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 Tangible Number Semifield, arithmetic and the real world
Treisaran
Posted: Jan 21 2016, 08:27 PM


Dozens Disciple


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Joined: 14-February 12



Sorry for the scant posting lately; it's not so much for being busy as it's for having had no new ideas. Here's something, maybe not new but possible to chew on.

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 √−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 √−1 is pure fantasy.

Doing searches on the web, I've found various interesting answers to my objections, such as:
  1. 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.
  2. Division by zero has actually been accommodated by such constructions as the Riemann sphere, where 0 on one pole is matched by ∞ (unsigned infinity) on the other pole. This keeps quite a few field axioms and operations intact, but not all.
  3. Mathematicians once considered negative numbers to be just as absurd as those we now call 'imaginary' (following Blaise Pascal's term of ridicule).

The last point is one I found especially intriguing. On one maths tutorial website it was remarked that fractions had been part of arithmetic long before negative numbers because it's easier for people to grasp the concept of having half a bushel than having minus two bushels.

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? smile.gif )

It is a semifield because not all operations are defined for it. Addition, multiplication and division are defined, but subtraction isn't, because yx for any xy 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 x≤0 (read: of nonpositive numbers) are out of range of the real numbers (log 0 is −∞, and logs of negative numbers give the imaginary number iπ).

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 √2, 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 π, 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 π).

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.
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Myrtone
Posted: May 15 2016, 11:18 AM


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QUOTE (Treisaran @ Jan 21 2016, 08:27 PM)
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.

Yes, he has been described as a finitist, like someone called Doron Zeilberger is today. He basically had the rational point of view, where an irrational "number" is no number at all, things like Cauchy sequences not defining new objects any more than infinite sets do.

QUOTE (Treisaran @ Jan 21 2016, 08:27 PM)
When I first learnt of the imaginary unit constructed for √−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 √−1 is pure fantasy.


One could actually object to the square roots of numbers like 2, 3 and 5 on the grounds that the square of any rational number is either less than or greater than any integer other than a perfect square.

QUOTE (Treisaran @ Jan 21 2016, 08:27 PM)
Doing searches on the web, I've found various interesting answers to my objections, such as:
[LIST=1]
[*]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.


Yes, they do with Gaussian rationals, that is the rationals extended by √−1, same with the rationals extended by square, cube and other roots of prime numbers.

QUOTE (Treisaran @ Jan 21 2016, 08:27 PM)
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.


Irrational numbers are actually quite abstract too, at least according to the rational point of view.

QUOTE (Treisaran @ Jan 21 2016, 08:27 PM)
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).


All the examples given are with rational numbers. Rational numbers are closed under all four arithmetic operations (division by zero not being a valid operation), and are dense.

QUOTE (Treisaran @ Jan 21 2016, 08:27 PM)
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 √2, 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 π, 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 π).


We intuitively think of polygons as having side-length because we can measure the sides of a physical model. Claiming that √2 is there as the length of the diagonal of a unit square assumes that this intuitive physical notion (length) corresponds to something precise and mathematical. The Pythagorean theorem, from which this "length" is "derived", is really a statement of the area of a square built upon the diagonal of another.
Put it another way, you are assuming that there is even such a thing as the (corner-to-corner) diagonal of a square. If you distinguish between a line segment and it's length, you can say the diagonal of unit square is the side of a square with area 2. This alone does not define the length of a side, only a line segment.
Another claim of yours is that transcendentals correspond to tangible entities. In case of π, it is said to correspond to two entities, one is the ratio circumference of a circle to the diameter, the other is the ratio of the area of a circle to that of a square with one corner at the centre of the circle and an adjacent corner on the edge. Now the circumference of a circle is an example of the length of a curve, itself another intuitive physical notion. In fact, the length of a curve seems to be just as difficult to define mathematically as transcendental "numbers", same with the area enclosed within it.
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wendy.krieger
Posted: May 15 2016, 12:51 PM


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Posts: 2,432
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Joined: 11-July 12



The way that numbers are treated by mathematicians kind of sidesteps a good deal of things, one area is the 'constructable numbers'.

The non-carry base corresponds to base 'x', but it affords the use of various 'digits' like integers, rationals, and reals. When you construct a non-carry base in x over some set, you are basically creating a system which preserves addition, subtraction, and multiplication.

If you suppose the set is over Z (integers, positive and negative), and apply some x^n = some expansion in lesser n, you are effectively saying 1,0,0,0 = -1,2,1 or 2 or something. In other words you are folding x into three or four columns.

While something like x^3-2=0 is an algebraic equation, it affords an integer system, since x^3=2 is actually a carry rule, that 1,0,0,0 = 2, and the numbers written in this space, actually are a ring in the mathematical sense.

More over, all non-carry + modulo (like the one above), are governed by particular rules.

1. Every integer divides some Z.
2. The intersection of the derived set, and the set Q = Z/Z, is Z. (ie no rationals except integers).
3. Any number h which divides some prime p, but for which p does not divide its power, will never do so, to the extent that one can not cojoin integer sets where h divides p.
4. The sets described by a^n = -1, are the cyclotomic numbers CZn. The set of sets of these is ZZ. The real part of CZn is Zn, the span of chords of a unit-edge n-gon.
5. The higher order roots (over the square root), are never members of ZZ. On the other hand, every square root can be constructed in ZZ.
6. The sets described by every algebraic root system, are the algebraic integers Y.
7, The intersection of the sets Y and Q is Z.
8. The class rules apply.

The class of equations over the same form, in rationals, is FY = Y/Z, is variously represented by

1. Polynomials in rationals, or rationals in the abacus.
2. Polynomials in Z, with a non-unit high place.
3. The closure of Y to division.

The closure of Zn and CZn to division gives these sets divided by Z.

The numbers beyond Y/Z are transendental.
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