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Some Problems Can Be Proved Unsolvable

Granville Sewell
Photo credit: Stephanie LeBlanc, via Unsplash.

Here are two mathematical problems for you to work on in your spare time, and one problem from biology:

  1. Find positive integers x,y, z and n>2, such that xn+yn=zn.
  2. Remove two diagonally opposite corner squares from a chess board, and cover the remaining 62 squares with 31 dominoes, each of which covers two adjacent squares.
  3. Explain how life could have originated and evolved into intelligent humans, through entirely natural (unintelligent) processes. 

You can spend a lot of time trying different solutions to mathematical problem #1. After a while you might begin to wonder if it can be done, but don’t give up, there is an infinite number of integers you can try for x,y,z and n. 

For problem #2, get out your chess board and some dominoes, cut out two diagonally opposite corner squares, and start covering. If your first try doesn’t work, keep working, there is a huge number of ways you can lay out the dominoes. 

New Theories Multiply

A number of theories as to how life could have originated through entirely unintelligent processes have been proposed, but none are plausible, and this problem is generally considered to have not yet been solved. But new theories are constantly being proposed, and it would be unscientific to give up and declare the problem to be unsolvable…wouldn’t it? Charles Darwin felt he had explained how intelligent humans evolved from the first living organisms though entirely unintelligent processes. Today his theory is doubted by an increasing number of scientists, and most of those who still support it would probably agree with microbiologist René Dubos that “its real strength is that however implausible it may appear to its opponents, they do not have a more plausible one to offer in its place” (The Torch of Life, 1962). Most of these doubters have proposed modifications to his theory or alternative theories of their own, but there are always serious problems with the alternative theories too. However, scientists should never give up, even if none of the theories proposed so far are plausible…right? French biologist Jean Rostand (A Biologist’s View, 1956) says, “However obscure the causes of evolution appear to me to be, I do not doubt for a moment that they are entirely natural. We have ample time to discover them; biology is in its infancy.” 

Well, mathematicians sometimes do give up, after we have proved a problem to be impossible to solve. How can you prove a problem is impossible to solve, if you can’t examine every possible solution? Often you say, assume there is a solution, then using that assumption you prove something that is obviously false, or known to be false. Pierre de Fermat claimed in 1637 to have a simple proof that problem #1 has no solution, but the proof was “too large to fit in the margin” of a document he was working with. Did Fermat really have a short, correct proof? Not likely, because no one else could find a rigorous proof until 358 years later, when Andrew Wiles produced a very long, complicated proof of Fermat’s last theorem. Whether or not Fermat’s short proof was valid may never be known, but at least his conclusion was correct. 

After trying a long while, you may get the idea that mathematical problem #2 is also impossible to solve, but if you try to prove it is impossible, you may start to think that this one may also take years to prove impossible. But in mathematics, you can often prove an apparently difficult theorem in a surprisingly simple way once you look at it from the right perspective. For this problem, all you have to do is notice that each domino will cover one black and one red square, so if you could solve the problem, you would have to conclude that the board consists of the same number (31) of black and red squares. But this conclusion is false: the original chessboard had an equal number of red and black squares, but the diagonally opposite corners you removed were the same color! 

A Simple Proof in Biology

Well, I have a very simple proof that the biological problem #3 posed above is also impossible to solve, that does fit in the margin of this document. All one needs to do is realize that if a solution were found, we would have proved something obviously false, that a few (four, apparently) fundamental, unintelligent forces of physics alone could have rearranged the fundamental particles of physics into libraries full of science texts and encyclopedias, computers connected to monitors, keyboards, laser printers and the Internet, cars, trucks, airplanes, nuclear power plants and Apple iPhones. 

Is this really a valid proof? It seems perfectly valid to me, as I cannot think of anything in all of science that can be stated with more confidence than that a few unintelligent forces of physics alone could not have rearranged the basic particles of physics into Apple iPhones. In the first half of my video “Why Evolution Is Different” I argue with a bit more scientific sophistication, and a bit more scientific detail, that problem #3 has no solution, but my arguments are still very simple. Unfortunately, most biologists don’t seem to be impressed by such simple proofs; they don’t believe it is possible to reject all solutions to a difficult problem without looking at the details of each. But mathematicians know that sometimes it is possible. 

For the first mathematical problem, it didn’t take too many years of failed attempts before mathematicians realized their time was better spent proving this problem unsolvable than continuing with attempts to solve it. Maybe after another 358 years of failed attempts to solve problem #3, someone will finally produce a proof that convinces even biologists that they didn’t fail because they just never hit on the right solution, but because the problem doesn’t have a solution. Or maybe someone already has.