As the first mathematician to deliver a McKay lecture, Jack V. Johnson predictably elected to discuss the connections among time, space, and infinity for the twenty-seventh lecture. After earning both the B.S. (1964) and M.S. (1966) degrees from Idaho State University, Johnson came to Church College of Hawaii in 1966, later completing his formal education with a D.A. from his alma mater in 1980. Though muscular dystrophy has limited his participation in some activities over the years, it has not curtailed his interests: Johnson actively supports a variety of extracurricular programs on campus. The recipient of several grants form the National Science Foundation, Johnson wrote a mathematics text used in University general education courses for a dozen years. In addition to many Church teaching positions, he has been elders quorum president, bishop's councilor, and bishop. Chair of the Division of Math, Science and Technology since 1985, he and his wife Cassie, who teaches math part-time at BYUH, have seven children: Leilani, Rebecca, Holly, Cassandra, Van, Wesley, and Rose.
Introduction
In producing this work, I have made a genuine effort to use nontechnical sources which are readily available to the reader. I have, for instance, relied heavily on Stephen W. Hawking's book, A Brief History of Time (1988) which M. Mitchell Waldrop described as, "perhaps the most challenging book ever to spend half a year on the New York Times best seller list" (1248). I know many in the audience have read Hawking's book, at least in part. For any who got all the way through it and would like to know the latest from this extraordinary mathematician, I refer you to the December 2, 1988, issue of Science magazine, from which I took the previous quotation.
The goal I have for this lecture today is to provide additional support for the thesis I introduced to many of you in my devotional address in 1985: The advances of science and technology tend toward overlapping or explaining, in effect supporting, the teachings of the restored Gospel of Jesus Christ (Johnson).
I was pleasantly surprised to observe within the 30 days immediately following my topic selection that two apostles and a presidential lecturer made reference to the same idea. Gerry Avant in the October 15, 1988, issue of the Church News, in a brief biographical sketch of Elder Richard G. Scott, specialist in nuclear energy and new member of the Quorum of the Twelve, quotes Elder Scott: "Truth in both the realm of science or religion match perfectly. All truth comes from the same source, from our Father in Heaven and His Son, Jesus Christ" (qtd. in Avant 6).
Additionally, in his Presidential Lecture on our campus November 1, 1988, Orson Scott Card stated basically the same feelings, and at the Book of Mormon Symposium, October 30, 1988, at BYU, Elder Boyd K. Packer declared,
Laws which govern both the temporal and the spiritual are ordained of God. After all of the tomorrows have passed and after all things have been revealed, we will know that those laws are not in conflict, but are in harmony. (5)
In trying to convey this message in my 1985 devotional, I used a large, fixed circle to represent all truth, and smaller, moving and expanding circles to represent man's knowledge, emphasizing then as I must now, that there is no way to make the scale accurate (figure #1). If a small circle is to be man's knowledge, the circle of all truth might be as big as this room, as big as our solar system or even infinite! As man sincerely pursues truth, his circles of knowledge grow and tend to overlap more and more the circle of truth.
From time to time the Lord reveals some truth through his prophets. How fortunate we are if we have the faith to accept what is revealed, even when it runs contrary to the current theories of men. The Word of Wisdom is an excellent example, but we have similar, more recent examples. For instance, the Church was down on loud music almost as soon as it began making itself heard, but it took almost 30 years for the scholars to catch on that it was causing deafness. One quote from Time, September 26, 1988, illustrates the present situation:
Fred Schneider, vocalist for the B-52's, stuffs tissue into his ears. Rock Promoter Bill Graham. . . keeps vats of earplugs available for everyone from security guards to roadies.Fans, too, are being urged to plug up. (Toufexis 78)
Is it not very interesting that with people using plugs both on and off the stage, no one seems to have the sense to just turn the amplifiers down?
Although R-rated movies do not similarly cause blindness, I suspect the long-term damage may be much more devastating to many partakers. The counsel from the Church is simple and easy, "Just say no." Some day the statistics will be in. I could continue for hours talking about how recently developed plate tectonics corroborate that the earth had only one land mass in the beginning (Gen. 10: 25; D&C 133: 24), how modern electronics give us the capacity to be seen and heard around the world, how advances in chemistry and physics enable us to change one substance to another, and many other examples which literally would have been miracles just a century ago (see figure #2).
I could go to Book of Mormon evidence and start with this notation from the National Geographic Society, Archaeological Map of Middle America (1968). The site is Monte Alban in southern Mexico, which I believe was the first principal Nephite city. The notation reads, "Monuments date from first occupation of site around 600 B.C., and reflect earliest use of the calendar and writing in Middle America" (Archaeological). Equally interesting is an entry for Tikal, Guatemala, which I think was the first major Lamanite settlement. This notation reads, "Huge platform of Early Classic temples conceals structures dating back to 200 B.C. Pottery atop bedrock here dates Tikal's first inhabitants at around 600 B.C." (Archaeological).
We could also look at calendar and numeration systems. While there are many differences between Central American and ancient Middle East calendars, there are also some very interesting similarities. For instance, of the 33 calendar systems which I reviewed (Richmond 31-121, the only two which used fixed rules for determining month and year length for their civil calendars were the Egyptian and the Central American (Richmond 98). Broughton Richmond in his book, Time Measurement and Calendar Construction (1956), makes this observation, when talking about Babylonian and Egyptian celebrations of the Pleiades, "Curiously enough, certain tribes in South America celebrate the New Year festival by observation of the Pleiades, and give to the year the same name as the constellation" (58).
The numeration system for the Mayan calendar uses the number 360 as a base position (Richmond 98). The widespread use in our culture today of numbers based on 60, such as 60 seconds in our minute, 60 minutes in our hour, 360 degrees in our measurement of the circle, comes down to us from the mathematicians and astronomers of the ancient Middle East, the same area from which Lehi departed (Richmond 128-129, 134).
Many of the ideas above could provide material for an entire lecture; I gave a thumbnail sketch to let you know some of the breadth of my claim that the results of scholarly efforts on many fronts tend towards the everlasting Gospel. The balance of this lecture must be spent in time, space and infinity.
Time
For thousands of years, time was a constant, as familiar as a family member and more reliable than the proverbial tax collector. "Sunrise, Sunset. Sunrise, Sunset" (Bock). A song even tries to suggest monotony. But in the days of our grandfathers, time began to take on a very different appearance. First, theory suggested and then observation confirmed that time does not march on tick by tick as the pendulum clocks suggested. I will use one of Hawking's examples:
Consider a pair of twins. Suppose that one twin goes to live on the top of a mountain while the other stays at sea level. The first twin would age faster than the second. Thus, if they met again, one would be older than the other. In this case, the difference in ages would be very small, but it would be much larger if one of the twins went for a long trip in a spaceship at nearly the speed of light. When he returned, he would be much younger than the one who stayed on Earth. This is known as the twins paradox, but it is a paradox only if one has the idea of absolute time at the back of one's mind. In the theory of relativity there is no unique absolute time, but instead each individual has his own personal measure of time that depends on where he is and how he is moving. (33)
I will add that if an object actually reaches the speed of light, the equations of Einstein's relativity call for time to stop altogether. This revolutionary step in the concept of time brings us nearer to an understanding of God. In the Doctrine and Covenants 130: 7, we read:
But they reside in the presence of God, on a globe like a sea of glass and fire, where all things for their glory are manifest, past, present, and future, and are continually before the Lord.
When that section of the Doctrine and Covenants was recorded, Einstein had not even been born; his father probably had not been born, and his revolutionary theory was almost a century in the future. Now we can at least say the scripture agrees with both theory and observation, or more accurately, that theory and observation have come to agree with the scripture.
There are some interesting questions raised by this concept of time. Certainly there is no doubt that God knows our future, for He is ever present in every moment of time. He can see our future now. What happens if we make it into His presence in the end, and as the scriptures say, "there shall be time no longer" (D&C 88: 110)? What does "future" mean in that context? Does God still know our future? Does He know His own future? I do not recommend that you get hung up on those questions, nor shall we here. We must leave time and enter space.
Space
As a high school student some 30 years ago, I won an essay contest by explaining my own personal concept of matter and space. I reasoned that everything we knew was made of smaller parts and, at the same time, was a part of something larger. The earth we knew to be made of molecules, which were made of atoms, which were made of protons, neutrons and electrons, which at that time were thought to be the basic particles of matter. On the other hand, the earth was a part of the solar system, which was part of the Milky Way galaxy, which I reasoned would be discovered to be part of some larger cosmic system. I refer you to the January 7, 1989, issue of the Smithsonian for a superb article on galaxies by James Trefil. Astronomers have determined that our galaxy is part of a collection of galaxies called the Local Group. According to the article,
The Local Group is not isolated in space, either, but is part of a still larger group called the Local Supercluster. The great majority of the galaxy clusters are arranged in such superclusters.Only in the past couple of years have we discovered that this rather regular, hierarchical structure is just the beginning of the story, rather than the end. All these galaxies, clusters and superclusters, are distributed as if they outlined the surfaces of huge bubbles. (Trefil 52)
In that high school essay, I reasoned similarly that the protons and neutrons would also be discovered to be composed of smaller particles, which would be composed of smaller particles, and on we go. Such seems to be the case today, and I will let Hawking clinch the issue with a statement near the end of his book: "We might indeed expect to find several new layers of structure more basic than the quarks and electrons that we now regard as 'elementary' particles" (167).
One of the biggest problems we have with space is its size. Light travels 186,000 miles per second. That means it can go almost to the moon, or around the earth more than seven times in just one second. The light from our sun comes in less time than we allow for our break between classes. Yet, we have observed stellar systems so far out in the universe that it takes over 12 billion years for their light to reach us. Furthermore, our theory says that matter cannot travel faster than light. We know from Doctrine and Covenants 131: 7-8 that spirit is matter: "All spirit is matter, but it is more fine or pure, and can only be discerned by purer eyes; We cannot see it; but when our bodies are purified we shall see that it is all matter."
Since our resurrected body contains our spirit, it can travel no faster than light according to the theory of relativity. That seems not to be an acceptable conclusion because God has a resurrected body, and how could He govern in a universe that would take Him billions of years to cross? Must we take the limit off the speed of matter then? That seems unlikely since any finite maximum would be unacceptable on similar grounds, and to have infinite speed would be equivalent to simply being there already, which is essentially the "fills the whole universe" concept (D&C 88: 12). We will look at two other possibilities.
One possibility is additional dimensions in space. We see space as having three dimensions, which are most obvious in the corner of the floor of a rectangular room. One dimension runs in the direction of the length of the room, another in the direction of the width, and the third goes up towards the ceiling. What would be the properties of a fourth dimension? To get some idea, let us see what happens when a third dimension is added to a world that had only two.
Try to imagine that you are a dot living in the two-dimensional world of a piece of paper. If in your travels you came to a line segment, you would have to go around it; you could not simply jump over it because that would utilize a third dimension which is not a part of your experience. If you came to a square, you would not be able to see more than two sides of it at a time, and you could never see what was inside (see figure #3).
However, if you were allowed to take a step into the third dimension with the rest of us, you would immediately notice that not only can you easily see what is inside the square, but you can see all four of its sides simultaneously. Similarly, if we in three dimensional space encounter a closed three dimensional object such as a rectangular box, we cannot see inside, nor can we see more than three sides of the box. If we could step into a fourth dimension, we could see all six sides of the box and could see inside without opening it! Furthermore, just as the dot could shorten his travel time by using the third dimension rather than going around the line segment, so could we shorten our travel time by using a fourth dimension, if we could but see it.
Since a picture is worth a thousand words, a movie must be worth a million words. I have obtained a movie from Texas titled Dimension and another movie about time from our Learning Resource Center (Mystery). The schedule of showings is printed on the strip of paper you were given with your program. That little strip of paper is more than a schedule though; it is a visual aid (figure #4). Think of this strip as being two separate universes since it has two surfaces and there is no way for a resident, such as the "D" in "Dimension," to get to the other side to visit the "S" in "Space" without going over an edge. By using the third dimension, I can join these two universes together in an interesting way.
Remove the protective paper from the tape and hold the strip as if you were reading the film schedule. Our goal is to place the black section on the right directly onto the tape in a very specific way; so please follow my instructions carefully and do not stick the paper to the tape until I tell you. Curve the paper strip into a ring so that both the tape and the black mark are facing up. Now twist the end with the black mark so the mark is facing down and join it to the end with the tape. You now have what is known in topology as a Mobius strip. We have changed our paper strip having two sides into a strip having only one side, for "D" can now move along the strip to visit "S" without crossing over an edge. Remember, though, that this marvelous change required the use of a third dimension, which "D" did not have initially.
The second possibility for quick space travel is by far the most intriguing and the one that I personally prefer because it provides possible answers for more than just the travel question. It is called a "wormhole." Really, I am serious. In fact, I wanted to title my lecture, "Time, Space and Wormholes," but it just left me with a bad taste. I am sure a decade from now it will sound perfectly nice. Wormholes are discussed in the article already referred to in Science magazine and in the January, 1989, issue of Science Impact (7), but the best treatment for our purposes is by Michael D. Lemonick in the January 16, 1989 issue of Time. The article is so good that it prompted me to drop the mathematical treatment I had planned to use for infinity and apply the information in the article instead, but first we must finish with space. The following is from the Time article just mentioned:
The idea of wormholes comes directly from the accepted concepts of general relativity. In that theory, Einstein argued that very massive or dense objects distort space and time around them. One possible distortion is in the form of a tube that can lead anywhere in the universe--even to a spot billions of light-years away. The name wormhole comes about by analogy: imagine a fly on an apple. The only way the fly can reach the apple's other side is the long way, over the fruit's surface. But a worm could bore a tunnel through the apple, shortening the trip considerably. A wormhole in space is the same sort of tunnel; it is a shortcut from one part of the universe to another that reduces the travel time to just about zero.Virtually instantaneous travel leads to the idea of wormhole as time machine. If it were somehow possible to move one end of a wormhole at nearly the speed of light, general relativity dictates that time at that end would slow down, and that portion of the tunnel would then be younger than the other end. Anything moving from the faster-aging end of the wormhole to the slower would essentially go backward in time. The mode of travel, however, could be nothing like the mechanical time machine, complete with saddle, envisioned by H. G. Wells. It is hard to conceive how a human being could move through a wormhole, since it would theoretically be narrower than an atom, and it would tend to vanish the instant it formed. (Lemonick 55)
I will merely add that I cannot imagine an opening too small for God to pass through, nor an instant too short for Him to make the passage.
Infinity
I imagine that you, as I, when thinking about the numberless creations of God, have thought of the immensity of the space we have talked about. I have thought about the billions of galaxies, each having billions of stars as possible suns for other earths, and have felt quite comfortable that there was plenty of room out there for all of God's creations. But I no longer believe that we have been thinking anywhere near big enough! I quote again from Lemonick's article:
This week at the American Astronomical Society's winter meeting in Boston, physicist Alan Guth of M.I.T. will announce the most mindnumbing wormhole-related news yet. Guth and two collaborators have determined, he says, that 'it would apparently be possible in principle for some advanced society literally to create an entirely new universe. . . . Such a baby universe would form a wormhole and escape, creating its own space and time in the process.' (55)
Here are some related quotes from Mitchell Waldrop's article in Science:
it is possible that empty space can spontaneously give rise to a whole, new, baby universe. . . . [I]t is entirely possible for this baby universe to have stars, galaxies, planets, and even life. . . . If we can imagine a wormhole appearing for an instant in one spot, says Hawking, then we can equally well imagine quadrillions of wormholes flickering in and out of existence at every spot in the universe. (1249)
Let me make sure you have the picture. At every point in space you could have a wormhole form. Each wormhole could close itself off from our universe and go on to produce an entire universe like the one we know. And the same thing could happen again at each point in each one of those universes! Do you see what I mean about thinking really big? I mean infinitely many infinities. Granted, these men are mathematicians and theoretical physicists; but, I want to remind all of you that so was Einstein 70 years ago, and soon we made the bomb. I will close Infinity with a brief discussion of two scriptures, which I believe relate to these multiple universes we have just discussed. Here is Doctrine and Covenants 88: 37: "And there are many kingdoms; for there is no space in the which there is no kingdom; and there is no kingdom in which there is no space, either a greater or a lesser kingdom."
When God says "no space," does he mean no cubic light-year of space, no cubic mile of space, no cubic centimeter of space, or no point of space? Everyone would agree there is space between us and the moon. Technically there is space between you and me. Does that mean there are kingdoms there as well? We will, of course, each come to our own conclusions. I now quote Moses 1: 35 and 38-39:
But only an account of this earth, and the inhabitants thereof, give I unto you. For behold, there are many worlds that have passed away by the word of my power. And there are many that now stand, and innumerable are they unto man; but all things are numbered unto me, for they are mine and I know them. (1: 35)And as one earth shall pass away, and the heavens thereof even so shall another come; and there is no end to my works, neither to my words. [Italics added]For behold, this is my work and my glory--to bring to pass the immortality and eternal life of man. (1: 38-39)
Does that sound to you as it does to me, that there are entire universes coming and going as God performs his works? We may be the only life in this universe. Our cousins just might be a wormhole away in universes of their very own.
Conclusion
On the lighter side, there are much easier ways to show how mathematics brings people closer to God. One could just quote Ernest L. Boyer slightly out of context. I do not have an exact quotation from the lecture he gave at the University of Hawaii about a year ago, but it went something like this, "No matter what the courts and legislatures do, as long as there are final examinations in mathematics, there will be prayer in the public schools!"
My final quotation is again from Stephen Hawking:
ever since the dawn of civilization, people have not been content to see events as unconnected and inexplicable. They have craved an understanding of the underlying order in the world. Today we still yearn to know why we are here and where we came from. Humanity's deepest desire for knowledge is justification enough for our continuing quest. (13)
Fortunate are we to not only know why we are here and where we came from, but to know who we are as well, and, I might add, to have some idea of where we are going. That we may place all our learning in context with these and other revealed truths is my prayer for all of us, in the name of Jesus Christ. Amen.
Works Cited
Archaeological Map of Middle America: Land of the Feathered Serpent. Washington D.C.: National Geographic Society, 1968.
Avant, Gerry. "He follows example of role model." Church News 15 October 1988: 6, 13.
The Bible.
Bock, Jerry. "Sunrise, Sunset."
The Book of Mormon.
Boyer, Ernest L. "The Challenges of Providing Quality Undergraduate Education in a Research University." Address. Center for Teaching Excellence. University of Hawaii-Manoa, Honolulu, 16 February 1988.
Card, Orson Scott. "Who Do You Think You Are?" Presidential Lecture. Brigham Young University, 1 November 1988.
Dimension. Van Nuys, CA: AIMS Media, 1969. (16 mm., 13 min., color.)
The Doctrine and Covenants.
Hawking, Stephen W. A Brief History of Time. New York: Bantam, 1988.
Johnson, Jack V. "The World in Which We Live." Devotional Address. Brigham Young University-Hawaii, 15 November 1985.
Lemonick, Michael D. "Wormholes in the Heavens." Time 16 January 1989: 55.
The Mystery of Time. Whittier, CA: Moody Institute of Science, 1968. (16 mm., 28 min., color.)
Packer, Boyd K. "The Law and the Light." Address. Book of Mormon Symposium. Brigham Young University, 30 October 1988.
The Pearl of Great Price.
Richmond, Broughton. Time Measurement and Calendar Construction. Leiden: E. J. Brill, 1956.
Toufexis, Anastasia. "A Fire Hose Down the Ear Canal." Time 26 September 1988: 78.
Trefil, Janies. "Galaxies." Smithsonian 19:10 (January 1989): 36- 53.
Waldrop, M. Mitchell. "The Quantum Wave Function of the Universe." Science 2 December 1988: 1248-1250.
"Where No One Has Gone Before." Science Impact January 1989: 7.