It is obvious from my experience that not everyone is interested in philosophy or philosophical questions. But there should be wide interest in an answer to the above question.
I was taking my car in for an oil change dressed up in a suit and tie. The attendant asked me what I do for a living and I said I was a professor and that I had just finished filming a lecture on epistemology, the study of human knowing. He asked what that meant, and I looked around the room, spotting one of the chairs in the waiting room. I said, “You see this chair here?” and I kicked the chair. “It is a physical thing in this room that people sit on. Now, you know this chair is here. But how does this chair, a physical thing in this room, get into your head so that you know it’s here?” He stared at me for a second and then his eyes got big and he made the universal sign/sound for “mind-blown.” Yes, my good man, a very interesting question.
Notice how the question above is phrased, it is not “Do we know immaterial truths from physical realities?” that would treat human knowing as hypothetical and potentially lead to skepticism (this is the beginning of the whole philosophical experiment beginning with Descartes and ending with the skepticism of Hume and later the agnosticism of Kant). This is an important question about the very project of epistemology itself, one that cannot be addressed in a short blog post without consuming it. So, we will have to be content to say that one need not begin a philosophy of human knowing by treating knowledge as hypothetical. It is possible to begin such a study by starting with the fact of human knowing and to be content with a philosophical explanation of that fact. To be clear, I am not assuming human knowing in order to give an explanation for it. Human knowledge is not an assumption to be proved (that is a Cartesian starting point), it is a curious fact that demands explanation, an explanation which is uniquely philosophical.
As with most blogs intended for mass consumption, this will have to merely assert without much elaboration an answer to the above question. I will try and supplement these assertions with some relevant resources so that the bold inquirer can follow up with his/her own research. With the prolegomena out of the way we will return to the original question: How can [humans] know immaterial truths from physical realities?
An explanation of this fact rides on two conditions: 1) That humans have a natural power for knowing that is, itself, immaterial, and 2) That there are things in the world which can be proportioned to this power. In what follows, I will address these conditions from the philosophical tradition of Thomas Aquinas, also know as Thomism, in a sub-discipline that is best described as philosophical anthropology (a philosophical view of the nature of humans).
For Aquinas, humans are a composite of an immaterial soul and a physical body (this is not substance dualism, which says that humans are two substances conjoined somehow; it is an extension of Aristotelian hylomorphism, which says that humans are one composite substance with immaterial and material facets). The immaterial soul has two purely spiritual/immaterial powers, intellection and volition, which arise from the faculties of mind and will, respectively. Again, we cannot take the space to delve into a full defense of this philosophy of mind. Suffice it to say, if the mind is able to receive into it that which is extrinsic and material in a way that does not require it to change into a material form, this indicates the nature of its operation is wholly immaterial. This is the natural power of knowing that is, itself, immaterial. Meeting condition #1 above.
Condition #2 gets to the crux of the matter. How can physical things in the world be appropriated to such a power? The answer gets into how something sensed becomes appropriated by the intellect.
Following from humans being a composition of soul and body arises two fundamental powers of knowing: Intellection and sensation. It is from the “mutual interpenetration” of these that arises the complexity of human knowing. Frederick Wilhelmsen summarizes,
Gilson reveals the foundation of realism in the unity of the knowing subject. This unity of sense and intellect within a knowing subject had been ruptured by the cogito [of Descartes]; the resulting discontinuity had been accepted by Kant as a fait accompli … Gilson and other Thomists rediscovered the true solution, to the scandal of the modern world, in the teaching of a thirteenth-century theologian. True, the senses can only grasp particulars and the intellect universals, but sense and intellect are not distinct entities at war with each other. They are powers of a single knowing subject, and through their mutual interpenetration the intellect ‘sees’ the universal in the singular.
There is one final piece to the puzzle, and it is important to note because I have heard it consistently misapprehended. Once we understand how the senses give rise to a phantasm (something more immaterial than the thing sensed but more concrete than a purely intelligible idea), and then the intellect pulls from the phantasm a purely intelligible universal concept, we are not done with the explanation of human knowing. The reason is that in order to know the truth of things, the intellect then needs to turn back to the world of particulars in order to truly know. This leads to Aquinas’ clarification that, “Man has prior knowledge of singulars through imagination and sense. Consequently, he can apply his universal intellectual knowledge to a particular; for, properly speaking, it is neither the intellect nor the sense that knows, but man that knows through both…”
Humans, existing in a world of material objects, have two powers of knowing: sensation and intellection. First by sensing things in the material world, these sensations give rise to phantasms (e.g. in the imagination), which then provides the intellect the material from which to abstract universal concepts. Take a simple declarative sentence, “This book is green.” The terms “book” and “green” are universal concepts but the intellect uses them to apply, through sensation, to a particular extra-mental object captured in its this-ness in sensation.
The explanation of human knowing lies in an analysis of anthropology, sensation, and intellection. The explanation for humans existing takes us down a metaphysical road that ends with the divine. We should not confuse or conflate these two analyses.
In the approach of presuppositionalism, for example, these two analyses are often conflated. It is asserted that God, in order to be exalted as God, must be the proximate ground for both man’s being and man’s knowing. But we have to step carefully here. This assertion fails to recognize these intermediate facets: “God” as a term in predication, “God” as a concept in the mind, and “God” as a metaphysical reality. The argument jumps to the last facet without pausing over the first two. But to assert anything about God entails all three facets. (more on God and predication in another post).
In classical empiricism (i.e. Aquinas) humans are capable of knowing truths about sensible reality and these allow him to reason to truths about spiritual reality such as the existence and nature of God. As Étienne Gilson notes,
The human mind cannot have God as its natural and proper object. As a creature, it is directly proportioned only to created being, so much so that instead of being able to deduce the existence of things from God, it must, on the contrary, of necessity rest on things in order to ascend to God.
Here, of course, Gilson is concerned only with natural knowledge and general revelation. God is not a natural object of human knowing (like trees and dogs) and His existence can only be sought metaphysically by starting with sensible objects and reasoning back to God as the necessary ground for their existence (e.g. Aquinas’ Five Ways). This is not to say that one cannot come to a knowledge of God via special revelation (e.g. Scripture), but special revelation entails something supernatural and what we want in our epistemology is to understand what man knows or can know apart from supernatural aid.
Man exists as a spiritual-physical hybrid. With his senses he engages with the individual particulars that surround him in the material world. With his intellect he “sees” things about them that abstract from their singular existence. The intellect goes beyond this world of sense to “say” things about the world that are logical and mathematical; metaphysical and theological. All this predication arises from the complex interpenetration of the senses by the intellect. Human’s know as a feature of natural union of these two powers. To be sure, this unity itself requires an explanation (and for Aquinas that ultimate explanation is God), but the question of human knowing itself has a much more proximate philosophical explanation.
 See Joseph Hassett, Robert Mitchell, and J. Donald Monan, The Philosophy of Human Knowing (Westminster, MD: The Newman Press, 1961), esp. chapters 5 and 8.
 See Henri Renard and Martin O. Vaske, The Philosophy of Man, (Milwaukee, WI: The Bruce Publishing Company, 1956) esp. chapters 2 and 6.
 F. Wilhelmsen, “Foreword” in Etienne Gilson, Thomistic Realism and the Critique of Knowledge (San Francisco, CA: Ignatius Press, 1986), 18-19.
 Aquinas, De Veritate (On Truth) Q.2, a. 6, ad 3.
 Étienne Gilson. Methodical Realism (Kindle Locations 687-690). San Francisco: Ignatius Press, 1990), 575-580, Kindle.