Could Euler do conceptual thinking?

Euler’s 1765 Algebra is very poor in conceptual thinking on whole numbers, addition of whole numbers and fractions.  It is worse than Euclid in some ways.

Euler was very good at procedural thinking.  He was a master at that. But at the concepts of elementary math, he was not so good.

If schools teach students in the Euler way using Euler type explanations, they won’t learn the 19th century answers as to what natural numbers are or what addition of natural numbers is or what fractions are.

Just teaching students to think like Euler doesn’t give them the right answers or to find the right answers on what natural numbers are or what addition of them is.

These things have to be taught.  Even if schools taught their students to be Eulers, they still couldn’t self discover what natural numbers are or the definition of addition of natural numbers using the right additive identities and recursion.

Some parts of the core standards are New Math concepts like a function as a set of ordered pairs in which no two have a common first element and different second element.  Euler struggled with the definition of function, but did not arrive at the 19th century definition with assurance.

Just teaching students how to think will not teach them 19th century abstractions or techniques.  Those have to be taught as specific knowledge.

What is taught for natural numbers and addition and fractions is wrong or at least incomplete.   These ideas run into problems and gaps.  They don’t really add up.  Just like Euler’s attempts in his 1765 Algebra book.   He makes some assertions, waves his hands, but accomplishes nothing on the foundations of natural number or addition.

Read Euler, he doesn’t know how to explain it.

If the students can’t self discover that what is taught about these is wrong or incomplete, then it shows they are not being taught conceptual thinking.




About New Math Done Right

Author of Pre-Algebra New Math Done Right Peano Axioms. A below college level self study book on the Peano Axioms and proofs of the associative and commutative laws of addition. President of Mathematical Finance Company. Provides economic scenario generators to financial institutions.
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7 Responses to Could Euler do conceptual thinking?

  1. Paul Salomon says:

    This is just completely outrageous. Euler was one of the greatest mathematician’s that ever lived, and to claim that he didn’t understand arithmetic is completely ridiculous. If every student had the mathematical mind of Euler, America would have the greatest mathematical culture in the history of the Earth. Don’t be ridiculous.

    You may prefer a certain type of rigor or foundation to arithmetic, but that doesn’t mean it’s the only way to understand things or be mathematical in the ways that are truly important.

    • Have you read the first few pages of the Euler 1765 Algebra book I linked to?

      • Paul Salomon says:

        Maybe (just maybe) I can agree that Peano and Dedekind and Euler’s successors (no pun intended) did a better job building out foundations for math. OK. I mean, Euler was working on the problem more than a century earlier, so what do you expect? It doesn’t invalidate Euler’s work. Do you hate Euclid’s Elements as well? It was written more than 2 millenia before Dedekind and Peano were inspired and fascinated by the gorgeous math they contain.


      “Whatever is capable of increase or diminution, is called magnitude, or quantity.
      A sum of money therefore is a quantity, since we may increase it or diminish it. It is the same with a weight, and other things of this nature.
      2. From this definition, it is evident that the different kinds of magnitude must be so various, as to render it difficult to enumerate them and this is the origin of the different branches of Mathematics, each being employed on a particular kind of magnitude. Mathematics, in general, is the science of quantity; or, the science which investigates the means of measuring quantity.
      3. Now, we cannot measure or determine any quantity, except by considering some other quantity of the same kind as known, and point out their mutual relation. If it were proposed, for example, to determine the quantity of a sum of money, we should take some known piece of money, as a louis, a crown, a ducat, or some other coin, and show how many of these pieces are contained in a given sum. In the same manner, if it were proposed to determine the quantity of a weight, we should take a certain known weight; for example, a pound, and ounce, &c., and then show how many times one of these weights is contained in that which we are endeavouring to ascertain. If we wished to measure any length, or extension, we should make use of some known length, such as a foot. ”

      It continues like this. Euler really doesn’t know how to start the numbers off. This poor attempt is why Grassmann invented the additive identities.

      x+0 = x

      x+y’ = (x+y)’

      using Dedekind notation.

      Euler is confused compared to this. Dedekind takes it much further. In some ways, Peano is a step back from Dedekind.

      Dedekind’s book is here, the second book in the unified volume given.

    • I do not hate Euclid’s Elements. But comparing texts from different ages and criticizing them is what great books programs are about. We don’t worship great books in a reading program, we have at them.

      When you compare Euclid to Euler to Grassmann to Dedekind to Peano and try to find what is missing or where the earlier one goes off or what it has that was kept later or transformed later you learn more than by just praising them.

      Think of Shakespeare and his contemporary playwrights trying to best each other in wars of wits. That is the spirit.

      • Paul Salomon says:

        I get what you’re saying, and I can understand analyzing the progress in the search for foundations of mathematics, but it’s complete nonsense to bad mouth Euler.

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