Here are a couple of difficult mathematical problems for you to work on, in your spare time:
- Find positive integers x,y and z, such that x3+y3=z3.
- Draw a 2D map which is impossible to color (such that countries which share a border have different colors) with fewer than 5 colors.
And here is a difficult problem from biology:
- 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 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 also. But 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 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 space shuttles. A very simple “proof”, but I cannot imagine how anything in science could ever be proved more conclusively, it is all the proof I need, at least. The video below (here is a Spanish version) presents this argument in a simple, clear, way, and connects it to the second law of thermodynamics (more on this connection here); but please note that the argument does not really depend whether or not what has happened on Earth technically violates the second law, it is much simpler than that.
Unfortunately, my proof is just too simple to be interesting to most scientists, they are generally not interested in an argument that is so simple you can understand it without a PhD. (Actually, mathematicians do prefer simple proofs to complicated ones, so maybe we are different.) If I could find a proof that was as complicated, and as inaccessible to the layman, as the proofs that the above mathematical problems were unsolvable, perhaps my arguments would be taken more seriously in the scientific community.
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 which 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 which have not yet been proved to be unsolvable.
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