I could go on for quite some time about the difference between dreaming and being awake. I could see myself picking carefully through every thought I have ever had about the significance of dreams, and I know I would end up with a proliferation of questions, rather than a clarification of anything. But I think that this is how it should be. We understand so little about our awareness, our consciousness, and what cognitive processes actually produce for us. All of this comes to mind, at the moment, because I just read an article in Plus Magazine based on a press conference given by Andrew Wiles at the Heidelberg Laureate Forum in September 2016. Wiles, of course, is famous for having proved Fermat’s Last Theorem in 1995. The article highlights some of Wiles’ thoughts about what it’s like to do mathematics, and what it feels like when he’s doing it. When asked about whether he could feel when a mathematical investigation was headed in the right direction, or when things were beginning to ‘harmonize,’ he said
Yes, absolutely. When you get it, it’s like the difference between dreaming and being awake.
If I had the opportunity, I would ask him to explain this a bit because the relationship between dream sensations and waking sensations has always been interesting to me. The relationship between language and brain imagery, for example, is intriguing. I once had a dream in which someone very close to me looked transparent. I could see through him, and I actually said those words in the dream. I would learn, in due time, that this person was not entirely who he appeared to be. But what Wiles seems to be addressing is the clarity and the certainty of being awake, of opening ones eyes, in contrast to the sometimes enigmatic narrative of a dream. This is what it feels like when you begin to find an idea.
Wiles also had a refreshingly simple response to a question about whether mathematics is invented or discovered:
To tell you the truth, I don’t think I know a mathematician who doesn’t think that it’s discovered. So we’re all on one side, I think. In some sense perhaps the proofs are created because they’re more fallible and there are many options, but certainly in terms of the actual things we find we just think of it as discovered.
I’m not sure if the next question in the article was meant as a challenge to what Wiles believes about mathematical discovery, but it seems posed to suggest that the belief held by mathematicians that they are discovering things is a necessary illusion, something they need to believe in order to do the work they’re doing. And to this possibility Wiles says,
I wouldn’t like to say it’s modesty but somehow you find this thing and suddenly you see the beauty of this landscape and you just feel it’s been there all along. You don’t feel it wasn’t there before you saw it, it’s like your eyes are opened and you see it. (emphasis added)
And this is the key I think, “it’s like your eyes are opened and you see it.” Cognitive neuroscientists involved in understanding vision have described the physical things we see as ‘inventions’ of the visual brain. This is because what we see is pieced together from the visual attributes of objects we perceive (shape, color, movement, etc.), attributes processed by particular cells, together with what looks like the computation of probabilities based on previous visual experience. I believe that questions about how the brain organizes sensation, and questions about what it is that the mathematician explores, are undoubtedly related. Trying to describe the sensation of ‘looking’ in mathematics (as opposed to the formal reasoning that is finally written down) Wiles says this:
…it’s extremely creative. We’re coming up with some completely unexpected patterns, either in our reasoning or in the results. Yes, to communicate it to others we have to make it very formal and very logical. But we don’t create it that way, we don’t think that way. We’re not automatons. We have developed a kind of feel for how it should fit together and we’re trying to feel, “Well, this is important, I haven’t used this, I want to try and think of some new way of interpreting this so that I can put it into the equation,” and so on.
I think it’s important to note that Wiles is telling us that the research mathematician will come up with some completely unexpected patterns in either their reasoning or their results. The unexpected patterns in the results are what everyone gets to see. But that one would find unexpected patterns in ones reasoning is particularly interesting. And clearly the reasoning and the results are intimately tied.
Like the sound that is produced from the numbers associated with the marks on a page of music, there is the perceived layer of mathematics about which mathematicians are passionate. And this is the thing about which it is very difficult to speak. Yet the power of what this perceived layer is may only be hinted at by the proliferation of applications of mathematical ideas in every area of our lives.
Best wishes for the New Year!