Let’s begin with some definitions of “mystery” taken from reference works in traditional Catholic theology. From the entry for “mystery” in Pietro Parente, Antonio Piolanti, and Salvatore Garofalo, Dictionary of Dogmatic Theology:
In the past century, the Church magisterium has fixed definitively the meaning of the term (Gregory XVI, Pius IX, Leo XIII). The Vatican Council (sess. 3, c. 4, DB, 1795 ff.) gives the definition: “The divine mysteries of their very nature so transcend the created intellect that, even when revealed and believed, they still remain veiled and obscure during this mortal life.”
In the strict sense, therefore, a mystery is a truth, whose existence can be known by human reason only by way of revelation, while its essence cannot be properly and fully understood, even after revelation. Thus, e.g., the mystery of the Holy Trinity… Human reason cannot demonstrate a mystery taken in the strict sense, but can illustrate it and defend it from objections.
And here’s a definition from Bernard Wuellner, Dictionary of Scholastic Philosophy:
mystery, n. 1. a hidden truth 2. something of which the fact is known, but the reason of the fact or its harmony with other facts and truths is not understood.
strict mystery, a truth so far exceeding the capacities of human reason that its full meaning cannot be comprehended by us nor a natural proof of its truth be discovered even after God has revealed that truth to men.
If you look at Dale Tuggy’s article “Trinity” from the Stanford Encyclopedia of Philosophy, you’ll see that he there distinguishes between “negative mysterianism” and “positive mysterianism.” The definitions just cited seem to fall into the “negative mysterian” category, and in the post of his I linked to previously, Dale himself rightly characterized my own view as a kind of negative mysterianism. (I do think Dale’s characterization of negative mysterianism needs some qualification, though. For example, he says that the view is that the Trinity “is not understandable because it is too poor in intelligible content.” But strictly speaking, what the negative mysterian holds is that the doctrine has content that is perfectly intelligible in itself – a sufficiently powerful intellect could fully grasp it – but that we cannot fully grasp it given the limitations on our intellects. No doubt Dale realizes this, but it is an important point to emphasize so as to avoid attributing to the negative mysterian the dubious view that there could be such a thing as a truth that was unintelligible in itself, whatever that could mean.)
As I’ve indicated before, Catholic theology regards “mysterianism” as an essential part of orthodox thinking about the Trinity, not some optional approach a Catholic theologian may or may not take as he sees fit. Aquinas holds that “trying to prove the Trinity by reason would injure the faith” (Summa Theologiae I.32.1, McDermott translation). And the Church herself has more or less said the same thing.
There are in Catholic theology several grades of theological certainty and several corresponding degrees of censure of theological error, which you’ll find described in old manuals like Ludwig Ott’s Fundamentals of Catholic Dogma. The highest degree of certainty attaches to propositions which have been directly divinely revealed, and to those which have been infallibly proclaimed by the Church. These are de fide, absolutely essential to the Catholic Faith; to deny them is to be guilty of heresy in the strict sense. Next come propositions which are “proximate to Faith,” sententia fidei proxima, i.e. propositions which are generally regarded as having the status of divine revelation but which have not been formally proclaimed as such by the Church. To deny them is to be “proximate to heresy” – slightly short of heresy proper but something no orthodox theologian would let himself get anywhere close to doing. Then there are propositions considered “theologically certain” – those which also have not been formally proclaimed by the Church, but which have such a tight logical connection with revealed truths that they cannot be doubted. Denials of such propositions might be classified as “suspect of heresy” or at least “erroneous,” and again, an orthodox theologian will steer clear of such denials. Next come “common teachings” or propositions that are generally accepted by orthodox theologians even though differing opinions about them are in principle legitimate; probable opinions; well-founded opinions; pious opinions (i.e. those that are at least consonant with the Church’s general way of thinking); and tolerated opinions.
Now, the doctrine of the Trinity itself is de fide; to deny it is strictly heretical. Many specific claims about the relations between the Persons are also de fide; others are considered at least theologically certain. (See pp. 50-75 of Ott for a useful summary.) That God’s nature is not fully comprehensible by us is also de fide. (Ott, p. 20) Moreover, the first Vatican Council declared: “If anyone shall have said that no true mysteries properly so-called are contained in divine revelation, but that all the dogmas of faith can be understood and proved from natural principles, through reason properly cultivated: let him be anathema” (Denzinger, Sources of Catholic Dogma n. 1816) and that “reason… is never capable of perceiving those mysteries in the way it does the truths which constitute its own proper object. For, divine mysteries by their nature exceed the created intellect so much that, even when handed down by revelation and accepted by faith, they nevertheless remain covered by the veil of faith itself, and wrapped in a certain mist, as it were,” at least in this life (n. 1796). But the Trinity has always been considered the chief of the mysteries of the Faith. Hence that the Trinity is a mystery, something knowable only through divine revelation, is considered in Catholic theology a proposition that is “proximate to Faith,” and its denial “proximate to heresy.” (Cf. Ott, p. 74) At the same time, Vatican I also teaches that “although faith is above reason, nevertheless, between faith and reason no true dissension can ever exist, since the same God, who reveals mysteries and infuses faith, has bestowed on the human soul the light of reason; moreover, God cannot deny Himself, nor ever contradict truth with truth. But, a vain appearance of such a contradiction arises chiefly from this, that either the dogmas of faith have not been understood and interpreted according to the mind of the Church, or deceitful opinions are considered as the determinations of reason.” (Denzinger n. 1797) If a formulation of the doctrine of the Trinity appears contradictory, then, it cannot be regarded as a genuine expression of the Church’s teaching on the matter.
So, “mysterianism” concerning the Trinity – and, it would seem, “negative mysterianism” specifically – is taken to have a very high degree of certainty in Catholic theology. It is just short of de fide, and denying it is “proximate to heresy.” Thus, it is non-negotiable.