Intuition and Mathematics


5


XiXiDu

While reading the answer to the question 'What is it like to have an understanding of very advanced mathematics?' I became curious about the value of intuition in mathematics and why it might be useful.

It usually seems to be a bad idea to try to solve problems intuitively or use our intuition as evidence to judge issues that our evolutionary ancestors never encountered and therefore were never optimized to judge by natural selection.

And so it seems to be especially strange to suggest that intuition might be a good tool to make mathematical conjectures. Yet people like fields medalist Terence Tao seem to believe that intuition should not be disregarded when doing mathematics,

...“fuzzier” or “intuitive” thinking (such as heuristic reasoning, judicious extrapolation from examples, or analogies with other contexts such as physics) gets deprecated as “non-rigorous”. All too often, one ends up discarding one’s initial intuition and is only able to process mathematics at a formal level, thus getting stalled at the second stage of one’s mathematical education.

The point of rigour is not to destroy all intuition; instead, it should be used to destroy bad intuition while clarifying and elevating good intuition. It is only with a combination of both rigorous formalism and good intuition that one can tackle complex mathematical problems;

The author mentioned at the beginning also makes the case that intuition is an important tool,

You are often confident that something is true long before you have an airtight proof for it (this happens especially often in geometry). The main reason is that you have a large catalogue of connections between concepts, and you can quickly intuit that if X were to be false, that would create tensions with other things you know to be true, so you are inclined to believe X is probably true to maintain the harmony of the conceptual space. It's not so much that you can imagine the situation perfectly, but you can quickly imagine many other things that are logically connected to it.

But what do those people mean when they talk about 'intuition', what exactly is its advantage? The author hints at an answer,

You go up in abstraction, "higher and higher". The main object of study yesterday becomes just an example or a tiny part of what you are considering today. For example, in calculus classes you think about functions or curves. In functional analysis or algebraic geometry, you think of spaces whose points are functions or curves -- that is, you "zoom out" so that every function is just a point in a space, surrounded by many other "nearby" functions. Using this kind of zooming out technique, you can say very complex things in short sentences -- things that, if unpacked and said at the zoomed-in level, would take up pages. Abstracting and compressing in this way allows you to consider extremely complicated issues while using your limited memory and processing power.

At this point I was reminded of something Scott Aaronson wrote in his essay 'Why Philosophers Should Care About Computational Complexity',

...even if computers were better than humans at factoring large numbers or at solving randomly-generated Sudoku puzzles, humans might still be better at search problems with “higher-level structure” or “semantics,” such as proving Fermat’s Last Theorem or (ironically) designing faster computer algorithms. Indeed, even in limited domains such as puzzle-solving, while computers can examine solutions millions of times faster, humans (for now) are vastly better at noticing global patterns or symmetries in the puzzle that make a solution either trivial or impossible. As an amusing example, consider the Pigeonhole Principle, which says that n+1 pigeons can’t be placed into n holes, with at most one pigeon per hole. It’s not hard to construct a propositional Boolean formula Φ that encodes the Pigeonhole Principle for some fixed value of n (say, 1000). However, if you then feed Φ to current Boolean satisfiability algorithms, they’ll assiduously set to work trying out possibilities: “let’s see, if I put this pigeon here, and that one there ... darn, it still doesn’t work!” And they’ll continue trying out possibilities for an exponential number of steps, oblivious to the “global” reason why the goal can never be achieved. Indeed, beginning in the 1980s, the field of proof complexity—a close cousin of computational complexity—has been able to show that large classes of algorithms require exponential time to prove the Pigeonhole Principle and similar propositional tautologies.

Again back to the answer on 'what it is like to have an understanding of very advanced mathematics'. The author writes,

...you are good at modularizing a conceptual space and taking certain calculations or arguments you don't understand as "black boxes" and considering their implications anyway. You can sometimes make statements you know are true and have good intuition for, without understanding all the details. You can often detect where the delicate or interesting part of something is based on only a very high-level explanation.

Humans are good at 'zooming out' to detect global patterns. Humans can jump conceptual gaps by treating them as "black boxes". 

Intuition is a conceptual bird's-eye view that allows humans to draw inferences from high-level abstractions without having to systematically trace out each step. Intuition is a wormhole. Intuition allows us get from here to there given limited computational resources.

If true, it also explains many of our shortcomings and biases. Intuitions greatest feature is also our biggest flaw.

The introduction of suitable abstractions is our only mental aid to organize and master complexity. — Edsger W. Dijkstra

Our computational limitations make it necessary to take shortcuts and view the world as a simplified model. That heuristic is naturally prone to error and introduces biases. We draw connections without establishing them systematically. We recognize patterns in random noise.

Many of our biases can be seen as a side-effect of making judgments under computational restrictions. A trade off between optimization power and resource use.

It it possible to correct for the shortcomings of intuition other than by refining rationality and becoming aware of our biases? That's up to how optimization power scales with resources and if there are more efficient algorithms that work under limited resources.