sleepyams
As a mathematician, I don't disagree with you, I do think the recent events have exposed a certain brand of elitism/gatekeeping etc within the mathematical community. While that may be true, it's not always about "finding the solution", oftentimes we first have to learn how to even frame the problem to begin with.
I believe that we can at least posit some kind of mechanism through which emergence can happen. In my opinion we should look at language and how language evolved. However, i also believe we should expand our study of natural language to things like network protocols, and observe the "protocol hourglass" structure that has emerged from the internet protocol stack.
Teachers increasingly must assume that students will not learn on their own outside of the time spent in class. This is difficult because we only see students for a handful of contact hours every week, and there is not enough time to both lecture AND ensure that a student has attained thorough understanding of the material. Teachers have adapted to this in various ways, but another stressor is that students are coming into college much less prepared. When I teach precalc I essentially have to assume that I will need to teach students how fractions work. The current system is not built to support this kind of learning. In order to make sure students are really learning, I now have to cut out major portions of the curriculum to make sure I have time to do active learning during class. It's obvious through students' performance that their understanding of things we work on in class is much more internalized and sophisticated than things I lecture about but assume they will learn on homework problems.
A novel I really love is Anathem by Neal Stephenson. While Anathem is an admittedly goofy work of fiction, I do think Stephenson's vision of the future is compelling: the purity of mathematics education and research must be protected from the hyper-technological outside world. In reality I'm not anti-technology of course, I feel like access to a non-internet-connected computer is fine. But, I wonder if such a model would work for an educational institution?
https://en.wikipedia.org/wiki/Cardinality
https://en.wikipedia.org/wiki/Measure_(mathematics)
https://en.wikipedia.org/wiki/Ultrafilter_(set_theory)
https://en.wikipedia.org/wiki/Euler_characteristic
https://golem.ph.utexas.edu/category/2008/02/metric_spaces.h...
https://golem.ph.utexas.edu/category/2006/10/euler_character...
(and of course there are certainly many that I'm missing)
For more fun, see these slides from John Baez: https://math.ucr.edu/home/baez/counting/counting.pdf
I also highly recommend this survey paper by John Baez and Mike Stay: https://math.ucr.edu/home/baez/rosetta.pdf
There are plenty of interesting results in category theory, in fact your comparison to abstract algebra is apt. There is only so much you can say about an arbitrary group in general, or an arbitrary topological space, just like there is only so much you can say about an arbitrary category.
Let C be the countable product of the set with ten elements, i.e. {0, 1, 2, ..., 9}. The space C naturally has the topology of a Cantor set (compact, totally disconnected, etc). Furthermore, for example, in this space the tuples (1, 9, 9, 9, ...) and (2, 0, 0, 0, ...) are distinct elements.
The space C can also be described in terms of a directed graph, where there is a single root with ten outward directed edges, and each child node then has ten outward directed edges, etc. C can be thought of as the space of infinite paths on this graph.
A continuous and surjective map from C to the unit interval [0, 1] can be constructed from a measure on these paths. For any suitable measure, this map is finite-to-one, meaning at most finitely many elements of C are mapped to a single element in the interval. For example there is a map which sends (1, 9, 9, ...) and (2, 0, 0,....) to the element "0.2".
The point is that all decimal expansions of elements of [0, 1] can be described like this, and we can instead think of the unit interval not as being composed of numbers _instrinsically_, but more like some kind of mathematical object that _admits_ decimal expansions. The unit interval itself can be described in other ways mathematically, and is not necessarily tied to being represented as real numbers. Hope this helps someone!
For example the feed, besides being a UI pattern, also claims that content on the web is for consumption and not exploration, and so we remain stationary on these aggregation platforms instead of 'browsing' the web as we used to. This is efficient but politically bad for society, and we would need to introspect on how we use the internet in tandem with the development of new tech like decentralized platforms.
I interpret Conway's idea (in part III) to be that this is not a structural randomness, but rather randomness only in response to structure that _we_ impose during an experiment (e.g. determining spin), and that this is similar to our free will which is not structured (if you subscribe that that theory).
EDIT: I feel like I should clarify: at play here is the question of how mechanistic the universe is. If we accept that the universe is mechanistic, then we are forced into a position where we have to say (if we wish to avoid determinism) that there is such a thing as a "completely random" mechanism, and furthermore we have to effectively absolve free will of any structural restrictions by requiring that it is somehow part of our vitality as humans (call this mind-body dualism or whatever -- there is a long history of this debate). Obviously this is all beyond physics, but it is important to realize what assumptions we are implicitly requiring in order to discuss randomness and free will in this case.
IDK much about octonions, but quaternions have similar algebraic properties to rotation groups and can be used to encode rotations in 3D graphics.
Reminds me of the concept of interpassivity: https://en.wikipedia.org/wiki/Interpassivity. We maybe be intrinsically motivated to _do_, but we have developed technology that can satiate this desire, e.g. watching someone else play video games on twitch.tv.