Numbers: Real, Imaginary, Absurd and Irrational

[BaalChatzaf]

The only numbers that have been greeted kindly are the positive integers. Why? Because positive integers are used to -count stuff-. We love them so much we say God created the integers. All other numbers have been bad mouthed and bad penned and otherwise denigrated at some time for another.

Negative numbers. At one time called fantastical numbers. And why not? Who in his right mind can subtract five apples from three apples. Negative integers gained respect when bookkeeping required accounting methods for tracking both debits and credits.

Irrational numbers. The Pythagoreans believed that all numbers were either integers or ratios of integers. What a nasty shock they had when they discovered there do NOT exist a pair of positive integers m, n whose ratio squared is equal to two. In geometric terms the diagonal of a square is not co-measurable with the side of the square. There does not exist a unit of length that goes into the side and the diagonal leaving no remainder.

Zero. How can Nothing be a number. Integers (counting numbers) have been around for ten thousand years. The zero, the nothing (so-called) has been around for less than two thousand years. The Greeks did not have the zero. The Babylonians did, but they did not consider it a number, but a place holder in their base 60 positional system. Zero is the Nothing that is Something. It is the number that can be added to any other number without changing it. Zero makes addition *general*. It also makes subtraction sensible.

Rational. These are beloved number. They are the integers and ratios of integers. You can use them to describe musical chords. The Pythogoreans loved them and eventually all the Greek mathematicians and geometers came to love them too. But as we have seen, rational numbers are not enough. The Greek mathematician Eudoxus showed how to handle ratios without presuming rationality (ratios). If Eudoxus had the zero, the Greeks might have invented algebra fifteen hundred years earlier.

Real numbers. It turns out that rational numbers are not enough. Not only can one not get at things like the square root of two but sequences of rational numbers cannot be guaranteed to converge. Real numbers were invented to permit the convergence of sequences of numbers under certain circumstances (see Cauchy Sequences).

Imaginary Numbers. These are numbers that popped up during the Renaissance. In the sixteenth century mathematical acrobats such as Cardano and Tartaglia attempted to solve quadratic, cubic and quartic polynomial equations by the extraction of roots. When negative numbers started appearing under square root signs, the initial reaction of the mathematicians was crazy, absurd, and ain't no such thing. Square roots of negative quantities can't exist! Everyone knows that squares of any numbers positive or negative are positive. So these numbers must be fantastical or imaginary.

Just as zero was introduced to make any linear equation solvable, the so called imaginary unit (designated i for imaginary) was invented to make it possible to solve any polynomial equations either by root extraction or by approximation. Numbers made from imaginary and real components are called complex numbers. It turns out that any polynomial in one variable with real co-efficients has a a solution in the domain of complex numbers.

Do you see what is happening? Strange numbers had to be introduced to make certain mathematical operations general. Expanding context and domain is the essence of progress in mathematics.

In set theory, the empty set (which suffered the same kind of discrimination initially as did arithmetic zero) was introduced to make the intersection of sets general. Given any two sets, the intersection is defined even if the sets are disjoint. Equivalently the empty set is the set that can be "added" (set union) to any other set without changing it.

Introducing strange new numbers or objects has been the historical story of mathematics. Objects which cannot be handled by common sense or ordinary intuition have been introduced for the sake of generality or completeness of known operations. It is all about extending context.

That is how non-Euclidean geometry and projective geometry came about. We know that two distinct points determine a line (actually a line segment). Wouldn't it be wonderful if any two lines (or line segments extended) determined a point of intersection. A world without parallel lines. It turns out that such a geometry exists. It is called projective geometry and is the geometry underlying perspective drawing which was invented during the Renaissance. It is the geometry that explains why a pair of parallel railroad tracks seem to meet Way Off, Very Far Away. Non-Euclidean geometry came about because attempts to prove Euclid''s Fifth postulate invariably failed. They must fail, else Euclidean geometry itself is inconsistent.

In the field of formal logic, that bastion and protector of Consistency there has arisen a class of logics than can cope with paradox and inconsistency in a limited fashion. These paradox tolerant logics are called Paraconsistent Logic. There is an article describing them in Wikipedia. Paraconsistency is a way of "saving the appearences" in the presence of contradictory statement pairs. So even in formal logic the absurd has been tamed and domesticated much the same way as the irrational, the zero and the imaginary has been tamed in mathematics.

Ba'al Chatzaf

Edited September 12, 2007 by BaalChatzaf

[Brant Gaede]

Zero. How can Nothing be a number. Integers (counting numbers) have been around for ten thousand years. The zero, the nothing (so-called) has been around for less than two thousand years. The Greeks did not have the zero. The Babylonians did, but they did not consider it a number, but a place holder in their base 60 positional system. Zero is the Nothing that is Something. It is the number that can be added to any other number without changing it. Zero makes addition *general*. It also makes subtraction sensible.

1 + 0 = 10

1 +0 + 0 = 100

This is so easy a caveman can do it.

--Brant

[Chris Grieb]

Bob; Wasn't zero something the Arabs had something to do with?

I wonder if anyone asked Ayn Rand about that idea?

[BaalChatzaf]

1 + 0 = 10

1 + 0 + 0 = 100

This is so easy a caveman can do it.

--Brant

I think no. Cavemen where not into positional notation. If they had numbers at all they did plain old tally.

Ba'al Chatzaf

[BaalChatzaf]

Bob; Wasn't zero something the Arabs had something to do with?

I wonder if anyone asked Ayn Rand about that idea?

The Arab, Hindu and Chinese mathematicians developed the zero. The Babylonians used it as a place holder. They did not regard it as a number, but as a notational trick. Strangely, the Greeks missed out on it. Even Archimedes did not hit on the zero when he wrote -The Sand Reckoner-. If he had we would have had positional notation 2000 years earlier.

Ba'al Chatzaf

Edited September 12, 2007 by BaalChatzaf

[Brant Gaede]

1 + 0 = 10

1 + 0 + 0 = 100

This is so easy a caveman can do it.

--Brant

I think no. Cavemen where not into positional notation. If they had numbers at all they did plain old tally.

Ba'al Chatzaf

Thanks for correcting my spacing.

--Brant

[Ellen Stuttle]

The Arab, Hindu and Chinese mathematicians developed the zero. The Babylonians used it as a place holder. They did not regard it as a number, but as a notational trick. Strangely, the Greeks missed out on it. Even Archimedes did not hit on the zero when he wrote -The Sand Reckoner-. If he had we would have had positional notation 2000 years earlier.

The Greeks used letters of the alphabet for numbers, which I expect is why they wouldn't have thought of the zero.

http://www.fargonasphere.com/piso/numcode.html

The site has a list of the numbers and their corresponding characters.

Ancient Greek had no cyphers (characters 0-9) as we have today. Instead, the alphabet itself was made to serve a dual purpose. Thus a given character had its corresponding number. For example, the Alpha character also stood for the number 1.

Ellen

___

[BaalChatzaf]

Ancient Greek had no cyphers (characters 0-9) as we have today. Instead, the alphabet itself was made to serve a dual purpose. Thus a given character had its corresponding number. For example, the Alpha character also stood for the number 1.

Ellen

In -The Sand Reckoner- Archimedes developed an exponential system of quantities. This is very close to positional notation for numbers. Alas Archimedes did not have the zero and he did not develop an algebra. That was to come 1200 years later with the Muslim scholars.

The mathematician Diophantis (thirid century c.e.) developed a shorthand notation, but it was not quite algebra.

He did not conceive of zero as a number either.

Ba'al Chatzaf

[Ellen Stuttle]

In -The Sand Reckoner- Archimedes developed an exponential system of quantities. This is very close to positional notation for numbers. Alas Archimedes did not have the zero and he did not develop an algebra. That was to come 1200 years later with the Muslim scholars.

The mathematician Diophantis (thirid century c.e.) developed a shorthand notation, but it was not quite algebra.

He did not conceive of zero as a number either.

Ba'al Chatzaf

Did either of them have an idea of negative numbers?

Ellen

___

[BaalChatzaf]

In -The Sand Reckoner- Archimedes developed an exponential system of quantities. This is very close to positional notation for numbers. Alas Archimedes did not have the zero and he did not develop an algebra. That was to come 1200 years later with the Muslim scholars.

The mathematician Diophantis (thirid century c.e.) developed a shorthand notation, but it was not quite algebra.

He did not conceive of zero as a number either.

Ba'al Chatzaf

Did either of them have an idea of negative numbers?

Ellen

___

AFIK, neither did. Negative numbers were invented during the Renaissance to handle debits and credits.

Ba'al Chatzaf

[Guyau]

More on Numbers

“The Nature of Numbers” by Merlin Jetton

http://objectivity-archive.com/volume1_number1.html#1

ABSTRACT

Jetton argues that the concept number is a certain way of specifying quantity. Regarding numbers as attributes, he takes numbers to lie in the concretes of the world independently of consciousness. Regarding numbers as entities, he takes numbers to be essentially dependent on conceptual consciousness, but with moorings to concretes. Jetton presents some of those interpretative moorings—sensitive to child development and the development of number concepts in human history—for these kinds of numbers: the positive and negative integers, zero, the irrationals, the imaginaries, and the reals. Orders of mathematical infinity are introduced, and the distinction of number from number base is delineated. Jetton criticizes the following philosophical perspectives of numbers: Platonism, formalist nominalism, and Kantian conceptualism. He evaluates these against his ontology and epistemology of numbers as attributes and as entities.

“The Complexion of Numbers” by David Ross

http://objectivity-archive.com/volume1_number3.html#101

“From the Ground Up” by Paul Enright

http://objectivity-archive.com/volume1_number5.html#151

[Brant Gaede]

In -The Sand Reckoner- Archimedes developed an exponential system of quantities. This is very close to positional notation for numbers. Alas Archimedes did not have the zero and he did not develop an algebra. That was to come 1200 years later with the Muslim scholars.

The mathematician Diophantis (thirid century c.e.) developed a shorthand notation, but it was not quite algebra.

He did not conceive of zero as a number either.

Ba'al Chatzaf

Did either of them have an idea of negative numbers?

Ellen

___

AFIK, neither did. Negative numbers were invented during the Renaissance to handle debits and credits.

Ba'al Chatzaf

By those pesky Jews? Not satisfied with the positive they had to go negative. Hence the world we live in!

--Brant

[BaalChatzaf]

More on Numbers

“The Nature of Numbers” by Merlin Jetton

http://objectivity-archive.com/volume1_number1.html#1

ABSTRACT

Jetton argues that the concept number is a certain way of specifying quantity. Regarding numbers as attributes, he takes numbers to lie in the concretes of the world independently of consciousness. Regarding numbers as entities, he takes numbers to be essentially dependent on conceptual consciousness, but with moorings to concretes. Jetton presents some of those interpretative moorings—sensitive to child development and the development of number concepts in human history—for these kinds of numbers: the positive and negative integers, zero, the irrationals, the imaginaries, and the reals. Orders of mathematical infinity are introduced, and the distinction of number from number base is delineated. Jetton criticizes the following philosophical perspectives of numbers: Platonism, formalist nominalism, and Kantian conceptualism. He evaluates these against his ontology and epistemology of numbers as attributes and as entities.

“The Complexion of Numbers” by David Ross

http://objectivity-archive.com/volume1_number3.html#101

“From the Ground Up” by Paul Enright

http://objectivity-archive.com/volume1_number5.html#151

Stephen, with all due respect. Numbers live in our heads, (In Here, not Out There). If there were no sentient beings in the cosmos, there would be no numbers. We must not confuse our imaginations with facts. The attraction that civilized humans have for mathematical truth is a kind of narcisism. We are in love with some of our ideas. We love some of our ideas so much we make them out to be gods.

Ba'al Chatzaf