On behalf of JohnnyL, I post the following letter he sent to me.

I find it interesting and provoking (but have no time myself for going into the details), what do you guys say?

I find it interesting and provoking (but have no time myself for going into the details), what do you guys say?

Quote:Hi, Henryk!

I'm one of occasional readers of tetration forum, I have mathematical education, but do not work as mathematician. As many other people, I noticed "new operation" at school. It is nice to see that so many people decided to investigate the question, but... below, you may find my opinion about the tetration theme. Certainly, I may be wrong. In any case, I do not pretend on the only truth and I do not intend to be rude or hurt you or anyone of tetrafolk

A. Short summary.

1. The analogy between definitions of multiplication by addition and power by multiplication had been noticed.

2. This analogy uses rule of "right braces". The rule has no clear interpretation and seems to be "arbitrary".

3. New operations had been constructed recursively by the noticed analogy for integer values.

4. Numerous kinds of extension for real and complex values had been constructed for first such operation - "tetration".

5. Not all proofs had been discovered to achieve mathematical correctness of corresponding solutions (because of proof complexity).

6. The equivalence of proposed solutions is unknown (because of question complexity).

7. A behavior of known solutions are quite complex.

8. Applications (practical or theoretical) of constructed operations and corresponding functions are absent (except Ackermann functions).

B. Conclusion.

The noticed analogy leads to operations and functions which are too complex to be useful. It is unclear why tetration is so different from exponentiation.

C. The question: Why so "natural" analogy leads to so "scanty" results?

Answer.

In my opinion, this analogy leads to poor results, because from very beginning there is no actual understanding of its essence. As consequence, obtained results have no reasonable interpretation and therefore they cannot express any useful properties we can understand and apply.

D. Yet another question: If the analogy is "false" and operations are not "natural" then what is reasonable analogy?

Answer.

There is different approach to the initial idea of extending arithmetical operations. This approach leads to different set of functions, but this set includes tetration. What is important – the approach is not based on arbitrary choice of arrangement of braces in formal expression, and allows, more or less, clear interpretation "what for" on every step of construction.

There are many things around us, too many. We cannot think about all of them simultaneously. To think about specific theme we need concentrate on several things, then on several other things, then on other and so on. Thus, our thinking organized and rough structure of this organization is sequence. Model for sequence is set of natural numbers. This is why this set is so useful in practice. Sequence gives us operation "get next element" i.e. successor function. Considering things we go forward on our sequence of thoughts - this gives us addition. Sequencing and addition is useful, but is not enough - we need consider things on different level of details to overcome complexity. To change level of details we can by grouping of things. And grouping leads us to multiplication. Group things, and then group groups of things and so on - this leads us to power operation. It is important to note, that the less level of details, the high level of abstraction. On high level of abstraction we think in natural numbers of corresponding units and take no care on more detailed units. Consider example. If you have only several apples - it's one style of life (quite risky) - you think about apples in apples. If you have several thousands of apples - this is completely another style of life (your life is more complex, because you need to trade, probably) - you think about apples, say, in containers (and no matter if you lose pair of apples). If you manage tens of millions of apples - you think about apples even in more abstract units (and no matter if you lose one or two containers). Actually, lives of people can be characterized by exponent (logarithm, abstraction level) of things they think about. Note, that exponent - new notion (relative to group) and it characterizes level in the hierarchy of complex grouping.

Well, the joke begins. Suppose we have 10^m apples and m - so much, that we cannot say how much exactly. In this case, exact value, actually, is not important, but it is important, that m ~ 10^n. Thus, we have scaling: 10^(10^n). Note, m - level in grouping hierarchy (pockets - by 10, boxes - by 10 of 10, containers - by 10 of 10 of 10 and so on). What means n in real life? Answer is funny: In real life, actually, we have no corresponding notion, because does not need it, because even number of particles in observable Universe is about 10^80 only. Well, but we are mathematicians, we may consider case n ~ 10^k, i.e. scaling 10^(10^(10^k)). What notion should express k? It's hard to say. Each level of scaling will add new notion, furthermore, this set of notions will give us basis to introduce even new notion - height of power tower. Can you say what sense has this height? At the moment, I cannot answer correctly. But I can say that in such way we can construct infinite set of notions and scaling functions through which can view the world, can talk on suitable level of abstraction - this is the essence of them. But all of these functions will have no sense, even for natural numbers, because in practice we do not use corresponding levels of abstraction and, I'm afraid, will never use... These scalings are instruments for another level of perception-consciousness, for another realm of reality.

Nevertheless, I suppose, constructed set of scalings may have some applications to number theory, because it proposes characteristics of numbers, reflecting its internal organization; consideration of numbers as hierarchical structures may be interesting by itself. Or this set may be used, probably, in theory of probability, because this theory also tries to abstract from not important details and get the whole picture.

In any case, in my opinion, worth results may come only from reasonable goals, not from abstract transformations of formulae. Therefore, I think, efforts must be applied to answer the question: What for we need solution of this task? I know, you consider tetration only as working object to develop methods of continuous iteration and so on - this changes matter, but not completely.

Sincerely yours,

Jhonny.