Argumentum Ontologicum and Argumentum Ornithologicum: Anselm of Canterbury and Jorge Luis Borges

Abstract

In this paper, the authors attempt to prove there is a relationship between Borges’ “Argumentum ornithologicum” and Anselm’s argument “Argumentum ontologicum”. We suggest Borges, using the image of a flock of birds, with oriental reminiscences, half joking, half serious attempts to prove the existence of God. We demonstrate the fallacies incurred by Borges and why his “Argumentum” has no place within the traditional set of ontological arguments. However, it would easy to forget that Borges’ claim is not philosophical, nor theological, nor apologetic, but rather ironic or paradoxical.

MARGARETE: Zur Messe, zur Beichte bist du lange nicht gegangen. Glaubst

du an Gott?

FAUST: (…) Wer darf ihn nennen? Und Wer bekennen ‘Ich glaub ihn’? Wer

empfinden, und sich unterbinden su sagen ‘Ich glaub ihn nicht’? Der Allumfasser, der AllerhalterFaflt und erhált er nicht dich, much, sich selbst? (Goethe, Faust)

1 What is the Ontological Argument?

An ontological argument proves God’s existence using a syllogism based on ontology. Unlike metaphysical arguments that demonstrate the existence of God through the study of being and its attributes, the ontological argument aims to reach this same goal based on the same concept of God by means of a concept “than which nothing greater can be conceived”. Throughout the history of philosophy many great philosophers have been concerned about this issue. There are many advocates of using this argument, though there are also many detractors. Even Christian authors such as Thomas Aquinas and William of Ockham have been shown their dissatisfaction with ontological arguments; though perhaps Emmanuel Kant provided strong opposition to these arguments with his famous aphorism “the definition of something does not mean its existence”.

The different variations of the ontological argument, and also the various reasons why it has been rejected, are themselves a proof, that in the ontological argument, philosophy faces a difficulty in resolving issues through agreement disagreement truth and falsehood. Like the phoenix, the ontological argument resurfaces again and again with new challenges to thinking at logical, gnoseological or metaphysical levels, and it is not uncommon in rejecting the argument, to retain the impression that something important is lost along the way. It is fair to note, moreover, that no supporter of the ontological argument has ever claimed that we have an adequate knowledge of the Divine Essence. The argument takes as its starting point our limited access to that Essence, i.e., a “cognitive reflection” of the same, or knowledge to show that the proposition “God exists” is immediately evident. Moreover, the question of the scope that a demonstration of God’s existence may have remains open. For more certainty than the test can confer, the “object” demonstrated is incommensurable with our experience, and does not provide access therefore, to the reality of God. It engenders the conviction that without the necessary existence of God, certain facts or natural evidence would not provide an ultimate explanation; this is not a minor matter, but not enough by itself for an act of faith, a conversion or to guarantee an epiphany. Except for the Hegelian philosophers, even those who believe that philosophy can talk about God admit that the name of God belongs rather to the realm of religion. It is both pre- and post-philosophical, as it compromises all dimensions of human existence, not only rational thought. Far from subtracting rationality from discussions about the problem of God, that conviction recognized a much higher discourse, that does not facilitate the work of the believer philosopher, but that makes this work more intense and demanding.

There is a constant in the ontological argument: it all started with an idea, a concept or a definition. Deductions are based on something abstract. Simplifying, the mechanism of any ontological argument is a necessary attribute, the characteristic of existence of the original idea. Thus, the existence of the entity referred to by this idea, concept or definition is a real or logical necessity. That concept is therefore compelled to exist. So according to Graham Oppy (2001, p. 4) “the premises of ontological arguments often do not deal directly with perfect beings than which no greater can be conceived, etc.; rather, they deal with descriptions of, or ideas of, or concepts of, or the possibility of the existence of, these things”

2 A Priori and A Posteriori Reasoning

It is common to argue that the traditional distinction between a priori and a posteriori demonstrates that the first moves from cause to effect, while the second does the reverse procedure from effect to cause. As God was not caused, proof a priori is excluded. As one might wonder whether it is possible to speak of God in a formal cause, in the sense of essence, and say that necessary existence is caused by the very essence of God. But while God’s name makes sense, it does not contain the essence of God, but only a reflection of it, so that His existence does not follow from His essence, as in saying that there exists what we understand by God. Furthermore, according to the same mode of reasoning, if existence is implied from essence, this would imply something absurd like something caused without existence. Therefore, any demonstration of the existence of God should be a posteriori, i.e., existence is argued from something whose ultimate cause is sought. It might be objected to this rejection of the ontological argument that it forgets that the argument has not divine essence as a premise, but our knowledge of it. Although the name “God” does not indicate the fullness of the divine essence, however, it does say something about that essence. Otherwise, in preaching about the existence of “God,” we would be preaching something else. So it’s always coming to a notion of God in which we see that a being cannot have just any form of existence, that is, it necessarily exists.

In modern thought, however, an a priori demonstration is understood as having its validity justified, regardless of experience. Knowledge “a priori” contains, not the efficient cause or other process on which the reasoning is possible, but the formal cause of knowledge and reason. I.e., a priori knowledge is not knowledge that would start from the formal cause or essence of the thing, but starts on the manner or organization of knowledge as such. If we understand a priori reasoning as that which is derived from the form or formal cause of knowledge and not their particular objects, then we must examine whether this form is capable of serving as a principle of reasoning. Clearly, the issue is also relevant to any rational argument, since the form of knowledge is present in all objects of intelligence, and in all operations of reasoning. Paradoxically, this issue cannot be resolved a priori, but depends on the mental operations involved. Ignoring the problem and going directly to particular formed objects of knowledge in order to obtain some conclusion, would, on one hand, remove the problem of knowledge and, on the other hand, not give an explanation of reasoning as such, which is a mental process and not a step from one thing to another in reality. Do not forget that a posteriori arguments are called mental operations and their force is not given by their objects, but by the reliability of reasoning itself, its principles and its form. That is, the operations always assume the principles of reason, under which they pass from an observed fact to a conclusion.

Transition from effect to cause, is given under the principle of causality, whose universal validity cannot originate in this transition because it is judged precisely as an “effect”. Therefore, the dynamic force of the argument is explained and justified by the origin and validity of the principles, which confer a need for the conclusion. In this second sense, a priori argument will not be deducted from the essence of God, but from the operations of reason, i.e., the formal cause of knowledge. The possibility of a priori reasoning depends on how something can be inferred from that possibility and, if this were possible, it would explain the legitimacy of the a posteriori arguments. The name of a priori is not forced, but very convenient. Mainly because it would not make sense that a priori means “from the foregoing in the order of being”, because that would just imply God, whose existence is disputed, and supposed to have been already known in the order of things. The very possibility of this type of argument would be ruled out from the start by a terminological question.

But can one derive something from the idea of a universal being? Examples are the first principles. If its validity came from a particular thing, its applicability would be restricted to the said thing and the principles would have at most an inductive value, but would not be necessary and therefore would not have universality. That’s why they cannot rely on any knowledge whether particular or general. Its universality comes from a different nature, there is no reason not to call it ideal.

The idea of being is essentially objective, and subsumes knowledge of the subject, even of the knower. If it was subjective in nature, and participated in the same contingency and limitation of the subject, it would not exist objectively and therefore would not need the status of universal knowledge. But if this universal knowledge is objective, everything that follows from it, regardless of what it refers to, is universal and necessarily true. For that reason the possibility of a universal and necessary knowledge of which the form of knowledge is objective and independent of the subject, and also thanks to it, the subject himself is known. A priori reasoning is that which is performed from the idea of a universal being, without any addition. And we have valid a priori reasoning which proves the existence of God, if from the idea of universal being we can prove that there must be a real infinite being. In other words, if it is possible prove that it is contradictory to the form and principles of reason to suppress the existence of an absolutely necessary being. Such an argument would not be properly ontological, but neither would one classify it as a posteriori, as it is not deducted from any specific finite thing or any experience of the world, strictly speaking. It could only be called a posteriori as the idea of being is not God, and thus it is understood as only an effect of God, as it would be a complete and subsistent real thing.

This form of intelligence, moreover, would not be solely composed of regulatory rules, that would confer form to objects of knowledge or judgment validly. More than being a function of intelligence, their interest would be that which the object (not just a thing) is, and from which something can be inferred. That the principles by which something can be inferred from it, have in turn their origin in the same form of intelligence is neither a circular argument nor petitio principii, but has the same characteristics as the idea of being. That is, the first rule of intelligence, its form, is at the same time, regulatory and objective. And that means it is both a source of the validity of any reasoning, and as something in itself, namely the ideal being, intelligible being, a being comprised in the idea.

3 The Ontological Argument of Anselm of Canterbury

The philosopher who developed this well known a priori argument was the Benedictine monk Anselm of Canterbury, a scholastic philosopher, theologian and Church Father, to the extent that it is often presented as owned by the author himself: “the ontological argument of St. Anselm.” Augustine’s influence had led him to try to prove from a rational standpoint, truths we know by revelation. The famous Augustinian aphorism “quaerens intellectum fides” (faith seeking understanding) pushed him to end the use of reason in defense of the truths of faith. Anselm concluded that the divine essence is “that Being which is altogether greater and better than whatever is not what it is.”¹ This method, which can be described as epistemological, helped him to establish the existence of God from a rational point of view. In his Monologio he proposed various a posteriori arguments based on the principle of causality like Aquinas with his famous “five ways”. In another of his works, the Proslogion Anselm proposed the ontological argument in the second chapter. Here, using the a priori method he descends from God to men. Since the end of the first volume had established that nothing greater than God can be conceived,² or in other words, God is that great being of which it is impossible to imagine another, which is greater. We see the notion of God as the greatest being that can be imagined, which is corroborated by the dictates of faith. Subsequently Aquinas established the importance of faith to confirm any statement made from reason, although as we shall see, the ontological argument is strictly rational.

Let’s look a little more closely at the ontological argument, in a simpler way:

• (1) Since the conclusion drawn from the first book of Anselm, Monologio, it is possible to claim that God is the greatest being that can be conceived.

• (2) Having accepted this premise can be reached through reasoning that when a person hears about God, he understands in his understanding; though not fully understanding the idea of God.

• (3) We must accept that what exists in reality is greater than what exists only in thought. So reality is more perfect that a concept in our understanding.

• (4) If the being that is greater than can be conceived exists only in our minds, this would be greater than what could be conceived, but this leads to an obvious contradiction.

• (5) So we have no other solution than to acknowledge that God exists. One might say that reason leads us to believe in God, or else we fall into the absurd.

It should be clear that this ontological argument is valid as we recognize that existence is an attribute of perfection. Anselm assumes that anything that exists in reality is more perfect than something that only exists in our intellect. Also keep in mind that God is immaterial and without needs, and so this argument is perfectly applicable, but it does not work for material and contingent realities.

Anselm’s demonstration uses a form of argument known as reductio ad absurdum. First, the truth of some proposition is assumed. Then it is demonstrated that this leads to a contradiction, and from there, since contradictions are false, the falsity of the alleged proposition is demonstrated. In more schematic terms, the ontological proof proceeds as follows:

$$\begin{array}{*{20}l} P \hfill \\ \vdots \hfill \\ {Q \wedge \neg Q} \hfill \\ { - - - - - - } \hfill \\ {\neg P} \hfill \\ \end{array}$$

In more intuitive terms, the scheme tells us that if a proposition implies a contradiction, such a proposition is false.

For Anselm, the statement “God does not exist” (which is unthinkable for him, and contradictory), can only be said, on condition that one distinguishes between two ways of thinking about something. A thing can be considered differently, one when one thinks what the word means, or two, when intelligence perceives and understands how the word is used. In the first sense we can think that God does not exist; and the second God does not exist. One who understands what God is cannot think that God does not exist, but can pronounce these words in themselves and without attributing any meaning, or attributing a meaning… For Anselm, the statement “God does not exist” is either a meaningless statement, or a statement whose meaning is vague. In the first case, what we do is to consider the name “God”, without attaching a specific meaning, which does not produce any genuine proposition that denies something. In the second case, we can say that God does not exist, provided we understand by “God” what Anselm meant by that name, that is, we can say that God does not exist provided that “God” has not its usual meaning, which does not produce a genuine denial. For Anselm, there is no alternative: either the proposition “God does not exist” is contradictory or its opposite is true or it is not contradictory but meaningless or ambiguous.

In Chapter II of Anselm’s Proslogion (Anselm 2008) the ontological argument is presented; however, it is not easy to identify precisely the argument within the Latin text. Some authors see in that chapter II two arguments: the main argument that concludes with the statement that God exists in reality and an auxiliary argument, where the burden of proof would be found, and which concludes with the assertion that God exists in the mind.

The premises consist of the definition of God as id quo maius cogitari nequit, the id being somewhat generic and not specific, and that when we give a definition of any concept, we make the definition abstract rather than individual.

Definition 1

The Divine is anything greater than which nothing can be conceived, i.e., anything that can think, and that is greater than anything that can be thought.

With this definition, we are indicating that for every element $\aleph$, $\aleph$ is divine, if and only if $\aleph$ can be thought, and that no other element $\aleph^{*}$ that $\aleph^{*}$ might be thought, and $\aleph$ is greater than $\aleph^{*}$. The biconditional is to express the logical sense of the definition of the divine, that is, that if something is divine, can it then be thought, and furthermore there is no other being which can be thought and is superior to him. If something can think and also there is no other being that can be thought and exceeds him, then, that something is divine.

Axiom 1

Any being can be thought and exists in reality.

With this axiom there is an implicit assumption in Anselm’s reasoning, and that is a worldview based on realism. He is telling us that there is at least one thing $\aleph$ such that $\aleph$ can be thought, and $\aleph$ exists in reality.

Axiom 2

Any being that can be conceived and exists in reality is superior to any other that may be conceived but that does not exist in reality.

This axiom means that for any two beings $\aleph$, $\aleph^{*}$ if one of them $\aleph$ may be thought, and also exists in reality, and the other $\aleph^{*}$, can be thought of, but does not exist in reality, then $\aleph$ is greater than $\aleph^{*}$.

Conclusion

Both must exist in the understanding and in reality, a superior being for which a greater cannot be conceived.

The conclusion is telling us that there is at least one element that is divine, that can be thought, and that also exists in reality.

Let K be the operator of knowledge, $\left| {\aleph D(\aleph )} \right.$ be “is divine” and $\left| {\aleph S(\aleph )} \right.$ be the operator “is superior to”. To formalize the above statements into a first-order language, we use the following non logic predicates: $\left| {\aleph D(\aleph )} \right.$, $\exists_{M}$ (It can be thought, conceived, etc., that is, exist in minds) and $\exists_{R}$ “exist in reality”. Let $\left| {\aleph D(\aleph )} \right.$, $\exists_{M}$ and $\exists_{R}$ be monadic predicates and S be dyadic predicate.

• (1) $\left| {\aleph D(\aleph )} \right.$ or $\aleph$

• (2) $\exists_{M}$$\aleph$

• (3) $\exists_{R}$$\aleph$

• (4) $\left| {\aleph S(\aleph )} \right.\,\aleph *$ or $\aleph > \aleph *$

Thus, the premises of the argument of Anselm will be as follows:

• For the definition 1: $\forall \aleph (\aleph \leftrightarrow \exists_{M} \aleph \wedge \neg \exists_{R} \aleph *(\exists_{M} \aleph * \wedge (\aleph > \aleph *)))$

• For the axiom 1: $\exists_{R} \aleph (\exists_{M} \aleph \wedge \exists_{R} \aleph )$

• For the axiom 2: $\forall \aleph \aleph *((\exists_{M} \aleph \wedge \exists_{R} \aleph ) \wedge (\exists_{M} \aleph * \wedge \neg \exists_{R} \aleph *) \to \aleph > \aleph *)$

• Conclusion: $\exists_{R} \aleph (\aleph \wedge (\exists_{M} \aleph \wedge \exists_{R} \aleph ))$

However, from the data provided it can be demonstrated that the conclusion does not follow from the hypothesis.

Theorem 1

$\{ Definition\,1,Axiom\,1,Axiom\,2\} \ne conclusion$

Proof

It is sufficient to find a model M where hypotheses are true and the conclusion false. Such a model is: $M = \left\langle {U,\left| D \right.^{M} ,\exists_{M}^{M} ,\exists_{R}^{M} ,\left| S \right.^{M} } \right\rangle$ where $U = \left\{ 1 \right\}$, $\left| D \right.^{M} = \emptyset$, $\exists_{M}^{M} = \exists_{R}^{M} = \left\{ 1 \right\}$, $\left| S \right.^{M} = \left\{ {\left\langle {1,1} \right\rangle } \right\}$

In this model all the hypotheses are true, however the conclusion is false:

$$M \ne \exists_{R} \aleph \left( {\aleph \wedge \left( {\exists_{M} \aleph \wedge \exists_{R} \aleph } \right)} \right)$$

because

$$\left| D \right.^{M} \cap \exists_{m}^{M} \cap \exists_{R}^{M} = \emptyset$$

Then Anselm’s ontological argument is false.□

3.1 Variations of Anselm’s Ontological Argument

We consider two variants of Anselm’s ontological argument. In the first, the conclusion is modified, taking as such a universal statement instead of the existential statement. The conclusion, although correct, does not prove the existence of God. In the second, the set of hypotheses includes an existential premise. The hypothesis is so strong, that this adds to the difficulty of accepting the desired conclusion. It is the fallacy known as begging the question.³

3.1.1 First Variation

We have considered the logical form of the conclusion, as an existential generalization $\exists \aleph conclusion$ and not as a universal generalization $\forall \aleph conclusion$, because it is the right expression of the conclusion of the argument. Furthermore, if the argument is intended to conclude with something valuable, one should reach the affirmation that there is at least one element in the intersection of the three sets corresponding to ${\text{D}}\aleph$, $\left| {\aleph D(\aleph )} \right.$, $\exists_{M} \aleph$ and $\exists_{R} \aleph$, the concept of the divine is not empty. Since arriving at the conclusion from the above premise that everything is divine must exist in thought and in reality, that is, $\forall \aleph \left( {\left| D \right.\left( \aleph \right) \to \exists_{M} \aleph \wedge \exists_{R} \aleph } \right)$ does not give us anything revealing, as an a priori demonstration of the existence of God. The empty set is accepted in logic as an important feature, and we know that $\forall \aleph \left( {\left| D \right.\left( \aleph \right) \to \exists_{M} \aleph \wedge \exists_{R} \aleph } \right)$ is true if and only if we interpret $\left| {D\left( \aleph \right)} \right.$ as $\emptyset$, i.e., if we say that there is no being that is greater than which nothing can be conceived, or what is the same, that the concept $\left| {D\left( \aleph \right)} \right.$ is not instantiated in any element. Currently such an argument proves nothing, since its conclusion is a conditional which states that if something is divine, then that something exists in the mind and in reality. But it does not ensure the existence of any being that is divine.

In the current approach to logic, from the definition of a concept, it does not follow that this concept is different from an empty one, demonstrates that it is not, and requires further demonstration of existence. We know, however, that in medieval logic, not knowing the empty quantification, to conclude that for all $\aleph$, if $\aleph$ is divine, then $\aleph$ exists in the mind and reality, could be accepted as a valid test for the demonstration of God a priori, due to the presence of an existential commitment to universal statements. Whereupon, the above reasoning is not an a priori proof of the existence of God to us, and could be accepted as a valid proof a priori to medieval logicians.

3.1.2 Second Variation

If we introduce a new premise, the ontological argument in Proslogion automatically becomes a new form of valid reasoning. The new premise is formulated as follows:

Axiom 3

The divine, i.e., that greater than that which it is conceivable; it may be thought, that is, exists in the mind. $\exists \aleph \left( {\left| D \right.\left( \aleph \right) \wedge \exists_{M} \aleph } \right).$

Then

$$\{ Definition\,1,Axiom\,1,Axiom\,2,Axiom\,3\} = conclusion$$

The problematic nature of this new premise lies in the fact that they have heard the words of the definition of Anselm: “than that which nothing is conceivable”, does not imply in the mind of the listener, such a thing, in the sense that is needed by the argument. Furthermore, the introduction of $\exists_{R} \aleph \left( {\left| D \right.\left( \aleph \right) \wedge \exists_{M} \aleph } \right).$ is for us ensuring that the set of the divine is not an empty set, as there is at least one element in the intersection of the sets of “the divine” and “able to be thought”, so that an element belongs, given a structure M, in both set $\left| {D\left( \aleph \right)} \right.^{M}$ as the set $\exists_{M}^{M}$. Having asserted this, we have ensured that the divine is an exemplifying concept, because there is at least one element that falls under it, and therefore, the only step remaining is to show that the predicate of existence in reality can be applied. That is why, in the introduction of the axiom 3 it can be seen as a petitio principii, since it is taking as a premise that there is at least one element that satisfies the property of being divine. With other assumptions, the only thing that one can do is try to ensure that this being falls under the concept of “exist in reality.”

3.2 Gaunilo’s Objection to Anselm’s ARGUMENT

The monk Gaunilo of Marmoutiers, used an ontological argument for the existence of the perfect island in the Innocents (Gaunilón 1952). This island, according Gaunilo, is not likely to exist. However, according to Anselm’s argument, in that case we would not be devising the largest and most perfect conceivable island, as it should have all the attributes of perfection and greatness that can be conceived, including existence. According to Gaunilo, this argument may seem absurd and contrary to reality, it is not more so than Anselm´s argument.

Guanilo´s argument is included in objections that the ontological argument results in claims for the existence of too many perfect things. There would be a saturation effect: it is not intended to show where or how the argument fails, but simply Guanilo reasons that if the ontological argument is accepted as a valid way of thinking, then it should accept that the conclusions of all those arguments that are formally analogous to it lead to absurdities and are contrary to experience. Such arguments indeed saturate the universe with an indefinitely large number of necessarily existing perfect islands, perfect rabbits, perfect pens, and perfect combs and so on. In addition, the counterargument of Gaunilo points out what has been seen as the main weakness of the ontological argument, that something is more perfect existing than not existing. This has been designated as elusive and meaningless, since it involves comparing something that exists with something that is not existing.

4 The Argumentum Ornithologicum of Jorge Luis Borges

The first to attempt a demonstration of God’s existence from a flock of birds was Jorge Luis Borges. This genial demonstration collected in El Hacedor under the name of “Argumentum ornithologicum” (Borges 1996, 165 p.):

I close my eyes and see a flock of birds. The vision lasts a second or perhaps less; I don’t know how many birds I saw. Were they a definite or an indefinite number? This problem involves the question of the existence of God. If God exists, the number is definite, because how many birds I saw is known to God. If God does not exist, the number is indefinite, because nobody was able to take count. In this case, I saw fewer than ten birds (let’s say) and more than one; but I did not see nine, eight, seven, six, five, four, three, or two birds. I saw a number between ten and one, but not nine, eight, seven, six, five, etc. That number, as a whole number, is inconceivable; ergo, God exists.

There are not too many works in the scientific literature dedicated to the study of Borges’ ornithological argument, possibly as Cantarino (1976) argues, due to its brevity. Anselm starts from the analysis of an idea and his proof is not a posteriori, as in the work of Aquinas and Duns Scotus. The argument presented by Borges starts from a visual image, that is to say something concrete, a flock of birds, and so it would seem to be an a posteriori test. But later, Borges works with the number of birds in that flock and from there a derivation is made that approaches Anselm’s argument. It is not the aim of the authors to make a linguistic or aesthetic analysis of the Borges text, but rather to express in logical-mathematical language the arguments presented by the brilliant Argentine writer. This is not to undervalue the wonderful Borgian irony of this text (Rios 2018).

The parts of the argument are as follows:

(1) I close my eyes and see a flock of birds.

Borges parodies the start of the third meditation of Descartes (1982):“Claudam nunc oculos, (…) avocabo omnes sensus, imagines etiam rerum corporalium omnes vel ex cogitatione mea delebo” [“I will close my eyes, (…) suspend my senses; until I will wipe from my mind all images of corporeal things.”]. Borges refutes Descartes: he does not write in the future but the present. He goes the other way: despite who closes his eyes, he does not suspend the “images of corporeal things.” This flock of birds is the starting point. One should note the contrast with Anselm and the rest of the philosophers who have sought arguments of this type. Borges does not take a concept or a definition but an image. This is a novelty in the history of ontological arguments. The flock of birds that crossed the mind of Borges is unique. Only he saw them, only he saw the birds on that occasion. In this sense it is singular. However, whoever studies “Argumentum” plays in imagination his own flock of birds. So that image has a universal resonance. Although Borges refers to a particular flock of birds, any flock of birds may help prove the existence of God as long as we do not know the exact number of birds. Borges provides the step from the singular to the universal.

(2) The vision lasts a second or perhaps less.

It seems Borges meant to suggest that this vision was given. The vision is instantaneous, so that the observer is unable to know the exact number of birds flying. Not knowing the exact number, Borges suggests that vision was not caused by him. The vision:

• (1) May have been induced by another.

• (2) Came casually.

There are three possible interpretations:

• (1) The vision is an inspiration. It is impossible, since Borges disbelieved the sweeping inspiration.

• (2) It is a product of chance.

• (3) God sent the image.

If the first is impossible, there are two possibilities: “coincidentally” a flock of birds has crossed the mind of Borges or, on the contrary, that God himself has intervened.

(3) I don’t know how many birds I saw.

From Image to the idea or concept, Borges brings up the concept of number, even implicit in this stage of “Argumentum”. It will strengthen his demonstration in the number problem, not in relation to the image of the birds. Although Borges seems to move at the intellectual level when he comes to the concept of number, we know that ultimately his argument is based on an imaginative representation. Unlike Anselm, for whom it is clear that there is “something beyond which nothing can be thought” or Descartes’ cogito, Borges lacks intellectually unshakable certainty. His only certainty is visual: no doubt there loomed in a few birds flying.

(4) Were they a definite or an indefinite number?

Borges does not know the number of birds and pretends that’s the main problem. He moves the problem to another plane: the nature of numbers. He was then asked whether the number of birds was definite or indefinite. We know that the number is finite because there are no endless flocks, because all things existing in reality make a finite universe. Although the birds are a picture, they constitute a fortiori a finite number, since nobody can imagine the infinite, as it is also impossible to imagine nothingness or eternity.

Undefined is “without fixed limits; indefinite in form, extent, or application.” Borges does not try to say that the number is infinite but is unknown. The problem is not in the finitude and infinitude of numbers but in their ignorance. As Enrique Garcia de la Garza. says:

Borges recognizes that the number is inconceivable… The reason for his statement is that the number, although it is in the finite set that goes from 1 to 10, does not correspond to any of the whole numbers that make up that universe: ‘but I did not see nine, eight, seven, six, five, four, three or two birds’. In other words: the number is inconceivable because, in the case of an integer less than 10 and greater than 1, it does not correspond to any of them. That number is therefore inconceivable. Let’s call it ‘5’. From the above, it follows that the number ‘5’ is not only undefined but also inconceivable. Our minds contain a strongly ingrained concept of inconceivable so that we understand when Borges writes: ‘the whole number is inconceivable.’ It is impossible to doubt that ‘5’ existed, at least in the mind of Borges. Since ‘5’ existed, it is possible that another indefinite and inconceivable being exists, like ‘6’ or even God. God is also an indefinite and inconceivable concept, less so in the Western tradition that embraced Borges. In Logic, possibilities imply need. It is possible, as it was said, that the inconceivable being that we call God exists. Then, it is necessary that God exists ‘Ergo, God exists.’ As you can guess, this procedure is closer to Anselm’s argument. But it is equally fallacious. Gaunilón was right

(5) This problem involves the question of the existence of God.

Borges no longer knows if God guarantees the truth of the statement or, on the contrary, the statement makes possible (or at least discloses) the existence of God. It is possible that Borges supports Gödel, and links in their own way, God with mathematics. It is interesting to notice that Borges here includes himself in the chain of mathematical attempts to reach God.

(6) and (7) If God exists, the number is definite, because how many birds I saw is known to God. If God does not exist, the number is indefinite, because nobody was able to take count.

In the Judeo-Christian tradition, one of the attributes of God is complete vision. Theologians resolved first that God has complete knowledge and so knows the world of real beings. The proof that God also knows human thoughts is, according to the unanimous consensus of theologians, evident in God’s knowledge of sins in thought. Through this certainty of offending God with a thought, we are certain that God reads human minds.

(8), (9) and (10) In this case, I saw fewer than ten birds (let’s say) and more than one; but I did not see nine, eight, seven, six, five, four, three, or two birds. I saw a number between ten and one, but not nine, eight, seven, six, five, etc.

Borges saw at least two birds but not more than nine. There is a number from one to ten indicating the exact number of birds. Given finite possibilities, Borges had to have seen a number between one and ten. That Borges does not know the number does not imply that this number is not (by itself) knowable. Borges is aware of having seen a finite number. Ignorance of this number leads him to deny every single possibility. He recognizes that it is possible to choose a number between those on the universe of one to ten, but when he stops to consider each of the individual possibilities, he hesitates. Consideration of each option takes him to the next. But ultimately, he ends up discarding all and each of the individual possibilities.

(11) and (12) That number, as a whole number, is inconceivable; ergo, God exists.

From the formal point of view, here evidently is an argument consisting of a series of logically related propositions such that given the premises the conclusion necessarily follows. That is, it is a syllogism. However, the complexity of this syllogism is the beginning of the dilemma (God exists = God does not exist), and where the second term led to their absurd conclusions demonstrating the truth of the first, thus reaches the conclusion-form solution of the dilemma (ergo God exists). The veracity of syllogistic⁴ reasoning used here depends on the legitimacy of the proposition that we consider in the dilemma, and the logic that deduced the conclusions which lead to the absurd end. Only if both legitimacy and logic pass the scrutiny of analytical reason, can we admit that the argument itself is legitimate and logical, and therefore worthy of being taken into account (Reinstadler 1937).

4.1 First Interpretation: Inconceivable (Ineffable) = Undefined

According to the Oxford Dictionary inconceivable is “that cannot be imagined”. Borges said that the number has to be finite, id est, defined. However, he explains that this number is “inconceivable” or “ineffable” (Usó-Doménech et al. 2015). By inconceivable (ineffable) we understand undefined. What is the basis for this statement? His ignorance of the number. This is the fallacy of modus tollendo tollens. Let $\aleph .$ be God and B be the set of birds such that $CarB = 10$ and $B_{1}$ is a subset such that $B_{1} \subseteq B$ and $CardB_{1} = n,\mathop {}\limits_{{}} 1 \le n \le 10,\mathop {}\limits_{{}} n \in N$. Let K be the operator of knowledge and S be the observer subject (Borges, in this case). That the number is defined means knowing the cardinal number of the subset of birds B₁. Then

$$\begin{aligned} & If\,\,\exists \aleph \Rightarrow KCardB_{1} \\ & \neg KCardB_{1} \Rightarrow \exists \aleph \\ \end{aligned}$$

The error lies in that of negation of the consequent, and affirmation of the antecedent does not follow.

(2) Borges explained on line 10 that the number of birds is a defined number. Founded in this apparent certainty, he shows that God exists. However, this reasoning is the fallacy of modus ponendo ponens:

$$\begin{aligned} & If\,\,\exists \aleph \Rightarrow KCardB_{1} \\ & KCardB_{1} \Rightarrow \exists \aleph \\ \end{aligned}$$

The error lies in that of affirming the consequent, affirmation of the antecedent does not follow.

Three premises can be established:

• (1) The flock of birds is formed by a (natural number) precise number of birds (line 4). So the cardinality of B₁ is known.

• (2) That number is unknown (line 3). So the cardinality of B₁ is unknown.

• (3) A couple of options (lines 6 and 7).

  • Option 1 If God exists, then God knows the number of birds. That number would be undefined for Borges but defined for God. That is, if God exists, the number is defined (line 6).

  • Option 2 If God does not exist, then the number is (absolutely) undefined, since neither Borges nor anyone else, can know that number (line 7).

The two possible minor premises are:

• (P1) The number is defined. The reason is that God knows it, because of the argument of omniscience. Here a petitio principii is committed, because who recognizes the divine omniscience previously accepts the existence of God.

• (P2) The number is undefined. The reason is in line 3: Borges does not know it. That is, the number is defined for God but undefined for Borges.

Whoever chooses option 1 will take naturally, almost by instinct, P1. And whoever chooses option 2 will take P2. Two syllogisms can be formed:

(1) Syllogism 1

Option 1::

If God exists, the number is defined

P1::

The number is defined

Then: God exists.

This syllogism may be symbolized as follows:

$$\begin{aligned} &If\,\exists \aleph \Rightarrow KCardB_{1} \\ & KCardB_{1} \Rightarrow \exists \aleph \\ \end{aligned}$$

This is a fallacy because it is a variation of modus ponendo ponens.

(2) Syllogism 2

Option 2::

If God does not exist, the number is undefined

P2::

The number is undefined

Then: God exists.

This syllogism may be symbolized as follows:

$$\begin{aligned} If\mathop {}\limits^{{}} \neg \exists \aleph \Rightarrow \neg \diamondsuit KCardB_{1} \hfill \\ \neg KCardB_{1} \Rightarrow \exists \aleph \hfill \\ \end{aligned}$$

This is a fallacy because it is a variation of modus tollendo tollens.

4.2 Second Interpretation: Inconceivable $\ne$ Undefined

4.2.1 According to Modal Logic

Borges acknowledges that the number is inconceivable. He is making the statement that the number, although it is in the finite set ranging from 1 to 10, does not correspond to any of the natural numbers that conform to our universe.

The number is inconceivable because, being a natural number less than 10 and greater than 1, it does not correspond to any of them. That number is as inconceivable to Borges (as it is to other people). Call it $\xi$. It follows that the number $\xi$ is not only indefinite but even inconceivable. i.e., $\exists \xi \,such\,that\,CardB_{1} = \xi ,\,and\,\neg KCardB_{1} \wedge \neg \exists_{M} \xi$.

It is impossible to doubt that $\xi$ has existed, at least in the mind of Borges. Since $\xi$ existed, it is possible that another undefined and inconceivable being exists, as $\psi$ or even God. i.e., $If\,\,\exists_{M} \xi \Rightarrow \exists \xi \Rightarrow \diamondsuit \exists \psi \wedge If\,\,\exists \xi \Rightarrow \diamondsuit \exists \aleph$.

Since God is an undefined and inconceivable concept, at least in the Judeo-Christian tradition in which Borges was educated. In logic, possibility implies necessity.

It is possible that the inconceivable being we call God exists. Then God must exist. “Ergo, God exists.” i.e., $If\,\,\diamondsuit \exists \aleph \Rightarrow \square \exists \aleph$

This reasoning is closer to Anselm’s argument. But it is equally fallacious. Gaunilo of Marmoutiers was right.

4.2.2 According to the Scholastics

The syllogistic method adopted here for Borges expressed his interest in staying within a scholastic line of argumentation. To examine whether Borges complies with these rules in their syllogism, it is desirable to reduce it to its logical scheme:

(A) If God exists, the number is defined. $If\,\,\exists \aleph \Rightarrow KCardB_{1}$

(B) If God does not exist, the number is definite-indefinite

$$If\,\,\neg \exists \aleph \Rightarrow KCardB_{1} \wedge \neg KCardB_{1}$$

(C) and as a definite-indefinite number is inconceivable

$$KCardB_{1} \wedge \neg KCardB_{1} \Rightarrow \neg \exists_{M} CardB_{1}$$

(D), ergo God exists.

$$\neg \exists_{M} CardB_{1} \Rightarrow \exists \aleph$$

The argument still is reduced to a schematic formula, which takes the following form:

$$\begin{aligned} &If\,\,\exists \aleph \vee \neg \exists \aleph \hfill \\ &If\,\,\neg \exists \aleph \Rightarrow KCardB_{1} \wedge \neg KCardB_{1} \hfill \\ &Absurd\,\,KCardB_{1} \wedge \neg KCardB_{1} \Rightarrow \exists \aleph \hfill \\ \end{aligned}$$

The logical value of the argument is essentially based on the following points:

(1) In the full and perfect alternative of the premises A and B. That is, that the assertion of one, imposes, logically and irrefutably, the negation of the other and vice versa.

(2) In the strict relationship of cause and effect, premise and necessary consequence between B and C.

(3) In the logical character of metaphysical absurdity and inconceivability of C.

Regarding the first point (or A or B) no doubt the logical rigor of the proposed alternative. These are two propositions, the first is affirmative, the other negative, of which one says exactly and strictly what the other denies. That is, it is composed of two contradictory sentences.⁵ These propositions cannot both be true or false at the same time, nor may have an intermediate proposition. This means that the truth of one necessarily follows the falsity of the other, and vice versa. From which we can infer that the premises A and B that Borges proposes are contradictory, if we accept the falsity of the proposition B, we have to accept the truth and inexorable logic of its contradictory A, and thus recognize the existence of God, “quod erat demonstrandum” according to the Scholastics.

The third point about the logical nature of metaphysics and absurdity and inconceivability of C is that Borges did not report specifically in his argument, although he did presuppose it. When talking about the “inconceivable”, Borges could refer theoretically to the human mind as a limit, with a radical and uncompromising idealism. I.e., what my mind cannot conceive, is not there nor can it ever be, and is therefore absurd. But it is not. Borges does not state that a certain “number” is not conceived and therefore is false, but rather implies that it is false-absurd and therefore inconceivable. The “inconceivability” of the absurd is based here in that the theoretical number would be, if God did not exist, definite and indefinite simultaneously, i.e., possess two contradictory qualities, which is obviously false and metaphysically absurd, since nothing, for scholastic logic, can be and not be at the same time any particular quality. The touchstone of the whole argument is reduced to the second point, the logical and ontological legitimacy of the relation of cause and effect, the premise and necessary consequence between B and C. That is, if the acceptance of B as irrefutable truth leads strictly logically to accept the fact ontologically of a false C, then B cannot be true.⁶ Borges’ case is clear: if B necessarily leads to the absurd C, B cannot be true, but it is true, by necessity, its contradictory proposition, ergo A.

The number, the quantitative definition of the flock of birds, is made concrete for a singular action and must be defined. However, consider the mere fact that the understanding of a nonexistent God would not have been able to capture it, nor can an “I” observer who ignores argumenti causa: can this fact ontologically prevent the definition of the number? It comes on the one hand, from the numerical definition quantitas, and otherwise the importance or necessity of its presence in beings and their actions.⁷

From the point of view of the individual, i.e., one who is “undivided in itself but perceived from multiple perspectives”, the numerical quantity is merely possible and thus a quality which is accidental to its essence. Being “bird” is given and then, perhaps, there are two, or three, or a flock of them. Now, if we refer to a multiple being in itself and for itself, then a multiplicity of numbers is essential, since it is based on the ontological unity of its multiple being. The flock of birds as such is one and unique, with essential oneness (the set), but in reality it is composed of parts, which are birds (the elements). If we apply this notion to the argument that we are studying, we must admit that Borges, interpreting the birds flying together and at a given time, has viewed the birds from multiple perspectives. As such these perspectives or this being, although it is undivided and indivisible itself in actu, in the words of the scholastics, it is not in potentia.⁸ And as a physical entity, it consists of parts, which have a number. For this reason, the quantitative-numerical determination is essential, both as a transcendental property unit and is, as the being, a necessary determination for their existence. The flock cannot be without a number of birds.⁹

The problem thus presented involves using Borges’ terminology, the relation between reality and the knowledge that one has of this. That is, an individual action, and as such, specific and determined, in the opinion of Borges does not get its definition and final determination by itself. Do not forget that the flock of birds had a certain numerical amount not three or four, seven or eight; they flew or three or four or seven or eight flew; but only a concrete and specific number, although the observer with closed eyes could not define it, nor could anyone determine it if God does not exist. Moreover, if God exists, He knows the number, which is determined and defined by His knowledge and makes it ontologically possible and conceivable.

However, Borges proposes simultaneously possible knowledge of God, and a self that is observer. Is it that Borges equates the knowledge of God and of man in the determination of reality? Not necessarily. In Borges’ argument, the self that is an observer does not serve as a function of possible alternatives to divine knowledge, as if he said: the number is determined when, or God, if He exists, or I, if I observe and count. Is it not the case, as that number is determined and defined in both cases, yes, but this is for different reasons. Furthermore, the numerical determination of the flock of birds specifies different ontological relations with the percipient mind as God or me. So that self as observer, if not identical to the God, whose existence he wants to prove, only serves to point out the problems and ensure objectivity of the argument: we do not know whether the number is not conceivable, since nobody has conceived it. If this would have conceived the self as observer, and as the scholastics say de jacto ad posse valet illatio, we would have to conclude that the number is concrete, real and conceivable. But recall that we were once without argument and unable to prove the existence of God. Now, God knows, if He exists; that He has knowledge of things; the reason is clear, as human knowledge is a posteriori and assumes the object, God’s knowledge, at least in theory, is a priori, i.e., prior to these same things, objects of divine knowledge.¹⁰

Thus, the Argumentum ornithologicum faces its greatest test because to preserve the logical line of his thought, it requires the knowledge that such a God, whose existence Borges want to demonstrate, is not external knowledge, but is chronologically prior to such, and is, instead, constitutive knowledge as God is creator of the same object.

Following the technique adopted by Borges, this could be expressed as:

(1) The knowledge that God has of things, either it is creative or, if it is not creative it assumes the object.

(2) If it presupposes the object, depends on Borges.

(3) If it depends on Borges it is not divine knowledge.

(4) Ergo the knowledge that God has of things, it’s creative.

For this reason, the knowledge that God knows the number of birds is what determines and defines if there were three, four, five or more, but exactly what God knew they were, these were, for His knowledge is the limit as He is aware of all action and all being.

5 A Possible Criticism

There was not the idea of the authors to make a comparison between the arguments of Anselm and Borges. Moreover, we are aware that many thinkers are convinced that there is no interesting connection between the two arguments. Some of the arguments can be stated as follows:

Borges’ argument concerns a vision—a “fleeting mental image”- of a flock of birds. Given the premise that God exists iff it is determinate how many birds belonged to the flock, we can conclude that God exists from the further premise that it is determinate how many birds belonged to the flock.

• (a) God exists iff it is determinate how many birds belonged to the flock (Premise).

• (b) It is determinate how many birds belonged to the flock (Premise).

• (c) Therefore God exists (from 1, 2).

While the logic of this argument is impeccable, the premises are both questionable. And it is difficult to see any philosophically interesting relation between this argument, and the argument of Proslogion Ch. 2.

In defense of the first premise, we are given this:

• (a) All that God’s omniscience require is:

  • (a1) If it is determinate how many birds belong to the flock, then God knows how many birds belong to the flock.

  • (a2) If it is indeterminate how many birds belong to the flock, then God knows that it is indeterminate how many birds belong to the flock.

• (b) Even if God does not exist, there could still be question about how many birds belong to the flock.

In defense of the second premise, we are given this: If Borges did not see a determinate number of birds, then, while Borges saw (say) fewer than ten birds, Borges did not see one bird, and Borges did not see two birds, and…, and Borges did not see nine birds. But that is inconceivable!

The obvious reply here is that the imagined scenario is not inconceivable. What Borges saw—the content of his vision—was a flock of birds of indeterminate number. Sure, if Borges sees a flock of birds in the sky, then the flock has a determinate number, even if no one knows what it is. But we should not suppose that “fleeting mental images” have determinate content in the same kind of way that genuine vision has (Dennett 1992).

6 Conclusions

(1) There seem to be not two, but three ways to argue the existence of God:

• (1) From the very essence of God.

• (2) From what we manifest in concrete finite beings.

• (3) From the form of reason, and the idea of universals.

And if we do not consider the ontological argument as a priori in the strict sense, but a simultaneo, as the existence of the essence of God would be inferred, although it is recognized that among these arguments there is no precedence, then we will have four modes of argument, regardless of their respective validity:

a simultaneo (ontological), a posteriori (from the effect), a priori in the first sense (from the cause) and a priori in the second sense (from the reason or form).

(2) Valid versions ending with a universal statement and found in Anselm, contribute nothing as a demonstration of God’s existence, as they are vacuously true. Valid versions concluding with an existential statement in Anselm, or a defined description in Descartes may be accused of committing the fallacy of begging the question, since the first get the conclusion by introducing axiom 3, whereby it is assumed that the concept of the divine is not empty (which is what it has to prove), and second, by introducing the concept of supreme defined as a self-description, which should be in the conclusion. Anselm’s ontological argument proves that we need the principle of the excluded middle in this formulation, as in the case of any other judgment this would lead to a reduction ad absurdum. The principle of the excluded middle cannot apply to infinite sets. This intuitionist strategy avoids the paradoxes where the formalists fell. But it is exactly these paradoxes that are the approach through which theology has attempted to speak about God. If we are to no longer accept the principle of the excluded middle, then its formulations about God will still not appear paradoxical, they will simply turn into some assertions which operate with the infinite; therefore, they will no longer be included in the sum of the sets that operate with the principle of the excluded middle.

(3) Many mathematicians have been personally concerned about the problem of God, like Descartes or Newton. Attempting a demonstration of the existence of God from mathematics implies the impossibility pointed out by Gaunilo of Marmoutiers ten centuries ago: passing from the mental to the real is forbidden. Mathematicians know that these problems belong to a field of knowledge other than their own. It is no coincidence that many mathematicians only recognize the “ontological argument” as valid proof of the existence of God. We do would not even venture to assert that the main purpose of Gödel in his “ontological argument” was to prove the existence of God.

(4) The argument used by Borges is: “either A or B; if B, then it is C; but since C is inconceivable… ergo A”. Assumptions A and B are two contradictory statements (one affirms what the other denies), formulated in absolute terms, i.e., without the possibility of including other possible substitute statements (Z, X…), so that.

• (1) There are two possibilities, A or B.

• (2) The truth of one (A) infers the falsity of the other (B) and vice versa.

The essential question for Borges, is whether C is possible ontologically. Thus, C is definite and indefinite simultaneously, so that, in effect, it is not acceptable, insofar as nothing can be both simultaneously. We can substitute equivalents, so that the resulting argument would be expressed as follows: “If God exists, the total reality can be known, because (since man has unsurpassed physical and intellectual limitations) only God can conceive that universal wholeness; if God does not exist, the total reality cannot be known”.

(5) Borges’ argument is literary and logical. Borges never clarifies that its argument is an a priori proof of God’s existence. However, all the ontological arguments are intended as demonstration. Borges intended to emulate the ontological argument, but he does not meet the typical apriorism they require. When leaving a visual image, his argument takes an a posteriori character. Borges by taking as the starting point a visual image, marginalizes the concept. However, this commitment prevents identification with apriorism. Borges’ demonstration is therefore a posteriori.

(6) The scholastic argument falters when applied to the question of for what reason God exists, God has knowledge of the number of birds in the flock, and this is constitutive of his being defined and not external and dependent on him. Borges here would have no choice but to apply to God the same rules of human thought: if it is external and alien to things, they may be later or simply never occur. Real objection to the validity of Borges’ argument would be in total denial of the need for an ontological true concept of the divine creator, the essence of things. That is, the negation of all cosmic and universal order, the product of a reason-wisdom, creative and organizer, to accept, however, an atomism of actions and reactions product of chance, as a universal and cosmic kaleidoscope, where figures are repeated, only at random, if they are repeated; or a total determinism of concatenation of causes and effects; in these cases, the problem of the number of birds remains determined and conceivably even without the existence of God, simply defined by its material causes. We can also, of course, deny the existence of reality beyond the perceiving mind, in which case, if God does not exist, closing his eyes the self-observer can say there were no birds at once without any problem.

(7) Borges has in common with the medieval scholastics, an acceptance of divine knowledge as an intellectual operation (Magnavacca 2009), their latent concept of “ontological truth” as the adequacy of reality to the divine intellect, contrary to the “logical truth” that represents an adaptation of understanding with reality; logical truth is exclusive to man, while knowledge of reality is only in God, whose knowledge is the cause of things and that God is the ground of all being in its transcendental meaning. Borges has in common with the medieval Jewish and Muslim philosophy, the importance given to the action of divine knowledge in the creation of the material world, although he does not develop his argument to point out the nature of divine action. Thus, Borges’ theological order is worthy of the most renowned scholastics during the Middle Ages, in which God, although cause and foundation of things, is distinct from things. That position is a emanationist hylomorphism¹¹ as we teach in Jewish and Muslim philosophy, in which the knowledge of God the Creator is first an essential characteristic and a form of being. The universe of Borges proclaims, for him, an ontological organization, a conscious and wise product of that God who with his claim he demonstrates exists.

(8) One last disquisition: why does Borges use birds as an argument to prove the existence of God? The mention of birds is continuous in the Scriptures, in the Bible, in their Jewish and Christian versions. They often serve as the basis for allegories or parables and made to carry the word of God or some mission entrusted by Him. In the Bible (Ecclesiastes, 10, 20) warns that birds take every word. The bird that says great truths, often starring or at least appearing in stories and Oriental fables (Frankel 1989, 1993). Among the powers that characterize the chosen of God, appears the language and comprehension of all creatures, particularly birds, as in the case of R. Isaac Luria.

Borges never ceased to emphasize the debt that Western literature has to Hebrew culture (Aizenberg 1986). The fascination Jewish mysticism exerted on it; studied carefully by Gershom Scholem, whom Borges called teacher, and boasted of being “the first and very imperfect translation of the work of Martin Buber”. The Kabbalah is one of the central motifs in identifying Borges with Judaism, as in his work on the Aleph and his admiration for the Jewish cult of Torah. Reading a book, talking about a book, and remembering a book were all fabulous experiences for him. Borges, like the Kabbalists, devoted his work to the infinite task of revealing the secret of cosmic creation. But precisely the impossible nature of this enterprise was what fascinated him: he argued that the Kabbalists had not written to facilitate the truth but to hint at and stimulate the search. And this may be a possible explanation.

(9) The interest expressed by Borges in mathematical logic is well known. As every reader of Borges knows, paradoxes are cited throughout his writings. Borges explicitly mentioned Bertrand Russell as among the most important of those who tried to solve them. Linking Russell’s explanation with Cantor’s theory of sets, Borges mentions the same issues concerning the treatment of infinity. But, in spite of his praise of Russell, it is enough to briefly look through the book that Borges mentions (Russell 1919) to ascertain the very limited perspective which he had on Russell’s thinking in general. Studying the contributions of Russell in the field of mathematical logic, requires a tremendous technical effort in mathematics and logic itself, as in the philosophical discussions associated with them. However, none of this appears, even by reference, in the writings of Borges. Although he could have learned it from Russell’s books (at least partially), Borges gives no sign of having understood the great transformation of logic since the beginning of the 20th century. In particular, Borges never mentioned what is undoubtedly one of the greatest achievements of modern logic, that is, the theorems demonstrated in 1931 by Kurt Gödel that implied a total and absolute change of our conceptions of system, deduction and the idea of “truth” in mathematics. This turns out to be surprising since these theorems are based on self-referencing reasoning of the type that took Borges’s attention. But given the very limited perspective with which Borges tackled the topic of self-reference and more generally of logic, it cannot surprise us that developments of this type have been beyond the limits of his interest, and perhaps his understanding.

(10) In the ontology of Borges there is a kind of “Platonism of individuals”, with Archetypes and individuals, but where the ontological status is indifferent. Or better, the distinction between Archetype and individual does not exist (or does not matter). We believe that there is such an Absolute Totality in Borges’ ontology, that it annuls any type of hierarchy, not only ontological, but even ethical and aesthetic. A hypothesis to investigate could be: there are Platonic Forms, there are also individuals. But the foundational relation between them does not exist, the individuals are not less than the Forms. Perhaps it is a Platonism where the Forms are individuals, and not Archetypes, where between the genres and the concrete things there is not an inexplicable relation of participation, but a total isolation from each other, and, in any case, there is the community regarding the fact of being. This is where we think we see Spinoza as a precursor of Borges. The Names with which it is designated today, for us, are God, Everything, World, Universe… For other languages, it can be Sphere or Library or Aleph or Pyramid. But whatever it’s called, that thing is One. Otherness is impossible, novelty too. Within that single Thing, individuals appear, as internal modifications. If it were possible to see that Unit from outside, that Parmenidean sphere, no movement would be observed.

Plato and the Plotinian hypostases, Augustine, Schopenhauer, Berkeley and Hume, are footholds on which the archetypal universe and the fictions of Borges can be supported, although there are also references in Borges to Keats, Coleridge, Yeats, Kafka and Thomas Mann. With idealism in its various branches, the Argentine author’s literature aims to rise for a moment (Nuño 2005). Unity and dispersion, the One and multiplicity, the leap from the intelligible to the sensible, are all drawn in the Borgesian stories. The terror of the copy and the mirror that, as in the cave fable, reproduces what happens before it is evoked in each mansion, sometimes in an almost caricatural way: “all men, in the dizzying moment of intercourse, are the same man “ The effort to circumvent the limits of perception to understand the functioning of the building is also present. The long corridor of time, which opens the possibility of infinity or a domain of idealistic purity, transcends almost everything. And next to it, the subject, the memory and the language that serves as a support between the sensible and the intelligible and that has ended up becoming a zone of danger and loss. Borges chooses to undertake the game of a knowledge anchored in abstraction, in speculation, in universalism and in the nostalgia of an intelligible world abandoned by critical solipsism, by rationality, by empiricism, and by finding that the visible and its reading, nature, language and its flares, have become part of the alterations and disorder of the contemporary world. Borges wants to avoid the solipsism in the opening of the Cartesian Discourse on Method, and how it is cast in the territory of memory. In his journey we notice, in effect, that the author prefers the reproductions of old passages and an individualism that adheres to the perceptions of intelligence; we also discover that he does not want to make an abstraction of identity by returning to Cartesianism and adopting the voice of the subject that the wounds of the years do not affect.