The Issue of Experiment in Mathematics
ABSTRACT: The issue of the status of mathematical knowledge a priori or a posteriori has been repeatedly considered by the philosophy of mathematics. At present, the development of computer technology and their enhancement of the everyday work of mathematicians have set a new light on the problem. It seems that a computer performs two main functions in mathematics: it carries out numerical calculations and it presents new areas of research. Thanks to cooperation with the computer, a mathematician can gather different data and facts concerning the issue of interest. Moreover, he or she can carry out different "tests" with the aid of a computer. For instance, one can study strange attractors, chaotic dynamics, and fractal sets. By this we may talk about a specific experimentation in mathematics. The use of this kind of testing in mathematical research results in describing it as an experimental science. The goal of the paper is to attempt to answer the questions: does mathematics really transform into experimental or quasi-experimental science and does mathematics vary from axiomatic-deductive science into empirical science?
For thirty years the computer has been used by mathematicians to solve some problems. Automatic proving of theorems, proofs obtained with the aid of the computer for the theorems whose traditional proofs are not known (e.g. the four colour problem), using computer graphics, observations of different systems behaviour with parameters changed, solving differential equations, integration — these are only a few possibilities of computer application in mathematics. Using the computer created new work conditions for a mathematician, at the same time bringing about severa...
... middle of paper ...
...objects. Because there can be shown an analogy between mathematics and natural sciences. Physical objects are recognized in the process of our experiencing materialistic reality. The experiment in natural sciences can be defined as a dialogue between the learning subject and the nature, which exists objectively. If we treat the experiment in mathematics in similar way, then there has to be two interlocutors: a mathematician and the field of mathematical objects, subjected to its own rules independent on the researcher's will.
Notes
(1) B.Mandelbrot in the context of using computer graphics states that: "The eye deserves to be made an integral part of the process of scientific thought" ("Opinions", Fractals 1(1993)1, p.120).
(2) Those examples are quoted by G.Polya in "Mathematics and Plausible Reasoning", vol. I, Princeton-New Jersey 1954, p. 90-100, 168.
Before World War II, Britain was strictly divided into classes: the upper class, the middle class, and the lower working class. Once born into a class, it was almost impossible to leave; people were bound to classes for life. The structure was stern and rigid. George Orwell even called England (and by extension Britain) “the most class-ridden country under the sun.” Classes tolerated each other, but the “upper and middle class people were brought up to believe the lower classes dirty and inferior,” creating an environment of stark inequality (The Class System). The small upper class held the majority of the wealth and employed much of the large lower class as servants, paying them menial wages. The middle class, who consisted of doctors, shopkeepers, lawyers, and people in similar professions, remained sandwiched in the center. On September 7, 1940 the blitz began and bombs started to rain down on London. However, the force of the bombs did little to blast away the walls that separated and distinguished the classes.
Although I do not enjoy contemporary Christian music and do not consider it Christian at all seeing as the music does not worship God in its entirety, I do recognize its attractiveness to some believers of Christianity. This type of music is attractive to some believers of Jesus Christ because of the roots of secular rock music. Contemporary Christian music is a middle ground for some people who are not necessarily willing to give up all secular music cold turkey. All in all, secular music has enormously transformed Christian music as a whole and made contemporary Christian music a bridge from the secular community to the community of evangelical
Gangs can be classified as a group of adolescents who are perceived to be a threat to society, are mostly recognized by their name and territorial power, and have been involved in numerous acts that violate criminal law procedures in North America. (Esbensen, Winfree, He and Taylor, 2001). The first theme that was present in the pieces of literature collected was the lack of opportunities. As previously stated before, becoming involved in a gang starts at a young age. An article titled “Youth Gangs and Definitional Issues: ‘When is a Gang a Gang, and Why Does It Matter?’” explicates what exactly constitutes a gang, starting with young adolescents. Using a survey conducted in the United States, Finn-Aage Esbensen, L. Thomas Winfree, Jr., Ni He, and Terrance J. Taylor (2001) surveyed over 5,000 students. The questions asked were based on how and why they chose to be in a gang (whether they were a part of it currently or before the survey was conducted). The authors concluded their research in deciding unanimously that there was a connection between a social learning theory, and the commencement of gangs. Correspondingly, Herbert C. Covey (2003) created an academic book entitled, Street Gangs Throughout the World, which gave an in-depth look at the different types of gangs across the world. Throughout several chapters, Covey looks at the root causes of how any why gangs are formed. The author noticed that there was a significant trend among young, deprived adolescents and gang membership. (Covey, 2003). Covey (2003) indicated that the more underprivileged a youth was, the more likely the chances would be of them joining a gang, which is a major concern.
...erself expanded gospel’s exposure when she appeared twice on “The Tonight Show with Jay Leno.” Meanwhile, television producer Bobby Jones reaches four and a half million viewers each week with his BET program, “Gospel Explosion.” However, the test for Gospel music reflects one that all Christian musicians must wrestle with: Can Gospel continue to increase its fortune in the mainstream marketplace while still maintaining its spiritual base? Despite what you believe the answer to be, African American Religious music will continually evolve. Since Thomas Dorsey first stretched the boundaries to create gospel music, choirs, quartets, and power vocalists have been singing the same song, albeit in different styles and places. As African American religious music continues to grow beyond even Dorsey’s expectations, one can only hope that it will be embraced regardless of how it is labeled by everyone who needs to be reminded of the good news it represents.
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...
On first thought, mathematics and art seem to be totally opposite fields of study with absolutely no connections. However, after careful consideration, the great degree of relation between these two subjects is amazing. Mathematics is the central ingredient in many artworks. Through the exploration of many artists and their works, common mathematical themes can be discovered. For instance, the art of tessellations, or tilings, relies on geometry. M.C. Escher used his knowledge of geometry, and mathematics in general, to create his tessellations, some of his most well admired works.
Moritz Schlick believed the all important attempts at establishing a theory of knowledge grow out of the doubt of the certainty of human knowledge. This problem originates in the wish for absolute certainty. A very important idea is the concept of "protocol statements", which are "...statements which express the facts with absolute simplicity, without any moulding, alteration, or addition, in whose elaboration every science consists, and which precede all knowing, every judgment regarding the world." (1) It makes no sense to speak of uncertain facts, only assertions and our knowledge can be uncertain. If we succeed therefore in expressing the raw facts in protocol statements without any contamination, these appear to be the absolutely indubitable starting points of all knowledge. They are again abandoned, but they constitute a firm basis "...to which all our cognitions owe whatever validity they may possess." (2) Math is stated indirectly into protocol statements which are resolved into definite protocol statements which one could formulate exactly, in principle, but with tremendous effort. Knowledge in life and science in some sense begins with confirmation of facts, and the protocol statements stand at the beginning of science. In the event that protocol statements would be distinguished by definite logical properties, structure, position in the system of science, and one would be confronted with the task of actually specifying these properties. We fin...
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.
Observation: Teacher goes over to student struggling with math worksheet. Brings over abacus and sits next to him. Begins to demonstrate. “Now how many do we take away?” child is the one to show the math on abacus. “Now how many are left?” prompts child to count the rings in order to figure out problem. Slides first number over, gets student to take away the right number. Then counts the remaining to get the right answer.
The history of computers is an amazing story filled with interesting statistics. “The first computer was invented by a man named Konrad Zuse. He was a German construction engineer, and he used the machine mainly for mathematic calculations and repetition” (Bellis, Inventors of Modern Computer). The invention shocked the world; it inspired people to start the development of computers. Soon after,
James Ford Bell Library. University of Minnesota Driven to Disover. 5 january 2010. web page. 30 April 2014.
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.
On a basic note, Gauss’s theorems and theories have enabled a smoother transaction in everyday life, whether known or not by individuals, his works have left an everlasting imprint on the development of mathematics in areas including technology and practical problem solving.
Between 1850 and 1900, the mathematics and physics fields began advancing. The advancements involved extremely arduous calculations and formulas that took a great deal of time when done manually.
The Nature of Mathematics Mathematics relies on both logic and creativity, and it is pursued both for a variety of practical purposes and for its basic interest. The essence of mathematics lies in its beauty and its intellectual challenge. This essay is divided into three sections, which are patterns and relationships, mathematics, science and technology and mathematical inquiry. Firstly, Mathematics is the science of patterns and relationships. As a theoretical order, mathematics explores the possible relationships among abstractions without concern for whether those abstractions have counterparts in the real world.