sleepyams

I think that whether we think of consciousness in terms of dualism, or if we think of it as emergent from natural behavior, it really doesn't matter functionally. Either way we are positioning it as something inaccessible from our realm or control.

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.

In my mind, the concepts of control and autonomy are what need to be revisited. We conflate the two: we are autonomous IF we are able to exert control on the environment. However, I think the reality might be that autonomous systems are more similar to an API in the sense that we can interact along the boundary but through the API we cannot exert control over the internal structure of the system (through knowledge or physical control).

I wonder if a solution for higher education is to enforce access to technology both inside and outside of the classroom? I've been teaching math at various institutions for the past 8 years after getting my PhD, and the conversation among my friends and peers has become increasingly bleak over the years. At first LLMs were not great at math, but at this point they can reliably solve the majority of problems one might see in an undergraduate math course. Recently I interviewed for a position at a university and spoke to the Dean of Arts and Sciences who admitted to me candidly that the university didn't really know how to handle AI.

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?

I think Terrence Tao makes at least 600k at UCLA. Not too bad if you ask me.

This is really awesome! I've been really interested in creating an interactive introduction to basic algebra where parts of the equation can be manipulated using drag-and-drop, but I couldn't really figure out the best way to do it. Maybe using Katex is the way to go?

What does "higher-order" mean?

This is a pretty amazing setup! I think in 2025 I would definitely prefer something like this. However, I think back in "the day" part of what made LAN parties fun was that everyone's PC was so individualized. I remember all of my high school friends and I coming of age and building our PCs. I helped a lot of my friends build their PCs and we all chose different things (such as the amount of RGB LEDs, which I thought were tacky...). I remember a friend of a friend had a water cooling system and I was so excited about checking it out. Also, things like the desktop wallpaper you chose, etc, contributed to this. There was something very magical about it all. Lugging our PCs to each others houses was a real labor of love.

I recommend the Mario 64 "Half A-Press" video if you haven't seen it:

https://www.youtube.com/watch?v=kpk2tdsPh0A

The utility behind complex numbers (for physicists at least) is really that they are a model for certain algebraic and geometric properties that are together very useful.

Good points, a large chunk of math is different formalizations of the notion of size, for example (in order of increasing absurdity):

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

This is cool! There is also Orca: https://100r.co/site/orca.html

Yep! More specifically I'm basically talking about geodesics on Riemannian manifolds.

I always wondered why that is for a long time (in basic physics that at least). It wasn't until grad school that I understood that position and velocity are enough to uniquely determine a trajectory on a manifold.

I think the field of chaos theory has branched into several other fields, so I would not necessarily agree that it is dead in the water. I personally have worked on projects within the last few years studying chaotic dynamical systems from the point of view of operator algebras.

One interesting result which you may find interesting: https://ncatlab.org/nlab/show/Lawvere's+fixed+point+theorem

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.

I think this is good advice but also not the only possible path towards completing a PhD in math. I get a lot of fruitful research done while binge-watching television, for instance. The key is that you have to find something that works for you. For some that may be logging hours but I know a lot of people for whom logging hours is a hindrance. There are mathematicians who have to actively hold back from just doing research all the time, and for these types there's no point in logging hours. My point is that if you don't think the work ethic described in this article is agreeable, don't let that dissuade you from pursuing a PhD in math.

Do you have any recommendations for a good youtube video to watch?

I actually find Pocket to be very useful as a cross-platform bookmarking app

There is a nice characterization of decimal expansions in terms of paths on a graph:

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!

That makes sense, for example Reddit has a very simple concept with the upvote/downvote, but also more advanced features if you need them. But I would also argue that UIs also implicitly convey a preference for how data on the web should be organized, and that affects usage.

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.

Will we ever solve UI? UI is the meeting place of tech and user, and in a sense it is THE problem for humanity right now. UI informs how we tend to use the internet, and for example we are currently involved with UIs that tend towards consumption (e.g. discovery mechanisms, feeds, etc). To solve UI would imply a certain optimal way to live our lives, if that exists. I agree with you that UI is unsolved and extremely complex, but I don't think it something to be solved as much as it is the political heart of technology.

Roughly speculative realism?

I think the insight is that there isn't really a built-in mechanism in particles for making choices, it's more that through experimentation we are _imposing_ a structure that forces a choice in an attempt to extract information. But we find that even when we force the system to make a choice, we get effectively no information and this appears to us to be a sort of complete randomness.

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.

In addition to the other replies, check out Clifford algebras which generalize these types of numbers:

https://en.wikipedia.org/wiki/Clifford_algebra

https://en.wikipedia.org/wiki/Quaternions_and_spatial_rotati...

IDK much about octonions, but quaternions have similar algebraic properties to rotation groups and can be used to encode rotations in 3D graphics.

Also FileCoin:

https://filecoin.io/

While I would like to think that humans are intrinsically motivated to _do_, experience shows me that most people are content to merely consume. Technology engineered to keep us scrolling only seems to exasperate the issue.

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.

I think this is actually a really great point about making a better product. One could venture to ask: is the reputation system within the scientific institution intimately linked the notion of natural truth? If so, does focusing on approaching science from the lens of product development signify a shift in the notion of what science is about?

I'm a PhD student in math currently and I often wonder what the difference really is between the creation and discovery of mathematical ideas. It seems the difference is really a matter of perspective: are we trying to discover something true or create something useful? Often these two perspectives intersect.

A forum would definitely be core functionality. I've also been curious about the merits of anonymous publishing?

What would a platform for independent researchers (or potential collaborators!) look like?

Perhaps a self-loop would represent time spent curating one's own profile (or crafting tweets) as opposed to consuming others?