Mathematical Causation

Some argue that intuitive judgements about mathematical statements lead us to believe in Mathematical Platonism. But the mathematical objects of platonistic theories are supposed to be non-spatiotemporal and disconnected from the world, specifically they are acausal. This has issues, if mathematical objects are disconnected from the world, then it seems like they make no difference to the world. The world would be the way it is even if mathematical objects did not exist. Additionally, people like Benacerraf (1973) argue that given a causal theory of knowledge, and given their acausal nature, we could never know about mathematical objects and so should not posit them. Elsewhere in the literature people offer indispensability arguments for mathematical objects (e.g. Baker, 2009). Our best scientific theories seem required to invoke explanations that quantify over mathematical objects, so we should believe in them in just the same way as we believe in the unobservable objects of science. We can then say we know that mathematical objects exist through an inference to the best explanation, based on our best scientific theories. But this does not tell us what mathematical objects are like, importantly it still does not answer “makes no difference” arguments. We may be forced to believe in them but this does not tell us whether or not they actually do anything. I want to discuss one way in which mathematical objects might do something. I think there is a useful and informative way we can talk about mathematical objects as being causal. I do this by discussing a case of mathematical constraint, as proposed by Marc Lange (2017). I will elaborate on the notion of mathematical constraint and talk about the constraint relation more generally. I then move on to discuss structural equation models and how they can be used to represent causal relationships. This fits particularly well with the Interventionist view of causation, which I will describe, and how it can be used as a test to determine which relationships are causal. I think mathematical constraint passes this test. With this framework in place, I will discuss a specific structural equation model which represents a constraint relation. This particular constraint relationship is also straightforwardly causal. I think the structure of this relationship naturally maps on to the structure of an archetypal example of mathematical constraint. Not only do they share a structure, but both relationships behave in the same ways under interventionist treatments. This should grant us reason to say that mathematical constraint is causal. For those interested in the epistemology of mathematics, this can allow a sidestepping of Benacerraf-style objections; it would be mysterious how we know about acausal mathematical objects, but given mathematical objects are causal we can explain our knowledge. For those sympathetic to, or concerned about, makes no difference arguments, we again have a response, mathematical objects are causal so it would make a difference to the world if they did not exist.