Post a Comment. Sunday, June 18, The older the better. One of the characteristic forms of his essays was the following of a metaphor through history. The archetypal example is "The Eternal Rose of Coleridge" which is self referential, because Borges compared excellent metaphors to that Rose "The Name of" which is the title of a best seller.

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Post a Comment. Sunday, June 18, The older the better. One of the characteristic forms of his essays was the following of a metaphor through history. The archetypal example is "The Eternal Rose of Coleridge" which is self referential, because Borges compared excellent metaphors to that Rose "The Name of" which is the title of a best seller.

I was struck by "Avatars of the Tortoise," well actually appalled. Borges considers many versions of Zeno's paradox. Zeno said consider the fleet Achilles who runs the hundred meters in 10 seconds, 10 times as fast as the slow tortoise this has to be the worlds fastest tortoise.

If he starts meters behind the tortoise quickly reaches the point where the tortoise started, but the tortois is 10 meters ahead of him. He makes up that distance, but the tortoise is then 1 meters ahead of him. He makes up that distance but the tortoise is 10 centimeters ahead of him. The series goes on forever, therefore the fleet Achilles never passes the slow tortoise. Zeno's error is obvious. It is not necessarily true that the sum of an infinite series is infinite.

Amazingly the great Borges took Zeno's paradox seriously. He argues that it is a hint of unreality that convinces us that the image we have made of the universe is an illusion.

I think it shows that people find the concept of infinity very very confusing. In fact, I think that behind every profoundly counterintuitive result in mathematics you can find a tortoise.

That they all have something to do with infinity. This thought came back to me when I considered the mother of all counter-intuitive theorems Godel't theorem.

Godel proved that, except for the simplest mathematical systems he called first order logic, it is not, in general, possible to determine whether an axiom system is consistent. That is we can not know if we can prove a contradiction from the axioms.

The connection with infinity is clear. If we have looked and looked a billion years for a contradiction, we don't know if we would find it after a billion years and a day. In a model universe which can only has a finite number of possible states, it is easy to check if axioms are consistent. Just see if they are true in any of the states. If they are, they are consistent.

If they aren't they can't all be true in that universe, so they are inconsistent. That is, all is ok, if we consider only models with a finite state space. Another case in which infinity makes infinite trouble is the halting problem. This is the question of whether a Turing machine will ever stop, that is, a question of whether a computer will ever finish running a program.

We confront this problem whenever our system hangs. Thus windows users are much more knowledgeable about this fundamental issue than our Mac users. The problem is clearly related to Godel's problem, and, in fact, the questions are equivalent. It is easy to prove that no computer program can tell, for any other program, if a computer will ever stop if it is running the other program. The proof is take the program A which determines if another computer program will stop running.

Rewrite A to B so that if A outputs "yes" add an infinite loop and if A outputs "no" B prints "no" and stops. Now use B as an input to B. If B stops, B doesn't stop. If B doesn't stop B stops. Now that is a paradox. There is an analogous problem which can be solved. A Turing machine has an infinite memory. The memory is in two forms, one is the "state" of the Turing machine of which there are a finite number, the other is a tape with 0's an 1's written on it.

Now consider a finite circular tape with N bits of information. There is a Turing machine C which can tell if any other Turing machine will stop if set on a circular tape with N zeros. The reason is that there are only finitely many possible states of the tape and Turing machine. If the Turing machine moves around for a while and gets to a state of it and the tape where it has already been, it will never stop.

Otherwise it will stop. In fact C can do this with another finite tape with many many more than N bits of information but still finite.

The halting problem does not exist if we consider only Turing machines with finite circular tapes. Now the question is might our universe have a finite state space?

Blake would disagree claiming one can hold infinity in a grain of sand and eternity in an hour, but Blake didn't know about Heisenberg. There is a smallest measurable interval of time and space. If the universe always has finite volume and lasts a finite period of time, it has a finite state space. That is, if the Universe is closed it is finite and all paradoxes are hints of reality that show that our images of the universe are as mistaken as Zeno.

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## Jorge Luis Borges and mathematics

JavaScript is required to view this site. Log in Sign up. Most recent Most popular Most recent. Filter by post type All posts. Grid View List View. We the indivisible divinity that works in us have dreamed the world. We have dreamed it resistant, mysterious, visible, ubiquitous in space and firm in time, but we have allowed slight, and eternal, bits of the irrational to form part of its architecture so as to know that it is false.

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## Jorge Luis Borges

To fall in love is to create a religion with a fallible god. Labyrinths The Lottery in Babylon The drunkard who improvises an absurd order, the dreamer who awakens suddenly and strangles the woman who sleeps at his side, do they not execute, perhaps, a secret decision of the Company? Library of Babel In adventures such as these, I have squandered and wasted my years. It does not seem unlikely to me that there is a total book on some shelf of the universe; I pray to the unknown gods that a man--just one, even though it were thousands of years ago!

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## avatars of the tortoise

Jorge Luis Borges and mathematics concerns several modern mathematical concepts found in certain essays and short stories of Argentinian author Jorge Luis Borges , including concepts such as set theory , recursion , chaos theory , and infinite sequences , [1] although Borges' strongest links to mathematics are through Georg Cantor 's theory of infinite sets, outlined in "The Doctrine of Cycles" La doctrina de los ciclos. He was also aware of the contemporary debates on the foundations of mathematics. In Borges' story, "The Library of Babel", the narrator declares that the collection of books of a fixed number of orthographic symbols and pages is unending. In his short story " The Book of Sand " El Libro de Arena , he deals with another form of infinity; one whose elements are a dense set , that is, for any two elements, we can always find another between them.