... The diameter of the Aleph would be of two or three centimeters, but the cosmic space was there, without size decrease... Jorge Luis Borges, The Aleph.

 

The Mathematical Infinite: ontologic interpretations

Luis Eduardo Zamudio Suárez - Graduate in Mathematics - Caracas, Venezuela

Among all the created ideas or discovered by the man along the history of the thought, the notion of Infinite without a doubt is one of the most enigmatic. The most lucid intellects in the philosophy, the theology and the mathematics have contributed different interpretations of the infinitud, a surrounded idea—like we will see—of mysticism and madness through the centuries. Next a simplified historical recount of the notion of Infinite is presented in its more rigorous version—in my opinion—, the mathematical Infinite.

 

Later on he will settle down the hypothesis that the categorical infinite—also call current infinite—he belongs together with an ontologic reality that involves the human being and their relationship with the Divinity. Such a hypothesis will allow to express a possible answer to the Aristotelian question on the man's last function.

Brief historical recount

The origins of the notion of Infinite in mathematics go back until Pitágoras (approx. 569-500 B.C.).  Como is known, the Pythagorean ones practiced philosophy, mysticism and mathematics, three areas that were inextricably related for them, while they sustained the firm conviction that all mathematical knowledge reveals an invisible angle of the Reality.

Among other beliefs defended by the Pythagorean ones on the meanings of each particular number it highlights the function that played the number one as inductive generator of the whole numeric system for them, that which allows to deduce that they had clear idea of the potential infinite call: given any number, for bigger than it is, we can always obtain simply a bigger number adding him the unit.

Approximately one century later Zeno of Elea (495-435 B.C.) it promulgated the paradoxes that have immortalized it. In them it tried to prove the impossibility of the movement leaving of the premise of an infinitely divisible space.

It is necessary to point out that this paradoxes can be been in a natural way (from the formalization of the limit concept) through the idea of convergence of an infinite series. Here also it is the notion of potential infinite the one that plays the protagonistic list in opposition to the current infinite, the one which traditionally was rejected for mathematical and philosophers until final of the XIX century of our era.

In fact, the mathematical tradition had always used the potential infinite in the form that Eudoxio inaugurated (408-355 B.C.) and Arquímedes (287-212 B.C.) who for the calculation of areas and volumes of geometric figures applied the potential character of the infinite, no longer only to manipulate arbitrarily big quantities, but also in the consideration of extremely small quantities, those that being positive can still be made spread to zero in the limit.

With the purpose of emphasizing the fundamental distinction among the potential and current infinites, as well as the perception that had the mathematical tradition regarding the inadequate thing of this last, let us see a small fragment of a letter that he wrote anything less than the usually considered bigger mathematician of the history, Carl F. Gauss (1777-1855), to their colleague Heinrich Schumacher:

• But concerning your proof, I protest above all against the use of an infinite quantity [Gröse] ace to completed one, which in mathematics is never allowed. The infinite is only to parler façon, in which one properly speaks of limits. Mentioned such which for Dauben in [5], p. 120.

Nevertheless, a lot before Gauss the first well-known appearance was given—although delicate—of the current infinite through one of the most complete scientists that registers the history, Galilean Galilei (1564-1642) who noticed of the existence of a correspondence one-to-one among the elements of the group of natural numbers {1, 2, 3, etc.} and those perfect square calls {1, 4, 9, 16, 25, etc.}. Such a correspondence can be represented by the function f(n)=n² 

This way to the number 1 correspond him the same one 1, to the number 2 correspond him the 4, at the 3 the 9 and so forth. A fact was demonstrated apparently paradoxical: so many perfect squares exist (which constitute a subset characteristic of the natural numbers) as natural numbers.

In terms of current mathematics, the correspondence one-to-one used for Galilean fulfills two fundamental properties that in fact serve to define and to identify to all the correspondences of this type, also calls functions biyectivas.

The first of such properties consists in that for all couple of elements n and m of the domain of the función (en the present example, for all couple of natural numbers) it is completed that: n?m implica f(n)? f(m) 

The second property consists in that all element of the group of arrival of the function (that would be the group of square numbers in our example) it is “reached” for an element of the domain through the function. This means in this case that if m is a square number, a natural number exists such n that f(n)=n²=m Aquí is evident that given squared m, the number n = √m is the natural number that reaches it through the función f.

When between two groups a function biyectiva exists it is said that both have the same cardinalidad, indicating with this that both groups have the same quantity of elements, like it is intuitive given the definition of correspondence one-to-one or function biyectiva.

Using this language, that that Galilean it published in 1638 it is translated in the statement that the group of natural numbers has the same cardinalidad that an own subset. In other words, he refused the principle that the everything is bigger than their parts; I not begin applicable in fact to be being fought with an all infinite. But maybe, it fits Galilean especular, para it was already enough with the problems that their cosmological theories had carried him with the Inquisition y finalmente he decided not to attack a deeper investigation on the meaning of their discovery, meaning that inevitably has taken it to declare the existence of the current infinite, a notion that could only be associated with the Divinity for the time.

Or perhaps, as some experts they affirm it, have been the power of the Infinite what stopped to Galilean.

Be like outside, without importance it was the first appearance of the current infinite that a giant of the philosophy and the mathematics, as it was it Gottfried W. Leibniz (1646-1716), he wrote later less than one century in their New rehearsals on the human understanding (it works monumental that practically summarizes the whole knowledge of the time), following:
Properly speaking, it is that is to say true that there is an infinity of things, that there are always more than those that we can designate.

But if they are taken as authentic all, then there are not infinite number, neither line neither any other quantity that it is infinite, like it is easy to demonstrate. The schools have wanted or due to say that, when admitting an infinite sincategoremático [potential infinite], but not the infinite categoremático [current infinite], to say it in their language. In rigor, the true infinite is only in the absolute thing that is previous to all composition and it is not formed by addition of parts.

Although Leibniz in its creation of the Infinitesimal Calculation used thoroughly—the same as Newton and its Greek predecessors—the potential infinite, its opinion is evidenced with regard to the current infinite, opinion that reflects the established knowledge of its time.

Therefore, in mathematics any other approach was not given to the concept of Infinite until the XIX century, when the priest and mathematical Bernhard Bolzano (1781-1848), influenced by the works of Eudoxio and Galilean, it glimpsed new lights on the nature of the infinitud.

Bolzano began to wonder if the seemingly paradoxical property that he had discovered Galilean with regard to discreet groups (such as the group of natural numbers where is certain which the successor of each elemento) también is it could be given in continuous groups on the real straight line.

He found indeed that a correspondence could settle down one-to-one among an interval of the straight line and a subintervalo included in the same one. In this sense it defined the function f(x)=2x, on the closed domain [0, 1]. It is clear that this function it is a biyección or correspondence one-to-one between their domain and their arrival group, the closed interval [0, 2] that in turn contains the domain [0, 1] as an own subset.

It arose this way in 1851 the publication posthumous Paradoxes of the Infinite where Bolzano became the first mathematician in defending the existence of the current infinite, expressing that this could be introduced in mathematics in a consistent way, free of contradictions.

The destination leaned, however to that the current infinite stayed far from the attention of the mathematicians for more than twenty years after the publication of Bolzano, until the revolutionary works of Georg Singer (1845-1918) they began to almost transform the mathematics's areas.

Among their first taxes Singer it formalized the definition of infinite group. From then on it is said that a combined X is infinite if and only if a correspondence one-to-one exists between X and some own subset S contained in x (ЅcX). is it in fact the negation of the principle that the everything is bigger than its parts, valid principle only in finite groups.

In these terms, Galilean and Bolzano had demonstrated formally that the group of natural numbers and the group of numbers real contents in any interval of the real straight line are combined infinite. But in 1874, Singer gave a decisive additional step demonstrating the unthinkable thing until that moment: the existence of several “sizes” or infinitud orders.

In a sample of penetrating lucidity Singer it proved that it is impossible to establish a correspondence one-to-one between the group of natural numbers and the group of real numbers between the zero and the one. In other words, the real numbers of this interval (and in fact, of any interval) they cannot be labeled in an indefinite list until the infinite in the way A1, A2, A3,... therefore, the cardinalidad of the continuum, the quantity of points in any segment of real straight line, is superior to the infinite quantity of natural numbers.

This discovery inevitably even unchained consequences beyond the properly mathematical territory, while Singer attacked the defense of the categorical infinite (or current) bigger than it has been given in the history of the western thought.

At least in the mathematical context, the existence of an entity can already be argued by its consistency with the rest of the ideas and mathematical constructions established. In this sense, for Singer it was perfectly consistent that the process of “to count” elements of finite groups were extended to elements of infinite groups, as complete and completed entities, through functions biyectivas.

For example, the group A = {w, x, y, z} of objects any (they are books, houses, animals or what is) different to each other, you can consider like a total entity whose cardinal number is the number four, since a correspondence exists one-to-one among the group TO and the subset of natural numbers {1, 2, 3, 4}.

Of the same combined manera, el of perfect square numbers {1, 4, 9, 16,... n²,...), verbigracia of infinite group, you can consider like an entirety whose cardinal number is superior to any finite number. Therefore, Singer denoted with an aleph (first letter of the Hebrew alphabet) and a subindex similar to zero to the number cardinal transfinito N0 de all the equivalent groups in quantity (susceptible of biyectar) to the group of natural numbers.
In this order of ideas, Singer's discovery that the cardinalidad of the group of real numbers (the continuum) it is bigger to that of the natural numbers it implies the existence of another number cardinal transfinito, N1 say, superior to No.

But, which was the reaction from the mathematical community to these discoveries?

In principle, as many other discoveries that affect the established paradigms, Singer faced a strong and persistent opposition, on one hand, and he enjoyed some few partisans, for another. Knowing that the opposition to its work came from different fronts, it began to determine which they had been historically the arguments against the current infinite and he prepared to refute them one by one armed with its new theory of numbers transfinitos. In particular, the classic Aristotelian argument that consisted on using arithmetic principles to contradict the existence of the current infinite criticized. In front of this Singer established that to reason the infinite mathematically it was necessary to create a special arithmetic that him same he took charge of being founded.

Years of intense investigation on the part of the best mathematicians in the world among final of the XIX century and principles of the XX one, they demonstrated that Singer's ideas would stay forever in the mathematics's Kingdom. But, according to their biographer Joseph Dauben, the deep motivation that impregnated Singer's soul, and that it impelled it to defend their theories in any land showing a strong conviction, era largely nun.

Singer not only found encouragement and support from his faith in God, but has also believed that there is was destined to put that knowledge into service for the greater understanding of God and nature.    

But these they are ideas that already touch to the ontologic hypothesis that next is presented.

Ontologic hypothesis

Accepted the existence of the current infinite in mathematics, in the first place, is necessary to wonder to what he/she belongs together in the world physical so enigmatic idea?, is the categorical infinite maybe representation of something in the material world?

The physiques and astronomers have determined that we live in a curved universe in permanent expansion from the Big Bang.

Also, strong indications are had that the universe is finite, although there is not absolute certainty in this respect. On the other hand, new theories spread to not only establish an atomism in the space but also in the time, what would imply the impossibility of dividing any matter fragment or tiempo.  infinitely

Indeed, the infinitud in the physical world has only become present in the infinite density that is supposed it existed in that point native of where the Big arose Bang. Beyond that moment primigenio, there are not evidences that in the universe entities that belong together with the mathematical infinite exist.

Nevertheless, the mathematics and the Reality—that includes evidently something more than the material world—they have shown along the centuries an inextricable relationship. It is perfectly feasible that the reality that belongs together with the mathematical infinite is beyond the physical world, in a superior link of the ontologic chain, chain outlined by diverse philosophers and that for the coarse current purposes with its image:

According to the mathematician, physique and astronomer Sir James Jeans (1877-1946), the Reality is intrinsically mathematics in a much deeper sense to which pointed out Galilean when he affirmed that the book of the nature is written in mathematical language. Simplifying the traditional way to make physical, this consists on observing the material world, to model it through equations and later on to verify the predictions carried out through the mathematical models empirically. In this sense, the mathematical formulas describe the reality, among other reasons because they are created to posteriori, jointly with relating empiric.

However, to reach the deepest sense that proposes Jeans, it is necessary to notice that the mathematics that Einstein used in the development of the general relativity at the beginning of the XX century, they were created independently of any empiric observation. Indeed, they were theories of pure mathematics (I eat the geometries for example no-euclidianas) invented by the thought abstract several decades before it was suspected at least that could have some application to the physical world.

For Jeans, the tremendous success manifested in the applications of these theories is test that the Reality has an inherent mathematical structure. It would be the only explanation that they have been able to check numerous predictions calculated through mathematical theories created independently to all empiricism, prior to all observation of the external world.  

Now then, given the existence in mathematics of the current infinite and considering the previous argument, if in the physical world there are not indications of the Infinite, it is not natural to suppose that their mathematical existence belongs together with a reality beyond the physical thing? It would be an idea that doesn't throw to the enclosures of the matter, but rather this transcends their being since it belongs to links ontologic superiors.

This is in fact the hypothesis that was wanted to express: the current infinite in its mathematical form belongs together with an ontologic reality that involves the man's Being and its relationship with the Divinity. We will already see on what this is based, but for the time being, notice you that we could be in presence of a mathematical notion that only represents realities tried by metaphysicians, theologians and until mystics.

But, on what it consists this ontologic reality? There are diverse possible interpretations, all closely linked.
Beginning with the metaphysical leibniziana, we have that Leibniz stops all the substances simple, called mónadas, they are related to each other, and in turn each one with the surrounding universe. These relationships are of such a nature that an infinite intellect could know the whole universe starting from the investigation of a single mónada, since this is an image or mirror that it reflects the whole universe.

Now then, this connection or accommodation of all the things created to each an and of each one to all the other ones, makes that each simple substance has relationships that express all the other ones, and that consequently it is an alive and perpetual mirror of the universe.

... And the author of the nature has been able to practice this divine and infinitely wonderful artifice for that each portion of the matter is not only infinitely divisible, as they have recognized it the old ones, but also subdivided at the moment to the infinite, each part in parts, of those that each one possesses some own movement: otherwise it would be impossible that each portion of the matter could express the whole universe.

We see Leibniz so, although in mathematics it only defended the potential infinite, he believed in the existence of the current infinite in the nature. In fact, it is the current infinite that he makes possible that a sphere limited by a finite diameter (without caring the small thing that it is this) it can be biyectada in a continuous way to the whole infinite three-dimensional space,  such and like Borges describes in his extraordinary story The Aleph. Remembering the epigraph to the present rehearsal, the Aleph is a small sphere limited by a diameter of two or three centimeters that, however, it contains to the whole universe. It evidences indubitable of the Infinite: although limited by their diameter, the sphere contains so many points like the infinite space that in turn contains to the sphere.

But Leibniz is not limited to speak of the matter. Plus still, he affirms that each mónada represents to the whole universe from its point of view, something that can be understood only through the current infinite, this time operating in an ontologic space, that which cannot be a surprise since—like Heinz affirms Heimsoeth—Leibniz “it is convinced metaphysically of the existence of the current infinite in the world.”

Also, Leibniz establishes a developing difference between the simple souls and the rational ones:
Regarding the rational soul or to the spirit, there is something more in her of what is in the mónadas or even in the simple souls. It is not only a mirror of the universe of the creatures but also an image of the Divinity...

This last statement allows to come closer to another interpretation possible of that ontologic reality that our hypothesis affirms it is represented by the current infinite. The romantic idealists who promulgated the existence of the Me infinite or the absolute Auto-conscience, they manifested adherence to the possibility of identification of the limited conscience (human) with the Infinite Conscience (Divine). Nevertheless, these ideas are not completely original of the romantic idealists. At least they have their roots in the neo-platonic philosophers, especially in Plotino (204-270 D. C.) and their descriptions of the experience of the ecstasy.
For Plotino the ecstasy is the abolition of the alteridad among which you go and the thing seen and the total identification and enthusiast of the human soul with God.

Maybe a metaphysical biyección? Wittgenstein would say that in this respect we only can and we should remain silent. But in any way, the ecstasy of Plotino can only be propitiated leaving of the principle that the human spirit possesses infinite potentialities characteristic of the Divinity, in spite of being limited by the matter, and even possibly for the intellect which sphere limited by its diameter. Now, such a premise or belief is also in diverse oriental traditions as the Hinduism or the Buddhism. According to Borges, for these religions, as well as for the rest of religions and philosophies of the Indostán, the doctrine of the Vedanta is fundamental pillar. The learned writer compresses it in the following way:

• The doctrine of the Vedanta is summarized in two famous sentences: Tat twuam this way (That is you) and Aham brahmasmi (I am a Brahmin). Both affirm the identity of God and of the soul, of one and the universe. This means that the eternal principle of all being that projects and it dissipates worlds, it is in full and indivisible each one of us.

Therefore, we have the imperium of choosing among the metaphysical leibniziana, the romantic idealism or the mysticism of neo-platonic or oriental court. That that here is affirmed it is that the current mathematical infinite is metaphor of a reality transmitted from ancestral times by philosophers and mystics, as much in occident as in east: the substance of the human spirit is the same one, in all its current infinitud that the substance of the Divinity. And how another image beyond the mathematical infinite could make comprehensible that undoubtedly limited beings are infinite in essence?

To take conscience that with all our limitations we are substantially infinite he would open a breach to respond to the Aristotelian question: the function of all man cannot be another that to discover its interior infinite and biyectarse—like he/she would say a Pythagorean one—with the infinite substance that is “the eternal principle of all being”. Because like Heimsoeth says paraphrasing the Doctor Sutilísimo: “The end for which God has created us is this way in agreement with our abilities.”

It emerges this way the terrible query for all Westerner: the intellect one of such abilities will be in agreement with the last end of all man?, or will the path maybe be, toward the unassailable soul for the weapons of the reason?

The attainment of the man's end in West

In 1884, amid their investigations on the hypothesis of the continuous one, —problem impossible to solve like it was demonstrated later years—Singer suffered his first “mental breakdown”—as they call it his biographers. In reason of he/she stayed it to it two months confined in a center of psychological recovery. One doesn't have certainty on the causes of their crisis, although commonly you speculates on a possible combination of tendency genetic, extreme difficulty and frustration facing the hypothesis of the continuous one and a lingering opposition to their ideas on the part of one of their former one-more remarkable professors.

Be which is the true combination of causes, to the mathematician and writer Amir he found Aczel unavoidable to imagine that Singer experienced the consequences of having sought to seize a knowledge inescrutable. 

• Trying to understand the real meaning of the various levels of infinity—trying to dissect the unreachable infinite and proved its innermost parts—may have cost him his sanity.

Indeed, as he/she went suffering numerous crisis and more and more lingering intermittent relapses, Singer was deteriorating mentally until he abandoned all mathematical investigation. Seemingly their thoughts in the final months of their life rotated around the belief that through him, God had communicated to the world good part of the essence of the Infinite.

Being based on the stranger coincidence that Singer's natural successor, the logical mathematical Kurt Gödel (1906-1978),when facing the Infinite years after their predecessor, it also experienced intermittent episodes of madness, Aczel concludes that the mathematical infinite involves a mystery that resists to be revealed. However, Aczel doesn't offer conjectures on meanings possible of such a mystery.

Our ontologic hypothesis tries to look for him sense to the enigmatic idea of the infinitud, aware that the mathematical infinite hardly allows to glimpse with the light of the reason a reality that concerns to the spirit.
The mathematics's unquestionable big madness of two could symbolize the limits of the reason in its search of the Infinite. Nevertheless, the western science—legitimate daughter of the reason—he/she has slope to decipher the true potentialities of the human mind. For what reason is that cerebral capacity that we don't not even use in our moments of more concentration there?

Contrary to becoming characteristic of East, in West the domain of the nature, and in general of the external world, it has deprived on the domain of the itself. But, if we dare to radically confront the common practice of transforming to the simple means into ends, living many times on the edge of the alienation because of it, it is probable that, in spite of having taken the longest road, let us be before the nobleman and fundamental task of discovering the infinite universe that palpitates in our interior.

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