Some Problems Can Be Proved Unsolvable - Evolution News & Views

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Some Problems Can Be Proved Unsolvable

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

  1. Find positive integers x,y and z, such that x3+y3=z3.
  2. Draw a 2D map that is impossible to color (such that countries which share a border have different colors) with fewer than 5 colors.
  3. Explain how life could have originated and evolved into what we see today, through entirely 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 are always other integers to try. You can also spend a lot of time drawing maps. If one map doesn't work, don't give up, there are always others you can try. I once told my 10-year-old son that if he could find such a map, he would be famous. He drew map after map and gave them to his older brother, who always was able to color them using four colors. He finally gave up. More than one mathematician actually thought he had found such a map, but it always proved to be possible to color them with four or fewer colors after all.

A number of theories as to how life could have originated through entirely unintelligent processes have been proposed, but none are convincing, and this problem is generally considered to have not yet been solved. But new theories are constantly being proposed, as it would be unscientific to give up and declare the problem to be unsolvable. Charles Darwin felt he had explained how life and even human intelligence evolved from the first organisms though entirely unintelligent processes. Today his theory is doubted by an increasing number of scientists. 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. Who knows what new theories future scientists will come up with, the problem will surely be solved eventually.

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 try 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. Andrew Wiles proved in 1995 that mathematical problem #1 did not have a solution (he actually proved something more general than this, called "Fermat's last theorem," 358 years after this famous theorem was first proposed). And in 1976, Kenneth Appel and Wolfgang Haken ended 124 years of uncertainty by proving that mathematical problem #2 could not be solved (they proved the "four color theorem" ).

The proofs that the above mathematical problems are impossible to solve were quite difficult, but there is a very simple proof that the biological problem #3 posed above is impossible to solve. 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.

In other areas of science, when one theory fails, scientists propose new ones, and usually a better one is eventually found which is successful. Thus it is not surprising that those of us who claim that the biological problem posed above is impossible to solve are criticized as not understanding how science works. But anyone who spends much time trying to explain how atoms spontaneously rearranged themselves into the first living things, and how genetic accidents then produced more and more complicated arrangements of atoms, and how eventually something called "intelligence" allowed some of these complicated arrangements of atoms to design cars and computers and Apple iPhones, finally starts to realize, or at least should start to realize, that this problem is different. Just as mathematicians who repeatedly tried and failed to solve problems #1 and #2 eventually turned their attention to proving that these problems were unsolvable, biologists should, after repeated failures on problem #3, begin to suspect that there is some fundamental principle involved here that cannot be overcome simply by working harder and producing better theories.

Although it is hard to find a claim that will be met with harsher criticism from the scientific community, I claim that this principle is in fact the fundamental principle behind the second law of thermodynamics. If the principle which dooms all attempts to solve problem #3 is not one of the human statements of this law, it is at least the fundamental natural principle behind this law, as I argue in my June 2013 BIO-Complexity article "Entropy and Evolution." But please note that the proof given above that problem #3 is impossible to solve does not really depend on whether or not what has happened on Earth technically violates the second law, it is much simpler than that.

Meanwhile, I don't spend much time trying to find positive integer solutions to x3+y3=z3, or trying to draw maps that require five colors, and I really don't feel I need to understand each new theory on the origin or evolution of life that is proposed. You can say this is a very unscientific attitude, and if you want to work on these problems I am certainly not going to try to stop you, but I would rather spend my time on problems that have not yet been proved to be unsolvable.


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