The Trinity is not a mere doctrine, but who God eternally is. God is the Trinity. We love and worship the Trinity. The Trinity is not something God has, but who God is. My definition of the Trinity is a modified version of the one given in James R. White’s book The Forgotten Trinity: The one true God exists eternally as three coequal and coeternal persons, namely, the Father, the Son, and the Holy Spirit who share equally and indivisibly the one divine nature or attributes of God and are distinguished by various personal properties.
The three pillars of God as a Trinity of persons are: 1. Monotheism: there is only one God and object of worship who is simple or indivisible; 2. Distinction: the one true God eternally exists as three distinct persons with distinct personal properties; 3. Equality: the three persons of the Trinity share equally the one divine nature or attributes of God so that the Father is God, the Son is God, and the Holy Spirit is God. But the Father is not the Son or the Holy Spirit and the Son is not the Father or the Holy Spirit. To prove the doctrine of the Trinity, we must demonstrate that there is only one God, prove that the Father, the Son, and the Holy Spirit are distinct from each other (but not separated from one another), and prove that the Father is God, the Son is God, and the Holy Spirit is God.
Monotheism is the belief that there is only one God. The word is derived from the Greek words monos “only” and theos “God.” The opposite of monotheism is polytheism or the belief that there are many gods. This one God is the creator of all things and is distinct from creation. Because there is only one God, he must be the sole object of our worship. Because God is simple or indivisible, the three persons of the Trinity are not “parts” of God, but three distinct persons who cannot be separated from one another. Christian monotheism is trinitarian rather than unitarian.
Trinitarians affirm that Jesus is God. He is the eternal Son of God who existed with the Father from all eternity and became human in the incarnation while remaining fully God. He is both fully God and fully man at the same time in one person. Jesus is not a lesser god or a created being, but the creator of all things and deserving of worship, honor, and prayer. He is equal in nature, being, attributes, and person to the Father. He came to live and die among us, “that all may honor the Son, just as they honor the Father. Whoever does not honor the Son does not honor the Father who sent him” (John 5:23). Only one who is fully God could satisfy the infinite justice of God on our behalf. Only a sacrifice of infinite worth could satisfy infinite wrath which results in eternal punishment. And only one who is fully man could die on our behalf since it was a man who sinned against God. Because we have the righteousness of Christ in justification, we have the righteousness of God (1 Cor 1:30; 2 Cor 5:21).
The Holy Spirit is God equal with the Father and the Son. He is the one through whom the Father and the Son carry out their work in the world. He is personal in nature rather than an impersonal force. He speaks, teaches, knows all things, can be sinned against and lied to, can be grieved, bears witness, creates assurance, brings regeneration, intercedes for, glorifies Christ, loves, and created all things. I have written about the deity and personality of the Holy Spirit in previous articles.
In contrast to trinitarianism are polytheism (many gods), modalism (the Father becomes the Son and then becomes the Holy Spirit), Arianism (the Son is not God), and binitarianism (the Holy Spirit is not God while the Son is). All human analogies fail when attempting to describe the Trinity. The analogy of ice turning into water and then into air is modalism, not trinitarianism. The analogy of a human father being both a father and the son of someone else is also modalism. Using a pretzel to depict the three persons of the Trinity is simply dishonoring to who God is. The Father, the Son, and the Holy Spirit are distinct persons, not holes in a piece of bread. Many evangelicals have a view of God that is functionally modalistic rather than trinitarian. I have never met a member of a church who was a polytheist or an Arian, but I have encountered more than one person who had a modalistic understanding of God.
For a helpful article on the Trinity, see this article by Matt Perman.