April | 2015 | Fewer Lacunae

Content Summary: 600 words, 6 min read

And now, an unprovoked foray into number theory!

Simple Continued Fractions (SCFs)

Have you run into simple continued fractions in your mathematical adventures? They look like this:

Let $A$ represent the coefficients $(a_0, a_1, a_2, a_3, ...)$ and $B = ( b_1, b_2, b_3, ...)$ . If you fix $B = (1, 1, 1, ...)$ you can uniquely represent $n$ with $A(n)$ . For example:

$n = \frac{415}{93} = 4+\frac{1}{2+\frac{1}{6+\frac{1}{7}}}$

$A(n) = (4,2,6,7)$

Let us call $A(n)$ the leading coefficients of n. Here we have represented the rational $\frac{415}{93}$ with four coefficients. It turns out that every rational number can be expressed with a finite number of leading coefficients.

Continued Fractions- Number Properties v1

Irrational Numbers

Life gets interesting when you look at the leading coefficients of irrational numbers. Consider the following:

$A(\phi) = (1, 1, 1, 1, 1, 1, 1, ...)$

$A(\sqrt{19}) = (4, 2, 1, 3, 1, 2, 8, 2, 1, 3, 1, 2, 8, ...)$

$A(e) = (2, 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, ...)$

$A(\pi) = (3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, ...)$

First note that these irrational numbers have an infinite number of leading coefficients.

What do you notice about $A(\phi)$ ? It repeats, of course! What is the repeating sequence for $A(\sqrt{19})$ ? The sequence $213128$ .

How about $A(e)$ ? Well, after the first two digits, we notice an interesting pattern $211$ then $411$ then $811$ . The value of this triplet is non-periodic, but easy enough to compute. The situation looks even more bleak when you consider the $A(\pi)$ …

Thus $\phi$ (golden ratio) and $\sqrt{19}$ feature repeating coefficients, but $\pi$ and $e$ (Euler’s number) do not. What differentiates these groups?

Of these numbers, only the transcendental numbers fail to exhibit a period. Can this pattern be generalized? Probably. 🙂 There exists an unproved conjecture in number theory, that all infinite, non-periodic leading coefficients with bounded terms are transcendental.

Continued Fractions- Number Properties

Real Approximation As Coefficient Trimming

Stare the digits of $\pi$ . Can you come up with a fraction that approximates it?

Perhaps you have picked up the trick that $\frac{22}{7}$ is surprisingly close:

$\pi = 3.14159265359$

$\dfrac{22}{7} = \textbf{3.14}285714286$

But could you come up with $\frac{22}{7}$ from first principles? More to the point, could you construct a fraction that comes yet closer to $\pi$ ‘s position on the number line?

Decomposing these numbers into continued fractions should betray the answer:

$A(\pi) = (3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, ...)$

$A\left(\dfrac{22}{7}\right) = (3, 7)$

We can approximate any irrational number by truncating $A(\pi)$ . Want a more accurate approximation of $\pi$ ? Keep more digits:

$(3, 7, 15, 1) = A(\dfrac{355}{113})$

$\dfrac{355}{113} = \textbf{3.141592}92035$

I’ll note in passing that this style of approximation resembles how algorithms approximate the frequency of signals by discarding smaller eigenvalues.

About π

Much ink has been spilled on the number $\pi$ . For example, does it contain roughly equal frequencies of 3s and 7s? When you generalize this question to any base (not just base 10), the question becomes whether $\pi$ is a normal number. Most mathematicians suspect the answer is Yes, but this remains pure conjecture to-date.

Let’s return to the digits of $A( \pi )$ . Here is a graph of the first two hundred:

Do you see a pattern? I don’t.

Let’s zoom out. This encyclopedia displays the first 20,000 coefficients of $A( \pi )$ :

So $A(\pi)$ affords no obvious pattern. Is there another way to generate the digits of $\pi$ such that a pattern emerges?

Let quadratic continued fraction represent a number $n$ expressed as:

Set $A = (1, 2, 2, 2, 2, ... )$ . Here only $B = ( b_1, b_2, b_3, ...)$ is allowed to vary. Astonishingly, the following fact is true:

$B\left(\dfrac{4}{\pi}\right) = (1, 3, 5, 7, 9, 11, 13, 15, 17... )$

Thus, continued fractions allow us to make sense out of important transcendental numbers like $\pi$ .

I’ll close with a quote:

Continued fractions are, in some ways, more “mathematically natural” representations of a real number than other representations such as decimal representations.

Part Of: Demystifying Ethics sequence
Prerequisite: An Introduction to Propriety Frames
Content Summary: 1400 words, 14 min read

Introduction

This article presents a theory of normatives. It is not a complete theory (it does not even describe implementation), nor does it answer a number of important objections. I sketch it, in its current state, in order to motivate the claim that explaining normatives is possible.

Hyperpoints

Consider again the problem of describing locations in a space. In a two dimensional space, any coordinate system I use (e.g., Cartesian or polar) require two pieces of data to uniquely specify the location. In a space with three degrees of freedom, three points of data are required. Call this symmetry between dimensionality and data the data-dimension bridge.

Your brain is embedded in a three dimensional space, and your visual system evolved in this context. It is just as mathematically meaningful to discuss locations in a 17D space, despite the fact that you lack the mental equipment to visualize it.

Take this argument to an extreme. The universe contains approximately 10^82 atoms. If we suppose there are 100 interesting facts we can name for every atom, this means that we need 10^84 data points to describe the universe. Given our data-dimension bridge, this means that the entire universe can be described by the location of one point in a (10^84)D space.

Such hyperpoints turn out to be a surprisingly versatile concept, appealed to frequently by mathematicians. They were first introduced in an inaugural lecture by Bernhard Riemann in 1854.

Propriety Frames

Main Article: An Introduction To Propriety Frames

Our everyday lives are immersed in highly structured interactions. Much of these community expectations are taken for granted; they are practically invisible to us. But if you travel to a culture sufficiently remote from your own, or spend much time around people suffering from Asperger’s syndrome, and you can begin to build an appreciation for these culturally-approved roles.

Consider a typical evening spent at a fancy restaurant. If I asked you to come up with a complete list of examples of things that would surprise you, how long do you think that list would be? Consider some representative examples.

The waiter states he is not in the mood to take your order.
Several guests are engaged in a foodfight.
An expensive item on the menu is marked as free of charge.
Instead of payment, the manager comes out to request that you wait tables next weekend.
The food is dumped directly on the table.
On taking a bite, you realize that your meal is actually plastic: an artistic creation designed purely for visual effect.

The length of the list corresponds to the volume of knowledge embedded within your brain’s social intuitions.

Social psychology likes to speak of frames as a useful way to bundle collections of facts together. We can say that our restaurant experience is defined as the intersection of three different frames: host-guest, place of business, eating. Each of the above surprises are produced by a violation of one of these frames.

Propriety Frames- Restaurant Example (3)

When a mother instructs her son to not yell in the store, the child installs an update to his Shopping frame. When a family exchanges gossip around a campfire, they are doing so in part to synchronize their propriety frames.

Mental Hyperpoints

Time to synthesize the above. Each social frame can be decomposed into a collection of facts. Thus, a person’s social intelligence can be fully described by a hyperpoint in frame-space.

In fact, we can represent any mental process in terms of hyperpoints. In the following graphic, we are representing the social memory, semantic memory, and perceptual memory.

Take a moment to appreciate this graphic. Every sentence you utter, every subtlety of your body language, every friendship you have ever formed: the totality of your life as an organism is captured as a single point in behavior-space.

Did you notice how Person A only encodes other people in behavior-space? This is because human beings do not have direct access to one another’s mental lives. That said, we can imagine that knowledge of another person’s behavior may let us estimate the social rules they hold as legitimate.

A Theory Of Normatives

Let’s switch gears. What does it mean for somebody to say “you should buy this stock” or “you should not blackmail your coworkers”?

We call statements with “should” in them normative language. These contrast with descriptive statements of reality. David Hume noticed this distinction clearly, and formulated what is now known as the is-ought gap:

In every system of morality, which I have hitherto met with, I have always remarked, that the author proceeds for some time in the ordinary ways of reasoning, and establishes the being of a God, or makes observations concerning human affairs; when all of a sudden I am surprised to find, that instead of the usual copulations of propositions, is, and is not, I meet with no proposition that is not connected with an ought, or an ought not. This change is imperceptible; but is however, of the last consequence. For as this ought, or ought not, expresses some new relation or affirmation, ’tis necessary that it should be observed and explained; and at the same time that a reason should be given, for what seems altogether inconceivable, how this new relation can be a deduction from others, which are entirely different from it.

With this backdrop, I would like to propose the following three-part theory of normatives.

First, normatives are nothing more than hyperpoints in frame-space, tagged by a separate module with a “should” label. Call these marked hyperpoints normative attractors. As evidence, consider that every statement of “ought” can be cast back into descriptive language (“you should do X” can be translated into “good people do X”).

Second, normative attractors are fed into our motivational subsystems. The word “ought” is not just a fun word to throw around, it evolved to promote behavioral change.

Third, normatives co-evolved with natural language. Recall the functional purpose of language: to allow humans to exchange mental information. Mental information about the world (human semantic memory) required no additional hardware to synchronize: perceptual psychophysics provides a sufficient reference point. But coordinating social frames had no such advantage, and required new hardware (the attractor).

Let us call this toy hypothesis the Frame Attractor theory of normatives. Here’s an illustration; where the yellow attractor represents Person A’s vision of optimal behavior (e.g., Buddha, Jesus, “being nice”, etc):

Disaggregated Attractors

It is a virtue of explanatory models to consume prior considerations under a more comprehensive framework. I have used this model to improve my understanding of why conflicts happen, and how to resolve them peaceably. This is left as an exercise to the reader. 🙂

In the above image, consider the plight of Person C. Is the sheer distance between their behavior and the norm motivating?

In Normative Therapy, I state the following:

Let me zoom in on one “population subset”: the sociopath. Such an individual is biologically incapable of meeting aggregate-level normative impositions such as “everyone should care about the well-being of those around them”. The argument here, is that normative impositions should be explicitly tailored to the individual situation; for people like sociopaths, this would be something like, “sociopath X should put herself in situations where her non-empathic behavior can be held accountable”.

This observation, and the argument underlying that post, suggests that it may be advisable to disaggregate normatives. If the goal of morality to more effectively move people towards the normative attractor, perhaps innovation in cultural technology will permit stable individuated attractors, like this:

The above example uses disaggregated normatives as lures to attract people towards The Goal. But this is not the only use of such a mechanism. Perhaps we could also train personal-attractors in a more diverse ethical setting with multiple Goals. But, for now, let me leave such a question for another day.

Takeaways

Thinking about complex ideas as locations in a high-dimensional space (hyperpoints) is a surprisingly powerful conceptual tool.
Complex community rules are most likely implemented via a particular memory system that encodes social frames.
The Frame Attractor theory of normatives is grounded in three principles
1. “Ought language” are hyperpoints in frame-space, tagged with a “should” label. Call these normative attractors.
2. Normative attractors are fed into our motivational subsystems.
3. Normatives co-evolved with natural language.
This theory can be used to explain other arguments, such as my argument in support of person-specific normatives.

Fewer Lacunae

Distilled, Integrative Research

Month April 2015

Getting Real With Continued Fractions

Bridging the Is-Ought Gap