## System Of Reasoning (S1E01): The Rules.

The Pilot.

This is an episode in a series on mathematical logic approached with some rigour. Here, we will be closely following the book by Peter B. Andrews: An Introduction To Mathematical Logic and Type Theory. In this episode, we will:

Part 1

2. Show the equivalence of the well-formed formula definition to that of a formation sequence on a formula.
3. Revisit the principle of mathematical induction and complete induction on the natural numbers.
4. Learn the principle of induction on the construction of a well-formed formula.

Part 2

1. Learn about substitutions in the context of propositional logic. And use this idea to derive one of De Morgan’s Laws.

The only knowledge this post will assume is a basic knowledge in set theory.

Part 1.

(The Prelude) Before we start with logic, let’s revise two ideas which may serve to be important later on. If one is familiar with the principle of mathematical induction and strong induction, one may skip this.…

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## Correlation vs Mutual Information

This post is based on a (very small) part of the (dense and technical) paper Fooled by Correlation by N.N. Taleb, found at (1)

Notes on the main ideas in this post are available from Universidad de Cantabria, found at (2)

The aims of this post are to 1) introduce mutual information as a measure of similarity and 2) to show the nonlinear relationship between correlation and information my means of a relatively simple example

Introduction

A significant part of Statistical analysis is understanding how random variables are related – how much knowledge about the value of one variable tells us about the value of another. This post will consider this issue in the context of Gaussian random variables. More specifically, we will compare- and discuss the relationship between- correlation and mutual information.

Mutual Information

The Mutual Information between 2 random variables is the amount of information that one gains about a random variable by observing the value of the other.…

## The Res-Net-NODE Narrative

This is second post in the blog series, and it is meant to give a broad narrative of the content for next two blog posts. Like the previous post, it will be more of an overview, but the two posts that will follow it will unpack and discuss deeply whatever appears in this one and slightly more.

Humble Beginnings: Ordinary Differential Equations

The story begins with differential equations. Consider $f$ such that $f:[0,T]\times \mathbb{R}^n\to \mathbb{R}^n$ is a continuous function. We can construct a rather simple differential equation given this in the following way. We let $\begin{cases} {y'(t)}=f(t,y(t))\\ y(0)=y_0\in \mathbb{R}^n \end{cases}$

A solution to this system is a continuous map that is defined in the neighbourhood of $t=0$ such that this map satisfies the differential equation.

Ordinary differential equations are well-studied, and we know that, for example, a solution to the given differential equation will exist whenever the function $f$ satisfies the following: $(\forall x,y\in \mathbb{R}^n)(\exists C>0 (\in \mathbb{R}))(||f(t,y)-f(t,x)||\leq C||y-x||)$

This property is known as Lipschitz continuity. A function that satisfies this condition is said to be Lipschitz.…

By | March 18th, 2020|Uncategorized|Comments Off on The Res-Net-NODE Narrative

## Deep Q Learning, Briefly!

This blog post is a direct translation of a talk that was given by the author on the 17th of February 2020. The ideas was to very briefly introduce Deep Q-Learning to an audience that was familiar with the fundamental concepts of reinforcement learning. If the person reading this is not familiar with these basics, then a very great introduction can be found here: An Introduction to Reinforcement Learning. Without the additional details from the talk, one will note that this post is rather brief, and should really be used as a tool to gain an overview for the method or a gateway to relevant resources. This will not the case for posts later in the series, because the intention is to deal more with the mathematical aspect of reinforcement learning.

Basic Reinforcement Learning Notions

The idea behind reinforcement learning is that there is an agent that interacts with the environment in order to achieve a certain task.…

## Curves for the Mathematically Curious – an anthology of the unpredictable, historical, beautiful and romantic, by Julian Havil – a review

NB I was sent this book as a review copy. What a beautiful idea. What a beautiful book! In studying mathematics, one comes across various different curves while studying calculus, or number theory, or geometry in various forms and they are asides of the particular subject. The idea however of flipping the script and looking at curves themselves and from them gaining insight into: statistics, combinatorics, number theory, analysis, cryptography, fractals, Fourier series, axiomatic set theory and so much more is just wonderful.

This book looks at ten carefully chosen curves and from them shows how much insight one can get into vast swathes of mathematics and mathematical history. The curves chosen are:

1. The Euler Spiral – an elegant spiral which leads to many other interesting parametrically defined curves
2. The Weierstrass Curve – an everywhere continuous but nowhere differentiable function
3. Bezier Curves – which show up in computer graphics and beyond
4. The Rectangular Hyperbola – which leads to the investigation of logarithms and exponentials
5. The Quadratrix of Hippies – which are tightly linked to the impossible problems of antiquity
6. Peano’s Function and Hilbert’s Curve – space filling curves which lead to a completely flipped understanding of the possibilities of infinitely thin lines
7. Curves of Constant Width – curves which can perfectly fit down a hallway as they rotate.

## The Objective Function

In both Supervised and Unsupervised machine learning, most algorithms are centered around minimising (or, equivalently) maximising some objective function. This function is supposed to somehow represent what the model knows/can get right. Normally, as one would expect, the objective function does not always reflect exactly what we want.

The objective function presents 2 main problems: 1. how do we minimise it (the answer to this is up for debate and there is lots of interesting research about efficient optimisation of non-convex functions and 2) assuming we can minimise it perfectly, is it the correct thing to be minimising?

It is point 2 which is the focus of this post.

Let’s take the example of square-loss-linear-regression. To do so we train a linear regression model with a square loss $\mathcal{L}(\mathbf{w})=\sum_i (y_i - \mathbf{w}^Tx_i)^2$. (Where we are taking the inner product of learned weights with a vector of features for each observation to predict the outcome).…

## Tales of Impossibility – The 2000 year quest to solve the mathematical problems of antiquity, by David S. Richeson – a review

NB I was sent this book as a review copy. Four impossible puzzles, all described in detail during the height of classical Greek Mathematics. All simple to define and yet so tempting that it has taken not only the brain power of many, many thousands of mathematicians (amateur and professional alike), but also two millennia to show that however hard you may try, these puzzles are just not possible. The puzzles are:

• Squaring the circle: With only a compass and a straight edge, draw a square with the same area as that of a given circle.
• Doubling the cube: With only a compass and a straight edge, draw the edge of a cube with volume twice that of a cube whose edge is given.
• Constructing regular polygons: Given a compass and a straight edge, construct a regular n-gon in a given circle for $n\ge 3$.
• Trisecting an angle: Given a compass and a straight edge, and a given angle, construct an angle that is one third of the original.

## What is mathematics?

Below you find some thoughts on this wide question, I encourage you to think about. What is your vision of mathematics? It will be most probably the result of your own experience with the subject, traumas that happened along the way and realizing that, could make you more conscious about your relationship with the subject and the walls you might have built against the subject or part of the subject. In a sense, by understanding the bias and blockage, by objectively thinking about its value, you could allow yourself to be able to equip yourself with the full set of skills mathematics gives you to build your own greatest life.

Many of thoughts in the following around this topic are taken from the paper I recommend to read: Teaching and Learning “What is Mathematics and why we should ask, where one should experience and learn that and how to teach it”, by Gunter M.…

## Reasoning and making sense: a pillar of mathematics?

An essential part of learning mathematics is about reasoning and making sense. What does this exactly mean?

When a student is given a problem, he needs to make sense of it, from his level of perceptive which is unique to each individual. This will come with big struggle, and the important next step is to stay motivated, curious, be extremely perseverant and not give up after the first few attempts. This might also require a good relationship with mistakes.

A students will have to develop his own strategy to solve a given problem. That might imply first to translate it in their own language, use their own words and knowledge background to get (understand) the actual question and problem they are attempting to solve.

They will have to build bridge in their mind to similar problem they have solve in the past even though they might seem different. This bridge will be easier and easier to connect with practice and experience and sometimes might not work and some other connections will need to be created until finding a suitable one.…

## The Wisdom of the Crowds

This content comes primarily from the notes of Mark Herbster (contributed to by Massi Pontil and John Shawe-Taylor) of University College London.

Introduction

The Wisdom of the Crowds, or majority rule and related ideas tend to come up pretty often. Democracy is based (partly) on the majority of people being able to make the correct decision, often you might make decisions in a group of friends based on what the most people want, and it is logical to take into account popular opinion when reasoning on issues where you have imperfect information. On the other hand, of course, there is the Argumentum ad Populum fallacy which states that a popular belief isn’t necessarily true.

This is idea appears also in Applied Machine Learning – ensemble methods such as Random Forests, Gradient Boosted Models (especially XGBoost) and stacking of Neural Networks have resulted in overall more powerful models. This is especially notable in Kaggle competitions, where it is almost always an ensemble model (combination of models) that achieves the best score.…