Addition is defined from counting by one

Right additive identities that define addition completely:

x+0 = x

x+y’ = (x+y)’

where 1 = 0′, 2= 1′, etc.

This defines addition from counting by one. So addition is derived from counting by one.  The term successor just means counting by one.  Successor may be too fancy a term that hides the meaning of the definition of addition in terms of successor.

There is no valid reason not to teach the above two equations to students in K-6.  In fact, it should be taught as soon as the plus sign is taught.

Letters can spell different words is taught.  It is no more complicated to teach letters can stand for different numbers.

Six and seven stand for 6 and 7.  If they can learn that, then they can learn that n can mean 6 or 7 and stand for either case.



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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44 Responses to Addition is defined from counting by one

  1. Paul Salomon says:

    “So addition is derived from counting by one.” Cool. I’m actually into that.

    But how is one “defined?” Or “counting?”

    • Paul Salomon says:

      I meant to say “how is ‘one’ defined?”

    • Thanks, I am glad we can agree on something, especially something so fundamental.

    • Counting by one is the successor, the next element. Successor is a function. So we have the ordered pairs (0,1), (1,2), (2,3), …,(n,n’),(n’,n”)…

      Any set that contains 0 and contains n’ if it contains n contains all natural numbers.

      • Paul Salomon says:

        “For Dedekind, he says the natural numbers start with 1.” Is that a definition? I’ll keep stress testing a minute: How do I find 1? Where is it? I know it’s the start, but where is that? Is it on my hand? Can I see one?

        As for ordered pairs, how do we define them? What about that “…” in the middle of the sequence?

        The thing that’s troubling about your position is that with one hand you fervently hold firm to formal definitions and axioms, but with the other hand, you speak informally and explain with loose and undefined terms. Your solution to the foundations of fundamental arithmetic is not a solution, because it can neither be explained nor justified in isolation.

        Understanding of number and arithmetic, and all other aspects of mathematics, and likely all other concepts is necessarily built up in a web of connected understanding. Though each strand may be somewhat vaguely understood, the web can stand very rigid and communication and agreement can still occur.

        How else could you be lead to adore the Peano Axioms so much if it weren’t for the “understanding” you alledgedly hadn’t built up before studying them?

        The Peano Axioms are great, they totally have their place, but there’s no need to push them in isolation of other conceptions, and there’s absolutely no need to bad mouth great mathematicians and teachers.

    • Comments to questions below.

      One or zero is in the PA or Dedekind version as case may be.

      It is standard to “explain” axioms with intuition and explanations that come from outside them. Especially to the buyer as you are. You have to be convinced of the axioms as a purchase and throw out all sorts of objections to buying which then have to be answered in the context asked.

      I agree with a web and circles and building up to understanding.

      The PA give us points of focus and a guide. Euler and Euclid both lack that.

      To repeat a comment from elsewhere, great book reading programs don’t worship great books they have at them. Part of great books reading is to compare them from different ages and see if there is progress and what it is.

      Euclid and Euler have parts left out of what we have now. They have some parts similar to what we have. Making the comparison helps us learn.

      Think of the projects you can assign students to compare Euler, Euclid and Dedekind.

      • Paul Salomon says:

        “It is standard to “explain” axioms with intuition and explanations that come from outside them.”

        There it is. The question then is not a matter of which axiom system is cleanest or shortest or furthest back in a chain of deductions, but instead the question is whether a starting place is the one that Kindergartner’s can attach to upon their intuition. What are they willing to convinced by and buy into. On that account, I have to say a typographic PA is probably not the right one for kids with almost no symbolic experience. Their intuition and number sense will be far from PA. They may think in terms of adding one and so on, that’s very natural and taught in all schools, but the right additivity properties and so on are just not central to their nascent connection to math.

    • Ordered pairs are defined (a,b) = {a, {b}}.

      The … is covered by induction, axiom 5. However, we can illustrate with them just like we use number lines.

      We can go back and forth from informal to formal.

      Musicians practice and it is not the same as the performance. Actors too. Lines can even be changed in practice.

      You as teacher can do it your way with the tools provided. Or a K teacher or 6th grade or whatever. The PA tools and framework is far superior because it does have answers and they fit together without known gaps.

      Euler and Euclid have known gaps. The David Joyce notes on Euclid or others point some of those out.

      • Paul Salomon says:

        “We can go back and forth from informal to formal.”

        Great! That’s what I needed to hear. I think I was getting the sense that you were a hard-nosed formalist from K up, and I need to hear that you see “the rules of the game” are not the game. Math is much more.

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