Paragraph 19 is where Euler defines the natural numbers as adding one starting from zero.
19. In the same manner, therefore, as positive numbers are incontestably greater than nothing, negative numbers are less than nothing. Now, we obtain positive numbers by adding 1 to 0, that is to say, 1 to nothing; and by continuing always to increase thus from unity. This is the origin of the sequence of numbers called natural numbers; the following being the leading terms of this sequence:
and so on to infinity.
But from this progression, Euler derives no definition of addition or multiplication or proofs of the associative or commutative laws of addition. This waited until Grassmann 1861 did it using induction and recursion.
68. When a number, as 7, for instance, is said not to be divisible by another number, let us suppose by 3, this only means, that the quotient cannot be expressed by an integer number; but it must not by any means be thought that it is impossible to form an idea of that quotient. Only imagine a line of 7 feet in length; nobody can doubt the possibility of dividing this line into 3 equal parts, and of forming a notion of the length of one of those parts.
Euler later gives the rules for fraction procedures but without using this “definition” of fractions in geometry at all. For example Euler paragraph 94.
Thus Euler leaves a huge gap from his geometry to justify fractions to his procedural rules for manipulating fractions. The same gap exists in modern K-12 teaching. The gap is filled by dropping the geometry and using algebraic reasoning and logic to define fractions and their rules. Area and length are then applications of fractions not their definition.
Modern K-12 teaching has the identical fallacies and misdirections as in Euler 1765 Elements of Algebra. Those were fixed in the 19th century by Grassmann, Dedekind, Peano and others. That is New Math. There is no going back. Teaching Euler’s 1765 fallacies all over again in 2012 does not help. It is the same as simply replaying a Khan Academy video that is known to contain errors. But this time it is done by mathematicians who should know better.
Telling students that the rules for manipulating fractions come from geometry is a deception in the middle of the progression from natural numbers to the rules of rational numbers. It cements not teaching them the Peano Axioms and it is a road block to their understanding algebra and the structure of math that gives rise to algebra. For this reason it can be called the Central Lie in Arithmetic. It is related to the tower of deceptions in teaching natural numbers. Those tend to be lies of omission rather than of commission.
Math moved on from Euler’s 1765 Elements of Algebra. But math teaching did not. Even though Grassmann was an elementary and high school teacher and his taking the next step forward came directly out of his realizing the gaps in Euler.
Because of Euler’s great prestige as a mathematician, his approach to teaching elementary algebra persists. Thus Euler’s dead hand lies heavy still on elementary math education. Even though Grassmann, Dedekind and Peano and others have found and filled the gaps. Despite this the identical fallacies and outright deceptive teaching in Euler 1765 is persisted in.
Students come across these same logical chasms in 2012 just as Grassmann and his brother perceived them in the years leading up to 1861. There is no road out of the Euler Alps except over the bridges built by Grassmann, Dedekind and Peano.
The only alternative to teaching the outright deception that the rules of fractions come from some geometric definition is to state explicitly that they do not and to say they must come at least in part as new mathematical creations. We can use them in geometry but we do not really derive the rules for fractions from any geometric definition. Anything else is perpetuating a fallacy that has devastating consequences for understanding the structure of arithmetic and thus the concepts of arithmetic and algebra both.
Until students are taught to use the bridges built by Grassmann, Dedekind and Peano out of the Euler Alps, students will continue to die like Hannibal’s elephants.
==Summer Sale on e-book
Pre-Algebra New Math Done Right Peano Axioms
is on sale for 2.99 at e-book vendors.