Groucho Marxism

An axiomatic system is a set of formal statements (axioms) used to logically derive other statements. Zermelo-Fraenkel set theory, henceforth ‘ZF’, is an axiomatic system proposed by mathematicians Ernst Zermelo and Abraham Fraenkel in the early twentieth century to formulate a theory of sets. Today, ZF is the standard form of axiomatic set theory and is considered to be the foundation of all mathematics. Most of the axioms in ZF are uncontroversial and coincide with our intuition about sets. For example, the ‘axiom of extensionality’ states that two sets are equal if they have the same elements; and the ‘axiom of pairing’ states that if x and y are sets then there exists a set which has x and y as elements. Axioms like these seem entirely reasonable.

Another axiom in ZF which is usually considered uncontroversial is the ‘axiom of infinity’, which basically states that there exists a set containing infinitely many members. However, unlike the other axioms, which are rooted in our real-world experience of how sets work, this axiom cannot be justified in this way. Sets with infinitely many elements simply do not exist anywhere in the world, or anywhere in the universe for that matter. So why is this axiom not considered more controversial? The reason, I think, is that this axiom, in conjunction with the other axioms in ZF, allows mathematicians to have fun creating all sorts of weird and wonderful infinite sets. In fact it’s probably fair to say that 99% of modern mathematics would not be possible without it.

Now, we are free to choose the axioms of mathematics however we like. We could therefore choose to replace the axiom of infinity with its negation – i.e., assume that there are no sets containing infinitely many members. In fact it can be shown that ZF implies neither the axiom of infinity nor its negation and is therefore compatible with either. As noted already, the reason mathematicians are unwilling to do this is not that the axiom of infinity makes intuitive sense; it’s because replacing the axiom of infinity with its negation would deprive them of the opportunity to work with infinite sets. This seems to run counter to the way the axiomatic method is supposed to work. It is choosing the axioms to fit the mathematics, rather than fixing the axioms then seeing what mathematics be derived from them.

If we want to root mathematics in the material world, it would make much more sense to assume the negation of the axiom of infinity (or equivalently, assume an ‘axiom of finiteness’). This raises a philosophical question about what mathematics is actually for. Is it a way of describing the world, or is it just a fun game that mathematicians play? Opinion seems to be divided on this matter. If it’s the latter than there is no problem with using the axiom of infinity. And to be clear, I have no problem with mathematicians treating mathematics as a fun game if that’s what they want to do. The problem arises when mathematicians try to justify what they do based on its real-world applications. This makes little sense when mathematics is based on an assumption that simply doesn’t apply in the material world.

At this point mathematicians will probably point out that the whole of physics is based on mathematics which uses infinite sets. For example, differential calculus, the cornerstone of Newtonion mechanics, is defined using infinite limits. And quantum mechanics is defined mathematically using infinite-dimensional Hilbert spaces. The mathematics of general relativity also relies heavily on infinite sets. Wouldn’t the replacement of the axiom of infinity with its negation deprive us of the ability to do physics? I don’t think so. All of the theories mentioned above were developed intuitively before they were developed rigorously using infinite sets, and the reason they were developed rigorously in this way was simply that this was the prevailing mathematical paradigm of the time.

It may actually be the case that many ideas in physics that seem strange when formulated using infinite sets make perfect sense in a finite framework. For example, the Heisenberg Uncertainty Principle states that there is a limit to the precision with which certain pairs of physical properties, such as position and momentum, can be simultaneously known. This seems counter-intuitive if we assume that time can be divided up into infinitely many segments. But if we assume time can only be divided up into finitely many segments, it seems a lot more intuitive. Position is then measured at one time point, whereas momentum is measured at two separate time points, so it makes sense that both cannot be known simultaneously.

In my view, mathematics took a wrong turn over 100 years ago with the widespread acceptance of the axiom of infinity. In adopting this axiom, mathematicians unwittingly committed themselves to working within a Platonic or Idealist framework entirely detached from the material world. This wrong turn highlights the importance of getting your philosophical underpinnings correct. To put mathematics back on a firm materialist footing, we should replace the axiom of infinity with its negation. Only then will we be able to say that mathematics can truly describe the material universe in which we live.