In my finite and imperfect way of thinking, I suggest that love (or Love) has to be central in any finite and imperfect formulation for the triune Godhead.

In our conceptions of God, we image bearers have to be content with metaphors and analogies, since the map is not the territory (i.e., the metaphors and analogies are not the reality to which the metaphors and analogies point). All metaphors and analogies break down at some point.

A soldier, for example, may be "a lion in battle" (or he may be lion-like in his battling skills), but he'll never be a lion. So it is with our descriptions of God.

We do, however, have a key word in description The Three-In-One God whom we worship and serve, and that word is love.

Where I enlist an analogy/metaphor is in thinking of the Triune God in terms of a husband and wife, who are co-equal (albeit in slightly different ways), and the love they have for one another. Their personhood is a given (or taken for granted--an axiom, if you will--thanks, @El'endia Starman), but the love which they share is LIKE a third person in that there is this bond between them which is vivified by loving feelings and actions. The Father loves the Son; the Son loves the Father; and the Spirit actuates that love relationship via affection and behavior.

Interestingly, in the human love relationship between a husband and a wife there is unity in diversity, and that unity is expressed in the Bible as follows:

"and the two shall be one flesh. Therefore, what God has joined together, man must not separate."

Maybe @Jon Ericson is on the mark when he suggests the Triune God is like the grammatical "subject/object/verb." God is one person (i.e., one sentence) who is exists eternally in a love relationship between Father and Son, energized, if you will, by the Spirit. The Spirit, in a sense, binds the Godhead together?

When you think about it, if marriage is a metaphor for Christ's relationship to the Church Universal, then the Spirit likewise is the "glue" which binds the Son to the Son's body, his church. In a sense, there is another three-in-one, which is a metaphor/analogy for THE Three-In-One.

Hey, I gave it my best shot!

Why is it three and not any other number?

The best answer I think is that God wanted it so naturally on the basis of his character, which can be concisely described as love. He picked an integer from the set of integers he invented such that the relationship of love could be fully modeled. You can see, citing Solomon whose description of Wisdom you have mentioned, scripture stating the nature of triune solidarity:

With one, there is no manifestation of love:

• Whoever isolates himself seeks his own desire; he breaks out against all sound judgment. Prov 18:1

With two it is manifest but exclusive -- and with three it is full, dynamic, in flux. Amazing and strong. (Consider also the example of Father, Mother, and Child below):

• Two are better than one, because they have a good return for their labor: If either of them falls down, one can help the other up. But pity anyone who falls and has no one to help them up. Also, if two lie down together, they will keep warm. But how can one keep warm alone? Though one may be overpowered, two can defend themselves. A cord of three strands is not quickly broken. Ecc 4:9-12

I'm going to provide a bit of scripture but also use physical science as a demonstration of God's "choice for trinity". First, we know that the trinity is certainly consistent with scriptural context:

• "Let us make man in our image" - Gen 1:26, Hebrew

• "In the beginning, God [Elohim - Hebrew plural God] created the heavens and the earth" - Genesis 1:1

• "In the beginning was the Word, and the Word was with God, and the Word was God" -John 1:1

• "Therefore go and make disciples of all nations, baptizing them in the name of the Father and of the Son and of the Holy Spirit" -Matthew 28:19

• "I am the God of Abraham, the God of Isaac, and the God of Jacob" -Matt 22:32; Ex 3:6

We see God's choice of 3 not only in scriptural doctrine but also the universe he chose to design according to his character:

• Nucleons (nuclear particles: protons + neutrons) are made of 3 quarks. Quarks are the "fundamental constituents of matter". Consider quantum chromodynamics -- quarks are described as red, green, or blue.

• Atoms are composed of 3 particles: proton (+), neutron (0), electron (-)

• Matter can be described in 3 macroscopic forms: solid, liquid, gas. (Plasma
  is effectively a gas of broken atoms)
  - E.g. Ice, Water, and Steam are all the same "thing" fundamentally.

• Father, Mother, Child: (familial) love does not exist without more than two subjects, yet is fundamentally sufficient at three. Four is not evil but there is no new role fulfilled with a second child. God is love; God is three, yet One.

• Noah's 3 sons propagated humanity in the earth.
  • Ham
  • Japheth
  • Shem

Is there any Biblical evidence as to why God "decided" to be a Trinity, as opposed to a Duo or a Quartet?

The Bible doesn't say much about transcendent nature of God and relations between Hypostases. It doesn't also say anything about God "deciding to be Trinity" or "why is God a Trinity". These are things beyoun human understanding. However God revealed himself as a Trinity (during the Baptism in Jordan or when Christ commanded to baptise in the name of the Father, and of the Son and of the Holy Spirit).

CS Lewis discusses this in Mere christianity! (excellent book) From what I remember:

There is the 'Father.' The 'I AM that I AM' The prime mover, the ultimate reality, the fountain from which all flows. He just is.

The son: Lewis talks about a thing being 'begotten not made': before our timeline begins, Jesus is 'the firstborn of all creation.' when you say 'Beget' you are saying that you are creating something like yourself, not unlike. When you 'make' something, you are creating something unlike yourself. Humans beget humans, humans make cars, televisions, etc.

Lewis argues that he can only imagine that there must only be one begotten son of God because its inconceivable to think that there be more then one son. How would you differentiate between multiple sons? Therefore, only one begotten son.

The spirit: Lewis asks: have you ever been in a meeting, or in a club where people walk away talking about the collective 'spirit' in the room? How, when united, multiple people of a like mind can form a unique atmosphere that takes the form of a distinct personality.

Lewis argues that The Holy spirit is the realization of the unity of mind and spirit (and ultimately, love) between the first 2 persons of the trinity.

(I would quote CS Lewis, but I don't have the book on me at the moment! hope this helps a bit)

We do not know. Without Divine revelation, we cannot know. We may never understand exactly "why" God is a Trinity of Persons, and the question might not make sense with respect to God.

The very fact that God is a Trinity is divinely revealed knowledge. In Catholicism (and I would guess many other denominations), it is confessed that this fact can never be known unless it is revealed. From the Catechism:

237 The Trinity is a mystery of faith in the strict sense, one of the "mysteries that are hidden in God, which can never be known unless they are revealed by God". To be sure, God has left traces of his Trinitarian being in his work of creation and in his Revelation throughout the Old Testament. But his inmost Being as Holy Trinity is a mystery that is inaccessible to reason alone or even to Israel's faith before the Incarnation of God's Son and the sending of the Holy Spirit.

Since the fact that God is a Trinity is "strictly" a mystery, the reason why God is a Trinity (if there is any reason) would also be a mystery that can never be known unless it is revealed.

The question might not even be valid with respect to God. When we ask "why is it like that?" or "why did you do that?", we are typically (in the theological context) looking for something that might make reference to the purpose of a created thing, or the purpose behind doing something or choosing to be a certain way. But God is not created (and He had no "inner need" to create anything). God did not at some point, for some reason, choose to become a Trinity. He is, was, and always will be a Trinity. Furthermore, He was in no way forced to be a Trinity. A reason (or the idea of being forced, or a choice, etc.) in the relevant sense implies something that preceded God, and there is, of course, nothing that preceded God.

It is important to remember that the Trinity as a doctrine was inductively defined. That is, the doctrine exists because it explains (most Christians would argue) the data present in the New Testament, namely that Yahweh, Jesus, and the Spirit all "are God," but that God is "one" (cf. Deut 6:4-5). In order to avoid sounding like polytheists, Christians had to come up with a way of reconciling their inductive, experiential belief that Jesus (and the Spirit) were "God," with the Second Temple Jewish ideal of monotheism.

In other words, it's tautological—which is shown in your title "Why is the Trinity a Trinity?"

As it turns out, the Trinitarian nature of God is actually an axiom.

In mathematics, an axiom is a statement that is taken to be true without a proof. For example, the statement that a straight line can be drawn between any two points is one of the five axioms that comprise Euclid's Postulates. To most of us, this statement is obviously true, but it can't be proven as true using any simpler axioms. This is because it is actually assumed to be true and then the math is carried out from there on that basis. It is indeed entirely possible to assume that some statement is actually false and then go on from there, which is how the non-Euclidean geometries (elliptic and hyperbolic) were discovered - the parallel postulate was taken to be false and it was discovered that the resulting geometric systems were self-consistent.

In this sense, the fact that God is a Trinity is true because it just is. God is uncreated and eternal, which means that He was always composed of three Persons - the Father, the Son, and the Holy Spirit. There are also no other gods, so the only god in existence is Yahweh, who is a Trinity. In effect, God couldn't have been anything else.

This idea that such a complex characteristic could be an axiom can be somewhat unsatisfying, and indeed, I'm still coming to grips with it. However, it can help to take the axiom as false and see what comes of it, just like Janos Bolyai and Nicolai Lobachevsky decided to treat the Parallel Postulate as false and see what happens.

• One Person (Unitarian) - God has no need for anything and no reason to desire anything. He is by Himself, and this is perfectly fine with Him. There is therefore no reason to create a universe with angels and men.
• Two Persons (Duonitarian?) - The two Persons would have a relationship with each other, but there would be no concept of an "another" relationship. They would be wholly satisfied with each other, so there wouldn't be any reason for them to create a universe with angels and men.
• Three Persons (Trinitarian, included for completeness) - Multiple relationships exist for each Person, which makes it possible for each Person to desire more relationships, leading to the creation of a universe with angels and men that they can form more relationships with.
• Four or more Persons ("Quadritarian" and up) - Again, multiple relationships exist, but this time, there are too many. Any Person could leave or be separated and there would still exist multiple relationships among the other Persons. Loss means less here. By contrast, in a Trinity, if any Person leaves or is separated, the remainder is a Duo, and that has already been established as insufficient.

In other words, if God was Unitarian or Duonitarian, the universe wouldn't have been created. On the other hand, four Persons or more is too many. Therefore, the Trinity is just right in that the concept of intimacy with multiple people exists and the loss of any Person (such as Jesus' death on the Cross, where He was forsaken by His Father) is a hugely significant event. Yahweh is a Trinity because the other options are insufficient. He didn't have to be a Trinity, but He is a Trinity.

Besides these three persons, no fourth in the divine nature can be asserted

Says St. Thomas Aquinas. And why? Because of the "proceeds"

The Three Persons of the Blessed Trinity are alike, except in their relation to one another

These relations of origin one must understand not as a procession which inclines to what is without - for what proceeds thus is not co-essential with its principle - one must understand them as proceeding within.

The relations are completed internally and nobody is going to add to them. Our existence, contrary to scientific opinion, is clearly just an accident (in the philosophical sense) of their love.

But the most compelling argument (I think) that St. Thomas makes is that

... in God it is not possible that there be more than two Persons proceeding

he says that if one person proceeds by way of the Intellect (the Word, i.e. the Son) or Love (the Holy Spirit) then they're cut off from the other two. And the only relationship that makes complete sense is to have one who does not proceed (God the Father), one who proceeds and is proceeded (God the Son) and one who proceeds from them both and is not proceeded (God the Holy Spirit)

The Pocket Aquinas

The Holy Spirit Proceeds from the Father and the Son

Jonathan Edwards' Ontological Argument

In his "Essay on the Trinity" (and private notebooks and public sermons), Jonathan Edwards suggested a form of ontological argument for each of the three persons of the Trinity. Anselm's argument starts from a definition of a hypothetical God who perfects all excellences and proceeds to show God must actually exist since existence is an excellence that God must have perfected. Edwards argues along the same lines for each Person of the Trinity:

And this I suppose to be that blessed Trinity that we read of in the Holy Scriptures. The Father is the Deity subsisting in the prime, un-originated and most absolute manner, or the Deity in its direct existence. The Son is the Deity generated by God's understanding, or having an idea of Himself and subsisting in that idea. The Holy Ghost is the Deity subsisting in act, or the Divine essence flowing out and breathed forth in God's Infinite love to and delight in Himself. And I believe the whole Divine essence does truly and distinctly subsist both in the Divine idea and Divine love, and that each of them are properly distinct Persons.

The Father perfects existence

Edwards did not seem to make the ontological argument with respect to the Father (though it is that Person normally called simply "God"). Instead, he follows Aristotle in considering the Father to be the prime mover or cause (in all senses) of all that exists. If you've ever played the "why" game with a toddler (Q: "Why does the sun shine in the morning?" A: "Because the Earth rotates." Q: "Why does the Earth rotate?", etc.), you know that the ultimate answer is always God. (If you don't happen to believe in some prime being such as God, the ultimate answer must be "Because that's the way things are.") Even the Son, who is conceived by the Father, and the Spirit, who proceeds from the Father, depend on the Father for existence.

The Son perfects understanding

Edwards assumes that the Father's end is to enjoy Himself and that in order to accomplish that, He must have an "image" or idea of Himself. The image must be distinct from the Father; it must be the object of God's affection. If the image is worthy of God's infinite enjoyment, it must also be perfect in representing God. Fundamentally, the Father's imagination is so powerful that He creates the Son. In fact, Proverbs 8 describes how God fathered Wisdom, which is just another way to represent the understanding of God, before the Creation.

The idea is not that different from what we mean by "a gleam in your father's eye". God contemplates Himself and the Son is that image. Unlike our imaginations, which remain safely in our minds without some sort of effort, God's understanding of Himself is made manifest. The Son existed before He was incarnated as Jesus because God has always been delighting in Himself. Edwards argues that a duplicity is necessary for God to be the object of His own delight.

The Spirit perfects action

Finally, the Spirit is the manifestation of God's own affection for Himself. Now it must be admitted that this is certainly a strange Person. But Edwards argues that this is exactly the sort of Person we see in the Bible. He concludes from 1 John 4:8, which states that God is love, that the embodiment of the love between the Father and the Son is the Spirit. We note, in passing, that Edwards affirms the filioque because the love, delight, honor, glory, and so on between the First and Second Persons is mutual.

According to Edwards, these are not feelings, but actions. The interaction between Father and Son is also perfect, so like God's understanding of Himself, it too is manifest as the Third Person. The essay notes that Paul passes on grace, peace, and mercy from the Father and Son, but never the Spirit. In addition, the Father and Son express their affection for each other, but never for the Spirit. One solution to these puzzles is to conclude that the Spirit is God's love.

Perichoresis

In order to explain why at least two Persons must exist, Edwards notes the (then) common notion that "God is infinitely happy in the enjoyment of Himself". From that idea, he deduces:

However, if God beholds Himself so as thence to have delight and joy in Himself He must become his own object. There must be a duplicity. There is God and the idea of God.

The argument is philosophical, by the way, not grammatical, as it marries the ontological argument with the conception of God's self-enjoyment. Edwards was not the first to extrapolate that the Spirit is the action of love between the Father and the Son:

If, as is properly understood, the Father is he who kisses, the Son he who is kissed, then it cannot be wrong to see in the kiss the Holy Spirit, for he is the imperturbable peace of the Father and the Son, their unshakable bond, their undivided love, their indivisible unity.—St. Bernard of Clairvaux, in Sermon 8, Sermons on the Song of Songs

Of course, these are mere illustrations of the intimacy (perichoresis) that the members of the Trinity enjoy. Going back to Paul himself, Christians have wrestled with this great mystery.

Conclusion

Jonathan Edwards was the first to point out that he did not propose to make the mystery of the Trinity unmysterious. The degree you accept that ideas perfected become "real" is the degree you are likely to accept this formulation. However, it does insist on exactly three Persons since it requires God to be object, subject, and verb of the sentence:

God delights in Himself.