The metaphysics of fractions

In learning whole numbers, the Peano Axioms or the 5 rules of counting are not taught explicitly.   This leaves the knowledge of whole numbers without structure.  This tends to make it learning by rote and drills.

When fractions are reached, the conceptual framework of whole numbers has not been properly laid.  This makes fractions even more a matter of rote learning.  The rules for adding, comparing, etc. of fractions are taught as pure rules.

Attempts at justifying these rules from pizza are half hearted.  The supposed invoking of pizza to justify fractions ends up being just repeating the rules without any attempt to derive them from pizza.

We see the same in Euler Algebra 1765. Euler says the line justifies fractions, and then states rules for fractions without any attempt to justify them based on the line.

If we think of whole numbers as a line of successive nodes governed by the Peano Axioms as rules of indexing the nodes in the line, then we need a way to fit fractions into such a conceptual metaphysics of indexing and labeling as the basis of counting and of numbers.

We can think of several of these lines, each given a separate label, which we call the bin index.  Along each line, we count nodes according to the Peano Axiom rules.  But we can also think of grouping the nodes into bins along each line.

So the bin 2 line has little imaginary circles that bin each successive pair of nodes.  The bin 3 line each triple.

As long as we work with a line by itself, we ignore the bin groupings.

When we compare lines, we find a common bin line by multiplying the bin indices of the two lines.  So to compare nodes on the bin 2 and bin 3 line we use the bin 6 line.

The index of a node on the bin 2 line is multiplied by 3 to get the corresponding index on the bin 6 line.  The node index on the bin 3 line is multiplied by 2 to get the node index on the bin 6 line.

We now can work with the two corresponding nodes as if they were native nodes of the bin 6 line. We can compare them for greater than, we can add them.  We can multiply them.  We could divide them if that was possible.

So 1/3 is compared to 1/2 by finding node 2 on the bin 6 line for 1/3 and node 3 on the bin 6 line for 1/2.  Since node 3 is bigger than node 2 on the bin 6 line, we define 1/2 to be larger than 1/3.

If we use yet a higher common bin multiple, the order is preserved. This can be proven as a theorem.

Within a bin line, we have all the rules of Peano Axioms including addition as counting on and multiplication as grouping.

We see that fractions and multiplication are linked by the same grouping.   This is very gratifying since we want to interpret 1/2 as an inverse to multiplying by 2.  Thus we want 1/2 to be related to grouping by 2 if we interpret multiplication as grouping by 2.

The fraction 1/2 is interpreted as node 1 in the bin 2 line.  Two times this node gives us node 2 in the bin 2 line.  This corresponds to one bin of 2 in the node 2 line.   Thus grouping by 2 in the bin 2 line leads naturally to seeing a single node as standing for the fraction 1/2 and having the property that 2 of these nodes corresponds to 1 bin ie one group of 2.  So we see the inverse relation.

Based on this, we want to define dividing by a whole number such as 6 to mean going to the bin 6 line, and forming bins of 6.   We then count whole bins and partial bins.  The partial bin of one node is the fraction 1/6.

The unit fractions emerge as a single node on a bin line.

The rule for forming a common denominator is the rule for going to a common bin line. The numerators of the common denominator fractions are the nodes on the common bin line.

This works for order and thus for addition.  Once we get to the common bin line, we apply the Peano Axioms as if these were native nodes.  Thus the Peano Axioms are needed as rules of counting to understand a bin line by itself.  So the Peano Axioms are needed as rules of counting nodes before we can understand fractions as involving nodes of corresponding lines, or of grouping nodes by different bin sizes.

By having parallel lines and matching bins as the number of nodes in each bin increases, we model the idea of smaller subdivisions of the same line.  We can think of the bins as a single line and we divide it into smaller fractions.  The unit fractions play a key role in this.

The rule for forming common denominators or bins and the resulting adjustment to the numerators of the starting fractions then gives us a rule for finding the node in the common bin line corresponding to the initial fraction.   This rule is fundamental to a model or metaphysical explanation of fractions.  This is a working model of fractions since it can reproduce their 4 operations.  That is done by incorporating parts of the fraction rules to switch lines and find the corresponding node together with the Peano Axioms applying along each line to the nodes on that line.

This corresponds to the numerator being worked according to rules of whole numbers.

When we change denominators it is different than going along a bin line changing the numerator.  There is a qualitative distinction in numerators and denominators.

The model of parallel Peano Axiom lines grouped into different bin sizes preserves the distinction between numerators and denominators.  Numerators are bins along a bin line governed by the Peano Axioms. Denominators distinguish the bin lines and we have a rule for finding the corresponding node in a common bin size line.  This preserves the distinction.

Dividing by a whole number corresponds to finding a line of that bin size and counting bins and partial bins.  If we apply this to a bin line of bin size greater than 1, then we apply this to the bin line of the product of the bin sizes.  So if we divide 1/2 by 3, we get 1/6 ie the bin line of size 6.




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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