Surreal Numbers
by
by
Also Sprake JHWH
All words in a language, all symbols, all operations that can be performed in the language are contained only within the language itself. That is to say, languages are self-defined; their elements are constituted by other elements of the language not by not-words outside the language. Language is therefore a circular affair. Or, more optimistically, language is helical. It refers to itself endlessly and gets more expressive as it does so as it builds upon itself.
Expressive of what? Of itself of course. This is what makes language so astounding. It makes the most amazing things out of... well out of nothing but sounds and marks. And it envelopes those who use it in a potentially infinite artificial universe. This infinite character of its creation is temporal as well as spatial. No one is sure when language started. Although we know it must have had a beginning in the development of life we call evolution, it nevertheless appears that it has always been there, waiting to be discovered. Before its discovery, nothing is thinkable. And it could exist forever, like viruses continuing in an inanimate state until encountering some organism which has been prepared for it. If language is ultimately annihilated, we will never know about it. There is literally no future, as there is no past or present, without it.
This appearance of eternal existence promotes the idea of language as divine. Indeed, for many it seems to logically necessitate the idea of god itself. And language can make a credible claim to creating the world. But it does so, as in other aspects of the evolutionary process, out of the chaotic material which is already available. The point at which words emerged from the chaos of not-words is indeterminate. But the Book of Genesis is probably as good an account of the event as any. Knuth thinks so anyway.
By analogy with the emergence of DNA, perhaps language appeared as a consequence of a chemical combination as a simple instruction set from which a linguistic edifice could be built. And perhaps it is within this instruction set that the first ‘word’ was chemically contained. The instruction set operated on this strange new chemical object and self-referentially produced other words that congealed around the first, analogously to a planetary system, into an extensive language.
All this is only a theory of course. But its a theory that fits very well with Knuth’s tale of the development of surreal numbers. These numbers are not surreal because of their properties, which are more or less the same as other numbers, but because of their origins. They are produced (or explained, the distinction is largely irrelevant) by a self-referential process in which all numbers are defined in terms of not axioms, or elemental definitions of ‘units’ but in terms of each other.
Knuth lays out two rules in his fictional account of surreal numbers:
Thus surreal numbers simulate the development of all language. They are the result of combinations of yet more primal numbers, which are in turn generated by yet more primal numbers, which may well include the numbers one started with. Circularity for sure but with that distinctive helical twist of increasing complexity characteristic of all languages. The key of course is in the instruction set, the algorithm, which ‘bootstraps’ its own development by ‘positing’ that which it then develops. And what it posits is, quite literally ‘nothing,’ the null set {0,0}.
Knuth’s two rules are exceptionally clever, but they may require a particular mathematical set of mind to grasp fully. The novel is clearly meant to help in easing the reader into that state. Even then the basic self-referentiality of Knuth’s rules may seem disconcerting. Because they are exactly that. Recursive logic shakes the foundations of all foundational thinking. It tears the fundament out of fundamentals.
This might be easier for most folk to appreciate in words rather than numbers. Take the word ‘fact.’ In simple terms ‘a fact is a thing known or proved to be true.’ If one were to follow each of these definitional terms back through the Oxford English Dictionary, and make substitutions along the way, the definition of ‘fact’ that would emerge is as something along the lines of: ‘a fact is that which is not contradicted by any other fact.’
This derived definition is no more or less correct than the initial one. It merely demonstrates the necessary circularity of language, a circularity which the derivation of surreal numbers in the language of mathematics makes obvious as a sufficient condition for all languages. It also indicates why an appeal to facts in an argument always begs the question of what constitutes such a thing.
Knuth’s fictional explication of surreal numbers is based on the original mathematics of John Horton Conway (JHWH, or Jehovah of the novel) who also wrote a dozen or so serious books, as well as at least that many very serious games including the cellular automaton called the Game of Life.
All words in a language, all symbols, all operations that can be performed in the language are contained only within the language itself. That is to say, languages are self-defined; their elements are constituted by other elements of the language not by not-words outside the language. Language is therefore a circular affair. Or, more optimistically, language is helical. It refers to itself endlessly and gets more expressive as it does so as it builds upon itself.
Expressive of what? Of itself of course. This is what makes language so astounding. It makes the most amazing things out of... well out of nothing but sounds and marks. And it envelopes those who use it in a potentially infinite artificial universe. This infinite character of its creation is temporal as well as spatial. No one is sure when language started. Although we know it must have had a beginning in the development of life we call evolution, it nevertheless appears that it has always been there, waiting to be discovered. Before its discovery, nothing is thinkable. And it could exist forever, like viruses continuing in an inanimate state until encountering some organism which has been prepared for it. If language is ultimately annihilated, we will never know about it. There is literally no future, as there is no past or present, without it.
This appearance of eternal existence promotes the idea of language as divine. Indeed, for many it seems to logically necessitate the idea of god itself. And language can make a credible claim to creating the world. But it does so, as in other aspects of the evolutionary process, out of the chaotic material which is already available. The point at which words emerged from the chaos of not-words is indeterminate. But the Book of Genesis is probably as good an account of the event as any. Knuth thinks so anyway.
By analogy with the emergence of DNA, perhaps language appeared as a consequence of a chemical combination as a simple instruction set from which a linguistic edifice could be built. And perhaps it is within this instruction set that the first ‘word’ was chemically contained. The instruction set operated on this strange new chemical object and self-referentially produced other words that congealed around the first, analogously to a planetary system, into an extensive language.
All this is only a theory of course. But its a theory that fits very well with Knuth’s tale of the development of surreal numbers. These numbers are not surreal because of their properties, which are more or less the same as other numbers, but because of their origins. They are produced (or explained, the distinction is largely irrelevant) by a self-referential process in which all numbers are defined in terms of not axioms, or elemental definitions of ‘units’ but in terms of each other.
Knuth lays out two rules in his fictional account of surreal numbers:
“This shall be the first rule: Every number corresponds to two sets of previously created numbers, such that no member of the left set is greater than or equal to any member of the right set. And the second rule shall be this: One number is less than or equal to another number if and only if no member of the first number's left set is greater than or equal to the second number, and no member of the second number's right set is less than or equal to the first number."
Thus surreal numbers simulate the development of all language. They are the result of combinations of yet more primal numbers, which are in turn generated by yet more primal numbers, which may well include the numbers one started with. Circularity for sure but with that distinctive helical twist of increasing complexity characteristic of all languages. The key of course is in the instruction set, the algorithm, which ‘bootstraps’ its own development by ‘positing’ that which it then develops. And what it posits is, quite literally ‘nothing,’ the null set {0,0}.
Knuth’s two rules are exceptionally clever, but they may require a particular mathematical set of mind to grasp fully. The novel is clearly meant to help in easing the reader into that state. Even then the basic self-referentiality of Knuth’s rules may seem disconcerting. Because they are exactly that. Recursive logic shakes the foundations of all foundational thinking. It tears the fundament out of fundamentals.
This might be easier for most folk to appreciate in words rather than numbers. Take the word ‘fact.’ In simple terms ‘a fact is a thing known or proved to be true.’ If one were to follow each of these definitional terms back through the Oxford English Dictionary, and make substitutions along the way, the definition of ‘fact’ that would emerge is as something along the lines of: ‘a fact is that which is not contradicted by any other fact.’
This derived definition is no more or less correct than the initial one. It merely demonstrates the necessary circularity of language, a circularity which the derivation of surreal numbers in the language of mathematics makes obvious as a sufficient condition for all languages. It also indicates why an appeal to facts in an argument always begs the question of what constitutes such a thing.
Knuth’s fictional explication of surreal numbers is based on the original mathematics of John Horton Conway (JHWH, or Jehovah of the novel) who also wrote a dozen or so serious books, as well as at least that many very serious games including the cellular automaton called the Game of Life.
posted by The Mind of BlackOxford @ February 19, 2020
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