01/21/2008, 02:08 AM

A remark Gottfried made about the matrix transpose notation got me thinking about notation in general. I really hate arrow notation for hyper-operators, but it is also the most common and well-known notation. I also don't like dot or prime notation (f') for derivatives, I perfer Leibniz notation the best (although some people confuse it with division).

So what is the relationship that notation has to our feelings? Well, when we want to express something, and communicate an idea, the failure to do so can make us feel bad. On the other hand, the asthetics of a particular notation can also make us feel bad. So if we were to express ideas in a notation that no one understands, but looks good, then we would feel good, and other people would suffer (this seems selfish). Conversely, if we were to use common notations that make us feel bad and help other people understand, then we suffer, but others will be able to understand (this seems more selfless). I think in the grand scheme of things, it is better for one person suffer, now, so that hundreds (or billions, I don't know) of people can understand what is written for a long time in the future.

A while back I wrote a questionnaire that I never sent. I was going to email it to everybody I knew studying tetration. Since this forum was started, many of the questions I had, are not worth asking now. Some of the questions remain, however, and I don't have a recommendation for them. Hyper-operator notation is most commonly expressed with Knuth's arrow notation (and the Bromer-Mueller arrow can help extend Knuth's arrow notation to mixed hyper-operators). The notation system developed by Barrow, Shell, Thron, and several others is already apt at expressing nested exponentials, but there is no standard for which letter is used (although I perfer T). The notation for iteration usually involves some kind of super-script (with optional decorators). Tetration is usually written with either iterated exponential notation (combination of exponential notation, and iteration notation), nested exponential notation, hyper-operator notations, or the notation exclusively devoted to tetration: Mauerer's left-superscript notation.

With all of these things worked-out, I still would like to know:

In general I feel that notations should be extensible, for hyper-operator notations, you should be able to write n instead of ... and a bracket (both Arrow and Box notations provide this). There should also be provisions for commonly used inverse functions (only box notation provides for hyper-logs and hyper-roots). And there should not be any confusion with existing notations (This applies to Knuth/Bromer-Mueller confusion, as well as E/Sigma confusion). Although it could be said that left-superscript notation is too confusing for dislexic people. Lastly, whatever notations are used in the FAQ, the collection should be consistent, ideally we should use Occam's razor to decide notations, but history has a way of circumventing this.

One simplification that was sort of an epiphany was Jay D. Fox's recomendation for the nth tetrate, which in my mind consolidated both iteration and tetration terminology. In this post, we came to the conclusion that for any binary operation called X-ation (just as an example) the one-variable function can be called the nth X-ate or the other one-variable function can be called an X-ational. Although each term must be assigned on a case-by-case basis. In the case of hyper-operations, we can assign all of them at the same time, since they are all defined by the same pattern in the first place. This not only simplifies tetration terms, but also the terms for pentation and beyond. This even makes it easier to talk about higher operations, like "iterated pentationals", and "nested hexationals".

I suppose I'm really thinking about the FAQ while I'm doing this. Anyone can use any notation, in any paper, in any journal (UXP is proof of this). So what I'd really like to know, is not what I should be using (I will only use T, not E), but what would be the most common, accepted, unambiguous terms and notation to be used in the FAQ so that minimal confusion will result. I personally think that the most confusing aspect of all of these notations is iteration notation.

I have seen several notations used for iteration:

The last example reminds me of Large numbers on Wikipedia. This article and related articles use section notation extensively (incomplete infix expressions are called sections in Haskell). Although this is quite intuitive for some people, there are not many places that talk about this kind of notation. Sections are the easiest way of writing powers and exponentials (if you don't want to use "pow" and "exp"), but there is an implicit convention that this is a shortcut for a function mapping, and as such, it should be treated as a notation rather than something that is assumed the reader knows. However, I would argue that section notation is far more extensible than using mnemonics like "pow", "exp", "spow", "sexp", since these can all be written \( ({\uparrow}n) \), \( (b{\uparrow}) \), \( ({\uparrow}{\uparrow}n) \), \( (b{\uparrow}{\uparrow}) \) respectively. What bothers me about this is that if we were to be consistent (which is something I am very concerned about), then we should also use this notation for exponentials, which up to this point have been written with "exp".

One advantage of using section notation is that this is the only way that you can represent hyper-logarithms and hyper-roots with Knuth's arrow notation, currently. Although you could represent them in box notation very easily, using Knuth's arrow notation, this would be accomplished through \( (b{\uparrow}{\uparrow})^{-1}(x) = \text{slog}_b(x) \) and \( ({\uparrow}{\uparrow}n)^{-1}(x) = \text{srt}_n(x) \) respectively. With this notation we could even talk about hyper-logarithms and hyper-roots in the FAQ, since we would be using the standard arrow (rather than box notation, or mnemonic notation, which would have to be introduced). I believe that this is very consistent, and would cause the least confusion.

To overview, the notations that are available for expressing tetration-related ideas range from boxes, arrows, chained-arrow (Conway-Guy), map-arrow ("\mapsto" looks different than "\rightarrow"), symbolic (for ssqrt), mnemonic (srt/slog/spow/sexp), left-superscript, towers (nested exponentials), iteration, and sections, to the more obscure notations based on specific authors, like uxp and Campagnolo's \( b^{[n]}(x) \) for iterated exponentials. But for the FAQ, I think we should limit the notations to: Arrow, Iteration, Section, and Tower notations. These are enough.

Andrew Robbins

PS. Attached is the original questionnaire I never sent.

So what is the relationship that notation has to our feelings? Well, when we want to express something, and communicate an idea, the failure to do so can make us feel bad. On the other hand, the asthetics of a particular notation can also make us feel bad. So if we were to express ideas in a notation that no one understands, but looks good, then we would feel good, and other people would suffer (this seems selfish). Conversely, if we were to use common notations that make us feel bad and help other people understand, then we suffer, but others will be able to understand (this seems more selfless). I think in the grand scheme of things, it is better for one person suffer, now, so that hundreds (or billions, I don't know) of people can understand what is written for a long time in the future.

A while back I wrote a questionnaire that I never sent. I was going to email it to everybody I knew studying tetration. Since this forum was started, many of the questions I had, are not worth asking now. Some of the questions remain, however, and I don't have a recommendation for them. Hyper-operator notation is most commonly expressed with Knuth's arrow notation (and the Bromer-Mueller arrow can help extend Knuth's arrow notation to mixed hyper-operators). The notation system developed by Barrow, Shell, Thron, and several others is already apt at expressing nested exponentials, but there is no standard for which letter is used (although I perfer T). The notation for iteration usually involves some kind of super-script (with optional decorators). Tetration is usually written with either iterated exponential notation (combination of exponential notation, and iteration notation), nested exponential notation, hyper-operator notations, or the notation exclusively devoted to tetration: Mauerer's left-superscript notation.

With all of these things worked-out, I still would like to know:

- Opinions on Romerio's box notation.

- Opinions on combining (Knuth arrow) and (Bromer-Mueller arrow).

- Opinions on the terms hyper-operator / hyper-operation.

- Opinions on the notation of iterated powers.

- Opinions on the notation of auxiliary super-roots.

- Opinions on the notation of hyper-logarithms and hyper-roots.

- Opinions on the Greek letter used for nested exponentials (E or T).

- Opinions on the "decorators" used with iteration notation.

- Opinions on Mauerer's left-superscript notation.

In general I feel that notations should be extensible, for hyper-operator notations, you should be able to write n instead of ... and a bracket (both Arrow and Box notations provide this). There should also be provisions for commonly used inverse functions (only box notation provides for hyper-logs and hyper-roots). And there should not be any confusion with existing notations (This applies to Knuth/Bromer-Mueller confusion, as well as E/Sigma confusion). Although it could be said that left-superscript notation is too confusing for dislexic people. Lastly, whatever notations are used in the FAQ, the collection should be consistent, ideally we should use Occam's razor to decide notations, but history has a way of circumventing this.

One simplification that was sort of an epiphany was Jay D. Fox's recomendation for the nth tetrate, which in my mind consolidated both iteration and tetration terminology. In this post, we came to the conclusion that for any binary operation called X-ation (just as an example) the one-variable function can be called the nth X-ate or the other one-variable function can be called an X-ational. Although each term must be assigned on a case-by-case basis. In the case of hyper-operations, we can assign all of them at the same time, since they are all defined by the same pattern in the first place. This not only simplifies tetration terms, but also the terms for pentation and beyond. This even makes it easier to talk about higher operations, like "iterated pentationals", and "nested hexationals".

I suppose I'm really thinking about the FAQ while I'm doing this. Anyone can use any notation, in any paper, in any journal (UXP is proof of this). So what I'd really like to know, is not what I should be using (I will only use T, not E), but what would be the most common, accepted, unambiguous terms and notation to be used in the FAQ so that minimal confusion will result. I personally think that the most confusing aspect of all of these notations is iteration notation.

I have seen several notations used for iteration:

- \( f_n(x) \) -- Many older texts on iteration.

- \( f^n(x) \) -- 90% of modern texts on iteration.

- \( f^{(n)}(x) \) -- Galidakis, can be confused with derivatives.

- \( f^{[n]}(x) \) -- Campagnolo et.al., and myself.

- \( f^{<n>}(x) \) -- Aldrovandi.

- \( f^{\circ n}(x) \) -- Trappmann.

- \( f(n, x) \) -- some people

- \( f(x, n) \) -- some people

- \( (f \circ)^{n}(x) \) -- no one, but it makes use of sections \( (a \times) \ :\ b \rightarrow a \times b \).

- \( (f {\uparrow}{\circ} n)(x) \) -- no one, but uses the Bromer-Mueller arrow.

The last example reminds me of Large numbers on Wikipedia. This article and related articles use section notation extensively (incomplete infix expressions are called sections in Haskell). Although this is quite intuitive for some people, there are not many places that talk about this kind of notation. Sections are the easiest way of writing powers and exponentials (if you don't want to use "pow" and "exp"), but there is an implicit convention that this is a shortcut for a function mapping, and as such, it should be treated as a notation rather than something that is assumed the reader knows. However, I would argue that section notation is far more extensible than using mnemonics like "pow", "exp", "spow", "sexp", since these can all be written \( ({\uparrow}n) \), \( (b{\uparrow}) \), \( ({\uparrow}{\uparrow}n) \), \( (b{\uparrow}{\uparrow}) \) respectively. What bothers me about this is that if we were to be consistent (which is something I am very concerned about), then we should also use this notation for exponentials, which up to this point have been written with "exp".

One advantage of using section notation is that this is the only way that you can represent hyper-logarithms and hyper-roots with Knuth's arrow notation, currently. Although you could represent them in box notation very easily, using Knuth's arrow notation, this would be accomplished through \( (b{\uparrow}{\uparrow})^{-1}(x) = \text{slog}_b(x) \) and \( ({\uparrow}{\uparrow}n)^{-1}(x) = \text{srt}_n(x) \) respectively. With this notation we could even talk about hyper-logarithms and hyper-roots in the FAQ, since we would be using the standard arrow (rather than box notation, or mnemonic notation, which would have to be introduced). I believe that this is very consistent, and would cause the least confusion.

To overview, the notations that are available for expressing tetration-related ideas range from boxes, arrows, chained-arrow (Conway-Guy), map-arrow ("\mapsto" looks different than "\rightarrow"), symbolic (for ssqrt), mnemonic (srt/slog/spow/sexp), left-superscript, towers (nested exponentials), iteration, and sections, to the more obscure notations based on specific authors, like uxp and Campagnolo's \( b^{[n]}(x) \) for iterated exponentials. But for the FAQ, I think we should limit the notations to: Arrow, Iteration, Section, and Tower notations. These are enough.

Andrew Robbins

PS. Attached is the original questionnaire I never sent.