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Hello Stripes! When I wrote earlier to suggest that perhaps integers are analogous to time, and real numbers to eternity, I was thinking in terms of the number line that is composed of both. The reals comprise both rational and irrational numbers, each of which can be imagined as having an infinite number of digits to the right of the decimal point. The digit series of a rational number presents itself as a sequence of elements that eventually will be found to repeat, ad infinitum. This is not the case for the irrational numbers, whose digit series is found to be not a sequence at all (at least sequence or periodicity has not been detected yet), but an infinite series of digits that appear to be distributed at random. The most famous example of an irrational number is pi of course, which does not appear to be a countable number at all, yet is a quantity fundamental to the development of Euclidean geometry. On the number line, pi appears in the “band” between the finite integers 3 and 4. With respect to the rationals and the irrationals, we seem to be looking at two completely different orders of infinity, e.g., one that is “quasi-finite” (the rationals), and the other “infinite” (the irrationals). The number line carries both types of numbers, integers and reals, both rational and irrational, all lined up “cheek to jowl,” so to speak. The “picture” one gets is that of a line smoothly progressing in time, with bursts of infinity interposed all along its extent pointing out of time to timelessness. And the number line itself is infinite, “in both directions” indicated by <0 and >0. Truly it appears that infinities come in different orders, or “sizes.”

I don’t know whether this makes any sense. I admit that when it comes to higher mathematics, I still have my “training wheels on.” So feel free to correct my understanding as needed: I have a lot to learn.

You write that there is “no number for infinity in the real number system,” thus the idea of infinity must be presented symbolically — roughly speaking, the symbol takes the form of a “snake biting its own tail,” with a good “twist” in the middle (so to speak). You point out that infinity is a direction. I don’t dispute this. Yet paradoxically, direction implies a process in time. In what way can we reconcile the idea of process in time with infinity, which you note corresponds to the idea of eternity (or timelessness)?

This problem has preoccupied philosophers since ancient times. Plotinus (204–270 A.D.) argued that time and the sequence of its movements are understandable only under the presupposition of a complete wholeness of time that he termed eternity, aion. As Wolfhart Pannenberg writes (Toward a Theology of Nature),

“The whole of time, according to Plotinus, cannot be conceived as the whole of a sequence of moments, because the sequence of temporal moments can be indefinitely extended by adding further units.… In his conception the total unity of the whole of life is indispensable in the interpretation of the time sequence, because it hovers over that sequence as the future wholeness that is intended in every moment of time, so that the significance of eternity for the interpretation of time in Plotinus results in a primacy of the future concerning the nature of time….”

Thus, “the infinite has priority over any finite part.” And in Kant’s development of Plotinus’ concept of time, this statement applies not only to the case of time, but also to the case of space — which to my mind at least seems to anticipate a crucial insight of modern relativity theory, and also of quantum field theory.

Pannenberg points out that this concept corresponds to “the Israelite understanding of eternity as unlimited duration throughout time,” i.e., the concept of time that runs throughout the Old Testament. (To my mind, this idea seems roughly analogous to the irrationals on the number line.)

It appears Pannenberg’s own model is a development from the Plotinian insight into the nature of time. He writes:

“I have developed this concept of eternity from the human experience of time, from the relativity of the distinction of past, present, and future corresponding to the relativity of the directions in space. In view of the relativity of the modes of time to the aspect of the human being experiencing time, this resulted in the assumption that all time, if it could be, so to speak, surveyed from a ‘place’ outside the course of time, would have to appear as contemporaneous.”

To borrow Alamo-Girl’s perceptive description here, in other words, from this perspective, the 4D block would appear, not as a progression of discrete events moving from past, present, to future, but as a “plane” or “brane.” Pannenberg continues:

“This assumption is confirmed by a unique phenomenonon of the human experience of time through the experience of an ‘expanded present’ in which not only the punctiliar now but everything on which a position may be taken still or already is considered as present…. Understood in the sense of the suggestions above, the concept of eternity [i.e., infinity] comprehends all time and everything temporal in itself….

“The worldview of the theory of relativity also can be understood in the sense of a last contemporaneousness of all events that for us are partitioned into temporal sequence.” [ibid, p. 100f.]

Of course, as Pannenberg himself notes, the perspective from which one could view such things “would not coincide with any position in the world process.”

In Pannenberg’s model, “creation can be conceived, on the ground of the theory of relativity, as an eternal act that comprises the total process of finite reality, while that which is created, whose existence happens in time, originates and passes away temporally.”

Thus, the way I figure it, eternity is not itself “duration;” rather it is the “matrix” in which ‘durations’ — temporal events (seemingly exhibiting the idea of, not only ‘duration,’ but also of ‘passing away’) — take place. Including scientific observations and measurements, which are often based on “abstractions” such as Planck time – the teensiest piece of “punctiliar time” that the human mind can measure or grasp.

Thus, eternity is the “Eternal Now” – which is not a datum of human sensory experience, for sure; rather it is a concept to which the human mind (and heart) can aspire and understand.

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Thank you so much, Stripes, for the discussion of Liebnitz’s understanding of his own work and what the academy has seemingly reduced it to. I was unaware that mathematics or the natural sciences could ever be exposed to the work of the deconstructionists, who ever seek to separate the “author” from his “text,” so as to make of the “text,” in effect, whatever they want. I guess I’ve been mistaken about this. Yet to me, the “author’s intention” is indispensable to the understanding of any “text.”

Just goes to show you how irredeemably “old-fashioned” I am.

Thank you so much for your informative and thought-provoking post.

Hello Stripes! When I wrote earlier to suggest that perhaps integers are analogous to time, and real numbers to eternity, I was thinking in terms of the number line that is composed of both.

The reason I wanted to post about numbers was that I found your idea of an essay about numbers and eternity quite interesting, but I found your approach a little trouble some, namely equating real numbers to eternity (i.e. infinity.) I assumed at some point you would talk about the infinte number of digits required to represent an irrational number in a decimal system of notation. So let me offer this for your consideration.

It is a problem that caused great scandal in the Pythagorean Brotherhood more that 2500 years ago. Reportedly it drove some Pythagoreans mad, even to the point of riot and murder.

Pythagoras is most famous for the Pythagorean theorem: the square of the hypotenuse of a right triangle is equal to the the sum of the squares of the other two sides. So the finite length of the hypotenuse is the square root of the sum of the squares of the other two sides.

Now take an isosceles right triangle. The legs which are at a right angle to each other are one inch long. What is the length of the hypotenuse? Exactly the square root of 2. The square root of 2 is an irrational number. It cannot be expressed as the ratio of whole numbers. The philosophy of the Pythagorean Brotherhood was based on the idea of rational numbers. When they discovered irrational numbers, some of them went mad.

Does that mean that the square root of two is infinite? Look at that isosceles right triangle again. What is the length of the hypotenuse? The length of that finite line segment is exactly the square root of two. If the square root of two were an infinite number, it would take you an infinite amount of time to draw that hypotenuse. But it doesn't. You can draw it quite easily without exhausting all the lead in your pencil.

It turns out that irrational numbers aren't very mysterious after all. The are very mundane things. We use them all they time. We can hold them in our hands. They are finite.

I think you have offered some other ideas about eternity in this thread that are very profound. I have also offered some suggestions that may have possibilities for further development: process, direction, division by zero, infinitesimals, etc. Why have mathematicians forbidden division by zero? What are they afraid of? Well, I think they are afraid of eternity. And maybe in some sense we should all be afraid of eternity. Or at the very least approach it with due respect.