Let's start by combining two strange, but true, premises:
- Georg Cantor, the first great prophet of set theory, believed (at least some of the time) that his insights into the transfinite had been provided to him by the divine nature (this is from Dauben[??], as cited in Wikipedia):
Letters (and the testimony of colleagues who knew him) reveal that Cantor believed he was chosen by God to bring the truths of set theory to a wider audience. He also regarded the successive waves of manic-depression that began to plague him in the 1880's -- peaks of intense activity followed by increasingly prolonged intervals of introspection -- as divinely inspired. Long periods of isolation in hospital provided opportunities for uninterrupted reflection during which Cantor envisioned visits from a muse whose voice reassured him of the absolute truth of set theory, whatever others might say about it. ... Elsewhere, Cantor actually described his conviction about the truth of his theory explicitly in quasi-religious terms:
My theory stands as firm as a rock; every arrow directed against it will return quickly to its archer. How do I know this? Because I have studied it from all sides for many years; because I have examined it from all sides for many years; because I have examined all objections that have ever been made against the infinite numbers, and above all because I have followed its roots, so to speak, to the first infallible cause of all created things.
- Not every theist is a dualist. For example, Peter van Inwagen is a Christian materialist.
- Conclusion: it is roughly possible to believe that I am a fully material being, that a being with absolutely infinite power can affect the contents (and even the structure) of my mind, and so that I, despite my material essence, can be "taught by God" about the "contents and structure of God's mind," to wit the transfinite things of set theory.
Now, to be sure, I don't know how e.g. van Inwagen applies his materialism to the issue of God's own nature. Tertullian, for example, thought of the Father as constituted out of His own kind of matter, with the Son and the Spirit as particular extensions of this matter. Or one might follow Newton in declaring that space is a sensorium of God, while somehow also trying to follow Descartes in declaring that space is res extensa:
... supersubstantivalism takes space as primary, and matter as secondary or derived from space (see Sklar 1974, 222). Descartes, on the contrary, takes matter or body as primary and space as a derived, abstract concept: “the same extension which constitutes the nature of body also constitutes the nature of space, and . . . these two things differ only in the way that the nature of the genus or species differs from that of the individual” (Pr II 11).
So, by making these or similar dialectical moves, one might try for a stabilization of a non-dualistic ante rem realism about mathematics. Granted, for such a theory to depend on something as ethereal as the concept of God can be quite troubling, but if we're just asking about general possibilities, then I suppose that this is one of them.
Another option is to go a neo-Kantian route of sorts. Start with what seems like in re realism, with a claim that mathematical objects are "embedded in" the transcendental will. If you want to bypass Frege's complaint about how people can't have their own copies of e.g. the number 2, each per a person's own mind, adapt this will to the image of a common agent intellect. (The universality of the laws of the will does duty for this notion in Kant anyway.) But now Kant also claimed that transcendental freedom is not really much a part of experience: its causal principle transcends that of physics. So how do we know the laws of this freedom? In the second Critique, Kant writes:
We can become conscious of pure practical laws just as we are conscious of pure theoretical principles, by attending to the necessity with which reason prescribes them and to the elimination of all empirical conditions, which it directs. The concept of a pure will arises out of the former, as that of a pure understanding arises out of the latter.
This is akin to the abstractionism mentioned by Conifold in his comment (or it's even just a version of it, I think). So now if mathematics is embedded/encoded in the seemingly in re will, which straddles the divide with the ante rem realm, and yet the will is not such as can be clearly declared to be a metaphysical substance per experience, then do we have a non-dualistic account of how ante rem mathematical knowledge is possible? Note also that Kant floated the idea of neutral monism in the first Critique:
The task of explaining the community of the soul with the body does not properly belong to the psychology of which we are here speaking; because it proposes to prove the personality of the soul apart from this communion (after death), and is therefore transcendent in the proper sense of the word, although occupying itself with an object of experience,—only in so far, however, as it ceases to be an object of experience. But a sufficient answer may be found to the question in our system. The difficulty which lies in the execution of this task consists, as is well known, in the presupposed heterogeneity of the object of the internal sense (the soul) and the objects of the external senses; inasmuch as the formal condition of the intuition of the one is time, and of that of the other space also. But if we consider that both kinds of objects do not differ internally, but only in so far as the one appears externally to the other—consequently, that what lies at the basis of phenomena, as a thing in itself, may not be heterogeneous; this difficulty disappears. There then remains no other difficulty than is to be found in the question—how a community of substances is possible; a question which lies out of the region of psychology, and which the reader, after what in our analytic has been said of primitive forces and faculties, will easily judge to be also beyond the region of human cognition.
