graph the answers to n^n=a where a=d^(e^f) from the sets {2,3,4}, {2,3,5} and {3,4,5}
Friday 27 December 2013
The light in a bulb
graph the answers to n^n=a where a=d^(e^f) from the sets {2,3,4}, {2,3,5} and {3,4,5}
Friday 20 December 2013
Functional Type Shifting Patterns - FAQ
Functional Type
Shifting Patterns
FAQ for mathematicians and scientists
I am happy with the
nice and easy ( … ) Why did you have to complicate it when the numbers produced
are too esoteric to be useful?
My answer is like this, if mathematicians accept Graham’s number, then it is reasonable to accept compositions of hyperoperations. The ironic thing is, understanding compositions of hyperoperations seems to make Graham’s number look even more unreal. As far as the linear notion of ellipsis ( … ) this is fine and the best way to do most of the normal problems of maths, so mathematicians don’t need to fear about interference with the usual techniques and notations.
But the formulae are useless as far as I can see.
They are mostly too unwieldy for the arithmetic used in number theory. Some borderline examples are laddered exponents (Conway and Guy) where comparisons can sometimes be made. The more complicated examples exist as well-defined concepts, and are constructible, and information flow patterns can be compared. They have their own internal logic and character, and can be studied from that point of view. The logic uses the natural continuation of the principle of iterated exponentiation on smaller natural numbers that leads to tetration and so on. The patterns are number classes, and the visually displayable patterns form a metaclass of classes of numbers. The number classes are interesting to understand and imagine, and usually one has the feeling of peering into math infinity via these number representations.
Is it math or pseudomath?
If you are a strict ultrafinitist and think that numbers beyond a googol or a googolplex are silly to consider there’s no need to take an interest in the topic. If you like big numbers you probably could like the pattern concept. If you are a set theorist, it is interesting to see a new kind of understanding of hyperoperations beyond the already known recurrence relations and current information on Wikipedia about hyperoperations. In a way, the patterns are a conceptual exploration of the so-called Frivolous Law of Arithmetic which says that almost all numbers are very, very large.
Are the patterns arbitrary with respect to their geometric form?
No. To fully explain the specific details about the patterns would take a lot of explanation, and the settled-upon form is the most common sensical and extensible.
Are there new results to do with these patterns?
Yes. The new results were discovered from a careful analysis of the properties of these patterns.
Should the patterns be taught at all?
They could be taught to postgraduate students with an interest in the foundations of maths.
Why do you need 19 animations?
The 19 animations cover aspects such as syntactical components, folding patterns, computational pathway, syntactics of pattern extensibility, pattern-formula correspondence, number class boundaries, transitional sequences, coordinate systems, relation to epsilon numbers and compositions of patterns.
My answer is like this, if mathematicians accept Graham’s number, then it is reasonable to accept compositions of hyperoperations. The ironic thing is, understanding compositions of hyperoperations seems to make Graham’s number look even more unreal. As far as the linear notion of ellipsis ( … ) this is fine and the best way to do most of the normal problems of maths, so mathematicians don’t need to fear about interference with the usual techniques and notations.
But the formulae are useless as far as I can see.
They are mostly too unwieldy for the arithmetic used in number theory. Some borderline examples are laddered exponents (Conway and Guy) where comparisons can sometimes be made. The more complicated examples exist as well-defined concepts, and are constructible, and information flow patterns can be compared. They have their own internal logic and character, and can be studied from that point of view. The logic uses the natural continuation of the principle of iterated exponentiation on smaller natural numbers that leads to tetration and so on. The patterns are number classes, and the visually displayable patterns form a metaclass of classes of numbers. The number classes are interesting to understand and imagine, and usually one has the feeling of peering into math infinity via these number representations.
Is it math or pseudomath?
If you are a strict ultrafinitist and think that numbers beyond a googol or a googolplex are silly to consider there’s no need to take an interest in the topic. If you like big numbers you probably could like the pattern concept. If you are a set theorist, it is interesting to see a new kind of understanding of hyperoperations beyond the already known recurrence relations and current information on Wikipedia about hyperoperations. In a way, the patterns are a conceptual exploration of the so-called Frivolous Law of Arithmetic which says that almost all numbers are very, very large.
Are the patterns arbitrary with respect to their geometric form?
No. To fully explain the specific details about the patterns would take a lot of explanation, and the settled-upon form is the most common sensical and extensible.
Are there new results to do with these patterns?
Yes. The new results were discovered from a careful analysis of the properties of these patterns.
Should the patterns be taught at all?
They could be taught to postgraduate students with an interest in the foundations of maths.
Why do you need 19 animations?
The 19 animations cover aspects such as syntactical components, folding patterns, computational pathway, syntactics of pattern extensibility, pattern-formula correspondence, number class boundaries, transitional sequences, coordinate systems, relation to epsilon numbers and compositions of patterns.
Online Papers are on the Eretrandre website ( "Mike Smith" )
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