Short answer
Three cannot be one in the way the Trinity needs, and its own creed shows why. The Athanasian Creed, an old Christian statement of belief, says the Father is God, the Son is God and the Holy Spirit is God. It also says the Father is one person and the Son another, and there are not three Gods but one. Everything turns on what "is God" means. If it means "is the one God", the Father and the Son are the same. Two men who are each the town's only mayor are one man. The creed denies that. If it means "has God's nature", three who have it are three Gods. Three who have human nature are three humans. The creed denies that too. Either way, one of its own lines is broken. The Qur'an's answer is simple: God (﷾) is One, with none like Him.
Full explanation
What the creed says
The Athanasian Creed is an old Christian statement of belief. It was written in Latin in about the 5th or 6th century, and the Western churches have used it ever since. The English words shown here come from the Book of Common Prayer, the Church of England's prayer book. Philip Schaff, a church historian, revised them in 1877. The creed says the Father is God, the Son is God and the Holy Ghost (the Holy Spirit) is God. Yet there are not three Gods, but one God (﷾). It calls the Father, the Son and the Spirit three "persons". The three persons must not be mixed together. The Father is one person, the Son another, the Spirit another. And each person, taken by himself, is God.
“So the Father is God: the Son is God: and the Holy Ghost is God. And yet they are not three Gods: but one God.”
“And the Catholic Faith is this: That we worship one God in Trinity, and Trinity in Unity; Neither confounding the Persons: nor dividing the Substance [Essence]. For there is one Person of the Father: another of the Son: and another of the Holy Ghost.”
“For like as we are compelled by the Christian verity: to acknowledge every Person by himself to be God and Lord: So are we forbidden by the Catholic Religion: to say, There be [are] three Gods, or three Lords.”
The test: what does "is God" mean?
Take the line "the Father is God". It can only mean one of two things. First, it can mean "is the one God", as "Tom is the mayor" means Tom is the town's one mayor. If Tom is the mayor, and Dick is the mayor, and there is one mayor, then Tom is Dick. In the same way, if the Father is the one God and the Son is the one God, the Father is the Son. The creed denies this. It even says the Father is not "begotten", that is, not coming from anyone as a son, while the Son is begotten. One and the same thing cannot be both. Second, "is God" can mean "has the nature of God": everything that makes God God. Christians call this the "divine nature". It works as "Tom is human" means Tom has human nature. Tom, Dick and Harry each have human nature, so they are three humans. In the same way, three persons who each fully have the divine nature are three Gods. The creed denies this too. Some reply that the divine nature is not a kind that many can share, like humanity, but one single thing. Then each person is that one thing, and we are back at the first meaning: the Father is the Son. Whichever meaning a Christian picks, another line of the creed is broken. The Bible's own evidence on the doctrine is set out on the page "Does the Trinity make sense?".
“So the Father is God: the Son is God: and the Holy Ghost is God. And yet they are not three Gods: but one God.”
“And the Catholic Faith is this: That we worship one God in Trinity, and Trinity in Unity; Neither confounding the Persons: nor dividing the Substance [Essence]. For there is one Person of the Father: another of the Son: and another of the Holy Ghost.”
“The Father is made of none: neither created, nor begotten. The Son is of the Father alone: not made, nor created: but begotten.”
The churches closed both ways out
The churches themselves reject both easy escapes. Three separate beings, like Abraham, Isaac and Jacob, would be tritheism, belief in three gods. One God (﷾) showing three faces, or modes of appearing, would be Sabellianism, named after its early teacher Sabellius. The Council of Constantinople, a meeting of church bishops in 381, condemned the Sabellians by name. The creed also forbids "confounding the Persons", that is, mixing them up. Philip Schaff, explaining the creed, rules out both. What is left, he says, is a real and eternal distinction, but an "ineffable" one: it cannot be put into words.
Schaff's note on clause 4 says that in modern usage a person means a separate and distinct individual. He says the three persons of the Trinity are not three beings, like Abraham, Isaac and Jacob, for that would be tritheism. Nor are they merely three aspects or modes of the one God, in the Sabellian sense. He calls what remains a real, objective and eternal, though "ineffable", distinction within the one divine being.
The council confirms the faith of Nicaea and anathematises every heresy, naming among them the Sabellians.
“And the Catholic Faith is this: That we worship one God in Trinity, and Trinity in Unity; Neither confounding the Persons: nor dividing the Substance [Essence]. For there is one Person of the Father: another of the Son: and another of the Holy Ghost.”
The simple answer: God (﷾) is One
The Qur'an tells Christians not to say "Trinity", for Allah (﷾) is one God. It calls the claim that Allah (﷾) is one of three a blasphemy. Surat al-Ikhlas, chapter 112 of the Qur'an, describes God (﷾) in four short lines. God (﷾) is One, Eternal and Absolute, neither begetting nor begotten, with none like Him. This belief needs no escape routes. It is also the first commandment as Jesus taught it: "The Lord our God is one Lord". And Jesus called the Father "the only true God".
“O People of the Book! Commit no excesses in your religion: Nor say of Allah aught but the truth. Christ Jesus the son of Mary was (no more than) a messenger of Allah, and His Word, which He bestowed on Mary, and a spirit proceeding from Him: so believe in Allah and His messengers. Say not "Trinity": desist: it will be better for you: for Allah is one Allah: Glory be to Him: (far exalted is He) above having a son. To Him belong all things in the heavens and on earth. And enough is Allah as a Disposer of affairs.”
“They do blaspheme who say: Allah is one of three in a Trinity: for there is no god except One Allah. If they desist not from their word (of blasphemy), verily a grievous penalty will befall the blasphemers among them.”
“Say: He is Allah, the One and Only; Allah, the Eternal, Absolute; He begetteth not, nor is He begotten; And there is none like unto Him.”
“And Jesus answered him, The first of all the commandments is, Hear, O Israel; The Lord our God is one Lord:”
“And this is life eternal, that they might know thee the only true God, and Jesus Christ, whom thou hast sent.”
Replies Christians make
Athanasian Creed — clauses 3-5
“And the Catholic Faith is this: That we worship one God in Trinity, and Trinity in Unity; Neither confounding the Persons: nor dividing the Substance [Essence]. For there is one Person of the Father: another of the Son: and another of the Holy Ghost.”
Cited for
God is one in being and three in persons. "One" counts the being and "three" counts the persons, so there is no clash. In short: one "what", three "whos".
The response
Then ask what each "who" has. Each fully has the one divine nature, and that is exactly what "is God" means in "the Father is God". Three whos who each fully have human nature are three humans. Some reply that the divine nature is not a kind that many can share, like humanity, but one single thing. Then each person is that one thing, and the Father is the Son: the first meaning again. Gregory of Nyssa, a 4th-century bishop who defended the doctrine, was asked this very question about Peter, James and John. He held that the three men share one human nature, as many gold coins share one gold. So, he said, calling them "three men" is a misuse of words. By his rule they should strictly be called one man, yet they are plainly three. He also argued that the three divine persons never act apart. Every work comes from the Father, through the Son, and is perfected in the Spirit. But even in his picture each has his own part: from the Father, through the Son, in the Spirit. One work done by three is still done by three. And the creed says each person "by himself" is God.
Where it lands — Three persons who each fully have the divine nature are three Gods, as three people are three humans. If the nature is one thing that each person is, the Father is the Son.
Athanasian Creed — clauses 15-16
“So the Father is God: the Son is God: and the Holy Ghost is God. And yet they are not three Gods: but one God.”
Cited for
The Father and the Son are the same God, but not the same person. "Same" always means the same something, so there is no clash. Nothing says they must be simply the same thing underneath.
The response
Then count the things that are there. If there is one thing, and the Father is it and the Son is it, the Father is the Son. If there is more than one, there is more than one God (﷾). Saying "the same God, but not the same" does not answer the count. It only moves the words. And the creed marks a real difference. The Father is not begotten, that is, not coming from anyone as a son. The Son is begotten. Two who differ like that are two. If two can still be "the same God", then "God" is shared the way "human" is shared, and that means three Gods.
Where it lands — Being "the same God" without being the same thing leaves the count unanswered, so the test stands.
First Vatican Council, Dogmatic Constitution Dei Filius (24 April 1870) — chapter 4, On Faith and Reason
“But although faith is above reason, there can never be any real discrepancy between faith and reason, since the same God who reveals mysteries and infuses faith has bestowed the light of reason on the human mind; and God can not deny himself, nor can truth ever contradict truth. The false appearance of such a contradiction is mainly due, either to the dogmas of faith not having been understood and expounded according to the mind of the Church, or to the inventions of opinion having been taken for the verdicts of reason.”
Cited for
The Trinity is a mystery revealed by God. It is above reason but never against it, so any clash is only apparent. Our minds are too small to see how the lines fit.
The response
A mystery and a contradiction are different things. A mystery is something we cannot fully understand, such as how God (﷾) knows all things. A contradiction says yes and no to the same claim at once. The First Vatican Council, a meeting of Catholic bishops, taught in 1870 that truth can never contradict truth. It said a seeming clash mainly comes from misreading the church's official teaching, or from taking opinion for reason. But the test uses the creed's own words. The creed itself reasons step by step: "we are compelled" to call each person God (﷾), "so are we forbidden" to say three Gods. On either meaning of "is God", one of its lines breaks.
Where it lands — Calling a contradiction a mystery does not remove it, and the churches themselves say truth cannot contradict truth.
Athanasian Creed — clauses 15-16
“So the Father is God: the Son is God: and the Holy Ghost is God. And yet they are not three Gods: but one God.”
Cited for
God is like water, which can be ice, liquid and steam. Or He is like one man who is a father, a son and a husband at once.
The response
Both pictures show one thing in three forms or roles. That is Sabellianism, also called modalism, from "mode", a way of appearing: one God (﷾) wearing three masks. The churches condemned it. The creed forbids "confounding the Persons", that is, mixing them up. The Council of Constantinople in 381 named the Sabellians among the heresies, the false teachings it condemned. Even at the one temperature and pressure where ice, water and steam exist together, each is a different part of the water. That brings in a different problem instead: parts.
Where it lands — The water and the three roles teach a doctrine the churches themselves condemned.
Athanasian Creed — clauses 19-20
“For like as we are compelled by the Christian verity: to acknowledge every Person by himself to be God and Lord: So are we forbidden by the Catholic Religion: to say, There be [are] three Gods, or three Lords.”
Cited for
God is like a shamrock: three leaves, one plant.
The response
The leaves are parts, and no single leaf is the whole shamrock. But the creed says each person "by himself" is God. It also forbids "dividing the Substance", that is, splitting up God's (﷾) one being. A God made of three parts would make the Father only a third of God (﷾).
Where it lands — The shamrock makes the persons parts of God, which the creed denies.
Athanasian Creed — clauses 15-16
“So the Father is God: the Son is God: and the Holy Ghost is God. And yet they are not three Gods: but one God.”
Cited for
God is beyond numbers. Counting belongs to created things, so God cannot be counted as one or three.
The response
The creed counts first. It says there is "one Person of the Father: another of the Son: and another of the Holy Ghost". It says there are "not three Gods: but one God". If numbers do not apply to God (﷾), both of those lines fall, and the doctrine falls with them. Islam has no such problem: it counts one God (﷾) and no persons within Him.
Where it lands — The Trinity is itself a statement about numbers, so it cannot step outside counting.
Athanasian Creed — clauses 3-5
“And the Catholic Faith is this: That we worship one God in Trinity, and Trinity in Unity; Neither confounding the Persons: nor dividing the Substance [Essence]. For there is one Person of the Father: another of the Son: and another of the Holy Ghost.”
Cited for
The Trinity is not 1 + 1 + 1 = 3. It is 1 × 1 × 1 = 1.
The response
Multiplying is arithmetic on paper, not a way of counting who is there. Three people in a room are three, whatever sum you write. The same sum would also turn three persons into one person. The creed forbids that: the Father is one person, the Son another, the Spirit another.
Where it lands — Choosing a sum that gives the answer one does not change the count, and the creed itself counts three persons.
Perspectives
Mainstream Christian
God is one being in three persons, the Father, the Son and the Holy Spirit. Each person is fully God, yet there is only one God. The doctrine is a mystery above reason, but not against it.
The Athanasian Creed says the Father, the Son and the Holy Ghost are each God. Yet, it says, there are not three Gods but one.
Rebuttal
The creed does say this, but it cannot make both halves true. If "is God" means "is the one God", the Father is the Son, which the creed denies. If it means "has the divine nature", there are three Gods, which the creed also denies.
God is one in being and three in persons: one "what" and three "whos". So "one" and "three" do not count the same thing.
Rebuttal
Each "who" fully has the divine nature, so three whos are three Gods, as three whos with human nature are three humans. If instead that nature is one single thing that each person is, the Father is the Son. Gregory of Nyssa, a 4th-century bishop who defended the doctrine, faced this very case. He answered that even "three men" is a misuse of words. By his rule Peter, James and John would be one man, yet they are plainly three.
The Trinity is a mystery revealed by God: above reason, but never against it.
Rebuttal
A mystery is something we cannot fully understand; a contradiction says yes and no to the same claim. The First Vatican Council, a meeting of Catholic bishops in 1870, teaches that truth can never contradict truth. It says a seeming clash comes mainly from misreading the church's teaching, or from taking opinion for reason. But the test uses the creed's own words, and on either reading one of its lines breaks.
The Islamic perspective
God (﷾) is One in every sense. The creed's lines cannot all be true together, and both the Qur'an and Jesus' own words teach a oneness with no three in it.
Take either meaning of "is God", and a line of the creed breaks. Either the Father is the Son, or there are three Gods.
The churches themselves condemn both ways out: three beings is tritheism, and one person in three modes is Sabellianism.
The Qur'an forbids saying "Trinity" and teaches that there is no god except One Allah (﷾).
Surat al-Ikhlas describes God (﷾) as One, neither begetting nor begotten, with none like Him.
Jesus called "The Lord our God is one Lord" the first of all the commandments. He called the Father "the only true God".