11/25/14

Some Tweet-a-Programs

You can tweet Mathematica code to @wolframtap and they will return the result! It's a great challenge to see what you can do with so few characters. Here is some of the code I submitted:

11/23/14

Different Occurrences of the Same Type of Assembly

Just as marks can look like each other so can assemblies of marks. When two marks look like each other we say they are of the same type. When we have defined a type of mark, usually by giving a canonical example of a mark of that type, we can identify and create new occurrences of that type of mark.

Here are some occurrences of the same type of assembly :


When there are only a few marks in a pair of assemblies it can be easy to identify correctly whether they are both occurrences of the same type of assembly or not. As the number of marks occurring in a pair of assemblies increases it becomes more and more difficult to identify with certainty whether they are of the same type. For example, are these assemblies of the same type? That is, are they similar to each other in the same way that a pair of marks are similar to each other when they look like each other?



In order to use assemblies and marks effectively and efficiently it must be possible to decide, with certainty, whether any pair of assemblies or marks look like each other and are, consequently, of the same type. Though it may be obvious that being able to clearly and exactly identify assemblies of the same type is desirable in practice, it is common in everyday life to ignore such details in favor of an approximate or tentative certainty that a pair of given assemblies are "mostly" of the same type. This vagueness is very interesting as it seems to be an essential part of the human thought process, but it is difficult, if not impossible, to deal with it on a formal level at this point in our description of math and physics. Furthermore, it is possible, to consider assemblies with a known structure and in this way approach the problem of identifying the type of an assembly with absolute certainty.

11/21/14

Events of Addition

The code below was used to construct these events of addition.




The idea is that a gif is used to represent a sequence of events. Ultimately, the act of addition is learned by studying the basic events of addition (addition of single digit numerals), and extending that procedure via a naive procedural and spatial intuition. Students learn to add by observing events of addition and becoming familiar with the form of an additive sequence of events. Later, in more abstract mathematics, a clever student might discover the principles which govern these forms so that they might prove such forms of sequences of events serve the relevant function.

11/20/14

Marks as Assemblies and Assemblies as Marks

In "Assemblies of Marks and Boxes" it was said that a detailed discussion of the foundation of marks and assemblies would be avoided at the moment, but it is for the benefit of utility and clarity that the seemingly "circular" relation between marks and assemblies be addressed in an informal way.

Any mark, when examined closely, may appear as if it is actually an assembly of marks. This is especially true of marks made with electronic displays or modern printers. On a display, or with a printer, there is a "smallest" mark that can be made. On a modern digital screen the smallest mark is called a pixel, and by turning on or off pixels in an assembly of pixels we can create the letters of the alphabet.



It is possible that there is a physical principle which states that there is a smallest mark that can be made, a so-called "quantum mark". A detailed analysis of what such a mark might be leads straight to the forefront of modern physics, and to follow such a line of thought would lead us far astray (at the moment). It is enough, for now, to note that the idea of an indivisible unit of mark is not a foolish approximation, but rather a fitting model of how things appear to be when investigated by experimental science.

As a concrete example of all that has been discussed so far, the following image shows how each of the marks we call capital letters can be represented as assemblies of a single mark (a small green dot).


Thus it is not uncommon, in practice, for an assembly of marks to be identified as a single mark, and not an assembly. Furthermore, and naively, every mark is an assembly of marks containing only itself, and, for those marks we use when writing with pen and paper, it is always possible to recreate each mark as an assembly of tiny dots.

Assemblies of Marks and Boxes

Marks alone are of little interest to anyone. A mark that is given a context and that stands in some naive spatial relation to other marks is capable of changing the world and delighting the mind. Any combination of marks that are near each other on a page form an assembly of marks. Here are some random assemblies of ASCII marks:


There are a few reasons that the general notion of assembly is not used in math and physics. The first is actually more related to marks than assemblies. In the following random assembly there are so many marks that it is impossible to tell which marks do or do not occur within the given assembly.


Since math and physics both deal with clear and exact descriptions of things, the inability to identify clearly and exactly whether a mark does or does not occur in an assembly places a huge restriction on the types of assemblies used in science. Outside of science, the general notion of assembly is used to make, sometimes beautiful, art.


When a mark is on a piece of paper in front of us we can point to it with our finger, and in this way we can pick out a unique mark that belongs to an assembly. In practice, we use bounding boxes to bring our attention to a mark on a page when our finger is not readily available. So for practical purposes, and for the purpose of using marks clearly and exactly, we only use assemblies composed of marks that can be selected uniquely by placing a bounding box around them. Furthermore, we use the same box method when bringing a specific assembly of marks to our attention.

Here is an assembly of visible ASCII marks


Here is an occurrence of each mark in that assembly being selected/box:


The vague idea is this: boxes let us select and organize marks and assemblies on a page. Ultimately it is much easier to use these things than it is to describe how they are used. It is for that reason, and the fact that their use is quite efficient, that we will tentatively avoid a more detailed discussion of the foundations of marks and assemblies.