Critical Thinking, Formal Logic, and Mathematics

According to Ennis (1990), “Critical thinking is reasonable, reflective thinking that is focused on deciding what to do or believe” (as cited in University of Western Sydney, n.d.). While according to Moon (2008), “Critical thinking is a capacity to work with complex ideas whereby a person can make effective provision of evidence to justify a reasonable judgment. The evidence, and therefore the judgment, will pay appropriate attention to context” (as cited in University of Western Sydney, n.d.).

However, both of these influential definitions seem to be too vague and narrow to be useful. In fact, they don’t even make a distinction between deductive and inductive reasoning. Deductive reasoning involves deciding (using any number of fixed rules of deductive logic) what conclusions must follow from a given set of premises/propositions. In a correctly performed deductive thread of thoughts, a conclusion is guaranteed to be true, if the premises from which it was deduced are true. On the other hand, inductive reasoning involves deciding (using few, if any, definite rules) what conclusions may follow from a given set of premises/propositions. Hence, no matter how well an inductive thread of thoughts is performed, the conclusion is never guaranteed to be true, even if the premises from which it was induced are known to be true. Most scientific theories are good examples of high-level inductive reasoning; while most, complete solutions to complex mathematical problems are good examples of high-level deductive reasoning. And this seems to pose a problem for the definitions of critical thinking given by Ennis (1990) and Moon (2008).

After all, solving complex mathematical problems definitely requires “reasonable, reflective thinking that is focused on deciding what to do or believe” (Ennis, 1990). Similarly, solving complex mathematical problems definitely requires “a capacity to work with complex ideas,” making “effective provision of evidence to justify a reasonable [or even undeniable] judgment,” and paying attention to context (Moon, 2008). Anyone who doubts that solving complex mathematical problems, even by simply following the rules/steps developed for their solution, requires all these skills, should only take a look at the following flowchart -  http://www.nature.com/protocolexchange/system/uploads/2626/original/flowchart.jpg?1372325178,  which graphically and textually represents a mathematical algorithm (i.e. a sequence of steps required for reaching the correct solution) “for the control of complex networks and other nonlinear, high-dimensional dynamical systems” (Cornelius & Motter, 2013). In this respect it is important to note that computers, who only run on algorithms, have long become unrivaled (with regards to speed) in solving complex mathematical problems, by simply following the rules/steps developed for their solution (i.e. the algorithms).

Thus, it seems possible to be considered a critical thinker, according to the definitions of Ennis (1990) and Moon (2008), despite being completely incapable of inductive reasoning (which humans actually use at every corner) and independent thought, like most computers.    

References

Cornelius, S. P., & Motter, A. E. (2013). NECO – A scalable algorithm for NEtwork COntrol. Protocol Exchange. doi:10.1038/protex.2013.063

 

University of Western Sydney. (n.d.). Develop your skills in critical thinking and analysis. Retrieved from https://www.uws.edu.au/hall/hall/critical_thinking. 

Is it possible to conduct a full, unbiased and unambiguous study of a topic when the resolution of scientific and/or technological uncertainty is not a research objective?

There are many examples in humanities and the arts where something similar to scientific method (i.e. collection and grading of evidence, attempts to theorize and draw conclusions from it) is habitually used to answer a variety questions. Thus, while such research, does involve attempts to resolve uncertainty on the topics under investigation, these topics are not on the subjects addressed by the natural/social sciences or engineering, while the attempts to resolve uncertainty in these topics, don’t utilize any knowledge from the natural/social sciences or engineering.

Also, research in a number of fields outside of natural/social sciences and engineering, has long been producing results, which are superior in their objectivity and certainty to anything produced by the natural/social sciences or engineering. In particular, derivation of a logically valid proof to any theorem in mathematics (or more broadly, in any system of formal logic), guarantees that this theorem is true in all the cases that it claims to address and that it will always be true. By contrast, scientific theories and empirical research results are frequently discarded, or at least modified, in response to new empirical evidence or development of more sophisticated theories with greater explanatory/predictive power.

However, it is important to keep in mind that research projects in many subfields of humanities and the arts never try to “conduct a full, unbiased and unambiguous study” in the first place; if only because the subject matter under investigation is open to a wide variety of interpretation; preventing which would only move us further away from, albeit unreachable, objectivity (since many perspectives would be deliberately neglected).

Moreover, there is an influential body of thought, which argues that artistic research is a unique method of research. And while attempts to outline it are rather complex; it is clear that according to its proponents, artistic practice (i.e. the creation of art itself) is an integral part of artistic research (Borgdorff, 2012, pp. 140 – 173). However, given that all other (“non-art”) academics generally find the concept of artistic research distasteful and confusing, attempts, to describe artistic research as something a lot like scientific research, have been made (Borgdorff, 2012, pp. 56 – 103). In fact, according to Lesage (2009), even artistic practice “can be described in a way more or less analogous to scientific research” (p. 5). Thus, an artistic project, “begins with the formulation, in a certain context, of an artistic problem” (p. 5). This leads to “an investigation, both artistic and topical, into a certain problematic, which may or may not lead to an artwork, intervention, performance or statement” (Lesage, 2009, p. 5). However, if the investigation does lead to “an artwork, intervention, performance or statement,” the artist uses it as a new reference point for looking at the initial artistic problem and its context (Lesage, 2009, p. 5).

References

Borgdorff, H. (2012). The conflict of the faculties: Perspectives on artistic research and academia. Leiden, NL: Leiden University Press. Retrieved from https://openaccess.leidenuniv.nl/bitstream/handle/1887/21413/file444584.pdf?sequence=1

Lesage, D. (2009). Who’s afraid of artistic research? On measuring artistic research output. ART&RESEARCH: A Journal of Ideas, Contexts and Methods, 2(2). Retrieved from http://www.artandresearch.org.uk/v2n2/pdfs/lesage.pdf

Validity, Reliability and Qualitative Research

In research, a reliable measure is the one that gives the same result over and over again (assuming there is no change in what is being measured) (Trochim, 2006a); while a valid measure is the one that gives the correct value (assuming there is such a thing as a “correct” value for the given measurement) (Trochim, 2006b). However, according to Cronbach (1975), all phenomena, even those that can be quantitatively measured, will, sooner or later, change. This is especially true in social and behavioral sciences, where (1) the studied phenomena change very rapidly, and (2) it is often impossible to tell whether two different measurements of the same phenomenon were made under identical conditions; because isolating the variables of interest, from all external influence, may be impossible (Cronbach, 1975, pp. 122-123; as cited in Lincoln & Guba, 1985, p. 115). Those social and behavioral phenomena which are primarily studied through qualitative methods (often because it is hard to study them through quantitative methods) are especially prone to such unpredictability. Hence, whatever is being “measured,” through qualitative research, is usually constantly changing. And, as a result, there is no way to known what its “correct” value is at any one time; because this value is also constantly changing.  Therefore, when it comes to qualitative research design, the concepts of validity and reliability seem to be inapplicable.

References

Cronbach, L. J. (1975). Beyond the two disciplines of scientific psychology. American Psychologist, 30, 116-127.

Lincoln, Y. S., & Guba, E. G. (1985). Naturalistic inquiry. Beverly Hills, CA: Sage.

Trochim, W. M. K. (2006a). Theory of reliability. Retrieved from http://www.socialresearchmethods.net/kb/reliablt.php

Trochim, W. M. K. (2006b). Reliability & Validity. Retrieved from http://www.socialresearchmethods.net/kb/relandval.php