Tina S. Chang|
"Unreasonable Effectiveness" reminds me of a classic Arthur C. Clarke style
short story. It has exactly enough mathematics done correctly and a twist that
boggles the mind at the end. To be fair it has a twist earlier too, one which was an idea I've about the universe, but could never incorporate into a story effectively. Kasman not only builds on the idea, adding the second twist, but he draws the reader in with two believable characters as well.
Mathematician Amanda Birnbaum (whose
research in algebraic topology has developed stunning applications in
biology) travels to an isolated house on a remote island which
subscribes to all of the world's research journals. There she learns the real answer to the old question "why does pure math research turn out to be so useful in science?" Even though the tongue-in-cheek answer provided the story is not to be taken seriously, a little bit of serious discussion about this question is included as well.
It should be noted that the title is intended to be a clear reference to two important papers that ask the question "why is mathematics so useful?" See Eugene Wigner's 1960 article and RW Hamming's 1980 follow-up.
It's difficult for me to describe this work objectively, because it is my first published story. So, I hope that other people read it and write in with their own comments.
The story appears in the collection Reality Conditions. It was originally published in the April 2003 issue of Math
Horizons magazine (pp. 29--31). By gracious permission of the editor, I am
allowed to distribute it for free here as a pdf
I liked the simple, Asimovian quality of the writing, as well as the selection of such a relevant topic for working mathematicians. The story is realistic, and it might even be true! Things sure work that way in my research. I make up things, and next thing you know, somebody's using it (no biologists or physicists yet).
I have read it a few days ago and I loved it. It is short, simple, and very very very interesting. I like the fact that the main character is a female. It is based on mathematics, which is excelent. Up until the end the reader doesn't know what will happen. And the end comes as a big surprise. Almost as a shock. Unbelieavable and fantastic. I recomend googling for "Unreasonable Effectiveness". I found out that such a problem really exists! I think that you won't regret reading Kasman's "Unreasonable Effectiveness". By the way, my professor mentioned to me that in the real life it would be impossible to collect all the copies of Amanda's thesis and replace them by new ones.
Thank you, Sonja, for the kind words. You can tell your professor that the idea of Amanda recalling the journal with her thesis due to an error is a literary reference to a famous, real event in the history of Mathematics. The famous mathematician Poincare published a prize winning paper on celestial mechanics in the equally famous journal Acta Mathematica in 1889...only to realize that he had made a significant error. He contacted them to halt the printing but learned that all of the printing was in fact done and that some of the copies had already been distributed. (see Ivars Peterson's article for more information on this interesting story.) This is the same situation as described in my story, with The Memoirs of the AMS taking the place of Acta Mathematica. (Moreover, you can point out to your professor that in the story, Amanda is not successful in replacing all of the copies of the article...which is exactly how she ends up discovering the strange little man on the strange little island!)
Warning: The following contribution (also from Sonja) contains spoilers. You are encouraged to read the story before you read it because it gives away some of the surprises.
Unreasonable Effectiveness is a story that deals with the usefulness of mathematics. There are only two characters in this story. One is Amanda Birnbaum. She made a mistake in her PhD thesis, which she found out and replaced all (but one) copies with the correct ones. She found out who has that incorrect copy of her thesis and that’s how she meets the other character of this story. She visits him at his home. He lives on a small island (unclaimed by any country). They talk about the usefulness of mathematics. They both agree that when high-dimensional topology, non-Euclidean geometry and the use of non-commutative rings and imaginary numbers in particle physics first appeared everybody thought that it is useless. But when Rieman, Clifford, Hilbert and Einstein started using it, it became useful. Amanda says that mathematics is used in engineering, physics, biology and economy. They have a discussion on what mathematics really is. There are two groups of people with two different answers to that question. But she is the only one that finds the third option. She claims that there is somebody who changes the universe in order to fit it to mathematics. That says a lot about the mathematics and mathematicians, doesn’t it? This story says that mathematics is nothing special and mathematicians don’t do anything important. They make up nonsense and if “they” like it, they change the universe. I’m thinking about the mathematicians in out history. Can you imagine what they had to go through? How many hours did they spend doing their researches? How much effort did they put in their theorems and proofs? I think that much more than we can imagine. The story tries to convince us that all their work was done for nothing. Of course it is fictional, but it presents mathematics in a different way, a way that is unknown to us. I think that mathematicians always had (still have) a reputation of geniuses, gifted people, special people. This story changes their reputation. The truth is, in my opinion, somewhere in between. Mathematics and with it also mathematicians are not so special and gifted, but on the other hand, they are important and the world needs them. After all, we are all the same and as much as we need economists, doctors, etc, we also need biologists, physicists and mathematicians.
Sonja, I'm not sure I agree that the work of mathematicians would be "wasted" if the universe were to alter to fit their discoveries. In some ways, you could say that mathematics in such a situation would be even more influential than in a universe where the only changes due to mathematics was in our own understanding. However, I also want to make clear that for many mathematicians, the discoveries they make in mathematics are real and interesting whether or not they have anything to do with the physical universe. I do not think that they would feel that their work was "wasted" unless it was later discovered that they had made an error and that their "theorems" were actually incorrect. This is an important distinction that many people do not appreciate: mathematics is not a science which studies the physical universe, it is a completely mental exploration of what is either (depending upon who you ask) an completely theoretical construction or an alternate non-physical reality. Either way, whether a theorem is true or not has nothing to do with the physical universe. In fact, this is why a mathematical fact remains true even when science undergoes a revolution. Consider, for instance, that Newton's mechanics has been surplanted by quantum physics and relativity. Still, Newton's calculus remains a completely accurate mathematical construction.
I think this is a very difficult idea to explain, and I'm not sure whether the paragraph above will do the job. Do you see what I'm trying to say?
Hm. It is hard to understand what you are saying. In a way it is true that if the universe would change in order to fit to our mathematics, then, yes, I gues you are right. Mathematicians would have a very important job. They would be the ones who would change the Universe. Their responsibility would be huge. I didn't see it that way. Thanks for telling me that there is another way to look at it!
As far as mathematics is concerned...it is true that it lasts for centuries, and it is never wrong. At least not that I would know of. Maybe there are some newer and better proofs from time to time. And some long unanswered questions get solved sometimes. If I understand you correctly, physicists get wrong sometimes and their ideas are completely replaced by new ones?!? So, why is mathematics different than physics? Why is mathematics always true and physics makes mistakes? Is it only because of the "relation" with the Universe? I mean, physics is connected to the Universe, and physicists have to guess a lot(since we don't know everything about the Universe). Mathematics is not related to the Universe and what is once proved is proved for ever. So, why did you connect mathematics with Universe in "Unreasonable Effectiveness"? Do you think that they are somehow connected or did you try to point out something else?
Yes, I think you've got it. From the point of view you've espoused above, it is not clear why there should be a connection between math and reality. Mathematics can be seen as something completely separate from science and any investigation of reality. It is like a big puzzle, a game. We can be certain of our answers because we created the rules ourselves...unlike science which investigates the real world in which we don't know what the rules are or where they came from. BUT, there obviously is some connection between math and reality. We use math to make startling predictions (i.e. precisely when there will be an eclipse or how many more times a player on Let's Make a Deal will win the car if they switch instead of staying with their first chosen door), to control reality (we were able to crash a probe into a comet millions of miles away from Earth using math) and to make discoveries (electro-magnetic waves were discovered by Maxwell merely because the equations for the interaction of electrons and magnets could be put in a form that resembled the equation for the vibration of a violin string.)
So, math appears to be "effective", but why?? That's the mystery! That is why the effectiveness of mathematics has been termed "unreasonable". It is a deep question, so I would not pretend to be able to do it justice in this brief summary or in the story either. Much has been written on it. But, I recommend that you read the two papers whose titles inspired the title of the story: "The Unreasonable Effectiveness of Mathematics in the Natural Sciences" by Eugene Wigner (1960) and "The Unreasonable Effectiveness of Mathematics" by RW Hamming (1980).
I found this great website, containing so much information about mathematical fictions by accident. Perhaps due to the fact that I am now a PhD student major in partial differential equations (PDE), I was touched when I finished reading this amazing and entertaining fiction. To my knowledge, PDE, one branch of mathematics, has been recognized as a useful tool for modeling the phenomenon in the real world. Therefore, someone in his or her sense might think of PDE as applied mathematics. So before I read over this fiction, I have always been believing the usefulness of mathematics. The viewpoint of why mathematics is so useful in other fields suggested in it is so science-fictional (really very original, I suppose) that now I get interested in your other works on mathematical fictions. Could I have the honor to read your other works freely :)? Anyway, I enjoyed this short and elegant fiction very much and I will introduce it (maybe I shall translate it into Chinese so that they won't have difficulty to read it) to my classmates (also major in mathematics)! Thank you for your superb opus:).
The narration is perhaps too straightforward, but I liked the examples of mathematics used in the text (they are not the classical number, calculus or euclidean geometry common places). The final twist is rather clever, I found it quite amusing.
I enjoyed this tale and was surprised by it. I anticipated that it was headed in one direction, but it resolved in a different fashion. I am often amazed by the usefulness of certain mathematical discoveries and the interconnectedness of seemingly disconnected branches of mathematical theory (as exemplified by Euler’s equation e^(iπ)=-1) I found this story vaguely reminiscent of a chapter in "The Restaurant at the End of the Universe" by Douglas Adams. Have you read this work? & Was it an influence in your writing?
Thanks, anonymous visitor. I most certainly am familiar with the entire Hitchhiker's Guide to the Galaxy opus. It may have influenced me, though I do not think I had it in mind when I wrote this story. Are you thinking of the scene where they meet the ruler of the universe, who is apparently an absent-minded old gent, more interested in his cats than in their problems?
Yes, this is precisely the story to which I was referring. As I pondered my remarks to you last night, before falling asleep, it occurred to me that there are probably more differences than similarities between "Unreasonable Effectiveness" and this chapter. However, the human mind tends to look for similarities and patterns even when none exist. I would love to read what others have to write about a comparison of these two works.
A delightful short short story, and I see from others' comments that they have entertained a similar (fanciful?) notion themselves, as have I. The way it's been realized is very neat.
Perhaps there is an analogy to the "consciousness causing collapse of wavefunction" conjecture here, in which the act of observing reduces possibilities to actualities. (Though of course, in no way approaching the intellectual achievement of proving a theorem.)
I had actually come to this website in the hope of rediscovering an SF short I read many years ago, with a physics-themed similar premise. Unfortunately I remember only a few vague details, it was based on the "Is the moon there when we are not watching" question. The premise was taken to rather absurd extremes but they worked well within the context of the story.
I think the author may also have extended the idea to imagining historical events and thus changing the course of history retroactively, but I am not sure about that part.
Anyway I skimmed through some of your categories but could not find it there. Perhaps one of your readers would have an idea?
Thank you for the kind words about my story, Eugene. I'm afraid that I do not recognize the story you are searching for, but will leave this remark here and let you know if anyone else does. -Alex