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.
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.
In first-order predicate logic, we talk about two kinds of things:
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
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.
Last bit went over my head 🙁
Haskell code doesn’t help 🙁