Degrees of Abstraction – Degrees of Knowledge

(Updated November 2015)

As Chesterton said in Philosophy for the Schoolroom, any argument should begin with the parties stating their “infallible dogmas” (=axioms, first principles, undisputable first facts), so that the discussion could proceed to the real basis of the point in dispute.

This is to say most arguments would come to a full stop there, as the disagreement is most often in the very first steps of grasping reality and reasoning about it.

Chesterton was alluding to real philosophical dispute, where the different parties speak the same language, but in the case where one or more of the parties ignore philosophy, the problem lies at a more profound level. There is another step before axioms, which is largely forgotten now that philosophy is unknown or deformed by most people. This step is the hierarchy of sciences based on the different degrees of abstraction.


I had an example of that problem while reading Leah Libresco (Unequally Yoked), who is currently following Chris Hallquist’s project of an atheist response to religious experts (mainly theologians). I perused Leah’s posts on the subject and some excerpts of the book in progress on Hallquist’s blog (The Incredible Hallq).

In a post titled “The First Thing We Do, Let’s Kill All the Philosophers” (, Leah alluded to the fact that mathematical tools do not seem to work in ethics and metaphysics.


…philosophy doesn’t have some of the advantages of science (natural, empirical science, I assume)…

…plenty of people have read Kahneman, know about Aumann’s Agreement Theorem, and still disagree about ethics, let alone more-inaccessible metaphysics. So we’ve all got the theologian problem…

I see in this, as well as in many other comments, a mixing up of the three degrees of abstraction by which we can gain different types of knowledge. (Cf. Jacques Maritain, The Degrees of Knowledge, a work on epistemology. In The Range of Reason, available online at Notre-Dame Maritain Center, the first and third essay touch on the subject of knowledge, in relation to belief and unbelief in God, the immortality of the soul and the supernatural origin of the moral order.)


Natural sciences (physica) are about things perceptible by external senses, which we can know at first from observation and experimentation, then we can abstract axioms and laws from them: this is the first degree of abstraction.

Mathematics (mathematica) are at the second degree of abstraction and are devoted to measures and quantities; for more precision they most often work on ideal plane rather than on natural beings directly.

Neither physica nor mathematica are directly of use to know or understand essence and existence, matter and form, potentiality and actuality (potency and act in scholastic terms), substance and accident, identity, finality, causality, etc. — in a word, metaphysica. (Cf. Maritain, Introduction to Philosophy and Preface to Metaphysics.)

The only parts of philosophy which use second-degree abstractions are logic and dialectic: these are useful tools to consider probabilities and form an opinion, but they are unable to demonstrate anything related to the proper object of metaphysics: being as such.

It is no coincidence that Descartes degraded metaphysics to a branch of physics-mathematics, and that Hegel and Marx discarded metaphysics and based their worldview on dialectic instead, so that knowledge is now reduced to mere opinion in all minds devoid of proper metaphysical thinking.

Blaise Pascal’s father was aware of this danger and prevented his brilliant son to learn formally mathematics and geometry before adolescence in order that he be a good philosopher first. The father knew mathematical and natural sciences would be a lure into materialism for the young genius. And he succeeded. Pascal became a good philosopher as well as a prominent physicist and geometer. He was the living proof that logic is not really learned best with mathematics, but with general logic learned first with life – particularly moral and religious life – and language.


If physics and mathematics are not of use in philosophy, metaphysics in turn is indispensable to other sciences in that it gives them their principles and their autonomy – there is a philosophy of nature (physics) and of mathematics, a philosophy of history…

Metaphysics is still more indispensable to natural theology (theodicy), morals, epistemology, and theology proper. No one ignorant of philosophy and no bad philosopher can be a theologian worthy of the name. No theology is real theology, that is, a science in its own right, if it does not have a coherent metaphysical basis, which serves as a conceptual frame for studying what in Revelation is accessible to natural reason aided by faith. Sacred theology is not indeed merely a part of metaphysics, like natural theology (theodicy), which studies what in the invisible order is accessible to natural reason without the help of faith.

Natural theology (theodicy) gave birth to the phrase “god of the philosophers”. Usually, it seems to mean that the god arrived at by natural reason unaided by faith and the God of Revelation are not the same, but it is not necessarily so. If the philosophy in question is true (Socrates, Plato and Aristotle’s were true to a great extent), it can only be about the one God, himself the source of all truth. (Cf. Peter Kreeft’s book Socrates Meets Jesus, or the theatrical adaptation of the same by Kevin O’Brien and the Theater of the Word.)


So, philosophy – metaphysics to be exact – is necessary to look at the problem of God. Chris Hallquist is a philosopher but, in his projected book, he is looking at the problem of theism-atheism through diverse theological views. The impression I got from what I read is that he risks to be mired in a large straw man argument.

My reasoning is maybe simplistic, but by Occam’s razor I think I might be right: if there is a God, there can be only one; if he revealed himself, he can only have done it and continue to do so in one organically coherent revelation through organically coherent representatives, which he endowed with means to truly be in a love relation with him.

Debunking heretical or marginal expressions of the one true religion and the one true God has been done over the centuries and continues to be done by the Catholic Church. If an atheist wants to know what is the present form of the original Judaeo-Christian monotheism, he should devote his effort to examining Catholic theology.

The Catechism would be a good place to start, if only to become aware that most nominally Catholic people are bad Christians. After the Catechism, Vatican II dogmatic constitutions are the next best reads, then the encyclicals from John Paul II and Benedict XVI. Along the process, to obtain brief and reliable explanations of technical terms before tackling something like Aquinas’ Summa Theologiae, we have that marvelous handy reference on the net: the Catholic Encyclopedia.

Do not enter Scripture as a tourist without a map. The necessary explanations and references to eventually grasp Scripture are found in the above-mentioned documents and in the notes and cross-references of a good Catholic edition of the Bible.