Strage Horizons is a fantastic speculative fiction publication. They're considered a professional market and pay their authors at professional rates. The archives are worth checking out. Clarkesworld is another similar publication and has won the Hugo award for best semi-pro zine two years in a row (http://clarkesworldmagazine.com/). If you enjoy this type of writing, support these places! They're run by small communities and often do quite a bit of independent publishing.
Only bleem is not the secret integer between 3 an 4 but the smallest real number greater than zero. Yes, I know real analysis and all about supremum and coverings and epsilons and all the other ways to hide its existence. John Conway gets close in "On Numbers and Games" (there is some of the secret in non-standard analysis) but the conspiracy pushed him off into knot theory when he got too close... Still if you drop the real number line from (0,1) something has to hit the ground first. Bleem.
No, no, no. See, I think it's between 0.99999999_ (repeating) and 1. I mean, something has to go between the two, or people will think they're secretly the same number. But they're not. Numbers can't be Superman. They can't just change costumes in the blink of an eye. They should always look the same, dammit.
Sure, you can multiply 1/3 (== 0.33333_) by 3 and make it look like they're the same, but there's really a tiny little epsilon of something that's getting zeroed out.
It's a conspiracy I tell you! They hide all the cool stuff with infinity and you can't see it any more, then it's like it never existed.
It took thousands of years to accept that the parallel postulate might not be required, and lots of neat geometry emerged. Why not have number systems where infinitesimals are allowed?
Actually, if you look at the original version of that page (http://en.wikipedia.org/w/index.php?title=Archimedean_proper...), it doesn't mention the word "axiom" at all. Someone edited the page at some point to introduce this concept! It seems to have been done in a somewhat anonymous fashion, lending credence to Natsu's theory that there's a conspiracy involved here.
Yep, exactly :). (Should have made it more clear that I was referring to other systems like surreals and hyperreals, but wasn't clear whether the parent was joking or not).
Thanks for the pointer! I enjoy discussions about .999... because it really makes us question what we mean about infinity (and assumptions about the reals).
In mathematics, a definition cannot be 'wrong'; it can lead to an inconsistency, but that isn't the same thing.
The definition of the real numbers (the set of numbers where 0.999... == 1) does not lead to any inconsistency. However, there are other consistent definitions of sets of numbers where (the equivalent of) 0.999... does not equal (the equivalent of) 1. Those sets of numbers have values mathematicians call 'infinitesimals', which do not exist in the set of the real numbers. The hyperreals are one such set of numbers.
That's a great short story. It reminds me of the movie Pi, in which a man obsessively searches for a secret number, to the point of debilitating paranoia. If you enjoyed this short story even slightly, then I highly recommend watching Pi.
I've always thought about stuff like this; a numerals system based upon quantities appearing in nature. 3.1415... is silly. Make that 1 or 4, and see where everything else "fits", or something similar based on a universal constant as the base.
But, in your number system where π is 4, what is e?
More to the point, we aren't just making up numbers here to enjoy infinite decimal expansions; the reason is that you can wrap the radius of a circle around its circumference more than 3 times (or, if you don't like numerals, as many times as letters in 'pie'), but fewer than 4 (not as often as letters in 'easy')—and no changing of units is going to get around that.
Very cool short story. One of the lessons I got from it is that you should never consider somebody crazy for having a crazy idea. Some of our most brilliant scientists, mathematicians, entrepreneurs, etc have been called crazy fools for having radical ideas (That ended up being right). Our society needs radical thinkers to shake things up, and revolutionize the world. Next time somebody tells you about their own "bleem" just consider it, and don't throw them to the street as a heretic.
Next time somebody tells you about their own "bleem" just consider it, and don't throw them to the street as a heretic.
Mathematicians are aware of bleem. It's what you get when you remove the induction axiom from Peano arithmetic: A number which is not in the set {0, 1, 2, ...}.
In general the higher up you go in academia the more open people are to "variant realities". Most of our analysis of curved space-time comes to us thanks to mathematicians who thought they were playing with entirely theoretical constructs ("what happens if we remove the parallel postulate?").
You actually do not have to remove any axioms from Peano arithmetic to prove that a model of PA with such a 'bleem' exists. It is a straightforward consequence of the compactness theorem: just enrich PA with a new constant c and an infinite series of axioms 'c != N' for every numeral N.
For any finite set of these new axioms a model exists (just set c to a large enough number), therefore by the compactness theorem there exists a model which satisfies all our new axioms.
Except the number you get this way is not between 3 and 4. It's essentially infinite with no functions in the model that separate it from the finite numbers (and this can be proven in PA: if a number isn't 0..N, then it's greater than N).
Generally, I don't think it's fair to characterize this story as something studied by mathematicians. Really, it's an exercise in reasoning about nonsense.
Inequality can be defined in Peano arithmetic in a total way. Maybe you're talking about defining some relation on the model? But what justifies calling such a relation an inequality?
I am reminded of Ramanujan's proof that the sum of all positive integers is -1 / 12. The Wikipedia article has a fascinating excerpt from his letter to Dr. Hardy:
This reminds me of a short story I read when I was a kid, where a guy discovered a shape with zero sides. I think at the end he made his boss disappear by folding him into the shape.
This story is in either "Fantasia Mathematica" or "The Mathematical Magpie", both edited by Clifton Fadiman. I don't have my copies here to check, but a look at the table of contents suggests that it might be "The Hermeneutical Doughnut", by H. Nearing Jr. (Even if it's not, it's worth picking up the books to look; anyone who liked this story is guaranteed to like them.)
Thanks I saved both of these stories off after reading them. This story gave me the chills. I have NO idea how the author made me feel like the narrator.
I was absorbed with the story and I could feel the silence of my companion as he traveled behind me gazing at the disappearing stars.
Sadly, it seems Igor only wrote that one, almost 12 years ago, and stopped writing on that website. You can read some more from him at http://www.stanford.edu/~teper/writing.html
Anyone that liked that story should pick up a copy of Mathenauts - an anthology of Mathematics themed short stories put together by Rudy Rucker. Many of the stories have a similar feel to that one. You can find it on Amazon here: http://www.amazon.com/Mathenauts-Mathematical-Wonder-Rudy-Ru...
I read the story above and I did like it. It has a blend
of math, insanity, and consciousness like the story in the main article.
A synopsis for others to decide if they want to read it:
Chloe is a beautiful humanoid companion. Deliberately implanted
in the programming of her mind is a minor reasoning defect.
Whereas an ordinary person could live with it, and perhaps
not even be aware of it, Chloe cries and has fits whenever
she becomes aware of the flaw in her chain of thought.
A psychologist is brought in to figure things out.
By the way, "Sample 27" is the title and not the
27th sample story from the author as I originally thought.
A suggestion to the author: You might want to either not
use HTTPS or buy a certificate. I don't think it's a barrier
to Hackernews readers but most people won't get past
the browser's security exception box.
As I was reading I imagined at the end Dr. Tomlin back home found the bag of jelly beans he emptied onto the table still in his pocket only he would find one jelly bean still in it.
