Is the Christian concept of the Trinity influenced by Philo's philosophy?

Although Philo's philosophy may have some similarities to the doctrine of the Trinity (there are other like examples), it did not contribute to this doctrine, as even Moses and Abraham "implicitly and under a veil" knew of the Trinity.

Answer

Original question: Was the Christian concept of the trinity inspired by Philo's philosophy?

No, the Christian concept of the Trinity is not inspired by Philo's philosophy. It seems most probable that Philo based some of his writings on the Trinity from his knowledge of Jesus and Holy Spirit in the events of the Gospels, some writings as a philosophical expansion on the previous renderings of the Logos, and some writings based on the Old Testament, as the word Logos in the Septuagint appears quite often. See this link. Additionally, you asked

How can it be shown that Christian theology was not influenced by Philo's philosophy?

"Influenced by" is quite distinct that "inspired by," unless you meant these words to be synonymous. "Inspired by" generally entails being the source of something, vs an "influence" just affects something else in any way, shape, form, or extent. I would say that Christian theology is/was influenced by Philo, probably even a much smaller extent compared to Arian. So the exact answer would be I don't think it can be shown, because it seems probable that Christian theology was influenced by Philo, though not inspired by.

Discussion

The exact answer to your question depends on what exactly you mean by the "Christian concept of the Trinity." If you are referring to the creeds formulated at Nicaea, then it is probable that those were affected by Philo's heterodox teachings. If you are referring to the doctrine of the Trinity as believed by ante-Nicene church fathers, then it is also probable that Philo's writings affected them, since ante-Nicene includes all of 1st century up until 325 AD. If you are referring to the doctrine of the Trinity itself, then no...see below.

An important element of this question hinges on the dating of Philo's works, since he lived 20 BC to AD 50. If the Christian concept of the Trinity is inspired by Philo's writings, then his relevant writings should be prior to when the Christian concept of the Trinity was established. I have unsuccessfully attempted to find any date given to Philo's writings, of which the ones including the Trinity and the Logos being the most relevant and central to this question. Philo spent a large portion of his writings and his life on Genesis and the Pentateuch rather than the Trinity.

The concept of the logos is familiar to Greek philosophy prior to Jesus' time and starting in the 5th century BC with Heraclitus, though with somewhat of an inconsistent and transient usage. Aristotle used logos in combination with ethos and pathos in his writings on persuasion. Read more here. Philo's discussion of the Logos could be a purely philosophically-based expansion on the previous philosophical/theological renderings of Logos, as the Logos had already had theological implications at least with the Stoics in the 4th century BC.

The New Testament writings were most likely all written after the death of Philo in AD 50; however, the events of the Gospels all happened by AD 30. The earliest church father I am aware of besides the New Testament writers, Ignatius of Antioch, was born in AD 35. I doubt he would have been able to write a treatise on Jesus as the Logos before Philo did. I don't know if we can know how the apostles applied the term Logos to Jesus before the Gospel accounts. In Eusebius' Ecclesiastical History, Peter and Philo supposedly met, which is relevant. It is also possible that Philo was familiar with the events occurring with Jesus, including the Holy Spirit, Pentecost, "breathing" the Holy Spirit on the disciples, etc., which explains some of Philo's writings on the Holy Spirit (pneumatology) and the Trinity. He probably (possibly?) would have been familiar with what the Christians taught and believed about Jesus, so that most likely would have affected his writings.

You could prove me false by demonstrating that Philo's writings on the Trinity pre-date the events of the Gospels (before ~AD 25). It is difficult to otherwise separate what Philo knew of Christian beliefs at the time of his writings. Jesus was born around 4-6 BC, so his ministry started around AD ~25-27, so if Philo wrote extensively on the Trinity (especially and specifically pneumatology) before the events described in the Gospels, then I would be wrong. Otherwise, I would think it is the Gospel events (not the books) that are Philo's primary basis for his doctrine. To have such sophisticated work on the Trinity would be extremely surprising based on the Old Testament alone.

Conclusion

In short, this is a great and difficult question; I don't think we have the data to come to a solid conclusion on this. And this was a terrible answer but just wanted to give you some information for your consideration in finding an answer to this question. Summary: I think Philo based his writings on the Logos from the Old Testament and philosophy, but his writings on pneumatology the events contained in the Gospels that he would have heard about.

Edit: This link (also linked in the Answer section) helped me see you could have some substantive writings on the Logos from the Old Testament alone, especially with a philosophical background because Logos appears a decent amount in the Septuagint, the Greek translation of the Hebrew Scriptures (Old Testament).

Therefore, it seems that Philo bases his ideas of the Logos on philosophy + Old Testament considering that he does quote it, see this answer, but the ideas of pneumatology on Gospel events since those seem to be very specifically from Jesus' ministry and teachings, and probably some overlap between the two. Jesus often referred to the Old Testament to reveal the Trinity, so Philo could have been aware of that.

P.S. if someone can give me dates for any/all of Philo's writings, that would be wonderful.

Philo was a Jewish scholar, possibly born about 20 years before Jesus, so he based his philosophical interpretations about God upon the existing Hebrew scriptures. He may have been dead before the Christian Greek scriptures began to be written.

The real question that arises is, "Was Philo's concept of God based upon already existing Hindu concepts of Trimurti?" The crucial point about such a pagan idea of God is that it requires three gods, not one, which is a triad. But a triad is not a trinity. It is also (like Philo's system of hermeneutics) a philosophical concept. Philo would be learned in many pagan belief systems. Check out the similarities between Philo's ideas (detailed in the question) and the Hindu ones as shown in this encyclopedia:

"Hindus have developed the lofty philosophical concept of Brahman as the One and the All, yet the visible manifestations of the Divine in the form of different gods and goddesses number millions... Sometimes the three gods Brahma (a personification of the impersonal Brahman), Vishnu and Shiva are grouped together in a triad called Trimurti, occasionally although wrongly described as a 'Hindu trinity'. In this representation of 'One God in three forms' Brahma is the creator, Vishnu the preserver and Shiva the destroyer. Well known from Indian sculpture, the triad has largely remained artificial and is unimportant to Hindu worship, which is much more strongly founded on the notion of istadevata, the idea that an individual or a family chooses a specific god as the main focus for devotion. Trimurti is worshipped in the form of the guru principle called Dattatrya." (Encyclopedia of World Faiths, p193, McDonald Orbis, 1987)

The point of detailing this is to flag up the dangers (for Christians) of becoming philosophical in their approach to hemeneutics. Philo was a Jewish philosopher and not a Christian. And, yes, his religious philosphising did come to affect some Christian thinking as time went by. Christians came to grasp the enormity of just who Jesus really was after his resurrection from the dead, and they came to experience the power of the Holy Spirit forty days later, and from thereon in. Their understanding was based on revelation, not philosophy, which is why the scriptures written after those two events provide direction (in conjunction with the Hebrew scriptures) that Christians should go by, to worship the one God "in spirit and in truth".

Philo's approach to the Hebrew scriptures is detailed here:

"One major difference between Alexandrian and Antiochene theologies revolved around hermeneutics (biblical interpretation). The Alexandrian pattern had been established in the time of Christ by the Jewish theologian and scholar Philo, who believed that the literal and historical references of the Hebrew Scriptures were of the least importance. He sought to discover and explicate the biblical narratives' allegorical or spiritual meaning. In other words, many passages of the Hebrew Bible seemed to be about one thing, while - according to Philo - they really referred to something else. As a person seeking to integrate Hebrew religion with Greek (especially Platonic) philosophy, Philo could not take literally much of what he read in the Prophets. He believed that through allegorical interpretation he could demonstrate the underlying unity of Greek ethical and philosophical thinking with Hebrew religion. Many early Christian thinkers borrowed their hermeneutical strategies from Philo, and that was nowhere truer than in Alexandria itself. Clement and Origen both sought to dig down through the various layers of meaning in the Bible to discover its hidden gems of spiritual truth separated from the crude literal and historical narratives and images...

Thus the Alexandrian and Antiochene theologies diverged at their very roots - biblical interpretation. The Antiochene historical-literal-grammatical method is the more influential one in modern, Western Christianity, whereas the Alexandrian allegorical-spiritual method tended to dominate much early Christian thinking and remained a powerful influence throughout the Middle Ages in both the East and the West...

Docetism - denial of Christ's true humanity - lurked in the background of Alexandrian theology and was a constant danger in both its biblical scholarship and its Christology." (The Story of Christian Theology, pp.203-4, Roger E. Olson, Apollos. Emphasis mine.)

The answer to your question is that although Philo's philosophy did influence later Christian thinking in various respects, the Bible alone provides the basis for the Christian Trinity doctrine. It is only when people start philosophising about the nature of God, Christ, and the Holy Spirit that corrupting pagan influences creep in. The whole battle of the early Church about the Trinity doctrine was against such influences (e.g. resisting Docetism, and Christ being created). This is why any serious discussion about it should be based entirely on what the Bible states and reveals, with non-Christian philosophers kept strictly out of it, because they have never had the glory of God revealed to them - they're too full of their own thinking and that of pagan or atheistic ideas for the Holy Spirit to influence them.

No wonder the Christian Greek scriptures warn that "the world by [its] wisdom knew not God" and "the wisdom of this world is foolishness with God" (1 Cor.1:21 & 3:19). "Beware lest any man spoil you through philosophy and vain deceit, after the tradition of men, after the rudiments of the world, and not after Christ" (Col. 2:8). Paul could have been speaking of Philo.

From the introduction of The Trinity, Fr. Réginald Garrigou-Lagrange, O.P., gives the following objection and responds to it:

The five following truths, then, are announced in the Prologue of St. John's Gospel: the Son of God is

1. distinct from the Father,
2. equal and consubstantial with the Father,
3. the Creator,
4. the author of both the natural and the supernatural life,
5. the Redeemer and the author of salvation.

In this way the divinity of the Word is proclaimed.

Objection. The rationalists and liberals say that this doctrine of the Word apparently stems from Philo, an Alexandrian Jew, born about 20 B. C., who tried to conciliate the monotheism of the Jews with the Neoplatonism in vogue at the time in Alexandria. Relying on the Old Testament, Philo admitted the existence of one personal God, the Provider, but in accord with the Greek philosophers of Alexandria he held that the most high God could not produce this finite world except through some intermediate being [Demiurge], which he called the λόγος (logos). As a Jew, Philo tried to reconcile two contradictory teachings, namely, monotheism and free creation with the pantheistic doctrine of necessary emanation. Thus, when he considers the λόγος under the Neoplatonic aspect he speaks of him as an intermediate being, but when he considers the λόγος in the light of the New Testament and Jewish monotheism he speaks of him as a divine attribute.

Reply. The Catholic reply to this difficulty is the following. A great difference exists between the λόγος of Philo and the Λόγος ["Word"] of St. John [ch. 1]. The Λόγος of St. John is neither a being beneath God nor a divine attribute, but He is properly the Son of God the Father, at the same time God, the Creator, and the Redeemer in the strict sense. Philo's λόγος, however, is in no way the Redeemer. St. John's teaching, therefore, is not derived from Philo, but from Christ's preaching, as explained by him, and as understood by the other apostles, as we see in the preaching of St. Peter and in the epistles of St. Paul. St. John could have found an adumbration of this mystery in the Old Testament, especially in the Book of Wisdom, "for she is a vapor of the power of God, and a certain pure emanation of the glory of the almighty God: and therefore no defiled thing cometh into her. For she is the brightness of eternal light, and the unspotted mirror of God's majesty" (7:25 f.).

As to the word "Λόγος" itself, St. John could have taken it from revelation, but it would not be derogatory to admit, as many do, that he derived it directly from Philo, for when the Evangelist was writing in Ephesus, Apollo was preaching there, and Apollo was widely versed in Alexandrian philosophy. Quite probably also the earliest heretics misused the word "λόγος" to designate a being midway between God and the world. St. John may have used the term to correct the current false interpretation, when he said, "The Word [Λόγος] was God." [John 1:1]