Hedonic Calculus

 This post is dedicated to BK. I've procrastinated this one because it's a huge topic and one quite important to me, learned from good teachers.

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 ).