REFLECTIONS ON SCIENCE AND THE HUMANITIES
Formulating a Natural Law
How do scientists discover or formulate natural laws? What kind of thinking are they doing? The following exercise is designed to put you in the shoes of scientists who are faced with a puzzle and trying to discern a pattern that can be stated as a law. (If you want to know more about what a scientist means by a law, first read What is Science?)
What is the next number in this sequence: 1, 2, 4, ... ?
Solving this puzzle requires formulating a law, like the natural laws that scientists attempt to formulate from data collected in experiments. A natural law is a rule that nature appears to follow, as far as we can tell. It can be used to make predictions about how nature will behave in situations we have not yet observed. In that sense, a law, and the process of testing it, can guide scientists to new knowledge.
Imagine
Astronomers discover a line that extends away from us into deep space, and on this line, they find the number 1; farther out, the number 2; still farther, the number 4. The line extends out of the reach of the best telescopes. Scientists want to build better telescopes to see additional numbers, wondering if they might find some meaning in the sequence, or learn something about the universe from it.
Meanwhile, theorists pore over the sequence, trying to perceive a rule that they might use to predict the next numbers. Such rules are called laws. Formulating laws from existing data is usually cheaper than building telescopes.
One possible law is that each number is two times the previous number. It is useful to state laws mathematically, to allow us to calculate the next members of the sequence. Let’s refer to any number in the sequence as N, and the number before it as N(previous). Stated in these terms, the law is
N = N(previous) x 2 (Law A) [I'm using "x" as a symbol for multiplication; "x 2" means "times 2".]
According to this law, the next member of this sequence will be 8. Its previous member is 4, so
N = N(previous) x 2 = 4 x 2 = 8.
Applying this rule repeatedly, the sequence would be 1, 2, 4, 8, 16, ....
But there is another possibility. Start with the first member in the sequence, 1. If you add 1, you get 2, the second member. If you add two to the second member, you get the third member, 4. Maybe the rule for finding the next members is that we add 3 to the third member, then 4 to the fourth member and so forth.
Can we state this as an equation? If we use n to designate the position of a number in the sequence (4 is in position 3, for example), the mathematical form of the law is
N = N(previous) + n(previous) (Law B)
In other words, to calculate the next member of the sequence, add to the previous member the number indicating its position in the sequence.
According to this law, the sequence would be 1, 2, 4, 7, 11, 16 ... (Can you check this?)
Which law is valid, A or B? We have no basis to decide. We need a bigger telescope to see farther along the number line. In other words, we need more data. So scientists write proposals and get funding for a bigger telescope. Using it to look farther, they clearly see, beyond the number 4, the number 7. Some proponents of Law B break out the champagne, but smarter scientists keep an open mind when trying to formulate laws. They want to look farther, out beyond 7. At the farthest reach of the new telescope, they can barely make out a blurry two-digit number. Some of them think the two digits look slightly different. Those drinking the champagne think they look just alike. More proposals are written to improve the telescope and apply image-enhancement methods to make the next letter clearer.
Meanwhile, theorists are looking again at the sequence (1, 2, 4, 7, ...) to try to formulate another law that would fit this sequence. Can you formulate another law that would generate this same sequence, but make a different prediction about the fifth member?
Around the time that the telescope is improved, theorists, imagining and trying out various rules on this sequence, are discussing other possibilities. One evening, before their work is complete, an e-mail from the telescope community arrives with shocking news: the fifth member has been resolved. It is 12. Oh, well, at least proponents of Law B got to enjoy their champagne before their pet Law B was overturned. It’s risky to celebrate when more data might come in (or maybe you should celebrate first, before new data cuts the party short).
So, up to this moment, the sequence is known to be 1, 2, 4, 7, 12, ... .
We need a Law C, which fits this sequence, and will make predictions about subsequent members.
Even if scientists formulate a law C, which fits the sequence 1, 2, 4, 7, 12, ... , how can they know that it will always predict correctly the next member of the sequence? They will eventually need to see that next number. Perhaps, no matter how far the telescopes can see, the line will not end, and thus even a law that continues to predict the right numbers will never be tested completely. Laws of nature are like that. In the meantime, if anyone has to make a decision or take an action based on current knowledge, they can only act on what they know up to now.
Laws of nature are not like legal or moral laws. They are not rules that nature is compelled to follow. Instead, they are rules that nature appears to follow, as far as we know now. The best of current laws are very useful in the realms where they have been verified, but scientists are always open to the possibility that experiments in new realms will uncover data that do not fit the best law so far. If that happens, re-interpretation of the data will be needed, and a new law must be formulated. Remember, natural laws are sought, not legislated. One way that scientists seek them is by looking for patterns in the results of experiments.
When scientists get results that violate a well-established law, is it a disaster? Quite the contrary, it is one of the most exciting moments in science. For this and other reasons, testing laws against new data is a never-ending process.
Is a newly formulated law a discovery or a human creation?
I believe that laws are created, not discovered, for a simple reason. The laws we perceive are always, in some way, flawed or incomplete. If so, then they could not have been lurking out there in nature, to be discovered. Scientific laws (and theories as well) are creative works, just as are works of art, literature, and music.
More on the Nature of Laws
Read this.
Footnote
Let’s give the name N(previous-previous) to the number before N(previous).
Here is a new law that fits all the data—at least, so far:
N = N(previous) + N(previous-previous) + 1 (Law C)
In other words, the number that follows 7 will be 7 + 4 + 1 = 12.
Law C predicts that the sequence will be 1, 2, 4, 7, 12, 20, ... (and then, what?)
See if you can verify that the law works for all members of the sequence discovered so far.
Just for fun, can you formulate a Law D, which predicts that the first 5 members are 1, 2, 4, 7, and 12, but that the sixth member is not 20? Then you, as a theorist, might be ready to explain things, if the next number revealed turns out to be a surprise.
