non-Euclidean (curved) geometries require new axioms, wherein Euclidean (flat) geometry becomes a special case. As I said earlier, I don’t believe new discoveries prove previous discoveries untrue; they just augment them.

 

But the very employment of axioms, begs a question that no one I listened to addressed: didn’t we humans invent the axioms? And if the axioms are the basis of all the mathematics we know, doesn’t that mean we invented mathematics?

 

Let’s look at some examples. As hard as it is to believe, there was a time when mathematicians were sceptical about negative numbers in the same way that many people today are sceptical about imaginary numbers (i = √-1). If you go back to the days of Plato and his Academy, geometry was held in higher regard than arithmetic, because geometry could demonstrate the ‘existence’, if not the value, of incalculable numbers like π and √2. But negative numbers had no meaning in geometry: what is a negative area or a negative volume?

 

But mathematical ‘inventions’ like negative numbers and imaginary numbers allowed people to solve problems that were hitherto unsolvable, which was the impetus for their conceptual emergence. In both of these cases and the example of non-Euclidean geometry, whole new fields of mathematics opened up for further exploration. But, also, in these specific examples, we were adding to what we already knew. I would contend that the axioms themselves are part of the exploration. If one sees the Platonic world of mathematics as a landscape that only sufficiently intelligent entities can navigate, then axioms are an intrinsic part of the landscape and not human projections.

 

And, in a roundabout way, this brings me back to my introduction concerning the numbers that we discovered as children, whereby we saw a connection between an abstract concept and the physical world. James Franklin, whom I referenced earlier, gave the example of how we measure an area in our backyard to determine if we can fit a shed into the space, thereby arguing the case that mathematics at a fundamental level, and as it is practiced, is dependent on physical parameters. However, what that demonstrates to me is that mathematics determines the limits of what’s physically possible and not the other way round. And this is true whether you’re talking about the origins of the Universe, the life-giving activity of the Sun or the quantum mechanical substrate that underlies our entire existence.

Footnote: Daniel Sutherland (Professor of Philosophy at the University of Illinois, Chicago) adopts the broad category approach that I did, only in more detail. He also points out the 'certainty' of mathematical knowledge that I referenced in the main text. Curiously, he argues that the philosophy of mathematics has influenced the whole of Western philosophy, historically.