Our tendency to accept claims in mathematics without rational grounds can perhaps be explained by emotion. In Judy Jones and William Wilson’s book, An Incomplete Education, there is a reference to Gödel's Theorem being used to “argue that a computer can never be as smart as a human being because the extent of its knowledge is limited by a fixed set of axioms, whereas people can discover unexpected truths”. (495) This is a good representation of how emotional qualities can work together with rational criticism in order to establish new truths which may lead to a more subjective approach to mathematics. To further illustrate my point, a panel of referees published Hale’s proof of Kepler’s sphere packing conjecture (by packing balls using the face-centred cubic method, it will create the highest average density) even though it was only 99% certain. (par. 13) The acceptance of even the smallest uncertainties, show that reason alone may actually be a hindrance to mathematics because we cannot, or simply do not have the time, to evaluate the truth of every knowledge claim – as established before, sometimes these truths cannot be provable. When emotions such as curiosity are present with the reasoning process, mathematicians are able to tweak pre-existing proofs with their own cognitive abilities and although complete certainty may not be achievable, high precision can be obtained. Although math once followed the concept of rigorous proof, modern math has changed. Due to the limitations of deductive reasoning, some mathematicians have claimed that instead of proofs, abstract concepts such as real life situations can be modelled with computer-run experiments. Certainty may still be possible without rigorous proof but as of yet, it is too early to identify the flaws embedded in computer technology.
An area of knowledge that is usually regarded as less certain than mathematics or the sciences is the arts. When one watches an episode of American Idol (a singing competition), one may wonder if claims related to a contestant’s singing can be criticized with appropriate grounds. Given the condition that participants on the show are required to create personal renditions of various songs; logically speaking then, the judges shouldn’t be able to justify their criticism as they themselves may not entirely understand the singer’s interpretation. From my experience in the arts, particularly in music, I have received numerous feedbacks about my piano performances from various adjudicators. From reading these comments, I found out that the main focus of their criticisms was based on my technical mastery, not improvisation. This suggests that before personal interpretation, one must consider the mechanics of art; in my case, the judges made knowledge claims about the fluency of my playing and the appropriateness of the tempo which are essential components to the practice of piano. These fundamental rules exist to provide artists with a foundation that allows them to develop original ideas. Therefore, knowledge claims in the arts – arguably, can be rationally criticized on the basis of technicality. Although these fundamentals are not required to produce art, one can still be judged as though one possessed them. Going back to the American Idol example, knowledge claims that concern a contestant’s singing, regardless of whether he had any training, can be justified through reasoning based on technical abilities such as the execution of various pitches.
The issue of subjectivity in the arts, however, becomes very problematic when we stray away from technical components and move towards personal interpretation. This is because grounds used as the basis of rational criticism may differ from person to person. Similar to mathematics, the other ways of knowing: perception, language, and emotion play a major role in the reasoning process. James O. Young, author of Art and Knowledge, claims that: “Frequently, audiences do not need to acquire new experience before they can recognise that the perspective presented by an artwork is right. A good work of art can be convincing just because it enables audiences to reinterpret their past experience.” (111) When we deduce that a piece of artwork has met the necessary requirements, we can determine using perception and our past experiences, as Young suggests, the features that are most appealing to us. The judges of American Idol use a similar process: when a number of contestants possess similar abilities, they employ emotions such as amazement or curiosity, to decide which singer has the personality of a “superstar”. One interesting point about art is that it can describe the world in a plethora of ways, from experiences in the physical world to the supernatural. Due to this abstract nature, many people have claimed that art can provide important insight into the way we understand the world through interpretation. This indicates that knowledge gained through subjectivity is not always unreliable, and perhaps there is an element of certainty gained instead. Although we can evaluate knowledge claims in the arts, other ways of knowing should be considered during processes of reasoning.
Pythagoreans thought that only rational numbers existed in mathematics. After taking the square root of two, they realized that there were also irrational numbers and as a result, one of their fundamental truths was broken. As can be seen, when emotion, language, and perception are combined with the reasoning process, we can acquire truths through not only objectivity but also subjectivity. Knowledge claims in mathematics and the arts can be open to rational criticism, but reason alone may actually prevent the progress of both areas. By referring to this information, it is likely that these trends will occur in the other areas of knowledge.
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Works Cited
"Is Arithmetic Consistent?." Mathpages. Mathpages. 12 Apr 2009 <http://www.mathpages.com/home/kmath347/kmath347.htm>.
Benson, Ophelia. "Praise to be Proofs." The Philosophers' Magazine (2008). Web.14 Apr 2009. <http://www.exacteditions.com/exact/browsePages.do?issue=3329&size=2&pageLabel=1>.
Jones, Judy, and William Wilson. An Incomplete Education. 1. U.S.: Random House, Inc., 1995.
Rehmeyer, Julie. "How to (really) trust a mathematical proof." Science News 14 Nov 2008. Web.14 Apr 2009. <http://www.sciencenews.org/view/generic/id/38623/title/How_to_(really)_trust_a_mathematical_proof>.
Young, James. Art and Knowledge. 1. U.S.: Routledge, 2001.
