I have long been trying to express a particular idea with respect to how I experience mathematics. Namely, Mathematics feels like a place, which I poetically like to call The Garden. This post is an attempt at articulating that thought, along with some recent reflections on what the future of mathematics might look like.

On placehood

Before talking about mathematics proper, I should specify what I mean by “feeling like a place”. I actually have something of a definition¹:

Something feels like a place (or has “placehood”) if it satisfies the following properties

• Partial Perceptibility You can perceive or experience a portion of a place, but usually not all at the same time.
• Traversability: You can “move around” a place, which is felt by a change in what you perceive of it.
• Coherence: As you move around in a place, your perceptions form a cohesive whole, e.g. there is no sharp discontinuity. You can come back to portion you’ve already experienced and find it mostly unchanged. You can build an accurate mental model of it that lets you navigate.
• Permanence: A place “feels like it exists” independently of you perceiving it. It “exists” through time, and you can observe it change. It has a “history”.

To give a few concrete examples:

• A house is a place: You can see the rooms in a house/apartment and physically move around in it. Your perceptions of the house are coherent and let you build a mental map of it. The house exists independently of you, and over time you can see it change (e.g. new furniture, decoration, …)
• A desert is not a place: A sand desert does not feel like a place, because its layout changes too fast to be coherent. The desert might look completely different on two different days.

Concretely, the properties I described are necessary to be able to build a mental model of something. In other words, a place is something that can be mapped. Of course, what I’m describing is a subjective feeling, not anything absolute, which is why I’m being vague. Permanence in particular should be interpreted as “a place does not change too fast”. It is essentially “coherence through time”.

Physical places are not the only things that have placehood. For example:

• A Discord server feels like a place: You can perceive the channels and threads of a server, and move around between channels. One channel will look mostly the same when you come back to it, and you can view the history of messages, new server emojis or stickers… Contrast with something like a twitter feed, which is more like a desert than a house.
• A Book can feel like a place: This is particularly the case for textbooks, which are not necessarily meant to be read linearly. You perceive the book when you read it, you can move around to specific chapters or sections, (and for physical books this is associated with the physical operation of your fingers on the pages). The contents of the book have a certain nonlinear structure that you can experience when reading. The book exists whether you read it or not.

Hopefully these examples have helped convey the class of subjective experiences I’m trying to point to. Certain things feel like they have a coherent structure that can be discovered and mapped out, and often a good map can let one infer possible paths through them before actually following those paths.

Mathematics feel like a place

The main point of this post is to explain that Mathematics feel like a place to me. Concretely,

• Mathematics are perceived by doing math, by reading a mathematical text, thinking about a problem (mentally or on paper)… Specific results or theories are “locations” within mathematics.
• Different parts of mathematics are in relation with one another, and elaborating these connections feels like movement. Going through a proof of a result is a form of “movement”. Another example is pedagogy, one can teach a topic by “approaching” from already known topics (For example, The Euler-Lagrange equations can be approached from a mechanics point of view, or from an optimization point of view).
• The relations between different locations of math let one have multiple viewpoints of one location, and those viewpoints are “coherent”. (See the example of the Euler-Lagrange equations above)
• Mathematics feel like they have a reality independent of the mathematician. The subjective experience of doing mathematics is often more like discovering some uncharted territory than inventing something. And in so far as mathematician invent anything, it is closer to naming landmarks than building them.

Poetically, I think of mathematics as a Garden. An infinite landscape of nature with many recurring plants and features. Parts of the Garden have been knowns for centuries and are well-trodden. We can see features in the distance which no one has reached yet, and we keep finding new paths between known locations. Some locations are like hills or mountains, in that they let us see entirely new parts of the Garden.

What drives most of my enjoyment of mathematics is that the Garden is beautiful. It is intricately connected and one can wander at random and expect to find something recognizable in a new context. It has a complex structure, but one can guess about the existence of a path between two locations and often be right.

Mathematicians as cartographers

If Mathematics are a place that can be explored, then mathematicians are best described as explorers and cartographers, drafting a map of the Garden. This explains why there is prestige attached to being the first to reach a location, or the first to draw a complete map of an area. Solving a mathematical problem (like proving Fermat’s last theorem) is very much like path finding through uncharted territory, or trying to reach the peak of a mountain.

In recent years, mapping out the Garden has been a bit of a cottage industry, with many mathematicians working on tiny portions of the garden to avoid competing with others. A side effect of this is that the known parts of the Garden are too numerous to be understood by a single person, and no complete map of the Garden exists.

Until recently, mathematicians-cartographers have been working mostly by hand, using simple methods that have existed for centuries, which sort of justified the diversification and specialization of labor. However, outsiders to the community have now come up with a tool that is much faster than humans at mapping out the territory, and has a much more complete map than anyone.

Much like satellite imagery dramatically changed cartography, and put many cartographers out of business, I think the activity of mathematicians will change profoundly in the years to come, for better or for worse. Only time will tell.