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?
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?
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.
Cool, thanks! I'm currently building a eurorack module where I need to estimate the frequency and phase of a sequence of input gate signals, and an issue I've run into is the delay inherent in the STFT algorithm. This seems like it might work better!
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.
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.
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.
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!
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).