The been instances in history of mathematicians disagreeing

The essential definition of quality, for the purposes of this essay, is the degree to which one can reach the same conclusion after numerous trials and tests. In other words, the rigorousness of testing that a given theory may be held to when it is accepted or declared false. It makes sense, then, that if an academic discipline has been developed for an extensive amount of time, the knowledge produced by such a discipline should be of higher quality than knowledge produced by a relatively newly studied area. However, knowledge is quantified in multiple ways across all disciplines. In math, a robust and quantifiable framework of knowledge has defined the area of knowledge for centuries, and only through rigorous testing and repeatable trials can mathematical, axiomatic knowledge be accepted. On the other hand, in history, a theory can never have a quantifiable amount of quality, as many historians and historical texts disagree with each other. With new evidence on a subject area introduces doubt in the theories of historians, and as such complete historical certainty can never be truly achieved. In this way, duration of study does not increase quality, as quality can never be achieved in history. Although duration of study can serve to decrease overall doubt, due to the fact that different areas of knowledge view quality differently, the claim that duration directly increases quality is not to be accepted for all knowledge. Axiomatic proofs are the basis of mathematical knowledge. However, there have been instances in history of mathematicians disagreeing about theoretical knowledge. For one example, take the real life situation of Cantor’s Theorem of infinite sets. In short, Cantor proposes that a set can be composed of an infinite number of values, resulting in a set larger than any set that any person could ever write out by hand. Essentially, the issue with Cantor’s Theorem comes from the idea that mathematicians of his time did not like the idea of mixing ‘philosophy’ with ‘mathematics’. They accepted the idea of infinity as a philosophical idea, not a logical, mathematical idea. The idea whether a set can hold an infinite value or not is still debated today. Carl Friedrich Gauss wrote on the subject of Cantor’s Theorem: “I protest above all against the use of an infinite quantity… which in mathematics is never allowed.” Gauss did not accept the use of infinity in mathematical knowledge. Where Cantor believed in the value of mathematics beyond empirical knowledge, Gauss rejected the notion that such knowledge has a place in mathematics. This real life situation brings to the foreground the issue in mathematics about how far it should be applied at a theoretical level rather than an axiomatic level. Truly, if a mathematical Theorem cannot be proven or disproven, ambiguity to whether mathematical knowledge holds such a rigorous standard is introduced. Thus, if such ambiguity is introduced, the very notion of quality in mathematics is tested at its core. Mathematics is seen by some as a pure and unshaken vessel towards knowledge, as they believe that mathematics cannot be defined by personal emotions or ideologies. One such philosopher is Bertrand Russell, who compared the pure and unshakeable knowledge of mathematics to artistic knowledge, essentially defined by the notions of emotion and personal views. Russell believes that mathematical knowledge is the same for all, regardless of who they are or what they believe. In this way, mathematics should stand the test of time, holding its value forever. To Russell, math can only ever be of high quality, regardless of how long it has existed, as it is a stone-like immovable form of knowledge in which the theories of the mathematician will only be stated as mathematical knowledge when proven, unmistakably, as true. Adversely, a branch of epistemology known as empiricism would disagree with Russell’s ideas. An empiricist would base their knowledge on what they can view as either true or false. In this way, mathematical knowledge would not be of high quality to an empiricist, as much of modern mathematics relies on theoretical ideas. For example, a true cartesian plane could not exist in the real world, as a completely flat plane cannot exist. An empiricist could argue that since mathematics proven on a cartesian plane do not exist in the real world, such mathematics are of low overall quality. Thus, an empiricist philosopher would take no note of the duration of the development of mathematical knowledge, rather defining the quality of such knowledge based on whether it can be proven without any doubt using real world mathematics. An empiricist would completely disregard any knowledge that cannot be proven in the real world, resulting in the belief that the quality of mathematical knowledge is, overall, not very high. In history, new data and historical ideas are constantly being updated and replaced. As a historian develops their understanding about a subject through research and discussion with peers, their knowledge on a given historical theory is constantly undergoing rapid change. Whether this change occurs due to a differing opinion on a topic or simply the introduction of new data, a historian’s knowledge is never concrete. It is important, then to address the fact that a historian can never have all the facts about a historical event, and thus the rigorous testing of historical knowledge may prove unsuccessful in determining the quality of historical knowledge. Consider the real life situation of historical studies of the causes of the Cold War. During the cold war, an American historian may choose to paint the cold war as a time where the ‘good and the evil’ fought for power over the world. In the same way, a Russian historian would likely paint their version of the cold war in a similar, yet opposite way. As such, personal bias will always influence the reliability of a historian’s views. With regards to objectivity of historical knowledge, Richard Rorty believes that the purpose of historical investigation is not to gain an objective idea of what is true in history, but rather to gain knowledge through ‘justification’ and sources. On the other hand, Herodotus, in his study of the Persian war of his time, treated his study of the war as an objective account of what happened, as he treated all data as important, without any bias as to what was right and wrong. One could argue that this way of researching history is the only method of gaining an unbiased and high quality account, although it could be argued adversely that this method has the weakness that those he gained accounts from had their own biases about what had happened, creating further uncertainty. Thus, bias will always be present in historical accounts, and the resulting knowledge will always be inaccurate, regardless of the duration of time taken to research and present sources. However, some believe that history is not even about personal accounts and bias. Rather, history is able to be likened to mathematics and science, as an area of knowledge with concrete and absolute facts at its core. J.B Bury wrote that “history is a science, no more and no less.”. Through this belief, duration indeed would directly affect the quality of historical knowledge, as concrete facts and ideas simply need to be found to advance historical knowledge. The job of the historian, in Bury’s eyes, is not to convey an idea of why an event happened, but how it happened. This is the only way that concrete facts about history may be found. But some philosophers believe it is the job of the historian to tell about the reasons why historical events happened, to see through the eyes of those involved in a historical account. Carl.L. Becker believed that history can only ever be subjective, and that historical writing cannot be scientific in its nature. It is not, therefore, the job of the historian to find the facts about what had happened, rather to build an idea of what those who are part of the historical account had believed and what caused them to do what they had done. History therefore should not be treated as a science, to be rigorously tested and found correct or incorrect, rather history serves to engage with society in order to gain an idea of what may have happened in the past, and what not to do in the future. In this way, duration of historical knowledge becomes irrelevant altogether, as any individual historian’s knowledge is of as high quality as any others, that is to say, an amount of quality that a reader, or indeed the world may assign to it. Such knowledge does not lend itself to be tested and prodded, rather it is created to explore the human condition. In this way, historical knowledge may be likened to the knowledge of the arts or ethics, where the viewer is the only one who has the ability to assign a degree of worth. In conclusion, duration, although it has its role in some areas of knowledge, to varying extents, it does not seem that duration directly has an effect on the quality of knowledge that an area of knowledge produces. This distinction must be made, as quality is determined differently across all knowledge, and it may not matter at all in some areas. Duration alone must not define quality, as indeed quality itself is a subjective term. Certainly, due to the fact that quality may only be defined after taking into account the factors affecting an area of knowledge, duration is not directly responsible for the degree of quality, if indeed any, that an area of knowledge holds.