The Richard Dedekind book from 1888 that contains New Math in all its original glory is available in English for free.
The following was typed into latex and then converted to pdf:
The Nature and Meaning of Numbers starts on page 26 of the pdf file above. This is numbered page 21 of the latex created pdf. Note this is a recent pdf from Latex and therefore has recent pdf features. This makes it the better one to use. However, page references do not correspond to the printed version from Dover or to the scanned version.
Project Gutenberg homepage on book
Google books scanned in a 1909 printing of the Wooster Woodruff Beman translation into English.
Page 57 of the above if one downloads the pdf there is where “The Nature and Meaning of Numbers” starts.
One can also buy the paperback from Dover to carry around.
This book should be studied closely, starting with this second essay, not the first. This is something to study as a Confession of Arithmetic Faith. Drills are works. Dedekind is a catechism that is worth learning almost to the point of memorizing parts of it.
Students in the primary grades need something to learn by heart that represents the faith component of arithmetic. Dedekind is exactly that. Faith is faith in something. For that one has to have a creed that is written down to memorize. Dedekind’s essay is a little long for most of us to memorize, although it is possible for some.
However, we should study it over many times just short of memorizing it. This will then become the faith part of faith and works in learning arithmetic. The study drills are the works part. Those are already available in abundance for the traditional parts of arithmetic.
Neither faith nor works alone are sufficient for learning arithmetic. It takes both. Faith without works is sterile and works without faith lacks meaning and inspiration. Both together are needed to learn arithmetic. This was realized by the creators of 1960s era New Math. They simply did not have the vision of learning arithmetic from the Peano Axioms through the place value notation algorithms as a unified logical whole. That is what 1960s New Math failed. It lacked a Confession of Faith that could serve as inspiration, guide and to unify the faithful.
The central part of the Dedekind essay are statements about what he calls chains. These are what we call closed sets under succession. So all numbers from 2 onwards is closed under succession because the successor of each member of the set is in the set. We call this the tail of 2 and it includes 2.
Dedekind goes through the properties of closed sets, he calls them chains, in detail. He does one small lemma after another. These have short proofs. These are not something to be skipped but to be studied closely. One does not know and has not learned the natural numbers until one knows these lemmas thoroughly.
Doing addition first to make it easy doesn’t work. This is because the proofs in Dedekind of these lemmas are the real explanations of the meaning. Proofs of the same lemmas using addition would be more cumbersome and less explanatory. This is because they would be putting the cart before the horse.
The meaning of arithmetic starts with the meaning of sets closed under a function like successor. With certain adaptations, these lemmas also work for predecessor applied to the natural numbers. We need to make the change that a set is closed under predecessor if for each number in the set, the predecessor is in the set, if the predecessor exists. This is because zero has no predecessor. The sets closed under predecessor are the initial segments or head segments. If we start from zero as the natural numbers they are sets without a gap that include zero and include every number from zero up to some maximum.
The art of Dedekind is to define these things in order so it is not circular. The above explanation used many concepts to explain it that are familiar but really come later in logic.
A gapless set can be defined as a set that contains the predecessor of each element except for one, and the successor of each element, except for possibly one element. If it has an element with no successor it is finite. The initial segments are finite gapless sets that contain zero. The tails are infinite gapless sets.
The logic of the natural numbers comes from understanding these 3 types of sets. Counting and addition are understanding functions between such sets. For example, if we have a head segment from 0 to i and then a gapless set from i’ to w and there is a 1 to 1 correspondence between a head segment from 0 to j and i to w, then w is the sum of i and j.
The zero of the head segment from 0 to j maps to the i. This lets us count from 0 to w. This gives us a sum. This is the logic of addition from one point of view.
This approach corresponds to taking two number lines. On the first number line we have 0 to i. We put the 0 of the second number line over the i of the first. Under the j of the second number line is the sum, w.
This type of explanation is what is in the book “Pre-Algebra New Math Done Right Peano Axioms”.
The Edmund Landau book Foundations of Analysis and books that make addition an axiomatic operation such as Peano did all fail to teach the inner logic of addition of natural numbers described above. Such books fail to be a guide to how to teach natural numbers and their addition using the number line. Without such an understanding the number line falls short in teaching the concepts of natural numbers and addition. Filling this gap is what the book “Pre-Algebra New Math Done Right Peano Axioms” does.