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Joshua Meyers (Sep 12 2020 at 17:52):If we want to figure out our values, it is essential to think about our capabilities. What are we capable of as a community? What is ACT capable of? What is math capable of? There is an underlying sense in many of us that ACT has revolutionary capabilities; what is the best precisification of this sense that you have seen, or can think of? Once we know our capabilities, we can articulate our values in terms of how we want to use such capabilities.
Morgan Rogers (he/him) (Sep 13 2020 at 09:16):Taking a generous view on the scope encompassed by CT, as Tom Leinster does, the most powerful thing that CT can do is lead us to unifying perspectives. Most disagreements arise from people holding incompatible views or priorities and having no means of adequately communicating those ideals in a common framework. This applies to international politics, different teams in a company or shared project, or at the level of individuals. That second option is where CT solutions are already starting to emerge, but I see no reason that it couldn't happen at other levels too.
Morgan Rogers (he/him) (Sep 13 2020 at 09:19):That's a little on the nebulous side as an answer to your question :sweat_smile: it doesn't include a detailed practical approach, but I would jump at the opportunity to develop such from the ground up.
Henry Story (Sep 13 2020 at 09:37):I was going to write something very much along the lines of @[Mod] Morgan Rogers .
I found at a certain point that the real difficult problems seemed to be communities that were trying to work towards the same goal not understanding anything the other communities they needed were doing. Sometimes these are related to people working in dual categories, and the unifying perspective of CT is one that allows dualities to exist without reducing them or failing to distinguish each side. (see Maruyama's thesis on duality. Finding such translations can be very helpful.
I also find it very helpful to get an idea of where I should look for answers for something.
Jules Hedges (Sep 13 2020 at 14:16):Figuring out our capabilities even very approximately is probably a multi-decade project
Jules Hedges (Sep 13 2020 at 14:17):(Personally, I estimate our capabilities are lower than most people around here seem to, although of course higher than a complete skeptic)
Nikolaj Kuntner (Sep 13 2020 at 16:14):Okay, so for capabilities: Is there an attempted list going through other math topics science subfields where influence of CT could be in demand and useful?
Nikolaj Kuntner (Sep 13 2020 at 16:19):Morgan linked Leinster's blog post.
Is there more sort of Manifest vibe texts?
Joshua Meyers (Sep 13 2020 at 22:13):Nikolaj Kuntner said:
Okay, so for capabilities: Is there an attempted list going through other math topics science subfields where influence of CT could be in demand and useful?
Yes! The NIST report
Joshua Meyers (Sep 13 2020 at 22:16):I started reading the Leinster piece that @[Mod] Morgan Rogers linked to, I really like his "second definition" of category theory, as contrasted with the first, "obvious" one:
- It’s highly conceptual, with almost no calculation. (Contrast other proofs.) You could easily explain it out loud to someone while walking in the park. The only point when you might be tempted to reach for pen and paper is the last step, calculating the constant of proportionality, but even that makes only a small demand on your powers of mental algebra.
- It solves the problem by moving to a higher level of generality — and getting the right level of generality, at that.
- The generalization wasn’t quite the most obvious one. Good generalization almost always begins by carefully preparing the ground. Here, this was the little observation that for events that occur at most once, expected value is the same as probability. A more obvious generalization would be to ask what the probability of crossing was for needles of arbitrary length.
- We see uniform behaviour across a class of objects (curved needles of arbitrary length). Under other circumstances this would be called functoriality or naturality.
- There is a piece of grit around which the pearl forms. You can never solve a specific problem by entirely general means: at some point, you have to make the link between the generalities and the specific situation that you started with. Here, the grit is the calculation of the constant 2/π2/\pi, got by thinking about a circular needle.
I also have the impression that category theory isn't really about categories per se, it's rather a way of doing math, an aesthetic sensibility.
Joshua Meyers (Sep 13 2020 at 22:20):[Mod] Morgan Rogers said:
Taking a generous view on the scope encompassed by CT, as Tom Leinster does, the most powerful thing that CT can do is lead us to unifying perspectives. Most disagreements arise from people holding incompatible views or priorities and having no means of adequately communicating those ideals in a common framework. This applies to international politics, different teams in a company or shared project, or at the level of individuals. That second option is where CT solutions are already starting to emerge, but I see no reason that it couldn't happen at other levels too.
I'm going to play devil's advocate here for the sake of the conversation: How exactly would it help if people with incompatible views or priorities were able to communicate them in a common framework? Wouldn't they still have incompatible views or priorities?
Joshua Meyers (Sep 13 2020 at 22:20):Nikolaj Kuntner said:
Morgan linked Leinster's blog post.
Is there more sort of Manifest vibe texts?
What do you mean by "Manifest vibe"?
Henry Story (Sep 13 2020 at 22:47):Joshua Meyers said:
I'm going to play devil's advocate here for the sake of the conversation: How exactly would it help if people with incompatible views or priorities were able to communicate them in a common framework? Wouldn't they still have incompatible views or priorities?
If two groups can find out that the views are dual, then in a way it would explain the difference and explain that any proof one group had in their category would, flipped around, be a proof in the opposite category. So at the very least it settles the discussions about which is best: they are a weird form of mirror image. Duality also does not identify both sides. Algebras and coalgebras have very different properties for example.
There is a whole philosophy of duality in this sense. See Maruyama's The Dynamics of Duality.
Nikolaj Kuntner (Sep 14 2020 at 00:36):Joshua Meyers said:
What do you mean by "Manifest vibe"?
Something people read and are like, "ah, yeah, now my next thingy I do I will run with under this banner"
Morgan Rogers (he/him) (Sep 14 2020 at 08:21):Joshua Meyers said:
I'm going to play devil's advocate here for the sake of the conversation: How exactly would it help if people with incompatible views or priorities were able to communicate them in a common framework? Wouldn't they still have incompatible views or priorities?
Eugenia Cheng gives a really nice example of how this helps in her public lectures. I would recommend everyone watch the whole thing, but the relevant part is here, a section that lasts maybe 15 minutes.
Having a common framework not only allows you to identify what you have directly in common with another party, it allows you to identify the common reasoning (as @Henry Story has nicely described) and analogies between situations, and these form an essential basis on which to build constructive discussions. They also highlight fundamental power or information imbalances that must be addressed in order for a fair discussion and conclusion to be reached.
John Baez (Sep 14 2020 at 15:14):Jules Hedges said:
(Personally, I estimate our capabilities are lower than most people around here seem to, although of course higher than a complete skeptic)
I think there's a big difference between what we actually know how to do fairly soon, and what we might eventually figure out how to do.
John Baez (Sep 14 2020 at 15:15):The latter is unknown, of course. And yes, I agree it will take several decades at least to figure it out.
John Baez (Sep 14 2020 at 15:18):My strategy is to just march ahead optimistically, focused on things I can do pretty soon.