Max_Daniel

Karma: 15

Max_Daniel 6 Aug 2026 20:44 UTC
6 points
−1
in reply to: Jacob_Hilton’s comment on: Jacob_Hilton’s Shortform
As a fellow former mathematician, I share your sense that “mathematics as poetry” seems like the most likely outcome, at least for pure maths and barring more fundamental AI-driven changes to the human condition (e.g., extinction, or humans being replaced by very different kinds of minds).
Some of your discussion reminded me of Field Medalist Bill Thurston’s 1994 note “On Proof and Progress in Mathematics”:
1. What is it that mathematicians accomplish?
[...]
Rather, as a more explicit (and leading) form of the question, I prefer
How do mathematicians advance human understanding of mathematics?
This question brings to the fore something that is fundamental and pervasive: that what we are doing is finding ways for people to understand and think about mathematics.
The rapid advance of computers has helped dramatize this point, because computers and people are very different. 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. On a more everyday level, it is common for people first starting to grapple with computers to make large-scale computations of things they might have done on a smaller scale by hand. They might print out a table of the first 10,000 primes, only to find that their printout isn’t something they really wanted after all. They discover by this kind of experience that what they really want is usually not some collection of “answers”—what they want is understanding.
Similarly, Thurston wrote on MathOverflow:
The product of mathematics is clarity and understanding. Not theorems, by themselves. [...]
I think of mathematics as having a large component of psychology, because of its strong dependence on human minds. Dehumanized mathematics would be more like computer code, which is very different. [...]
In short, mathematics only exists in a living community of mathematicians that spreads understanding and breaths life into ideas both old and new. The real satisfaction from mathematics is in learning from others and sharing with others. All of us have clear understanding of a few things and murky concepts of many more. There is no way to run out of ideas in need of clarification. The question of who is the first person to ever set foot on some square meter of land is really secondary. Revolutionary change does matter, but revolutions are few, and they are not self-sustaining—they depend very heavily on the community of mathematicians.
To be clear, I do think that AI comes with bigger challenges for, and will have bigger impacts on, the maths community than the kind of ‘automatic computation’ or ‘computer code’ Thurston mentions in these quotes because LLMs can do a much wider range of maths tasks, including communicating about maths in many of the ways human mathematicians do, and so their use likely won’t be limited to narrow use cases outside an exclusively human-driven core activity of generating and transmitting ‘understanding’ to each other. And so in particular I agree with the broad picture you outline where AI use is a pervasive feature of a “mathematics as poetry” world.

Max_Daniel 14 Feb 2020 19:19 UTC
7 points
0
on: [Link] Beyond the hill: thoughts on ontologies for thinking, essay-completeness and forecasting
[Thoughts on what to do if there is an ontological mismatch between one’s thinking and a tool]
- When I saw Jacob present a version of the OP in person, the discussion focused on cases where the correct response is to use a different tool, ideally one that matches the natural ontology of ones thinking. E.g. when using a whiteboard rather than a Google doc to express thoughts most naturally expressed as a mind map.
- But I think it’s important that there are other cases where it can actually beneficial to ‘learn how to think in a different ontology’. I think this is quite common in pure maths, but also shows up in more everyday situations: e.g. initially I found it quite counterintuitive to use, say, Emacs org mode or LaTeX, but after I had payed the fixed cost of adapting to the ontologies imposed by them I actually think that it made me more efficient at some tasks.
- Similarly, I think it’s useful to be able to translate between different ontologies. To learn this, it can be useful to deliberately expose oneself to ontologies that seem unnatural/bad/cumbersome initially.

Max_Daniel 14 Feb 2020 19:18 UTC
3 points
0
on: [Link] Beyond the hill: thoughts on ontologies for thinking, essay-completeness and forecasting
[Thoughts on the term “ontology”.]
- In philosophy, ontology refers to the subfield aiming to answer the question “Which things exist?”.
- Perhaps as a consequence of this, hearing “ontology” makes me think of questions like: What are the primitives or building blocks here? E.g. in a spreadsheet, there would be cells; in a graph there would be vertices and edges etc.
- But I think the important things Jacob is talking about show up as answer to the different question of which relations there are between these primitives. E.g. the important thing about a spreadsheet is that each cell is unique identified by a pair of numbers, that you can perform computations/functions on the values of cells; the important thing about a (certain type of) graph is that every edge has exactly one vertex as source and exactly one vertex as target.
- Brainstorm for alternative terms: structure, conceptual structure, conceptual apparatus, conceptual scheme, conceptual toolkit, relations, relational constraints, conceptual landscape, …
- (The math version of my complaint is that “ontology” makes me think of set theory, whereas I think the important bits are more naturally visible from a category theory point of view.)

Max_Daniel 21 Jun 2017 10:26 UTC
2 points
0
in reply to: username2’s comment on: S-risks: Why they are the worst existential risks, and how to prevent them
Thank you for your feedback. I’ve added a paragraph at the top of the post that includes the definition of s-risk and refers readers already familiar with the concept to another article.