The blog Dy/dan discusses an additive recursion formula. It is apparent from the discussion that they are familiar with recursive and explicit as terms.
It is clear from the discussion that the Peano Axioms are not any harder. Addition itself is defined recursively in terms of counting is a conceptual jump not present in this discussion.
The recursion is
z = 9
z[n+1] = z[n] + 2
As an explicit formula we can write
z[n] = 9 + 2 * n
Addition is defined recursively as
x+0 = x
x+y’ = (x+y)’
These are the right additive identities.
Here ‘ means next or successor. It can also be read as count one. So 2′ means count one starting from 2. That gets you to 3.
So the right additive identities state
Count 0 from a number keeps you at the number.
The sum of a number and zero is the number.
Given two addends, count 1 from the right addend gets you to one more than the sum of the two starting addends.
Given two addends with a sum, count 1 from the right addend equals counting one from said sum.
All of these statements are as easy to understand as the recursion of adding 2 at a time starting from 9 being discussed at Dy/dan.
Addition is defined recursively. This means we can’t reduce addition to a formula that itself uses addition and multiplication.
The recursion discussed at Dy/dan
z = 9
z[n’] = z[n] + 2
has the explicit solution
z[n] = 9 + 2 * n
The explicit form uses addition and multiplication. So the recursion that defines addition has no explicit form in this sense.
This is an important point to point out to build an understanding of the difference between defining addition recursively in terms of counting by one and of solving this recursion after addition and multiplication have been defined.
This conceptual discussion is absent in the dy/dan thread. However, it could be learned by these teachers and then taught to students in K-6. However, this would require overcoming the apathy potential barrier. That requires quantum mechanics and time.
Even though these teachers can understand and engage in this discussion of recursion given that addition and multiplication are already defined, the conceptual step to defining addition recursively in terms of counting and then defining multiplication recursively in terms of addition is nowhere present in their discussion.
This would take a huge step for the teaching profession and perhaps for individuals as well. My books cover this huge step divided into baby steps. More numerical examples of the Peano Axioms are in my books than likely exist in all other materials combined.
You can read them with a free computer ap without buying the e-reader. They are well formatted and were never pdf or word docs at any stage in preparation. They were uploaded to e-vendor as pure html.
The discussion at these two blogs illustrates the need to explicitly solve the problem and then explicitly discuss it in detail down to the final nit picks. That the solution of the recursion problem at Dy/dan requires that addition and multiplication are already defined whereas the definition of addition by recursion obviously can not do so is something that has to be pointed out explicitly in writing.
This is why the anti-textbook tendency of progressive math is wrong. This has to be spelled out in words and symbols and with examples for teachers, students, tutors, college math ed profs, and parents to get it.