Criticism, as proposed by Karl Popper, is “the lifeblood of all rational thought.” At a first glance, one may agree with this because by critically questioning or evaluating the validity of a knowledge claim through reason, it can provide one with certainty and truth. However, the assertion that: “All knowledge claims should be open to rational criticism” gives us an alternative judgement as the word “should” is not definitive and this perhaps suggests that it is necessary to consider other viewpoints.
Through inductive and deductive reasoning, we can test knowledge claims and indicate the grounds of which the claim is based on. Yet, as evidenced by Victor Johnson’s hedonic tone theory and the notion of altruism, emotion plays an important role in our reasoning process which asks the question of whether rational criticism is free of these emotional motives. In mathematics, people tend to accept knowledge claims like: the sum of a triangle’s internal angles is equal to 180 degrees, without a rational basis. A growing number of people believe that the arts are subjective or based on personal taste because of its abstract nature, which may suggest that these knowledge claims are not open to rational criticism in the first place. Although we can examine various knowledge claims using inductive and deductive reasoning, this process might not be applicable to all areas of knowledge.
For most, certainty is possible in rudimentary arithmetic as few doubt that 1 + 1 = 2; although this claim can be rationally criticized, many of us do not question its validity because the definition of two is two ones. This possibility of certainty, however, does not apply to all areas of mathematics, especially in complex theorems that need to be vigorously tested before publication. Although mathematics may require the use of various syllogisms like logic, the validity of deductive reasoning is based upon the logic of the argument and not the truth of its foundation. This truth is assumed to be correct: for mathematics, however, this truth is compulsory in order for us to continue with the deductive process.
Kurt Gdel, a prominent mathematician, proposes that “it is impossible to prove the consistency of arithmetic, which is to say, [there is] no rigorous proof that the basic axioms of arithmetic do not lead to a contradiction at some point.” (“Is Arithmetic Consistent?”) So, when different branches of mathematics are used in order to prove something more abstract such as modelling real life phenomena, there exists difficulty in detecting which claims are made from falsely assumed truths or contradictions. One can find truth in mathematics using deductive reasoning; however, this truth may or may not be properly proved.
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 Gï¿½del’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.