Read Online Gödel, Escher, Bach: An Eternal Golden Braid by Douglas R. Hofstadter - Free Book Online
Authors: Douglas R. Hofstadter
Tags: General, science, Biography & Autobiography, music, Computers, Artificial intelligence, Genres & Styles, Philosophy, Art, Science & Technology, Mathematics, Individual Artists, Classical, Logic, Symmetry, Bach; Johann Sebastian, Metamathematics, Intelligence (AI) & Semantics, G'odel; Kurt, Escher; M. C
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are mere illusions of the sense. Some have mocked his views; but I will show the absurdity of their mockery. My argument is quite simple. I will illustrate it with two characters of my own Invention: Achilles )a Greek warrior, the fleetest of foot of all mortals), and a Tortoise. In my tale, they are persuaded by a passerby to run a footrace down a runway towards a distant flag waving in the breeze. Let us assume that, since the Tortoise is a much slowerrunner, he gets a head start of, say, ten rods. Now the race begins. In a few bounds Achilles has reached the spot where the Tortoise started.
ACHILLES: Hah!
Zeno: And now the Tortoise is but a single rod ahead of Achilles. Within only a moment, Achilles has attained that spot.
ACHILLES: Ho ho!
Zeno: Yet, in that short moment, the Tortoise has managed to advance a slight amount. In a flash, Achilles covers that distance too.
ACHILLES: Hee hee hee!
Zeno: But in that very short flash, the Tortoise has managed to inch ahead by ever so little, and so Achilles is still behind. Now you see that in order for Achilles to catch the Tortoise, this game of “try-to-catch-me” will have to be played an INFINITE number of times –
and therefore Achilles can NEVER catch up with the Tortoise.
TORTOISE: Heh heh heh heh!
ACHILLES: Hmm… Hmm… Hmm… Hmm… Hmm…That argument sounds wrong to me.
And yes, I can’t quite make out what’s wrong with it
Zeno: Isn’t it a teaser? It’s my favourite paradox.
TORTOISE: Excuse me, Zeno, but I believe your tale illustrates the wrong principle, doe sit not? You have just told us what will come to known, centuries hence, as Zeno’s “Achilles paradox” , which shows (ahem!) that Achilles will never catch the Tortoise; but the proof that Motion Is Inherently Impossible (and thence that Motion Unexists) is your
“dichotomy paradox”, isn’t that so?
Zeno: Oh, shame on me. Of course, you’re right. That’s the new one about how, in going from A to B, one has to go halfway first – and of that stretch one also has to go halfway, and so on and so forth. But you see, both those paradoxes really have the same flavour. Frankly, I’ve only had one Great Idea – I just exploit it in different ways.
ACHILLES: I swear, these arguments contain a flaw. I don’t quite see where, but they cannot be correct.
Zeno: You doubt the validity of my paradox? Why not just try it out|? You see that red flag waving down here, at the far end of the runway?
ACHILLES: The impossible one, based on an Escher print?
Zeno: Exactly. What do you say to you and Mr. Tortoise racing for it, allowing Mr. T a fair head start of, well, I don’t know –
TORTOISE: How about ten rods?
Zeno: Very good – ten rods.
ACHILLES: Any time.
Zeno: Excellent! How exciting! An empirical test of my rigorously proven Theorem! Mr.
Tortoise, will you position yourself ten rods upwind?
(The Tortoise moves ten rods closer to the flag)
Tortoise and Achlles: Ready!
Zeno: On your mark! Get set! Go!
Chapter 1
The MU-puzzle
Formal Systems
ONE OF THE most central notions in this book is that of a formal system . The type of formal system I use was invented by the American logician Emil Post in the 1920's, and is often called a "Post production system". This Chapter introduces you to a formal system and moreover, it is my hope that you will want to explore this formal system at least a little; so to provoke your curiosity, I have posed a little puzzle.
"Can you produce MU ?" is the puzzle. To begin with, you will be supplied with a string (which means a string of letters).* Not to keep you in suspense, that string will be MI . Then you will be told some rules, with which you can change one string into another.
If one of those rules is applicable at some point, and you want to use it, you may, but-there is nothing that will dictate which rule you should use, in case there are several applicable rules. That is left up to you-and of course, that is where playing the
ACHILLES: Hah!
Zeno: And now the Tortoise is but a single rod ahead of Achilles. Within only a moment, Achilles has attained that spot.
ACHILLES: Ho ho!
Zeno: Yet, in that short moment, the Tortoise has managed to advance a slight amount. In a flash, Achilles covers that distance too.
ACHILLES: Hee hee hee!
Zeno: But in that very short flash, the Tortoise has managed to inch ahead by ever so little, and so Achilles is still behind. Now you see that in order for Achilles to catch the Tortoise, this game of “try-to-catch-me” will have to be played an INFINITE number of times –
and therefore Achilles can NEVER catch up with the Tortoise.
TORTOISE: Heh heh heh heh!
ACHILLES: Hmm… Hmm… Hmm… Hmm… Hmm…That argument sounds wrong to me.
And yes, I can’t quite make out what’s wrong with it
Zeno: Isn’t it a teaser? It’s my favourite paradox.
TORTOISE: Excuse me, Zeno, but I believe your tale illustrates the wrong principle, doe sit not? You have just told us what will come to known, centuries hence, as Zeno’s “Achilles paradox” , which shows (ahem!) that Achilles will never catch the Tortoise; but the proof that Motion Is Inherently Impossible (and thence that Motion Unexists) is your
“dichotomy paradox”, isn’t that so?
Zeno: Oh, shame on me. Of course, you’re right. That’s the new one about how, in going from A to B, one has to go halfway first – and of that stretch one also has to go halfway, and so on and so forth. But you see, both those paradoxes really have the same flavour. Frankly, I’ve only had one Great Idea – I just exploit it in different ways.
ACHILLES: I swear, these arguments contain a flaw. I don’t quite see where, but they cannot be correct.
Zeno: You doubt the validity of my paradox? Why not just try it out|? You see that red flag waving down here, at the far end of the runway?
ACHILLES: The impossible one, based on an Escher print?
Zeno: Exactly. What do you say to you and Mr. Tortoise racing for it, allowing Mr. T a fair head start of, well, I don’t know –
TORTOISE: How about ten rods?
Zeno: Very good – ten rods.
ACHILLES: Any time.
Zeno: Excellent! How exciting! An empirical test of my rigorously proven Theorem! Mr.
Tortoise, will you position yourself ten rods upwind?
(The Tortoise moves ten rods closer to the flag)
Tortoise and Achlles: Ready!
Zeno: On your mark! Get set! Go!
Chapter 1
The MU-puzzle
Formal Systems
ONE OF THE most central notions in this book is that of a formal system . The type of formal system I use was invented by the American logician Emil Post in the 1920's, and is often called a "Post production system". This Chapter introduces you to a formal system and moreover, it is my hope that you will want to explore this formal system at least a little; so to provoke your curiosity, I have posed a little puzzle.
"Can you produce MU ?" is the puzzle. To begin with, you will be supplied with a string (which means a string of letters).* Not to keep you in suspense, that string will be MI . Then you will be told some rules, with which you can change one string into another.
If one of those rules is applicable at some point, and you want to use it, you may, but-there is nothing that will dictate which rule you should use, in case there are several applicable rules. That is left up to you-and of course, that is where playing the
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