Hedonic Calculus

This post is dedicated to RA and this meme he sent me. This post is a little longer than usual, but hopefully it seems small in proportion to its topic.   In The Fault In Our Stars, Hazel-Grace boldy proclaims that "some infinities are bigger than others" citing the fact that there are more numbers between 0 and 10 than between 0 and 1. If I were marking her work then she would get a drawing of a donut because, although her conclusion is correct, she used woefully misinformed reasoning to reach it. Let's dig in.

 In the  Ant on a Rubber Rope post  I mentioned a discrete solution that I found very pleasing and I plan to present it here. I would normally recoil from anything that smells of 'discrete maths' but I think turns out to be far more enlightening as the gut feeling that the ant shouldn't reach the end of the rope is the same feeling that the harmonic series should converge.

I actually wrote this over a year ago but this seems like the right platform so please enjoy an alternative motivation of the extension from the real numbers to the complex numbers.

In the previous post we mercilessly mocked an ant who was condemned to spend its entire life falling and so couldn't distinguish time from height. In this post about the next paradox we similarly toy with an ant, but this time the ant prevails in the end!

 This post is dedicated to KP and our relentless debate on this matter.

Just as the last paradox was an excuse to talk about proof by induction, this is an excuse to talk about some basic logic. Let's lay some foundations so that we're all swimming in the same pool. We're going to be playing with statements and statements can have one of two values: True or False. Then one relationship 2 statements can have is the 'implication' relation, or more succinctly the word 'if'. Fact: all thumbs are fingers. This sounds like a statement that can't be decomposed but it's actually an implication: "if something is a thumb, then it is a finger". Notice that this isn't symmetric; "if something is a finger, it isn't necessarily a thumb". We would say here that A implies B but B does not imply A. We can define an equivalence relation  on the set of statements by saying that A and B are equivalent  if A implies B and B implies A. I hope that seems sensible. The implication is the perfect way of describing (an...

 This post is more for my own understanding than making maths accessible, as the topic requires a bit of Partial Differential Equation (PDE) knowledge. My 4th year project concerns, succinctly, the equation common to wave propagation and heat flow. I plan to write a post on my project at some point and another motivating the wave and heat equations but I'm going to start at the end here and provide physical intuition for something called the Robin boundary condition (RBC), what my project focuses on specifically. I have read a large number of sources that refer to the RBC as describing a "partially insulated boundary" or an "elastically supported boundary" but it took me a very long time to find a source providing any intuition as to why this equation describes those physical phenomena so thank you to the textbook of Strauss.

 I'd like to start a little series on some paradoxes, maybe for the pure joy of the paradox or perhaps to talk about some related idea. Today's uses an abuse of proof by induction,  a handy little tool that is great for proving (verifying really) facts that we're told but awful at telling us why they're true. Induction is used to prove a fact about all  whole numbers (perhaps after a point) and we start by showing that something is true for the smallest case, usually n = 1. Then we follow by showing that if the statement is true for some n, then it must be true for n + 1. And in fact, it's true for n = 1, so this proves it's true for n = 2. Now we know it's true for n = 2, we can conclude it for n = 3, and so on. A simple example might be: we can reach any rung on a ladder because we can reach the first rung ( base case ) and from any rung we can reach the next rung ( induction step ).

Now it's time to address the title of this trilogy: how can we solve a probability problem without using probability? We translated our original problem into one about area and took our answer to the area problem to be the same as the answer to the probability, but how can we justify this? There clearly is a relationship between the two, as we'll see in the next example, but that if that relationship isn't a perfect correspondence then our solution rests of shaky ground. 

 Yesterday I deprived you of a derivation and I now plug that hole. On reflection, I much prefer the method of   Michael Penn  who spends sometimes half a video arming himself with a set of 'tools' before facing the task in hand, deploying his arsenal where necessary. This feels more pedagogically elegant and in future posts "like this" will precede posts "like yesterday's".

 I'd like to share a problem which was covered in  3blue1brown's lockdown math lecture series  and I found particularly delightful. This post is adapted from a talk I've given to year 12 further maths students in my sixth form, as the solution contains only A level maths content and I wanted to give a flavour for the abstractions of university maths. I promised myself not to recycle content from 3B1B as I believe he's the best maths educator on the internet and there's nothing I can add but this problem wasn't covered in full animated detail and I'd like to talk about measure theory. The problem goes as follows: "2 numbers are randomly (independently) chosen between 0 and 1. We divide them and round down the number to the integer below. What is the probability that the resulting integer is even?" Or, for the maths symbol lovers like myself (rookie latex error italicising the mod): Firsly I'd like to clarify what I mean when I say "randomly...

Following the problem from yesterday, I thought it'd be an insightful activity to model the situation in python. Here is the code I wrote which uses some rudimentary numpy. In no way am I suggesting that my code is optimal. The first block of code is simply importing several packages I'll need to use. Matplotlib is used to make images, numpy makes it much easier to do operations with lists of numbers and cmath means I can use complex numbers 1 + 2j rather than lists [1,2].

 Over the summer I spent a week or so playing with a problem Steven Strogatz proposed on an episode of the 3B1B podcast , which goes as follows: 4 dogs sit on the corners of a square. At some instant, each dog chases the dog in the next clockwise corner, each with equal speed. The dogs chase each other in a spiral and eventually meet in the centre. How far does each dog travel before the collision?  The value of the problem extends far beyond its solution and it has expanded to fill 3 posts. The first (which you are reading) shares some problems with similar solutions before giving the deceptively simple solution to the title problem. The second compares 2 tasks, a programming simulation of the problem and an analytic approach to explicitly finding the paths taken by the dogs. The third will hopefully be more philosophical, comparing the skills of proof writing and coding. Before solving this it is helpful to think of another problem with a similar solution, which I encountere...

 I've toyed with the idea of writing a blog for a while, encouraged largely by SA but his introducing me to Neel Nanda and  Nanda's post on blogging has given me the necessary push. The friction that prevented me from starting this was a fear of sounding pretentious; the assumption that I have something to say that is so important that other people should give up their time to read it. So I begin this with zero intention that others will read it and I instead take a wholly selfish approach. These posts are for me: they are an opportunity for me to clarify my own thoughts on topics important to me, as well as a chance to improve my writing ability. If you happen to stumble across these posts then that is your own mistake and I don't presume that anything I write is original or worth your time . And that said, the title of the blog refers to the fact that if you do decide that my thoughts might make clement inhabitants of your brain, a post should not outlive the time it t...