Finite representations of infinite objects

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1 How many dots are red?

2 Ratio of integers as counts

3 Infinite rational approximation

4 Finite rational approximation

5 Finite representation

Computer/Information science can be defined as the search for finite representations of (potentially) infinite objects.

Is this possible? Does it make sense?

How many dots are red? $Finite representations: how many dots are red?$ What is the ratio of red dots to blue dots?

The ratio of red dots to blue dots is one third or 1/3. $Finite representations: ratio of integers as counts$ How is this represented in decimal notation?

The decimal notation for 1/3 is infinite. $Finite representations: infinite rational approximation$ How can we finitely represent this infinity?

Mathematical repetend notation can be used to show that the finite representation is "repeated". $Finite representations: rational approximation$ How can we show this in graph notation as might be used in computer science?

We could print out this finite representation to any desired precision as long as we eventually stop. Lazy evaluation only continues as needed. $Finite representations: finite representation$ Do real numbers exist? Or are they just in the human imagination as a way of approximating results.

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1 Look and do

2 Look not do

3 Do not look

Many people think of two choices.

1. Look and do it.
2. Look but do not do it.

A third choice is at the heart of computer science and statistics.

3. Do not look. There is a cost to looking. And it may not matter.

It may already be known what is there, or it may have been determined that it is not necessary.

This idea is used all of the time in computer science and programming. A simple example is that of conditional evaluation of expressions (i.e., no side-effects during evaluation).

In the expression A and B, if A is false, then one need not look at B.
In the expression A or B, if A is true, then one need not look at B.

It may be the case that evaluating B may cause an error.

In discussions, it may be the case that B is a "red herring" and does not matter. The only thing that matters is A. Here are some tautologies often used in programming (as algebraic transformations).

(A and false) = false
(A and true) = A
(A or false) = A
(A or true) = true
(A = true) = A
(A = false) = false = (not A)

More: Red herring: look at me distraction deception

More: Nothing is something: actions