George Boole

Boole stood out from the list of mathematicians because of his relations to the Boolean concepts in computer science.  His father, a shoemaker, was interested in science and taught him his first lesson in mathematics. This interest in science led to involvements with a social group in their English town of Lincoln called Lincoln Mechanics' Institution. The group promoted education and held discussions about science.  After a while, Boole’s father became the librarian for the group and provided the conditions to allow Boole to discover and learn foreign languages and mathematics.  Given access to books, combined with encouragements of his father Boole’s self-starting personality lead the way.    
Boole had to support his parents and siblings early on. For a few years, he did this by teaching at village schools and when he turned twenty, he opened his own school in Lincoln. Sources I looked at about him wan to point out that his work and responsibilities left him with little time to further his study but my personal feeling is that the daily grind with students and re hashing the basics probably contributed to insights that worked in his advantage.

Foundations of Boole’s interest in math were as a tool to solve mechanical problems in instrument making. This practical focus evolved into his many papers; one in particular one on differential equations which gained him the Gold Medal of The Royal Society of London. His talent in breaking down a problem into smaller parts and using algebraic formulas to move toward a solution is the basis for Boolean logic we use in computers today. This allowed for logical problems of sentences and words to be presented as algebraic problems, which are eventually solved mechanically. The general idea of classifying objects into sets and replacing the given set with a symbol somehow (my understanding of it is very limited) made 0 and 1 a special set of numbers called idempotent numbers, or numbers that do not changed when multiplied by themselves. In what he referred to as “The Rule of 0” in his book The Laws of Thought Boole states that an argument is valid if and only if after writing it as an equation and restricting the values for the symbols to only 0 or 1 we arrive at a valid equation in algebra. Boolean algebra correlates the operation of multiplication to the word “AND” and addition to the word “OR”.
With the industrial revolution came the use of electricity, a perfect fit for the Boolean logic system. Dividing an abstract problem into its elemental parts that can only be allocated values of 0 or 1 and evaluating the parts in order to validate the larger problem is time-consuming and cumbersome. But, the mechanical system of evaluation combined with speed of electrical circuits make Boolean logic viable. George Boole’s self-educated research method affords us this luxury.

Works Cited
Encyclopedia Britannica, George Boole. 11/29/2013 <http://www.britannica.com/EBchecked/topic/73612/George-Boole>.
Stanford Encyclopedia of Philosophy, George Boole. (2010). 11/29/2013 <http://plato.stanford.edu/entries/boole/>
Reville, William, University College, Cork. “The Greatness of George Boole” (1996). 11/29/2013 <http://understandingscience.ucc.ie/pages/sci_georgeboole.htm>.

Counting and Recording Without Writing

17 November 2013
I can count to 10 on my figures without taking my eye off the job. Perhaps I use the beads and the string to make a strand of 10 beads. This way every time I count from 1 to 10, utilizing the fingers of my hands, I move one of the beads along. This allows me to count up to hundred without losing count. After the counting process is completed, I would use two sticks to record the account. On one stick, I would carve a grove all the way around. One for every bead I had counted, therefore each grove would mark 10 sheep. On another stick, I would mark straight notches to record the counts on my fingers, the ones that were not transferred to the beads. The owner would take the two sticks as the record of his stock.

Vectors

I have always wanted to know about vectors. So far, I only knew that in the computer world if I was dealing with a vector-based image then I could manipulate he size if that image without it becoming pixelated. There is a lot to learn in this area. My basic understanding is that in a simplest form a vector-based line is different from an image of a line. An image of a line is hard coded information about a selection of pixels that form that line. However, a vector-based line is information about the direction and the length of that line. This is the reason that they –vector-based graphics- are always clear and not pixelated. Once we know the direction and the length of the line, we are free to draw it in way that suite our local environment. How this leads to functions, arrays, and what happens with shapes (collections of vectors) should be a fascinating study.

Early History of Numeration

Why did people first need numbers?

Probably the need for numbers is closely tied with people’s sense of tracking items and events in their lives. The number of nights since they last saw the moon in full. The number of livestock that were in pasture yesterday. Number of sunny days required to have a successful harvest. The answers to these questions as well as many others had direct effect on people’s lives. The symbols that are place holders for these answers eventually evolved to become what we currently call numbers and the method of evaluating them is numeration.

What were some of the ways people first kept numerical records?

The easiest method would have been replacement of real object or event with mare accessible substitute. Early human’s fingers were perhaps the most accessible and portable way of substitution. To compensate for the limitations of 10 fingers or 20 fingers and toes tally sticks and knotted strings were used. This led to further refinement and efficiency by grouping. A group could then be represented by a symbol pressed in clay or painted on a surface.

What were the number systems of ancient cultures like?

Once groping and symbolism combined to add efficiency to the system two general systems began to emerge. The literal systems such as the Egyptian and Chines systems used symbols that were noticeable in their daily lives. For instance Egyptians denoted one million with a figure of an astonished man. More abstract systems such as the Roman or Babylonians used more abstract symbols that resembles the alphabet.

Charles Lutwidge Dodgson

What other name did he use?

To come up with a pen name, Dodgson took his given name “Charles Lutwidge” and translated it to Latin: “Carolus Ludovicus”, reversed the first and last name: “Ludovicus Carolus”, and retranslated it into English: Lewis Carroll.

What are some of his published works under that name?

The Alice books are probably the best known of his publishing. His first book under his pen name was “Solitude” published in 1856. The Lewis Carroll society’s website has a long list of publications including “A Tangled Tale”, “Through the Looking Glass”, and “Rhyme? & Reason?”.

What was he known for under his real name?

He was a mathematician and lectured at Oxford England. He liked playing chess and designed many puzzle games. He spent 24 years perfecting his art of photography. He also served as Anglican deacon at Oxford.

What did he do for a living?

Lewis Carroll’s day job was teaching mathematics in college of Christ Church at Oxford University. He also published textbooks and academic papers on math and logic.

How he is specifically related to what we have studied?

Fifty eight of over two hundred items published by him were scholarly works on to mathematics and logic. He wrote nearly two dozen texts for students in arithmetic, algebra, plane geometry, trigonometry and analytical geometry. Many of his lectures and diaries were also published posthumously.
The Lewis Carroll society
http://lewiscarrollsociety.org.uk/pages/aboutcharlesdodgson/life.html
Modern Mechanicx
http://blog.modernmechanix.com/lewis-carroll-mathematician/
10 Wonderful facts about Lewis Carroll
http://www.youtube.com/watch?v=Rz8SiiTpODI
Alice In Wonderland - w/ historical commentary
http://www.youtube.com/watch?v=pxZ38Dq754Q&feature=share&list=PLWWN2YAAAlpfSOohWhn1SL_eklz3lPErj

Reasoning by Transitivity

The basic definition of reasoning by transitivity is easy to neglect. It is remarkable how regularly we use this concept of logic in our everyday life. The classic example a=b, b=c, therefore a=c is quick to say and easy to understand. But as I thought about the subject I realized that this concept is far more prevalent. I thought of various ways that we measure things in our daily lives and it became obvious. Looking at the ruler on my desk I realized that I do not have a tangible idea for what an inch is. But someone else, some trusted authority, does and has marked my ruler accordingly. So as I measure a line on a page all I know for sure is that the line I drew is equal to the distance between 0 and 1 on my ruler. An additional premise is that the distance between 0 and 1 on my ruler defines 1 inch. Therefore, by the power of Reasoning by Transitivity I can declare that the line I drew is one inch long. This means the validity of my declaration is completely related to the accuracy of my ruler’s manufacturer.

Evolution of Logic in Relation to Evolution of Numeration

Study of chapter four added perspective to the mecanical routine of dissecting word problems, applying memorized formulas, carrying out the underlying aritmatic, and discovering the answer to the prblem; namely practice of mathemathics. We regularly call math the language of science. The simpleast reason may be that if we can express an idea mathematically we have a bether path to suport or deny logical validity of such idea. This makes me wonder how cloesly did the early man, perhaps subconsciously, dabbled in logic as it started to form numbers and keep records.
An older, more learned member of clan may remember that clan members equal to the number of fingers on both hands went hunting and returned with food. Another day members equal to number of fingres on one hand went hunting and returned empty handed. Fairly shortly after realizing numbers.