Archive for the ‘Abstraction’ Category

When viewed at the appropriate level of abstraction, many seemingly different questions are equivalent

February 18, 2008

Many questions in many subjects are equivalent when viewed at the appropriate level of abstraction.

For example, the mathematical structures that one creates to study and model motion are identical, mathematically, to the structures that model aspects of economics, population growth, traffic flow, fluid flow, electricity, baseball, planetary motion, and countless other topics.

By looking at the mathematics itself, we strip away the extraneous  features of the questions and focus on the underlying relationships and structures that govern and describe our world.

Change and Motion: Calculus Made Clear by Professor Michael Starbird

Heroic Oversimplification for Modeling our World

October 20, 2007

The invention of deliberately oversimplified theories is one of the major techniques of science, particularly of the “exact” sciences. The biophysicist employs simplified models of the cell, the cosmologist uses simplified models of the universe, and so on.

It may be useful to examine one successful scientific abstraction, to see what it is like and for the sake of the hints it may give us. We choose one which is surely an example of a heroic oversimplification.

Let us assume that we may, in order to study their motions, replace each of the major bodies of the Solar System by a point; that each point experiences a mutual attraction; that we may estimate the attractive force by multiplying the mass of one point by the mass of the other, after which we divide that product by the square of the distance between the points; and that we may neglect all else.

The fact is that this theory, the Theory of Gravitation, has been adequate for predicting the motions of the planets for two and one-half centuries – and this in the face of constant checking by positional astronomers, who, it can fairly be said, carry precision to extremes. The worst strain has come from the orbit of Mercury, which unaccountably drifted from the predicted place by one-fifth of a mil (a foot, at a distance of a mile) per century, thus showing that the theory is rough after all, just as it looks. The improved theory, by Einstein, accounts for this discordance.

The Compleat Strategyst by J.D. Williams

Ascending the staircase of abstraction …

October 3, 2007

Abstract thinking tends most often to strike during moments of quiet repose. As for example the early morning, especially if you wake up slightly before your alarm goes off, when it can suddenly and for no reason occur to you that you’ve been getting out of bed every morning without the slightest doubt that the floor would support you.

Lying there your mind engages in this train of thought:

It appears at least theoretically possible that some flaws in the floor’s construction could make it buckle.

Am I truly justified in my confidence about the floor? Yes, I have gotten out of bed in the morning thousands of times so far, and each time the floor supported me. It’s the same way I feel justified in believing that the sun will come up. Because they’ve happened over and over before.

What principle am I using? The principle involved is really the only way I can predict any of the phenomena I just automatically count on without having to think about them. And the vast bulk of daily life is composed of these sorts of phenomena; and without this confidence based on past experience I’d be unable to function because I’d have to stop and deliberate about every little last thing. It’s a fact: life as I know it would be impossible without this confidence.

Still, though: Is this confidence actually justified, or just highly convenient?

Notice in the above description the staircase that I am ascending in my thoughts. With each step I am going to higher levels of abstraction:

  • I started by considering whether the floor would hold me, then
  • … step … I moved up to thinking about how I assume things will occur in the future because they have occured in the past, then
  • … step … I moved up to thinking about the principle being used in this assumption, and then
  • … step … I moved up still another step in my thinking by asking whether this principle was justified.

— Extracted from Everything and More by David Foster Wallace

The “not discovered here” approach to discovery

September 29, 2007

Five steps to making discoveries and inventions:

  1. Define: define the problem you want to solve and define a goal that you want to attain that would solve the problem. At this point you know “what” you would like to see attained, but don’t know “how” to attain it.
  2. Abstract: The problem and goal that you identified will be expressed in terms specific to the area that you are working in. Your next step is to raise it up, abstract it, so that it is expressed in terms independent of your particular area of endeavor. Get the fundamental concepts identified.
  3. Already Discovered: This may be the most important step: assume that someone has already solved the problem (made the discovery) [1], although it may be in a different area of endeavor.
  4. Search: search the internet and read voraciously, looking for people working on analogous problems. You will likely need to abstract their work, and then compare their abstract problem statement with your abstract problem statement to determine if they are the same.
  5. Apply: once you’ve found someone who has already discovered what you want to do, apply their ideas and techniques to solving your problem

So, the problem of making a discovery comes down to these factors:

  • How good are you at searching?
  • How widely read are you? Do you have varied interests?
  • How good are you at recognizing that your problem is (abstractly) the same as someone else’s problem?

With this approach you acknowledge that, with near certitude, someone else has already solved the problem (made the discovery). The only “discovery” you have to make is to find their discovery, recognize that it’s essentially the same thing that you’re working on, and then apply their ideas and techniques to your particular situation.

This is the “not discovered here” approach to discovery.

[1] “Most claims of originality are testimony to ignorance and most claims of magic are testimony to arrogance.” [James March, Stanford University]

Scientific theories are like maps – there are course-grain maps (and theories) and fine-grain maps (and theories)

September 3, 2007

Scientific theories are like maps.

“Maps are approximate pictures of an underlying reality; a map of Oskaloosa, Iowa is only an approximate representation of the real Oskaloosa.  The only perfect map of Oskaloosa is Oskaloosa itself, which is too big to fit into the glove compartment of your car and thus not very useful.  Just as map makers idealize and leave out certain features of the terrain, scientists simplify and idealize their theories.  What is included or left out will depend on the purpose of the map or theory.  If you are driving across the country, you might just need a course-grained map that shows the major highways.  If, on the other hand, you were going to visit your great-aunt on Ford Avenue in Oskaloosa, you would need a fine-grained map that shows the street grid of Oskaloosa, but not all the highways in the country.”

“The course- and fine-grained maps (and theories) must agree with each other and the observations of the underlying reality.  If a highway map places a river in a particular location, the river must be in the same location on the local map, and must agree with observations of where the river actually is.  One cannot just move roads and rivers around for the purpose of making the maps easier to draw.”

“Science requires different levels of abstraction for different phenomena.  Scientific theories can be big picture and course-grained like a highway map, or fine-grained like a local street map. Both are equally valid; they just need to agree with each other and conform to reality.”

The Origin of Wealth by Eric D. Beinhocker

Knowledge for its Own Sake

July 14, 2007

“No more impressive warning can be given to those who would confine knowledge and research to what is apparently useful, than the reflection that conic sections were studied for eighteen hundred years merely as an abstract science, without a thought of any utility other than to satisfy the craving for knowledge on the part of mathematicians, and that then at the end of this long period of abstract study, they were found to be the necessary key with which to attain the knowledge of one of the most important laws of nature: the law of planetary motion.”

— Alfred North Whitehead