An aspect of the division of the sciences

St. Thomas divided the objects of study by their relation to sensation (which presupposes that all have some relation to sensation).

All sensible objects are studied by natural science. This includes all we now call science, even including things like politics so far as we simply asking what the state or nation is.

Mathematics studies things that are not in the sensible world, but have a substantial likeness to the things in it. One can never point to what the geometer means by a circle, nor can he point to that mysterious unity that the mathematician calls, say, “five” (though five distinct things are easy enough to point to). At the same time, the mathematicals have a substantial likeness to sensible things since quantity and sensible quality are said of some underlying thing.

Metaphysics studies things that are neither in the sensible world, nor have a substantial likeness to things in it.

Mathematics and metaphysics both have a unity and likeness to the sensible world, but in different ways. Mathematics has a likeness and unity by way of substance, whereas metaphysics has it by way of causality. The objects of these sciences, however, are not given directly to sensation, nor is the science assumed to be about anything real prior to some demonstration of its existence.

Because neither mathematical things nor metaphysical things are directly given in sensation, the objects of both sciences have an interiority and immanence within us that the objects of natural science do not have. Mathematics is the first muse we have instructing us that we cannot identify objectivity with being given in sensation, and she makes her point quite emphatically. Though her science treats of nothing we can point to in the sensible world, it has more rigor, precision, and ease in learning than any other science (in this regard, we have taken exactly the wrong conclusion from Einstein’s observation that the physical world cannot be understood in three dimensional terms. We should have taken it as another confirmation that mathematical reality is simply different than sensible reality, but instead most people have taken this as a reason to deny the reality of mathematical things. But, for example, Euclid’s science is not a branch of physics, and so it is sheltered from any change in physics.)

The objects of metaphysics are even more removed from the physical world than those of mathematics, but because we relate them to the world by way of causality, we are forced to speak of them as subsistent, whereas quantity- which shows itself as an accident- need not be thought of as subsistent. We do not come to the existence of metaphysical objects because we approach the world with n principle of causality that we assume must apply to the world- we simply take metaphysical causes where we find them, for example, when we are explaining what is required for motion, or physical causality, or contingency, or sensible goodness, or the existence and activity of natural things.

Metaphysics requires in different ways a prior knowledge of phyics and mathematics. From physics, it gahters that which it must distinguish itself from, and from mathematics it gathers greater confidence that objectivity does not require being given in the sensible world.

One of the gravest errors we can make in metaphysics is thinking that the subsistence of its objects must consist in them being “out there” like the objects given in sensation. The sense we understand God as “existent ” must be- and can only be- is a way opposed to the what things are given in sensation. All of Kant’s paradoxes about the first cause are rooted in his identification of objectivity and being given in sensation. This is a scheme he forced on reality a priori for the sake of his system, as opposed to simply taking truth where he could find it and then arrange it later. Much of modern thought suffers from the desire to impose some pre-fab criterion for knowledge on the mind rather than letting it simply go out, see what it can find, and then sort through the pieces later. Such an approach is antithetical to simply grounding sceince on experience, as Aristotle and St. Thomas do.


  1. Peter said,

    November 26, 2008 at 4:29 pm

    Speaking of Euclid’s science, what do you make of the common criticisms of the first proposition of the first book of the Elements? The criticisms I’m familiar with are summed up here, near the bottom:

    I suppose adding the implicit postulates would be ok, but I tend to think that such difficulties are more the result of over-thinking the problem than anything else; and so we shouldn’t place too much emphasis on proving what should be manifestly clear. (“Why does point C exist?” I’m tempted to reply, “uh, because it does: it’s right there!”)

  2. a thomist said,

    November 27, 2008 at 1:00 am

    … my puncuation keys are being irrational, only commas and periods work… remember this.

    St. Thomas says that the one who demonstrates does not lay down all teh axcioms from which his point follows, but only the important or most significant ones. These objectors to Euclid either disagree with St. Thomas altogether, or they disagree about what constitutes a significant or important axiom. The dispute is about the stucture of demonstration, and so is not properly geometrical but logical, best dealt with in some subset of the Posterior Analytics.

    The case of the objectors to Euclid is pretty simple. Something only actually follows from somethign actually said. But some axioms the Geometer uses are not actually said, therefore etc. If one objects that the axioms are obvious, they can respond that most axioms are obvious, and in fact many of the propositions that Euclid proves would not be doubted if they were stated to a reasonable person. How can we justify leaving out an axiom, then! St. Thomas is wrong and Euclid is too. QED.

    St. Thomas never proves his case, but I think it is the stronger claim. Euclid is writing, and therefore communicating, and the particular sort of communication he is involved in is teaching. All that he includes or leaves aside is subordinate to a teaching function. Some axioms are easier to learn if they arise concomitantly with the proof, others need to be stated explicitly. Sometimes we must prove straightforward propositions to show the rigor of Geometry, other times we leave them out. Is it not the case that some obvious axioms are better seen than put into words…question mark… sometimes we use simple propositions to move the mind forward toward a point, otehr times we let the truth be simply seen in the picture.

    In a word, complete rigor does not require that one actually include everything required to come to the conclusion- if this even is possible. It would be, among other things, bad teaching.

    And an ugly science. Can you imagine how ugly and silly a science would look if it included everything distinctly that one needed to know…quest… Science, being true, cannot shake its relation to beauty and goodness, and an such a total science where everything is said distinctly would have to do so.

    So then how do we determine what should be included and what should not be…q… The best way is to follow what a master teacher does. Like Euclid, for example.

  3. Peter said,

    November 27, 2008 at 5:35 am

    Good point.

  4. Joe said,

    July 26, 2010 at 4:35 pm

    Thanks for this post (I know it’s an old one). I’m putting together something on the nature of Mathematics, how it can properly dispose one to being open to God, and its place in a Catholic curriculum. Do you mind if I quote this article in it? This is the quote I’m looking at taking in particular:

    “Metaphysics requires a prior knowledge of … mathematics … [F]rom mathematics it gathers greater confidence that objectivity does not require being in the sensible world.”

    Thank you,

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