On the concept of “infinity,” David Grandy, a former professor of philosophy at BYU Provo and Hawaii, wrote the following which one found interesting:
Inexhaustible Giving
To me, the very least of all saints, this grace was given, to preach to the Gentiles the unfathomable riches of Christ.—Paul, Ephesians 3:8 (NASB)
Coming back down to Aleph null or ground-level infinity, we find rich theological insight. Bessy points out that infinity, as understood mathematically, makes plausible the scriptural promise contained in Doctrine and Covenants 84:38: “And he that receiveth my Father receiveth my Father’s kingdom; therefore all that may Father hath shall be given unto him.” Normally it is not possible to give all the one possesses without suffering loss, or to give all that one possesses more than once. But the scriptural implication is that this giving can happen numberless times, and always without diminution of God’s kingdom. So does “give in this case just mean to share without actually giving, or is there a deeper manner of giving, based on an economy of infinity that enables transfer of one’s entire kingdom without loss?
Suppose, says Bessy, that you have an infinite number of gold coins which you wish to give to an infinite number of people while retaining an infinite number yourself. Is this possible, and, if so, by what arrangement? Thanks to Galileo and later mathematicians, it is easy to see that you could, without loss, divide your wealth infinitely among a finite number of people. Assigning a positive integer to each coin, you could give all the even-numbered coins to another person, thereby splitting the original infinity into the same-size infinities—one or yourself and one for the other person. Or you might give every first coin to one individual, every second to another individual, and keep every third coin for yourself. But doing this, or something like it, would not allow you to realize your wish of optimizing the potential largesse of an infinity of gold coins.
The solution, according to Bessy, inheres in the following arrangement:
You keep 1 3 6 10 15 21 . . .
You give 2 5 9 14 20 . . .
You give 4 8 13 19 . . .
You give 7 12 18 . . .
And so on. The key lies in successively increasing by one the interval between the coins you keep, thereby bringing another person into the embrace of your giving, ad infinitum, without shrinking in the least your own possession.
Even such a modest infinity as the countable integers enables infinite or endless giving. Who knows what awaits us as we learn the mysteries of the uncountable infinities, the first of which are the irrational numbers. In retrospect we may regard them as breathing holes in the neatly-ordered, tightly sealed vision of reality once entertained by the Pythagoreans. The most famous of all irrational numbers is pi (π), the relation of a circle (its circumference) to its diameter, approximately 3.14. Though pi’s decimal expansion has been calculated to billions of digits, the “sequence of digits . . . looks like gibberish,” says Gregory Chudnovsky, a mathematician who, along with his brother David, has spent years exploring pi. But looks are deceiving, adds Chudnovsky: “Pi is a damned good fake of a random number. I just wish it were not as good a fake.” There may be an order, a beauty, to its mind-boggling complexity. In fact, like other irrational numbers, pi may be “a powerful random-number generator” (Preston, “Mountains of Pi,” p. 64), an algorithm for keeping the quest for pattern and meaning alive. If so, we have much to look forward to.
Leaving aside the metaphysical question of what pi is all about, however, we can say this much about the larger infinity of which it is a member. Difference is the primordial intrigue of irrational numbers because it is the engine that prevents each one from suffering from the fate of terminal exactitude, which is precisely which keeps rational numbers in their finite, potentially redundant place. Irrational numbers have no end because they do not lend themselves to closure; they stay alive, as it were, by generating irreducible difference with each new digit. Contemporary culture likes to put things in their place, permanently so, but irrational numbers suggest a larger world in which things forever unfold in new and surprising ways. Hence there is no redundancy in this larger world, and no prospect of finality. (David Grandy, Worlds Without Numbers: An LDS Perspective on Infinity [2018], 96-99)
