A Conjecture

The following is from Mathematics for Elementary Teachers, a textbook that I am using this semester:

You may be surprised to know that some problems in mathematics are unsolved and have resisted the efforts of some of the best mathematicians to solve. One such problem was discovered by Arthur Hamann, a seventh-grade student. He noticed that every even number could be written as the difference of two primes. For example,

2 = 5 – 3

4 = 11 – 7

6 = 11 – 5

8 = 12 – 5

10 = 13 – 3

After showing that this was true for all even numbers less than 250, he predicted that every even number would be written as the difference of two primes. No one has been able to prove or disprove this statement. When a statement is thought to be true but remains unproven, it is called a conjecture.^[1]

Sounds fascinating, doesn’t it? A seventh grader has managed to stump brilliant mathematicians. Actually, there is a very simple reason why it has not been proven that every even number is the difference of two prime numbers: there is no pattern to prime numbers.

Let me show you what I mean. We will begin with an example of the even numbers. What are the even numbers? An even number is an integer that is divisible by 2. Note that we are going to use this as our definition of an even number, meaning that we will act like we do not know that a number is even until we can demonstrate that it is divisible by 2. See, even numbers are common enough to us that we take them and their definition for granted. For example, we all know that the even numbers are

…, -6, -4, -2, 0, 2, 4, 6, ….

Just so there is no confusion, when we defined even numbers as integers, that includes the negative numbers, not just the positive numbers. Also, the ellipses (…, the triple periods) note that the sequence continues onward in both directions.

Now, note that the sequence given above does not directly follow from the definition of even numbers. It is true that the numbers in the sequence meet the definition of even numbers, but the definition itself does not tell us the sequence. We might consider this trivial: after all, the sequence “just makes sense” when you know what an even number is. Why belabor the point that it does not follow the definition?

Because to a mathematician, nothing is taken to be true unless it can be shown in a proof. That proof has to be so universal that it encompasses every possible option. In other words, saying something like, “Well, if you have an even number and add 2 to it, you get another even number, so the sequence of even numbers is just all of the numbers separated by 2” is not enough. There needs to be a formal proof, like this one:

Let e be an even number. Since an even number is divisible by 2, let e = 2a where a is some integer. Now let us take e + 2. Since

and since a + 1 is also an integer, e + 2 is also an even number. Thus, if we add any even number by 2, we get another even number and the sequence 0, 2, 4, 6, … is justified.

A similar argument can be made but substitute e – 2 for e + 2, and that would be the proof that the sequence …, -6, -4, -2, 0 is justified.

Notice that the formula e = 2a applies to every even number by definition, so by showing that e + 2 is divisible by 2, and therefore is an even number itself, applies to every even number.

Now, let us turn our attention back to the opening quote and the prime numbers. What is the definition of a prime number? It is a whole number that is only divisible by itself and by 1. The sequence of prime numbers are

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 51, ….

Now, we can test each of these numbers in order to check and see if they are prime numbers, but is there any kind of pattern to them? Can we define an equation to describe all prime numbers, like we did with the even numbers? The answer is no, to both questions. There is no pattern to the list of prime numbers. The spacing between them is erratic so there is no pattern of addition between them. Since they are, by definition, not multiples of other numbers, we cannot have a pattern of multiplication between them. These observations mean that we cannot write a single equation to describe every prime number. In other words, while even numbers can be described in a sequence or by an equation, prime numbers cannot. Put another way, while a mathematician can “know” all even numbers, in the sense that every one of them can be simply represented, mathematicians cannot “know” all prime numbers: each one can only by found by testing, not by calculation.

Since the prime numbers cannot be described with a single equation, it is impossible to tell if every even number can be written as a difference of two prime numbers. Since we do not know all of the prime numbers, we simply cannot know if that statement holds true in all cases. That is why Arthur Hamann’s observation will remain a conjecture, not because mathematicians aren’t “good enough” at math, but because there is no pattern available to test Hamann’s observation for all prime numbers.

Now, I am going to make a sharp swerve here to wrap up this post. Just so we are clear, the concept of a proof in mathematics is as stringent, if nor more so, than I have presented it here. Testing an idea on the first 250 numbers does not constitute a proof, not even close.

What about with science? We like to say things like, “Scientists prove such and such” or, if we think we are clever “Scientists show such and such” (as if by avoiding saying “prove” we don’t actually mean prove). But…

Can scientists actually prove anything?

I would contend that science is much more like prime numbers than it is like even numbers. If a scientist makes a universal statement (“There are no uncontested transitional fossils”), he cannot back up that statement with an equation that applies to all circumstances. Rather, he has to go out and check every circumstance (i.e. check every available fossil and test it as a transitional fossil). In other words, he has to treat his observations like prime numbers (check every one) rather than like even numbers (use a single equation). Thus, I would contend that science is, by necessity, conjecture and can never prove anything.

Just so we are clear, I do not believe that science is useless. Prime numbers are not useless. In fact, they are very helpful when simplifying fractions, finding least common denominators and greatest common factors, and so forth. However, the usefulness of prime numbers is limited to the prime numbers that we know about and verify. We cannot extrapolate with prime numbers. Similarly, extrapolating with scientific knowledge becomes more and more uncertain, because it is a conjecture. In short, science has an inherent limitation: it works best for things that have been observed, and the further we move away from what has been observed, the less certain science becomes.

By the way, you may notice that this post echoes a post from a little while back, about the stupidity of man. That post took a different approach but reached a similar conclusion: science alone cannot prove anything to be true.

I know, my statement that science is conjecture is itself conjecture, but I don’t want to go down a recursive rabbit trail. Instead, I will simply stand by my statement that science is conjecture until it can be shown otherwise.

Thoughts from Steven

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^[1]Bennett, Albert; Laurie Burton; L. Ted Nelson; Joseph Ediger (2016) Mathematics for Elementary Educators: A Conceptual Approach 10th Edition, McGraw Hill, New York, New York, pg. 3-4