My essay for a mathematics course.
In this short essay, I will offer some personal reflections on the question “what is mathematics?” My first impression of mathematics was its precision. We can be absolutely certain that an “adequate” proof which follows the correct definitions and logical steps cannot be wrong. However, this is not the case for most of the humanities: unlike the case of mathematics, we can never be certain of the grade for a philosophy paper. I believe the precision of mathematics comes from its axiomatic structure. Starting from clear definitions, we use the strict rules of logic to deduce consequences the theorems. Any results derived from correct definitions and the logical rules of inference are inherently true.
To think mathematically is to recognize such an axiomatic structure, which requires the sequential reasoning of drawing inferences by applying rules. Mathematical reasoning seems to be very different from ordinary reasoning: in ordinary life, we tend to employ the parallel reasoning to form associative links across concepts, and to contemplate the meaning of the concepts on the basis of their similarity. The following example explains the two different kinds of reasoning systems[1].
Rate the convincingness of the following arguments from 0-10: [2]
(A)
All birds have an ulna artery.
Therefore, all robins have an ulna artery.
(B)
All birds have an ulna artery.
Therefore, all penguins have an ulna artery.
On average, people rate the convincingness of argument A and B of 9.6 and 6.4 respectively. A sense of the prototypical bird drives subjects’ responses: robins are more “bird-like” than penguins. To sequential reasoners such as mathematicians, any given animal will simply be in or out of the category “bird”. There is no middle ground. Therefore, both arguments are equally convincing because they have the same logical form. It shows that most people do not apply rules to ordinary concepts; instead, they judge “robins” as more central to the category “bird” than “penguins” by applying similarity metrics. Perhaps this result explains why most students have unhappy memories of struggles with meaningless mathematical formulae in school. Most of us simply have another reasoning system than that of a mathematician.
Mathematics is also different fom other disciplines in its subject matter. For example, music is the study of pleasurable audible sound, sociology is the study of human group behavior, and biology is the study of living organisms. These are the common sense answers. But I could not come up with an immediate answer for mathematics. Based on my study and experience, mathematics is “the study of pattern”, “the study of number” and “the study of order”. Clearly, patterns, numbers and order are entities which exist independently of the external world. Pure mathematics seems to be self-referential and takes least from the outside world. Further evidence that mathematics is a pure form of thought comes from our innate capacity for simple calculation. For example, most people understand the meaning of 1+1=2. This truth can be contemplated by the human mind even thought the identity itself can never be “found” empirically.
Although mathematical objects do not exist independently of the real world, as other entities such as animals, sound, and light do, most mathematical objects reflect the order or disorder of the reality. Most of us learned number counting in kindergarten our first experiences of the subject. It is plausible that numbers were invented to denote the abstract notion of magnitude counting is used to establish a one-to-one relationship between numbers and objects in the real world. Another example is the intermediate value theorem, which can be used to capture the concept of growth in height. A child who was 4’3” and has grown to a young man of 6’1”, must have had the height of 5’0” at a certain moment of his life. A further example is the mathematical induction, which is typically used to establish that a given statement is true of all positive numbers. It is done by proving that the first statement in the infinite sequence of statements is true, and then proving that if any one statement in the infinite sequence of statements is true, then so is the next one. It seems that the axiom of mathematical induction is motivated by the idea of “domino theory.” If the first domino falls, and if each domino that falls causes the next one to fall, then each domino will fall.
As we can see, mathematics is a fascinating subject which involves the creation of abstract notions to describe the patterns of the real world. It provides a systematic approach to express and measure concepts in reality, such as the concepts of “space”, “change”, and “order”.
[1] Steven Sloman, "The Empirical Case for Two Systems of Reasoning"; quoted in Psychological Bulletin (1996, Vol. 119, No. 1, 3-22)
[2] The minor premises of the two arguments (All robins/penguins are birds) are omitted intentionally to test how people categorize robins and penguins as birds.
