How does one actually go about discovering a scientific law? When science expresses laws as mathematical formulas, how do these expressions relate to the reality of these laws? Are they true? Are they unchangeable? Are the "discovered" or "selected"?
Aside from the fact that any scientific experiment faces the same obstacles that empiricism in general faces, additional issues arise when trying to draw conclusions from a given set of obtained data. Take, for example, an experiment used to validate Newton's Second Law of Motion. According to Newton's second law, a body will accelerate in the direction of any unbalanced force, and the magnitude of this acceleration will directly proportional to the magnitude of that unbalanced force. The validity of this law may be tested by tying one end of a string to a known mass (20 g) and the other end to a glider. The glider can be placed on a Stull-Ealing linear air track as shown below, reducing the friction of the glider on the track to a negligible value.
The time value can be measured with a power source which sparks every 1/60 of a second. A wired is attached to the glider which fires onto the spark tape. The mass of the glider in this particular experiment was 504.6 grams. Thus we can arrive at a theoretical value as follows:
∑F = ma
W = (m+M)a
mg = (m+M)a
a = mg/(m+M)
a = (20.0 g)(980 cm/s2)/(20.0g + 504.6g)
a = 37.4 cm/s2
Let's suppose that the experiment, repeated a few times, yielded the following results:
|Displacement (x)||time (t)||acceleration (a=2x/t2)|
|5.0 cm||0.5 s||40.0 cm/s2|
|13.0 cm||0.8 s||40.6 cm/s2|
|55.0 cm||1.4 s||56.1 cm/s2|
|60.0 cm||1.7 s||41.5 cm/s2|
|100.0 cm||2.2 s||41.3 cm/s2|
Obviously, there is a notable difference between the actual a theoretical values. Perhaps the forces of friction on the air track and the pulley weren't negligible after all. Perhaps the air flowing out of the track exhibited an additional unbalanced force, however slight, upon the glider. Perhaps the timed power source was inaccurate, or certain marks did not record properly onto the spark tape. In any case, most scientists would feel that Newton's second law has been validated by the results of this experiment.
What do we make of the third measurement, which yields a result containing an error rate of 40%. While the honest scientist would include this result in his reporting, it would be discarded as an anomaly for all practical purposes. This is a problem in what Thomas Kuhn refers to a "normal" science. Experiments are expected to yield certain results, and those that don't must be either explained away or ignored. Thus we must conclude that science is not "unbiased". Indeed, biases are necessary in science in order to have any progress or ability to build upon previous discoveries.
But the scientist is faced with yet another problem in trying to analyze the results of his data. Typically, the scientist would plot his data representing points on a graph, as shown below.
By connecting the points with a curve, the scientist would validate the law he was testing. However, we can see that the curve does not pass through all of the "points" plotted. This is due to variable errors on X-Y axis, which if taken seriously, would prohibit an experimenter from plotting data as points in the first place. Instead, each plot would have to be a rectangle in order to account for errors in both the X and Y units, as follows.
Granted, there are ways to get more accurate measurements than can be obtained by the crude setup shown above, but that would only result in smaller rectangles. Since an infinite number of curves can be drawn through the proposed rectangular data, the chance of choosing the correct "law of motion" is 1/∞, or zero. So when we see Newton's second law of motion expressed in the form ∑F = ma, we can conclude that the law, however useful, is false.
The role that scientific paradigms play in expressing scientific laws can be further understood by realizing that ∑F = ma is no longer the accepted expression of Newton's second law. Instead, it is:
...at least until the next scientific revolution. Certainly the law itself hasn't changed (or else it would not be a law). Gordon Clark explains:
"It may be a fact that gold is heavier than water, but it is not a scientific fact; it may be a fact that the longer and the farther a body falls, the faster it goes, but Galileo was not interested in this type of fact. The scientist wants mathematical accuracy; and when he cannot discover it, he makes it. Since he chooses his law from among an infinite number of equally possible laws, the probability that he has chosen the "true" law is one over infinity, i.e. zero; or, in plain English, the scientist has no chance of hitting upon the "real" laws of nature. No one doubts that scientific laws are useful: By them the atomic bomb was invented. The point of all this argument is that scientific laws are not discovered but are chosen." (Gordon H. Clark, Science and Truth, The Trinity Review; May, June 1981)
This does not in any way undermine the importance of science, but only to show what it can and cannot tell us. Science is useful, but can never be validated as truth.