Norman attempted to define some new"operations" by "induction". In fact it is usually described as "by analogy". This is a common problem , when a term supposedly rigorous is used synonomously, extending the use of the term, but creating varying degrees of confusion. This is a normal etymological language development, how language grows and murphs, and begs the question: what is the value of logical precision in pragmatic terms?
At one time the term "Numeros" meant quantity, count of magnitude, and arithmos meant figurate magnitude, now we subsume the 2 in a hird concept called "number".
Equally recursion was recognised in the "latin" expression of it, but "iter" or "again" is the subjective latin experience of it. For a long time recursion developed a monstrous reputation of complexity, which iteration did not; so i replace the notion of recursion with iteration.
Similarly Deduction in the early 18th century, and the symbolic logic developed an uneven reputation of infallibility, nd all victorious application, so that much that we hold dear by induction is not readily understood to be arrived at by a slow process of validation by previous results. All seek the fame of deducing that which is new to human insight, but often is just dressed up common sense.
When Benoit Mandelbrot used the most powerful iteration tools men have ever devised to discobrt yjr Mandelbrot set, it was not as you would imagine a straightforward event, but instead it was the most arse over tit, backward and reverse thinking that one could imagine, and it was defined recursively and on vectors, which at the time were thought of as complex numbers.
The point is simply this, Fractals, almost self symmetry, and iteration are the very things my experiential continuum, my "reality" consists in, and yet to get to this simple, everyday pragmatic observation, i have been led down the most convoluted mish mash of utter "nonsense" it has eer been my privilege to experience. But i guess, that to dig away the foundational compacted earth and expose the roots is the most terrifying thought for a system, for how will it survive?
A system will survive, providing in its iterative scheme it leaves room for seed production, and the seeds of this monstrous plant are ot all survivable. The simplest seed development is likely to be the longest survivor and the generator of new adapted systems. These adaptations will of course develop through iteration and induction. This is my induction.
Zn+1 = z2n + Z0 for n= 0,1,2,3,4,…; Z0 € [-2,0) for example, where n are vectors with direction e'; and e'' + e' = 0 vector so "-" is an abbreviation for e''
One aspect of this fractality is the inevitable tautology at the beginning of the process. This tautology, especially when it is written down has the formal sense of "completeness". But this completeness is not "derived" it is arrived at by trial and error, by tinkering with the notation until it gives the right results.
Deduction can take one only so far. The nature of reality requires oe to take leaps of faith and try different things out until they succeed. This principle is so prevalent it actually occurs in many different disguises: for example, when trying to balance an unstable object. This principle is enshrined in Quantum mechanics as the uncertainty principle, in dynamics as the stability principle, in kinematics as the equilibrium principle, and in mathematic as the convergence principles, and it underpins the abstract notion of continuousness in the very face of these regios of instability of continuum. It is essential for the principle of least action.
Trial and error is the alternative description to an iterative process that tends to the region of least action, greatest stability, final equilibrium, and induction is that validation of achieving that goal once again, every time one looks.
The analogy between induction and iteration is complete, stable and testable…utterly empirical.