by: Brian Vuyk, student at Redeemer University College [1]

Mathematics, as defined by the Encyclopedia Britannica is the ‘Science of structure, order, and relation that has evolved from counting, measuring, and describing the shapes of objects.’ For thousands of years many scholars have struggled to understand the nature of the world around us, and the reality we are presented with. Because the nature of this study is to understand and define what we understand as reality, there is a certain measure of philosophical implications present, concerning not only the nature of our world, but also the nature of mathematics as a system of study and reality.

During the course of study of geometry, we are introduced to two main systems of geometry: Euclidean and hyperbolic geometry. Both of these systems are equally consistent, relying on each other for proof of the consistency of the other.1 Only one of these geometries may hold in any given system; there are theorems in each that would contradict if both geometries were to hold with equal authority2. This leads, however, to the question of which of these geometries holds true in the world around us?

Euclidean geometry seems to be the natural fit for the world around us, although this may very well be due to the fact that most elementary classes in geometry teach a strictly Euclidean model for geometry, emphasizing such theorems as the existence and uniqueness of parallel lines. And it appears, at least to our eyes, that we can see such shapes such as rectangles in the world around us. Engineers and architects handily use trigonometry and other mathematics in their work, resulting in robust designs. Therefore, it seams quite intuitive to assume that Euclidean geometry is the model for nature.

This may not be the case, however. Humanity as a whole plays a minuscule role in the organization of the universe; that which we may observe can hardly be any fair representation of the universe as a whole. What we interpret as Euclidean results may only appear so due to the trivial size of the line we are considering. Indeed, we have no way of observing any large3 geometric phenomena with relative abstraction from the triviality of our natural observations.

As of yet, we have no way to prove which of the two geometries may hold within our universe. Let us take take as an example the experiment done by F. W. Bessel, concerning a rectangle made by three stars. Bessel measured the angles of this celestial triangle, using the angle of parallax of a distant star. The results of his experiment were considered inconclusive. The reasoning for this was that no matter how precise Bessel’s measurements are, there is always a finite amount of error within them. Therefore, the only conclusion which could be drawn from his experiments is that if space is hyperbolic and earthly triangles have defects, they must be extremely small.4

Another area of philosophy within mathematics concerns the nature and reality of numbers and other mathematical objects. Numbers are something we work with day in and day out. The average person uses them as tools; mathematicians use them to study the phenomena around them. But what is the nature of numbers? How do we understand them as students of math?

There are three major schools of thought: Platonism, formalism and constructionism. Each of these schools represents a completely different approach to numbers.

Platonists believe in numbers as tangible, corporeal entities from outside of our reality. This stems directly from the writing of Plato and his concept of ideal forms. Plato held that there were two separate realities: the material and the immaterial. For instance, a drawing of a square belongs to the material realm, while a square itself is immaterial. The class of material objects is characterized by imperfection, impermanence, inconsistency and change, while the class of immaterial objects is characterized by perfection, permanence, consistency and immutability. For instance, when we draw a square, it’s angles can never be perfect, the lines never perfectly straight, and it could be erased or erstwhile destroyed. Thus, it is part of the material realm, while a square itself is perfect, with perfect angles and lines, and is permanent in every fashion, as part of the immaterial realm. A true square has certain intrinsic properties which may be deduced with infallible rigor.5

A Platonist assumes that there are certain metaphysical rules and properties associated with numbers and other mathematical objects. He understands the study of mathematics to be the discovery of these properties. Every question about a mathematical answer is presumed to have a definite, unambiguous answer. A Platonist does not presume to invent anything in the field of mathematics; rather all he can do is discover these answers, much as any other empirical scientist may make discoveries about his field.6

The formalists school of thought takes a very different viewpoint to the Platonists. A formalist defines mathematics as a series of rigorous proofs7 , or a science of formal deductions8. A formalist will take as a starting point a set of unproven axioms or assumptions, and by the application of logic, derive a consistent series of results. It is important to understand that the starting axioms are not necessarily true. Rather, the concentration in formalist mathematics is that all results are derived in a logical fashion from the beginning axioms.

Because the primitive terms are undefined in formalist mathematics, statements derived are meaningless, unless supplied with an interpretation. When the primitive terms are defined, then a statement has a tangible meaning.9

Both Platonism and formalism can coincide in relative harmony with respect to the basis of the principles of reasoning. Constructivism, however, contradicts both of these views, and restricts the realm of mathematics only to that which can be obtained by a finite construction.10

L.E.J. Brouwer, a Dutch mathematician and constructivist, opposed the attempts made by formalists such as Bertrand Russel to reduce mathematics to pure logic. He held that logic had nothing to do with pure mathematics; rather, it dealt only with the language of mathematics.11

Brouwer also rejected a lot of the concepts held by the Platonists. For example, he rejected the Trichotomy Postulate, which states that numbers are either negative, zero, or positive. While this postulate can be accepted in Platonism, it cannot be proved through construction.

A constructionists will reject any concept of infinity, and the concept of limits, as they cannot be found through construction within a finite amount of steps. In general, constructionists only accept that which is intuitive to our human nature.

In light of these views, we must begin to determine how as Christians we are to view mathematics as a field of study. As Christians, we can understand that the Lord created this world with a certain natural order and consistency in mind. Therefore it is easy to understand mathematics in a view similar to that of the Platonists. The world has a certain set of properties and rules which will always hold, and mathematics is the process of discovering these constraints. By it’s very nature, Platonism implies a supernatural reality; a metaphysical universe of being. It is merely ours to suggest that the immaterial reality contains the Lord creating his laws for reality, rather than what we perceive as reality being a reflection of immaterial forms.

Yet, there is also room for formalism within our philosophy of mathematics. For the logical progression and flow of deduction provides a solid basis for determining truth in reality. Because of the logical and consistent way in which the Lord created the world, the application of logic will allow us to discover the properties he has placed in the world.

Constructivism, however must be rejected. Constructivism rejects all that cannot be defined by a finite construction. The Lord’s power goes beyond the that which we may intuit; by extension, it is reasonable to believe that his power in creating the world and the constraints he placed upon it go beyond that which is intuitive to us. Therefore, it is not possible to reconcile constructivism with our Christian philosophy.

Mathematics as a field of study contains many philosophical implications which affect the way we perceive and study it. Our realization of the philosophical assumptions we make with respect to mathematics leads to an improved understanding of the field, as it allows us to look somewhat beyond our philosophical bias, and to reformulate our personal philosophy to better enable us in our studies.

1. Greenberg, 290

2. Venema, 189

3. large, with respect to the universe as a whole

4. Greenberg, 291

5. Anglin, 58

6. Davis and Hersh, 318

7. Davis and Hersh, 339

8. Davis and Hersh. 340

9. Davis and Hersh, 340

10. Davis and Hersh, 319

11. Beth, 71

Bibliography

1.Greenberg, Marvin Jay, Euclidean and Non-Euclidean Geometry. New York: W.H. Freeman & Co. Publishers, 1993.

2.Venema, Gerard A., Foundations of Geometry. Upper Saddle River: Prentice Hall, 2006.

3.Davis and Hersh, The Mathematical Experience

4.Beth, Evert W., The Foundations of Mathematics. New York: Harper & Row, 1959.

5.Anglin, W. S., Mathematics: A Concise History and Philosophy. New York: Springer-Verlag, 1994.