The One and the Many

“Numbers are related to an old philosophical problem, called the problem of the one and the many. We can also describe it as the problem of unity and diversity. How do unity and diversity fit together? It is worthwhile understanding a little about the problem.

The Philosophical Problem of One and Many

Philosophers in ancient Greece already confronted the problem. How does the multiplicity of things that we observe relate to the unity of one world and the unity belonging to every member of a particular class? How does the unity of the class of cats relate to the particularity of Felix the cat and each other cat? Parmenides and later Plotinus said that the one was prior to the many. But if we start with one thing, and it has no differentiation, how can it differentiate later or lead to the observed dif- ferences among things in the world? Heraclitus and the atomists said that the many were prior to the one. But if we start with many things, how can they then be related to one another, and why do they exhibit the common characteristics of belonging to one class (like the class of cats)?

Medieval philosophy continued to consider the question. On one side of the dispute were philosophical realists. These people said that universal categories like the category cat or horse were real. (This kind of realism should not be confused with other modern views called by the same name.) Like the followers of Plato, they thought that the categories existed prior to any particular cats or horses. The categories were like original patterns or archetypes. They were the universal patterns that explained why all cats share common features. Each cat, when it came into existence, conformed to the prior pattern of the universal category, which might be called catness.

Medieval realism started with the unity of a category. So how did it explain diversity? The medieval philosophers believed in God, so they believed that God creates each cat. He uses the same pattern, namely catness. But if he uses the same pattern, why does each cat not come out exactly the same, like candies made using the same mold (the same pattern)? Even candies made with the same mold show minute differ- ences, which may be due to imperfect mixing of the ingredients, or slight differences in the making process. So a person could try to say that the cats are different because the matter used to make them is different, or the making process shows slight differences. But this explanation just pushes the problem back in time. What generated the differences in the matter? What generated the differences in the processes? The processes presumably have a universal category to describe the unity that belongs to them. So what leads to the differences when we compare two instances of the same process?

Opposite to the medieval realists were the nominalists. They said that the many was prior to the one. We start out with many cats in the world. Then we give them a common name, the name cat. According to the nominalists, the name is nothing but a name. (The word nominalism is cognate to the Latin word nomen, which means “name”). A name like cat does not label a universal category that is out there in the world. The category of catness is only in here in our minds. We have invented it. And its invention depends on the prior existence of the many cats out there. Clearly, nominalists think that the diversity of cats is first, and the unity of the category is derived.

Nominalism had the opposite problem from realism. Its problem was to account for the unity. We start with many cats. Why is there anything in common between the many cats, any commonality that would lead us to group them all under a single category of “cat”? Nominalism suggested that the category is our invention, corresponding to nothing out in the world. It is simply an idea. It is an illusion. Or, if a nominalist did not want to go this far, he could say more guardedly that the unity is a secondary construction, based on the primary reality of the diversity of cats. But if we start with pieces that are purely diverse, how can we later create unity? Even if the unity is pure illusion, we need to explain where the unity in the illusion came from. Moreover, it is not plausible to claim that there is nothing “really” similar about the different cats.

Unity and Diversity in the Trinity

According to Trinitarian thinking, the unity and diversity in the world reflect the original unity and diversity in God. First, God is one God. He has a unified plan for the world. The universality of the truth 2 + 2 = 4 reflects this unity. God is also three persons, the Father, the Son, and the Holy Spirit. This diversity in the being of God is then reflected in the diversity in the created world. The many instances to which 2 + 2 = 4 applies express this diversity: four apples, four pencils, four horses, etc. God is the original, while the unity and diversity in the created world are derivative. So we may say that God is the archetype, the original pattern, while the instances of unity and diversity in the created world are ectypes, derived from and dependent on the archetype.

We can put it in another way. God governs the world by speaking (chapter 1). God has both unity and diversity. So when he speaks— through the Word of God, who is the second person of the Trinity—his speech has unity and diversity. The unities in God’s speech specify the unities in the world that he has made; its diversities specify the diversities in the world that he has made.

We can also illustrate unity and diversity in a third way. The unity of God’s plan has a close relation to the Father, the first person of the Trinity, who is the origin of this plan. The Son, in becoming incarnate, expresses the particularity of manifestation in time and space. He is, as it were, an instantiation of God. Thus he is analogous in his incarnation to the fact that the universal law 2 + 2 = 4 expresses itself in particular instances like four apples.

What is the role of the Holy Spirit? In addition to other roles, the Holy Spirit expresses by his presence the fellowship between the Father and the Son (John 3:34–35). His role in fellowship has been termed the associational aspect.3 The Holy Spirit is the archetype for the associational aspect. A universal law like 2 + 2 = 4 and the particular instances, like four apples, also enjoy a relation of association. The one inheres in the other. In general terms, the associational relation between the one and the many that instantiate the one is an ectypal associational relation.

The Numerical Character of the World

God’s plan is the source for the numerical character of the world, as it is the source for every aspect of the world. God’s plan is consistent with his character and reflects his character. He is Trinitarian in his character, and so his plan exhibits unity and diversity, and the unity and diversity in the world arise as a result.

In God we find the foundation for numbers. In the world that God has created, we sometimes deal with one, two, three, four, or more apples. Why? Because there are many apples in the world. The apples have diver- sity. They also have unity. They all belong to one class, the class of apples.

When we have four apples on the table, and we wish to count how many there are, we have already made the decision to treat all the apples on the table as members of one class, the class consisting of the apples on the table. This class has its own unity and diversity. It has the unity of being one class, and the diversity of the four apples in the class. The four apples belong to one class, exhibiting the associational aspect. Counting is possible only when we have the unity of one class (the four apples taken together), the diversity of members in the class (each apple), and an as- sociational relation of belonging: that is, the individual apples belong together with the other apples on the table, and they all belong to the same class.

In sum, in our everyday experience, the very idea of number de- pends on features of the world that embody unity, diversity, and asso- ciations. God is the archetype for unity and diversity and association. What we see in the world is the effect of God’s word, expressing his plan and his character. He has made the world with ectypal unity and diversity. The combination of these gives us the numerical character of the world.

We have collections of one, two, three, four, or more apples. And we have collections of one or more pears or peaches or pencils. Every class of four apples is an instantiation of the idea of “having four mem- bers.” The number four expresses the commonality among all instances of four apples, peaches, and the like. In this respect, the number four is the one, showing the unity belonging to all the instances. The instances are the many, showing the diversity. The relation between the unity of the number four and the diversity of four apples or peaches or pencils is an associational relation. Thus the number four depends on the unity and diversity in the Trinity.

The same, of course, is true of any other natural number: one, two, three, four, and so on. Each number, such as 114, is a unity, and the col- lections of 114 apples or 114 peaches are diverse instantiations of the unity.

Now we can notice another unity in diversity and diversity in unity. All the natural numbers together have a unity. They are all natural num- bers! And they have a diversity: each one, such as 114, is distinct from the others.

All of this is so natural, so ordinary, that we are accustomed to taking it for granted. But we can thank God for it. God made it so. Because God is stable, faithful, and consistent with himself, the numbers are stable and the relations of unity and diversity are stable. We live in a world, rather than an absolute chaos. More specifically, God made it so by his word, specifying that it would be so. God speaks. He speaks according to his Trinitarian character. Numbers reflect his character. By reflecting his character, they show us who God is:

For what can be known about God is plain to them, because God has shown it to them. For his invisible attributes, namely, his eter- nal power and divine nature, have been clearly perceived, ever since the creation of the world, in the things that have been made. (Rom. 1:19–20)”

(Poythress, Redeeming Mathematics)

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