The Limits of Experimentation and Mathematics
Our second consideration concerns what is meant by mathematical formulations and their role in experimentation. We have already mentioned that,
Not only did Galileo prepare his trough and roll the marbles down, but what is more, he varied the length of their descent. This means he measured the distances; he measured the times also. And he made these measurements for the purpose of manipulating them mathematically.”
But Clark continues here to say,
One of [Galileo’s] mathematical analysis showed that varying the velocity with the distance presupposes infinite speed; and since this is impossible, he assumed that the velocity increases with the time. From this assumption, this law, this mathematical equation, it can be deduced that a body falls four times as far in two seconds as in one. Later scientists used more mathematics; indeed, they invented more mathematics to use” (Clark, 44).
Thus, these mathematical formulations are the results of the creation of laboratory conditions and have not stemmed from a real world observation. The trough was prepared, the lengths were chosen, the marble was crafted, and even mathematics needed to be invented to keep up with the plethora of experiment. It is quite curious then, given these pristine conditions, that one could venture to apply the results of these experiments to real world nature. Has the scientist “proven” anything? He has done well in tracking the velocity and motion of a marble on a trough, a bob on a pendulum, but would it perhaps be too much to ask that the scientist find us these perfectly crafted and prepared materials in nature?
Clark then is correct in noting that, rather than discovering these scientific laws, the laws are more accurately seen as constructions. Brilliantly constructed no doubt, but constructed no less. Mathematics then is useful in observing and tracking our marbles and bobs, but our problem is that we have not shown that mathematics is useful in constructing a worldview. And thus the claim that mathematics, upon which modern science is built, is the basis for studying all of reality is a claim that cannot be sustained. Or as Clark says: “Therefore, since the Newtonian laws do not describe the actual workings of nature, they cannot be used as a satisfactory demonstration of the impossibility of God and miracles” (Clark, 55).
The problem with mathematical formulations does not stop there. While physics depends completely on mathematics, we must also demonstrate, leaning heavily on Clark, that mathematics gives us no absolute answers even in pristine conditions. If we are correct in saying this, what, then, are the prospects for physics?
Let us look at the technique of experimentation in pursuit of a scientific law. Gordon Clark writes: “Whatever the experiment may be, all measurement is the measurement of the length of a line.” He explains what he means by giving some examples:
If a scientist is attempting to determine the boiling point of fluid, he measures the length of the mercury in a thermometer. If he is interested in the specific gravity of this or that, he measures the distance between a zero mark on a balance and another mark on a scale.”
As a good proponent of the scientific method, the scientist will conduct the experiment numerous times, each time ending with another measurement. He would perhaps even say that the more experiments he does, the more measurements he takes, the more accurate will be his findings. He then concludes his experiments and finds that his measurements are all different from each other. Indeed he has a collection of differing numbers! Clark puts it like this: “a list of different readings is an inescapable result of measuring” (Clark, 55).
Truly then, the scientist cannot stop here. For no law can be written which depends on readings that differ from each other. Clark reveals what is done in the next step toward a law of physics. “[The scientist] adds [the readings] and divides by the number of readings; that is, he computes the arithmetic mean.” But why, Clark asks, is the arithmetic mean chosen as the next step? Are there not two other types of averages; namely, the mode and the median? On what ground is the mean considered and not the other two? “Whatever answer is attempted, clearly nothing in the observational data has dictated its choice” (Clark, 55). That is to say, for all the praise that the scientists lavish on themselves for being completely “empirical” and “sticking to the facts,” it seems that they have already contributed something non-empirical to their experiment.
And thus, there is another example of the scientist constructing, but not discovering, an aspect of their experiment. Not only have they created an unnatural condition, but they have also begun to look for conclusions based on a method that they themselves, apparently at their whim, have imputed into the data.
But Clark is not done yet. Each reading must be subtracted from the average mean so as to discover their differences. And each difference must be added together and divided by the number of readings so as to discover the mean of the differences. At this point, to clarify, the scientist has computed both the mean of the readings as well as the mean of the differences between the mean of the readings and the readings themselves. So again, the scientist has, at his own discretion, added to his experiment a step which has not been justified by the observed data. That is to say, on his way toward a law of physics, he has twice chosen an average based on his own desires. So much for absolute empiricism.
But again, Clark has more to say. The scientist must plot his values on a graph so that he can make a law out of them. Clark uses the following example of a value: 19.31 ± .0025. The 19.31 is the original mean. The ± .0025 is known as the variable error. Thus, to plot these results of the experimentation is to make a mark at the 19.31 point. But the variable error suggests that the result might not be right on that 19.31 point, but rather could be twenty-five ten-thousandths greater or lesser than that. How can this be plotted? With a line. To continue with our example, Clark suggests that we say we are looking at the force of gravity between two bodies. And thus a line which represents the readings has with it another line on the other axis that represents the “corresponding range of… force” (Clark, 56). We have just drawn a rectangular area on the graph. And each of our experiments has its own area on this graph, and we therefore have a graph with as many rectangular areas as experiments.
We are finally ready to produce a law of physics. The scientist “passes a curve through these areas, and this curve he calls a law” (Clark, 56). But where do we draw the curve? Once we draw a curve, we realize that there was room for us to draw it elsewhere. In fact, to quote Gordon Clark, “it is to be particularly noted that through a series of areas an infinite number of different curves may be passed” (Clark, 56). Empirically, no one can say that we must draw one specific curve. So which do we choose? Clark continues by writing:
“In other words, so far as observation is concerned, the scientist could have chosen a law other than the one he actually selected. Indeed, his range of selection was infinite; and out of this infinity he chose, he did not discover, the equation he accepts” (Clark, 56).
How can the scientist choose the correct “law” out of an infinite number of choices? Truly this is not possible. And if it is not possible, we should agree with Clark when he concludes, “therefore, all the laws of physics are false.” The laws of physics are false and these laws are not based on empirical observation alone, contrary to the claims of the scientists. Mathematical formulations then, while wonderful for keeping track of our laboratory experiments, are not at all able to lead us to absolute laws. Proving that God does not exist seems a bit more problematic now. And surely we have made headway on our quest toward a philosophy of science; namely, that the scientific method cannot create a comprehensive and consistent worldview.
[i] Both books, (The Philosophy of Science and Belief in God and Behaviorism and Christianity) are included in Clark’s Modern Philosophy. Thus, all quotations, unless otherwise noted, are taken from “Clark, Gordon Haddon. Modern Philosophy. Unicoi, TN: Trinity Foundation, 2008. Print.”