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‘… that the will is infinite
[but] the execution confined,
that the desire is boundless
[but] the act a slave to limit.’
This is Shakespeare, and although he was expressing frustration not about mathematics but another human essential (can you infer which?) I feel much the same having taken on the labour of trying to define mathematics. It’s a tricky one to tackle but I hope we (you’ll be working on it too!) can do it some justice. To begin with, take some time to consider how you would define mathematics, in a paragraph or just a few words.
I think of mathematics as a beast, as I am reminded of the Indian parable of the six blind men who seek to comprehend an elephant. 1
Each of us (the blind men) focus on the aspect that we are grappling with and see it in light of our earlier experiences. And as John Godfrey Saxe concludes in his telling of the tale:
‘And so these men of Indostan
Disputed loud and long,
Each in his own opinion
Exceeding stiff and strong,
Though each was partly in the right
And all were in the wrong!’
The definition I would like to share is one put forward by one of the foremost geometers of our time, William Thurston. Writing in his essay, On proof and progress in mathematics, Thurston says,
‘Mathematicians generally feel that they know what mathematics is, but find it difficult to give a good direct definition. It is interesting to try. For me, "the theory of formal patterns" has come the closest, but to discuss this would be a whole essay in itself. Could the difficulty in giving a good direct definition of mathematics be an essential one, indicating that mathematics has an essential recursive quality?
Along these lines we might say that mathematics is the smallest subject satisfying the following:
• Mathematics includes the natural numbers and plane and solid geometry.
• Mathematics is that which mathematicians study.
• Mathematicians are those humans who advance human understanding of mathematics.’
I would add ‘interpreting data’ to Thurston’s set of seeds (number and geometry) but what I would like to focus on is that ‘mathematics is what mathematicians do’, and more importantly, Thurston’s definition of mathematicians – those who advance human understanding of mathematics . This includes mathematics teachers and learners as being mathematicians (as each learner could be advancing their own understanding). Traditional views would exclude these communities. When I left mathematics research to move to full -time teaching, I stopped calling myself a mathematician, until I came across Thurston’s essay. The traditional view does need to be up-ended, if we were to apply it to music, possibly only composers would be called musicians!
But there is one problem with Thurston’s elegant (indeed mathematical) definition of ever-expanding mathematics – it doesn’t tell us much about the ‘doing’ of mathematics. How exactly do we advance human understanding of mathematics?
Many mathematicians would say that mathematics is not simply defined by objects but by the relations and connections between objects. These connections then become new objects - and then there are connections between them ... 2
This exponential growth can quickly get overwhelming! But worry not, in the next post we will consider a five-fold path to mathematical wisdom, and that could help tame the multi-dimensional ever-expanding beast that is mathematics. And in the spirit of ‘I do and I understand’, we will apply the five-fold path to understanding √2, the square root of two, a quintessential mathematical object.
Homework – what does √2 mean to you?