The Concept of Infinity
The concept of infinity has been evaluated many times throughout history. Only recently, in the nineteenth century, has major progress evolved in the field. The chapter "Beyond Infinity" answers the questions, "what is mathematics and why should I study it?" by reviewing several mathematician's theories of infinity.
First, the author mentioned Galileo who theorized that a line which measured 3 inches long contained the same amount of points as a line twice it's length. The author also referred to Bernhard Bolzano, a mathematician who later on attempted to define infinity as well, but failed to do so. Archamedis was also referred to, for he developed a system for infinity called 'myriad'. Using this, he was able to estimate the number of grains of sand there are on a beach.
The chapter was focussed on matmatician named Cantor who became well known towards the end of the nineteenth century. Cantor believed that one set of numbers is equivalent to another set if they can be paired together. This was referred to as his "stepping-stone" process. He also used this to define numbers beyond infinity. Cantor assigned the first letter of the Hebrew alphabet, along with a subscript, to represent the number of elements in an "ordinary infinite set". The first letter (Aleph) with zero for a subscript represented real numbers. Aleph with a numeral one stood for real and irrational numbers.
Cantor recognized infinity as a verb, rather than a noun. This was uncommon, for it contradicted previous Platonic theories. Therefor, many mathematicians' dismissed his theories at first. Also, he was criticized for not having an "absolute infinity". Some mathematicians eventually started to accept Cantor's theories. He then went on to prove that there are an infinite equal amount of fractions and whole numbers, and that the set of irrational numbers is larger than whole numbers and fractions. Cantor applied his theories to Geometry as well, thus demonstrating that there are the same infinite amounts of points in every space, despite dimensions. He also showed how points on a line can be paired with points on a plane, which can be paired with points in a volume, and so on.
In the short story “Where is Here” by Joyce Carol Oats the stranger discusses the idea of infinity. Infinity is an abstract concept that something is without a beginning or ending. The stranger gives three examples of this idea. All three can be represented of a different type of infinity.
... relationship in one problem that doesn’t appear in others. Among all of this, there is such vastness in how one person might approach a problem compared to another, and that’s great. The main understanding that seems essential here is how it all relates. Mathematics is all about relationships between number and methods and models and how they all work in different ways to ideally come to the same solution.
Zeno of Elea was a Greek philosopher and a mathematician. Zeno is particularly known for his paradoxes that helped build both mathematics and logic, they specifically targeted the concepts of continuity and infinity. Zeno was born in 495 BCE and died in 430 BCE. In his lifetime he contributed some great things to the subject of math. He studied at the Eleatic School, a leading school in Greek philosophy. He is said to have been a good friend of the philosopher Parmenides. After his studies he went on to write a book containing 40 paradoxes! Unfortunately none of Zeno’s writing has ever been found. Zeno contributed to mathematics greatly and he will always be remembered for this.
Newton’s inventive years with mathematics were from 1664 to 1696. Even though his companions also had likely various elements of the calculus, Newton summed everything up and included these ideas of his while developing new and more exact methods. The necessary elements of his thought were on hand in three tracts, De analysi (On Analysis), which went unpublished until 1711. In 1671, Newton developed a more absolute account of his course of infinitesimals, which appeared nine years after his death as “Methodus fluxionum et serierum infinitarum”.
In the construction of the Large Hardon Collider, physicists seek and hope to unlock the mysteries of the universe by analyzing the attributes of the most miniscule particles known to man. In the same way, theologians have argued back and forth over the course of human history with regards to the divine attributes of God, seeking and hoping to unlock the mysteries of the metaphysical universe. Although these many attributes, for example omnipresence, could be debated and dissected ad nauseum, it is within the scope of this research paper to focus but on one of them. Of these many divine attributes of God, nothing strikes me as more intriguing than that of God’s omnipotence. It is intriguing to me because the exploration of this subject not only promises an exhilarating exercise in the human faculties of logic, it also offers an explanation into the practical, such as that of the existence of evil, which we live amidst every day. So with both of these elements in hand, I am going to take on the task of digging deeper into the divine attribute of omnipotence in hopes of revealing more of the glory of God, and simultaneously bringing greater humility to the human thinker. In order to gain a better understanding on the subject of divine omnipotence, I am going to analyze four aspects of it. First, I am going to build a working definition of what we mean when we say that God is omnipotent. Second, I am going to discuss the relationship between divine omnipotence and logic. Third, I am going to discuss the relationship between God’s omnipotence and God’s timelessness. Last, I am going to analyze God’s omnipotence in relation to the existence of evil in the world. Through the analysis of these four topics in relation to om...
...thing had to be equal and be able to fit in the entire triangle. He had to take all the numbers that was there out them in order and then create the study. This is why Pascal loved math because he got to always work with numbers. Everything that he worked with or did a study with was numbers. Such as the triangle with the numbers in the center he made that as I stated before. I know that math is something that deals with numbers but Pascal always had a lot of them to work with. He did a lot of cool inventions and things he had made helped a lot of people. Least the things that he made was very useful to people. He loved math he was very dedicated to it. Even though he loved math he was very religious. He had a degree in religion to. Many people thought that was a very boring subject and didn’t enjoy studying it. But Pascal really enjoyed it.
The bridges of the ancient city of Königsberg posed a famous and almost problematic challenge a few centuries ago. But this isn’t just about the math problem; it’s also a story about a famous Swiss mathematician named Leonhard Euler who founded the study of topology and graph theory by solving this problem. The effects of this problem have lasted centuries, and have helped develop several parts of our understanding of mathematics.
The great field of mathematics stretches back in history some 8 millennia to the age of primitive man, who learned to count to ten on his fingers. This led to the development of the decimal scale, the numeric scale of base ten (Hooper 4). Mathematics has grown greatly since those primitive times, in the present day there are literally thousands of laws, theorems, and equations which govern the use of ten simple symbols representing the ten base numbers. The field of mathematics is ever changing, and therefor, there is a great demand for mathematicians to keep improving our skills in utilizing the numeric system. Many great people, both past and present, have made great contributions to the field. Among the most famous are Archimedes, Euclid, Ptolemy, and Pythagoras, all of which are men. This seems to be a common trend in mathematics, for almost all classical mathematicians were male.
Pythagoras held that an accurate description of reality could only be expressed in mathematical formulae. “Pythagoras is the great-great-grandfather of the view that the totality of reality can be expressed in terms of mathematical laws” (Palmer 25). Based off of his discovery of a correspondence between harmonious sounds and mathematical ratios, Pythagoras deduced “the music of the spheres”. The music of the spheres was his belief that there was a mathematical harmony in the universe. This was based off of his serendipitous discovery of a correspondence between harmonious sounds and mathematical ratios. Pythagoras’ philosophical speculations follow two metaphysical ideals. First, the universe has an underlying mathematical structure. Secondly the force organizing the cosmos is harmony, not chaos or coincidence (Tubbs 2). The founder of a brotherhood of spiritual seekers Pythagoras was the mo...
The history of mathematics has its roots on the African continent. The oldest mathematical object was found in Swaziland Africa. The oldest example of arithmetic was found in Zaire. The 4000 year old, Moscow papyrus, contains geometry, from the Middle Kingdom of Egypt, Egypt was the cradle of mathematics. The great Greek mathematicians, including Pythagoras, Thales, and Exodus all acquired much of their mathematics from Egypt, including the notion of zero. This paper will discuss a brief history of mathematics in Africa. Starting with the Lebombo bone and the Ishango Bone, I will then present Egyptian mathematics and end with a discourse on Muslim mathematics in African. “Most histories of mathematics devote only a few pages to Africa and Ancient Egypt... Generally they ignore the history of mathematics in Africa … and give the impression that this history either did not exist or, at least …is not knowable.”
Historically speaking, ancient inventors of Greek origin, mathematicians such as Archimedes of Syracuse, and Antiphon the Sophist, were the first to discover the basic elements that translated into what we now understand and have formed into the mathematical branch called calculus. Archimedes used infinite sequences of triangular areas to calculate the area of a parabolic segment, as an example of summation of an infinite series. He also used the Method of Exhaustion, invented by Antiphon, to approximate the area of a circle, as an example of early integration.
Fractal Geometry The world of mathematics usually tends to be thought of as abstract. Complex and imaginary numbers, real numbers, logarithms, functions, some tangible and others imperceivable. But these abstract numbers, simply symbols that conjure an image, a quantity, in our mind, and complex equations, take on a new meaning with fractals - a concrete one. Fractals go from being very simple equations on a piece of paper to colorful, extraordinary images, and most of all, offer an explanation to things. The importance of fractal geometry is that it provides an answer, a comprehension, to nature, the world, and the universe.
Irrational numbers are real numbers that cannot be written as a simple fraction or a whole number. For example, irrational numbers can be included in the category of √2, e, Π, Φ, and many more. The √2 is equal to 1.4142. e is equal to 2.718. Π is equal to 3.1415. Φ is equal to 1.6180. None of these numbers are “pretty” numbers. Their decimal places keep going and do not end. There is no pattern to the numbers of the decimal places. They are all random numbers that make up the one irrational number. The concept of irrational numbers took many years and many people to discover and prove (I.P., 1997).
The history of math has become an important study, from ancient to modern times it has been fundamental to advances in science, engineering, and philosophy. Mathematics started with counting. In Babylonia mathematics developed from 2000B.C. A place value notation system had evolved over a lengthy time with a number base of 60. Number problems were studied from at least 1700B.C. Systems of linear equations were studied in the context of solving number problems.
Sometimes there are theories that have to do with machines that do not exist and usually have things in them that are infinite and they usually work with letters and numbers.