Theory of Knowledge in Math and Science

Name 4 Name Professor Class Date: Theory of Knowledge “Without the assumption of the existence of uniformities there can be no knowledge.” One of the presumptions of acquiring knowledge, particularly knowledge in a scientific or mathematical context, is that there must be causal relationships that can be observed or intuited between different phenomena. Human beings base their behaviors on this presumption on a regular basis.

We wake up in the morning to the sound of an alarm clock and assume we can shut off that alarm using that particular button because we did so on previous occasions. We do not assume that every experience with an alarm clock is a new encounter. Similarly, we assume that the laws of gravity will secure ourselves in place to the earth and we will not go flying off into space.

In science and math, the presumption that natural and mathematical laws have a consistent and abiding existence (even with certain exceptions) and that causality can be deduced ensures that when the same cause is generated between two distinct phenomena, the same result will occur again. Knowledge may be observed in an inductive fashion, in other words, different phenomena in a variety of contexts may be observed. But deductive logic is still required to arrive at certain basic principles.

“Deductive reasoning, or deduction, starts out with a general statement, or hypothesis, and examines the possibilities to reach a specific, logical conclusion,” and goes from the general to the specific (Bradford). In contrast, with inductive reasoning, “there is data, then conclusions are drawn from the data” (Bradford). But in both instances, some presumptions of uniformities must hold fast.

When there is a general principle from which there is assumed to be some consistency, such as the law of gravity or relativity, then assumptions about new phenomena in science can be made; after observing consistency on an inductive basis (for example, similarities of responses to a particular drug), then an assumption can be made regarding the scientific utility of the pharmaceutical. Inductive logic, rigorously performed, is the primary assumption behind the scientific method, whereby a hypothesis is generated based upon past research about a new scenario.

After observing the scenario, the hypothesis is either proven correct or incorrect. Regardless of the outcome, the experiment can and must be able to be performed by another individual. In other words, the uniformity of knowledge must be able to be replicated by another individual. The challenge of consistency is one of the objections raised to qualitative versus quantitative research in science. “The quantitative types argue that their data is ‘hard,’ ‘rigorous,’ ‘credible,’ and ‘scientific.’ The qualitative proponents counter that their data is ‘sensitive,’ ‘nuanced,’ ‘detailed,’ and ‘contextual’” (Trochim).

Still, even if qualitative data is based upon an observation of a smaller group of individuals, it still can be called science if generalizations can be drawn based upon coding the data (Trochim). Quantitative research, where there is a control group in most instances to isolate variables that could affect results and more conclusively establish causality, is considered the gold standard of research.

In other words, provided that circumstances, either in a small or a large population, can be assured to be uniform, scientific causal inferences may be possible to be drawn in a way that can either advance knowledge or have utility for human beings. “Uniformity is of both practical and functional importance, necessary to increase structural sophistication and realize the promise of nanostructured materials” (O’Brien). The same is true of mathematical formulas. A very simple example of this are basic mathematical principles such as commutative, associative, and distributive principles.

Every time someone embarks upon solving a new algebraic equation with variables, there is an assumption that the same variables can be integrated in a particular manner, just as there is an assumption that numbers can be integrated in a particular fashion. Without consistency in behavior of numbers, basic algebra would be impossible and mathematics would have to be an entirely concrete and less useful discipline.

The very notion of numbers itself, even on a simplistic level, reflects abstraction in terms of how physical entities are represented as either one, two, and so forth. Mathematics ensures that sets of numbers can be conceptualized and solved more easily than the painstaking need to represent everything. Even something like negative numbers, which cannot be physically rendered, can be conceptualized in abstractions, based upon an awareness of consistent principles. Thus consistency.