What makes the existence of life and the universe seem so improbable? Without question, the incredible complexity of all things is at the heart of the improbability dilemma. And it requires some form of explanation. In order to properly examine improbability, we must first address complexity. How can complexity be explained?

The complexity of the universe is staggering, in some ways beyond human understanding. For many, this fact alone can’t be accounted for without a design, particularly when the only alternative considered is chance. With this comparison, design usually wins over chance, and design implies a designer. Ancient civilizations observed a universe that was much simpler—in their eyes—than the universe we know exists today. Nevertheless, it would have appeared complex enough to invoke a designer. Even a number of natural phenomena that are easily explained today were attributed to gods.

Our present understanding of the universe reveals a universe that is far more complex than the ancients could have imagined. We have the opportunity of looking back in time for answers—back to a time when the universe wasn’t nearly as complex. Through a series of scientific discoveries, simple origins were found to be the precursors of the present universe.

Darwin opened our eyes, albeit slowly, with his insights on evolution. As it pertains to life, Darwin showed us a different way of thinking about the emergence of life. His theory of evolution by natural selection broke down the complexity of life into incremental steps. He managed to shift the focus from the finished product (or the present product) to the steps that led to it. According to Darwin, and verified by other more recent discoveries, life has evolved from simple beginnings—simple relative to its present state. It all began with single cell organisms, and perhaps only one. Now we have a world full of diverse and complex life forms, some containing trillions of cells. Darwin showed that from simple origins, complexity could arise over time, and by a natural process.

Even the life that we see today starts simple, and grows in complexity. For example, a tree begins with a single seed, and grows to a complex structure of roots, branches and leaves. When I look at a seed I find it difficult to imagine that a tree can come out of it, and yet it does so naturally. Like the seed of a tree, a human being also has a simple beginning—we were all initially a single cell. You could make the argument that a cell is complex on its own, and it is, however, millions and trillions of cells working in unison is several orders of magnitude more complex. Keep in mind that what we classify as the origin of life—a single cell—is somewhat arbitrary. Even a cell has to be constructed from simpler chemical processes, which at some point we call life. Although life, especially the origin of life, is an amazing and mysterious process, we can clearly see that it moves in a direction from simplicity to complexity.

Now let’s turn our attention to the universe as a whole, and see if the same principle applies. After Darwin had provided an explanation for the evolution of life, it was not automatically assumed that the universe evolves by a similar process. In fact, the idea that the universe was eternal and unchanging was a long-held belief by the general population and scientists alike. This idea took some time to overthrow. But by the mid-twentieth century, new discoveries were pointing directly towards an evolving universe; one which had a beginning.

The big bang is analogous to a cell. Just as a single cell can be viewed as the origin of life, the big bang can be viewed as the origin of the universe. And as I mentioned earlier, a cell can also be thought of as complex, but nowhere near as complex as the life that arose from it. The universe can also be viewed in a similar light. Although the big bang was not necessarily a simple event, it was nonetheless simpler than the universe that emerged from it.

Scientists theorize that a substantial amount of activity occurred at the initial moment of creation. The basic forces of nature emerged (gravity, electromagnetism, and the strong and weak nuclear forces), as well as a host of elementary particles (such as photons, protons, neutrons and electrons). Space and time as we know it were also created. All that and more happened in a tiny fraction of a second. On the surface, this seems to present a problem as far as a simple beginning is concerned, however, there is more to consider.

In spite of this initial creative activity, for the first 300,000 to 500,000 years the universe was nothing more than an enormous cloud of hot expanding gas. Complexity would then increase gradually over time—in a sort of cosmic natural selection. It took one billion years before stars and galaxies formed. A few more billion years before supernovae explosions (the death of stars) created and distributed the heavier elements necessary for life. Simple life on earth emerged 9.9 billion years after the big bang. And from there it would take over 3 billion years of evolution to arrive at modern humans. From this simplified timeline, we can see that the early universe was much simpler than it is now—the result of 13.7 billion years of cosmic evolution.

There is another point worth noting that relates to the discussion. The big bang theory is a theory that describes the universe a fraction of a second after the universe came into existence. The big bang theory is silent on the cause of the creation event. Although scientists speculate on what the cause may have been, the big bang represents the edge of our present ability to understand the universe, a theoretical time barrier that we have not yet crossed. I like the way Bill Bryson wraps up the discussion regarding the cause of the big bang. In *A Short History of Nearly Everything*, he writes:

“… it may be that space and time had some other forms altogether before the Big Bang—forms too alien for us to imagine—and that the Big Bang represents some sort of transition phase, where the universe went from a form we can’t understand to one we almost can.”

Like a cell, which is created by more elementary processes, the big bang could have been a transition phase that was precipitated by a simpler pre-existing cosmos. Some scientists even suggest that the universe may have been created out of nothing. And by nothing, I don’t think they really mean nothing, but perhaps something very small that we don’t completely understand. Physicists now believe that you have to incorporate aspects of the quantum world in order to understand the big bang. And if you go by quantum theory, particles can spontaneously come in and out of existence from nothingness. That is the nothing that scientists are talking about. Bryson writes: “It seems impossible that you could get something from nothing, but the fact that once there was nothing and now there is a universe is evident proof that you can.” Therefore, if the universe was created from nothing or very little, you can’t get much simpler than that. And if this is even remotely correct, the principle of things moving from simplicity to complexity definitely applies to the universe as a whole.

Having said all that about complexity, let’s insert improbability into the equation. Both life and the universe evolved from simple origins, and through incremental steps, have grown in complexity. Although this does not explain how the simple origin came about, it does show that complexity can be achieved by gradual steps, even if the finished product seems improbable—improbable by means other than design. Also, an after the fact approach of looking only at the finished product can be deceiving, that is in terms of what improbability entails. If something is improbable, does it mean that it can’t happen? And because the existence of life and the universe appears improbable, does it mean that it came about by design?

Let’s begin with a simple exercise. Do you remember what you did yesterday? I mean everything you did yesterday. If you went to work, think about the route you took, and the exact location of the cars you passed. What about the people you met and the exact time you met them. Then there are the phone calls or emails you received. Where did you have lunch, what did you eat, and with whom? What tasks did you perform? And what about after work, what else happened? You get the idea. Although you may think you had an ordinary day, the fact is that the exact details of your day will never happen again. Yesterday, just as it occurred, was extremely improbable. And today, tomorrow, and every other day will unfold in a way that is also improbable.

Now let’s look at another example, something more profound than an *ordinary day*—your own existence. In order for you to have a life, an almost endless series of events had to occur. Think about the coupling of your parents, and their parents, and every ancestor that came before that. In order for you to exist, every combination of ancestors had to mate, and possibly at the exact time that they did. I will spare you the trouble of going any further down the evolutionary line, but the basic idea is that your life is extraordinarily improbable. And so is my life and everybody else’s. Just because something is improbable, does not mean it can’t happen. The fact is that as long as you have a universe, something has to happen, and that just about everything that happens is improbable.

Therefore, if improbable things happen all the time, does it have to come about by design? I am certain that many would say that it does. They could also argue that the existence of life seems so improbable that it implies a higher order to the universe. Although that may be true, it does not necessarily mean that life was designed. The universe’s enormous scales of time and space allows for limitless opportunities to create. Given the mind-boggling numbers that are involved, what seems improbable or impossible does not necessarily apply to the universe.

We know that the universe allows life, because we find ourselves on a planet that allows life. On the other hand, on all the planets that don’t allow life, there is no one to count the failed attempts, or whether any attempts were made—no one to contemplate why it wasn’t designed to allow life to exist, or if it was designed at all. Although we can’t definitely confirm that life exists elsewhere, we know that life is rare relative to the size of the universe. If life was plentiful, we probably would have found some elsewhere by now. This means that vast regions of the cosmos are without life. And if we could closely observe those regions, we wouldn’t think that they were anything special. We would see planets orbiting stars and swirling galaxies, but this would go on for eons, without any conscious experience. Keep in mind that the process that led to life here on earth is essentially the same process that led to the lifeless regions. Of course, there are a few exceptions. One of which is the earth’s special location.

The location of the earth is an example of something that appears improbable, and thus appears designed. The earth’s location has been called the *Goldilocks Zone*, taken from the fairy tale *Goldilocks and the Three Bears.* The obvious reason being that its location is just right (just the right distance from the sun to support life). Of all the possible locations that couldn’t support life, why here? Again you could say that it is by design. But it doesn’t have to be, simply because improbable things can happen, especially with large scales like the universe. With a universe as vast as ours, it is inevitable that some planets will be located in *Goldilocks Zones*. It may be that we just happen to be here. Not necessarily because it was designed that way, and not merely by chance. But rather by an evolutionary process on a cosmic scale, which moves in a direction from simplicity to complexity. It is a process that creates stars, galaxies, and planets. Sometimes when the conditions are just right, it creates life.

Goldilocks Zones are not only applicable to planets, but the same principle is also present in nature. For instance, let’s examine something that is closer to home, such as the life cycle of a tree. A mature tree can produce at least several thousand seeds in a growing season, which are eventually deposited on the ground. The vast majority of these seeds will never become trees. Usually, only a very small percentage will germinate and grow to become trees. They are seeds that fall in Goldilocks Zones. In this context, a Goldilocks Zone would include fertile soil, sufficient water, sunlight, shade, etc. The probability of any one specific seed becoming a tree is very remote; however, when all the seeds are taken into account, probabilities can be viewed in a different light. We know that some seeds will become trees, because they will benefit from conditions that are just right. What we don’t know is which seeds will be selected by this process.

There is another analogy that I have heard a few times, which deals with the improbability question. This analogy has been used in support of design, and it goes something like this: the world’s oceans, with the comings and goings of its tides and waves could never construct a sand castle. The argument being that it requires a design for something constructive to emerge, and this applies to all the complexity we see today. The problem with this view is that it evaluates design against only one other alternative—whether chance alone could construct the sand castle.

There is another way to look at this analogy, which in my opinion, better shows how seemingly improbable things emerge. I agree that the ocean could not directly construct a sand castle, but it could do so indirectly. Life emerged from the ocean, and gradually made its way on land, and over billions of years evolved into more complex forms. One of these forms, a child, walked on the beach and built a sand castle. Consequently, the sandcastle came about from a complicated natural process that can’t be broken down into simplistic explanations, such as the polar opposites of design or chance. If we could go back in time a few billion years, we would think that the likelihood of a sand castle appearing on any of the world’s beaches would be very low. And yet today, sandcastles regularly appear (and disappear). Therefore, whether we are talking about living planets, trees, or sandcastles—and even if the finished product seems improbable—it doesn’t mean it can’t happen.

References: Bill Bryson, *A Short History of Nearly Everything *(London: Black Swan, 2004), 31, 32.

indeed is true can stated prof dr mircea orasanu and prof drd horia orasanu and followed specially for measurable and therefore splits and so the last two terms can be combined to so (3) becomes which is the right hand of inequality (2).

So, it follows by a routine induction argument than a finite union of measurable sets is measurable.

Now, what about the intersection? If and are measurable, so are their complements (and vica-versa; the definition is symmetric). Now recall that = (note: the outer “ ” denotes set complement as I couldn’t get the LaTex command for the outer “tilde” to work) and the result follows.

2. Now we show finite additivity of disjoint measurable sets :

We need to show that

Clearly, the statement is true for . Assume that the statement is true for all integers up to .

Now by disjointness, and .

Now splits therefore

which gives a recurrence relation for the coefficients:

.

We may now build up the coefficients from the term. Starting from we find

.

Now putting gives

and so on. We see that the coefficients of all the even powers of x are given in terms of and we obtain the solution to the ODE as

.

The series specifies the J0 Bessel function:

.

So the solution to the ODE which we have discovered is a constant times the J0 Bessel function

.

Thus far this is quite good; we have discovered a new function which solves the above differential equation. But it is a second order differential equation and therefore, as with the previous SHO equation, there should be two independent solutions. Where is the other solution?

When we examined the solution of the wave equation for a drumhead we found the separated radial equation took the form of the zeroth order Bessel equation. And at that stage we simply noted that Mathematica gave, as independent solutions to that equation, the two zeroth order Bessel functions J0(x) and Y0(x). We plotted the functions and the behaviour of the functions in the vicinity of x = 0 gives us an important clue about the “other” solution.

J0(x) and Y0(x) Bessel functions

The J0(x) function goes to 1 as x goes to 0. This we see on the plot and we have discovered this in the series solution. The Y0(x) function, on the other hand, looks as if it is heading for minus infinity as x goes to 0. That is the problem.

Recall the point made when we introduced the power series method. A series

will only work when the function is “well behaved”.

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this subject so simplicity to complexity shown many aspects considered prof dr mircea orasanu and prof drd horia orasanu as followed the arounds stated as in more . from these is evident that The two last mentioned problems—that of Fermat and the problem of the three bodies—seem to us almost like opposite poles—the former a free invention of pure reason, belonging to the region of abstract number theory, the latter forced upon us by astronomy and necessary to an understanding of the simplest fundamental phenomena of nature.

But it often happens also that the same special problem finds application in the most unlike branches of mathematical knowledge. So, for example, the problem of the shortest line plays a chief and historically important part in the foundations of geometry, in the theory of curved lines and surfaces, in mechanics and in the calculus of variations. And how convincingly has F. Klein, in his work on the icosahedron, pictured the significance which attaches to the problem of the regular polyhedra in elementary geometry, in group theory, in the theory of equations and in that of linear differential equations.

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