***
Another example of combination is found in a deck of cards. If you are playing cards and are dealt the ace of clubs for your first card, you might think yourself lucky. If the second card is the two of clubs you might feel less lucky. However if you continue to receive the three, four and five of clubs you might be suspicious about the shuffling of the deck and wonder what everyone else was dealt. The dealer may tell you that these things happen. Actually that sequence is just as likely as any other; it just catches the eye. There are 2,598,960 five-card hands you could be dealt. The trick to a game like five-card draw (aside from betting and bluffing) is to have the least likely hand. So a royal flush (4 out of 2 million) beats a pair (1 million out of 2 million). The pair is a high-entropy combination.
Can you shuffle a deck and have all the cards return to their original order? This problem is related to the future of the universe and is called the recurrence time. When you shuffle the deck the cards can take on one of 8 × 1067 arrangements. Given that the age of the universe is 4 × 1017 seconds, it is not an everyday occurrence.
Another famous problem in randomness states that if enough monkeys were to randomly type for long enough they would type out all the books in the British Museum. If ten billion monkeys (a bit more then the human population of the Earth), were to type ten key strokes a second for the age of the universe they would type about 1029 characters or 1026 pages. Instead of waiting for them to type out the whole contents of the museum, let us only consider Hamlet, which contains about 180,000 characters. If the monkey’s keyboard had only 26 letters and a space bar, the probability of randomly typing Hamlet is 1 in 27180,000 or 1 in 10257,645. This is a ridiculously large number and so the probability of the monkeys getting it right is incredibly small. 10257,645 is a number with over 257,000 digits. When we introduced scientific notation with numbers like 3.0 × 108 m/s we said that we were dropping less significant information and focusing on the 3 and 8 only. Now we are looking at the number of digits only and disregarding what those actual digits are.
One of the classic numbers with a name is the googol. A googol is 10100, which is bigger than our count of quarks or photons, but dwarfed in comparison to combinations of pennies or letters. The mathematicians Edward Kasner (1878–1955) and James R. Newman (1907–1966) described the invention of this name in their book, Mathematics and the Imagination (1940):
The name “googol” was invented by a child (Dr. Kasner’s nine-year-old nephew) who was asked to think up a name for a very big number, namely 1 with one hundred zeros after it. He was very certain that this number was not infinite, and therefore equally certain that it had to have a name. At the same time that he suggested “googol” he gave a name for a still larger number: “googolplex.” A googolplex is much larger than a googol, but is still finite, as the inventor of the name was quick to point out. It was first suggested that a googolplex should be 1, followed by writing zeros until you got tired. This is a description of what would actually happen if one actually tried to write a googolplex, but different people get tired at different times and it would never do to have Carnera a better mathematician than Dr. Einstein, simply because he had more endurance. The googolplex then, is a specific finite number, with so many zeros after the 1 that the number of zeros is a googol.
Incidentally, the name googolplex was originally proposed as the name of that famous internet search engine. It was shortened to googol and then misspelt as Google. The company with that name has only about 1015 bytes of data in its system.
A googolplex is which is a big number that is also difficult to write. Numbers like this are sometimes referred to as power towers because of the way the exponents stack upon each other. Since they are difficult to write, and because this really is just the tip of the iceberg for truly large numbers, there are several alternative notations. Here I will describe one: the Knuth up-arrow notation. It starts out looking simple:
Despite the simplicity of this notation these are already huge numbers. Since 1010 is 10 billion, 10 ↑↑ 3 is a one followed by 10 billion zeros. Likewise 3 ↑↑ 4 = 37,625,597,484,987, which is 7 trillion threes multiplied together, which yields a number with 3.6 trillion digits.
This double-arrow notation is clearly about pretty big numbers, and it also is a extension of exponents or powers. Also, it is clear where this notation came from. The single-arrow notations looks like:
But what about a number like 3 ↑↑↑ 2? Double-arrows and triple-arrows are just arithmetic operations like multiplication and addition, just a bit more challenging. We might do well to step back and review what we think we know about arithmetic.
The first operation of arithmetic is addition:
3 = 1 + 2 or c =a +b.
The second operation of arithmetic is multiplication (subtraction is just negative addition):
So we can write the second operation (multiplication) in terms of the first operation (addition) repeated many times.
The third operation of arithmetic is exponentiation:
Again, exponentiation is the repeated application of the previous operation, namely multiplication.
The fourth operation is called the tetration, and brings us to our power tower:
At this point they are also usually called hyperoperations instead of just operations.
I will just mention the fifth hyperoperation to give us a feeling for how big these numbers are becoming. Consider:
As we saw before, 3↑↑3 has 3.6 trillion digits. That means that 3↑↑↑3 is a power tower with 3.6 trillion tiers going up!
The reason I have mentioned the up-arrow notation is that there are problems whose solution require it, much like the way Archimedes had to invent a number system to count those grains of sand. There is a famous problem in mathematics that involves cubes in higher dimensions and the lines that connect their vertices. Ronald Graham was working on this problem in the 1970s. He did not solve the problem, but he did put an upper bound on it. Martin Gardner, the mathematics writer for Scientific American, named this upper bound, Graham’s number, and for many years it held the distinction of being, according to Gardner, the “largest number ever used in a serious mathematical proof.”
Graham’s solution started with the number g 1 = 3 ↑↑↑↑ 3, which is a hefty number already. But in the next step he blows that number away.
That is, a number with g 1 arrows. But it gets worse.
And the cycle continues until we get to Graham’s number:
Despite its size, that number is finite. In principle, you can count that high and name that number.
***
One final word on combinations. In the study of dynamical systems there is something called the Poincaré recurrence theorem, which was alluded to when we looked at shuffled card decks. The theorem says that a dynamical system, given long enough, will return to a state very close to its initial conditions. This theorem has caught people’s imagination because it seems to say that given enough time the universe will arrange itself, atom by atom, as it is now. And then history will repeat itself, exactly. Actually the theorem has a number of caveats that means it may not apply. Poincaré’s theorem requires a constant phase space, which is a map of where things are and where they are going. In an expanding flat universe, this condition may not be met. Still, there was a recent calculation of the recurrence time for all the particles in the universe that arrived at:
This is about , a number with a mere googolplex digits. I like how the author wrote the time unit as “Planck times, millennia, or whatever.” Planck time is 10−44 seconds, a millennium is a thousand years or 1010 seconds. In other words, being off by 54 digits does not really matter.
***
Yet there is something larger than Poincaré recurrence time for the universe or even Graham’s number, and we use it every day. This is infinity. Since throughout this whole book I have plotted objects on a logarithmic scale, I will try to do that here too (see Figure 14.1). But even on a logarithmic scale, infinity is a long wa
y off. If I take everything we have talked about and shrink them to a single pixel, infinity is off the page, and so the best I can do is draw an arrow to show that the line continues.
Figure 14.1 Everything is small compared to infinity. The Planck length and the size of the universe are on top of each other in a logarithmic plot that tries to also show zero and infinity. Zero is still infinitely far to the left and infinity is infinitely far to the right.
Right in the middle is a dot that is labeled “Plank length” (10−34 m) and “size of observed universe” (1027 m) and they appear on top of each other. That dot also contains numbers such as Poincaré recurrence time times the speed of light (to get meters) and Graham’s number of meters (or Planck lengths, or parsecs). The axis stretches off to the right, off the page, beyond the Earth, beyond our galaxy and even beyond the edge of the observable universe. The other end of the axis also points off forever, even on a logarithmic scale, but not towards negative infinity. Rather, it points off to ever smaller and smaller sizes towards zero. Infinity and zero are somehow related.
One of the first times that infinity and the infinitesimal had a bearing on physics was in Zeno’s paradoxes, which were mentioned in Chapter 8 when we talked about the flow of time. The most famous of Zeno’s paradoxes was a race between Achilles and a tortoise. Zeno could have used a photon and continental drift to make his point, but instead he used Achilles, the athletic Greek warrior who chased Hector around the city of Troy three times in the Iliad.
In the paradox Achilles and the tortoise are to race. The swift Achilles gives the tortoise a head start, confident that he can easily catch him. A minute after the race starts Achilles has run to where the tortoise started, but the tortoise has moved forward a short distance. A few seconds later Achilles has moved to the tortoise’s one-minute mark, but the tortoise has already moved forward a further finger’s width. By the time Achilles has crossed the finger’s width, the tortoise has traversed an additional hair’s breadth, and so forth. Zeno was led to the conclusion that Achilles could never catch the tortoise; that it was logically impossible for him to do so. Therefore, Zeno summarized, motion was an illusion; a view supported by his mentor Parmenides.
Most physicists see this as a sort of calculus problem. An infinite number of infinitesimal steps leads to real, finite motion. In fact this is the basis of the calculus of Newton and Leibniz. We can treat a function or a trajectory as if we can divide it into an infinite number of parts, each infinitesimally wide. Whereas Newton’s own opus, Principia, tended to not rely on calculus, it is hard to conceive of modern physics without it.
But what really is infinity? A good way of trying to get a handle on an abstract concept like infinity is to make a list of collections that have an infinite number of members (see Table 14.1). For instance, there are an infinite number of natural numbers (positive integers). There is also an infinite number of odd integers, an infinite number of even integers, an infinite number of perfect squares and so forth. It was Georg Cantor (1845–1918), a German mathematician, who first really figured out how to understand infinity.
First, a collection is called a set and the number of members of a set is called its cardinality. For example, the cardinality of the alphabet is 26. We can also talk about the cardinality of a set that has an infinite number of members. In fact the cardinality of the set of natural numbers is ℵ0. This symbol is aleph-naught, aleph being the first letter of the Hebrew alphabet.
Table 14.1 Examples of sets that have an infinite number of members, or cardinality of ℵ0
Cantor pointed out that in all these examples there was always a one-to-one relationship with the natural numbers. This means all of these sets have the same number of members, or cardinality, as the natural numbers. I do not actually need an equation, I just need to be able to order them. For example, according to Euclid’s theorem there are an infinite number of prime numbers. Since I can match each natural number with a prime number, 1 ↔ 2, 2 ↔ 3, 3 ↔ 5, 4 ↔ 7, 5 ↔ 11,…and so forth, there is a one-to-one relationship. The cardinality of all of these examples is .
What makes infinite sets unique is that one set, for example, the perfect squares, has the same cardinality as natural numbers, yet it is missing members. This does not happen with finite sets, and so it is a way of recognizing infinite sets.
Cantor went beyond this; natural numbers are just the starting point. The cardinality of ordinal numbers , so these are a different type of infinite. The real numbers, which include irrational numbers such as π and are continuous, and the cardinality of the real numbers is denoted as [beth]1 (beth-one where beth is the second letter of the Hebrew alphabet). It can also be shown that these are very different to, and much larger than, . Apparently there are an infinite number of types of infinity.
***
So how important is infinity to the universe? True, we use calculus to chart the smooth curve of a planet in orbit or the trajectory of an electron. But is that the way that nature really works? Remember, down at the Planck scale, at 10−34 m, we have reasons to think that nature might be broken into tiny, but finite bits. Real physical space and time might be quantized and not made up of the infinitesimal.
What is clear is that while the physical world may be bracketed by the Planck length and the edge of the observable universe, mathematics knows no such bounds. The mathematical universe is not only growing with each generation of curious minds, but it seems to me that the growth is accelerating into a truly limitless space.
15
Forces That Sculpture Nature and Shape Destiny
I promised you at the beginning of this book that by the end we would be able to turn the question “How big is big?” into an answer: “This is how big ‘big’ is.” The problem was not just learning a number like 4.3 × 1026 m, the distance to the edge of the universe, or 1.6 × 10−35 m, the Planck length, but really appreciating what that 1026 means. What is one hundred septillion?
We have tripped through analogies, skirted across various number systems, touched on similes and descriptions. But almost always we have come back to that logarithmic plot, the scales of nature plot. This is because I think we understand how big something is by comparing it to things that are a bit bigger or a bit smaller then it. We can say that there is a factor of 1042 between a proton and the size of the universe, but that is not how we picture them in our heads or how we understand these sizes. We need to take steps, going down in size—periwinkle, hair, cell, virus, molecule, atom and then that leap to nucleus—or going up—whale, redwood, kilometer, Wales, Earth, to the Moon, to the Sun. There are still big leaps, such as the one from an AU to the Oort cloud, or the jump from interstar distances to galactic scales. But the jump from the Sloan Great Wall to the size of the observable universe is only a factor of a hundred and not so hard.
We understand how big something is by where it is located on our scale, and what its neighbors are. So I think this one plot of scales becomes central to our understanding of “big.” Therefore I would like to look at it one more time with a critical eye and ask if the way I have drawn it somehow distorts our view. Is there a bias? I will address this in two parts: first looking at the axes that are drawn and secondly looking at the objects that are plotted.
When drawing the axis, I have usually plotted object sizes expressed in meters and incremented major tick marks by a factor of ten. To see that this does not introduce a bias, at the start of this chapter I have plotted the same objects on three scales. The first scale is using the ancient unit of stades instead of meters. The second scale is the usual one, marked in meters. The final scale is also meters, but the tick marks are based on doubling, base 2, instead of factors of 10. This will change the words we use. The size of an atom is 5 × 10−13 stade, or 2−33 m. The numbers were simply multiplied by a conversion factor, but the spacing between objects did not change. No matter what scale we use, the solar system, as bounded by the orbit of Neptune, is still 6500 times bigger than the Sun. The relative spacing of
the objects does not change and this comparison of objects is really what is important.
What about the choice of objects I have plotted? Have I omitted legitimate objects, or have I grouped together things that should not be seen as a class or a type?
Yes I have omitted real and interesting things. This book has not been encyclopedic and has not tried to be. This is especially true when dealing with the scales of our everyday life. I have not discussed the sizes of flowers and islands. I once read a proposal for a classification system that would label hills, mountains, massifs, ranges and so forth based on their heights, prominence and expanses. Humans have a nearly inexhaustible desire to organize things and to name them. That landscape naming system did not even start to address what are a fells, hummocks, alps, tors and so forth. But it is not clear to me that these are necessarily distinct and fundamental objects of the cosmos.
Finally, have I lumped things together that should not be in the same group? This is an ancient problem that even the Greeks wrestled with. I will, however, follow the lead of naturalist Carl Linnaeus, who wrote Systema Naturae (1735) and tried to organize all living things. He introduced a hierarchy to the taxonomy of plants and animals: kingdom, class, order, genus and species, based primarily upon the appearance of the reproductive part of a plant or animal. His system has been greatly modified over time, with the addition of phylums, families and subdivisions, but it was a starting point. The most important change came after Darwin. Classifications are now not based upon appearance, but on evolutionary history.
How Big is Big and How Small is Small Page 26