Bitcoin

Some Thoughts and Ideas on Consensus, Proof-of-Work and Distributivity

Note: like most articles on my personal blog, this one assumes a fair degree of domain familiarity on the part of the reader. If you are new to blockchain technology, I have listed at the end of this article the resources that I’ve found to be the most clear and helpful introductions, and that I would suggest consulting if you want introductory material. Feel free to post more specific questions as comments.

Marc Zuckerburg has made headlines again for announcing a dedication to fixing everything wrong with Facebook. Included in the post was a personal reflection on decentralisation:

Continue reading

Advertisements
Standard
Maths, statistics

Column Vectors and other Notational Nonsense

Apparently the Harvard Business Review once called data science the sexiest job of the century. Math notation, by contrast, is not a very sexy topic. Perhaps this explains the general disdain that data scientists seem to have for correct mathematics. For example:

“I prefer clarity well above mathematical correctness, so if you’re an academician reading this, there may be times where you should close your eyes and think of England.” – John W. Foreman in Data Smart.

Mathematical correctness is pretty much the study of being clear. In the early 20th century some paradoxes in math lead to an entire foundational crisis, with people like Russel and Whitehead refusing to use natural language in math at all because of its potential for ambiguity. If ‘correct’ mathematics is still unclear, then something has gone very wrong.

Perhaps the author meant he preferred notation that is easier to understand? On that count there’s at least room for debate.

Continue reading

Standard
Maths

The Knaster-Tarski Lemma

The Knaster-Tarski Lemma is one of my favourite proofs. The logic is really simple but the ideas around the proof go from concepts in mathematical morphology through to analysis, order theory, algebra and even Galois connections. The proof also uses a synthetic construct that can make it difficult to appreciate the bigger picture. The motivation behind the lemma is of course also the powerful solution that it provides to the Cantor-Shroder-Bernstein theorem. Here is a video that I made which explores this proof:

 

Standard
Economics, Maths

Abstract Mathematics and the Commons

One of the principle objects of abstraction is economy of effort. Any 4 year old will quickly learn that adding 3 apples to 2 apples will yield a comparable result to adding 3 cars to 2 cars. Once she has conceptualised the idea that “2+3=5” regardless of what is being added, her answer when asked what 2 trees added to 3 trees is will be forthcoming much quicker than before this realisation. No longer will she have to go out and find a forest in order to determine the result. Likewise, when elementary algebra is first introduced in high school, a great saving of time is achieved when a student is able to say that x + x = 2x for an infinite variety of x values, without having to confirm this for each individual case.

Continue reading

Standard
Maths, Physics, statistics

Convolution

{(f \ast g)(t) = \int_{R} f(\tau)g(t - \tau) d\tau}

What what?

1. Introduction

A statistics tutor once mentioned to my tut not to worry about convolution because “Prof doesn’t understand it either”. The thing certainly looks confusing and appears to be aptly named. But convolution is a really nice concept. It’s probably the only interesting operation on functions other than composition that you will have in your examples box for thinking about functional analysis. And it’s damn useful. This paper is just a simple explanation of how convolution works with a very simple example.

Continue reading

Standard
Debating

Splitting Hairs and Rolling Chairs: Representation in Debating Adjudication

“Consequently, as far as is humanly possible, judges assess the persuasiveness of speeches according to a set of shared judging criteria, rather than according to their own views about the subject matter.” –WUDC Manual

“In short, the judges personal/political values and sensibilities cannot be excluded from the interpretive process or adjudication. Not because judges are weak and give in to political temptation, but because the exclusion called for by the traditional rule of law ideal is quite simply impossible.” – Professor Karl Klare: Legal Culture and Transformative Constitutionalism

 

Continue reading

Standard