Saturday, July 24, 2010

Invention, Discovery and Creativity in Mathematics

Non-philosophers I meet sometimes ask: do I think mathematical facts are invented or discovered? IMO, this is a weird question - and not one that comes up much in the phil math literature- because the contrast between invention and discovery is not very well defined. For example, did Alexander Gram Bell *invent* the telephone, or did he *discover* that putting components together in a certain way would build a telephone? Intuitively, one might say both.

Maybe what people mean to be asking by this question is just this: do mathematicians bring new mathematical objects into existence, or do they discover already existing objects? For, paradigmatic cases of invention typically do involve creating a new physical object, while paradigmatic cases of discovery involves visiting an already existing physical object. So e.g. Columbus discovered America (because it already existed and he went to visit it) whereas Bell invented the telephone, by physically creating the first prototype.

However, the contrast between invention and discovery can't really just track the distinction between cases where a new object is made vs. not. This is because making a new thing isn't required for invention *or* discovery. Consider an imaginary scenario where Bell just thought up a plan for a telephone, and told someone else who physically constructed the first one years later. Bell would still have invented at telephone, if he though up the plan and then worked out from known principles that the plan would work, but never made one.

While we are talking about invention and discovery, I think there's a third notion -artistic creation (e.g. what happens when someone composes a story or a poem)- which bears an interesting relationship to mathematical discovery. When a writer writes a story, they are putting down a sequences of sentences which already exists as an abstract object.

I mean, suppose that the story teller composes a story today. If a linguist said yesterday 'no intelligible sequence of English sentences has property P', the and the sequence or sentence which the story teller writes down today has property P, then then the linguist's claim yesterday was false. The domain of potential counterexamples to linguistics claims today, already contains all sequences of English sentences which literary ingenuity could ever devise. Note also that to compose a story or poem doesn't require writing it down anywhere, (the person in the Borges story who has time stop so he can finish writing a poem before he gets shot still counts as creating the poem). For this reason the task of literary "creation" doesn't really seem to involve creating anything, (neither a physical artifact, nor an abstract string of sentences), but rather directing your attention to an abstract object that already exists - carefully sorting out which string of sentences will combine certain varied and subtle properties in the right way.

Now, if I'm right about this- the creativity of a poet or novelist doesn't need to involve creating any new object, but rather amounts to discovering a pre-existing string of sentences which has a certain property - this suggests a potential confusion about the relationship between mathematical creativity and ontology. Arguably, mathematical creativity is much like literary creativity. But, if mathematical creativity is like literary creativity, it does not follow from this that the mathematician creates the mathematical objects he describes, or that he creates anything else. For (if the above is right) literary creativity isn't a matter of bringing new objects into being, but rather a matter of discovering, amid the combinatorial explosion of possible sequences of English sentences, one that has a certain special features.


  1. I think - like in many arts - creativity in math occurs at different scales. When anyone is problem solving, looking for a solution in a situation where they do not know how to get to it, the act of finding a solution is creative. The greats of the field take this further, creating whole new structures and concepts to solve problems. Sometimes these new structures are stumbled across by finding phenomena with new qualities or properties, and sometimes they are constructed to serve a purpose. Both of those are creative to me. Others may have seen the discovered structure before, but not been able to recognize it for what it was.

    Nice post!

  2. I don't follow your poet/writer example. For the poet/writer to be 'drawing our attention to a particular string' we must assume one of two things, with the first being false and the second unreasonable. If language and grammar was static (as say a formal system in mathematics is) then this viewpoint would make sense, however language is demonstrably non-static and it is often these acts of creativity by writers and poets that push its boundaries and change the language. Hence, for creative works they are often not just creating a string of characters consistent with yesterday's English, but shaping the English of tomorrow. Alternatively, you could just expand your ontic space to include all possible languages as 'real' but then you are are taking an unreasonably strong Platonist stance. Although such views re common among mathematicians, I find that they make for boring philosophy. I usually try to keep my ontological commitments as minimal as possible.

    I think you were on the right track with your opening, the question isn't quiet clear. Even if you look at everyday usage of the words among mathematicians, you will see that both discovery and invention metaphors are used to talk about different parts of mathematics. I highly recommend Tim Gowers' "Is Mathematics Discovered or Inventd" for a nice essay that gives the careful thoughts of a great mathematician on this topic. I think any philosophizing on the topic has to at least reflect or talk about many of the viewpoints expressed by Gowers.

  3. Sorry for the late reply (I stopped blogging for a few years). But if you are still out there somewhere, thanks for this excellent paper recommendation.