Sunday, February 14, 2016

Bauer and Banach-Tarski

Dan Piponi had a short and thought-provoking post in G+ the other day. To me the best bit was
"Conversations between people who view mathematics as being about beliefs, and people who view mathematics as being about practices, can often be at cross purposes.
I think I am firmly on the side of the ones who think that it's about practices, not beliefs. One thing that surprises me about Philosophy and philosophical questions is the notion of cross-purposed-ness, how one can be interested in a subject, yet, totally not engage with certain questions and manners of approaching it.

The whole Piponi's post is below. In response to an ensuing discussion with Rodrigo Freire on  whether intuitionistic practices of Mathematics are self-contradictory, given that the majority of the mathematicians do believe in Excluded Middle and Choice, Victor Botelho posted the link to the following video from Andrej Bauer talking in Princeton in 2013.

Five Stages of Accepting Constructive Mathematics

I loved it!! If you haven't seen it yet, do yourself a favour and have a look. It touches on lots of the questions that aren't at cross-purposes for me. Says some things I'd say too, but it has much more than I know, so I learned lots, and hope to follow up with some more learning on the subject of constructive mathematics. Including some on what to do with Banach-Tarski and the constructive "theorem" that every function is continuous. 

Philosophy is hard, I don't want to venture there much, but it would be nice to get enough od a feeling for the place, to avoid it.

Dan Piponi on Beliefs vs Practices in Mathematics
Many people think of religions primarily as systems of belief. I think this may be a skewed view because of the predominance of Christianity and Islam, both of which make creeds prominent. For example, although Judaism does have something like a creed, it tends to place more emphasis on practice than belief.

This reflects my view of mathematics. I think that for many, mathematics is a matter of belief. For them, mathematics is a way to find out what is and isn't true. I tend to see mathematics as a set of practices. As a result, I find myself bemused by debates over whether 2 really exists, or whether infinite sets exist, whether the continuum really is an infinite collection of points, whether infinitesimals exist, whether the axiom of choice is true, and so on. I find some ultrafinitists particularly confusing. They seem to believe themselves to be expressing skepticism of some sort, whereas to me, expressing skepticism about mathematical constructions is a category error. So to me, these ultrafinitists are surprising because of what they believe, not because of what they don't. This doesn't just apply to ultrafinitists. In an essay by Boolos [1], he seems confident in the self-evident truth of the existence of integers, say, but expresses more and more doubt as he considers larger and larger cardinals. Many mathematicians seem to have a scale of believability, and ultrafinitists just draw the scale differently.

Conversations between people who view mathematics (or religion) as being about beliefs, and people who view mathematics (or religion) as being about practices, can often be at cross purposes. And members of one group can often find themselves dragged into debates that they don't care for because of the framing of questions. (I don't want to debate the existence infinite sets, not because I can't justify my beliefs, but because I'm more interested in how to use such sets. I don't think belief is a precondition for use.)

Of course you can't completely separate belief and practice and I certainly do have some mathematical beliefs. For example I put a certain amount of trust in mathematics in my daily job because I believe certain practices will allow me to achieve certain goals.

[1] Must we believe in Set Theory? (I hope I'm not mischaracterizing this essay, but even if I am, the point still stands.)


  1. Google posts have permalinks — just click on the date. Somehow, modern websites treat that as the "obvious" and "standard" interface (see also: Twitter, Facebook). Here's the link to Dan Piponi's post (it took me a while to find it from your link):

  2. Thanks Paolo, but I think I prefer to recopy the stuff, at least some times. see my previous post on the Internet Archive ( As we say in Brazil, the "scalded cat is afraid of cold water". it is not clear if Google will "sunset" G+ this month, this year or never. at least now we have two copies of Piponi's post ;)