Numbers are a vital part of everyone's life, especially at this time of the year when calculators are broken out and taxes are computed. Few people, however, realize what an achievement our system of numbers represent. Fewer still ever think about how it came about—what in reality necessitated the development of numbers and counting. This ability to unconditionally accept and use our number system underscores the incredible conceptual power it represents. If I asked a layman how numbers came about, his response would most certainly be: "Who cares? They work and that's all I need to know." If I asked a philosopher the same question, although his answer would be more eloquent, it would still fail to miss the significance of numbers. So, why are they so important?
Without getting ahead of myself, numbers are important because they illuminate philosophical processes and form the underpinnings for mathematics. Epistemology is the branch of philosophy that deals with how we know what we know. The procedure of arriving at the concept number illustrates the entire process of concept-formation, an important epistemological function. So, in studying number systems, we are really gaining insight into how reason and our mind operate. Since knowledge of our mind's operations is of inestimable value to our daily life, the more we know about our mind the better. Similarly, before we can learn algebra, calculus, and trigonometry, we must have a thorough understanding of how our number system works. In other words, arithmetic is the foundation of these higher mathematics. We can thus say that understanding how numbers came into being and operate allows us to understand more difficult subjects.
Historically, there have been two schools of thought regarding number theory: formalism and intuitionism. Or: analytic versus synthetic. Or: mind versus body. Or: rationalism versus empiricism. This dichotomy has been the undercurrent of philosophy since the time of Plato and Aristotle. Proponents of the formalist/analytic/mind side of the dichotomy believes that concepts are of the mind only and have no correlation with reality (if they believe there is one). It can take the form that Plato gave it, where acquiring knowledge is a process of remembering, or it can take the form of Immanuel Kant, where acquiring knowledge requires a direct line to the noumenal world and sensory data is of an imperfect likeness of the noumenal called the phenomenal world. All variants, however, share the same essential characteristic: an abandonment of reality.
In the philosophy of mathematics, the prime exponent of this side of the dichotomy is Gottlieb Frege. He claimed that all arithmetic truths are analytic—that is, they need not refer to facts or observations—and, further, that they only need the fundamental rules of logic for proof, not any sort of first principles of mathematics. He also dispensed with the notion of intuition, saying that because we can rely on the fundamental rules of logic there is no need for a numerical intuitionism. His major achievement in this field, however, was the conception of number as object. Frege thought that was needed was the definition of an object, rather than the definition of the concept of an object. This way he could get to the heart of the object, without having to step through epistemological considerations. He believed that numbers were distinct from any pair of empirically-given objects, but not mental constructs. Thinking about numbers comes from thinking about definitions, definitions of objects. How did this come about? Frege believed that it is only in the context of a sentence that a word has meaning. As Frege said, "we must never try to define the meaning of a word in isolation, but only as it is used in the context of a proposition…." If a word in isolation could be defined, it stands to reason that its referent would be ostensive. So, if we look at the context of the word in a sentence, it is possible to define an object even if it is not possible to point to it. Further, so long as the truth value of any proposition is determinate, the name will serve as defined. As is obvious, Frege has severed any ties that mathematics has to reality. It appears that, under his system, one could go straight up to calculus without ever having to open one's eyes. Where do all these concepts and propositions come from? Who knows? Somewhere. These are the inherent problems of the rationalist systems.
The intuitionist/empiricist/body model takes the reality side of the dichotomy and sloughs the role of the mind. It spurns any sort of logical system, dismissing it as "just abstractions." The only valid sort of knowledge is sense perception; knowledge garnered from the immediate facts presented to the observer. Integrations of percepts, or concepts, are rejected because they are still at least a step away from the perceptual level. The quintessential empiricist is David Hume, who went so far as to question the rising of the sun because it seems to disappear at night. Whatever the details, the essence remains: the abandonment of the mind.
In the philosophy of mathematics, the greatest exponent of the empiricist/intuitionist tradition is Bertrand Russell. Russell belonged to the school of philosophy called the Logical Positivists, a school which had, in his words, "a robust sense of reality." The logical positivists believed, like John Locke, that sense experience is the only route to knowledge. Since numbers are not part of this reality, any knowledge that uses them cannot be said to be real. Numbers, to Russell, are merely linguistic conventions. In fact, he thought Frege's contextual definitions revealed the sterility of reason qua logical fictions and not reason qua sense perceptor. How was knowledge—that is, sense perception—to be used? Or useful? In what form could one hold percepts and still be able to compare and relate them in an effective manner? Russell had no answer, because he could not answer: it required concepts, or as he termed them "universals."
The rationalist misses the trees while seeing the forest; the empiricist misses the forest for the trees. Both miss the point: the forest is composed of trees and a collection of trees is a forest. Both fail to recognize that concepts are products of the interaction between reality and the mind. They are not of reality, singularly, because concepts would not exist were it not for the human mind. Nor are they of the mind, for a consciousness conscious of nothing is a contradiction. They are of both the mind and reality. This point cannot be stressed enough.
Before we go further, it is crucial that the importance of philosophy be understood. As Ayn Rand said, "Philosophy is a necessity for a rational being: philosophy is the foundation of science, the organizer of man's mind, the integrator of his knowledge, the programmer of his subconscious, the selector of his values." Philosophy is man's comprehensive guide to life. Metaphysics tells him about the nature of reality; epistemology tells him how he can know about the nature of reality; ethics tells him how to lead a rational, good life; politics tells him to react with others rationally and justly. Epistemology is the foundation of philosophy because it tells, as I said earlier, how we know what we know. As such, it is the most important branch. If our knowledge is settled on an unsteady ground, like a house with a shaky foundation, it will topple in uncertainty and doubt. Number theory is applied epistemology, so it is a chance to see abstract ideas in practice.
The metaphysics which epistemology presupposes consists of three axioms which represent the foundation of all concepts and knowledge. One, existence exists. In other words, Parmenides' words to be exact, "that which is, is." This axiom simply recognizes that there is something that exists independently of our consciousness. It states that reality exists. It tells nothing of the nature of this sum of existents, past, present, or future, collectively called existence--it just underscores that they exist. It also implies a corollary axiom: consciousness is the faculty of perceiving that which exists. Again, this axiom says nothing about the nature of consciousness nor its method. It sets up a relationship between reality and consciousness that gives primacy to the latter and passivity, in the metaphysical sense, to the former. Later, when we discuss concept-formation, this relationship will play a important epistemological role. The final axiom is the Law of Identity: to be is to be something. It is a corollary of the first two axioms. The Law of Identity identifies the fact that, if something exists, it has a nature. Its characteristics constitute its nature. Again, the Law of Identity says nothing about the nature of an existent's nature. It simply states that the existent has one. These axioms cannot be proven, since they are the basis of proof. They are perceptual self-evidencies. For example, if one is looking at a car, there is no proof that it exists except that one can perceive it. There is no proof that it is something, except that one perceives it. There is no proof that one is perceiving it, except that one is perceiving it. In essence, metaphysics simply identifies the fact of existence and its corollaries.
The most important axiom to epistemology is, obviously, axiom two regarding the relationship between consciousness and reality, From it, one can derive the epistemological foundations of numbers. What, then, is the nature of this relationship? The senses take in data in the form of percepts and the mind, through a willful choice of focus and direction, integrates this multitude of percepts into concepts. These concepts are called first-level concepts, for their referents are existents in reality. Higher levels of knowledge can be achieved by integrating these concepts to form new concepts. These latter concepts are abstractions from abstractions, since their referents are other concepts. Ultimately, all knowledge can (or should) be reducible to the existential level--no matter how long the conceptual chain. In this respect, we can say that concepts are objective; they are a unique product of the mind and reality.
With all the background information behind us, it is time to journey down the road to forming the concept of a number. First, I will put forth the theory of concept-formation which I believe eliminates the problems that occurred under Frege's, Russell's, and any other of their ilk's systems. Then, I will apply the principles thus elaborated to the specific situation of numbers and, a little more broad, mathematics.
What is a concept? As Ayn Rand stated, "A concept is a mental integration of two or more units which are isolated according to a specific characteristic(s) and united by a specific definition." The units may be anything in reality: entities, attributes, qualities, actions, et cetera. They are to be integrated into a new unit, one that can be treated singularly but can always be broken down back into its constituent units. The new unit is given a perceptual concrete to identify it and differentiate it from all other concepts--it is identified by a word that will hereafter denote concept formed. The word transforms the concept, heretofore a collection of perceptual relationships, into a mental entity. Then a definition is discovered, one which names its essential characteristics, giving it identity. Without a word, the act of summoning up a concept would be impossible: one would have to attempt to recall the sum of its referents. Without a definition, the concept would be meaningless; it would be noise.
Concept-formation starts with differentiation. The units with which I am to work are mentally isolated from the rest of the existents. I do this because of the perceived similarities that distinguish this isolated group of units from the rest of my awareness. Similarity is the relationship between two or more existents with the same characteristic that varies only in degree or measurement. It is discerned perceptually and need not use explicit measurement. The perceived similarities are abstracted from the multitude of characteristics and differences that permeate my awareness. I select this group because I sense, in whatever form, that they are similar, yet different from some other things.
Then, by omitting the particular measurements of the various entities thus separated, I integrate them into a single, inseparable mental unit: a concept. This concept then subsumes all concretes of its kind: past, present, and future. The integration is accomplished using a commensurable characteristic, or the Conceptual Common Denominator. The Conceptual Common Denominator, hereafter CCD, is a range of measurements of a specific characteristic that is commensurable among all the existents compared. Then, it is noticed that some of the existents are closer together on the CCD then the others. That is the technical solution to the ambiguity of similarity.
Next, a word is attached or fixed to the concept. This serves to render it easily recallable: instead of recalling all the referents of the concept, one need merely remember the word. Finally, a definition is contrived that reflects the essential characteristic that differentiates it from all other existents or concepts. This definition serves as an economization for all the referents and characteristics of the concept. In other words, it serves to identify the concept. However, a concept and a definition are not synonymous. The concept subsumes all of the characteristics of the mental integration, not just the essential one(s). Thus, the analytic-synthetic dichotomy is averted. This conception of concept-formation explicitly shows the economy of conceptualization. A vast number of referents in reality can be grouped into one integration that is instantly recallable through the use of language and definitional skills.
Number theory, according to Objectivism (whose epistemology I have just enunciated), amounts to an application of the principles explicated above. First, a group of similars are isolated from their general placement in the world. For example, a group of apples of various sizes and kinds are isolated from the trees on which they grow. Then, the specific measurements are omitted. In my example, the exact number of apples is omitted. What is abstracted from this is that there exists some degree of repetition of apples, without specifying the specific number. So, we have the concept of quantity. Next, a group of similars are isolated from their general placement in the world. To stay with my previous example, a group of apples of various sizes and shapes are isolated from the trees on which they grow. Then, the specific measurements are omitted. In this case, everything about the apples is omitted save the fact that they are each units, or ones (which is the basic mental symbol of "unit"). Then, given what we know about quantity, we can view the apples like so: |||||. Or: a repetition of "unit" (or "one") five times. We fix a word to this quantity: "five." To put it formally, "A 'number' is a mental symbol that integrates units into a single larger unit (or subdivides a unit into fractions) with reference to the basic number of 'one,' which is the basic mental symbol of 'unit.'"
This theory of numbers dispenses with a lot of the baggage attendant with the other number theories I have presented. It shows how numbers have an objective basis. It shows how numbers fulfill an epistemological need of unit-economy. Finally, this theory, in using an open theory of concepts, leaves implicit the whole of mathematics to be discovered by mathematicians.
- Tiles, Mary. Mathematics and the Image of Reason. London: Routledge, 1991. 38.
- Ibid., 46-47.
- Ibid., 82.
- Ibid., 84.
- Rand, Ayn. "From the Horse's Mouth." Philosophy: Who Needs It. New York: Penguin-Signet, 1982. 82.
- Peikoff, Leonard. Objectivism: The Philosophy of Ayn Rand. New York: Penguin-Dutton, 1991. 7.
- Rand, Ayn. Introduction to Objectivist Epistemology. Ed. Harry Binswanger and Leonard Peikoff. 2nd Ed. New York: Penguin-Meridian, 1990. 10.