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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#darwin, #id, #math, #science, #theory

Are allergies trying to protect us from ourselves?

This article is more than actual so I should repost it, again.  Sorry )


“I have a love/hate relationship with spring, thanks to the aggravating bouts of hay fever that transform me into a faucet for pretty much the entire season. So I’ll admit I was a little skeptical when my editor at Scientific American asked me last week if I wanted to write about a new paper coming out in Nature suggesting that allergies may actually be a good thing. But always curious, I said sure.

I have a love/hate relationship with spring, thanks to the aggravating bouts of hay fever that transform me into a faucet for pretty much the entire season. So I’ll admit I was a little skeptical when my editor at Scientific American asked me last week if I wanted to write about a new paper coming out in Nature suggesting that allergies may actually be a good thing. But always curious, I said sure.

Turns out it’s a fascinating—and pretty convincing—read. It’s dense, but the lead author, Yale immunobiologist Ruslan Medzhikov, was kind to take a good two hours out of his day on Monday to explain some of the gnarlier concepts to me. (Medzhikov is fascinating—you can read more about him in this profilepublished in Disease Models & Mechanisms.)

Medzhikov’s basic argument is that there is a convincing body of research suggesting that allergies have beneficial effects. They break down the toxic components of bee, snakescorpion and gila monster venom, for instance, and our allergic reactions to tick saliva prevent the parasites from feeding.

Ultimately, all allergic responses work towards a common goal: avoidance and expulsion, Medzhitov argues. As I explain in my piece,

More generally, hated allergic symptoms keep unhealthy environmental irritants out of the body, Medzhitov posits. “How do you defend against something you inhale that you don’t want? You make mucus. You make a runny nose, you sneeze, you cough, and so forth. Or if it’s on your skin, by inducing itching, you avoid it or you try to remove it by scratching it,” he explains. Likewise, if you’ve ingested something allergenic, your body might react with vomiting. Finally, if a particular place or circumstance ramps up your allergies, you’re likely to avoid it in the future. “The thing about allergies is that as soon as you stop exposure to an allergen, all the symptoms are gone,” he says.

Obviously, Medzhitov’s theory is just a theory, and it involves a lot of speculation (albeit informed speculation by a really smart guy). But some research suggests an association between allergy severity and cancer risk, in that people with more allergy symptoms are less likely to develop certain cancers. (One shouldn’t read too much into this though; some other factor may drive the association. Perhaps people who eat lots of eggs are more likely to have allergies but less likely to have cancer.) But all in all, I think Medzhitov’s idea does make sense and is well-supported, and most of the outside experts I spoke with agreed, though they did raise questions about some of the specifics.

One aspect of the theory that I didn’t mention in my piece is that it could explain a medical mystery: penicillin allergies. Medzhitov argues that in addition to protecting against venoms, vector-borne diseases and environmental irritants, allergies also evolved to protect against a class of toxins called haptens: proteins that bind to extracellular or membrane-bound proteins in the body, rendering them useless and ultimately causing all sorts of problems. As it turns out, in some people, the penicillin molecule undergoes transformation into a hapten. This transformation is very slow and inefficient—very few penicillin markers turn into haptens, which is a good thing because haptenated penicillin could be dangerous—but nevertheless, some people may develop allergic responses to these few haptenated penicillin molecules, and this can result in an allergic hypersensitivity to the drug, Medzhitov posits.

In the case of something like a penicillin allergy, management is fairly simple (though medically inconvenient): avoid penicillin. The problem today is that there may be millions of allergens in the form of environmental pollutants and irritants, and they may simply be unavoidable. This idea could help explain why allergic diseases have become more common in recent decades: We’re exposed to many more pollutants now than we were 50 years ago, and this chemical flurry could be dialing up our innate defense systems to a constant level of 11. An allergy may be protective, but “if it’s taken to an extreme, it is pathological,” Medzhitov says. I wonder, then, if we may have built ourselves a world that will forever make us sick.”


Palm, N., Rosenstein, R., & Medzhitov, R. (2012). Allergic host defences Nature, 484 (7395), 465-472 DOI: 10.1038/nature11047

Medzhitov, Ruslan (2011). Innovating immunology: an interview with Ruslan Medzhitov Disease Models & Mechanisms, 4 (4), 430-432 DOI:10.1242/dmm.008151

Akahoshi M, Song CH, Piliponsky AM, Metz M, Guzzetta A, Abrink M, Schlenner SM, Feyerabend TB, Rodewald HR, Pejler G, Tsai M, & Galli SJ (2011). Mast cell chymase reduces the toxicity of Gila monster venom, scorpion venom, and vasoactive intestinal polypeptide in mice. The Journal of clinical investigation, 121 (10), 4180-91 PMID: 21926462

Wada T, Ishiwata K, Koseki H, Ishikura T, Ugajin T, Ohnuma N, Obata K, Ishikawa R, Yoshikawa S, Mukai K, Kawano Y, Minegishi Y, Yokozeki H, Watanabe N, & Karasuyama H (2010). Selective ablation of basophils in mice reveals their nonredundant role in acquired immunity against ticks. The Journal of clinical investigation, 120 (8), 2867-75 PMID: 20664169

Sherman, P., Holland, E., & Sherman, J. (2008). Allergies: Their Role in Cancer Prevention The Quarterly Review of Biology, 83 (4), 339-362 DOI:10.1086/592850

#allergy, #conditions-and-diseases, #health, #research, #science, #theory