The Traversal of the Infinite
By: Dr. Sam Vaknin
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Finiteness has to do with the existence of boundaries. Intuitively, we feel that where there is a separation, a border, a threshold – there is bound to be at least one thing finite out of a minimum of two. This, of course, is not true. Two infinite things can share a boundary. Infinity does not imply symmetry, let alone isotropy. An entity can be infinite to its "left" – and bounded on its right. Moreover, finiteness can exist where no boundaries can. Take a sphere: it is finite, yet we can continue to draw a line on its surface infinitely. The "boundary", in this case, is conceptual and arbitrary: if a line drawn on the surface of a sphere were to reach its starting point – then it is finite. Its starting point is the boundary, arbitrarily determined to be so by us.
This arbitrariness is bound to appear whenever the finiteness of something is determined by us, rather than "objectively, by nature". A finite series of numbers is a fine example. WE limit the series, we make it finite by imposing boundaries on it and by instituting "rules of membership": "A series of all the real numbers up to and including 1000" . Such a series has no continuation (after the number 1000). But, then, the very concept of continuation is arbitrary. Any point can qualify as an end (or as a beginning). Are the statements: "There is an end", "There is no continuation" and "There is a beginning" – equivalent? Is there a beginning where there is an end? And is there no continuation wherever there is an end? It all depends on the laws that we set. Change the law and an end-point becomes a starting point. Change it once more and a continuation is available. Legal age limits display such flexible properties.
Finiteness is also implied in a series of relationships in the physical world: containment, reduction, stoppage. But, these, of course, are, again, wrong intuitions. They are at least as wrong as the intuitive connection between boundaries and finiteness.
If something is halted (spatially or temporally) – it is not necessarily finite. An obstacle is the physical equivalent of a conceptual boundary. An infinite expansion can be checked and yet remain infinite (by expanding in other directions, for instance). If it is reduced – it is smaller than before, but not necessarily finite. If it is contained – it must be smaller than the container but, again, not necessarily finite.
It would seem, therefore, that the very notion of finiteness has to do with wrong intuitions regarding relationships between entities, real, or conceptual. Geometrical finiteness and numerical finiteness relate to our mundane, very real, experiences. This is why we find it difficult to digest mathematical entities such as a singularity (both finite and infinite, in some respects). We prefer the fiction of finiteness (temporal, spatial, logical) – over the reality of the infinite.
Millennia of logical paradoxes conditioned us to adopt Kant's view that the infinite is beyond logic and only leads to the creation of unsolvable antinomies. Antinomies made it necessary to reject the principle of the excluded middle ("yes" or "no" and nothing in between). One of his antinomies "proved" that the world was not infinite, nor was it finite. The antinomies were disputed (Kant's answers were not the ONLY ways to tackle them). But one contribution stuck: the world is not a perfect whole. Both the sentences that the whole world is finite and that it is infinite are false, simply because there is no such thing as a completed, whole world. This is commensurate with the law that for every proposition, itself or its negation must be true. The negation of: "The world as a perfect whole is finite" is not "The world as a perfect whole is infinite". Rather, it is: "Either there is no perfectly whole world, or, if there is, it is not finite." In the "Critique of Pure Reason", Kant discovered four pairs of propositions, each comprised of a thesis and an antithesis, both compellingly plausible. The thesis of the first antinomy is that the world had a temporal beginning and is spatially bounded. The second thesis is that every substance is made up of simpler substances. The two mathematical antinomies relate to the infinite. The answer to the first is: "Since the world does not exist in itself (detached from the infinite regression), it exists unto itself neither as a finite whole nor as an infinite whole." Indeed, if we think about the world as an object, it is only logical to study its size and origins. But in doing so, we attribute to it features derived from our thinking, not affixed by any objective reality.
Kant made no serious attempt to distinguish the infinite from the infinite regression series, which led to the antinomies. Paradoxes are the offspring of problems with language. Philosophers used infinite regression to attack both the notions of finiteness (Zeno) and of infinity. Ryle, for instance, suggested the following paradox: voluntary acts are caused by wilful acts. If the latter were voluntary, then other, preceding, wilful acts will have to be postulated to cause them and so on ad infinitum and ad nauseam. Either the definition is wrong (voluntary acts are not caused by wilful acts) or wilful acts are involuntary. Both conclusions are, naturally, unacceptable. Infinity leads to unacceptable conclusions is the not so hidden message.
Zeno used infinite series to attack the notion of finiteness and to demonstrate that finite things are made of infinite quantities of ever-smaller things. Anaxagoras said that there is no "smallest quantity" of anything. The Atomists, on the other hand, disputed this and also introduced the infinite universe (with an infinite number of worlds) into the picture. Aristotle denied infinity out of existence. The infinite doesn't actually exist, he said. Rather, it is potential. Both he and the Pythagoreans treated the infinite as imperfect, unfinished. To say that there is an infinite number of numbers is simply to say that it is always possible to conjure up additional numbers (beyond those that we have). But despite all this confusion, the transition from the Aristotelian (finite) to the Newtonian (infinite) worldview was smooth and presented no mathematical problem. The real numbers are, naturally, correlated to the points in an infinite line. By extension, trios of real numbers are easily correlated to points in an infinite three-dimensional space. The infinitely small posed more problems than the infinitely big. The Differential Calculus required the postulation of the infinitesimal, smaller than a finite quantity, yet bigger than zero. Couchy and Weierstrass tackled this problem efficiently and their work paved the way for Cantor.
Cantor is the father of the modern concept of the infinite. Through logical paradoxes, he was able to develop the magnificent edifice of Set Theory. It was all based on finite sets and on the realization that infinite sets were NOT bigger finite sets, that the two types of sets were substantially different.
Two finite sets are judged to have the same number of members only if there is an isomorphic relationship between them (in other words, only if there is a rule of "mapping", which links every member in one set with members in the other). Cantor applied this principle to infinite sets and introduced infinite cardinal numbers in order to count and number their members. It is a direct consequence of the application of this principle, that an infinite set does not grow by adding to it a finite number of members – and does not diminish by subtracting from it a finite number of members. An infinite cardinal is not influenced by any mathematical interaction with a finite cardinal.
The set of infinite cardinal numbers is, in itself, infinite. The set of all finite cardinals has a cardinal number, which is the smallest infinite cardinal (followed by bigger cardinals). Cantor's continuum hypothesis is that the smallest infinite cardinal is the number of real numbers. But it remained a hypothesis. It is impossible to prove it or to disprove it, using current axioms of set theory. Cantor also introduced infinite ordinal numbers.
Set theory was immediately recognized as an important contribution and applied to problems in geometry, logic, mathematics, computation and physics. One of the first questions to have been tackled by it was the continuum problem. What is the number of points in a continuous line? Cantor suggested that it is the second smallest infinite cardinal number. Godel and Cohn proved that the problem is insoluble and that Cantor's hypothesis and the propositions relate to it are neither true nor false.
Cantor also proved that sets cannot be members of themselves and that there are sets which have more members that the denumerably infinite set of all the real numbers. In other words, that infinite sets are organized in a hierarchy. Russel and Whitehead concluded that mathematics was a branch of the logic of sets and that it is analytical. In other words: the language with which we analyse the world and describe it is closely related to the infinite. Indeed, if we were not blinded by the evolutionary amenities of our senses, we would have noticed that our world is infinite. Our language is composed of infinite elements. Our mathematical and geometrical conventions and units are infinite. The finite is an arbitrary imposition.
During the Medieval Ages an argument called "The Traversal of the Infinite" was used to show that the world's past must be finite. An infinite series cannot be completed (=the infinite cannot be traversed). If the world were infinite in the past, then eternity would have elapsed up to the present. Thus an infinite sequence would have been completed. Since this is impossible, the world must have a finite past. Aquinas and Ockham contradicted this argument by reminding the debaters that a traversal requires the existence of two points (termini) – a beginning and an end. Yet, every moment in the past, considered a beginning, is bound to have existed a finite time ago and, therefore, only a finite time has been hitherto traversed. In other words, they demonstrated that our very language incorporates finiteness and that it is impossible to discuss the infinite using spatial-temporal terms specifically constructed to lead to finiteness.
"The Traversal of the Infinite" demonstrates the most serious problem of dealing with the infinite: that our language, our daily experience (=traversal) – all, to our minds, are "finite". We are told that we had a beginning (which depends on the definition of "we". The atoms comprising us are much older, of course). We are assured that we will have an end (an assurance not substantiated by any evidence). We have starting and ending points (arbitrarily determined by us). We count, then we stop (our decision, imposed on an infinite world). We put one thing inside another (and the container is contained by the atmosphere, which is contained by Earth which is contained by the Galaxy and so on, ad infinitum). In all these cases, we arbitrarily define both the parameters of the system and the rules of inclusion or exclusion. Yet, we fail to see that WE are the source of the finiteness around us. The evolutionary pressures to survive produced in us this blessed blindness. No decision can be based on an infinite amount of data. No commerce can take place where numbers are always infinite. We had to limit our view and our world drastically, only so that we will be able to expand it later, gradually and with limited, finite, risk.
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