Platonism, Formalism, Rationalism, and Axioms

->Infinite coin purse<-

Over the summer vacation from university, I was given the assignment of reading a (relatively) recent mathematical paper, writing a report on it, and preparing a presentation. Of the papers on offer, I chose to read Jan Mycielski's 2006 paper, A System of Axioms of Set Theory for the Rationalists. I chose it largely on the basis that I'm interested in set theory and I'm taking a course on it this year. As I started reading more into the background material behind the paper (especially Gödel's 1947 paper What is Cantor's Continuum Problem? and its 1964 revision), however, I started to appreciate the philosophical arguments around these topics much more. I thought I'd write a blog post discussing some of the ideas behind the three main philosophies discussed by Mycielski: platonism, formalism, and rationalism.

It's worth pointing out that each of these philosophies appreciates that applied mathematics can be (and is) built to model and reason with aspects of the real world. So, constructs like random variables in probability are built to reflect the nature of random systems such as radiation, and euclidian geometry can be used to model the layout and shape of physical objects in the real world. They differ, though, in their treatment of pure mathematical objects (such as infinite sets), which don't have such clear origins in the real world. These differences affect the choice of axioms for fields such as set theory.

Platonism

Platonism, a view held by Gödel (among many others), sees pure mathematical constructs and rules as descriptions of the actual universe, independent of humanity and its mathematicians. Platonism acknowledges that certain statements about these objects seem obviously true to almost anyone who considers them. For instance, the statement that 'if two sets have exactly the same elements, they are the same set' (or {% m %}\forall X \forall Y (\forall a (a \in X \Leftrightarrow a \in Y ) \Rightarrow X = Y ){% em %}). Platonists explain this by claiming that these statements seem true because they are literally true about structures and objects which really exist.

These statements, then, are safe to be chosen as axioms for reasoning about the structures in question. This gives rise to Zermelo-Fraenkel set theory with the axiom of choice1 (ZFC), the most commonly accepted system of axioms of set theory today. (Indeed, the earlier given example is the 'axiom of extensionality', one of the axioms of ZFC). But what of other statements, which are consistent with the 'self-evident' axioms, but aren't so obviously true? A platonist would not want to take any such statements as axioms, because there's always a risk that they might be objectively wrong, even if they are consistent with the current theory. (Of course, in practice there are versions of platonism which allow this flexibility).

  1. A note about the axiom of choice: this is an axiom which probably doesn't count as a 'self-evident' axiom, and certainly isn't one that everyone agrees should be an axiom (at least historically). However, it is largely accepted by mathematicians nowadays as a lot of modern mathematics depends on it. See Wikipedia for more about this.

Formalism

->Formalism<-

Formalism, on the other hand, treats mathematics as simply a set of symbols, and a set of rules for manipulating those symbols. According to formalism, pure mathematics has no meaning beyond that which we give it. This means that we can choose any system of axioms that we like, as none is more correct than any other. Some have argued that, as a result, formalism contains no justification for its choice of axioms. In practice, however, formalists do usually have some non-arbitrary criteria for their choice of axioms (in a 1927 lecture published as The Foundations of Mathematics, David Hilbert said that his formalist program was to develop pure mathematics in such a way as to express the "technique of our thinking"). For more about this, see this Wikipedia article on the Brouwer-Hilbert controversy.

Rationalism

Mycielski describes a third view: rationalism. Rationalists account for the self-evident nature of the ZFC axioms by claiming that the structures concerned do exist -- but in our minds. He suggests that we evolved to think of these structures to help us reason about the world around us.

->Rationalism<-

For example, consider infinite sets. It is unintuitive to say that these exist in our minds. Just try to imagine infinitely many £1 coins. It's impossible! (Or should be, anyway - if you managed it, send me an email!). But now try to imagine a coin purse from which, no matter how many coins you have already pulled, you are always able to take out another coin. Obviously such an infinite coin purse doesn't exist in our universe, but we are able to imagine one. In this way, the 'infinite sets' in our minds are not actually infinite, but only potentially infinite (which is effectively equivalent, mathematically!). Mycielski gives a more mathematical discussion of how our interpretation of quantifiers can be reconciled with this view of structures existing in our minds.

This justifies the choice of self-evident statements as axioms: they are true (but of structures naturally existing in our minds, rather than of the objective universe independent of us). However, it also allows for the flexibility to choose other axioms as well. Mycielski suggests that we should accept those statements which satisfy any of these criteria:

  1. Natural rules of thought (ZFC axioms)
  2. Rules independent of ZFC which reduce unnecessary complexity in set theory without removing interesting objects
  3. Rules independent of ZFC which enhance set theory with interesting new objects

This is suggested with the ackowledgement that, as mathematics changes and develops, the axioms which seem useful might change over time.

An example of an axiom chosen by Mycielski with the above justification is the generalised continuum hypothesis (GCH), that for all ordinals {%m%} i {% em %},
{% math %} {\LARGE 2^{\alephi} = \aleph{i+1}} {% endmath %} This is an interesting and controversial hypothesis with a compelling history. I hope to write another blog post just about the GCH at some point in the future! Mycielski argues that it simplifies many set-theoretic proofs without restricting the strength of set theory, and so is worth using as an axiom.

Ultimately, Mycielski feels that rationalism takes the best of platonism (strong justification of basic axioms) and formalism (flexibility in axiom choice), leading to a choice of axioms based on what produces the best mathematics.

Personally, I find Mycielski's arguments that humans have evolved to think of pure mathematical constructs as part of our nature interesting. It's a bold claim, however, and I would be curious to see whether it could stand up to investigation by psychologists.

Further reading

If you're interested in these topics, you might want to check out some of these papers (as well as the links in the above blog post):

  • Kurt Gödel, What is Cantor's Continuum Problem? (revised), 1964
  • David Hilbert, The Foundations of Mathematics (1927)
  • Jan Mycielski, A System of Axioms of Set Theory for the Rationalists (2006)