Aryaman Gupta

This is the third article in a 3 part series introducing the subject of Mathematical Philosophy. Links to the first two articles can be found at the bottom of the page.

Although nowadays we might think it is ‘obvious’ that maths is based on logic, this idea, surprisingly, only arose relatively recently, and whether or not this is the case was actually one of the central debates of the subject during the early 20th century. Perhaps even more surprisingly, by the end of this time, it was found that a purely logical foundation for mathematics isn’t even possible; a discovery that defined almost every major subsequent development in the philosophy of mathematics.

Historically, from Aristotle until the late 19th century, logic was seen, first and foremost, as a subject of the philosophy of language and rhetoric, of forming convincing, reasoned arguments. Mathematics, on the other hand, being involved with more abstract matters of the realms of arithmetic and geometry, was not initially connected to the former. 

So how, then, did this change? The first major argument for placing mathematics on logical foundations was given by Frege, who wished to use it to guarantee for mathematics the absolute certainty and clarity that it allowed for. To achieve this, he endeavoured to show that all of arithmetic – and thus, from there on, all of the rest of mathematics – could be reduced to a set of logical laws as its foundation.

To do this, Frege introduced the idea of extensions – collections of all x that satisfy a certain logical statement of the form ‘x is y’ (eg. ‘x is a prime’, ‘x is a square’ etc) – as the basic logical ‘structures’ upon which to build all subsequent mathematical objects. Then, he suggested that all mathematical objects could be built up using nothing more than extensions and the appropriate ‘logical’ laws (for example that a statement and it’s negation cannot both be simultaneously true) which should govern the statements that extensions are ‘built’ from. For example, Frege defines each number n using this method to be the extension of the statement ‘x is an extension where the elements pair one-to-one to the set {0,1, … , n – 1}’ (with 0 being defined as the extension of the statement ‘x is not identical to itself’).

However, although Frege published this theory with full expectation that he had successfully grounded mathematics in logic, another philosopher – Bertrand Russell – sent him a letter just as the final volume of his treatise on the subject (Die Grundgesetze der Arithmetik) was to be published, which demonstrated that his system was self contradictory. Russell showed this by constructing a paradox with Frege’s system, which is now known as Russel’s paradox, and is as follows:

Consider the extension of the statement ‘x does not belong to itself’. Does this extension belong to itself? If x belongs to it, x does not belong to itself. If it does not belong to itself, it satisfies the condition to belong to itself. So we have an extension both containing and not containing itself, which is clearly a contradiction.

The effects of this were devastating, both to Frege’s logicist project and to his own personal life, causing him to be temporarily admitted into a sanatorium, after which he said he no longer believed that logic could act as the foundation of maths. Attempting to do so in his stead, Russell wrote the Principia Mathematica (along with fellow philosopher Alfred North Whitehead), in which they attempted to create a logicist foundation of mathematics that avoided this paradox (by using sets – which are like extensions, except that each set must be the subset of another set, thus preventing them from falling into Russell’s paradox).

However, despite this work being seminal in it’s achievement of basing all mathematics on almost purely logical axioms, it fell quite short of the actual aim of Logicism, since Russell and Whitehead were forced to include the axiom of infinity (that an infinite set exists) to construct out the rest of mathematics – despite the fact that classical logic cannot make any claims of how many things exist. This is because classical logic, by being purely formal and structural, could not say anything about ‘things’ and, say, how many of them could exist – it had to be absolutely universal, even if, say, the world had one, or two, or no things. As a result of this, the aforesaid axiom is not a logical one. This all meant that, even whilst avoiding Russell’s paradox, it seemed as though a logicist foundation to mathematics was quite out of reach.

And then along came Godel to put an end to the whole project of Logicism once and for all. His two incompleteness theorems, which demonstrate that the broader goal – to create a foundation of mathematics that is perfectly consistent and complete (i.e. where every statement has a proof and every proof is correct), thus allowing for every mathematical statement to be clearly decided to be true or false – is mathematically impossible.

The first of the two theorems states that if a logical system is capable of simulating arithmetic, then it will have true statements which can neither be proven nor disproven. As such, even if we could hypothetically reduce mathematics to purely logical laws, these laws would fail to allow us to classify all mathematical statements as either true or false, since there are some statements for which this simply can’t be demonstrated.  

The second, in turn, states that no logical system capable of simulating arithmetic can prove its own consistency. So if we want to prove that such a system is consistent, we would need axioms outside the system to prove it. But then that larger system needs more additional axioms to be proven consistent, and so on until infinity (indeed, even infinitely many axioms wouldn’t be sufficient!). 

The consequences of this are actually quite terrifying – we can never be 100% sure that a system we are using is consistent. At any time, we could chance upon a contradiction that blows all of our prior work up (since if a contradiction is found, all statements in the system will be true by the law of excluded middle). Clearly, the absolute certainty desired by Logicism – or indeed, any certainty at all – is simply impossible. Worse still, this isn’t only a problem with Logicism – it just affects it particularly badly due to its commitments to absolute categorisation of all statements. Even ‘less ambitious’ axiom systems – including the ZFC axioms, which are what most mathematicians now use – are susceptible to this. At any point, everything could just collapse in on itself!

So in conclusion, although the logicist project showed that it was possible to derive all of mathematics from basic axioms regarding sets, it failed in its own goals. Not only in the sense that it was unpopular – rather dishearteningly, it was shown to be mathematically impossible. Most philosophy of mathematics after Godel is deeply informed by this gaping realisation that logical perfection is impossible – and indeed, most later theories (like Wittgenstein’s reverting to formalism, or constructivism and psychologism) deliberately try to distance themselves from the grandiose claims to absolute epistemological perfection professed by the principia, due to it’s proven untenability. 

Looking beyond the realm of mathematics, there is almost a sense of cultural necessity that a totalizing project like Logicism was bound to fail, for this failure is eerily parallel to the transition from modernism to postmodernism. In the case of the latter, after the attempts at a universal, totalizing description and rallying of human nature by modernist writers and philosophers was abandoned in the violence and destruction of the second world war, subsequent postmodernist writers deliberately distanced themselves from such claims of totalization. Indeed, with the case of Russell in particular, it is more tempting to make the connection because he was friends with many of the great writers of the modernist period, like Woolf and Conrad, and was also a proponent of modernist philosophical trends such as Marxism. Not only, then, does this failure tie more broadly to mathematics as a whole (as a hanging sword of Damocles), it is also deeply connected to the broader course of philosophy, culture and literature after the second world war. 

Article 1: An Introduction to Maths and Philosophy – Platonism, Formalism and Intuitionism

Article 2: Mathematical Philosophy: The Decidability Problem