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Mathematical ideas are conventionally expressed using notation and terminology developed using static media.
Suppose, however, that mathematics had been invented after modern computers. This is perhaps difficult to imagine — after all, mathematics helped lead to computers — but let's do the thought experiment anyway.
Might mathematical notation have developed in a different way? Would we instead have developed a dynamic, interactive notation more powerful than the static mathematical and linguistic notations in common use today?
As part of my exploration of this question, I recently developed a simple prototype system, which I call Magic Paper code.
To get a feel for what the system does, below is a video of me using Magic Paper to explain the very simple! Note that you do not need to be deeply familiar with neural networks to follow along; an introduction to the basics of neural nets may be found here.
You may wish to fullscreen the video, for easier viewing:. The design of Magic Paper was inspired in part by a system developed by Ken Perlin, known as Chalktalk.
I built Magic Paper to explore two questions not directly addressed by Chalktalk. The first question is whether such a visual medium can be used to do serious mathematical exploration, the kind a researcher might do when working on a research problem.
In other words: can such a system be used as a medium for creative thought? While Chalktalk is a fun medium to explore in, it's not obvious to me that it's suitable for the kind of exploration that arises in mathematical work.
Of course, it's too ambitious for such a quickly-built prototype medium to be useable across all of mathematics. And so I decided to narrow the question, and ask whether such a medium could be used to explore at least one non-trivial mathematical question; concretely: can neural nets be used to approximate any function?
While focusing on one problem is much easier than building a general-purpose exploratory mathematical medium, it's still an extremely demanding and thus informative constraint.
Furthermore, it seems likely that patterns useful in solving one mathematical problem will be reuseable in solving others. In fact, even this is too demanding.
The idea for a proof presented above wasn't actually discovered in this medium. Despite this, I believe this context — explaining the idea underlying the proof of a relatively recent mathematical result — is a good one for understanding what is required for a general-purpose exploratory mathematical medium.
Note that while the video of Magic Paper perhaps makes it appear to be explanatory or educational in intent, that is not the case.
It is intended primarily to be a medium to think and explore in, and only secondarily to explain or to educate in. While these goals are sometimes congruent, they are also often in tension.
As an example of this tension, explanatory media are often highly polished, presenting a finished product from which distracting elements have been eliminated.
By contrast, exploration involves distractions, wrong paths, and dead ends. One sees this in myriad small ways in the video: the frequent readjustment of the position and size of the glyphs; the various unnecessary weight and bias parameters that are sometimes shown; and so on.
In a polished presentation these would have been eliminated, showing just what the viewer needs at any given moment.
But such superfluous elements are inevitable in exploration. Another example of the tension between education and exploration comes from the programming language Logo.
Logo is often touted as an example of discovery-based learning, and in many respects it's a good exploratory medium for mathematics. However, while Logo has had considerable success in helping children understand ideas about differential geometry, differential geometers do not, as far as I'm aware, use it as a medium for research.
An interesting project would be to create a single environment which can be used for both education and research. I am optimistic this is possible; after all, beginning chess players use the same board and the same moves as the world champion.
The second question motivating Magic Paper was whether it is possible to reify deep ideas about mathematics inside the interface. That sounds cryptic, but has a straightforward meaning.
To explain that meaning, let's briefly switch topic, and recall an idea from elementary physics, the principle of conservation of energy:.
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