###### tags: `one-offs`

# Some Intuition Boosters

In this note, I collect some snippets of mathematical intuition which have become clearer to me in recent memory (loosely interpreted). Many of the observations are essentially quite basic, but are still arguably deserving of repetition. I've not been particularly consistent on giving specific examples here, but do let me know if any of the claims are too vague to provide any insight, and I can then try to reinforce them with an example.

1. In probability and measure theory, martingales and filtrations are fundamentally useful as a mathematical formalisation for keeping track of information, and the way in which it is introduced into a system gradually. Trees (and directed acyclic graphs more generally) often play a similar role.

2. When thinking about some abstract space of objects, be equally prepared to think about the points which concretely make up that space, and about the functions which map that space into other spaces (and even just into the reals). Spaces of functions are very nicely structured, and thinking functionally about a space is very useful for suggesting analogies and abstractions.

3. In geometric settings, having curvature which is bounded uniformly away from zero makes life very nice in many ways. Actually, just having curvature bounded below by zero is still pretty great, though usually a little bit of extra work is needed to clarify that this is the case. Perhaps surprisingly, even just having the curvature bounded below by anything at all - even something negative - can be very useful news. Conversely, when looking for negative results, counter-examples, and so on, a space with unbounded negative curvature can often be a sensible first port of call.

4. If some object is known to exhibit some sort of interesting behaviour 'at infinity', then there should be meaningful hints of that behaviour before arriving at infinity. The best asymptotic results don't force you to wait all the way until the end.

5. If it's interesting for an object to have a given property, then it's probably also interesting for an object to almost have that property. A good definition of some property being almost-true can be the basis for interesting and robust developments of a main theory.

6. If a system exhibits nonlinearity in some sense, then this is usually a signal that interactions play a non-trivial role in the system. Interactions are often interesting and challenging because they propagate to higher orders, e.g. in a dynamical system involving binary collisions, one often needs to think about the higher-order effects of collisions which chain together. When any of these phenomena are in play, then a series expansion which enumerates these higher-order interactions is often just around the corner. In this way, nonlinear dynamics are often allied surprisingly strongly with combinatorics.

7. When we consider dynamical processes which evolve in time, we are very often interested in processes with stable and non-explosive behaviours (partially because even explosive and vanishing systems can be rescaled into such a form). With this in mind, analysis of dynamical systems should often aim to replicate this stability, and quantitative estimates should be designed to not blow up with the time horizon.

8. Objects which exhibit lots of symmetry tend to lose a bit of that symmetry once we make incomplete measurements of them. As such, while symmetric systems are a nice way to gather understanding, in practice, one quickly needs to be able to handle inhomogeneous, conditioned, and controlled systems. Fortunately, many methods and analyses can extend quite nicely to the inhomogeneous settings; keep this in mind when learning new tools.

9. When analysing some structure (perhaps discrete or combinatorial in character), it can be worthwhile to identify what the 'elements' / 'connected components' / 'building blocks' of that structure are.

10. When working with discrete objects, do not be tricked into thinking that only discrete operations are allowed. One is always allowed to think about the 'conceptual convex hull' of some scaffolding, and one often should. In a way, probability is obtained as the convex hull of combinatorics, or of deterministics.

11. Sometimes, one person might say that concepts A and B are 'the same', and another person might insist that those same concepts are meaningfully different. It is a useful exercise to hear both of these people out. Which perspective turns out to be more relevant can basically depend on the day; having both characterisations in one's back pocket is a useful skill.

12. Don't be shy about making an assumption from time to time. We would all like to prove the most useful thing, but we can start off by proving a true thing, and building up from there. One true result at a time, you can build foundations for a subject, and it helps to get the wheels of development spinning early.

13. Life can get very exciting upon realising that a problem has more { axes / limits / ... } than was initially apparent. Keep an eye out for these axes, and learn to assess which axes are most relevant to your interests. In this way, one can often realise that existing tools apply to a much wider range of problems than was initially intended.

14. Enduring mathematical tools are often effective in very humble ways. When learning new tools, it pays to witness those tools being used in simple, native, 'inevitable'-feeling contexts, where they solve a natural problem very well, and take them as they are; try not to expect magic (though don't be afraid to try to make them work magic once you're a bit more advanced).

15. A strength of abstraction is that it allows you to reason about concepts which would be challenging to 'instantiate concretely' in whatever sense. Some tools (often, but not always, more symbolic in character) can be disproportionately effective at scaling up intuition in this way. This applies particularly strongly if you care about 'high-dimensional systems' in whichever form.

16. An equality which suggests a useful - maybe even tight - inequality makes everybody happy. Similarly, understanding extremal cases in a practical inequality can be very insightful.

17. Examples give life, pictures give life, new questions give life, new connections give life. Embrace the bigger picture and strive to fit your ideas into it. Hinting well at these connections make it easier to spark creative readings.

18. There are questions and tasks out there which turn out to be genuinely 'impossible to solve' in some more-or-less meaningful way, and it's certainly not your fault that you might have run into one. It is nevertheless healthy to try to learn some of the warning signs eventually.

19. If an object can 'go forwards', then it can often also 'go backwards'. This doesn't always mean simply undoing what you did on the way forwards (an inverse is not generally the same thing as an adjoint). In any given context, it's typically worth understanding what the right 'backwards' equivalent is, as it usually doesn't exist by accident.

20. Try not to judge one area's tools by their ability to solve another area's problems. Sometimes this pans out, but an area can be perfectly valuable developing methods for its own purposes. Being 'cool and interesting' is not the same as 'useful for every application', and that's fine.

21. In the vaguest possible terms, it can be the case that i) the ways in which a method works best and ii) the ways in which a method fails worst, can bear no resemblance to one another, even qualitatively.

22. It is surprising just how often 'dynamic; methods can be used to prove things about 'static' objects, just by introducing some fictitious dynamics which shed light upon the static structures at play.