1Using language

2Using mathematics

  • Understand that mathematical notation is an abbreviation for natural language.
  • Read mathematical notation.
    • Mentally expand a prose that contains mathematical symbols into a prose in natural language without mathematical symbols.
  • Understand that "to understand X" means "to have a useful internal mental model of X" or "to be able to manipulate X".
  • Understand that a theorem is a technology, because it increases productivity: it reduces the effort required to prove something.
  • Understand naïve set theory, functions, logic, variables, algebra, arithmetics.
  • Understand function composition: \( (f\circ g)(x) = f(g(x)) \).
  • Understand the relationship between set theory and logic.
  • https://en.wikipedia.org/wiki/Change_of_variables

2.1Real numbers

  • Understand real numbers.
  • Understand sequences.
  • Understand the limit of a sequence.
  • Understand the interval notation1.

3The language of mathematics?

3.1Mathematical language

Even if there is nothing you don't already know, at least glance through this chapter, just to be aware of our idiosyncrasies.

If you program computers, you can think of mathematics as a domain-specific language for concise and precise thinking.

Humans have invented a mathematical notation that greatly increases information density; thus such notation also increases information transfer rate. Such notation speeds up thought, after an initial learning curve.

3.2Dispelling the fear of mathematics

Fear mathematics not, because it is just abbreviated English. For example, instead of repeating the cumbersome "where the ball is at a given time", we may write the much shorter "\( h(t) \)". Instead of writing the long and hard-to-parse phrase "a number that equals zero when multiplied by itself and then subtracted by one", we write "a number \(x\) such that \(x^2 - 1 = 0\)".

We have just seen mathematics speed up our thinking!

Mathematics originated as a way of modeling reality.

People did math because they wanted to do something in the real world. Shepherds want to avoid losing cattle, so they count their cattles. Carpenters want to cut woods for the diagonal braces of a roof, so they use the Pythagorean theorem. Merchants want to profit, so they subtract expenses from income, and use exponentials to calculate interests. Train operators want to profit, so they calculate the quantity of coal they should carry for a given distance. Nations want to avoid famine, so they calculate how much crop they should plant. And so on. People originally did math to avoid wastage, mistakes, and pain.

People also did math to plan. For example, if a bush can feed one person, and there are three people in my family, then I would need to forage three bushes to feed my family.

3.3The time required

Even with the necessary background knowledge, we often take any time from several seconds to several minutes in order to read an equation.

It does not help if we can 600 words per minute, because the difficulty of understanding mathematics is not in translating the notation into English, but in reconstructing the writer's understanding back from the notation.

The writers have a picture in mind when they write an equation. The readers have to reconstruct that picture, given only the equation.

3.4Digression: From "because we must" to "because we want"

(This may be false. Perhaps farming did not give people more leisure time, but it enabled people to build bigger and denser settlements.)

At first we did math because we had to survive.

We counted the things that determine our survival: animals, plants, people, weapons. We had numbers, but they were tied to units: we understood "one cow", but we did not understand "one".

We found linear relationships between the number of family members and the rate of resource consumption.

We wanted to survive, so we thought about optimization: to get maximum result with minimum effort. We built tools, farmed crops, trapped animals, built houses, settled down, simplified survival, and got much leisure time. It was futile to work harder than what was necessary for survival, because the surplus harvest would be wasted; Nature does not reward material possession beyond what is necessary to survive. We were wealthy in the sense that we had everything we wanted without working, because everything we wanted was to survive; crops gave themselves for us to eat; they didn't fight back like animals; what else could we want other than free food? It was heaven; we didn't know what else there was to want. We didn't know what to do with all that leisure time, so we began doing things for fun: painting cave walls, making statues, etc.

With so much leisure time, we began doing things because we could, not because we had to, because there was nothing we had to do.

Curiosity, not necessity.

We have moved from doing what we must to doing what we want.

3.5Expressions

An expression is something like \(1+2\), or \(x + y \cdot z\), and so on.

3.6Equations

An equation \(x = y\) (read "\(x\) is equal to \(y\)") means that every occurrence of \(x\) can be replaced with \(y\), and also the other way around: every occurrence of \(y\) can be replaced with \(x\).

3.7Sets

A set is a collection without duplicates.

Example: \( \Set{1,2,3} \) is a set of three things.

Example: \( \Real \) is the set of all real numbers. (Perhaps for now it suffices to know that \(\Real\) at least contains every number that you can type into a simple calculator.)

3.8Functions

(Should we just use the domain-codomain-pairing triplet formalism?)

A function \(f\) is usually defined by an equation like \( f(x) = \text{something} \). See the following example.

Suppose that we have defined \(f(x) = x+1\) and we want to evaluate \(f(2)\). We do this by assuming \(x=2\) (because we want to evaluate \(f(2)\)). Here is how we do it:

\[\begin{align*} f(x) &= x+1 & \text{by definition} \\ f(2) &= 2+1 & \text{by assuming \(x = 2\)} \\ f(2) &= 3 & \text{because \(2+1 = 3\)} \end{align*} \]

Therefore, \(f(2) = 3\).

As you become more proficient in math-speak, you will be able to skip the intermediate steps.

Note that, in the above example, the function is \(f\), not the expression \(f(x)\). People often mistakenly say "the function \(f(x)\)". Do not confuse a function and its application.

Sometimes we write \(f(x)\) as \(fx\).

We rarely do these, but we can write \(f(x)\) as \(f~x\), and we can write \(f(x) = x+1\) as \(f = (x \mapsto x+1)\). (This probably only makes sense to functional programmers.)

A function can represent the relationship between two quantities in which one quantity determines the other quantity.

3.9Integrals

See file:integral.html.

3.10Algebra

A letter (a variable) represents a number (something) that is not yet known.

Example: \(x+2 = 3\) means "What number, if added by 2, equals 3?".

4Theorizing and reasoning

4.1Theories

  • Understand that a theory is a logical formula.
  • Understand that theories usually have assumptions.
  • Understand reasoning (how to create theories).
    • Understand abduction: given observations E, find the simplest theory T that implies E.
      • Invent concepts if necessary.
      • Name things that you need to talk about often.
    • Understand induction (hasty generalization): observing \( p(x) \) for some \(x\), hypothesize \( p(x) \) for all \(x\).
    • Understand relaxation (generalization):
      • Remove an assumption.
      • Given a theory \( A \wedge B \to C \), produce theory \( B \to C' \), such that \( (B \to C') \to (A \wedge B \to C) \).
    • Understand subsumption:
      • Given theory X and theory Y, find the simplest theory U that subsumes X and Y. Find U, derive X from U, and derive Y from U. By "derive X from U", we mean "show that U implies X".
      • Find a unified theory \(U\).
      • Show \(U \to X\).
      • Show \(U \to Y\).
      • Understand that if X and Y contain incompatible assumptions (e.g. \(X \wedge Y\) is false), then those assumptions may have to be relaxed first.
  • Understand how to uncover the assumptions of a theory with corner cases (extreme values of model variables).
  • Examples of theory creation:
    • Relaxation:
      • From Newton's model of point-mass motion to continuum mechanics.
    • Subsumption/unification:
      • From the law of reflection and Snell's law of refraction to Fermat's principle of least time.
      • From the gas laws to the ideal gas law.
      • From various laws of electromagnetism to Maxwell's equations.

4.2Making theories: measure-model-abduce

This is how we make a theory:

  1. We begin with a measurement (an observation).
  2. We create a phenomenological model.
  3. We abduce a metaphysical principle while assuming as little as possible23, and derive the phenomenological model from that principle.

The motto is "measure-model-abduce".

The cycle is "measurement-phenomenology-metaphysics".

For example:

  1. We measure the motion of things by sampling their positions at various points in time.
  2. We model it phenomenologically with Newton's laws of motion.
  3. We abduce the principle of stationary action, and derive Newton's laws of motion from that principle.

A principle is a reasoned assumption.

We must always remember that models and principles are not the reality, and that falling in love with them will halt progress. We must always be ready to discard them.

Can experiments test principles?

Criteria for evaluating a scientific theory http://www.nytud.mta.hu/depts/tlp/gaertner/publ/schoemaker_huygens_fermat.pdf

4.3Which one of two theories is more general?

Let \(T\) be a theory.

Let \(U\) be a theory.

We write \( T \leq U \) to mean "\(T\) is derivable from \(U\)" or "\(T\) is implied by \(U\)".

We write \( T < U \) to mean "\(T\) is less general than \(U\)" or "\(T\) is subsumed by \(U\)" or "\(T\) is included in \(U\)".

We choose that notation to make it easier to remember the relationship between subsumption and derivability: \[ T < U \iff T \leq U \wedge U \nleq T \] which reads "\(T\) is less general than \(U\) iff \(T\) is derivable from \(U\) but \(U\) is not derivable from \(T\)".

For example, Snell's law is derivable from Fermat's principle, but Fermat's principle is not derivable from Snell's law. Therefore, Snell's law is less general than Fermat's principle. (Really?)

Derivability imposes a partial order on theories.

It may be useful to order physical theories ascending by derivability, and make a learning sequence according to that order.

Theories form a poset. Do theories also form a lattice?

If the lattice of theories is bounded, then the maximum of the lattice is the grand unified theory; otherwise there is no such grand unified theory.

We say "\(T\) and \(U\) are equivalent" to mean "\( T \leq U \) and \( U \leq T \)". (For what?)

4.4Which one of two models is more accurate?

We write \( T \le U \) to mean "every correct prediction of \(T\) is also a correct prediction of \(U\)".

We write \( T < U \) to mean "\(T\) is less accurate than \(U\)".

4.5Unification

By "X unifies Y and Z", we mean "X provides an unified explanation for Y and Z", that is, "both Y and Z are derivable from X".

5Meta-learning

  • Understand how to uncover gaps in your understanding.
    • Create a curriculum (a sequence of competencies).
      • A gap in your understanding will show up as a gap in your curriculum.
    • Teach.
      • The parts you find hard to teach are your gaps.
      • If you don't know how to teach it, perhaps you don't understand it.
    • Creating a curriculum is one way to find out our unknown unknowns (what we don't know we don't know).

  1. <2020-01-26>↩

  2. https://en.wikipedia.org/wiki/Occam%27s_razor↩

  3. https://en.wikipedia.org/wiki/Ontological_commitment#Ontological_parsimony↩