Aryaman Gupta

What is the Philosophy of Mathematics? What are the ‘big questions’ in the field? And how have the answers to those questions affected the subject of Mathematics? This is the first in a series of 3 short articles on the Philosophy of Mathematics, which will try to address the three questions listed above, as well as convince you that the subject has a much bigger effect on your life than you ever thought possible…

Let’s begin with the obvious – what even is the Philosophy of Maths? Well, it is a subfield of Philosophy (and Maths too) that deals, basically, with the nature of Maths and mathematical objects. Typical questions include things like the ontology of maths (i.e. what sort of ‘things’ are mathematical objects?), whether maths is discovered or created, what counts as a sufficient formal proof, and what the foundations of mathematics are, or ought to be. 

It’s been speculated upon since the very beginning of Philosophy itself, by people like Pythagoras (who claimed absolutely everything, up to human ideas and emotions, could be grasped by maths) and Plato (on who’s academy’s gate was inscribed ‘let no one ignorant of geometry enter’).

In this article, we will look at the first question – that of the ontology of maths – since that is, in some sense, the ‘most fundamental’ one, upon which our concerns for all the others must be based. In particular, I will walk us through the three biggest schools of thought – Platonism, Formalism, and Intuitionism.

Given that, as Russel (a great Philosopher of Mathematics himself) concedes, all western Philosophy ‘consists of a series of footnotes to Plato’, we start with Platonism. By far, it is the oldest position in the Philosophy of Mathematics, and the one that was held for the longest amount of time, reigning virtually unchallenged from classical times until the enlightenment. 

Broadly speaking, Mathematical Platonism (deriving from Plato’s broader theory of ‘forms’) is an ontology of mathematics according to which mathematical objects are abstract, timeless entities existing objectively independent of the circumstances of the physical universe in a separate, abstract realm. We gain access to this abstract realm only through the faculty of reason, which interfaces our subjective consciousness with this objective world of forms. As such, mathematical truth, by this theory, is something objective, eternal and independent of human cognition – evidently, then, very close to what  most of us would perhaps assume by default. It captures all of our desires for mathematics to be objective, eternal and independent of physical reality in a neat little metaphysical bow.

However, what’s really clever about this system is that, following Plato’s original theory of forms, it even explains why our actual, physical universe seems to obey the laws of mathematics so well – despite placing mathematical objects entirely outside the physical realm. In particular, since forms are the ‘original blueprint’ of the universe, manifested in a partial, imperfect form (like a bed is constructed from it’s ideal blueprint by a carpenter, according to Plato’s allegory), the mathematical structure of their abstract, perfect form gets reflected (albeit not to the same degree of perfection) in their concrete manifestations! 

However, being dominant for so long, while the rest of Philosophy went on into radically different directions, a huge number of very severe problems eventually came to be identified. For one, with the maturation of empiricist Philosophy – which claimed our senses were the primary source of our knowledge, and that all supposedly abstract truth was to be treated as suspect – the problem of access became a point of repeated critique. The problem itself was this: how do we have access to objective, non-temporal and non-spatial entities, as beings who can (presumably) only perceive spatial and temporal things? If we can access this abstract realm, is it even possible to know whether our perception of it is the same as what it is in itself? 

Secondly, what, precisely, is the relationship between the physical world (which does seem to conform well to mathematical laws) and the abstract one? How can the former (which is non spatial and temporal) influence a spatial, temporal one? Does this influence mean that physical objects are influenced by non-physical entities? Again, this one was inspired by empiricism, since, driven by the frankly ridiculous success of Newton’s mechanistic description of the universe, they came to view the physical universe as being solely driven by physical causes, and were thus compelled to argue against any abstract influence of the sort that Platonism inescapably contends.

Thirdly, we don’t ‘see’ mathematical objects in external, objective reality – the only place we can ‘find’ them is our own minds. So how do we even know they exist outside of ourselves? 

In all these critiques, the broader issue in question was the fact that, simply put, Platonism carries too much metaphysical baggage – that is to say, there is ‘too much’ that one has to prove to exist (an abstract realm, mathematical forms within said realm etc.) in order to argue for Platonism. As such, both of the other two other ‘main’ positions try to distance themselves from Platonism in this regard.

The first of these is Formalism, which developed at the end of the 19th century as a response to the fascinating developments of non-classical forms of mathematics like non-euclidean geometry. These new non-classical forms showed that the seemingly ‘absolute’ rules of something like geometry – rules that were assumed exist unchangingly in a realm of their own – could, in fact, be chosen at one’s will, to an extent. 

Mathematical Formalism is a theory for the ontology of mathematics according to which mathematics is a sort of game of symbols and rules, where new theorems are nothing more than new configurations of said symbols by said rules. So, for example, the axioms of geometry, combined with the postulates, form a ‘geometry game’, where certain ‘tokens’ (representing geometric shapes’ can be manipulated in determinate ways (e.g. with two point tokens, a straight line token can be added between them). Crucially, these rules can be freely chosen as one wills at the start (so long as they aren’t contradictory, of course), as the free choice of parallelism axioms in non-euclidean geometry had shown.

Since these symbols are just arbitrarily assigned mathematical meaning, there’s none of the metaphysical burden of Platonism – that is, there’s nothing that needs to exist in a certain way for this theory to be true. So there’s no problem of access, or interface with the physical universe. The game is a purely formal one, whose truth or falsity refers to nothing more than it’s own configurations. 

But, remarkably, there is no loss of objectivity for this reduction in metaphysical requirements – this criteria of truth and falsehood under formal rules makes the criteria for a proof absolutely objective. Indeed, this is arguably even more so than Platonism, where proofs can sometimes have varying degrees of preciseness, and the criteria for a ‘proper’ proof may not be entirely clear. Here, we have a clear list of rules ensuring that we have an absolute standard to judge truth.

However, beyond these, the primary advantage of this view is that, if mathematical theorems are conceived as certain moves of a game, this formal view allows for one to ‘mechanically’ enumerate all demonstrations within the proof system by listing out every possible sequence of moves – similarly to how, say, a chess computer can simulate all the possible outcomes of every possible subsequent move. The hope with this sort of process was that you could pretty much let a computer prove all true mathematical statements, by churning out every possible proof in turn (although most particular methods for this sort of truth enumeration are more sophisticated, however – see the computability article [COMING SOON] for more). 

The study of computability also shows, unfortunately, the failure of Formalism – namely, that every formal system of rules will have statements that are neither provable nor unprovable. Thus, not only is the ideal of proving everything by formal processes proven untenable, the proof enumeration is also, as a result of this, no longer possible.

On a more mundane level too, however, the seeming advantage of proving things step by step demands such long-winded and exhaustively rigorous arguments as to make such proofs effectively impossible to perform for most complex mathematical statements. Indeed, even when such absolute proofs are attempted, it is typically long after the result itself has been discovered, and can only be done with the assistance of a computer. As such, even with what advantages it retains despite issues of unprovability, in practice, they are too cumbersome to actually take advantage of.

Conveniently, this brings us to our third big theory, Intuitionism, which is effectively entirely opposite to Formalism, insofar as it has difficulties with asserting objectivity, but does so with the strength of almost perfectly reflecting the real experience of mathematical activity. 

Intuitionism basically asserts that mathematics is a language-less activity of human minds which is structured by the ‘categories of experience’ – broadly, lenses by which the mind views the world, and structures its experience of it. The two primary ‘lenses’ that Intuitionism typically deals with are those of space and time, with the former giving us the structures of geometry, by giving us an idea of space, proportion and distance; and the latter giving those of arithmetic by providing the mind with a concept of sequence, succession and numeration.

From these basic structures, we can ‘construct’ more complex mathematical entities out of them by intuitively conceiving and defining them, in the same way that, say, a musician constructs ‘subjective’ music using notes that are, in some sense, the foundational units that enable hearing. For a statement to be ‘true’, then, under Intuitionism, it must be intuitively ‘constructed’ out of these rudimentary perceptions and their subsequent intuitive demonstrations.

As I stated previously, the single biggest advantage of this theory is that it perhaps best encapsulates what mathematics actually ‘feels like’ to perform at an experiential level. No one actually doing mathematics will be mentally making moves of a game, and neither will they be ‘seeing’ concepts with reason – rather, they will typically be constructing loose, intuitive ideas of the objects and arguments they want to use; and only afterwards ‘translate’ these into absolutely formal terms. When, for example, trying to construct a topological argument, one will typically imagine some sort of 3 dimensional shape, and will try to twist and contort it in their mind to convince themselves of whatever it is they’re trying to show. Evidently, the shapes imagined will not be perceived as an abstract collection of points, as the Formalists claim, nor as an abstract form, as the Platonists do – rather, it will be experienced as an object (and structuring aspect of) perception.

Another crucial tenet of Intuitionist Ontology is a recognition of the temporal nature of our progression of mathematical knowledge over time. Since the truth of a statement consists in its construction, it is only at the moment of said construction that a statement becomes true. As such, unlike Platonism and Formalism, where there is an awkward need to extrapolate the derived truth or falsity of a statement to all of time, even before this was discovered, Intuitionism allows for the reality of mathematical progress (where statements can be conjectured whose truth or falsity are not known at the moment) to be acknowledged within the theory, rather than ignoring it.

Indeed – rather surprisingly for such a supposedly ‘subjective’ theory – it even explains why we see mathematics reflected in the order of the universe. Simply put, the structures that govern our perceptions of the universe and those that are the foundations of mathematics (i.e. the categories of experience) are the same, so naturally the two seem to agree, since they are structured by the same fundamental building blocks. That is to say, the universe appears to operate under certain structures not because they exist in the world, but because they structure our perception. And perhaps even more impressively, it acknowledges the seeming presence of mathematical structures in the universe without resorting to the sort of transcendent metaphysical entities that Platonism does. In other words, it doesn’t suffer from any of the epistemological issues of Formalism, nor the metaphysical baggage of Platonism.

However, even outside of the obvious charge of subjectivity that can be leveled against this theory, it suffers from rather unique kinds of problems, which, much like it’s advantages, have more to do with mathematical experience than metaphysics. For one, due to it’s constructive nature, we lose out on vital mathematical tools like the law of excluded middle, since it is a logical law, which cannot be derived from a view of mathematics where true demonstrations must be explicitly constructed out. As such, in some sense, Intuitionism leaves us with less to work with, an obvious problem for a mathematician who just wants to find things out about maths!

Furthermore, the ‘language-ness’ of mathematics according to this theory leads to awkward questions regarding the accuracy of mathematical communication, since there’s no guarantee that everyone means the same thing by the same terms. Since the actual meaning of mathematical terms is something in our intuition, there is no guarantee that all mathematicians are ‘speaking the same language’, it opens the possibility that each of them have their own private world of meaning, which is only agreed upon in the public mathematical symbolic language by pure chance!

So these are the three big ontologies of mathematics – most other positions, like Empiricism, Psychologism, or Logicism can be more or less categorized as combinations and variants of the primary three. Evidently, as I hope I have shown, despite us not typically thinking about it when we do maths, what these objects actually are is a fascinating area of debate and controversy, which remains contentious even to this day.

In the next two articles, I will talk about some of the consequences that arise from particular sub-schools of thought, and how debates around them defined a lot of mathematical discussion in the early 20th century and beyond. If you thought this article was wild, trust me, it’s only the beginning…

Article 2: Mathematical Philosophy: The Decidability Problem

Article 3: Mathematical Philosophy: Russell, Godel and Incompleteness