Plato and the working mathematician on Truth and discourse

Plato’s writing and philosophy are widely studied in colleges, and often turned to as founding texts of western philosophy. But if we went out looking for people that embraced the philosophy — if we went out looking for actual Platonist — then I think we would come up empty-handed. Or maybe not?

A tempting counter-example is the mathematician.

It certainly seems that to do mathematics, it helps to imagine the objects that you’re studying as inherently real but in a realm that is separate from your desk, chair and laptop. I am certainly susceptible to this thinking. Some mathematicians might even claim that they are mathematical platonists. But there is sometimes reasons to doubt the seriousness of this claim. As Reuben Hersh wrote in Some Proposals for Reviving the Philosophy of Mathematics:

the typical “working mathematician” is a Platonist on weekdays and a formalist on Sundays. That is, when he is doing mathematics, he is convinced that he is dealing with an objective reality whose properties he is attempting to determine. But then, when challenged to give a philosophical account of this reality, he finds it easiest to pretend that he does not believe in it after all.

What explains this discrepency? Is mathematical platonism — or a general vague idealism about mathematical objects — compatible with the actual philosophy attributed to Plato? This is the jist of a question that Conifold asked on the Philosophy StackExchange almost 4 years ago.

In this post, I want to revisit and share my answer. This well let us contrast mathematical platonism with a standard reading of Plato’s thought. After, I’ll take some helpful lessons from postmodernism and consider an alternative reading of Plato. Hopefully this PoMo Plato can suggest some fun thoughts on the old debate on discovery vs invention in mathematics, and better flesh out my Kantian position on the Church-Turing thesis.

Tensions between mathematical platonism and Plato

Mathematical platonism — or platonism more generally (with the lower case ‘p’ to distinguish it from the Platonism of Plato) — holds the following three theses about mathematical objects: they (i) exist, (ii) are abstract, and (iii) are independent of intelligent agents. This is typically all that mathematicians mean when they say they are platonists, and the differ in their further commitments. Plato would hold these three premises as well. In that way Platonism is compatible with platonism.

But Plato’s metaphysics is traditionally taken to extend further, committing him to stances that most mathematicians do not hold. Thus, I would say that mathematical platonism is not compatible with Platonism.

For a first example, let us consider ethics and the existence of moral truths. Obviously, for Plato, forms like the Good and Justice have the same sort of existence as Triangle. For mathematicians, there is a lot of variation.

Some, like Godel, will commit themselves to a thorough metaphysical platonism that extends to ethics, hence his ontological argument for the existance of God, for instance. Although note an important difference, Plato would not go to the same extent as Godel to see if some Form exists or not. Or, at least, Plato would not be as formal as Godel. Plato would extract the Form through dialogue, not derivation.

Others, like Russell, do not think there was an objective Good; to see this dear reader, you just have to read over his very pithy “Is there an Absolute Good?”. But with Russell it is also a little difficult to see if he held mathematical platonism and his moral positions at the same time, given that he went through at least three distinct stages and mathematics is mostly early Russell while ethics is middle and late Russell. However, from my personal experience, many mathematicians would not be platonists with respect to ethics.

For a second example, let us consider epistemology. For Plato, we simply ‘remembered’ the Forms, we do not discover them. A lot of mathematicians might object to this perspective. Although again, there is some heterogeneity here, for some we simply ‘see with our intuition’ (not that different from Plato’s remembering) a mathematical truth and then construct a proof to guide others to what we saw, much like Socrates might guide the slave boy.

For a third example, let us consider particulars. For Plato — most famously in the allegory of the cave — particulars are of the same ‘type’ as the Forms but are degraded by being corporeal. The Forms are, in some way, causes of the particulars.

I don’t think that many mathematicians would subscribe to this view.

I think that most mathematicians would implicitly subscribe to a form of dualism similar to Frege’s — another notable mathematical platonist — and think of separate worlds of the abstract and of the empirical with the effectiveness of mathematics in the sciences being either a selection-bias or unreasonable. At the times that I am a mathematical platonist, I think that I fall into this camp.

For a fourth example, let us consider structure. The ideal forms of a mathematician do not exist in solitude. The mathematician’s forms are interconnected. The universe mathematicians imagine is structured in some ways and they use that structure to navigate that universe. For example, by reducing facts about complex structures to simpler pieces.

It is not clear that Plato’s Forms have this structure, under most readings the various forms are rather isolated from each other. And the imagery of ascending and laying bare the Form of the Good certainly doesn’t seem to resonate well with how mathematicians proceed to understand various mathematical forms.

PoMo Plato: Culture as the World of Forms

But maybe I am giving an overly nuanced reading of mathematicians and not a nuanced enough reading of Plato.

Let me go back to the beginning. Let me return to my claim that Plato would ascent to the following the theses about Forms: they (i) exist, (ii) are abstract, and (iii) are independent of intelligent agents.

This is certainly how Plato’s world of Forms is presented in a typical introduction to philosophy. This is how you’d see Russell present Plato’s Forms in his history. But this is not how I understand Plato’s Forums.

Let’s take a brief detour.

One of my favourite compliments for TheEGG came from the author of the no sign of it blog on my post on truthiness:

You have touched on many of the issues that post-modernists frequently fret over without lapsing into their relativist despair.

Postmodernism is hard to characterize. In fact, many view evading characterization as a central aspect of postmodernism. But some features are more salient than others. In this context: postmodernism questions the tendency of Western philosophy to universalize, and to identify clear eternal objective Truths. Instead, the postmodern philosopher focuses on how our society constructs truth through discourse.

In this way, the postmodernist often sees herself as riling again Plato and the tradition that is all but footnotes to his work. And if we take the popular interpretation of Plato that I started this post with then this contrast is clear.

The issue, though, is that there is no reason to hold to the popular interpretation of Plato. I think that Plato can be taken back from the enlightenment and the modernists and read as a ‘postmodern thinker’.

In particular, discourse as a means to construct the Forms is a central aspect of Plato. The boundaries of language and culture are also essential in his dialogue. In this way, I like to imagine Plato’s Forms as existing in a cultural space that is in some ways similar to language. It has rules that we follow and that structure our perceived reality, but that we don’t necessarily explicit know. The role of the philosopher is then to extract and explicate that latent knowledge that exists and shapes the space between and within persons.

Through so doing, the philosopher can then also change that eternal realm of Forms and how we engage with it. That is why Socrates is treated as a radical and a corrupter of the youth.

Now, as with all social constructivist notions of Truth, it is important to not mistake this for a whilly-nilly notion that ‘Truth is anything you want’. There is an inescapable structure to Forms, much like there are inescapable structures to languages, or our perceptions of space and time.

If we take this reading then Plato would not longer accept thesis (iii). PoMo Plato would not accept that the Forms are independent of intelligent agents. PoMO Plato would also not accept the Forms as unstructured.

And this can brings us back to mathematics. In particular, it can be a fun lens through which to look at the common debate about whether mathematics is discovered or invented. PoMo Plato lets us take both options: mathematics is discovered in that we cannot individually escape the categories that define our thinking. But mathematics is invented in that as a community we can constantly build on and refine the categories that shape our social reality.

When I write about ideas like Transcendental Idealism and Post’s variant of the Church-Turing thesis, it is PoMo Plato’s view that I hold. I equated Post’s variant of the CT-thesis to the belief that the the Turing Machine or other equivalent forms of computation capture what is thinkable by us, and express the restriction of our finite understanding. Here I wasn’t using the Royal ‘our’. I think that what is thinkable by us is shaped both by what’s between our ears and by our social world.

I present this final position without much defense. Although if you want to start reading some defenses of similar positions, I would recommend Abeba Birhane’s article on why Descartes was wrong. But, I want to leave on this incomplete thought in the hope that it provides you, dear reader, with some fun fuel for your own reflection.

I’d be eager to hear your disagreements. Then we can build a new understanding through discourse.

About Artem Kaznatcheev
From the Department of Computer Science at Oxford University and Department of Translational Hematology & Oncology Research at Cleveland Clinic, I marvel at the world through algorithmic lenses. My mind is drawn to evolutionary dynamics, theoretical computer science, mathematical oncology, computational learning theory, and philosophy of science. Previously I was at the Department of Integrated Mathematical Oncology at Moffitt Cancer Center, and the School of Computer Science and Department of Psychology at McGill University. In a past life, I worried about quantum queries at the Institute for Quantum Computing and Department of Combinatorics & Optimization at University of Waterloo and as a visitor to the Centre for Quantum Technologies at National University of Singapore. Meander with me on Google+ and Twitter.

One Response to Plato and the working mathematician on Truth and discourse

  1. Pingback: Cataloging a year of social blogging | Theory, Evolution, and Games Group

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