I doubt I'll be able to post anythign substantial next week, as I'll be in Nashville at the American Maritain Association convention on "Nature, Science and Widsom." In preparation, I've been reading Jacques Maritain. His Science and Wisdom is a wonderfully brief and lucid explanation of many things I've been trying to convey here. I highly recommend it. His Degrees of Knowledge is longer but includes more complete explanations.
Below is a very helpful selection from Degrees of Knowledge on the three degrees of abstraction. The diagram is my contribution. The a axis parametrizes displines by whether their objects can exist without matter, while the b axis parametrizes disciplines by whether their objects can be conceived without matter. (Maritain's diagram from Degrees of Knowledge is here [new window]; it is more detailed, but for that somewhat less clear on the main point of this post.)
The mind can consider objects abstracted from, and purified of, matter but only to the extent that matter is the basis of diversity amongst individuals within a species.... In this way, the object remains; and remains to the very extent that it has been presented to the intellect, impregnated with all the notes coming from matter, and abstracts only from the contingent and strictly individual peculiarities, which science overlooks. The mind thus considers bodies in their mobile and sensible reality, bodies garbed in their empirically ascertainable qualities and properties. Such an object can neither exist without matter and the qualities bound up with it, nor can it be conceived without matter. It is this great realm that the ancients called Physica, knowledge of sensible nature [i.e., natural philosophy and modern science], the first degree of abstraction.
Secondly, the mind can consider objects abstracted from, and purified of, matter insofar as matter is the general basis for the active and passive sensible properties of bodies. In this case, it considers nothing more than a certain property which it isolates within bodies—a property that remains when everything sensible is left aside—quantity, number or the extended taken in itself. This is an object of thought which cannot exist without sensible matter, but which can be conceived without it. For, nothing sensible or experimental enters into the definition of the ellipse or of the square root. This is the great field of Mathematica, knowledge of Quantity as such according to the relations of order and measure proper to it—the second degree of abstraction.
Finally, the mind can consider objects abstracted from, and purified of, all matter. In this case it considers in things only the very being with which they are saturated, being as such and its laws. These objects of thought which not only can be conceived without matter, but which can even exist without it, whether they never exist in matter, as in the case of God and pure spirits, or whether they exist in material as well as in immaterial things, for example, substance, quality, act and potency, beauty, goodness, etc. This is the wide domain of Metaphysica, knowledge of that which is beyond sensible nature, or of being as being—the third degree of abstraction.
Maritain further explains in Science and Wisdom how the degrees of abstraction are inhomogeneous in degree:
The way in which things are organised in the thought of Aristotle is well known. The theory of the three degrees or the three orders of abstraction became classic in the schools.
In the first degree of this process, the mind knows an object which it disengages from the singular and contingent moment of sense perception, but whose very intelligibility implies a reference to the sensible. This first and lowest degree of scientific abstraction is precisely the degree of physics and of the philosophy of nature. It defines the field of sensible reality. Above it comes the degree of mathematical abstraction, in which the mind knows an object whose intelligibility no longer implies an intrinsic reference to the sensible, but to the imaginable. This is the domain of the mathematical praeter-real. And finally, in the highest degree of intelletual vision, the metaphysical degree, the intelligibility of the object is free from any intrinsic reference to the senses or to imagination. This is the field of the trans-sensible reality.
Thus Aristotle did not only lay the foundations of physics. At the same time he threw light on the difference which distinguishes physics from metaphysics—a matter of capital importance. The division of the three orders of abstraction is an analogical division. The three orders are not part of the same genus: they constitute fundamentally different genera. They are not set at stages one above the other in the same generical line: there is a true noetic heterogeneity between them.
Jacques Maritain, "The Philosophy of Nature," Science and Wisdom, trans. Bernard Wall (New York: Charles Scribner's Sons, 1954), 37-39.
Jacques Maritain, The Degrees of Knowledge, trans. Gerald B. Phelan (New York: Charles Scribner's Sons, 1959), 35-36 (diagram, 39).