Recent content on Eigil Fjeldgren Rischel
Forecasting is predicting whether something will happen, like who will be the next US president or whether a natural disaster will happen or what the economy is going to be like in a year. It’s notoriously difficult to think about. Typically who study this sort of thing think you should make quantifiable predictions with specific probabilities assigned to them, make them public, and then let…
Introduction The category of bimorphic lenses and its many generalizations has been widely studied and utilized in applied category theory. We will not give a review of the literature here, but see e.g. Riley’s paper on “optics” (one of the many generalizations) Categories of Optics, which includes a decent if somewhat out of date overview. Dispensing with a point of notation, we…
(Epistemic status: contains some mildly sloppy math, but is essentially true). Suppose someone offers you the chance to bet some money on a coinflip. On a heads, you multiply your stake by \(3.75\), but if you lose, you lose your stake. Should you bet, and how much? Clearly this bet has positive expected value, and higher expected value the more your bet, but equally clearly, the actual decision…
A number of different people have thought about ways to bring notions of approximation into category theory. There seem to be essentially two notions that one would like to express here: The idea that a digram, while it may not quite commute, commutes up to some specified tolerance \(\epsilon\) The idea that a mapping, while it may not quite preserve the relevant structures, preserves them up to…
Coarse geometry The idea of a coarse space is to formalize a sense in which the inclusion of metric spaces \(\mathbb{Z} \hookrightarrow \mathbb{R}\) is an equivalence. These two spaces have the same “large-scale structure”, in the sense that any function into \(\mathbb{R}\) can be approximated up to uniformly bounded error by one into \(\mathbb{Z}\). Say two functions \(f,f': X \to Y\)…
Astral Codex Ten: Bounded Distrust, also Against That Poverty And Infants EEGs Study In Search of Visual Texture How should you talk to think better? satisfying: a tool for replicating odd shapes and imperfections pic.twitter.com/PhEGFTGtXj — Visakan Veerasamy (@visakanv) January 12, 2022
Dordle, a Wordle variant where you have to guess two words at the same time. Intercats, a new seminar from the Topos Institute on “categorical interaction”. I’m scheduled to speak here (in June, so don’t get too excited yet) How To Become A Magician. See also Becoming A Magician. Postmortem on RatVac. I have the highest level of respect for everyone who made their own…
Here I am working with nonstandard analysis in the sense of Robinson, taking an ultrapower of the real numbers and building things out of that. But in general I am going to be a bit sloppy and not worry too much about the details. Recall that a standard function is differentiable if, for every standard \(x\) and for every infinitesimal \(\epsilon\), \(f(x+\epsilon)-f(x)/\epsilon \approx a\), where…
A sequence of bits \(N \to \{0,1\}\) is Martin-Löf random (or algorithmically random) if, roughly speaking, there is no computable pattern to it. There are three equivalent definitions: Kolmogorov complexity definition Let \(K(x)\) be the kolmogorov complexity of a binary string (finite). Say \(x\) is $c$-incompressible if \(K(x) \geq |x| - c\). An infinite string is Martin-löf random if there…
I wrote a long post on one of the papers I read this week: Example of my reading process: Cellular sheaves of lattices and the Tarski laplacian. See that post for the details! Some other stuff: Jade Master: The Open Algebraic Path Problem This paper came out in 2020, so it’s practically ancient history. But it’s really cool! It’s about The problem of finding a path between two…
There’s a lot of sort of “implicit” skills that are important in various fields of science, that you really only learn by just hanging around older people that already know them and picking things up by osmosis. This is one of the things that make it hard to just learn things by reading textbooks, as opposed to actually going to a university and getting a degree. I think we…
John Baez wrote a regular blog/column called “This Week’s Finds in Mathematical Physics” circa 1993-2012. The entries are really a treasure trove of cool mathematical nuggets, covering everything from hardcore theoretical physics, group theory, climate models, category theory, and more. Imitation being the sincerest form of flattery, I decided to shamelessly steal this format,…
Also contains links from February. Fantastic Anachronism: Two Paths to the Future Ansuz: What color are your bits? What do the notions of “random number” and “copyrighted music” have in common? They’re not about the specific bits under consideration, but about their color. mike_hawke: Some random parenting ideas. Zvi: Why I Am Not In Charge I was very taken with Lucy…
Also read Hillel Wayne: Why Python Is My Favorite Language. Whenever I need to write a program to do some dumb bullshit, I usually whip out Python. This is because I usually don't have to look up a bunch of shit to remember how to make it work, and I don't have to remember how the build system works either. More languages like that please — Eigil - abstr/acc 💎 (33322/50000 words) (@Ayegill)…
Alvaro de Menard: Are Experts Real?, and the followup, Unjustified True Disbelief. The former: There’s a superficial uniformity in the academy. If you visit the physics department and the psychology department of a university they will appear very similar: the people working there have the same titles, they instruct students in the same degrees, and publish similar-looking papers in…
Practical Foundations for Programming Languages (PFPL), by Robert Harper, is an introduction to the theory of programming languages. I recently finished reading through it. My read was fairly cursory - I stopped to think about ideas which seemed important or interesting, but I didn’t read everything deeply, and I didn’t do a lot of exercises. In this post I’ll summarize the…
Alternative title: Wolves hate him!! Shepherd compares the size of large sets with this one easy trick! Previously: Recognizing Numbers Let’s do a thought experiment. I place an empty box in front of you. Then, while you’re watching, I put these objects into the box: Then I remove these things from the box: You’re surprised! Why? Because what I took out is not a subset of what I…
A list of some of the things I found interesting in December. Low-Tech Magazine: “Low-tech Magazine questions the blind belief in technological progress, and talks about the potential of past and often forgotten knowledge and technologies when it comes to designing a sustainable society. Interesting possibilities arise when you combine old technology with new knowledge and new materials, or…
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(The idea for this post is due to this tweet by @sarah_zrf) Hilbert’s Nullstellensatz Consider the ring of complex polynomials in \(n\) variables, \(\mathbb{C}[x_1,x_2,\dots x_n]\). The elements of thing ring can be viewed as functions \(\mathbb{C}^n \to \mathbb{C}\). Given a polynomial \(f\), we can think of it as an equation in \(n\) variables - a solution to the equation is a tuple…
Thermodynamics is really weird. Most people have probably encountered a bad explanation of the basics at some point in school, but probably don’t remember more than Energy is conserved Entropy increases There’s something called the ideal gas law/ideal gas equation. Energy conservation is not very mysterious. Apart from some weirdness around defining energy in general, it’s just a…
Credence - and chance - without numbers (and with the Euclidean property) is a philosophy paper by Tim Maudlin. In it, Maudlin discusses the closely related notions of credence, the subjective likelyhood that a specific agent associates to some outcome, and chance, the objective likelyhood that the event happens. He argues that The traditional approach of measuring these outcomes with numbers is…
I recently got a reMarkable 2. I’ve had it for one week. This is my review of it so far. TLDR It’s very very good and I’m happy I bought one. If your relationship with working “on things” is like mine, I recommend it. It’s expensive and you can get an iPad + accessories + Apple Pencil for the same price, which may be better There are some annoyances Would I buy…
Another day, another post about dynamical systems. Today, I want to think about open dynamical systems. You can think of an open dynamical system as a system where The dynamics are parameterized by some variable (which is supposed to vary with time) And some function of the state is exposed (maybe to parameterize other systems). I want to describe two types of open dynamical systems: continuous…
This blog post largely retraces ideas from Lawvere: Functorial remarks on the general concept of chaos. I saw this in this tweet from Jade Master, which this blog post is basically an extended version of. Hat tip to her. Let’s try to apply category theory to the study of “dynamical systems”. What is a dynamical system? There are a lot of different versions: A discrete dynamical…
A Chu space over \(S\) consists of a pair of sets \((X,U)\), and a function \(e: X \times U \to S\). A map of chu spaces \((X,U,e) \to (Y,V,e')\) is a pair of maps \(X \to Y, V \to U\) so that the diagram commutes. This defines a category of Chu spaces, called \(Chu(Set,S)\) You can think of a Chu space as a normal-form game. \(X\) is the set of choices available to one player, and \(U\) is the…
Linear logic is a weird sort of logic. It’s most commonly explained by saying that the “weakening” rule: Sorry, your browser does not support SVG. and the “contraction” rule Sorry, your browser does not support SVG.. In other words - you have to use all the assumptions, and you can’t use an assumption more than once. This is usually interpreted in terms of…
Continuing the train of thought from this tweet, I compare and contrast two perspectives on the philosophy of category theory: that it’s about describing how things can be composed of other things (“compositionality”), and that it’s about describing things in terms of their transformations into other things (“universal properties”). Some uses of category theory…
Let’s prove a classical theorem (Emily Riehl’s favorite!) from category theory: Right adjoint functors preserve limits. So let’s assume we have categories \(C,D\), functors \(F: C \to D, G: D \to C\), and a natural bijection \(C(G(a),b) \cong D(a,F(b))\). Let’s also fix a diagram \(X: I \to C\) from some index category \(I\). Now recall that a limit of \(X\) is an object…
Context: Krause and Nikolaus: Group Theory for Homotopy Theorists (pdf). Krause and Nikolaus develop group theory using model categories (well, one model category). This is obviously a joke, but I think it is a very useful pedagogical joke. So I’m going to go through it and try to explain what’s happening. Group presentations If you’ve taken a course on group theory, you’ve…
See: Giorgios Bakirtzis and Christian Williams: Turing Categories. Turing Categories describe computability. I want to find a category to work with complexity instead. This is a stab at it. Fix a universal Turing machine and an encoding of the natural numbers. Of course, this lets us speak of computable functions \(\mathbb{N} \to \mathbb{N}\) (and these don’t depend on the choice of Turing…
See also: Jade Master: Dynamical Systems With Category Theory? Yes!, This tweet by me. Discrete dynamical systems A discrete dynamical system \((S,T)\) consists of a set \(S\) and a time-step function \(T: S \to S\). It’s clear that this is exactly the same thing as a \(\mathbb{N}\)-set, i.e a set with an action of the monoid \((\mathbb{N},+,0)\)1. A “morphism of discrete dynamical…
Terry Tao: A cheap version of nonstandard analysis. MathOverflow: Does Cheap Nonstandard analysis take place in a topos? (Answer: Yes, but an elementary topos, not a Grothendieck topos). This is partially a summary of Tao’s blog post, partially a small discussion of way LEM fails for cheap nonstandard reals. What is “nonstandard analysis”? In “normal” nonstandard…
Let \(\mathsf{FinStoch}\) be the category of finite sets and stochastic matrices. Given two stochastic matrices, \(f_1,f_2: X \to Y\), we can define their Jensen-Shannon distance as \(d(f_1,f_2) := \sup_x \sqrt{\operatorname{JSD}(f_1(x),f_2(x))}\), where JSD is the Jensen-Shannon divergence. It’s a standard result that the root of JSD defines a metric on the space of probability measures -…
Transfer learning is the idea that, after a machine learning system (or a non-machine learning system, for that matter, like a human) has learned to solve some problem, it should be able to transfer this knowledge to solving similar problems. Humans are pretty good at this, at least compared to current ML systems, which tend to suck. Why do we expect transfer learning to work? It seems that, in…
Stalks and points Recall that a point of a topos \(\mathcal{E}\) is a geometric morphism from the topos \(Set\). My preferred way to think about this is to consider the sheaf topos \(Sh(X)\) on some (sober) topological space \(X\). Then given \(x \in X\) and a sheaf \(S\), we can form the stalk at \(x\) \(S_x := \operatorname{colim}_{x \in U \subseteq X \text{ open}} S(U)\) This determines the…
The Ax-Grothendieck theorem says the following: Let \(f: \mathbb{C}^n \to \mathbb{C}^n\) be a polynomial function. If it’s injective, then it’s surjective as well. Here’s how to prove it: The statement can be formulated as a first-order statement in the language of fields If a statement like that fails for \(\mathbb{C}\), there’s a disproof in the first-order theory of…
My notes from Persistent Homotopy Theory by JF Jardine. The goal of the paper is to study “filtered spaces”. By this is meant in general something like an assignment \(s \mapsto X_s\) of a “space” or simplicial set to each nonnegative real \(s \in [0,\infty)\). A prototypical example is the Vietoris-Rips complex of a metric space, \(V_s(X)\). The idea being pointed towards…
Computationally bounded probability theory Probability theory is about how to manage incomplete information. One way to interpret a statement like “the probability of event \(X\) is \(p\)” is in terms of betting odds - you think the probability of \(X\) is \(p\) if you value a lottery ticket that pays out $1 if \(X\) happens at \(p\) dollars. From this interpretation, all the laws of…
TLDR This is a post about my paper The zero-one laws of Kolmogorov and Hewitt-Savage in categorical probability, joint with Tobias Fritz. This is a “companion piece” where I try to explain those ideas in a more understandable language. There are essentially three ideas in this paper: “Markov categories for synthetic probability theory” - this is only treated briefly, since…
TLDR I give a very short introduction to the idea of “causality” in statistics, then talk about two ways to infer causal structure for two variables - i.e, without using conditional independence statements. The ideas here are mostly taken from Peters, Janzing, and Schölkopf: Elements of causal inference: foundations and learning algorithms. What is causality? A “causal…
Here’s what goes into a website. A server (hardware). The software which runs on the server, also called a server. A domain (optional) An SSL certificate (technically optional but highly recommended) I will explain what these terms mean, and how to get your own stuff set up. Please note that this is a guide for people who want to do everything from scratch. There are plenty of easier ways to…
Time-saving blurb: This essay thingy eventually ends without any useful conclusion (I don’t manage to figure out how to make something compose). Also, it’s not clear that what’s here is particularly deep even if it could be made to work, which it hasn’t. P-values For convenience I’ll only work with finite sets - I’m not aware of any serious problems extending…