During the 1800s, we find the theme of independence, or freedom from outside constraints, in the development of two different frontiers. We find it in the American West through Manifest Destiny, freedom from caste, and in the chance that homesteaders had to acquire virtually free land. We find independence in math through in the building of stronger theoretical foundations, non-Euclidean geometries, and Cantor's infinities.
Independence involves breaking from the commonly accepted, traditional views in order to explore the new. It is not necessarily individual people working alone. We can see independence in a community of thought as well as in the work of a single person.
Independence is an important part of the Western culture as a concept. We find this in the concept of Manifest Destiny. This concept began when the Puritan immigrants "interpreted their victories over the Indians as part of God's plan." (Hine, 65) O'Sullivan first defined this term. He "coined one of the most famous phrases in American history when he insisted on 'our manifest destiny to overspread the continent.'"(Hine 199) Although Manifest Destiny meant different things to different people, the general definition was that God ordained the United States should expand to cover some undefined area. (Merk p.24) Some thought that the United States should cover the entire continent, and perhaps even South America, but others were more conservative in their views. Manifest Destiny was exemplified by the politicians. Stephen A. Douglas of Illinois declared that he "would blot out the lines on the map which now marked our national boundaries . . .and make the area of liberty as broad as the continent itself." (Hine 199)
The concept of Manifest Destiny is also seen in the actions of the common people. These common people included the family of my great grandmother's grandmother, Mary Rocell. In the 1880s, Hulda Rocell and her daughter Mary emigrated from Sweden to the United States. Abe Lincoln had just been shot. Mr. Rocell had to stay in Sweden because of his tuberculosis. Nevertheless, Mr. Rocell said, "Go to the United States. It is strong enough that Lincoln's assassination will not plummet the nation into chaos." Although he did not place this optimism under the title of Manifest Destiny, the idea that the United States is strong, and will continue despite opposition, is a part of this concept.
Hulda married, and her family settled in a sod house in northern Minnesota. Her husband showed independence and determination for the family to survive by planting fruit trees on the farm. It was highly unusual to attempt to plant fruit trees that far north.
The family's independent spirit, and courage was a vital part of surviving in the harsh conditions they encountered. In 1881, there was a terrible blizzard. The snow was so high that it covered the fence posts. Father tied a rope around his waist and the porch post to tend the animals in a nearby shed, so that he would not get lost in the blizzard. However, many neighbors froze to death right outside their own front doors. During the storm, the wind blew the door of the sod house open. The younger children got in the trundle bed to keep warm, while the parents and older children shoveled snow out of the house for the rest of the day, so that they could shut the door.
There were challenges in the summer as well as in the winter months. In the summer of 1885, a dark cloud approached the farm. It was not an impending storm, but a looming cloud of grasshoppers. Hulda tied her apron around her rose bush to protect it. The grasshoppers came, covering the ground with a living blanket that was a few inches thick. The voracious grasshoppers ate the apron, along with the rose bush. They also ate the fruit trees and the crops.
However, independence and courage are not invincible. After this disaster, the family's courage failed. They sold the farm for a chest of jewels, later discovering that the jewels were the worthless dime store variety. Thus we see that although independence and courage were a major part of the pioneers' lives in the West, there are times when these ideals suffered under the harsh glare of reality. (Pearl Losey and Cheryl Logan, personal communication)
Travelers also had challenges to face. Two of these challenges were mosquitoes, and wagon rides. As one traveler describes in a diary written in 1867,
Here our severest troubles commenced with armies of mosquitoes and buffalo gnats which beggar description . . . Our bars, tents and blankets and every other available means of fortification against the swarming hordes proved ineffectual. The sting of these little insects is fairly poisonous, and cause men and horses to roll, and fret, and pitch, and kick, and groan all night. Our horses are almost as poor as skeletons, and our own faces look like the last run of the measles. (Armstrong, p.176)
Mosquitoes were not only annoying, but they brought diseases such as malaria. Wagon rides were also a challenge. One traveler describes a 21 mile trip on Thompson's "dead axle" wagon saying, "My head was continually bobbing up and down in the frosty air like a churn dasher, while the rest of my body was bouncing all over the bottom of the wagon. (Armstrong p.182) Some pioneers endured not 21 miles, but thousands of miles of travel in the bumpy covered wagons as they went west. When faced with challenges of weather, mosquitoes, and transportation, pioneers had two options. They could quit, or they could continue. Independence and courage were vital for those who chose the second option.
Freedom from caste
We see independence in the concept of the American West through the freedom from caste that the new land offered. In moving west, an individual could virtually lose his own identity, and rebuild it to be in whatever form he desired. The republicanism of Manifest Destiny meant the "government of a classless society, as contrasted with that in a monarchy, which was dominated by an arrogant aristocracy and headed by a hereditary king." (Merk p.29)
Freedom from caste is seen in the story of the Rocell family. In Sweden, Mr. Rocell was a schoolteacher who taught boys in school, and his daughters at home. He was put in jail because teaching girls to read was prohibited. However, his work was instrumental in changing this law.
The daughter's education continued after the family immigrated. Although it still may have been unusual in the United States for girls to get an education, it was not prohibited. There was more of a chance for girls to succeed through access to education. This reflects a freedom from caste and a break from the traditional that was present in the West.
The experience of the runaway slave, Bernard Freeman, in the historical fiction of Karen Cushman, exemplifies both the freedom from caste and the limitations of this ideal that were present in real life. Bernard Freeman escaped his master, Mr. Sawyer, and headed west to California. While Bernard claims, "I ain't no slave out her", the prejudice against colored men was still present in the west. Also, Bernard was afraid to live in town, because if he did, some white man might make a slave again. (Cushman, pp.93-96) Eventually, Bernard went to San Francisco, where colored men owned "laundries, restaurants, shops [and] boarding houses."(Cushman, pp.175-176) This represents another step in the breaking of the caste system. There was a chance for Bernard to become free and independent, not tied down to the caste system of slavery.
The chance that settlers had to get land also reveals freedom from caste. The ideals of independence and courage are evident in the Homestead Act, also known as the "Act to Secure Homesteads to Actual Settlers on the Public Domain". This act went into effect January 1, 1863. It allowed both men and women over the age of 21, who declared their intention to become citizens, to file for up to 160 acres of surveyed land on the public domain. If the homesteaders cultivated and improved the land for five years, they could receive a full title to the land for a $10 fee. If the homesteader did not want to wait five years to own the land, they could stay on the land for six months, and purchase it for $1.25 an acre. Although unscrupulous cattlemen and lumbermen misused this plan, the Homestead Act did provide farms for a total of more then four hundred thousand families. (Hine pp 333-336)
Development of Math
We find the theme of individuality both in the West and in math. Individuality is in the Manifest Destiny, freedom from caste, and freedom to own land. The progression of math towards the abstract and theoretical also reflects the independence of thought. In order to understand this, we must learn what theoretical math is. To accomplish this, we will develop the analogy of a building.
Developing and learning math is like constructing an intricate and very unusual building. This building has a foundation, floors, stairways, and strangely shaped towers stretching to touch the sky. The foundation of this building is the unprovable assumption that deductive reasoning, or reasoning from general to specific, is valid. An example of deductive reasoning is: all people have heads and Sue is a person, therefore, Sue has a head. Deductive reasoning is not provable because we are not allowed to circular reasoning. In other words, we are not allowed to use deductive reasoning in the proof that deductive reasoning is valid. If we were to ask the question, "Why is the sky blue?" an example of circular reasoning would be the answer "because it is" This reasoning is not valid because it uses the fact that the sky is blue to explain why the sky is blue.
The next level of the building of mathematics is still underground and usually out of sight. The elements in this level have undergone revisions throughout the history of mathematics, but currently, this level of the foundation contains primitive notions, undefined objects, and multiple sets of axioms that expand the ideas in the primitive notions. (Enderton p.11,12) Axioms are statements that we accept without proof, and statements that we use to build most of the rest of the building. Thanks to Godel, we know that there will always be parts of the building not connected to the foundation. He proved that in any system at least as complicated as arithmetic with finitely many axioms, there are true statements that are unprovable from the axioms. In our analogy of the building, this is equivalent to having floating furniture in the building, furniture that is not connected to the building in any way.
The rest of the building has many floors, stairways, and towers. The floors of the building are theorems, statements that tell us more about its structure. These statements must be proved using previous theorems, axioms and the rules of logic. Proving statements is like building stairways between the floors. The towers and parts of the foundation represent abstract math. These towers are far removed from normal arithmetic and algebra. In building some of these abstract towers, you can use very little of your intuition or visualization to understand them.
The last part of this building is the siding. This is the part most often seen by the general public and beginning mathematics students. It contains the practical applications of the structure, including the applications of arithmetic, algebra and calculus. As siding is not the foundation, the floors, or the stairways of a building, so the practical applications are really outside the main structure of mathematics. The reason for this is that in building mathematics, consistency with the axioms is all that is required. It is outside the realm of pure mathematics to ask if the structure of mathematics is real or can be used in the real world.
During the 1800s, people intensified explorations of this building. Instead of working mainly with the siding, the practical applications, mathematicians did something any reasonable architect would do, they worked on connecting the floors to the foundation. They accomplished this through putting calculus on a better foundation. Mathematicians also developed shapes and space inside this building through exploring non-Euclidean geometries. Finally, they looked at a particularly tall tower, one named infinity. These explorations involved individualism for the community of mathematicians. Developing these areas involved breaking away from commonly accepted ideas and intuition. "Mathematical freedom will forever be the legacy of the nineteenth century."(Dunham p.248)
Stronger theoretical foundations
Independence of thought was present in the process of strengthening the foundations of calculus. Independence was a necessary part, because the process demanded that mathematicians ask the question why, and that they not rely on an intuitive understanding of the concepts. Newton and Leibniz developed calculus in the 1600s. However, little was done for a century after this toward logically strengthening the underpinnings of calculus. (Eves p.132) To understand what logical underpinnings needed strengthening, we need to understand the limit, a foundational idea in calculus. A limit is a number that we approach but do not land on. Imagine that we have a square of paper. Cut the paper in half, and keep one piece. As you repeat this process, the paper you are working with will get smaller and smaller. Regardless of the number of times you repeat this process, you will never run out of paper. The amount of paper will approach, but not hit, zero.
Before the nineteenth century, mathematicians used vague language to explain limits. It is not possible to precisely define terms such as 'approach' and 'get close to.' Karl Weierstrass (1815-1897) clarified this ambiguous language considerably. He introduced epsilon delta notation to explain limits.(Eves p.137,139) Instead of vague terms, we now have definitions such as this, translated from the Greek:
For all delta greater than zero there exists an epsilon greater than zero such that if the distance between x and a is greater than zero and less than delta, then the distance between f(x) and L is greater than O and less then epsilon.
These unambiguous, very precise definitions were a part of the process of strengthening the foundational concepts of calculus and the building of mathematics. " If the 1800s did not belong to a single mathematician, they did have a few overriding themes. It was the century of abstraction and generalization, of a deeper analysis of the logical foundations of mathematics that underlie the wonderful theories of Newton and Leibniz and Euler." (Dunham p. 245)
Geometry
The development and acceptance of non-Euclidean geometries is another example of how math became more independent from the real world (Dunham p.246). As we shall see in Cantor's work on infinities, geometry contains a problem whose solution lies in abstracting away from the intuitive manner in which we perceive the real world. This trend away from commonly accepted intuition and towards abstraction reveals the independence of the mathematical community.
Euclidean geometry appears to capture the way we see the world. (Delvin p.163) It matches with common sense quite closely. The angles of a triangle add up to 180 degrees, and intuitive notions of area and congruence hold true. In Euclidean geometry, shapes are congruent when they are the same as each other. The ancient Greek philosopher Euclid laid the foundation for Euclidean geometry and meticulously developed this geometry in his famous book The Elements. Euclid started with definitions, common notions, and postulates. "Common notions are truths common to all knowledge, to all the sciences, whereas postulates are truths within the field of geometry." (Witter p.232) In Euclid's five postulates he tried to capture self-evident truths about the fundamental patterns of nature. (Delvin p.143) Although Euclid's work attempted to base the geometry on axioms rather then intuition, he still believed that his foundational concepts were true, and represented reality. He also used intuition in his proofs.
While Euclid's work greatly advanced the field of geometry, there was a postulate in his work that raised a major problem. The root of the problem was the assumption that the geometry Euclid tried to axomatize was the geometry of the world. (Delvin p.144) The solution to the problem came when mathematicians broke from the belief that Euclid's geometry was the only valid one, and began to explore other geometries. Euclid's famous fifth postulate basically says that lines will meet if and only if they are not parallel. This is equivalent to saying, if we have a fixed line and a point not on that line, there is only one line passing through the point that is parallel to the fixed line.
This information about parallel lines was intuitive, but seemed complex enough that it should be provable from the other postulates. To use the analogy of the house, it seemed that the fifth postulate should be in one of the floors of the house of geometry, not in the foundation.
Euclid's statement of this postulate began a long, colorful, frustrating story of multiple attempts to prove it from the other axioms and postulates. The most common problem in attempts to prove this was that the mathematicians used intuitive ideas, or axioms which are equivalent to the parallel postulate. This breaks the rule in logic says circular reasoning is not allowed in logical arguments.
The development of proofs for the fifth postulate culminated in the work of the 1800s. Mathematicians realized that the parallel postulate is independent from the others. Since it is independent, it leads to a logically consistent, although somewhat strange geometry, if it is replaced.
Riemannian Geometry
Since the parallel postulate is independent of the others, we can replace it. Riemann, Gauss and other mathematicians developed a geometry based on the sphere by replacing the parallel postulate with the statement that, given a line and a point not on the line, there are no parallels passing through the point.
The shape upon which this geometry is built is basically a sphere, with the exception that we declare opposite points on the sphere to be the same point. This strange twist that defies visualization was needed in order to satisfy Euclid's other postulates, and definitions. Since any two lines on a sphere meet, it makes sense to replace the traditional, fifth postulate with the postulate that there are no parallel lines. Since a line is the shortest distance between two points, lines in this geometry are curves. More precisely, they are part of a great circle. A great circle is a slice passing through the center of the sphere. Passengers on airplanes regularly experience the results of this geometry. The shortest route for the airplane to follow is not a straight line shooting off into space, but a curved one, which ends at their destination. (Delvin p.168)
This geometry is consistent, but it yields different results then the Euclidean geometry. For example, in Riemannian geometry, the angles of a triangle always add up to more then 180 degrees. (Devin p.168-169)
Another geometry developed when the parallel postulate was replaced with the postulate that there were infinitely many lines passing through a given point that were parallel to a given line.
In this geometry, the angles of a triangle add up to less then 180 degrees. (p.59 Dunbar) This trend away from intuition and towards abstraction rose to yet a higher level with the discovery and development of group theory. In group theory, Klein generalized geometries even further by exploring common patterns of different geometries. (Delvin p.199)
Cantor's work on infinities
As in developing the limits in calculus, and working with non-Euclidean geometries, working with infinities breaks the sense of intuition, and demands separation from the real world, as we know it. Early work with infinities was greatly hindered because people required what they discovered to match with their intuition. If the results did not match, they assumed that the results were inconsistent, and ignored them. To show the non-intuitive nature of the subject, let us do a though experiment for one minute. Imagine that we have a container, numbered ping-pong balls, and a stopwatch. Start the stopwatch at one minute. Put ten balls into the container, and take one out. Repeat this process at 30 seconds, 15 seconds, 7.5 seconds and so forth. Soon, you will be working at unimaginable speed, but do not worry; this experiment will be over in a minute! If you look in the container at the end of the minute, how many balls will you find? Intuition tells you that you should find some balls in the container; because you were putting balls in so much more rapidly then you were taking them out. However, there are no balls in the container. If you claim there are, I will ask for the number that is on the ball, and then I will tell you exactly when that ball was taken out.
In order to understand how and why the infinities of set theory break the intuition, we need to understand the idea of a set. A set is a basic concept in the set theory that forms the rock bottom foundation for all mathematics.( Eves p.147). "A set is a collection of things the collection being regarded as a single object." (Enderton p.1) For example, we can have sets of numbers, such as the set of all whole numbers between 1 and 4. We could write this {1,2,3,4}, or {x ½x is a whole number and 0 < x < 5} In words, this says the set of all x such that x is a whole number and x is strictly between 0 and 5.
Suppose we had a set of 50 students and a set of research papers, and we wanted to see if the two sets were the same size. In other words, we want to know if each person had turned in a paper. There are two different ways we could do this. We could count the students and the papers, and see if the numbers were the same. But, what could you do if you could not count to 50? In this case, you could give each student a paper, and if any students ended up without a paper, you would know that the set of students and the set of papers were not the same size. This second approach allows us to find out if two things are the same size without resorting to numbers to describe them.
Matching up papers with students may seem a trivial way to solve the problem, because we can count to 50. However, it is very useful and leads to some surprising results in infinite sets. For example, suppose we had an infinite number of students and papers -- a grading nightmare! Then, it may be possible to match each student with a paper, even if there seemed to be fewer papers then students! Another example of this is if we match each natural number (1, 2, 3 . . .) to a number from the set (100,101,102). We can match them up, 1 matches with 100, 2 matches with 101, and so on. Therefore, the sets are the same size of infinity. But, intuitively, it seems that there are 100 more numbers in the first set then in the second.
From this example, you might think that all infinities are the same size, but strange as it may seem, there are an infinite number of sizes of infinity! For example, there are infinite sets, such as the set of all real numbers between 0 and 1, which cannot be matched up with the natural numbers. (Eves p.165) In order to work with these different sizes of infinity, we define a transcendental, or not finite, number that represents the "size" of an infinite set. To take the abstraction a level further, it is possible to develop an arithmetic using these transcendental numbers and the different "sizes" of infinity. For more information, refer to Lecture 34 in Eves Great Moments in Mathematics [After 1650], or pp.139-144 in Enderton's Elements of Set Theory, or one of the many other books on set theory.
Although the work of mathematicians forced them to use infinite sets before the 1800s, Bernhard Bolzano (1781-1848) was the first to accept the existence of a completed infinite set, or a set with a size. Accepting this idea involved breaking a barrier in mathematics that was similar to the barriers broken in the geographical expansion of the settlers in the West. The settlers had to break the barriers of blizzards, grasshoppers, and other calamities. They also had to cross natural barriers of mountains and oceans. In the world of mathematics, in order to accept the idea that infinite sets exist, Bolzano had to accept the non-intuitive concept that an infinite set can be the same size as a piece of itself. (Eves 160) George Cantor explored, and expanded these ideas, developing different sizes of infinity. Eventually, an arithmetic based on different sizes of infinities was developed. (Enterton 138) Thus, the concept of infinities in math became less intuitive, more abstract, and less connected to the real world as we perceive it.
The nineteenth century witnessed the development of a growing independence. This independence influenced the development of the Western identity through the Manifest Destiny, freedom from caste and freedom to own land. Growing independence in the mathematical community is found in the abstraction in the fields of calculus, set theory and infinities, and geometry. The parallel themes of independence as seen in both fields reveals a willingness to break past the familiar and explore the new frontiers.
Works Cited
Armstrong, Moses Kimball. The early empire builders of the great West. Compiled and enlarged from the author's Early history of Dakota Territory in 1866. St. Paul, Minnesota: Porter, 1901.
Cushman, Karen. The ballad of Lucy Whipple. New York: Clarion Books, 1996.
Delvin, Keith. The language of mathematics: making the invisible visible. New York: W.H. Freeman and Company, 1998.
Dunham, William. Journey through genius: the great theorems of mathematics. New York: Wiley, 1990.
Eves, Howard Whitley. Great moments in mathematics [after 1650]. Washington, D.C.: Mathematical Association of America, 1981.
Enderton, Herbert B. Elements of set theory. New York: Academic Press, a subsidiary of Harcourt Brace Jovanovich, Publishers, 1977.
Hine, Robert V. The American west: a new interpretation. New Haven [Conn.]: Yale University Press, 2000.
Merk, Fredrick, and Lois Bannister Merk. Manifest destiny and mission in American history; a reinterpretation. New York: Knopf 1963.
Witter, George E. The structure of mathematics an introduction, second edition. Lexington, Massachusetts, Xerox College Publishing, 1972.
References
Armstrong, Moses Kimball. The early empire builders of the great West. Compiled and enlarged from the author's Early history of Dakota Territory in 1866. St. Paul, Minnesota: Porter, 1901.
Carruccio, Ettore. Mathematics and logic in history and in contemporary thought. Trans. Isabel Quigly. Chicago, Aldine Pub. Co. 1964.
Conference on the History of Western America (2nd: 1962: Denver Colo.). The American west, an appraisal; papers. Ed. Robert G. Ferris. Ed. Advisors: LeRoy R. Hafen, Allen D. Breck, Robert M. Utley. Santafe, Museum of New Mexico Press, 1963.
Cushman, Karen. The ballad of Lucy Whipple. New York: Clarion Books, 1996.
Delvin, Keith. The language of mathematics: making the invisible visible. New York: W.H. Freeman and Company, 1998.
Duberman, Lucile. Social inequality: class and caste in America. Philadelphia: Lippincott, 1976.
Dunham, William. Journey through genius: the great theorems of mathematics. New York: Wiley, 1990.
Enderton, Herbert B. Elements of set theory. New York: Academic Press, a subsidiary of Harcourt Brace Jovanovich, Publishers, 1977.
Eves, Howard Whitley. Great moments in mathematics [after 1650]. Washington, D.C.: Mathematical Association of America, 1981.
Heath, Sir Thomas. A history of Greek mathematics vol.1 from Thales to Euclid. New York: Dover Publications, Inc., 1981.
Hine, Robert V. The American west: a new interpretation. New Haven [Conn.]: Yale University Press, 2000.
Kline, Moris. Mathematics in western culture. New York: Oxford University Press, 1953.
Leight, Robert L, Alice Duffy Rinehart. Country school memories: an oral history of one-room schooling. Westport, Conn: Greenwood Press, 1999.
Medlin, William K. The history of educational ideas in the West. New York: Center for Applied Research in Education, 1964.
Merk, Fredrick, and Lois Bannister Merk. Manifest destiny and mission in American history; a reinterpretation. New York: Knopf 1963.
National Park Service, Division of Publications. Exploring the American west, 1803-1879. Washington D.C.: U.S. Dept. of the Interior, 1982.
Paxson, Frederic L. History of the American frontier, 1763-1893. Boston, New York:
Resnikoff, H.L. Mathematics in civilization. New York: Dover Publications, 1984.
Stephanson, Anders. Manifest destiny: American expansionism and the empire of right. New York: Hill and Wang, 1995.
Weinberg, Albert Katz. Manifest destiny, a study of nationalist expansionism in American history. Baltimore: The Johns Hopkins Press, 1935.
Wilder, Raymond Louis. Mathematics as a cultural system. Oxford; Pergamon Press, 1981.
Witter, George E. The structure of mathematics an introduction, second edition. Lexington, Massachusetts, Xerox College Publishing, 1972.
Notes
The West
(Manifest Destiny) The beginnings of the manifest destiny happened when the Puritans "interpreted their victories as part of God's plan. The beginnings of the Manifest Destiny are evident whey John Mason wrote "Thus the Lord was pleased to smite our Enemies in the hinder Parts, and give us their land for an Inheritance."(Hine, 65)
(Manifest Destiny) "O'Sullivan coined one of the most famous phrases in American history when he insisted on "our manifest destiny to overspread the continent."(Hine 199 quote)
(Manifest Destiny) Stephen A. Douglas of Illinois declared he "would blot out the lines on the map which now marked our national boundaries . . .and make the area of liberty as broad as the continent itself." (Hine 199)
(Manifest Destiny) It meant expansion, prearranged by Heaven, over an area not clearly defined." (Quote p.24 Merk)
(Manifest Destiny) Manifest destiny also included geographical predestination. Nature determined natural boundaries.(Weinberg p.43)
(Manifest Destiny) In Europe, boundaries hemmed nations in. They prevented nations from expanding and separated them. America used the idea that nature determined boundaries, as a reason to expand. Safety came through space and land. (Weinberg 43-46)
(Caste) "A free, confederated, self-governed republic on a continental scale-this was Manifest Destiny. It was republicanism resting on a base of confederated states. Republicanism by definition meant freedom. It meant government by the people, or, rather by the people's representatives popularly elected. It meant more. It was a government of a classless society, as contrasted with that in a monarchy, which was dominated by an arrogant aristocracy and headed by a hereditary king. It meant, moreover, freedom from established churches headed by monarchs. "( Merk p.29)
(Challenges) Traveler describes a bone jarring wagon ride (Armstrong p.182)
(Challenges) Mosquitoes (Armstrong, p.176)
Mathematics
(Precedent) In Egypt, the following formula was given. "If you are told; A trucated pyramid of 6 for the vertical height by 4 on the base by 2 on the top. You are to square this 4, result 16. You are to double 4, result 8. You are to square 2, result 4. You are to add the 16, the 8, and the 4, result 28. You are to take a third of 6, result 2. You are to take 28 twice, result 56. See, it is 56. You will find it right. (Dunham p.3)
(Precedent) The authoritarian society of Egypt certainly was consistent with an approach in math that did not expect people to question authority by demanding carefully constructed proofs. (Klinne, p.23)
(Precedent) The fact that the priests monopolized learning helped create authoritarian society, which in turn influenced the lack of proof. I might be able to trace authoritarian versus democratic societies and how they relate to the development of math. There are two problems, simultaneousness does not prove cause and effect, and other factors such as economic, practical, search for beauty, and development for its own sake.
(Precedent) The independence of the Greeks, which resulted in analysis rather the blind acceptance, was one of the causes of Thales work in early demonstrative mathematics. He asked why, and developed the rigorous, logical proof.(Dunham pp. 5-7)
(Precedent) The careful reasoning of the Greek mathematicians is also seen in philosophy, and the idealism of the sculpture and architecture of the period. (Klinne, p.11)
(Math)"The 1800s many people advanced the development of math. (Dunham 245)
(Math) "Mathematical freedom will forever be the legacy of the nineteenth century." (Dunham p.248)
(Math) Hillbert, Godel, Cantor, and Henri Poincare were important mathematicians of the 1800s.
(Sets) "A set is a collection of things the collection being regarded as a single object." (Enderton p.1)
(Sets) "Set theory is, in a sense, a rock bottom foundation for all mathematics." (Eves p.147)
(Infinity) Pingpong ball example
(Infinity) There is an arithmetic based on transcendental cardinal numbers. (Enderton p.138)
(Euclidean Geometry) In Euclid's five postulates, he tried to capture self-evident truths about the fundamental patterns of nature. (Delvin p.143)
(Euclidean Geometry) the problem was in the assumption that the geometry Euclid tried to axomatize was the geometry of the world. (Delvin p.144)
(Euclidean Geometry) It does appear to capture the way human beings perceive the world. (Quote Delvin p.163)
(Euclidean Geometry) "The distinction between common notions and postulates was made on the basis that common notions are truths common to all knowledge, all the sciences, whereas postulates are truths within the field of geometry.(Witter p.232)
(Geometry)Klein generalized geometries even further by exploring patterns of different geometries. (Delvin p.199)
(Geometry) Make pictures of the parallel postulate and it's replacements.
(Calculus) Use idea of tearing a piece of paper in half for the limit.
(Calculus) Newton and Leibniz developed calculus in the 1600s.(Eves p.21)
(Calculus) Little was done for a century after this toward logically strengthening the underpinnings of calculus. (Eves p.132)
(Calculus) Karl Weinstraous (1815-1897) introduced epsilon delta notation. ( Eves p. 137, 139)