March 24, 2015 / Adam Robbert
[Image: Mohammad Reza Domiri Ganji]
I just came across Massimo Pigliucci’s interesting review of Mangabeira Unger and Lee Smolin’s book The Singular Universe and the Reality of Time. There are more than a few Whiteheadian themes explored throughout the review, including Unger and Smolin’s (U&S) view that time should be read as an abstraction from events and that the “laws” of the universe are better conceptualized as habits or contingent causal connections secured by the ongoingness of those events rather than as eternal, abstract formalisms. (This entangling of laws with phenomena, of events with time, is one of the ways we can think towards an ecological metaphysics.)
But what I am particularly interested in is the short discussion on Platonism and mathematical realism. I sometimes think of mathematical realism as the view that numbers, and thus the abstract formalisms they create, are real, mind-independent entities, and that, given this view, mathematical equations are discovered (i.e., they actually exist in the world) rather than created (i.e., humans made them up to fill this or that pragmatic need). The review makes it clear, though, that this definition doesn’t push things far enough for the mathematical realist. Instead, the mathematical realist argues for not just the mind-independent existence of numbers but also their nature-independence—math as independent not just of all knowers but of all natural phenomena, past, present, or future.
U&S present an alternative to mathematical realisms of this variety that I find compelling and more consistent with the view that laws are habits and that time is an abstraction from events. Here’s the reviewer’s take on U&S’s argument (the review starts with a quote from U&S and then unpacks it a bit):
“The third idea is the selective realism of mathematics. (We use realism here in the sense of relation to the one real natural world, in opposition to what is often described as mathematical Platonism: a belief in the real existence, apart from nature, of mathematical entities.) Now dominant conceptions of what the most basic natural science is and can become have been formed in the context of beliefs about mathematics and of its relation to both science and nature. The laws of nature, the discerning of which has been the supreme object of science, are supposed to be written in the language of mathematics.” (p. xii)
But they are not, because there are no “laws” and because mathematics is a human (very useful) invention, not a mysterious sixth sense capable of probing a deeper reality beyond the empirical. This needs some unpacking, of course. Let me start with mathematics, then move to the issue of natural laws.
I was myself, until recently, intrigued by mathematical Platonism . It is a compelling idea, which makes sense of the “unreasonable effectiveness of mathematics” as Eugene Wigner famously put it . It is a position shared by a good number of mathematicians and philosophers of mathematics. It is based on the strong gut feeling that mathematicians have that they don’t invent mathematical formalisms, they “discover” them, in a way analogous to what empirical scientists do with features of the outside world. It is also supported by an argument analogous to the defense of realism about scientific theories and advanced by Hilary Putnam: it would be nothing short of miraculous, it is suggested, if mathematics were the arbitrary creation of the human mind, and yet time and again it turns out to be spectacularly helpful to scientists .
But there are, of course, equally (more?) powerful counterarguments, which are in part discussed by Unger in the first part of the book. To begin with, the whole thing smells a bit too uncomfortably of mysticism: where, exactly, is this realm of mathematical objects? What is its ontological status? Moreover, and relatedly, how is it that human beings have somehow developed the uncanny ability to access such realm? We know how we can access, however imperfectly and indirectly, the physical world: we evolved a battery of sensorial capabilities to navigate that world in order to survive and reproduce, and science has been a continuous quest for expanding the power of our senses by way of more and more sophisticated instrumentation, to gain access to more and more (and increasingly less relevant to our biological fitness!) aspects of the world.
Indeed, it is precisely this analogy with science that powerfully hints to an alternative, naturalistic interpretation of the (un)reasonable effectiveness of mathematics. Math too started out as a way to do useful things in the world, mostly to count (arithmetics) and to measure up the world and divide it into manageable chunks (geometry). Mathematicians then developed their own (conceptual, as opposed to empirical) tools to understand more and more sophisticated and less immediate aspects of the world, in the process eventually abstracting entirely from such a world in pursuit of internally generated questions (what we today call “pure” mathematics).
U&S do not by any means deny the power and effectiveness of mathematics. But they also remind us that precisely what makes it so useful and general — its abstraction from the particularities of the world, and specifically its inability to deal with temporal asymmetries (mathematical equations in fundamental physics are time-symmetric, and asymmetries have to be imported as externally imposed background conditions) — also makes it subordinate to empirical science when it comes to understanding the one real world.
This empiricist reading of mathematics offers a refreshing respite to the resurgence of a certain Idealism in some continental circles (perhaps most interestingly spearheaded by Quentin Meillassoux). I’ve heard mention a few times now that the various factions squaring off within continental philosophy’s avant garde can be roughly approximated as a renewed encounter between Kantian finitude and Hegelian absolutism. It’s probably a bit too stark of a binary, but there’s a sense in which the stakes of these arguments really do center on the ontological status of mathematics in the natural world. It’s not a direct focus of my own research interests, really, but it’s a fascinating set of questions nonetheless.