Thinking like a Mathematician

Implications of Condorcet’s jury theorem

There once lived a man named Marie Jean Antoine Nicolas de Caritat, Marquis of Condorcet (1743–1794). A mathematician and philosopher, his work was mainly focused on advancing social progress towards a more egalitarian society. For instance, he strongly advocated for gender equality as early as in 1787, when he wrote:

“The rights of men stem exclusively from the fact that they are sentient beings, capable of acquiring moral ideas and of reasoning upon them. Since women have the same qualities, they necessarily also have the same rights.”


El Farol on Canyon Road in Santa Fe, New Mexico

Game Theory

“Oh that place. It’s so crowded nobody goes there anymore.” — Yogi Berra

The El Farol Bar Problem, sometimes known as the “Santa Fe Bar problem” (SFBP) is a constrained resource allocation problem for non-cooperating agents defined by economist William Brian Arthur (1945-) in 1994. The problem deals with ways of achieving an optimal collective resource allocation in situations where, if everyone uses the same pure strategy that strategy is guaranteed to fail no matter what it is.

Definition

The problem begins with explaining that on Thursday night every week 100 people decide independently whether or not to go to a…


An Illustrated Five Step Guide

Earlier this week Reuters broke the news that the U.S. House of Representatives Judiciary Committee has called on Amazon.com founder Jeff Bezos to testify on

“Allegations that the online retailer uses data from its own third-party sellers to create competing products”

The allegations are, and have been for some time, that Amazon engages in uncompetitive practices when it comes to how they compete with third-party sellers on the Amazon Marketplace.

Amazon’s platform for buying and selling consumer goods has since 2018 has enjoyed a 50% market share of all e-commerce in the United States. Starting in 2009, Amazon entered this…


Open Problems

Are there infinitely many primes p such that p + 2 is prime?

The twin prime conjecture states that:

There are infinitely many twin primes

A twin prime is a prime that differs from another prime by two. A set of two primes that differ by two are called a twin prime pair. The first twin prime pairs are:

(3,5), (5,7), (11,13), (17,19), (29,31), (41,43), (59,61), (71,73), (101, 103), (107, 109), (137, 139), ...

The prime pair (2,3) is not considered to be a twin prime set because they differ by one instead of two, thus they are more closely spaced than other all other twin primes.

Origins

Although Euclid in 300 BC proved…


Resistance map of E.coli bacteria to 3rd Generation Cephalosporin antibiotics around the world (CDDEP, 2020)

As of the writing of this article, 6,820 have died of corona infections in Italy alone. That number rose by 743 yesterday, and is expected to continue to rise as the Italian health care system struggles to keep up with new critical patients among its close to 70,000 confirmed cases. Italy’s nearby neighbor Spain in the same period lost 2,696 people, a number which began surging about a week ago, and is currently rising by over 500 new deaths every day. As of today, 50,000 people are infected in the country.

Lessons from H1N1

According to former head of the Center for Disease…


Left: Richard Feynman at 41 years old, from his Cornell Lectures in 1959. Right: Richard Feynman 27 years later at his office at the California Institute of Technology in 1986, aged 68 (Photos: Wikimedia Commons)

The Feynman Essays

“We have a different kind of awareness about what might happen than we have of what probably has happened”

In physicist Richard P. Feynman’s fifth lecture at Cornell University in 1959 he entertained for a moment the question of ‘what distinguishes the future from the past?’. His lecture sets out with the following primer:

It is obvious to everyone that the phenomena of the world are evidently irreversible. I mean, things happen that do not happen the other way. You drop a cup and it breaks, and you sit there a long time waiting for the pieces to come together…


Left: Claude Shannon (1916–2001). Right: Shannon’s notable 1949 paper “Communication Theory of Secrecy Systems” in Bell System Technical Journal 28(4) pp. 656–715.

CRYPTOGRAPHY

A Shannon cipher, invented by its namesake Claude Shannon (1916–2001) is a simplified cipher mechanism for encrypting a message using a shared secret key. A cipher is generally defined simply as an algorithm for performing encryption or decryption, i.e. “a series of well-defined steps that can be followed as a procedure”.

Example (Boneh & Shoup, 2020)
Suppose Claude and Marvin want to use a ciper such that Claude can send an encrypted message that only Marvin can read.
Then, Claude and Marvin must in advance agree on a key k ∈ K. Assuming they do, then when Claude wants to…


The Feynman Essays

“I would like to make a number of remarks on the relation of mathematics and physics”

During Richard Feynman’s Messenger Lecture Series on “The Relation of Mathematics & Physics” held at Cornell University in 1965, “The Great Explainer” addressed what he found to be the key differences between mathematics and physics. His thoughts are summarized below.

Differences in Epistemology

“Mathematicians prepare abstract reasoning that’s ready ‘to be used’ even though they don’t know what it’s being used for”

First, Feynman addresses the differences in the epistemological level of analysis between those studying mathematics, in particular singling out metamathematicians:

Mathematicians are only dealing with…


The Feynman Essays

In a lecture held by Nobel Laureate Richard Feynman (1918–1988) on September 26th, 1985, the question of artificial general intelligence (also known as “strong-AI”) comes up.

Audience Question

Do you think there will ever be a machine that will think like human beings and be more intelligent than human beings?

Below is a structured transcript of Feynman’s verbatim response. With the advent of machine learning via artificial neural nets, it’s fascinating to hear Feynman’s thoughts on the subject and just how close he gets, even 35 years ago.

Estimated reading time is 8 minutes. Happy reading!

Richard Feynman’s Answer

First of all, do they think…


Thinking Like a Mathematician

How to Ensure Fairness as a Mechanistic Outcome

In the context of economics and game theory, envy-freeness is a criterion of fair division where every person feels that in the division of some resource, their share is at least as good as the share of any other person — thus they feel no envy. For n=2 people, the protocol proceeds by the so-called divide and choose procedure:

The Envy-Free Cake-Cutting procedure states that if two people are to share a cake in way in which each person feels that their share is at least as good as any other person, one person ("the cutter") cuts the cake into…

Jørgen Veisdal

Author of Privatdozent. Editor-in-Chief at Cantor’s Paradise.

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