Peano Axioms help learn recursion in programming

When learning a computer programming language, there are often chances to write recursive functions. Learning the Peano Axioms makes this easier.

This can help learn the classic text Structure and Interpretation of Computer Programs by

Hal Abelson and Sussman and Sussman.

This text was used at MIT 6.001 intro computer programming course from approximately 1983 to 2006. Then they switched to Python. Sussman did a video saying the reason was all the new packages being created for Python for download.

The book uses Scheme which was invented at MIT by Sussman and Guy Steele.

This link criticizes the MIT choice to switch out of Scheme and thus Lisp as the foundation of its undergrad Computer Science program. MIT is no longer MIT. For decades, MIT was the home of Lisp and this dominated software language development in the United States. Now it is turning into another Python Java coding school for industry. Not very MIT at all.

Here at New Math Done Right, we support languages like Lisp and Scheme which are based on fundamental mathematical logic like recursion. If you want to learn these languages and ways of thinking, learning the Peano Axioms and recursive logic will greatly help.


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Peano Axioms 2015 Never more important

Learning the Peano Axioms has never been more important than in 2015 and the years to come.  The Peano Axioms mean that when we do elementary math at any time, we are thinking in terms of foundations of math and logic. They remind us of ordered pairs, sets, functions, recursion, definition by induction, proof by induction, and so on.  These are the basic concepts of math, logic and programming.  So we are always getting a reminder of these every time we use numbers when we learn the Peano Axioms.

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Learning different multiplication methods is a sound idea

Schools are sometimes ridiculed for teaching different ways of doing multiplication.  However, if this was so bad, then how can we teach applications of multiplication at all?  Isn’t another method to do multiplication an easy application of a first method?

If a student can’t apply one method of multiplication to another, then how can he apply it to something else?

What allows us to apply multiplication or even addition to some application?

What allows it is that the application satisfies the laws of addition or multiplication so that it behaves the same way as addition or multiplication does.

So to understand that we can apply multiplication to a problem, we have to be able to show that the problem satisfies the properties of multiplication.

For whole numbers, this means showing that

n*0 = 0

n*m’ = n*m + n

For addition, we have to show that

x+0 = x

x+y’ = (x+y)’

For successor, we have to show that x’ satisfies the 5 Peano Axioms.

If we don’t show or even understand these proofs or that they are required, then an application is really applying rote learning.

One reason it is hard to apply one multiplication method to another or translate them, is that the definition of multiplication of natural numbers in terms of repeated addition is not formalized.

n*0 = 0

n*m’ = n+n*m

These two equations define multiplication in terms of addition.  To apply this to area, we have to show that areas have this property, for whole lengths of the sides.  This is a matter of intuitive geometric understanding at the earlier grades.

If we structure math ed using the Peano Axioms, we can understand why people can’t easily go from one method of multiplication to the next. The reason is they didn’t learn the definition of multiplication, or its properties.  That is, they didn’t learn its structure. So they can’t apply the structure even to another method of multiplication, and recognize it has the same structure.

When we use the two equations to define multiplication, it is easy to verify or at least make plausible that another method of multiplication or an application such as area satisfy the same two rules.

This teaches abstraction and concepts.  If this is not done for natural numbers, it is difficulty for the student to form any proper concept of fractions.  Fractions are a mathematical structure with data and rules.  The data are natural numbers and the rules are methods of manipulating natural numbers.

So if the understanding of structure of natural numbers is completely absent, fractions are learned by rote if at all.

We can tell if structure of natural numbers is absent by testing if the student can apply or relate one method of multiplication to another.  If they can’t, they won’t understand the math structure of fractions either.

Rote learning experiences diminishing returns in math. As one ignores structure at one level, naturals, it becomes harder to learn the next level, fractions. After fractions, there are infinite decimals.  There is also algebra.  If showing two methods of multiplication have the same properties is not possible to the student, they will fail at algebra other than as rote learning. This shows up when they can’t do algebra word problems. They didn’t learn the structure of numbers let alone algebra, so they can’t translate a word problem to symbols.

Teaching math structure has to start with counting and relating counting to the 5 rules of counting by one, ie the Peano Axiom.

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Structure and abstraction fight the math is random drills feeling

Many students and teachers complain that math is a sequence of random drills with no connection or structure.  What is the plan? How does it fit together?

The answers to these questions comes from structure.  Structure is how parts of math fit together.

Abstraction is the architecture of the structure of math.  The nodes and a pair of nodes as a link are abstractions.

Natural numbers are a topic in math.  The structure of the natural numbers is given by the Peano Axioms.

Induction is itself an abstraction.  Hermann Grassmann recognized that induction is the key abstraction to define addition and multiplication of rational numbers and prove the properties of these functions.

Dedekind then rebuilt the structure Grassmann started and filled in gaps.  Peano then picked out some results in the middle of Dedekind’s book and made those a starting point as a set of axioms that would capture the essence of natural numbers.

The structure of the natural numbers as a sequence shows up over and over again.  There are other ways to constructively set up such a sequence such as the von Neumann structure of natural numbers.

These ideas lead to a better understanding of fractions as ordered pairs of natural numbers.  Understanding the abstractions in natural numbers is fundamental to understand a fraction as an ordered pair of natural numbers.

This takes away the randomness and substitutes meaning instead. The meaning is the abstraction.  The abstraction is key to the structure.

We have removed the applications.  The applications are not the meaning of natural number.  They are possible as applications because the abstractions of natural number apply to that situation.

We understand an application by recognizing that it is suitable to the abstractions in the natural numbers. That includes induction.

The motivation is the insight of how it fits together.  The motivation is going from seeing it as random drills to seeing it as a structure that has parts that fit together.

The applications can illustrate the abstractions and structure of natural numbers.  This way they teach the natural number abstractions and structure.

If we do applications willy-nilly, it is likely that the wrong concepts are taught.  Ones that are pre-Grassmann.   Thus emphasizing applications can lead to wrong teaching that creates problems as abstractions like fractions as ordered pairs are taught later.

Good teaching materials based on the Peano Axioms are the only way to prevent teaching the wrong concepts of natural numbers.

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Same 5 rules over and over again in Peano arithmetic

The same 5 rules of the Peano Axioms repeat over and over again as we develop elementary arithmetic of whole numbers. The 5 rules apply not just to the natural numbers, but can be restated to apply to the head sets as well.

So the sequence 0,1,2,3,… has the 5 rules.

  1. Start node.
  2. Out of each base node a single arrow consisting of a base node and tip node.
  3. The start node is not a tip of any arrow.
  4. No node is the tip of two arrows.
  5. If a path contains the start node and the tips it points to, then it contains all the nodes.

The sequence of head sets also follows these 5 rules




ie the head sets of natural numbers.

So do the sets




ie the head sets of the counting numbers.

The 5 rules apply to the unit fractions, i.e. to the sequence




They apply to any sequence 0/n, 1/n, 2/n

They apply to any sequence   m/1, m/2, m/3,…


Showing this teaches meaning to students.  This has not been pointed out by math profs or math literature or math education.  This is because they don’t do a good job explaining Peano Axioms.  This is because they don’t fully understand it.

Math textbooks put the easy stuff in the problems.  They put the think stuff in the problems. So the textbook writers don’t write the explanations of this.   You can’t improve what you don’t write.  Which is the result of the practice of putting a lot of steps and easy stuff into math problems instead of the text. The textbook writers never write the explanations and never improve them.  So they don’t get developed.  This is why the above points have never been articulated before.

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Head sets v tail sets the unknown controversy

Head sets of natural numbers starting from 0 are sets like





Tail sets are all the numbers starting from and after that number.

0 to infinity, including 0

1 to infinity including 1 but not 0

2 to infinity including 2 but not 0, 1


Dedekind in his 1888 book teaches tail sets first because the logic is easier. However, for students head sets are more concrete. Modern axiomatic set theory books teach head sets first, calling them initial segments. However, those books typically don’t do as good a job isolating head sets as a concept and teaching their properties.

If math profs understood their own literature from Dedekind onwards, they would consider whether it was better to teach head sets first or tail sets first. However, they are unaware that Dedekind teaches tail sets first and modern treatments like von Neumann construction teach head sets first.

This tells us that even math profs don’t understand this very well and don’t understand the historical literature. This then carries over to math profs not knowing how to teach math majors or even math grad students this material. If math profs don’t even know that Dedekind taught tail sets first and axiomatic set theory books teach head sets first, it shows they themselves don’t know Peano axioms very well.

This carries over to the whole teaching of pre-service math teachers in college. This is done by math profs who don’t know math foundations very well or how to teach math foundations even to math majors/math grad students. So they don’t know how to teach the substance to pre-service math teachers. They also don’t know how to teach the pre-service teachers how to teach it to the students.

The same applies to the textbook makers or to worksheet makers or to Khan Academy or other video makers. At least until Khan starts copying from my webpage. Which seems the only way that anyone will ever use it.

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Logarithms and calculators

Base 10 logs

log 2 = .301

log 3 = .477

log 5 = .699

log 7 = .845

How to remember these

log 2 = 2 * .15

log 3 = 3 * .16

log 5 = 5 * .14

log 7 = 7 * .12

So it starts at .15, goes up to .16, down to .15 at log 4 and .14 at log 5 and .13 at log 6 and .12 at log 7.

These are approximations. Using a calculator like Google can check how accurate they are.

Just remember .16 for 3. Then count down by .01 in either direction per whole number. This approximately works up to 10. You can straight line interpolate between whole numbers using the same slope. You can then work on improvements to this rule.

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Why good theory must inform pedagogy

If we understand the theory first and then rework it into slogans for children to learn, then we give them a path to understanding. If we invent slogans first and they don’t line up exactly with theory, then there are logical gaps and misdirections. These confuse the student and hinder an understanding of logical structure.

When logical structure is not developed this way, even without their knowing it, they are not ready for fractions or algebra.

It is very hard to work out rules of counting and understanding the counting of small sets. If we count a herd of cows in a different order do we get the same number? This is a hard thing to work out correctly why we do. One tends to assume the result in naive explanations.

To work out why we get the same number if we count a set in a different order requires theoretical understanding. This has to be written down exactly correct, or as best as possible.

Once this is done, we can then work the correct logic into slogans, chants, puzzles, games, etc. that teach this logical structure.

If we just invent the slogans and chants without understanding the actual logical structure, then we will create gaps and confusions. These accumulate and show up in walls that can’t be scaled at fractions and algebra.

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Students failing algebra rarely recover

“Students failing algebra rarely recover”

Jill Tucker
Published 10:22 pm, Friday, November 30, 2012

Read more:

If student’s don’t get algebra the first time, repeating it doesn’t work. The article specifically says teaching it again the same way doesn’t work.

The researchers challenged school officials to rethink when and how students are assigned to certain math classes and come up with new ways to teach the topics, especially to those who didn’t get them the first time around.

Take students out of the standard algebra curriculum and put into alternative logical foundations. They don’t get what algebra is about in the current approach. Current algebra teaching is procedural and it doesn’t form memories they can retain and use.

Current teaching in math is mostly memorization.
There are large gaps in the concepts. These gaps were filled in 19th century New Math of Boole, Grassmann, Dedekind, Peano.

However, this good new math was dropped in favor of an incomplete version that is mostly useless. 19th century new math was invented for arithmetic’s logic. this part is ignored in teaching new math and common core.

Common core takes Euler 1765 algebra and sprinkles a few words from set theory that have nothing to do and so are just ignored. Set, relation and function are not used in the way that Dedekind does. Is addition a function? Yes. What function is it? Can you write it in function notation?


f(x,0) = x
f(x,y’) = f(x,y)’

From this we can prove

f(0,y) = y
f(x’,y) = f(x,y)’

Are you completely lost? Think about it. Addition as a function is something you never heard of and never saw written before. However, this was developed in the 19th century to fill the logical gap in what addition is. yet modern teachers and students have never heard of it and are stupefied when they see addition written as a function.

Do you have any idea how to prove the second set of equations from the first set? This is fundamental to what addition is and why it works.

These two equations define addition

f(x,0) = x
f(x,y’) = f(x,y)’

Where ‘ means the next node in a chain from a starting node, 0.
This is counting along the chain. The f function here is the counting on function for the chain. this is addition. Addition is the counting on function for a chain of nodes on a line, i.e. satisfies the 5 linear chain axioms, i.e. the Peano Axioms.

Addition and multiplication of whole numbers are functions for counting along a chain of nodes. They depend on the chain of nodes being linear from a starting point. If we had a different geometry of the chain of nodes, we would get different functions for addition and multiplication by imposing these equations.

Let’s call f(x,y) as S(x,y) now. The S(x,y) function lets you move along the chain of nodes faster or in jumps like a Queen in chess.

The function S(x,y) has properties that make it what we expect of addition. Take S(x,y’) if you increment one of the inputs, the output increments by one. In fact,

S(x,y+z) = S(x,y)+z

If we increment an input by z, the output increments by z.

S(x,S(y,z)) = S(S(x,y),z)

Which functions solve this? Addition does. The addition function, S(x,y) satisfies

S(x,0) = x for all x
S(x,1) = x’ = S(x) for all x

Addition is a way to move along a chain of nodes by more than one node at a time. Addition is equivalent to going one node at a time.
Addition is repeated successor.

Learning to think in these terms is what algebra means and how it ties into arithmetic, a subject you may think you understand, but don’t. Those gaps were filled by Dedekind and others in the 19th century. My materials at this webpage and ebooks fill those gaps and are easier to understand than Dedekind or axiomatic set theory books.

This essay may seem cryptic to most. My materials in my ebooks help make it less cryptic. This is really what is going on with addition of whole numbers. This is what Grassmann, Dedekind and Peano showed when they filled in the logical gaps in Euler’s 1765 algebra textbook.

Yet current teaching in algebra is the same as Euler’s 1765 textbook. All the logical gaps found in the 19th century and filled in are ignored by modern teaching. They just sprinkle in the word set a few times and think this is the same as teaching the careful concepts in the Dedekind 1888 book. It isn’t.

Euler’s 1765 Algebra is a procedural text, i.e. a cookbook. Current math texts for K-12 are the same style of procedural cookbook. They don’t teach concepts and they don’t explain. They pretend to, but don’t. We know that from the Dedekind book and from axiomatic set theory books like Patrick Suppes.

We know some students can’t learn the recipes for algebra problems in current classes. Those classes try to go up a steep learning curve of procedural problems going to polynomial functions and rational functions including factoring.

This is not as important as understanding counting by one and how counting by one leads to the addition function for whole numbers. This is the logic of recursion and induction. This is mathematical thinking. This is what the struggling students need. This gives them insights that give them hooks to remember.

The best students can memorize procedural activities and repeat them on exams. They still don’t understand the underlying logic, but they can pass the procedural exams given. The books and materials are little more than rote memorization. But good students can slog through this amazing barrier of rote memorization material.

The struggling students can’t memorize an algebra book of procedures. So why not try teaching them the concepts of arithmetic using algebra and set theory? Then they might catch on to the meaning of what they are trying to learn.

Procedural math for factoring polynomials is not as important as conceptual math of how addition of whole numbers is defined as a function using recursion. That is indicated above in the equations for f which is renamed S. My materials here and in ebooks do this at a slow pace with many problems. They are the slowest paced set of material to teach this subject. They also have the most examples and worked out problems for Peano Arithmetic.

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If your students can’t do the associative law proof they have gaps

If your students can’t do the proof of the associative law of addition of whole numbers, then they have gaps in their knowledge.  Moreover, they don’t know they have them.  They will think they know common core math principles but will have hidden gaps in their understanding.  Those will catch up with them in fractions, decimals, sets, and algebra.

When you can teach them the associative law of addition, you will have mastery of the math behind common core math.  So will they.  This will make fractions a snap.  It will also get them over using a letter to stand for a variable.  This is supposed to happen in first grade in common core.  So you better know it well to explain it to first graders.

The younger the student you teach something to the better you need to know it. This way you can come up with the right answers to their questions. They have the same questions you do or math grad students do, and you need to learn the right answers. Those are in the Peano Axioms.

Hermann Grassmann, Dedekind and Peano proved that the only way to fill the logical gaps in Euler’s 1765 Algebra was with the Peano Axioms.  The same applies to common core math today.

Handwaving is what Euler did at critical points.  Handwaving by you the same way may end up with your students being befuddled for good reasons. The wrong answers don’t teach. That is why Grassmann was not satisfied with Euler’s 1765 Algebra book and found the right answers.  Dedekind and Peano finished the job of putting the right answers in a proper structure.

Now you just have to learn it and teach it. There is no other way to teach the concepts of elementary math because these are the concepts of elementary math.  Without Peano Axioms you are doing rote learning at critical points without realizing it.  This creates conceptual gaps that can open up like sink holes in fractions, sets and using a letter to stand for the unknown.




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