(Edits added 2/10/19)
Cornelius Van Til rightly distinguishes in God between the unity of singularity and the unity of simplicity. The first refers to God's numerical oneness. "There is and can be only one God." (The Defense of the Faith, 4th ed., p. 31) The second refers to God's absolute simplicity or lack of compositeness: ". . . God is in no sense composed of parts or aspects that existed prior to himself." (ibid.) Van Til apparently thinks that divine simplicity is a Biblical doctrine inasmuch as he refers us to Jer. 10:10 and 1 John 1:5. But I find no support for simplicity in these passages whatsoever. I don't consider that a problem, but I am surprised that anyone would think that a doctrine so Platonic and Plotinian could be found in Scripture. What surprises me more, however, is the following:
The importance of this doctrine [simplicity] for apologetics may be seen from the fact that the whole problem of philosophy may be summed up in the question of the relation of unity to diversity; the so-called problem of the one and the many receives a definite answer from the doctrine of the simplicity of God." (ibid.)
That's an amazing claim! First of all, there is no one problem of the One and the Many: many problems come under this rubric. The problem itself is not one one but many! Here is a partial list of one-many problems:
1) The problem of the thing and it attributes.
A lump of sugar, for example, is one thing with many properties. It is white, sweet, hard, water-soluble, and so on. The thing is not identical to any one of its properties, nor is it identical to each of them, nor to all of them taken together. For example, the lump is not identical to the set of its properties, and this for a number of reasons. Sets are abstract entities; a lump of sugar is concrete. The latter is water-soluble, but no set is water-soluble. In addition, the lump is a unity of its properties and not a mere collection of them. When we try to understand the peculiar unity of a concrete particular, which is not the unity of a set or a mereological sum or any sort of collection, we get into trouble right away. The tendency is to separate the unifying factor from the properties needing unification and to reify this unifying factor. Some feel driven to posit a bare particular or bare substratum that supports and unifies the various properties of the thing. The dialectic that leads to such a posit is compelling for some, but anathema to others. The battle goes on and no theory has won the day.
2) The problem of the set and its members.
In an important article, Max Black writes:
Beginners are taught that a set having three members is a single thing, wholly constituted by its members but distinct from them. After this, the theological doctrine of the Trinity as "three in one" should be child's play. ("The Elusiveness of Sets," Review of Metaphysics, June 1971, p. 615)
A set in the mathematical (as opposed to commonsense) sense is a single item 'over and above' its members. If the six shoes in my closet form a mathematical set, and it is not obvious that they do, then that set is a one-over-many: it is one single item despite its having six distinct members each of which is distinct from the set, and all of which, taken collectively, are distinct from the set. A set with two or more members is not identical to one of its members, or to each of its members, or to its members taken together, and so the set is distinct from its members taken together, though not wholly distinct from them: it is after all composed of them and its very identity and existence depends on them.
In the above quotation, Black is suggesting that mathematical sets are contradictory entities: they are both one and many. A set is one in that it is a single item 'over and above' its members or elements as I have just explained. It is many in that it is "wholly constituted" by its members. (We leave out of consideration the null set and singleton sets which present problems of their own.) The sense in which sets are "wholly constituted" by their members can be explained in terms of the Axiom of Extensionality: two sets are numerically the same iff they have the same members and numerically different otherwise. Obviously, nothing can be both one and many at the same time and in the same respect. So it seems there is a genuine puzzle here. How remove it? See here for more.
3. The problem of the unity of the sentence/proposition.
The problem is to provide a satisfying answer to the following question: In virtue of what do some strings of words attract a truth-value? A truth-valued declarative sentence is more than a list of its constituent words, and (obviously) more than each item on the list. A list of words is neither true nor false. But an assertively uttered declarative sentence is either true or false. For example,
Tom is tired
when assertively uttered or otherwise appropriately tokened is either true or false. But the list
Tom, is, tired
is not either true or false. And yet we have the same words in the sentence and in the list in the same order. There is more to the sentence than its words whether these are taken distributively or collectively. How shall we account for this 'more'?
There is more to the sentence than the three words of which it is composed. The sentence is a truth-bearer, but the words are not whether taken singly or collectively. On the other hand, the sentence is not a fourth thing over and above the three words of which it is composed. A contradiction is nigh: The sentence is and is not the three words.
Some will say that the sentence is true or false in virtue of expressing a proposition that is true or false. On this account, the primary truth-bearer is not the (tokened) sentence, but the proposition it expresses. Accordingly, the sentence is truth-valued because the proposition is truth-valued.
But a similar problem arise with the proposition. It too is a complex, not of words, but of senses (on a roughly Fregean theory of propositions). If there is a problem about the unity of a sentence, then there will also be a problem about the unity of the proposition the sentence expresses on a given occasion of its use. What makes a proposition a truth-valued entity as opposed to a mere collection (set, mereological sum, whatever) of its constituents?
4) The problem of the unity of consciousness.
At Theaetetus 184 c, Socrates puts the following question to Theaetetus: ". . . which is more correct — to say that we see or hear with the eyes and with the ears, or through the eyes and through the ears?" Theatetus obligingly responds with through rather than with. Socrates approves of this response:
Yes, my boy, for no one can suppose that in each of us, as in a sort of Trojan horse, there are perched a number of unconnected senses which do not all meet in some one nature, the mind, or whatever we please to call it, of which they are the instruments, and with which through them we perceive the objects of sense. (Emphasis added, tr. Benjamin Jowett)
The issue here is the unity of consciousness in the synthesis of a manifold of sensory data. Long before Kant, and long before Leibniz, the great Plato was well aware of the problem of the unity of consciousness. (It is not for nothing that Alfred North Whitehead described Western philosophy as a series of footnotes to Plato.)
Sitting before a fire, I see the flames, feel the heat, smell the smoke, and hear the crackling of the logs. The sensory data are unified in one consciousness of (genitivus objectivus) a self-same object. This unification does not take place in the eyes or in the ears or in the nostrils or in any other sense organ, and to say that it takes place in the brain is not a good answer. For the brain is a partite physical thing extended in space. If the unity of consciousness is identified with a portion of the brain, then the unity is destroyed. For no matter how small the portion of the brain, it has proper parts external to each other. Every portion of the brain, no matter how small, is a complex entity. But consciousness in the synthesis of a manifold is not just any old kind of unity — it is not the unity of a collection, even if the members of the collection are immaterial items — but a simple unity. Hence the unity of consciousness cannot be understood along materialist lines. It is a spiritual unity and therefore an apt model of the divine simplicity.
Back to Van Til
He is wrong to suggest that there is only problem of the One and the Many. The One and the Many is itself both one and many. Whether he is right that it is "the whole problem of philosophy," it is certainly at the center of philosophy. But what could he mean when he claims that the doctrine of divine simplicity solves the problem? It is itself one form of the problem. The problem is one that arises for the discursive intellect, and is perhaps rendered insoluble by the same intellect, namely, the problem of rendering intelligible the unities lately surveyed.
God is the Absolute and as such must be simple. But divine simplicity is incomprehensible to the creaturely intellect which is discursive and can only think in opposites. What is actual is possible, however, and what is possible might be such whether or not we can understand how it is possible. So the best I can do in trying to understand what Van Til is saying is as follows. He 'simply' assumes that the God of the Bible is simple and does not trouble himself with the question of how it is possible that he be simple. Given this assumption, there is no problem in reality of as to how God can be both one and many. And if there is no problem in the supreme case of the One and the Many, then there are no problems in any of the lesser cases.