Link: On Proof and Progress in Mathematics
Bill Thurston, a famous mathematician, writes about the importance of the social aspect of mathematics. Many mathematicians focus on the rigor while neglecting the importance of understanding.
For example, in the essay, Thurston shows that there are many different ways to understand a derivative, such as:
- infintesimal
- symbolic
- logical
- geometric
- etc.
See the paper for explanations of these perspectives
Each of these perspectives has something new to offer to a learner, and can add meaning and understanding.
NOTE
The divisions of human thinking that Thurston lists on page 4 remind me of IB’s Ways of Knowing.
In mathematics, it can be hard to convey an idea to someone that is not in your subfield because there is so much jargon and assumed foundational base involved that people are often lost within the first 5 minutes.
In addition, ideas can be lost overtime as the group that discovers an idea retires. This is because their writing and language doesn’t always capture their ingrained mental model, so later mathematicians have a hard time acquiring it. This can be combat in two ways:
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Younger generations of mathematicians can discover and rediscover the ideas, building their own mental models of the idea and contributing new perspectives to it.
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Mathematicians can invent names for ideas they discover in order to replace circumlocution with a good handle for an insight. For example, it’s much easier to discuss the nature of a “group” than the nature of “a system of substitutions satisfying…“.
This also allows for multiple perspectives to be applied to the same idea. A category can have different definitions in category theory and graph theory that are just two ways of saying the same thing in two different systems.
This idea reminds me of the concept of nimi sin.
TODO(unfinished): start from section 4 (page 8)