The mathematician Leopold Kronecker is known for the quote, 'The integers were made by God, all else is human creation'. He saw integers as the key to interfacing mathematics and reality. I too am interested in that interface, but my point of view is different from Kronecker's.
When I first learnt of the imaginary unit constructed for âˆšâˆ’1, I thought of it as a sleight of hand. 'Why not construct a new type of number that enables division by zero?' I thought, and also objected to the imaginary unit on the intuitive grounds that a negative number multiplied by itself always yields a positive number, meaning âˆšâˆ’1 is pure fantasy.
Doing searches on the web, I've found various interesting answers to my objections, such as:
- What matters is mathematics is that all the field axioms and derived operations (addition, multiplication and so on) should hold. With complex numbers they do, whereas division by zero breaks everything down.
- Division by zero has actually been accommodated by such constructions as the Riemann sphere, where 0 on one pole is matched by âˆž (unsigned infinity) on the other pole. This keeps quite a few field axioms and operations intact, but not all.
- Mathematicians once considered negative numbers to be just as absurd as those we now call 'imaginary' (following Blaise Pascal's term of ridicule).
Negative quantities are as much a part of reality as fractions are; however, fractions are tangible while negative numbers are abstract. Negative numbers crop up when talking about debit balances and coordinate reversals, therefore requiring abstract thought before translation into reality, while the correspondence between fractions and reality is automatic.
In my search for arithmetic closely corresponding to tangible reality, I've found my own phrase to match Kronecker's: only the positive reals are real; all other numbers are really unreal. That's not accurate, though, because all numbers are just mental entities, and the difference I've had in mind is between automatic and translated correspondence to reality (see above).
The positive real or tangible number semifield is better represented as a circle rather than a line. That circle looks like the real projective line, but with a crucial difference: the bottom of the circle is marked with 1 rather than 0. There is no 0 in the positive real semifield; fractions get smaller down to infinity, which is unsigned, thereby meeting the limit of the large numbers like a snake eating its tail. (Cosmic Uroboros, anyone? )
It is a semifield because not all operations are defined for it. Addition, multiplication and division are defined, but subtraction isn't, because yâˆ’x for any xâ‰¥y is out of range (that is, it yields zero or a negative number). The positive real semifield is closely related to logarithmic scales in that logarithms of xâ‰¤0 (read: of nonpositive numbers) are out of range of the real numbers (log 0 is âˆ’âˆž, and logs of negative numbers give the imaginary number iÏ€).
Positive reals correspond to the existence of the numbered object. When all the particular objects have been taken away from you, they no longer exist for you - zero apples is equally tangible as zero aircraft carriers. A debt of objects, too, is just an idea, while the debtor lacks the objects, meaning they no longer exist for him or her. When you divide an object, you still have something, no matter how many times you divide it, although repeated division can yield so small a quotient that it is almost as if you now have nothing at all (zero approximated by approaching infinity).
Irrational numbers, the bane of the Pythagoreans (who reputedly executed by drowning the one who proved their existence), are no problem in corresponding automatically to tangible entities. We may not be able to numerate âˆš2, but it's there as the length of the diagonal of a unit square. Root extraction by whatever method can be repeated, converging at the limit of 1, the bottom and fundament of the positive real circle. Transcendentals such as Ï€, no matter how difficult they may be to define, correspond to tangible entities (for example, the ratio of circumference to diameter in the case of Ï€).
The fields of integers, real numbers and complex numbers enable one to get to grips with the non-intuitive arithmetic equivalents of black holes and quantum fluctuations; for those who consider Newtonian physics or small-scale flat earth geometry as adequate approximations for their humble needs, the semifield of positive reals is enough. Although one can employ the entire field of real numbers for anything that can be done with the semifield of positive reals, the latter has the virtue of enforcing a sanity check on arithmetic operations.
Or so I think, anyway.