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Mathematica December 8, 2000 Research paper. Includes parenthetical citations and Works Cited in proper MLA format,
as specified in A Pocket Style Manual and online guides. Correct format is a major issue on this paper:
Do it right. The paper should explicitly relate some aspect of The Big Questions [our textbook] to the
research area of your choice. 1,000 words maximum. If you do not have trouble writing a paper this short, you
have not done your job.
- Prof. Harrison
Never in my life have I had a paper limited by a maximum - this is a new feeling for me. And I honestly did not expect to have any problems with it, but here is the final result, in all 1083 words of its glory! In an attempt to express certain basic concepts of mathematics precisely, one should consider a handful of different accepted and developed conceptions. Pythagoras, in the Fifth Century B.C., believed that the ultimate elements of reality were numbers; therefore the explanation for the existence of any object could only be explained in number. Gottlob Frege stated, in an idea referred to as logicism, that mathematics could in some sense be reduced to logic. The views of Plato state that we "know" these rules of mathematics at the intuitive level rather than the conscious level. Plato also believed that these forms existed previously in their perfect forms; humans know them in their imperfect forms through concept and imagination. Humans did not invent mathematics, but rediscovered these transcendent but real forms. Almost a century ago, Bertrand Russell wrote in The Problems of Philosophy that "philosophy should not be studied 'for the sake of definite answers to its questions, since no definite answers can, as a rule, be known to be true.'" For the problems mentioned here, however, it seems possible to give and justify answers. Certainly the effort should be made. Perhaps, through Pythagorean ideas, logicism and Platonism, a firmer understanding can be known of the grasp that mathematics has on the world. Due to the secrecy of the society in which Pythagoras, it is difficult to distinguish between any original works of Pythagoras from those of his followers. However, it is not the author that is important, but rather the notions presented. According to the view of the Pythagoreans that "all is number," the first four numbers have a special significance in that their sum accounts for all possible dimensions. In dot notation, "one" forms a point, which can be defined as the generator of dimension. "Two" forms a line of one dimension, "three" forms a triangle of two dimensions, and "four" forms a tetrahedron of three dimensions. The sum of these numbers represents the sum of all objects. The number ten, the tetractys, therefore represents the universe. Some contend that this early argument for the number ten is why, along with the number of hands and toes we have, we conduct mathematics in base ten. The landmark work on mathematical logic and the foundations of mathematics is Principia Mathematica, written in 1910 by Alfred North Whitehead and Bertrand Russell in defense of the logic of Gottlob Frege. Whitehead and Russell owe an amazing debt to Frege, whose Grundgesetze der Arithmetik ('Basic Laws of Arithmetic) provided the stepping stone for their collaborative work. Whitehead's and Russell's work succeeds in providing its intended purpose, but two ideas in particular are arguably non-logical in character: the idea of infinity and the idea of reducibility. The axiom of infinity states that there exists an infinity of objects, an assumption generally thought to be empirical rather than logical in nature. The axiom of reducibility is introduced as a means of overcoming the theory of types, discussed in Principia, and to avoid the paradoxes (such as Russell's Paradox discovered in 1901). Although technically feasible, the axiom was thought by many to be too convenient and exclusive to be justified philosophically. "There is often a possibility of human error in the formulation of the more complex logical mathematical truths, even if there is no possibility that a particular claim will be falsified by events in the physical world." As a result, the question of whether mathematics could be reduced to logic remained open. Plato took the step to say that "the mathematics of humans is imprecise." He believed that we know the concepts of mathematics through distant recollections of some previous existence, through intuition, and through our imaginations. This seems to be the predominant view today, although many are hesitant to accept it. It seems that Platonism is used today only to serve as an excuse not to believe in Pythagorean ideas or logicism. Humans see and interact with objects around them. These objects are in some fundamental respect disconnected from other objects near them - that is how humans can recognize them. They are seen as being physically separated, therefore to possess an identity. How do we justify these Platonic concepts epistemologically or metaphysically? There are at least two problems in justifying that human-perceived objects are truly separated from others, and it is maintained that these arguments cannot rationally be justified, but only accepted by faith, if at all. The first problem is that the perception that one object is distinct from all other objects is presuming a metaphysical truth that cannot be substantiated by observation. There are many concepts of existence in which all of our so-called objects are considered one entity. Such an assertion is not provable, but it is plausible. The second problem is more languishing: all human-created boundaries are arbitrary. Any attempt to affix a physical boundary to represent an imaginary boundary, such as the imaginary boundary dividing a piece of bread into halves, always leaves more or less than it should. One is left with fuzzy boundaries or no boundaries at all. Besides, there is no argument that could ever be given to convince anyone that the bread was torn exactly along that preset boundary. Man has many powers, known and yet unknown. It is quite certain that these include the power to solve the problems with mathematics and philosophy and apply them to life. It is less certain that man has solved them, and still less certain that the solutions will be applied. However, the attempt to bring the problems into light is well worth the effort, just as all attempts to solve these problems are appreciated. "Of course the proofs in mathematics are more conclusive than the philosophical proofs for someone who accepts the axioms from which they begin." Those axioms are continually being challenged, but if they are to be justified, it shall not be within the context of mathematical activities. Now we must turn to the philosophy of mathematics, "to the great debates between the formalists, the intuitionists, and the Platonists." These debates cannot be settled by mathematical proofs, however. The certainty of mathematics is merely conditional; it rests upon assumptions that cannot be proven within mathematics, but only within the philosophy of mathematics. Exactly the same problem applies with respect to the primary problems of philosophy. We can easily give practical arguments that seem very convincing, but when we analyze these arguments philosophically, we often find that the simple conventions of ordinary argument cannot be regarded as adequate. © III Enterprises & neotope.com |
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