It is often the case that one of the crucial factors underpinning wide scale adoption of any computational procedure is the availability of easy to use packages containing that procedure, usable by people who have only a passing familiarity with the innards.
Topological data analysis is one such toolbox, the inner workings of which have often seemed too hard to penetrate for anyone but the most mathematically well trained scientists (a very small population, indeed). Yet, the tool has many appealing characteristics, given how centrally relevant its core questions are to hierarchical representations, multi-scale modelling and so on.
In this context, it is nice to see this web based interface for persistent homology calculations, a key tool in the TDA toolbox:
Through my colleague, Prof. Andrew Ranicki, I came upon this interesting interview with another distinguished colleague, Sir Michael Atiyah: https://www.quantamagazine.org/20160303-michael-atiyahs-mathematical-dreams/. The interview contains interesting reflection upon many things including the notion of beauty in mathematics. Indeed, Atiyah has co-authored a paper based on a very interesting neuroscience study on neural correlates of beauty: http://journal.frontiersin.org/article/10.3389/fnhum.2014.00068.
The key conclusion of this paper is that,
… the experience of mathematical beauty correlates parametrically with activity in the same part of the emotional brain, namely field A1 of the medial orbito-frontal cortex (mOFC), as the experience of beauty derived from other sources.
This in itself is cool, but it made me wonder – if we have a parametric signal in our brains, after a certain level of expertise has already been acquired, which correlates with how ‘beautiful’ something is, then is this useful as a heuristic in the search for new such objects? One of the limitations of most computational intelligent systems is precisely that they perform so poorly when it comes to exploratory search in open ended domains. Does this signal then help us prune search in our heads, a bit along the lines of José Raúl Capablanca’s famous, “I see only one move ahead, but it is always the correct one”?
I came across this while looking for some other information today. It is a humbling list in that escaping all these traps all of the time really does require discipline (I interpret ‘proof’ not just as mathematical reasoning but any scientific argument, including experimental work):
Gowers and Nielsen have written a nice opinion piece (Nature 461, 879-881, 15 October 2009) on The Polymath Project, an open-source and collaborative attempt at solving an unsolved math problem – to find a new proof of a result in ergodic theory called the density Hales-Jewett theorem using only ‘elementary’ building blocks. The protocol for the collaboration was that each participant (who could be anyone – from beginning student to Fields medalist – with an interest in the topic, from anywhere in the world) could post one nugget of an idea at a time, to the weblog. There was no requirement regarding the individual contributions. Indeed, it seems like some of the posts were just comments clarifying small points. In this first trial, it took a group of 27 people 37 days and approx. 800 serious comments to get the desired result.
The authors ask, who would have guessed that the working record of a mathematical project would read like a thriller? But, of course, both of them know very well (as does every serious scientist) that this is exactly the nature of research – this is why we do what we do.
To me, there were two important messages to take away:
(a) The protocol begins to demystify the process of creativity. I have always been turned off by the macho posturing by researchers who deny that big ideas are, in the end, just clever compositions of carefully chosen smaller ones. Instead, this project strongly suggests that the collaborative effect of multiple incomplete but properly diverse viewpoints is what it takes, much like Minsky’s Society of Mind.
(b) As the article notes,
“Although DHJ Polymath was large compared with most mathematical collaborations, it fell short of being the mass collaboration initially envisaged. Those involved agreed that scaling up much further would require changes to the process. A significant barrier to entry was the linear narrative style of the blog. This made it difficult for late entrants to identify problems to which their talents could be applied.”
With my AI researcher hat on (i.e., as someone who looks at this project as inspiration for the design of corresponding ‘intelligent’ computational systems), I find that this is the exactly the challenge that an autonomous agent must come to terms with when trying to learn useful skills in a lifelong sense. It is not that hard to devise learning procedures that can do the equivalent of making nuggets of suggestions. The harder problem is to learn context and measures of appropriateness for the individual components. However, if we begin to understand how people really do this it shouldn’t be impossible to get machines to follow (although, as someone used to ask me in response to such bold statements, ‘…famous last words?!’).