The liturgy as the source of the intellectual life

When the first mass was said by Portuguese missionaries as they arrived in Brazil, a group of native indigenous men thoughtfully observed what was happening. After a while, another group of natives arrived at the place surrounded by priests and lay and were asking the others what was happening. To explain what they had understood, the first group of natives pointed to the altar and then to the sky. There was a clear understanding that what was taking place was not merely some theatrical animation or some mere dialogue. Those who did not understand the Latin language of the mass, nor had any knowledge of the Catholic faith already had a right intuition of what had happened in front of their eyes. This should be of great interest for us Catholics today.

Our Holy Catholic faith contains without a doubt the most intellectual tradition of any of the religions in the west. I do not say this out of any normalcy bias but rather out of the sheer evidence that is laid out every century where the Church is allowed to flourish. Think of the 12^th and 13^th centuries with the beginning of the university system where laity had finally the hope of receiving a higher form of education despite the barbarian invasions which had threatened the very foundations of civilisation and sought to destroy the classical literature which had once again begun to be read through the minds of many. It was the men of the Church that had embraced the Socratic Method, the Histories of Herodotus, the Euclidean geometry, the Roman law, the Aristotelian metaphysics and the Homeric poetry. All of this was included in the greatest system of education known to the west which was known as the quadrivium.

It was in the First Vatican Council which condemned the notion that the truths of the Catholic Church cannot be known by the light of human reason and the positive sciences. It anathematises he who claims that God cannot be known by the power of human reason. This was to oppose the errors of agnosticism which claimed that there was no rational argument for the existence of God. Faith rather was merely an article that cannot be explained through reason. This position came to be known as fideism and had as its defenders the German thinker Heinrich Jacobi and the Danish Søren Kierkegaard. We know by faith that the truths which the Church proclaims can be shown to be either rational or beyond the scope of reason, but never irrational. This helps us engage in the wide range of philosophical and theological arguments with others whilst having confidence in the infallible truth that our faith is reasonable.

However it does not simply stop there. In our engagement with arguments and debates with others, we can also demonstrate the faith in action. The highest form of contemplation as St Vincent Ferrer tells us, is the Holy Sacrifice of the Mass. The mass is indeed the centre of our Faith. It is at the altar after the consecration that Our Lord is truly present. Moreover, we receive Him! What a moment of supreme dignity that is! Even for the unbeliever, the concept of having a being of infinite superiority to all things in the universe made present in front of us should inspire wonder and atleast some awe. How much more should we Catholics be in bliss knowing that divine truth! The same who shed His blood for us on the Cross is now at the altar and offering himself for us. No one can ever express entirely the mystery of this truth. This is why many saints and mystics entered into ecstasy during Holy Mass. But for the non-believer how on earth, can we convey that the entirety of a solemn mass does not focus merely on getting together to express a public act of worship but rather on the Word Incarnate Himself? How can the liturgy convey the divine truths which lie at the foundation of our understanding of everything?

Thankfully, these questions are not unanswered in our Catholic tradition. When we walk in to what is believed to be the House of God, do we expect it to look like a friend’s house or like any other random commercial building? Indeed in the very Old Testament, for the holiest site regarded by the Israelites – the Holy of Holies – was solemnly ornamented and offered with incense. This was the ritual involved in the sacrifice of the lamb for the expiation of the sins of the jews. The Mass is equally the sacrifice of the unblemished lamb, of Christ on calvary. The Holy Council of Trent declares with regard to the particular liturgical celebration:

And whereas such is the nature of man, that, without external helps, he cannot easily be raised to the meditation of divine things; therefore has holy Mother Church instituted certain rites, to wit that certain things be pronounced in the mass in a low, and others in a louder, tone. She has likewise employed ceremonies, such as mystic benedictions, lights, incense, vestments, and many other things of this kind, derived from an apostolical discipline and tradition, whereby both the majesty of so great a sacrifice might be recommended, and the minds of the faithful be excited, by those visible signs of religion and piety, to the contemplation of those most sublime things which are hidden in this sacrifice.

The statement “and the minds of the faithful [are to] be excited, by those visible signs” catches my attention in this particular article of the Council of Trent. It should be therefore through the Holy Sacrifice of the Mass that we can best demonstrate to others the solemnity of the Faith, for it is the very heart and life of it. If the liturgy is transformed into anything else that does not convey the divine presence, it has ultimately become banalised and serves not to glorify God but to worship man. This is why symbology in the liturgy is extremely important. The proper manner of celebrating mass in the Roman Rite is by using the Latin language, which is of the Church and ad orientem, which is the direction which we all await the coming of Our Lord. It was in the protestant service that ministers began to face the congregations, transforming the ritual into an assembly, the focus of the celebrated onto the celebrant and above all, replacing God by man. Sadly as well with the introduction of the Novus Ordo Missae under the Papacy of Pope Paul VI in 1969, much of the spirit of Protestantism was introduced into the liturgy. Thankfully however, the immemorial mass, the old rite, which was a Latin translation of the Greek liturgy in the year 210 AD, is still celebrated and even encouraged by many groups in the Church.

If our intellectual life is still to be consistent with our prayerful life, then we ought therefore to seek the recovery of the mysterious and be open to that which is beyond our grasp of reason. Whether it is in the silence of the old mass during the most sacred moment – the consecration – or whether it is the Gregorian chant which elevates the spirit to the contemplation of the divine or the eastern direction which we all face awaiting the true presence of Our Lord on the altar, all of these liturgical traditions vitalise our mind and strengthen our conviction of the truths which we believe in. Our faith therefore does not merely end with conversations and dialogues with those outside of the Church, but is rather lived through the Holy Mass. When we talk of the Alpha and the Omega making Himself present at the altar, how do we expect Him to be treated? According to the same solemnity the Lord God commanded to the Israelites during the temple sacrifice or in a spirit of naturalism and human emotions? The liturgy ought to convey the presence of Jesus reigning supreme over all on that Cross rather than become merely an event which we go to feel better or fulfil some obligation which bears no truth or meaning. Let us seek therefore the restoration of the solemnity of the Mass which has sadly become banalised in the world in perhaps the greatest liturgical crisis the Church has ever faced.

This video offers perhaps a small comparison between the Old Mass (Tridentine) and the Novus Ordo being celebrated. Have a look for yourself here.

On Infinity

The rise of modern sciences has not given men an exemption from making claims which are not only counter-intuitive, but also patently irrational. The 20^th century physicist Hugh Everett had developed an interpretation of quantum mechanics which is known as the many-worlds interpretation. To simplify the general theory, it claims that apart from our universe composed of quasars, galaxies, black holes all the way down to leptons and quarks, there are many other universes which exist with parallel space-time realities. The theory was mostly formulated in order to solve the problem of any hidden variables in measurements at the quantum level and to deny the collapse of the wavefunction. Some physicists have gone as far as claiming that this means that not only are there many universes but that there are an infinite number of them. One of the other models of a parallel universe proposed by American physicist Brian Greene is known as the quilted model. In the book Hidden Reality: Parallel universes and the deep laws of the cosmos, Greene proposes a model for a parallel universe which is infinitely large and infinitely inflating. It must be mentioned that the parallel multiverse Greene proposes is not the exact thing as the many-worlds interpretation of quantum mechanics. He admits that it is purely speculative, yet nonetheless entertains such thought as a possibility. Here we will not delve in much on the criticisms of the many-worlds interpretation of quantum mechanics proposed by other physicists[1] but rather focus on the notion that there could ever be an infinite number of universes or that anything composed could be infinite.

Let us return to the very first of the 24 Thomistic theses. It states that:

1 – Potency and Act so divide being that whatsoever exists either is a Pure Act, or is necessarily composed of Potency and Act, as to its primordial and intrinsic principles.

Existence, as defined by St Thomas Aquinas is the act of being. In other words, that which is, is by virtue of being actualised. Now, Aristotle[2] claims that the infinite qua infinite is unknowable, so that what is infinite in multitude or in size is unknowable in quantity and what is infinite in variety is unknown in quality. Similarly, he also claims that it is impossible for a body to be infinite in composition[3]. The reason given is that if something were infinite in composition, we would run into to several contradictions. Composition presupposes a numerical identity and a certain number of attributes and things coming into being. If thing were infinite in composition, they would never come to be or their whole would never truly exist. Let us re-examine therefore the claim that there could be an infinite number of universes and ours just being one among the infinite clusters. If the amount was infinite, how could we have a number for it? If it was infinite, how could it even have an amount? Quantity and magnitude presuppose finitude and therefore measurability. So even if there is a plurality of universes, something which is still largely debated among physicists, there could never be an actual amount of infinite universes. Infinity exists therefore only in potentiality.

In Aristotelian physics, the principle of motion is caused through a transferring of potentiality to actuality. That principle stands as the basis of the whole of philosophy of nature and philosophy of science. It is true and virtually every observable experiment in the natural sciences confirms it. When a magnetic bar approaches a coil of wires, Faraday’s law tells us that there will be an induction of an electro motive force which is directly proportional to the rate of change of the magnetic flux. The stronger the magnet and the more coils the wire has, the greater the electromagnetic induction. In other words, a coil of wire is always potentially inducing a greater or smaller amount of electro motive force than what it actually is depending on the change of a few other potentialities. The potentialities are infinite yet the actuality of the force induced is one. It is always in this (actual) state rather than that (potential).

The other critique pointed out by Aristotle of the notion of an actual infinity is through mathematics. When we think of the concept of a line and its divisibility, we are setting finite points in it and removing them (subtracting) from its original length.

For any line A, if it is subtracted by B₀, it will still remain finite in size. The theorem is as follows:

A₀ = A – B₀

A₁ = A₀ – B₁

A₂ = A₁ – B₂

And Aristotle distinguishes between those theorems which are infinitely divisible in potentiality only to those which are actually divisible which would be by definition finite. The series infinite in potentiality is never actually completed, it is always possible to subtract more from the length of the line A or A_n­.

Georg Cantor had developed a theory of transfinite cardinal numbers, which argues that for any given set N, it will have an infinite cardinality. Cantor himself distinguished between a potential infinity, a transfinite and also an absolute. A set of rational numbers with an infinite cardinality is one which starts with {a₁,a_2­,a­₃,…a_n} thus having a never ending sequence. This theory lead to the other objection to the actual infinity was raised by the mathematician David Hilbert. The analysis given was in terms of an infinite hotel with infinite guests in it yet as one more guest arrives to check in, one would expect all him not to be able to. However, if all of the guests move one room number ahead of theirs, the first will be available and the new guest can check in. Now, if an infinite group of guests arrives to check in, all that is necessary is for all of the current guests in the hotel to move to the even number rooms and leave the odd-numbered ones available and equally all can go in. The cardinality of the odd-numbered rooms is equal to that of all the rooms in the hotel because both are infinite. This is patently contradictory. The symbol utilised by mathematicians to describe a sets with the lowest infinite cardinality is א₀, others similarly follow as א₁, א₂ and א_n. Yet as we know the cardinality of objects and sets is not to be found in material objects or anywhere in the physical universe. Infinite cardinal sets therefore exist in potentiality. Equally that which can be numerically identified has number and is therefore not infinite.

This is a brief overview of what philosophers since the birth of thought have debated over. But it is relevant inasmuch as the concepts still permeate popular culture with books by famous cosmologists which claim that we live not only amongst a plurality of universes but also in an infinity of them. There are problems with this claim which flies in the face of common sense and reason. Saying that there is an infinite amount of anything is as absurd as claiming that triangles have four sides. Never let anyone get away with telling you such things.

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[1] If you wish to read a more rigorous discussion of this, I would allude to the papers written by Adrian Kent and Shan Gao which can be found here: http://arxiv.org/pdf/gr-qc/9703089v1.pdf, http://philsci-archive.pitt.edu/9494/1/aa-mwi_further_v9.pdf.

[2] Aristotle – Physics (translated by W.D. Ross Oxford Clarendon Press 1908), 197a-b

[3] Ibid. Op. cit., 206a

Trying a Vocation

 

This is an extract from the booklet by Fr William Doyle SJ on the importance of considering whether or not one has a supernatural call to the religious vocation. Fr Doyle was a holy Irish Jesuit who served as a Chaplain during the Great War for the Royal Dublin Fusilers, 16th Irish Division. He died during the battle of Ypres 1917. He was known for his retreats, missions, penances and great intimacy with the Blessed Sacrament. We can all learn from this great priest the importance of this call to the religious life. The rest of the booklet can be found here.

 

 

Chapter 6 - Trying a Vocation

 

 

Spiritual writers tell us the evil spirit strives in every possible way to hinder all the good he can. If he cannot turn one away completely from the determination of giving oneself to God, he will work, might and main, to defer the moment as long as possible, knowing  that a person in the world is constantly exposed to the danger of losing both the grace of God and “the pearl of great price,” his vocation. He knows that until the doors of the monastery have closed behind the young Levite he has every chance of snatching away that treasure. He will lay traps and pitfalls, stir up doubts and fears; he will make the attractions of a life of pleasure seem almost irresistible, causing the bravest heart to waver: “I never realized how dear the world to me until I had to leave it” has been the agonizing cry of many.

      Under one pretext or another he induces them to put off their generous resolution from day to day. “O Lord,” exclaims St. Augustine, “I said I will come presently; wait a moment; but this presently never came, and this moment did not end.  I always resolved to give myself to You on the morrow, and never immediately.”

      How fatal this delay in responding to the call of God has been those can best tell whom age or altered circumstances have hindered from carrying out their first intention.

      If the vocation is doubtful, there is need of deliberation, and it must be serious, for hastiness and want of reflection would be unpardonable in such a matter;  but so enormous are the advantages to be reaped from a life devoted to God’s service, it would be a far greater calamity to miss a vocation through excessive prudence than to mistake a passing thought for the Master’s call.

      It is well to remember that a person who felt he had no vocation would not sin by embracing the religious state, provided he had the intention of fulfilling all its obligations and serving God to the best of his ability. For, in the opinion of the Angelic Doctor, God will not refuse the special graces, necessary for such a life, to one who sincerely desires to promote His glory.

      Our Lord tells us to learn a lesson from “the children of this world, who are wiser in their generation”; there is no hesitation about accepting a tempting offer of marriage, which binds one, perhaps to an unsuitable partner, for life; it is worldly wisdom not to delay about such a step when there is a chance of being well settled; and yet St. Ignatius teaches that there is more need for deliberation about remaining in the world than for leaving it.  He says: “If a person thinks of embracing a secular life, he should ask and desire more evident signs that God calls him to a secular life than if there were question of embracing the Evangelical Counsels.  Our Lord Himself has exhorted us to embrace His Counsels, and, on the other hand, He has laid before us the great dangers of a secular life ; so that, if we rightly conclude, revelations and extraordinary tokens of His will are more necessary for a man entering upon a life in the world than for one entering the religious state.”

      Endless harm has been done by well-meaning people, who, under pretext of “trying a vocation”, keep their children from entering a religious house for years.

      They urge that getting “to know the world” will develop their faculties and enable them to understand their own mind better; that such a process will broaden their views and help them to judge things at their proper value; finally, that a vocation which cannot stand such a trial, the buffeting of dangerous temptations, and the seductive allurements of worldly pleasure, to which it has been unnecessarily exposed, is no vocation and had far better be abandoned.

      “Is the world the place for testing a vocation?” asks St. Vincent de Paul. “Let the soul hasten as fast as possible to secure asylum.” The Church, realizing well the necessity of such a trial, prescribes at least a year of probation in every novitiate before admitting candidates to the religious profession. There, safe from the contagious atmosphere of a corrupt world, with abundant time for prayer and thought, with liberty to remain or leave at will, each one can test for himself the sincerity of the desire he felt to abandon all things and follow Christ, before he binds himself irrevocably by his vows.

      “One could not give a more pernicious counsel than this,” writes Father Lessius. “What is it in reality except the desire to extinguish the interior spirit, under the pretext of a trial, and to expose to the tempest of temptation  him who was preparing to gain the port of safety?

      “If a gardener were to plant a precious seed, requiring great care, in stony ground, covered with thorns ; if he exposed it to the rays of the sun and every change of climate to try would it grow in that unfavourable spot, who would not look upon him as a fool? Those who advise people called to religious life to remain, for a while, in the world have even less sense. A vocation is a divine fruit for eternal life. It is planted in the human heart, a soil little suited to its nature, and requires great care and attention. Watch must be kept that the birds of the air, the demons, do not carry it away; that thorns, the concupiscences and solicitudes of the world, do not  choke it; that men with their false maxims should not trample it under foot. Whosoever wishes to preserve and see grow in his heart the seed which the Divine Sower has cast there, ought to fly from the world and reach  a safe refuge as soon as possible.”

The implications of Gödel's incompleteness theorems

The problem of the foundation of the abstract science of mathematics was originally recognised by the mathematician David Hilbert in the presentation of the 23 axioms in the 1900 International Congress of Mathematicians. The core problem was revealed to lead into one of the greatest crises mathematics has ever gone through, namely knowing whether arithmetic and geometry have a solid foundation. Numerous schools of thought emerged and adopted logical tools which had their beginning with the mathematicians George Boole, Giuseppe Peano and Georg Cantor. This new language expressed mainly in logical axioms was known as formal logic and set theory. Various philosophers and mathematicians adopted this terminology in order to work out the foundations of mathematics. Those included Gottlob Frege, Bertrand Russell, David Hilbert and others. Of the 23 axioms, the 2^nd and 6^th of Hilbert’s problems should interest us here. The former dealt with the question of whether or not arithmetic could be proven as consistent and the latter with whether or not physics can be reduced to mathematics.

The Austrian mathematician Kurt Gödel had published in 1931 an essay regarding the incompleteness of second-order logic that would put the nail on the coffin of those who attempted axiomatising logically rules of arithmetic and Euclidean geometry. Gödel had realised that most logicists and formalists were taking the existential assumptions of logic for granted without attempting to prove the consistency or completeness of logic itself. One could in essence create a meta-logic to describe the metalogical principles within logic yet this would do nothing more than to appeal to another set of axioms and rules that would also suffer the same scrutiny of having to prove its own completeness. This process of justification in essence could go ad infinitum and we would never be able to prove the consistency of basic mathematical statements. The reason why this proves to be a major difficulty for logicists and formalists is because their own methodology - second-order logic - cannot be proved, disproved or be demonstrated as consistent. The other issue is that if a system of logic is demonstrated to be inconsistent and/or incomplete, it could not only be able to prove the axioms of mathematics such as the first axioms of Archimedes but also prove anything else. It was the final realisation that the problem of the foundation of mathematics cannot be appealed to by simply postulating another discipline in order to formalise it or analysing the semantic structures of equations and geometry. The mathematician John von Neumann commenting on the impact of the incompleteness theorems stated: “your result has solved negatively the foundational question: there is no rigorous justification for classical mathematics.”[1]

Here is an example of how the incompleteness theorem can be used. Suppose we have a simple formula

[1] 1 + 0 = 1

When we appeal to logical principles (Peano arithmetic) in order to demonstrate that this is true, we must invoke the law of addition to prove a basic rule of arithmetic.

[2] a + 0 = a

[3] a + S(b) = S(a + b)
[4] a + 1 = a + S(0) = S(a + 0) = S(a)

Yet precisely what Gödel’s incompleteness theorems demonstrate is that this method of logical proof is either inconsistent or incomplete. It can never be complete and consistent. This discovery has had a profound impact in the natural sciences and even in the philosophy of science. From two quite distinct perspectives, Fr Stanley Jaki OSB and Stephen Hawking have demonstrated that the incompleteness theorems proves to be a major problem for those who wish to either explain reality in terms of physics or to reduce everything to pure axioms of mathematics. Mathematics is finally demonstrated as being ω-inconsistent. An ω-inconsistent system is one which lacks ω-consistency and in other words it cannot prove that within infinite combinations of sentences, it will eventually be contradictory. As the Italian mathematician Piergiorgio Odifreddi points out commenting on the incompleteness theorems, the axioms of proof cannot be demonstrated by appealing to themselves. This was the major contribution of the incompleteness theorems. The analogy given by Odifreddi is that mathematics is not like a completed book but rather a library, constantly needing to have new volumes and works to sustain its life and its ragione d’essere. There is essentially no system that can demonstrate the completeness of mathematics.

Stephen Hawking places the smart money on the incompleteness of mathematics rather than on the inconsistency of it. However, he also sees the incompleteness theorem as a major problem for physicists who wish to reduce all phenomena to mathematics. This is directly linked with Hilbert’s 6^th problem. Hawking spots the problem with the positivist philosophy of science which states that a physical theory is a mathematical model. This means that if mathematical theories cannot be demonstrated as consistent, physical theories cannot be predicted or known[2]. This of course is due to the fact that mathematical formulae are derived from physical events such as we can observe with the equation derived from Newton’s second law of motion F = ma, where m is seen as the mass of an object and a as its acceleration or momentum over a time t. However if the positivist is right, even basic principles such as an equation which utilises multipliers cannot truly give us with certitude a full description of the physical phenomenon due to the fact that the axioms of the equation might be incomplete. Whilst the physical equations which explain the universe might be perfectly consistent, which is what the whole of science relies on, it cannot equally be complete. There is always a missing picture or explanation. In turn with the incompleteness theorem, not only is the positivist and logicist philosophy of science put at stake but also many scientific hypotheses. Fr Jaki argued that the neglect of Gödel’s theorems in science has lead physicists to wrongly think that they can develop a final theory of fundamental particles whilst knowing that it is complete[3]. The incompleteness theorems also demonstrated to be a major problem for those who wished to arrive at a physical theory of everything. It does not render such theory impossible but it equally does not allow physicists to fully know the coherence of a mathematically rigorous theory of everything. We would eventually never conclude or be able to prove logically or mathematically whether such theory was ever consistent or complete. It would simply just be a philosophical assumption. Fr Jaki also claims that what incompleteness means is that no physicist could ever construct a mathematical and/or physical theory that would somewhat undermine the contingency of our universe.

Our mathematical intuitions are derived from the physical phenomena; they are abstractions that we make of the world around us. We draw into our minds concepts of lines and icosahedrons from our ordinary sense experience. Although the incompleteness theorems only demonstrate the problem with using logical operators in an attempt to prove mathematics, it indicates to us that foundations of reality are not to be found in mere axioms but rather there are ontological and existential assumptions which are not provable in a system which takes them for granted. This invites us all to return primarily to metaphysics in order to explain reality and acknowledge the limits of mathematical enquiry and branches of natural science. Gödel showed to us the internal limitations of formalisms. We are still left with problems of being which cannot be reduced to axioms.

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[1] John von Neumann – Selected letters, Letters to Kurt Gödel, (Berlin 1930), P. 124

[2] Stephen Hawking – 2002 Dirac Lecture: Gödel and the end of the universe

[3] Stanley Jaki OSB – A late awakening to Gödel and physics, p. 8-10

The Relevance of the works of St Alphonsus Liguori

The name of this holy Doctor of the Church had come to me many times over the years yet I have never truly come to appreciate his contribution to the Church militant. It was only after listening to the awesome sermons of Fr Isaac Mary Relyea, a Franciscan whom I would consider the St Jerome of our times, that I came to know more about the moral theology of St Alphonsus Liguori. In his own day, St Alphonsus was considered by the laxed to be rigorous and by the rigorous to be laxed. Truly this Bishop of the Church, a man of prayer, penance and humility was with the church as opposed to the rigorous Jansenist heretics and those who wished to lax the Church’s teachings on morality and piety. Blessed Pope Pius IX declared him a Doctor of the Church in 1871. The Church teaches that his opinions on moral theology can lead no soul into error. It was only after reading the moral works of St Alphonsus that the Curé of Ars became a great preacher, excelling all French priests of his own time. It was also only after reading the works of St Alphonsus that Padre Pio became a good confessor. How much more can we sinners profit from his writings! St Alphonsus Liguori, pray for us and enlighten us all with the teachings which thou hast defended so thoroughly throughout thy life.

The works of St Alphonsus Liguori can be found on e-book format here.

On Plato's forms

It is widely defended throughout the early Socratic woks, the transition phase and the mature period of Plato¹ the doctrine of the forms or ideas. The doctrine states that apart from the material world, there exists a realm which transcends the senses and acts as the principle on which things are participating². An example of things which participate in the forms can be given in terms of things around us. This constitutes the ground of Platonic metaphysics. In fact, an entire school of metaphysics has been constructed around the theory of the forms known as Platonic realism. Ever since the 5^th century BC, there have appeared more theories and ontologies which have challenged Platonism. This on-going debate is known as the debate of universals and particulars. By the name the theory of forms would fall under the category of universals whereas tangible beings have often been seen as particulars. If we observe, say for example, a dog, we shall see that despite being numerically identical only to itself and unique, it still has something in common with other dogs. This is the obvious property of being a dog. The dog in this case is the particular. The principle which unites all of these together and brings them into being is the very form of dog itself which is beyond the limit of all dogs in which all participate. Therefore it can be said that the form of “dog” is the universal. In this essay we wish to demonstrate that the theory of forms does nothing more than to duplicate the material world and also to show the other objections which were presented by Plato’s contemporaries as possible solutions to the problem of universals.

Plato gives the description of the form of beauty in a manner which is best demonstrated in the Symposium³:

“and neither comes to be nor perishes, neither waxes nor wanes; next, it is not beautiful in part and in part ugly, nor is it such at such a time and other at another, nor in one respect beautiful and in another ugly, nor so affected by position as to seem beautiful to some and ugly to others.”… “the earth or sky or any other thing; but existing ever in singularity of form independent by itself, while all the multitude of beautiful things partake of it in such wise that, though all of them are coming to be and perishing, it grows neither greater nor less, and is affected by nothing.”

Similarly, in the Hippias Major⁴:

“Nothing is easier than to answer and tell him what the beautiful is, by which all other things are adorned and by the addition of which they are made to appear beautiful.”

It can be seen that this theory is an attempt to unify or to synthesise the properties of things which are observable. It is to argue that they have a ground of being in a transcendental realm of forms which are self-subsistent and of which all material entities have their objective ground of being. The philosopher and interpreter of Plato, A.E. Taylor argues that the fundamental forms of Plato are shapes, integers and mathematical objects⁵. Plato more specifically suggests a hierarchy of genera, species and being on which the ontological reality of the forms is divided and exist. They are further united and organised by an absolute principle, called the demiurge (Δημιουργός)⁶ which is best understood as a Supreme Being or God. This however might just be the fatal flaw in the theory of forms. If God is the uniting principle of the forms, and He is the one who created all of them, it begs the question as to whether or not they are necessary at all. It surely would seem possible that God could be the formal cause of all things without having to have forms as intermediaries or formal causes which transcend particulars themselves. Indeed this is the solution we propose in this work. To argue that the forms are what unite particulars truly does nothing more than to create or duplicate the reality of particulars to a reality of universals. What Plato did was not offering a proof for the existence of the forms in any substantial sense. It might be argued that his eloquence in explaining the uniting principle of particulars transcends the particulars themselves, yet they need not imply the existence of an actual universal in ontology. Here we side with Aristotle and St Thomas Aquinas who argued that Plato offered no substantial proof for the existence of the forms. In other words, the existence of particulars does not prove the existence of subsistent and transcendent universals⁷.

St Thomas states:

“According to Plato the Ideas are prior both to sensible things and to the objects of mathematics. But according to him the Ideas themselves are numbers; and they are odd numbers rather than even ones, because he attributed odd number to form and even number to matter. Hence he also said that the dyad [or duality] is matter. Therefore it follows that other numbers are prior to the dyad, which he held to be the matter of sensible things, and identified with the great and small. Yet the Platonists asserted the very opposite of this, that is to say, that the dyad is first in the class of number.”

This is one argument which shows not only the improbability but the inherent contradiction the Platonist faces when attempting to defend the realm of forms and ideas as concrete realities. It must be re-emphasised that the reason why Plato attempted to formulate an ontology around the forms was specifically to answer the problem of having particulars without universals. Yet there is a further dilemma which the Platonist faces which is known as the problem of the Third Man. The argument demonstrates that the multiplicity of forms leads to an infinite regress which renders the theory problematic if not entirely useless. This is because in order to explain the forms, there must be a principle which unites the forms. For there to be such a principle, there would be another which unites the principle and so on ad infinitum. In essence Plato attempted to explain the efficient causes of things by arguing that their ground of being was in this transcendental realm. However as has been demonstrated, the theory renders itself useless and problematic, conflating the existence of material things with real abstract properties. We must therefore solve the problem of universals without running into nominalist presuppositions in another manner.

Aristotle and St Thomas Aquinas recognised that particular corporeal beings are divided into form and matter. It is of particular interest to us to expand more on the notion of the formal causes which can eloquently explain the existence of beings in the world without positing the abstract realm of forms. The formal cause of being is that which gives it its essence, in other words, its genera and species. The formal cause of a human being is man itself. This is derived through the abstraction of the particular and individual human to a grasp of the essence which is common amongst all men. Hence when we use the predicate “is a man”, we presume that all beings who are men have a common property which can be known through the intellect via abstraction rather than participation in an actual form.

St Thomas Aquinas states⁸:

“For human nature exists in the intellect in abstraction from all that individuates; and this is why it has a content which is the same in relation to all individual men outside the soul; it is equally the likeness of all of them, and leads to a knowledge of all insofar as they are men. And it is from the fact that the nature has such a relation to all individuals that the intellect discovers and attributes the notion of the species to it.”

We therefore conclude by reaffirming that the Platonic theory of Ideas does not much in terms of self-explanation. Rather it attempts to solve the problem we encounter when attempting to explain the vast number of things of the same genera, species and properties. It suffices to say that Plato's argument for the transcendental forms are not proofs in the concrete sense but rather an appeal to what he deemed as the best explanation. This essay has demonstrated that this explanation is not only not the best but also faces serious problems when attempting to explain the infinite possibility of forms (third man argument) and also the fact that forms themselves do not have causation-like properties. It would require God to unite particulars and make them participate in the forms. This however would make the notion of the forms obsolete since if God exists, He could simply create individual beings without having to unite them to any transcendent entities.


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1 Frederick Copleston SJ – A history of Philosophy (Newman Press 1962), vol. 1, part III, p. 139-140

2 Republic, 596a

3 Symposium, 211-212

4 Hippias Major, 289d

5 A.E. Taylor – Plato: The man and his work, p. 512-514

6 Timaeus, 30a

7 St Thomas Aquinas – Commentary on Aristotle’s Metaphysics, Bk. I, Lesson XIV

8 De ente et essentia, a. 60

God's mercy according to St Anselm of Canterbury

This is from Chapter IX of St Anselm's Proslogion. A most beautiful work of prayer, meditation and of reason where he is famously known for saying "I do not seek to understand that I may believe. I believe that I may seek to understand" (Fides quarens intellectum). This work of St Anselm is also famous for its exposition of what is known as the ontological proof for God's existence. This chapter is to me the most beautiful of all of them. The saint here eloquently explains the infinite depths of God's mercy, even to sinners like us! Sancta Trinitas unus Deus, miserere nobis!

How the all-just and supremely just God spares the wicked, and justly pities the wicked. He is better who is good to the righteous and the wicked than he who is good to the righteous alone. Although God is supremely just, the source of his compassion is hidden. God is supremely compassionate, because he is supremely just. He saves the just, because justice goes with them; he frees sinners by the authority of justice. God spares the wicked out of justice; for it is just that God, than whom none is better or more powerful, should be good even to the wicked, and should make the wicked good. If God ought not to pity, he pities unjustly. But this it is impious to suppose. Therefore, God justly pities.
BUT how do you spare the wicked, if you are all just and supremely just? For how, being all just and supremely just, do you anything that is not just? Or, what justice is that to give him who merits eternal death everlasting life? How, then, gracious Lord, good to the righteous and the wicked, can you save the wicked, if this is not just, and you do not anything that is not just? Or, since your goodness is incomprehensible, is this hidden in the unapproachable light wherein you dwell? Truly, in the deepest and most secret parts of your goodness is hidden the fountain whence the stream of your compassion flows.
For you are all just and supremely just, yet you are kind even to the wicked, even because you are all supremely good. For you would be less good if you were not kind to any wicked being. For, he who is good, both to the righteous and the wicked, is better than he who is good to the wicked alone; and he who is good to the wicked, both by punishing and sparing them, is better than he who is good by punishing them alone. Therefore, you are compassionate, because you are all supremely good. And, although it appears why you do reward the good with goods and the evil with evils; yet this, at least, is most wonderful, why you, the all and supremely just, who lacks nothing, bestows goods on the wicked and on those who are guilty toward you.
The depth of your goodness, O God! The source of your compassion appears, and yet is not clearly seen! We see whence the river flows, but the spring whence it arises is not seen. For, it is from the abundance of your goodness that you are good to those who sin against you; and in the depth of your goodness is hidden the reason for this kindness.
For, although you do reward the good with goods and the evil with evils, out of goodness, yet this the concept of justice seems to demand. But, when you do bestow goods on the evil, and it is known that the supremely Good has willed to do this, we wonder why the supremely just has been able to will this.
O compassion, from what abundant sweetness and what sweet abundance do you well forth to us! O boundless goodness of God how passionately should sinners love you! For you save the just, because justice goes with them; but sinners you do free by the authority of justice. Those by the help of their deserts; these, although their deserts oppose. Those by acknowledging the goods you has granted; these by pardoning the evils you hate. O boundless goodness, which do so exceed all understanding, let that compassion come upon me, which proceeds from your so great abundance! Let it flow upon me, for it wells forth from you. Spare, in mercy; avenge not, in justice.