User:Physis/Proportion and ratio

Proportion
A proportion is a proposition/statement about four quantities.


 * $$\mathrm{proportio}(a, b, c, d) :\Leftrightarrow a\cdot d = b \cdot c$$

Behavior on unfamiliar algebraic structures
an advantage of this kind of definition is that it behaves very well on unfamiliar structures. E. g. the even natural numbers do not have a neutral element, still, we can pose statements about "unit" proportions.

Theorems on the analogy of those of equivalence relations
Not the same algebraic properties like reflexivity, symmetry, transitivity, but "straightforward" analogia of those:


 * "Reflexivity":
 * $$\mathrm{proportio}(a, b, a, b)$$


 * "Symmetry"
 * $$\mathrm{proportio}(a, b, c, d) \Leftrightarrow \mathrm{proportio}(c, d, a, b)$$


 * "Transitivity"
 * $$\mathrm{proportio}(a, b, c, d)$$ and $$\mathrm{proportio}(c, d, e, f)$$ implies that $$\mathrm{proportio}(a, b, e, f)$$

While proving them, we notice, that we do not need to refer to unit (neutral) element, and everything wors on structures without such (even natural numbers). All that is needed is commutativity of multiplication.

Abuse of notation
Each is a therorem about four qunatities, why do we call them "reflexivity", "transitivity", "simmetry", which are therorems about only two quantities? Because we can notice a pattern:


 * "Reflexivity":
 * $$\mathrm{proportio}(\underbrace{a, b}, \underbrace{a, b})$$


 * "Symmetry"
 * $$\mathrm{proportio}(\underbrace{a, b}, \underbrace{c, d}) \Leftrightarrow \mathrm{proportio}(\underbrace{c, d}, \underbrace{a, b})$$


 * "Transitivity"
 * $$\mathrm{proportio}(\underbrace{a, b}, \underbrace{c, d})$$ and $$\mathrm{proportio}(\underbrace{c, d}, \underbrace{e, f})$$ implies that $$\mathrm{proportio}(\underbrace{a, b}, \underbrace{e, f})$$

The above theorems, and possibly also some similar kinds, seem to allow the following abuse of notation, trying to "justify" the pattern:


 * "$$a:b = c:d$$" should denote that $$\mathrm{proportio}(\underbrace{a, b}, \underbrace{c, d})$$

We shall see whether this abuse of notation can be justified, filled with meaningful content, it is not a misnomer, and leads to further mathematical objects and theories.

Specific theorems
Of course, there are also interesting thorems, that are rather specific, and cannot be pushed fitting the analogz of properties of equivalnce relations.

Theorem ("inside" exchange property):
 * $$\mathrm{proportio}(a, b, c, d) \Leftrightarrow \mathrm{proportio}(a, c, b, d)$$

Proof:


 * $$\mathrm{proportio}(a, b, c, d)$$
 * $$ad = bc$$
 * $$ad = cb$$
 * $$\mathrm{proportio}(a, c, b, d)$$

The other exchange theorem ("outside" exchange property):


 * $$\mathrm{proportio}(a, b, c, d) \Leftrightarrow \mathrm{proportio}(d, c, b, a)$$

Proof:


 * $$\mathrm{proportio}(a, b, c, d)$$
 * $$ad = bc$$
 * $$ad = cb$$
 * $$\mathrm{proportio}(d, b, c, a)$$

The two exchange theorems have no standalone significance on their own, because the equivalence-motivated theorems "fuse" them together.

Ratio
Ratio is a relation between two quantities: quantities $$x$$ and $$y$$ are said to stand in a relation $$a:b$$ iff the statement
 * $$\mathrm{proportio}(a, b, x, y)$$

holds.

The notion raises possible inconsistencies, but this can be proven not to exist, see the above theorems and the remark below the introduction of the abuse of notation.

A simple, but nice theorem, just to play with the notion:

Theorem: $$x$$ and $$y$$ stand in the relation $$a:b$$, that is the statement as if saying, $$a$$ and $$b$$ stand in the relation $$x:y$$.

Main theorem
The "motivating" theorem we expect to hold is


 * $$\mathrm{ratio}(a, b) = \mathrm{ratio}(c, d)$$ iff $$\mathrm{proportio}(a, b, c, d)$$

that would justify also our dept, the above abuse of notation. This can be achieved simply by the familiar set-theoretic notion of relations:


 * $$\mathrm{ratio}(a, b)$$ is defined as relation $$\left\{\,\left\langle x,y\right\rangle \mid \mathrm{propositio(a, b, x, y)}\,\right\}$$

Didactics
Proportion is a simpler, more profound notion that ratio. Still, ratio is simpler, more profound than fraction. Fractions nned not be introduced beforehand, on the contary, prposrtions, aand then ratios can be used as foundational pre-notions for fractions. Like ethe example of even natural numbers shows, propostions and ratios can be spoken about also in rather wild algebraic structures, which surely do not have fractions.

The notion of ratio can be used also as a motivating example for the abstract introduxtion of relations.

Rations are often defines as a fraction. This seems for me to be an afterthought. For me, propprtion is basic, ratio is intermedate, and fraction is rather sophisticated.

For me, fraction is the reification end-tip of a very long introductionary, preliminary journey. Nearest analagy: Reification (computer science).

Direct proportionality
Definition:

A function $$$$ is said to be a direct proportionality iff on its entire domain, for all $$x_1, x_2$$ choices


 * $$\mathrm{proportio}(x_1, x_2, f(x_1), f(x_2))$$

Equivalent theorem:


 * $$\mathrm{proportio}(x_1, f(x_1), x_2, f(x_2))$$

Proof of the equivalent theorem: use the appropriate exchange theorem (the "inside" one).


 * $$\mathrm{proportio}(x_1, x_2, f(x_1), f(x_2))$$
 * $$\mathrm{proportio}(x_1, f(x_1), x_2, f(x_2))$$

It is the exchange theorem said in Note: we chose the "impure" property for theorem. Motivations:


 * Argument of being epinymous:
 * One more motivation for this choice: the very name "direct proportionality" refers to this pure variant.


 * Argument of being intuitive:
 * Third motivation: when we fill in proportionality tables in a quiz, intelligence test or school task, it is this property we use intuitively, almost unconscious, instinctly. See the example in Korányi 1987 I.


 * Argument referring to history of science:
 * Each of both ratios is "pure" (in the Euclid recorded sense). That's why we chose that property for defining property, not the (otherwise equivalent) one.


 * Arguments 1 and 2: I think, the underlying intuitive notion is something like "a proportional change", "proportional consequence", "proportional punishment". These refer indeed to the property that we chose for the defining property. See also the Rabbinic argument for balancing punishments for deeds (for a milder crime the punishment should not be harder). Use this Rabbinic tale (in a context for children) for an overall main motivating example for the entire topic of proportions, ratios and proportionality!
 * Counterargument: phrases like "a proportional figure", "a proportional distribution/plan" may refer more to the property in the "equivalent theorem", and not so much to the property we chose for defining property.
 * A to-do: a very basic intuition sees direct proportionality as "taking the like paces in the like periods". Also Korányi 1987 I mentions this. Although this is something of restricted scope (allowing only special domains, lacking the most general discussion), but maybe this sweeping idea can be transferred and extended in a straightforward way to profound generalizations and abstractions.

A theorem of restricted-scope: ...

Note: We do not use the proportionality factor at all in the definition, because it excludes unfamiliar structures for the domain (for natural numbers, factor can be a "time travel to later concepts", and for even natural numbers, the lack of neutral element makes it interesting).

Inverse proportionality
Definition:

A function $$$$ is said to be an inverse proportionality iff on its entire domain, for all $$x_1, x_2$$ choices


 * $$\mathrm{proportio}(x_1, x_2, f(x_2), f(x_1))$$

Equivalent theorem:


 * $$\mathrm{proportio}(x_1, f(x_1), f(x_2), x_2)$$

Proof of the equivalent theorem: ????? (surely use the appropriate exchange theorem, or both, but some other theorems are needed too).

Note: we chose the "impure" property for theorem here again. Motivations are the same as at the direct proportionality part (etymology, intuition, history).

A theorem of restricted-scope: ...

Note: We do not use the proportionality factor at all in the definition. Motivation is again the same as the correspondent one mentioned in the direct proportionality part (cases of unfamiliar structures).