Mathematics and Christian Education

Nothing could be more distinctive of the age in which we live than the overpowering prominence of mathematics. All through the Catholic centuries, arithmetic and geometry constituted all the mathematics that an educated Christian was asked to learn. Even these two subjects were treated from a more contemplative point of view, which made them far more harmonious with other liberal studies. Arithmetic consisted in the study of the properties of numbers; geometry in the study of shapes and figures. When not overdone, and when counterbalanced by the proper correctives from the other types of knowledge, geometry and arithmetic, as they used to be taught, cultivated a few desirable virtues of the mind like clarity and precision, and sharpened the mind for the perception of harmony, rhythm, and pattern in the study of nature and of Holy Scripture. But even then, many saints and sages warned against the excessive preoccupation with such studies, and especially against the seductive clarity of mathematics; for it is not enough for the mind to be accurate and clear; we are bound to ask “accurate and clear about what?” Since in mathematics accuracy and clarity are achieved at the price of the reality and the goodness of the object, it is a danger of the mathematical mind to continue to sacrifice reality and goodness for the sake of clarity in every other field in which man must seek and find the truth.

But in our time, education is overwhelmed by mathematics and on more than one score. For, while a contemplative interest in the properties of shapes and numbers is almost completely extinct, an illiberal and utterly inhuman form of mathematics dominates the years of learning of our boys and girls, almost completely from the very first year of the primary school to the very last year of college. In place of arithmetic and geometry, whose relation to reality is definite and understandable, there is now an indefinite confusion of branches which go by the name of mathematics, the nature of whose objects nobody understands! Such topics as topology, non-Eudidean geometry, Boolean algebra, transfinite numbers, projective geometry; not to speak of other more recognizable subjects like algebra, trigonometry, integral calculus, vector analysis and the theory of equations. These new subjects are not only more confusing but much more difficult to acquire, and therefore much less likely to leave the mind at leisure for other liberal studies. But the predominance of mathematics today is not restricted to those courses which go by its name, because mathematics, in some form or other, in matter or in method, has crept into every other corner of the curriculum. According to the modern positivistic conception, mathematics and not wisdom is considered as the prototype of science. In subjects ranging from physics to education, covering every field of human learning, there is an evident tendency to assimilate all knowledge to mathematical knowledge and to resolve all realities into mathematical formulas. This trend reaches its apex in the development of symbolic logic, in which guise mathematics invades even the field of philosophy, to distort all the basic conceptions of the mind, and to deflect all the activities of thought from attaining their fulfillment in true wisdom which consists in knowledge about God, by keeping them whirling endlessly around the nihilistic circle of sheer mathematical emptiness.

Now in an attempt to determine the influence of mathematics on the mind of a Christian, it would be folly to ignore the fact that after twenty centuries of Christian living, it is impossible to name one single patron saint for mathematics. There are Catholics indeed who occupied themselves considerably with mathematics and as far as we know kept the faith; but I know of no mathematician whose faith burned so brilliantly as to earn him a place among the stars of sanctity. Nor is this a mere coincidence, for any one of us can look into his own mind to find that there is no other kind of human knowledge or human experience which offers less in terms of value for the Christian message than mathematics. Almost all that one needs in the way of mathematics in order to learn all of Holy Scripture and all the Doctors of the Church, does not exceed the ability to count up to a thousand and to distinguish between a vertical and a horizontal line. Whatever it is you talk about in mathematics, it is never anything you can carry over to your meditations, or employ in your prayers; it gives you no courage in your moments of despair, and no consolation in your loneliness.

In the field of philosophy, mathematics has always been fertile grounds for sophistry. There is hardly any other intellectual interest which has contributed more to confuse men about fundamental truths regarding God, man, and the universe, than mathematics. Just to mention the names of Thales, Pythagoras, Plato, Descartes, Spinoza, Whitehead and Russell, would suffice to convince one even slightly acquainted with the history of thought about the great number of minds that were deceived by the mirage of mathematics, and misled to accept fraudulent substitutes for the saving truth. I believe that an unprejudiced consideration of the nature of mathematics and of the nature of its objects would reveal clearly that all these charges leveled against the mathematical mind are rooted in the very nature and essence of things.

But what kind of a science is mathematics? Is it a practical science which envisages the achievement of a good, or a speculative science which envisages the attainment of truth? A practical science, like medicine or ethics, would be eliminated by the elimination of the corresponding good. For example, if men were indifferent to health and its opposite there would be no criterion for distinguishing between a right prescription and a wrong one, and consequently, medicine would cease to be a science. In a similar way, if men per absurdum were suddenly to become neutral to the attainment of happiness or its opposite, that would be the end of ethics. But what good, if ceasing, would determine the end of mathematics? None whatever, for the simple reason that mathematics prescinds from all good and all value. Mathematics talks the language of a speculative science. It utters propositions which must be either true or false. Now a proposition is true or false depending on whether it is or is not in conformity with reality. Just as a practical science envisages a good to be achieved, which good functions as the criterion for right and wrong precepts in that science, so a speculative science considers some part or aspect of reality, which stands as the measure of truth and falsehood in that science. If there were no stars there would be no astronomy; and theology would be sheer nonsense if God did not exist. But what part of reality would destroy mathematics by being eliminated? What does the mathematician talk about? Is the object of mathematics a creature or a creator? Is it a substance or an accident? Is it something actual or merely potential? Is it changing or changeless? Temporal or eternal? Material or spiritual? Tangible or intangible? If one were to compose an inventory of all the subsisting realities of the whole universe, including God, the angels, men, animals, plants and minerals, would the objects of mathematics be on this list?

Am I asking too many questions? Well, here are a few answers whose reasons will either be supplied later, or be left to the reader to discover for himself. Mathematics is a speculative science whose value can only be in the practical order. It has no speculative value, because it does not convey any essential knowledge about any subsisting reality. It is not contemplative knowledge and therefore not essentially good for man, because it occupies the intellect with objects which the will cannot love. It is knowledge which does not proceed from understanding nor does it resolve in wisdom. It does not proceed from understanding, because the mathematical expression of any reality, never conveys any understanding of it. It may however convey the means for the control of that reality. You are not one inch closer to the penetration of the mystery of light and color when you know the number of Angstroms in each of the colors of the spectrum; nor about  the nature, cause, or purpose of gravity when you resolve its laws into mathematical formulas. And it does not resolve in wisdom, because neither is mathematics concerned with the First Cause, nor does it lead to the First Cause. The manner by which mathematics deals with its objects abstracts completely from any dependence upon God, and as a matter of fact, attributes to these objects a species of eternity and turns them into quasi divinities completely independent in themselves. This explains the autonomous nature of mathematics, according to which, left to itself, it never leads to anything non-mathematical. A mathematician might be led to think about God by an accidental non-mathematical reason, but never from the very needs of mathematics.

As for the object of mathematics, it is not a physical entity but a mental entity; it is not real but ideal. There is nowhere in the world, outside of the mind of a mathematician, a point without dimensions, a line without width or thickness, or a square root of minus one. But these fictions of the mind are founded on reality, and their foundation consists of the accident of quantity and its properties and relations. Arithmetic is founded on discontinuous quantities or multitudes; geometry on continuous quantities or magnitudes; while algebra is founded on abstract quantity considered generically, prescinding from whether it is number or magnitude and therefore potentially capable both of an arithmetical as well as of a geometrical interpretation. Other mathematical objects, more distantly removed from this real foundation of mathematics, are rooted in these simpler elements and in the relations which hold among them. Having experienced the three dimensions of bodies in space and having represented these three dimensions by the three variables of an algebraical equation, nothing prevents the mind from creating the fiction of a space corresponding to an algebraical equation of four variables – hence four-dimensional space.

But what do we know about this accident of quantity, on which is founded, proximately or remotely every object of mathematics? We learn from philosophy that quantity is an accident of material substances, and that in contrast with the accident of quality, quantity manifests the material and not the formal aspect of these substances. Therefore the real foundation of mathematics is found in the material aspect of material things. Further, an accident when conceived as an accident always brings you back to its substance; but in mathematics the accident of quantity is conceived as if it were a substance. Further, a material substance concretely considered, has a nature through which this substance moves to the attainment of an end, but the mathematician considers quantity as a substantialized material accident devoid of any principle of change and abstracted from any movement to attain an end. The concrete material substance manifests itself through its sensible qualities by means of which it is known, but the object of mathematics, without being a spiritual substance like an angel, prescinds from all sensible qualities and can be known only by the intellect and not by the senses. Hence we have the apparent paradox that while the only foundation for the mathematical object is the material aspect of material things, still mathematics represents its object such as matter could neither be nor be known. For matter is nothing but a principle of change, while mathematics prescinds from change; and matter can only be known through the senses while mathematics prescinds from sensibility.

The object of mathematics is therefore an accident parading as a substance, a material reality pretending to be immaterial, an ideal entity which poses for something real. At the basis of all these antinomies is the fact that mathematics arises only when an intellectual mind, directs the light of its spiritual intelligence, not for the purpose of contemplating being, but for the purpose of controlling potency. The mathematical object is the shadow that matter casts on spirit. For when spirit knows spirit, there is not even the foundation for mathematics; when material cognition (sensation) knows material things, the objects of mathematics cannot arise; even when a spiritual being knows matter contemplatively it understands a material substance through its form and its qualities. It is only when a spiritual being concerns itself with matter and for the purpose of sheer control that mathematics finally finds its grounds.

But how about the truth in mathematics? If the objects of mathematics are mental entities (entia rationis) what is it that determines the truth or falsehood of a mathematical proposition? What reality stands as the measure to the judgment of the mind? In the classical branches, arithmetic and geometry, the foundation in reality was close enough to preclude any statements that are not justified by the real properties of multitudes and magnitudes. But as mathematics branches out and develops into newer mathematics, and higher mathematics, and purer mathematics, that control becomes less and less until finally the mind remains its own measure. Consistency and not conformity becomes the touchstone of validity.

Apart from mathematics, there used to be three other distinct types of knowledge: physical, logical, and ethical. All three led ultimately to God – the physical sciences under the aspect of Ultimate Cause; the logical sciences by way of the Prime Truth; and the ethical sciences by way of the Supreme Good. But in mathematics, the mind reigns supreme, lord of all it surveys. The mind finds in itself a sufficient cause for the kind of being the mathematical entity enjoys. It is the only ultimate measure for the truth of its judgments. It prescinds completely from the aspect of goodness. Of all the intellectual pursuits, mathematics alone does not lead to God.

It is like the web of a spider, it proceeds from the very substance of the spider and ends up being its own jail. It gets more involved and more intricate the more it is extended, and finally, when the web is intricate enough, the new threads do not have to measure up to any real independent distances of walls or furniture, for when the new-thrown thread fails to meet a point of support, it sticks on another thread of the same fabric.

From the spider of mathematics, may God deliver us.

  • Bill Strom

    I think that mathematics is not taught well in the USA and to disparage it as “against the faith” –because that was the tone, seems… is a dangerous  overgeneralizing. Most good paying jobs deal with mathematics and a the stronger your background in maths the better your possibilities in finding a good paying job. I have friends who a physicists who have worked for NASA and they are VERY devote Catholics! I found the tone of this article irresponsible. It seems to have been as an emotional reaction to some bad experience with some mathematician,

  • I agree with that article. In the times being there is a sort of tyranny of the mathematics that prevent a lot of people with “well shaped rather than well filled brains” (Montaigne) to have their places in the Society. If you don’t handle the mathematics easily you must be rejected from having access to responsibilities posts. Certainly there is something important lacking to politics and corporates manager who cannot conceive the world but in terms of winning and losing, ratios and balances, when one sees the dangerous world they have created, they are ruling and which we are living in.

  • If you click on the author’s name, Br. Francis Malouf, MICM, under the title, you’ll see his bio. It’s unlikely that he was just reacting to some bad experience, in my opinion. The article is hard for me to fully appreciate, having almost no memory of my studies of the subject. But I do know a college student with a math and physics undergraduate degree, who is being paid by the college to do her masters in biology, with the purpose of coming up with formulas and theories to eventually save the planet against global warming. I explain it very badly, but I think I see some relationship between her experience, and what’s being pointed out by Br. Francis. 

    That college friend is working with theoretical math, which is all completely abstract, to the point that the word ‘population’ doesn’t even signify what population. So, she’s already learning to distance a word like population, from something like human or animal. She’s an agnostic, and at one time accepted evolution as true, and her colleagues all seem to be of the same group that way. 

  • BGT

    This short and insightful article (or speech) composed by Bro. Francis contains serious food for thought. He has compiled a considerable list of densely-packed sentences which apply his mastery of philosophy to a frank consideration of aspects of mathematics that you would be hard-pressed to find elsewhere compared. To whose writings could anyone look to find such a thing as the following, but those of Brother Francis: “… a material substance concretely considered, has a nature through which this substance moves to the attainment of an end, but the mathematician considers quantity as a substantialized material accident devoid of any principle of change and abstracted from any movement to attain an end. ”  I am happy to see that the Center preserves this short work of Bro. Francis, for it could be found a useful reference sometime in the future when someone is looking for a unique observation to bring some sanity into a crazy world devoid of same. This short work is rather admirable in its profundity and outspokenness. He obviously wasn’t trying to make any points with the Mathematics Department at any of the world’s educational institutions!

    The way I see it, we ought to keep mathematics at arm’s length, and not let our affections be deluded and misdirected by way of its intricacies and precision. For as accurate and detailed it can be, mathematics can equally misdirect us when only one small error is made and not noticed. When you’re an astronaut approaching re-entry, your very life depends on the precise calculation of the angle and velocity of your orbit; for too steep, and you will plunge to earth too fast and become a firey meteor, but too shallow, and you will therefore bounce off the fringes of the earth’s atmosphere and be uncontrollably hurled into outer space at escape velocity from earth’s gravitational pull. Your only chance for survival is correct use of mathematics. But the vast majority of us will never face that series of events. An astronaut who does his math wrong might therefore not survive, but if he’s in the state of grace, his death will not be to his eternal demise. Making a mistake in math is not a sin. But it can get you into a lot of trouble in this temporal, material world. As Brother Francis explains, doing the math right might save your life, but it will never evoke your salvation. Likewise, doing the math wrong might cost your life, but it cannot cause your damnation. In our age of materialism, all too often people get their priorities all wrong. This is a much more serious error than getting your answers wrong on a math test. Mathematics can be well used as a tool, and as such can be a beautiful thing, in its own, limited way. But when it is too removed from practicality to be of any use but for itself, it seems to take on the danger of becoming a sort of idol, good for nothing but something men can look at and think about and perhaps appreciate. Dare I say “love?” He may have been unwilling to say it, but maybe there are or have been people who have fallen into the trap of loving mathematics, and it seems rather unlikely that such a one would be able to use his love for this theoretical abstraction without object, to increase his love of God. On the contrary, appreciation for, or even love of mathematics per se, poses a greater threat to one’s holiness than a help. It is more of a distraction than an aid to contemplation of God. How could non-Euclidean geometry or even integral calculus aid anyone in his contemplation of any one of the 15 mysteries of the Rosary? — or, for that matter, any one of the “other 5?” About all it’s going to accomplish is getting you hung up on details, rather than focusing on the mystery inherent in the Rosary. One may be tempted to imagine something that isn’t even there, in order to have some math to think about, such as the relative heights of the Child Jesus compared to the elders in the Temple at the Finding, or the pattern of weave in the seamless garment worn by Our Lord to His crucifixion. You might qualify for a better job when you do math well, but the point is, a better job, alone, does not necessarily bring you closer to eternal salvation. Therefore, we must keep our priorities in order by recognizing mathematics for its particular usefulness, and focus our appreciation on more worthy objects of admiration. 

  • Charlie Patin

    I’m totally with Bill Strom on this.  Of course I’m a former very, very active RC who’s now an atheist so I know my view won’t count here among those who believe the myths of the Bible.  To even think that Mathematics should be kept at “arms length” is typical of those who are afraid of the truth.  

  • David of Glasgow

    If one were to compose an inventory of all the subsisting realities of the whole universe, including God, the angels, men, animals, plants and minerals, would the objects of mathematics be on this list?

    An interesting point. Certain numbers have an immaterial reality insofar as they are universals, but what about the propositions of non-Euclidean geometry?

    I understand Brother Francis to be criticising pure mathematics as opposed to mathematics applied to the physical universe, namely physics. But, of course, there has been the temptation here too to prefer mathematical clarity to reality and assume that physics offers us a complete explanation of causes which is properly the object of metaphysics.

  • Brendan Foley

    ” In place of arithmetic and geometry, whose relation to reality is definite and understandable, there is now an indefinite confusion of branches which go by the name of mathematics, the nature of whose objects nobody understands!”
    Please don’t confuse “objects that you don’t personally understand” with “objects nobody understands”.

    Things we don’t understand are not necessarily evil either (take the Holy Trinity, for instance, not even St. Augustine could fully comprehend the Holy Trinity, yet the Holy Trinity is the basis of all that is good, is the definition of good!). It seems you may have some confusion about many concepts in mathematics. Take this for instance:
    “There is nowhere in the world, outside of the mind of a mathematician, a point without dimensions, a line without width or thickness, or a square root of minus one.”

    Yes, there is somewhere in the world: everywhere. There are an infinite number of points without dimensions, infinite number of lines without width of thickness, and the square root of negative one is a real thing (though, it is rather inappropriately called an imaginary number). These things exist in the real world, and have practical application because they exist.

    It’s funny you mention that the physical sciences lead to God though, and yet mathematics does not. Both higher level physics and math are almost indistinguishable. They are often considered parts of the same field in all respects. Thus you will find mathematicians analyzing physical evidence in a car crash, where they are doing the math (or physics) based on the evidence to discover how a collision occurred, what speeds were achieved where the objects were at the time of the collision, how dangerous it was, and so on. Calculus itself is entirely designed for working out problems in physics (originally, gravity, to be specific), and most physicists are really just doing a particular set of applied mathematics, even experimental physicists do to analyze their data. In other words, if physics leads to God, why doesn’t mathematics? Even when you don’t consider physics, of course, you can understand that mathematics deals with a form of order and logic, which comes from God as well.

    And there is no reason to believe that mathematics has any conflict with spirit or Faith. It’s a broad field of methods of reasoning, and just as Vatican I declared there can be no conflict between Faith and Reason, there can be no conflict between Mathematics (properly understood) and Faith.

    Sure, there probably are a number of philosophers that engaged in mathematics that have fallen into error. However, not only is that kind of argument for mathematics being bad a logical fallacy (bad by association, or a form of ad hominem), but the same could be said about far more theologians. I’m sure you are aware of the large number of heretics who simply focus on theology.

  • Jon Bannon

    I appreciate this post, but must disagree with its content wholeheartedly. As a practicing mathematician, I note that mathematical research can be regarded a form of prayer. The need for insight in a mathematical question requires contemplation of nature, even if simply the nature of the human mind and will, which is certainly in touch with God deeply. God certainly is aware of the freedom and consistency of logic and information to its depths, and so contemplating these possibilities in a spirit of communion with the Totally Other in a discipline of contemplation is very much akin to prayer or contemplation. If one appreciates the prayer of St. John of the Cross, one appreciates the gift of one’s mind to God. To bend one’s mind in discipline to the structures of logic, especially those that appear in nature, requires a great humility and openness to what is true. To go beyond simply pushing symbols around, this requires some kind of mystical grace. To see what new is needed, what is living and true, is inaccessible to austere logic (this, oddly, was proved by Kurt Godel) so mathematicians rely on a sense of beauty that can only be described as a sort of prayer. Please avoid disparaging what you do not understand.

  • Larry_Dickson

    Mathematics is nothing other than logic. It says “If this is true, then this is true.” Its object is a Platonic form to which things in the real world approximate. Brother Malouf has forgotten that Platonism (not neo-Platonism) is one of the philosophies that can be made consistent with Christianity. His wholesale condemnation of mathematics is a wholesale condemnation of logic. By claiming that Christian religion is incompatible with logic, Brother Malouf plunges into a kind of obscurantism more appropriate for Wahhabi Islam.
    I have been a papal Catholic all my life, and am a PhD mathematician. There was never any conflict between these facts – unless you assume that time spent doing good work has to be time spent in opposition to God.

  • Ron

    I am a mechanical engineer who works in the chemical industry. As an engineer, I have always viewed mathematics as an indispensable tool, but have also from time to time admired the elegance of certain solutions. While I cannot say that a knowledge of mathematics would by itself lead one to God, I can say that a misunderstanding or careless lack of precision when applying mathematics can quickly lead to an intimate meeting with God. This depends on where one is standing when the high speed machinery comes apart.

    My patron saint is St. Thomas Aquinas, but I have a particular devotion to St. Joseph. He provided for his family by making beneficial things using the best math and geometry available to him at the time.

  • Logic and mathematics are two distinct disciplines. Logic is the art and science of correct reasoning. Mathematics, important as it is in many respects — and as much as it requires logical rigor — is the science that deals with the accident of quantity.

    Only in so-called “symbolic logic” is logic reduced to mathematics. That is a modern and bad innovation.

  • Larry_Dickson

    Every syllogism of logic can be represented as symbolic logic. Therefore they are equivalent, and symbolic logic is not a “bad innovation” but (as its name indicates) an expression of the same thing through symbols. And mathematics is the science that deals with deductive rigor. One of its APPLICATIONS is to quantity, treated as a Platonic form, not as an accident. (Aristotelianism is not the compulsory philosophy required of Christians!)

  • Symbolic logic can better be performed by a computer than a human being. Computers only do math (quantities 1’s and 0’s). But they cannot think; they cannot reason.

    Logic, if it is the art and science of correct reasoning, cannot be done by a computer. It is a human art, a human science.

  • Larry_Dickson

    We seem to have a difference of terminology. Computers don’t do math, any more than typewriters do novels or chisels do sculptures. The human being uses tools. The possibility that the tool may be getting too dominant, and the human art less significant, is a danger to every human art. Both Platonists and Aristotelians can agree on that.

  • Computers can solve problems in symbolic logic. But they cannot distinguish between the real (ontological), the logical (in the mind), and the linguistic orders. They cannot think. Therefore, they cannot solve real problems in logic, which include such things as supposition, modality, and predication. Logic is beyond math.

  • Brother Francis was a Catholic Philosopher. His fully developed philosophical ideas may be found in his lectures:

    and books:

    Articles on philosophical topics may be found on this site:

  • Larry_Dickson

    You misunderstood my point again. Mathematics, like other human arts, is done by human beings using tools. Computers must be programmed with great care by a human being before they can do symbolic logic or anything else. Anyone with experience in programming knows that computers’ striking out on their own in math is about as plausible as shaking watch parts in a jug and having them assemble themselves into a watch. Logic (i.e. Venn diagrams) is a part of math, but can be shown to be equivalent to the whole of math. Most computer code is actually a slightly incorrect approximation to logical or mathematical results.

  • While the mathematical sciences have a certain logic to them, and while they require logical thinking, and are a great discipline for the mind, math is not logic. In other words, the proposition “Math is logic” is false. Logic attends to so much of reality that math does not attend to.

  • Larry_Dickson

    The three classic laws of thought are the law of identity, the law of non-contradiction, and the law of excluded middle. These are all part of mathematics, and can be expressed in math notation (see “Law of thought” in Wikipedia), and in turn they, plus axioms, imply all of math (that’s what a mathematical proof means). If A includes all of B and B includes all of A, then A equals B (a simple logical proof). You seem to be confusing “math” with “math as applied to physical objects,” which is indeed more restricted.

  • A few questions, if I may:

    How does mathematics account for the supposition of a term?

    Does mathematics study the modes of the copula of a proposition?

    Are mathematicians trained to distinguish the predication of a proposition’s predicate?

    Are mathematicians (qua mathematicians) attentive to the distinction between the ontological (the real), the logical, and the linguistic orders?

    Is the objective of mathematics to achieve clear ideas, true judgments, and valid inferences?

    Lastly, is it true that symbolic logicians say that all universal statements about the “null-class” are true because no one could produce a case to the contrary as a refutation of them?

  • Larry_Dickson

    Your first three questions relate to the philosophical branch of linguistic analysis, not logic, and may I say that linguistic analysis has been far more hostile to the Faith than any mathematics. The fourth question is answered yes because mathematicians must know enough to avoid ontological questions and linguistic ambiguity. The answer to the fifth is “Certainly,” but always assuming the axioms. And the last is definitely true, as shown by logical quasi-paradoxes that lead to apparent contradictions because they refer to an empty class (there is even a mathematical “proof” that 1 = 0 by sneaking in a division by zero).

  • Thank you. You have proven (to me at least) that your idea of logic is simply as a mathematical discipline and not at all as a philosophical discipline that studies reality.

    My first three questions all pertain to the three integral parts of a logical proposition: the subject, the predicate, and the copula. As errors in these areas (e.g., in supposition) are among the classical logical fallacies, any “logic” that does not deal with them is completely inadequate to the task and unworthy of the name. Modality is very important because we generate authentic rules for immediate inference from them, as we also do by using the traditional Square of Opposition — not the hacked up symbolic-logic version of Boole.

    Supposition, modality, and predication are all essential to the study of logic. A science that does not consider them is a science other than logic.

    Your answer to my fourth question surprised me a bit. I will accept that, in your case at least, this trifold distinction is important. But I will note that mathematicians in general seem to be devoid of ontology. Most of them seem to think that Math is the supreme science, which it assuredly is not.

    Your answer to my fifth question (regarding the null set) confirms what I’ve been taught by my mentor, Brother Francis. Here is an excerpt from a work of his that I happen to be editing now:

    ***The great difference between traditional logic and the mathematical parody of it called Symbolic Logic can now be brought to light in terms of this matter of supposition. Of the many objections that can be made of Symbolic Logic, the following is given by way of an illustration: “All the centaurs are members of the United States Congress.” Centaurs are merely mythological beings, so no entities exist anywhere corresponding to the term centaurs. A scholastic philosopher would say that the statement is wrong. Why? For lack of supposition. There is no place where one can go to verify the statement; therefore, it is wrong — or even worse than that, meaningless — to suppose it. Not so in Symbolic Logic. In the language of Symbolic Logic, a class or set with no members, e.g., centaurs, is a Null-class. Symbolic logicians say that all universal statements about the Null-class are true. How is that? Because no one could produce a case to the contrary as a refutation of them. Thus, they argue: “A statement is true until it can be refuted.” The only way to refute a universal proposition is to find one instance to the contrary. In the Null-class, the class that has no members (nothing in it), a case cannot be produced to refute anything.***

    Any logic that says, “All the centaurs are members of the United States Congress” is true is undeserving of the name. Mind you, much that is bad can be predicated of members of the Unites States Congress, but that’s another issue entirely.

    The medievals perfected logic. Leibniz, Boole, Whitehead, Russell, and all these other post-“Enlightenment” figures who mathematized logic messed it up. At Saint Benedict Center and IHM School, we are “traditional” and “classical” in our approach to philosophy, including logic. Those fellows are the enemy.

    And if you really consider that one can be proven to equal zero, then what can I say? This “logic” is all reduced to alphanumeric trickery on paper and has nothing to do with reality. It is not “the art and science of correct reasoning.”

  • Larry_Dickson

    This detailed answer is food for thought. I need references showing examples from your second and third paragraph, although I notice that the Square of Opposition, as shown in Wikipedia, is perfectly equivalent to Boolean logic and Venn diagrams.

    I suspect the difference will lie in what is implied by your long 6th paragraph. Your “supposition” seems to mean “there is at least one”, but that is just a language question (e.g. “is zero a number?”). Thus the classic “all centaurs are members of Congress” is equivalent to the modern “there exists at least one centaur AND all centaurs are members of Congress”, and so is false. The modern expression just divides it up a little finer. Using expressions like “undeserving of the name” about this difference in terminology is political mud-slinging, not science or logic.

    You misunderstood my reference to “1 = 0” – you should have noticed that “proof” had quotation marks around it. Of course such a quasi-paradox is not a true proof, because it includes an incorrect step, a division by zero. The search for errors like this is an important part of mathematical (i.e. logical) checking, and that is why such quasi-paradoxes are part of mathematical education. Again, your heated expression “alphanumeric trickery” does not do justice to the work.

  • Pax. I can “de-escelate” the rhetoric (to use a neologism). Sorry I missed the quotes around the word prove.

    Here is some more food for thought:

    Those are notes from Brother Francis’ course in Minor Logic, which this little exchange has inspired me to post on the site.

    God bless and Mary keep you.

  • GarretKadeDupre

    Do irrational numbers prove math is illogical? :P (half-serious, I really am curious)

  • GarretKadeDupre

    I think St. Thomas’ version of Aristotelianism actually is required to understand the Real Presence.

  • GarretKadeDupre

    I love this article, not just because it’s pure genius overall, but because it implicitly dismisses Einstein’s Relativity theory as “fiction,” and boy do I despise Einstein’s theory, lol.

  • GarretKadeDupre

    Judging by the genius demonstrated in this article, the Brother understands the content perfectly.

  • Hylomorphism is necessary to understand what the Council of Trent says concerning the “form” and “matter” of all the sacraments. Further, the Ten Categories, at least their major division of “substance” and “accident” is necessary to understand what that same Council (and others) say regarding the Holy Eucharist. (I assume that your comment was inspired by this second point.)

  • I suppose irrational numbers could easily get set aside for that very reason, but “irrational” does not mean “without reason,” but rather, “without [mathematical] ratio,” meaning that the number cannot be expressed as a simple ratio of two integers. (See: , and , which I embarrassingly had to consult to remind myself of what an irrational number is!)

    Pi (π) is a classic example. I think that, because π actually quantifies something real, namely, the ratio of a circle’s circumference to its diameter, it is not illogical. The fact that π has been calculated to over a quadrillion decimal places with no discernable pattern to the numbers (and no end it sight) is a bit of a mystery in the natural order.

    As long as we recall that mathematics is the study of the accident of quantity, we can keep things in perspective. Accidents, too, are real, even though they are less important than substances. (Sanctifying Grace, however, is an accident (specifically a “quality”): don’t leave home without it!

  • GarretKadeDupre

    Oops. I’ve been using that “math is illogical because irrational numbers” argument for a long time … :P Thanks for pointing that out! The pi argument really drives the point home.

  • GarretKadeDupre

    Yes, that’s what I thinking of (the last one). When I learned about hylomorphism it was in the context of physics. My understanding is that everything physical is prime matter (calling it prime seems redundant) with a certain form, which I guess could be called shape.
    Not sure how this applies to the sacraments, though. I don’t see how confession could have a shape, or how it’s composed of matter since it’s not physical. Maybe that’s the next topic I will delve into when I’m feeling philosophical :)

  • Glad to help. I’d hate to see you have egg on your face, or Pi on your face, as it were.

  • Prime matter isn’t really redundant, as the term refers to matter that is totally without any form (which does not exist). Bread, for instance is the matter of the Eucharist, but it is not prime matter.

    “Morphe” literally means “shape” in Greek, as “Houle” means wood, i.e., that common material used for building. Aristotle used them in analogous senses. Aristotle’s “substantial form” has a deeper meaning than shape. Brother Francis’ book/lectures on Cosmology are very helpful here.

  • Garret Kade Dupre

    I’ve had correspondence with a Catholic physicist who advocates the hypothesis that space/prime matter/luminiferous aether are all the same thing. So there would be a bunch of prime matter between us and the moon! I personally find it quite convincing, but of course there’s no room to go into his theories here.

    Would you be able to point me to Brother Francis’ cosmological works? I love reading about cosmology from Catholics, as long as it doesn’t endorse those modern notions that seem theologically dubious (to say the least), like infinitely large cosmos/time-travel/quantum randomness/etc.

  • The top three options here should show Brother Francis’ Cosmology book, and the lectures (in both CD and MP3 formats):

  • Piotr Suwara

    “I don’t understand math, its connection to the real world and how it points to God, so math has no sense, no connection with reality and does not lead to God.”
    A stunningly weak argument!