# Math

At its most fundamental level, math is the manipulation of symbols that represent numbers. The existence of symbols (ie. variables) is precisely what makes mathematics applicable to a wide range of areas.

"Math has a tendency to reward you when you respect its symmetries."

Mathematics includes the study of:

• Arithmetic, number theory
• Algebra - functions and related structures
• Geometry - shapes and spaces in which they are contained
• Calculus, analysis - quantities and their changes

## Why learn math?

We don't learn things like integrals so that we can use them in everyday life. Instead, we learn them and practice them for the same reason that a soccer player does zig-zag drills and lifts weights. You will never see them do those things on the soccer field, but they will use the strength, speed, insight and flexibility that they built up by doing those drills. Therefore, learning those drills is a part of soccer.

• And if we don't want to be our proverbial professional soccer player, that's fine! We can still enjoy a game of soccer with friends, and we will be healthier for doing so. Mathematics is the same way. You may not be aiming for a a math-focused career, but math is woven into the way we reason. Knowing mathematics is like wearing X-ray specs that reveal hidden structures underneath the messy and chaotic surface of the world.

We learn math because it influences our habits of thought. In this sense, it is no different from learning about biases and fallacies: they help us recognize patterns in the world and influence our decision-making for the better.

• someone well-versed in math is always asking: "what assumptions are you making? And why are they justified?"
• ex. someone unfamiliar with Survivorship Bias will hear a story like Abraham Wald's Aircraft Bullet Holes and will make a set of assumptions. Their naive assumption is that all planes are returning from battle, and that is therefore the denominator in their calculation. Of course, this isn't true, and it causes them to reach a very wrong conclusion.
• "math is like an atomic-powered prosthesis that you attach to your common sense, vastly multiplying its reach and strength"

### Abstraction

Mathematics is all about abstraction. We observe the fingers on our hand and abstract over them the concept of natural numbers. We see how you can take away values from a set and abstract out the concept of negation. We take the concepts of numbers, addition, and negation and abstract them to groups. And so on, and so on.

• "Mathematicians don't just care about finding the answer but developing general problem solving tools and techniques."
• It's all about boiling down specific applications to their most basic and fundamental properties.

To be an effective at math, you need to be able to look past all of the empirical application of the problem you are trying to solve and understand the fundamental mathematics that lies behind. In this sense, we focus on the common skeleton shared by different problems that look very different on the surface.

• ex. if you are able to do this, it becomes apparent when biases are being assumed.
• sometimes only paying attention to the abstract part of the problem can cause you to ignore features of the problem that really matter

### The problem with how math is taught

In school, math is taught as a set of rules that one must obey or you fail. But this is not mathematics. Mathematics is the study of things that come out in a certain way because there is no other way they could possibly be.

### Intuitiveness of math

As humans, we have built-in mental systems for assessing the likelihood of an uncertain outcome. But past rudimentary mathematics, those systems are pretty weak and unreliable (especially when it comes to events of extreme rarity).

Addition is the most fundamental of mathematics and also the most intuitive. Without being taught the mathematical rule that $a + b = b + c$, kids know that $5 + 3$ is the same thing as $3 + 5$

By the same token, one could not say that the mathematical rule of $5 \times 3 = 3 \times 5$ is as intuitive, but there is intuition that can be gained once it can be applied to a real-world circumstance.

• ex. Consider a grid of 8 rows by 6 columns.

Not everything in math can be intuited. For instance, one cannot do calculus by common sense. But calculus is still derived from our common sense. Newton took our intuition about how objects move in straight lines, formalized it, and then built on top of that formal structure a universal mathematical description of motion.

# Prime numbers

The primes are the atoms of number theory, the basic indivisible entities of which all numbers are made.

• this is because a prime has only 2 factors: 1 and itself
• All whole numbers greater than 1 are either primes or products of primes.
• aside from 2, all primes are even

# Theorem vs Proof

• theorem is a statement that is expressed in a mathematical language and can be said with certainty to be either valid or invalid.
• For example, the theorem “there are infinitely many prime numbers”. To prove this theorem, it is not enough to point out an additional prime number for a specific given list.
• What we need to do is to present a convincing argument why this is the case, rather than just demonstrating the next number in the sequence.
• a proof on the other hand, requires some definitive reason why the fact that is posited is indeed the truth.
• We can prove that prime numbers are always continuing, because we know that "all whole numbers are either prime or composite (made up of 2 primes)". We are able to prove this by picking numbers at random, and verifying that this is true. With each sample, we will have zero variation, and will always be able to either: multiply 2 primes to reach the result, or declare the number a prime

An equation is an abstraction that allows the user of the formula to derive a result that they can then use.

• An equation is a function, functions are basically I/O machines, and abstractions by definition allow us leverage a simpler system to improve productivity.

# Functions

### Domain

the set of values that a function is able to accept. In other words, the function can only accept as inputs (arguments) the values which are included as part of the domain

• ex. in relational databases, the domain (ie. the boundary/scope) would be the type (int, bool, string etc)

### Classical Mathematics

The underlying idea that underpins everything in classical mathematics is that everything is straight, and made up of simple shapes (circle, square, triangle, pyramid etc.)

• This philosophy does not lend itself well to the idea of fractals

Therefore, classical mathematics is only really well suited to study the world that we as humans have created, and it fails to account for the geometry of the natural world.

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