Mathematics and Science in A Liberal Arts Education

Historically the liberal arts included arithmetic, geometry, astronomy, fine arts and history, grammar, rhetoric and logic. As broad general education, Liberal Arts programs are alternatives to training in a trade or craft.

Some students think of Liberal Arts as the history of (Western) culture and the themes, styles and movements in literature and the fine arts--an encyclopedia of cultural facts; lists of historical particulars

General (liberal arts) education has always been more than just particular truths in a narrow range of fields. It aims to reveal the relevance of these truths, the connections and relations among the particulars, and the subsumption of particulars under abstract general principles. Students should understand not just the truths but the search for truth; not just knowledge but the methods by which we acquire and confirm knowledge. Facts are important, but the interpretation of what the facts mean is crucial.

Logic is central to this understanding.

Western culture is the result of developments in mathematics and the physical and social sciences as much as it is a product of "merely historical" accidents or of changes in artistic or literary directions. Human creativity and awareness are as evident in logic, mathematics, and the sciences as they are in philosophy and the fine arts. For those who develop the understanding, sensitivity, and taste, the great logical and mathematical proofs and the deep and subtle theories of the sciences are as beautiful and as admirable as any product of the human spirit.

Logic and Rationalization

"Logic" has been defined as the science of right reasoning.

Freud and Marxists and the existentialists encourage a common confusion about the relationship between logic, rationality, and rationalization. Their idea is that people use logic to "explain away" behaviours and attitudes whose real explanations are non-logical. Freudians claim that one's beliefs are not grounded on logical reasons but have their source in the sub-conscious; Marxists blame ideology; existentialists emphasize "bad faith."

The science of logic begins with a value-judgment (what is right reasoning). The identification of logic with rationalization is based on a relativistic view of values. The claim is that there is no one standard of right reasoning, but that "right" (like any value) is a matter of taste. Standard canons of logic are decided by whatever social group (class, gender, etc.) has the power to impose its standards of "right reasoning" on the rest of society. So Marxists claim that logic is a bourgeois requirement. Some feminists claim that it is something that males impose on the world.

This text rejects such relativism.

Formal Science

In this course we mostly study pure (or formal or theoretical) mathematics and logic, more than applied mathematics and logic. We study math and logic from a theoretical point of view. Practical applications (such as using logic to persuade somebody or using mathematics for utilitarian calculations) are secondary. The aim is to develop some understanding of what these two sciences are about, of their methods, and of their beauty and interest for their own sakes.

So we study the principles of these two fields of study, not how one applies them. In logic, the course does not aim to teach rhetoric. In mathematics, this means that our ability to perform calculations will not be emphasized. Unlike most college mathematics courses, this course does not emphasize applying the techniques of trigonometry and differential and integral calculus. We look at the basic assumptions behind the two fields, the way mathematicians and logicians arrive at their basic assumptions, and the way they arrive at conclusions based on those assumptions. As Dagobert D. Runes says:

The traditional definition of mathematics as 'the science of quantity' or 'the science of discrete and continuous magnitude' is today inadequate, in that modern mathematics, while clearly in some sense a single connected whole, includes many branches which do not come under this head. Contemporary accounts of the nature of mathematics tend to characterize it rather by its method than by its subject matter.

According to a view which is widely held by mathematicians, it is characteristic of a mathematical discipline that it begins with a set of undefined elements, properties, functions, and relations, and a set of unproved propositions (called axioms or postulates) involving them; and that from these all other propositions (called theorems) of the discipline are to be derived by the methods of formal logic. On its face, as thus stated, this view would identify mathematics with applied logic. It is usually added, however, that the undefined terms, which appear in the role of names of undefined elements, etc., are not really names of particulars at all but are variables, and that the theorems are to be regarded as proved for any values of these variables which render the postulates true. If then each theorem is replaced by the proposition embodying the implication from the conjunction of the postulates to the theorem in question, we have a reduction of mathematics to pure logic....[1]

Runes then goes on to elaborate other views.

Scientific Method and the Formal Sciences

The steps of the method include (1) gather data; (2) narrow the field of inquiry; (3) classify and categorize; define terms (develop a technical vocabulary); (4) propose conjectures and hypotheses; (5) test the conjectures; (6) prove the hypotheses.

Gather data. The scientific understanding of good and bad reasoning is based on a collection of examples of reasoning. It starts with a non-theoretical appraisal of good and bad reasoning. It then aims to discover the common characteristics of all good reasoning (if any). Logic aims to be "the science of the rules of right reasoning."

Restrict the field. In logic, for example, we deal with reasoning only as it manifests itself in discursive verbal argument.

Classify the examples (data) into categories. All of the example cases that are similar in some respect should be classified together. We distinguish kinds of argument (including analogy, for example). We distinguish between inductive and deductive argument. We collect cases of good and bad reasoning of each type. We look for sub-classifications within the classifications. We name the kinds of good and bad argument.

Define terms. We start with vague general-purpose concepts and definitions devised for non-scientific usage. We create a more-precise language when ordinary usage leads to problems of vagueness and ambiguity. Good definitions must not be too broad or too narrow, not ambiguous, and not circular. We use terms precisely to draw distinctions (e.g., "rationality" and "rational" vs. "logic" and "logical"; "deduce" and "deduction" vs. "induce" and "induction." We invent new terms (e.g., "conjunct" and "disjunct").

Propose conjectures and hypotheses. We propose theories (theoretical rules or laws that would explain the generalizations).

Test the theories. We predict further consequences and see if the predictions or conjectured patterns are reliable. We scrap theories that give unreliable predictions, discard patterns which prove not to be universal. If testing reveals new kinds of cases, we review all the previous steps and adjust our classifications and generalizations. We review our definitions and adjust them.

Finally, in the formal sciences of mathematics and logic, we prove our conjectures or theories. The idea is to show not only that the conjecture (theory) holds (is true) universally, but that it necessarily holds for all cases.

We can then go on to take our theories as data and develop super-theories that explain the patterns that we used to explain the particulars. We try to apply tested and proven theories to new areas.

The principles of mathematics and logic (what this book is about) are the fundamental notions that regulate, systematize, and organize mathematics and logic.

Patterns in Sciences and Arts

For a Liberal Arts student, the main relation between the sciences on one hand and the fine arts and literature on the other is that both study patterns. Empirical sciences study the patterns in nature. Logic studies the patterns of human reasoning. Mathematics studies the patterns to be found in patterns. Douglas Hofstadter says:

... I am a relentless quester after the chief patterns of the universe ... central organizing principles, clean and powerful ways to categorize what is "out there." Because of this, I have always been pulled to mathematics.... [M]athematics, more than any other discipline, studies the fundamental, pervasive patterns of the universe. However, as I have gotten older, I have come to see that there are inner mental patterns underlying our ability to conceive of mathematical ideas, universal patterns in human minds that make them receptive not only to the patterns of mathematics but also to abstract regularities of all sorts in the world.... Gradually, over the years, ... my interest has turned ever more to Mind, the principal apprehender of pattern, as well as the principal producer of certain kinds of pattern.

To me, the deepest and most mysterious of all patterns is music, a product of the mind that the mind has not come close to fathoming yet. In some sense, all my research is aimed at finding patterns that will help us to understand the mysteries of musical and visual beauty. I could be bolder and say, "I seek to discover what musical and visual beauty really are."... how could anyone hope to approach the concept of beauty without deeply studying the nature of formal patterns and their organizations and relationships to Mind? How can anyone fascinated by beauty fail to be intrigued by the notion of a "magical formula" behind it all, chimerical though the idea certainly is? And in this day and age, how can anyone fascinated by creativity and beauty fail to see in computers the ultimate tool for exploring their essence?[2]

Aesthetics and appreciation of the arts involves feeling and responding to the patterns in nature and in works of art. Artistic creation is a matter of creating or reproducing or elaborating patterns. Empirical science is the discovery, description and analysis of patterns in nature. The formal sciences (logic and mathematics) study, describe, and create patterns of patterns. The beauty of structure and pattern is as central to the study of logic and mathematics as it is to literature, music or painting. Some recognition of this centrality is the main thing I hope students will develop through this course.