We begin our voyage of discovery with an elementary physics problem (elementary, when studied as an introductory university course in physics) that might seem trivial at first sight. But please bear with me, for it will lay the introduction that will set the ground stage to something that is more profound that the reader might realize at this point in time.
Our problem was taken directly out of the book “Physics for Students of Science and Engineering” coauthored by David Halliday and Robert Resnick. The statement of the problem which appears on Chapter Four of the first of two volumes is simple enough:
Problem # 6: “Calculate the minimum speed with which a motorcycle rider must leave the 15º ramp at A in order to just clear the ditch.”and is referenced to the following figure:
In practice, if a would-be daredevil is considering an attempt to actually make such jump, a problem such as this one must be solved exactly, for if the rider makes a small mistake in his calculation, there is a good chance that his speed might not be the minimum speed required to make the jump and he will fall down into the ditch.
Arriving at the right answer to this problem turns out to be an issue that is not trivial. It can be shown that the solution to this problem will be given by the following formula [v0 represents the initial velocity required at the point of departure from the ledge, θ represents the slope of the ledge from which the vehicle jumps, which is 15 degrees in this case, g represents the downward acceleration pull due to gravity which is 32 feet per square second, x represents the horizontal distance from the starting point to the end point which is 10 feet, and y represents the height difference between the starting and end points of both ledges, which in this problem is 5 feet]:
Notice that the time required to make the jump from one end of the ditch to the other does not appear explicitly in the formula given above. It is irrelevant. We can, if we so desire, derive another formula that will give us the time of flight. But it is not required explicitly in order to determine the minimum speed required to make the jump successfully. And why is the passage of time completely irrelevant for the correct solution of this problem, except as a medium that allows the jump to take place? Because the shape of the trajectory is the critical issue at stake here. If we could slow down the passage of time and watch the jump in “slow motion”, the path traveled by the rider as he clears the ditch would be exactly the same as the path he would travel if we were to witness the jump through the lens of a high-speed camera accelerating each frame. In the end both paths must be exactly the same. The path that the rider must travel to make the jump successfully is a curve commonly known as a parabola. And for a given angle of departure and the distances involved, it can be shown that there is just one and only one parabola that connects the point of departure and the landing point. As long as the rider manages to travel along this parabola, the jump will be a successful jump. If we imagine ourselves our rider as a point object and visualize the parabola that describes his path, we may be able to see that for any other speed different from v0 the parabola traveled by the rider will also be different. Thus, the solution to this problem really requires us to determine, out of the many possible parabolas we can draw that will pass through the given point of departure and will have the same slope of departure, which is the parabola that will also pass through the required landing point, and as we have already stated there is just one such parabola. This physics problem of motion in a plane is therefore at its very core just a problem in geometry. And being a problem in geometry, we can consider it to be essentially a problem in mathematics rather than a problem in physics. So where then does physics enter here in the panorama? It enters with the “natural, universal, physical gravitational constant” g of 32 feet per second per second, the downward acceleration pull due to gravity. This physical constant must be measured, and its value will be entirely dependent upon the system of units we are using to make the measurement (meters instead of centimeters, feet instead of meters, yards instead of feet, and so forth). Mathematics alone cannot predict this value and make it fit within one of our many systems of measurement. But we might take notice that since our systems of measurements are entirely arbitrary (after all, the actual true downward pull of gravity remains the same regardless of how we measure it), it could be possible that the physical constant g being used in the statement of the problem could perhaps be derived mathematically from a combination of other truly universal mathematical constants (such as the number e, the foundation of natural logarithms, the square root of two, etc.) and referenced to some other universal invariable number yet to be discovered, in which case the problem would then be entirely a mathematical problem with no physics involved whatsoever, with no need to make any type of measurements. Nevertheless, once the value of g has been fixed, the problem of the rider clearing the ditch becomes entirely a problem in geometry, making the passage of time an irrelevant issue except as the medium in which the geometry comes alive.
Substituting numerical values into the above equation, we arrive at the correct answer, which is v0 = 14.9 feet per second. [In the formulation and the solution of this problem, some simplifying assumptions have been made. First, the resistance presented by still air against the forward motion of the motorcycle -which from the perspective of the rider would be indistinguishable from a wind blowing directly against him- has been assumed to be negligible. In actual practice, for the distances considered above, this air drag will make a small difference, and would complicate the analytical solution of the problem requiring more sophisticated mathematical tools. Second, the jump is assumed to be carried out on a day with no winds in the forecast. On a very gusty day, even taking into account the maximum wind speeds that can be expected and adjusting the above formula accordingly, the rider would be wise to stay at home and leave the jump for another occasion.] If the rider is smart, and his motorcycle is not capable of attaining this speed under the best possible conditions, he will either procure another motorcycle capable of giving a higher speed than the one we have just calculated, or he will give up on his attempt to clear the ditch.
There is another central issue on the solution of this problem that we have not yet discussed. There can be no counterargument whatsoever that the successful and exact solution to a problem such as this one requires some form of intelligence. The solution to a problem like this one cannot possibly be the result of mere chance alone, and if this problem is to be confronted by the reader during a final examination at a university, any mistake in the derivation of the formula or any math miscalculation on his/her part will be punished with a lower grade score (if taken into actual practice, the rider may not be able to reach the other side, and will most likely fall helplessly into the deep ravine maybe killing himself.)
As stated before, the exact solution to this problem requires some form of intelligence. Most readers can surmise by now that this problem cannot be solved on paper by most elementary school students (unless the student happen to be a “whiz-kid”) or perhaps even by many high school students. Indeed, in order to solve this problem successfully, the person who solves it requires some mastery of the following tools:
- Algebra
- Basic kinematics
- Trigonometry
- Arithmetic (perhaps a hand-held calculator)
The above knowledge is something that does not come easy to most people during their lifetimes. Acquiring this knowledge necessarily implies that the person has enough information processing capabilities in his/her mind to fully understand the nature of the problem and to develop the tools that are required in order to solve it. Nobody in his right mind would accept the argument that a problem such as this one can be solved by mere chance alone on a consistent and repetitive basis. Arriving at the final equation required to solve the problem is something that just cannot be accomplished by mere chance alone. And this knowledge must precede any attempt at the correct solution of the problem. It cannot possibly be otherwise.
But there is yet another central issue to the physics problem just discussed. In the problem, the angle of jump is already fixed. And the distances from ledge to ledge are fixed also. The downward acceleration due to the gravitational pull from earth is also fixed. The only variable we have at our disposal to carry the rider safely to the other side is his velocity of jump. The minimum speed v0 required by him to make the jump safely is the initial condition required by the rider in order to make it to the other side. Anything less will just not do. And if the minimum speed required to make a safe jump is anything more than the maximum speed that the motorcycle is capable of giving, our would-be Evel Knievel will not even attempt to make the jump, unless he is intent on committing suicide. Thus, a safe jump necessarily requires knowing the right initial condition, and this in turn does not come by mere chance alone, it requires some form of intelligence. If the rider goes ahead with full confidence making the jump and his effort succeeds, then there can be no doubt whatsoever in our minds as we witness the jump that, as far as we are concerned, he has been preceded by knowledge and intelligence in the manner in which that knowledge was used.
Pushing the above arguments a little bit further, if we arrive some time after the rider has made his/her jump, even if we did not see the jump take place from ledge to ledge, if we can determine by other means that such a jump has indeed occurred (for example, by noticing the tracks made on the starting ledge and the tracks made on the landing edge), then we have more than enough information to determine that some kind of intelligence must have been at work prior to the actual jump, an intelligence which was of paramount importance in the successful conduction of the experiment.
And this will be our key to unlocking the possibility of whether an event could have been the result of mere chance alone or whether the event was programmed beforehand by some form of intelligence to happen in a prescribed manner: the initial conditions themselves, if they can be traced backwards in time from the evidence they have left behind, may carry most of the information we need to know in order to determine whether something was due to random chance alone or whether some intelligence was at work when it all started. We do not have to be present and aware at the very beginning when something took place in order to determine what the initial conditions were. In our elementary physics problem, just knowing that the rider made it safely to the other side should be more than enough to convince us that in order to make the jump he must have chosen and used the right initial condition, even if the event happened many years ago. Thus, initial conditions may be capable of making quite a distinct mark on the present, even though the event may have taken place a long time ago.
Unfortunately, if we are not witnesses to the actual jump, then it is always possible that the jump could have taken place in a manner completely different from the one we have assumed. For example, it is possible that in the event of some minor miscalculation of the initial condition or in the event of some last minute mechanical problem with the motorcycle, the rider could have cheated and could have retrofitted his machine with some rockets that would enable him to make the jump safely to the other side. We would like to be actual witnesses to the event, but if for some reason we are unable to actually be present, then we have no other choice than to make certain assumptions on what could have taken place when the jump was carried out. In the process of starting out with a limited amount of information and going from some very specific details to a bigger picture, we are actually carrying out a logical process of induction, trying to determine or induce something bigger by gathering together some (perhaps very few) pieces of a jigsaw puzzle. This is completely opposite to the process of deduction, the way in which science usually likes to work, whereby we start out from the very beginning with some generally accepted facts or axioms or postulates, and from those “self-evident” truths we arrive at certain conclusions by rigorous application of logic, as was done by Euclid himself in formally deriving and proving many theorems of geometry in his book The Elements. Critics of the process of induction point out (and their observation should be well taken) that the process of induction is fraught from the very beginning with the possibility at arriving at wrong conclusions, loaded with too many assumptions, and historically many big mistakes and faux passes in science are due to the blind trust placed in those occasions on the process of induction. However, even deductive reasoning may be highly vulnerable if just one of the basic assumptions is later proven to be wrong or incomplete, perhaps bringing down an entire pyramid of knowledge because just one of the bricks in the edifice was later found to be faulty (the Earth was not flat nor was it standing still with the heavens revolving around it, spontaneous generation of life –also known as abiogenesis- would never be proven in any laboratory, and the phlogiston never did exist).
Nevertheless, there are occasions in which we have to use inductive reasoning simply because we have no other choice. For example, even though many of us would like to jump into a time machine and travel backwards in time to witness the very moment of creation of the universe, we know this will not be possible, for even if with the discovery of some new physics principle we could be able to travel backwards into time, the enormously high temperatures and the almost infinite densities of matter just a little bit after the universe came into being would not allow us to sustain our own lives for very long. But even if we are forced to use inductive reasoning, our efforts may not be in vain. Other productive fields of science such as archeology have been able to grow and prosper by working backwards into time with only the remnants of bygone eras, aided by powerful new techniques such as carbon dating and DNA typing (besides using the shovel to dig up old bones.)
Keep in mind that we have only dealt with a very simple physics problem that does not come even close in difficulty with other problems that could be posed in such areas as quantum mechanics or general relativity. And there are many, many other problems, which must be solved exactly, before there is any hope of seeing a Universe evolve with any kind of primitive life, or for that matter, even a Universe (as we currently know it) actually evolve. The precise solution of any given problem has a footprint called intelligence, and as the problem gets tougher to solve it will demand increasing levels of wisdom in order to surrender its solution.
A counterargument that might be posed against the thesis that the correct determination of an initial condition as in the case of the rider who tries to clear the ditch is proof of intelligent abstract thinking would be the observation that all of our mathematical models are but symbolic idealizations of something we call reality, idealizations that in many cases can become much more complex than the reality they are supposed to represent, and that on final analysis occurrences that take place in everyday life can do very well without such complications. After all, on a time when there were no computers or calculating machines and when the science of mechanics was non-existent, William Tell was able to shoot an arrow from a distance targeting the arrow so precisely that it would land right on the apple placed above his son’s head, and after this he was able to shoot another arrow targeting the arrow so precisely that it would land on the tyrant who ruled his country. All this without having to add up a single number, almost as if William Tell knew what to do without the need of any mathematical models and without any prior knowledge of physics. By the same token, in order to be able to place a “hole in one” in a golf match, a master golfer such as Tiger Woods would try to pull off this feat by first looking around and “sensing” his environment, getting a “feel” for the wind direction and velocity, and watching how far the target appears to be from the place where he is standing. Logic alone tells us that in order to get that “hole in one” he must actually measure the wind speed and velocity and the distance between him and the target, perhaps to an accuracy of at least four significant digits (considering that the target hole which is many yards away from him is less than three inches wide), and once this is done he must use a computer and very precise mathematical models to determine the speed with which he must strike the golf ball, as well as the angle of stroke and the precise point of impact, thereupon adjusting his swing accurately to match the parameters required to make that “hole in one”. Yet, no master golfer has ever required such elaborate methodology to get to the PGA championship tournaments. Carrying out the above observations even further, if we look at a pond filled with fish and other living life forms, even though a snapshot of the activity taking place may reveal us that life in the pond can be described by an enormously huge and complex set of interrelated mathematical formulas the fish in the pond appear to be quite content and oblivious to such abstractions, going around their daily business of survival without even having to go to elementary school or kindergarten, almost as if the fish knew exactly what to do in order to survive without the need of any of the tools of mathematics.
The above counterargument is seriously flawed because it fails to take into account that, in order to use his bow and arrow with such a precision, William Tell had to acquire first a lot of experience by the time tested method of trial and error. Without this, he would have done no better than an amateur, and History might have been written differently. The master golfer must also undergo through a similar process, regardless of how good his “gut instincts” may be. There are reasons to believe that in cases such as these, the experience gained through trial and error allows our brains to build up an increasingly refined database incorporating the recollection of past failures and successes, and when a show of mastery is about to be displayed both the arrow marksman and the golfer subconsciously compare their current situation with previous ones, and drawing upon their experience database they adjust themselves to emulate those similar conditions that in the past proved to be successful. The disadvantage of using a memory acquired through trial-and-error versus intelligent abstract thinking is that experience alone may not be enough to solve situations in which there is no past experience. Space exploration is filled with many examples of first-of-its-kind events (the Apollo space program, the Sputnik satellite, the Voyager missions) where the luxury of trial-and-error must be ruled completely out of the picture because of budgetary constraints. None of these multimillion dollar programs would have taken off the ground if the launch in each case had been entrusted to “gut instincts” alone. Considering the distance from the Earth to the Moon, even a very small miscalculation of just one or two degrees would have resulted in one of the manned Apollo missions missing the target by thousands of miles, and without the gravitational pull of the Moon required to send back the team to Earth by turning the spacecraft around, the mission would have been lost forever in the vast emptiness of space. Thus, long before the rocket is rolled out to the launch platform, a lot of careful planning and abstract intelligent thinking must precede the launch. All of the required initial conditions must be in place just before the boosters are fired up. Once the countdown goes down to zero, there is no turning back, and the launch will be either a success or a failure. It will be the initial conditions, or more precisely, the intelligence behind the precise determination and the gathering together of all the required initial conditions, that will either make or break the project. There can be absolutely nothing random about something like this; nothing can be left to chance.
Besides, there is a widespread belief that in the case of those two historical shots plucked out of his bow by William Tell, luck was also on his side. And in the case of a “hole-in-one”, it is widely accepted that whenever a master golfer is able to elicit such a miracle from his club then, besides his mastery, luck also had to be on his side; and this is confirmed to us by the fact that such a feat is an oddity rather than the rule in masters tournaments [Nevertheless, a good game of golf can always be improved with some knowledge of the physics involved, even without going through the entire detailed math before each strike. See for example, the book "The Physics of Golf" by Theodore P. Jorgensen.]. In general, and this goes for many sports, luck is almost an essential ingredient in order to beat the odds. The same can be said for life itself. But for many other things, luck is no substitute for careful thinking and planning. The Eiffel Tower was not built based upon the hope that such a fanciful arrangement of metal beams and rivets would somehow not collapse; it had to be designed from the very outset as a giant structure that would withstand all the way down to its base the enormous tonnage that makes such structure, and it was designed to do so for many years to come. In the case of the Eiffel Tower, the initial conditions for such a monumental undertaking were all completely specified in the blueprint long before the first brick was laid down, long before its first iron beams had been manufactured. Likewise, any digital computer ever built by Man is no better than the software (computer programs) that is being run in the entrails of the machine. As any computer programmer will tell you, even the smallest mistake when writing down a computer program will most likely result in a crash sooner or later. In order to perform reliably 100% of the time, the computer demands software with no errors whatsoever; it demands perfection. No computer program was ever designed counting upon Lady Luck as a helpful aid. Interestingly enough, once a good computer program has been written in a so-called “high level language” (examples of such languages are COBOL, C++, Fortran, Lisp and Pascal) it can be run on any machine equipped to handle such language, regardless of the internal architecture of the machine, regardless of whether the computer was built using high-density integrated circuits, vacuum tubes, or using pistons activated by steam engine. The “hardware” will only follow blindly and faithfully the instructions it has been given by the “software”. MIT’s Raymond Kurzweil makes this point clearer in his book The Age of Intelligent Machines:
“So the message of science can be bleak indeed. It can be seen as a proclamation that human beings are nothing more than masses of particles collected by blind chance and governed by immutable physical law, that we have no meaning, that there is no purpose to existence, and that the universe just doesn’t care … And yet, the message doesn’t have to be bleak. Science has given us a universe of enormous extent filled with marvels far beyond anything Aquinas ever knew … Indeed, far from being threatening, the prospect is oddly comforting. Consider a computer program. It is undeniably a natural phenomenon, the product of physical forces pushing electrons here and there through a web of silicon and metal. And yet a computer program is more than just a surge of electrons. Take the program and run it on another kind of computer. Now the structure of silicon and metal is completely different. The way the electrons move is completely different. But the program itself is the same, because it still does the same thing. It is part of the computer. It needs the computer to exist. And yet it transcends the computer. In effect, the program occupies a different level of reality from the computer. Hence the power of the symbol-processing model: By describing the mind as a program running on a flesh-and-blood computer, it shows us how feeling, purpose, thought, and awareness can be part of the physical brain and yet transcend the brain.”
If the software is no good, the end results coming out of the computer will be no good either (in computerese, the saying goes “garbage-in, garbage-out”). Call it finicky behavior, if you like; the fact is that without a thinking mind overseeing the start up parameters of many things around us, such things would not be possible today.
Since the performance of a digital computer depends much more critically upon the “software” than upon the “hardware”, the hardware being just the medium that allows the software to execute, what are the initial conditions here in order to be able to accomplish a major computing task? Must we consider both the software and the hardware both part of the essential initial conditions? Or are the true initial conditions limited just to the software? Let us go back to the case of the rider who is trying to clear the ditch. Should we consider the motorcycle also an indispensable part of the initial conditions? At first, we might be inclined to say yes. But on second thought, we must come to realize that the problem to be solved remains exactly the same regardless of whether the motorcycle used to make the jump is a Harley-Davidson, a Suzuki, or a Yamaha. As a matter of fact, the vehicle in principle could even be an ordinary bicycle! The formula does not make any specific requirements upon the medium in which the jump will take place; the choice is up to the person making the jump. If the rider could run fast enough, then no vehicle would be necessary, since the jumper himself would be the vehicle. The minimum jump speed required at takeoff remains exactly the same. The only way in which we could justify drawing in the vehicle as part of the initial conditions would be regarding the design of the vehicle itself: Was the vehicle designed to be capable of attaining the required minimum jump speed? And, on final analysis, any motorcycle we might try to use could not have sprung out of nowhere; long before it was actually built it had to be a concept in the mind of somebody, it had to exist as an idea. Yes, iron ore had to be mined from somewhere, foundries had to do the work of purifying the metals used and cast them into a suitable shape, gasoline had to be extracted and purified from raw petroleum, rubber had to be molded into a toroidal shape, but each and every one of these activities was also preceded by an idea whose time had yet to come. Without a smart mind hard at work from which the idea will arise, all else remains just as a mere possibility that might never be realized, and all the materials will lay dormant with no useful purpose. The initial conditions contain all of the information necessary to accomplish a given goal. Indeed, even the setting of any given goal must be considered part of the initial conditions. Once the design or the solution to a problem exists in thought, then its full realization will depend solely upon the availability of materials and the willingness of the designer to make it real. But the materials themselves are incidental. Luck may have a part in the actual realization of an idea, especially if there are not enough resources to bring an idea or the solution of a problem into fruition, or in case something completely unexpected happens at the last minute. However, failure to anticipate unexpected events that will make everything end up as a big mess is generally considered to be poor planning, and poor planning seldom bears the mark of a bright mind (however, even bright minds do make mistakes, as in the case of Napoleon’s invasion of Russia in 1812, and in such cases the punishment that has to be paid for poor planning has generally been quite high). Returning to the case of the digital computer, if we are to draw in the computer as a part of the initial conditions then we must draw in the true initial conditions that allowed the machine to be built in the first place, and those initial conditions are usually found scribbled in a notebook, drawn on a big blueprint, or carried along the mind of someone who is watching an apple fall down from a tree.
There are many other types of problems that require an exact determination of at least one initial condition in order for a certain result to be achieved. If a rocket does not have the minimum escape velocity required to break off from the Earth’s gravitational pull, its mission will be doomed from the very moment it is launched and the rocket will come back to Earth to meet its fate (the minimum escape velocity can be shown to be about 11,200 meters per second, with the weight of the rocket itself making little or no difference). And by exact determination we mean precisely that: something determined with knowledge and intelligence after a close study of the problem at hand, especially if a solution picked out entirely at random will almost certainly result in failure.
