NON-DIOPHANTINE
ARITHMETIC

Imaginary arithmetic

Ah! why, ye gods, should two and two make four? (Alexander Pope)

"Arithmetic is the grammar of numbers" (Wittgenstein. Philosophical Investigations).


Concept

Conventional arithmetic, the well-known and commonly used arithmetic, is often called "Diophantine arithmetic", after Diophantus of Alexandria, for his important contributions to this branch of mathematics (including his famous treatise on arithmetic). This arithmetic has always been considered the "only true" one. However, it is perfectly licit to raise the existence of other alternative arithmetic (called "non-diophantine"), which are as consistent as the classical one, not only as a mere intellectual exercise of abstraction, but also to have practical utility in real problems.

This situation is analogous to that of Euclidean geometry, which was always considered the only true −evident, according to Kant−, until in the 19th century Gauss, Lobachevski and Bolyai put forward alternative geometries modifying the axiom of parallels (the famous fifth postulate), as consistent as Euclidean geometry.

Lobachevsky called his geometry "imaginary", but it was later discovered that real physical space has no Euclidean geometry. The discovery of non-Euclidean geometries is considered one of the great achievements of mathematics. It contributed greatly to the understanding of mathematics as an open discipline, based solely on axioms to be established. It also helped to improve our understanding of the physical world. Arithmetic, the last refuge of an absolute mathematics, is also subject to this philosophy.


Examples

Here are some examples raised by advocates or supporters of alternative arithmetic:
  1. Joining one drop to another drop does not yield two drops, but only one drop.

  2. By mixing one liter of alcohol and one liter of water, approximately 1.8 liters of vodka are obtained.

  3. The price of one kilo of rice is 3 € and the price of 2 kilos is 5 €.

  4. If a car is worth more than €20,000, an increase of up to €100 is considered nil or negligible, i.e. the price of the car is considered unchanged.

  5. Adding sugar to milk produces sweetened milk with the same initial volume, provided that the amount of sugar is less than 1/20th of the weight of the milk.

Justification

Although there is no universally accepted conception of mathematics −there are many competing conceptions−, there is universal consensus in arithmetic at the theoretical level. Another issue is the problem of implementation of arithmetic.

There is a pure mathematics and a mathematics applied to the physical world, mathematics that Morris Kline [1985] calls "outer mathematics" and "inner mathematics," respectively. Plato (in Philebus) already stated that "Arithmetic is of two kinds, one is popular and the other philosophical."

At the mental or theoretical level, traditional arithmetic is valid. Arithmetic is the most basic and oldest branch of mathematics. Without arithmetic we could not develop our daily life and we could not do science and technology. Arithmetic is the basis of modern number theory. If arithmetic were changed, a new number theory would be needed, because the properties of numbers would change. For example, the definition of prime number is based on the definition of division, which in turn is based on multiplication, and multiplication is based on addition.

The emergence of non-diophantine arithmetic is not due to doubts about classical theoretical arithmetic, as some authors claim, but is motivated by several reasons:
  1. Because of physical implementation problems, which force arithmetic to work differently, because of precision errors, or because of memory limitations. The set of numbers that can be operated on is finite. For example, in a computer, integers are usually limited by the number of bits used for their representation. If n bits are used, integers from −2n−1 to 2n−1 can be represented (the sign occupies 1 bit). Floating point numbers are also limited by the length of the mantissa and by the exponent. Then overflow or underflow can occur, depending on whether the exponent exceeds or falls below the allowed limit. It can (and does) happen that the simple operation of 0.5+0.5 not 1, surprisingly. Errors occur in the implementations of arithmetic, never at the theoretical level.

  2. Many physical phenomena and processes do not follow diophantine arithmetic, especially quantum and relativistic ones. In relativistic physics, the arithmetic is non-diophantine, since the imaginary expression: c+v = c (c is the speed of light). In quantum physics, there are quantities that do not satisfy the commutative and distributive properties.

  3. In clock arithmetic (Gaussian), numbers are limited between 1 and 12. For example, 9+5 = 2 (remainder of the division of 14/2). In general, in modular modulo arithmetic m, a+b;is the remainder of (a+b)/m.

  4. To implement qualitative concepts such as "much greater than" or "much less than" that can be formalized as imaginary arithmetic:

    If a≫b (a much greater than b) or b≪a ( b much smaller than a), then a+b = a (since b is negligible against a).

Mark Burgin's generalized arithmetic

Mark Burgin [2010] has invented a family or class of non-diophantine arithmetics, making them dependent on a functional parameter f(x), as in Lobachewski geometries. That is, it generalizes traditional (diophantine) arithmetic, this being a particular case of this class of arithmetic when f(x) = x: If, for example, f(x) = ax + b, then If a=1 and b=0, i.e., if f(x) = x, we have the traditional arithmetic.

Another possible non-diophantine arithmetic is: If, for example, f(x) = ax + b, then
f−1(x) = (x − b)/a   and   f−1(f(x)) = x

x + y = f−1(ax + < i>ay + 2b) = x + y + b/a

x * y = f−1(a2xy + abx + aby + b2) = axy + bx + by + (b2 − b)/a

If a=1 and b=0, we again have traditional arithmetic.


Specification in MENTAL

From the point of view of MENTAL, non-diophantine arithmetics are really imaginary arithmetics. This justifies that non-Euclidean geometries were initially called "imaginary geometries".

The imaginary unit i (i2 = −1) gives rise to a non-diophantine arithmetic: the arithmetic of complex numbers (a + bi). The dual numbers are also imaginary: they are defined as a + bε, where ε2 = 0; they also give rise to a non-diophantine arithmetic. The element ε (infinitesimal) is the foundation of differential and integral calculus.

Actually, in MENTAL you can easily define alternative arithmetic by means of generic (parameterized or not) substitution and/or conditional expressions, which specify the new arithmetic laws. The definition of imaginary arithmetic expressions depends on the imagination (and never better said) of the user.

When using parameterized generic expressions, we can additionally have "imaginary algebra", "imaginary logic" or combination of the two.

With MENTAL it becomes clear that non-diophantine arithmetic is the natural consequence of the possibilities of language in the domain of arithmetic. With MENTAL it becomes clear that mathematics is degrees of freedom and that we can use those degrees of freedom in many ways.


Examples
  1. (3+4 = 0)
    (1+2)+(3+4) // ev. 3

  2. ⟨( (3+n = 3 )⟩
    3+7 // ev. 3

  3. ⟨( (1000+n = 1000) ← n>0 ← n<100 )⟩
    1000+5 // ev. 1000
    1000+105 // ev. 1105

  4. ⟨( (n1+n2 = (n1 +° n2) ← n1&>n2 )⟩

    In this case, the addition is not performed if the first operand is greater than the second.

    34+7 // ev. 34+7 (self-evaluates)
    7+34 // ev. 41

  5. ⟨( (n1 a n2) = (f(n1) + f(n2)) )⟩
    ⟨( f(n) = 2*n )⟩
    (3 to 4) // ev. 6+8 = 14

  6. ⟨( (n1 + n2) = (f(n1) + f(n2)) )⟩
    ⟨( f(n) = 2*n )⟩
    (3 + 4) // ev. 6+8 ev. 12+16 ev. 24+32 ...

    In this case, the summation expression is recursive. Hence the need to define the operation as in the previous example.

  7. Recursive definitions of sum and product

    ⟨( (n + m) = (n + 1 + (m−1))←(m>1) )⟩

    ⟨( (n*mb>) = (n + n*(m−1))←(m>1) )⟩

    could be generalized by introducing, for example, a functional parameter:

    ⟨( (n + m) = (f(n) + (f(m) − 1) + 1)←(m>1) )⟩

    ⟨( (n*m) = (f(n)*(f(m) − 1) + f(n))←(m>1) )⟩

    These definitions would carry over to higher order arithmetic operations (power, hyperpower, etc.).

  8. Modular arithmetic modulo m:

    ⟨( sum(n1 n2 m) = (n2+n2 − ((n1+n2)÷m)*m) )⟩
    sum(3 11 12) // ev. 2

  9. Conditional or restrictive sum:

    ⟨( sum(n1 n2 max) = (k = n1+n2) (k>max → k=max) k)! )⟩

Coding the above examples

In the case of the examples raised by the proponents of alternative arithmetic, their specification further requires the use of quantities and functions.
  1. (1*drop + 1*drop = 1*drop)

  2. ( (1*drop)(alcohol) + (1*drop)(water) = (1.8*drop)(vodka) )

    A functional notation is used: magnitude(liquid).

  3. ( ( price((1*kilo)(rice)) = 3*€ ) →
    ( price((2*kilo)(rice)) = 5*€ ) )⟩

  4. (( ( price(car) > 20000*€) )→
    ( price(car) + n = price(car)) ← (n < 100*€) ) ) )

  5. ( (volume(milk + sugar) = volume(milk)) ←
    (weight(sugar) < (120.)*weight(milk)) )

Addenda

A little history

Perhaps the first to question the validity of conventional arithmetic was Herman von Helmholtz in "Counting and Measuring", a work of 1887, when he detected the problem of its applicability to certain physical phenomena.

During the 20th century, several authors (such as Kolmogorov, Littlewood and Kline) also detected problems where classical arithmetic was not applicable, so they suggested the need for alternative arithmetic. Other authors introduced different types of natural numbers, including numbers of qualitative type (e.g., small, medium, large, superlarge), although always within ordinary arithmetic.

The first mathematician who formally posed the problem of constructing alternative arithmetic was P.K. Rashevsky [1973].

More recently, Brian Rotman [1997] has presented a number of examples where classical arithmetic did not work, raising the need for other arithmetic.

The great promoter of non-diophantine arithmetic is Mark Burgin. The first family of non-diophantine arithmetic defined by this author are projective arithmetic [Burgin, 1977], which depend on a functional parameter f(x). When f(x) = x, one has traditional diophantine arithmetic. Another family is the family of dual arithmetic [Burgin, 1980].


Bibliography