Thurston, W. P. (1994). On Proof and Progress In Mathematics (arXiv:math/9404236). arXiv. https://doi.org/10.48550/arXiv.math/9404236
Summary
Abstract
In response to Jaffe and Quinn [math.HO/9307227], the author discusses forms of progress in mathematics that are not captured by formal proofs of theorems, especially in his own work in the theory of foliations and geometrization of 3-manifolds and dynamical systems.
Mathematics is defined recursively: one primary goal of mathematicians is to advance human understanding of mathematical subjects, and communicate it effectively.
Understanding mathematics requires a number of cognitive abilities: human language; spatial and kinestheic awareness; logic and deduction; intuition, association, and metaphor; response to stimuli; and understanding processes over time.
Communication in a subfield is efficient because of shared patterns of thinking. Formalism provides a language for understanding mathematics to individuals outside the subfield.
Atomic notes
- Mathematics is what mathematicians do
- Mathematical communication is most effective in subfields due to shared patterns of thinking
Selected concepts and passages
What do mathematicians accomplish?
Mathematicians advance human understanding.
For instance, when Appel and Haken completed a proof of the 4-color map theorem using a massive automatic computation, it evoked much controversy. I interpret the controversy as having little to do with doubt people had as to the veracity of the theorem or the correctness of the proof. Rather, it reflected a continuing desire for human understanding of a proof, in addition to knowledge that the theorem is true.
The subject of mathematics is recursively defined.
In other words, as mathematics advances, we incorporate it into our thinking. As our thinking becomes more sophisticated, we generate new mathematical concepts and new mathematical structures: the subject matter of mathematics changes to reflect how we think.
Could the difficulty in giving a good direct definition of mathematics be an essential one, indicating that mathematics has an essential recursive quality?
How do people understand mathematics?
Mathematical thinking is requires modular cognitive facilities.
Our brains and minds seem to be organized into a variety of separate, powerful facilities. These facilities work together loosely, “talking” to each other at high levels rather than at low levels of organization.
Human language communicates math.
We have powerful special-purpose facilities for speaking and understanding human language, which also tie in to reading and writing. Our linguistic facility is an important tool for thinking, not just for communication.
Mathematical symbols are related to universal grammar.
The mathematical language of symbols is closely tied to our human language facility. The fragment of mathematical symbolese available to most calculus students has only one verb, ”=“. That’s why students use it when they’re in need of a verb.
Capacity for spatial awareness exceeds capacity to communicate it.
On the other hand, they do not have a very good built-in facility for inverse vision, that is, turning an internal spatial understanding back into a two-dimensional image. Consequently, mathematicians usually have fewer and poorer figures in their papers and books than in their heads.
Formal logic is based on intuition, and later encoded.
Mathematicians apparently don’t generally rely on the formal rules of deduction as they are thinking. … In fact, it is common for excellent mathematicians not even to know the standard formal usage, of quantifers (for all and there exists), yet all mathematicians certainly perform the reasoning that they encode.
Formal logical structure does not reflect linguistic action.
It’s interesting that although “or”, “and” and “implies” have identical formal usage, we think of “or” and “and” as conjunctions and “implies” as a verb.
How is mathematical understanding communicated?
Communication is most effective in a subfield due to shared patterns of thinking.
In contrast, communication works very well within the subfields of mathematics. Within a subfield, people develop a body of common knowledge and known techniques. By informal contact, people learn to understand and copy each other’s ways of thinking, so that ideas can be explained clearly and easily.
Trasmission of mathematical knowledge in a subfield scales with time.
Mathematical knowledge can be transmitted amazingly fast within a subfield. When a significant theorem is proved, it often (but not always) happens that the solution can be communicated in a matter of minutes from one person to another within the subfield. The same proof would be communicated and generally understood in an hour talk to members of the subfield. It would be the subject of a 15or 20-page paper, which could be read and understood in a few hours or perhaps days by members of the subfield.
Universal mathematical language does not convey nuanced ways of thinking.
Mathematics in some sense has a common language: a language of symbols, technical definitions, computations, and logic. This language efficiently conveys some, but not all, modes of mathematical thinking. Mathematicians learn to translate certain things almost unconsciously from one mental mode to the other, so that some statements quickly become clear.
Subfields have shared internal language cues for thought patterns.
People familiar with ways of doing things in a subfield recognize various patterns of statements or formulas as idioms or circumlocution for certain concepts or mental images. But to people not already familiar with what’s going on the same patterns are not very illuminating; they are often even misleading. The language is not alive except to those who use it.
Understanding mathematics is distinct from writing mathematics.
A group of mathematicians interacting with each other can keep a collection of mathematical ideas alive for a period of years, even though the recorded version of their mathematical work differs from their actual thinking, having much greater emphasis on language, symbols, logic and formalism. But as new batches of mathematicians learn about the subject they tend to interpret what they read and hear more literally, so that the more easily recorded and communicated formalism and machinery tend to gradually take over from other modes of thinking.