Posted in books

“Good Math” Notes

Full Title: Good Math: A Geek’s Guide to the Beauty of Numbers, Logic, and Computation

Author: Mark C. Chu-Carroll

Here are some notes I took while reading this book. Overall I felt it was interesting, but there were large jumps in difficulty in some of the later chapters.

Continuous fractions

This was the most fascinating part of the book for me. I hadn’t heard of these before!

For example, the square root of 2 in decimal form is approximately 1.4142135623730951. But if you represent it as a continued fraction, you get [1; 2, 2, 2, 2, 2, …]. All of the square roots of integers that are nonperfect squares have repeated forms in continued fractions.

Interesting how continuous fractions give a new and clean way of looking at previously confusing numbers like sqrt 2 and other irrational numbers. Some nice parallels with how multiplication was hard in the Roman numeral system but drastically improved in tha arabic system.

Another great example is e. If you render e as a continued fraction, you get e = [2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10, 1, 1, 12, 1, …]. In this and many other cases, continued fractions reveal the underlying structure of the numbers.

First Order Predicate Logic

This chapter was not easy. But the section on prolog looked neat. Every statement is essentially a proof that the language satisfies. Now we’re into CTL i.e computational tree logic maybe?

FOPL has no notion of time, so it’s not easy to make logical statements and assertions with it when there is a time context e.g employee (me, Cisco, 2020) is cumbersome.

FOPL is interesting because it allows us to reason with statements and prove things without knowing a thing about the actual context. The proofs come purely through logic.

Set theory plus FOPL form the foundations of maths.

FOPL summary

In first-order predicate logic, we talk about two kinds of things: predicates and objects. Objects are the things that we can reason about using the logic; predicates are the things that we use to reason about objects.

predicate is a statement that says something about some object or objects. We’ll write predicates as either uppercase letters or as words starting with an uppercase letter (A,B,Married), and we’ll write objects in quotes.

Every predicate is followed by a list of comma-separated objects (or variables representing objects). One very important restriction is that predicates are not objects. That’s why this is called first-order predicate logic: you can’t use a predicate to make a statement about another predicate. So you can’t say something like Transitive(GreaterThan): that’s a second-order statement, which isn’t expressible in first-order logic. We can combine logical statements using AND (written ) and OR (). We can negate a statement by prefixing it with not (written ¬). And we can introduce a variable to a statement using two logical quantifiers: for all possible values , and for at least one value.

Naive set theory

This is what Cantor used for his diagonal trick to measure different sizes of infinities, is limited by things like Russel’s paradox. If you use FOPL to make theories about naive sets, you eventually hit a contradiction that challenges the foundations of logic. In summary It allows you to create logically inconsistent self referential sets. The next chapter has a better alternative: axiomatic set theory.

Axiomatic Set Theory

It uses axioms to give a consistent form of set theory based on some axioms. The one in this book is Zermelo-Frankel set theory with choice, commonly abbreviated as ZFC.

First we define a set by asserting that 2 sets are equal if you pair their objects and those are equal. Ths gives us a mechanism to get and compare elements, and defines a set and it’s main operations.

Once we define an empty set, we automatically get a new one which is the set containing the empty set. Then you define an enumeration axiom that allows you to append 2 sets.

Then the default infinite set is created, out of which other infinite sets are derived. This axiom carefully ensures that these sets are not self referential, thus avoiding paradoxes.

A powerset of A is the set of all possible subsets of A.

Using a powerset axiom, we now provide the ability to take an infinite set and build a second order set that’s larger than it.

Anyway once you have the final ‘axiom of choice’, you have this set theory combined with fopl to create all of maths. Integers come naturally. Axiom of pairing can be used to get the rational numbers. Dedekind cuts can be used to get the reals. And so on.

Todo add a note on what a dedekind cut is. From what I remember, you can define 2 sets, one that has all elements lesser than sqrt(2) and one that has all elements greater. That gives a clear definition for sqrt(2) itself.

Continuum hypothesis

The first infinite set larger than aleph0 (set of natural numbers) has a size equal to aleph0’s powerset (the set of all subsets of aleph0), and this is also the size of all the reals.

Unfortunately it is neither true or false. You can treat it as either and all of zfc maths will still work.

Here we have a hypothesis that is not provable, whereas in Russel’s paradox we had an inconsistency.

Group theory

Last bit went over my head 😦

Mechanical math

Haskell code doesn’t help 😦

Posted in Quote

On Patriotism

“Sometimes as I reached a gathering, a great roar of welcome would greet me: Bharat Mata ki Jai– Victory to Mother India! I would ask them unexpectedly what they meant by that cry, who was this Bharat Mata, Mother India, whose victory they wanted? My question would amuse them and surprise them, and then, not knowing exactly what to answer, they would look at each other and at me. I persisted in my questioning. At last a vigorous Jat, wedded to the soil from immemorial generations, would say that it was the dharti, the good earth of India, that they meant. What earth? Their particular village patch, or all the patches in the district or province, or in the whole of India? And so question and answer went on, till they would ask me impatiently to tell them all about it.

I would endeavour to do so and explain that India was all this that they had thought, but it was much more. The mountains and rivers of India, and the forests and the broad fields, which gave us food, were all dear to us, but what counted ultimately were the people of India, people like them and me, who were spread out all over this vast land. Bharat Mata, Mother India, was essentially these millions of people, and victory to her meant victory to these people. You are parts of this Bharat Mata, I told them, you are in a manner yourselves Bharat Mata, and as this idea slowly soaked into their brains, their eyes would light up as if they had made a great discovery.”

From Jawaharlal Nehru’s “The Discovery of India”

Posted in links

Coronavirus links round up

I’ve been gathering interesting links on the Coronavirus epidemic. Here are the ones that stand out from the rest:

Coronavirus: Why You Must Act Now – A well-circulated summary with some great graphs. The main takeaway is that the real count is far higher than what is being reported, so it is better to act now rather than wait. And that travel restrictions will at the most delay things by a few days, but social distancing is the best way to contain things.

CORONAVIRUS: LINKS, SPECULATION, OPEN THREAD – Some good graphs on mortality rates, exponential growth etc on SlateStarCodex. This is from March 2, and things are changing fast though.

Seeing the Smoke – A rationalist argues that it makes sense to prep NOW rather than wait for the slow engines of the government to move. This post inspired me to start working from home without waiting for an official mandatory recommendation from my company.

Don’t Panic: The comprehensive Ars Technica guide to the coronavirus [Updated 3/12] – ArsTechnica, as always, with comprehensive coverage.

Information

How quickly does this virus spread, compared to other diseases? The R0 (R-naught) number gives this estimate. Ed Yong has a good explainer on what the number is estimated to be, and why it is not very easy to measure.

Misinformation

Persistence of coronaviruses on inanimate surfaces and their inactivation with biocidal agents – I’ve seen some forwards talk about the virus being active for only about 12 hours on surfaces. But this paper shows that it can remain infectious for up to 9 days, depending on things like the temperature and material.

I’ve seen the usual conspiracy theories doing the rounds about this being a human-engineered bioweapon. This paper, specifically was getting some traction. if any one sends that to you, tell them that the paper has already been retracted.

Posted in books

Hate Inc. Review

Full Title: Hate Inc.: Why Today’s Media Makes Us Despise One Another

Author: Matt Taibbi

I’ve been reading Taibbi for a while now and this was a nice refreshing step back from his usual acerbic style. He covers the media industry’s complicity in creating a toxic political environment. Towards the end is where the book gets really interesting: as an inward-looking view into how the left-leaning media is as bad in creating filter bubbles as the usual suspects in the right.

Here are some snippets I saved from the book.

On Facebook’s curation of news

Chomsky: Take a look at the Facebook phenomenon. Where are they getting their news from? They don’t have reports.

They’re just getting it from the New York Times, so it’s the same sources of information. They’re just putting it out in trivialized form, so that people with a ten- year-old mentality can handle it. It’s a very dangerous thing. They’re not doing any of the things that the media do. They don’t frame things. They don’t select. They don’t send reporters out. They don’t investigate, you know, they just collect information and hand it over to kids to look at in ten minutes so you don’t believe the newspapers.

Red flags to look out for

Tricks used by the government to feed news via allied countries:

This is one reason to always have ears up when you start hearing bits and pieces of important intelligence cases happen to have been uncovered within the borders of America’s closest intelligence allies, particularly England, Australia, the other “Five Eyes” nations, and key NATO members.

Regarding unnamed Ferguson sources:

What is the purpose of the anonymity? Is it to protect someone’s job or freedom? Or to insulate the person against political consequence if the story goes sideways?

Who initiates the communication?

Incidentally: it’s a red flag if the call is coming from the official, as opposed to the reporter calling the officials. The average intelligence official wouldn’t stop to tell you if your child was on fire. When they start cold-calling agencies, and/or rotating scoops by doling them out to different outlets and papers each week, that’s a huge red flag.

Journalist histories:

When you see one of these stories, check to see if that reporter has a history of national security pieces. If he or she does not, if this transmission of classified scoops is taking place in the context of a new relationship, be extra wary.

U.S. Wars

In addition to actions in Syria, Afghanistan, Iraq, Somalia, Libya, and Niger, we’d been aiding the Saudi bombing of Yemen for nearly 1,100 consecutive days on December 11, 2017, when the Pentagon submitted its latest “where the hell we’re currently at war” summary—also known as a section 1264 report, which has to be delivered to Congress every six months under the National Defense Authorization Act.