I want to address something with EA, but I'm between classes and I'm afraid I'll forget to
So, can the existence of God be philosophically demonstrated? If God's essence is His existence, and His essence remains in principle philosophically unknowable to us, how could it be demonstrated? In fact, Aquinas claims that it can be demonstrated that there is a god, and that there is only one god. That God's essence remains in principle philosophically unknowable to us is the basis for Aquinas' denial that the existence of God can be demonstrated a priori. And any reliance upon knowledge of the essence that is only known to us by faith would by that fact cease to be properly philosophical. However, we have seen that Aquinas relies upon the distinction between nominal definitions of terms and essential definitions of the subjects referred to by those terms. To demonstrate the existence of a god one may use nominal definitions that appeal to a god as the cause of various phenomena. This is to argue a posteriori. The appeal to these nominal definitions forms the basis for Aquinas' Five Ways (Summa Theologiae, Ia.2.3) all of which end with some claim about how the term ‘god’ is used.
Again, some will claim that Aquinas isn't really interested in proving the existence of God in these Five Ways. After all, he already knows the existence of God by faith, and he is writing a theological work for beginners. What need is there of proving the existence of something he already knows exists? The Ways are very sketchy, and don't even necessarily conclude to a single being, much less God or the Christian God. In addition, Aquinas claims that God's essence is his existence and that we cannot know His essence, so we cannot know His existence. Aquinas must really intend the Five Ways as less than proofs; they are more like incomplete propaedeutic considerations for thinking adequately about God in Sacred Theology. In effect, Aquinas doesn't think philosophy can in fact demonstrate the existence of God.
But as elsewhere these claims are ambiguous and suffer at the hands of Aquinas' own texts. In the first place, the objection that he already knows by faith that God exists has some merit in it, if we understand it as directed at a reading of Aquinas that would have him attempting a foundational enterprise of grounding religious faith in what is rationally demonstrable by philosophy. But that reading is anachronistic, and does not attend to the context of Summa Theologiae. There is no reason to think that Aquinas thinks the proofs are necessary for the rationality of religious faith. They are part of the enterprise of showing that Sacra Doctrina meets the condition of a science as described by Aristotle in the Posterior Analytics, an issue that is different from the question of the broad rationality of religious faith.
In addition, the objections end up denying what Aquinas writes immediately before the Five Ways—that the existence of a god is “demonstrable.” (Summa Theologiae, Ia.2.2) And his introduction of the Five Ways begins by saying that the existence of a god can be “proved” in Five Ways. To counter the objection that he must mean something informal here by “demonstrate” and “prove”, one need only recognize the explicit use of Aristotle's Posterior Analytics to sort through the question. He cites Aristotle's distinction between demonstrating the existence of some subject, and going on to demonstrate properties of that subject by appeal to the essence of the subject as cause of those properties. The first kind of demonstration is called demonstration quia, the second demonstration propter quid. In order to have any science at all, the subject matter must exist. So demonstration quia must precede demonstration propter quid. If you want to have a science of unicorns, you have to show me that there is at least one unicorn to be studied. There is no science of what does not exist. So there are two demonstrative stages in any science, the demonstration of the existence of the subject (quia), and the demonstration of the properties of the subject through its essence (propter quid). Aquinas' denial that the essence of God can be known philosophically is a denial that one can have propter quid scientific understanding of God through philosophy. It is not a denial that there can be demonstration quia of the existence of a god. There is no reason to deny that Aquinas thinks the Five Ways are proofs or demonstrations in the most robust sense, namely that which he appeals to as set out by Aristotle in the Posterior Analytics.
Notice however the back and forth between the use of ‘God’ as a proper name and the use of ‘god’ as a common noun. One source of the ambiguity in the objections come about because it is claimed that Aquinas does not think one can demonstrate the existence of God. But in terms of the Posterior Analytics one cannot demonstrate the existence of anything under a proper name. One can point at Socrates, and say “see, Socrates is alive.” One cannot do that with God. In addition, one cannot give a formal argument for Socrates existence using ‘Socrates’. One can only demonstrate in the relevant sense using common nouns, since such nouns are the only ones that have definitions, either nominal or essential. So strictly speaking it is true that Aquinas doesn't think one can demonstrate the existence of God in the Five Ways. But he doesn't claim that one can. He recognizes the difference between ‘God’ used as a proper noun, and ‘god’ used as a common noun. (Summa Theologiae Ia.13.9) The ambiguity is pronounced in Latin which lacks the indefinite article ‘a’, where in English we can disambiguate between ‘God’ and ‘a god’. The situation is exacerbated by translations that simply translate ‘deus’ in the Ways as ‘God’ in English. In the Five Ways, he does not use ‘god’as a proper name, but as a common noun having five different nominal definitions. So each of the ways concludes that there is “a god.” So it is also true that the Five Ways do not as such prove that there is only one god. But it is for that reason that Aquinas himself thinks one must actually argue additionally that a god must be utterly unique, and thus that there can be only one, which he does several questions after the Five Ways(Summa Theologiae Ia.11). Of course, once the utter uniqueness of a god has been shown, one can begin to use “God” as a proper name to refer to that utterly unique being.
It is the utter uniqueness and singularity of a god that undermines the objection that whatever the philosophical arguments terminate in, it is not the god of Judaism, Christianity, and Islam, who is only known by faith. That is simply to deny Aquinas claim that the god Jews, Christians, and Muslims believe in can be known, but only partially by philosophical analysis. If the demonstrations work, as Aquinas thinks they do, what other god would the Jew, Christians, and Muslims believe in?
Finally, the sketchy character of the Ways reflects the fact that they are directed at beginning students. However the audience of beginners that Aquinas has in mind are not beginners in Philosophy. They are beginners in Sacra Doctrina. As we have seen, in the medieval educational setting such beginners would be thoroughly steeped in the philosophical disciplines before ever being allowed to study Sacra Doctrina. So Aquinas could expect his readers to know the much more extensive and complete arguments he was gesturing at with the Five Ways, arguments to be found in detail in other figures like Aristotle, Avicenna, and so on, as well as in other works of his own, the Summa Contra Gentiles for example. In short, even if the Five Ways are judged to be unsound demonstrations, a judgment that requires close analysis and examination of the filled out arguments, there is no reason to suggest that Thomas took them any less seriously as demonstrations or proofs in the fullest sense.