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.
of infinity in his time. A better understanding was achieved only in the late 1800s, when
For centuries, mathematicians tried to contradict Euclid's Postulate V, and determine that there was more than one line parallel to that of another. It was declared impossible until the 19th century when Non-Euclidean Geometry was developed. Non-Euclidean geometry was classified as any geometry that differed from the standards of Euclidean geo...
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...
Fundamentally, mathematics is an area of knowledge that provides the necessary order that is needed to explain the chaotic nature of the world. There is a controversy as to whether math is invented or discovered. The truth is that mathematics is both invented and discovered; mathematics enable mathematicians to formulate the intangible and even the abstract. For example, time and the number zero are inventions that allow us to believe that there is order to the chaos that surrounds us. In reality, t...
... 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.
To begin with, there are several theories and perspectives that can explain the causation of crime. Each perspective has its own opinion about why people commit crimes. It analyzes three different perspectives which are social, biological and classical. Let’s take a look at all three and determine which perspective would be best fitted with the two types of sentencing models which are called determinate and indeterminate.
The concept of impossible constructions in mathematics draws in a unique interest by Mathematicians wanting to find answers that none have found before them. For the Greeks, some impossible constructions weren’t actually proven at the time to be impossible, but merely so far unachieved. For them, there was excitement in the idea that they might be the first one to do so, excitement that lay in discovery. There are a few impossible constructions in Greek mathematics that will be examined in this chapter. They all share the same criteria for constructability: that they are to be made using solely a compass and straightedge, and were referred to as the three “classical problems of antiquity”. The requirements of using only a compass and straightedge were believed to have originated from Plato himself. 1
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.
bottomless concepts such as infinity. As it turns out there are many different kinds and orders of Hartmann 2 infinity that were documented by George Cantor, who opened up this area of math for the world.
To better attempt to understand Aristotle’s view on mathematical truths, further inquiry will be made in regards to a fictionalist versus a literalist view point of mathematical objects. Both literalism and fictionalism have been attributed to Aristotle
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.
Sir Isaac Newton came up with many theories of time and space. Euclid said that there can be a concept of a straight line but Newton said nothing could ever travel in a straight line, see illustration below.
There are many people that contributed to the discovery of irrational numbers. Some of these people include Hippasus of Metapontum, Leonard Euler, Archimedes, and Phidias. Hippasus found the √2. Leonard Euler found the number e. Archimedes found Π. Phidias found the golden ratio. Hippasus found the first irrational number of √2. In the 5th century, he was trying to find the length of the sides of a pentagon. He successfully found the irrational number when he found the hypotenuse of an isosceles right triangle. He is thought to have found this magnificent finding at sea. However, his work is often discounted or not recognized because he was supposedly thrown overboard by fellow shipmates. His work contradicted the Pythagorean mathematics that was already in place. The fundamentals of the Pythagorean mathematics was that number and geometry were not able to be separated (Irrational Number, 2014).
The 17th Century saw Napier, Briggs and others greatly extend the power of mathematics as a calculator science with his discovery of logarithms. Cavalieri made progress towards the calculus with his infinitesimal methods and Descartes added the power of algebraic methods to geometry. Euclid, who lived around 300 BC in Alexandria, first stated his five postulates in his book The Elements that forms the base for all of his later Abu Abd-Allah ibn Musa al’Khwarizmi, was born abo...