What's a number?

[warpus]

In that negative numbers represent debt (or going below the zero of a scale, e.g. temperature).

Fractions used when cutting cakes or pies into equal sized pieces.

Yeah, the basic properties of numbers often have corresponding parallels found in nature, but that's not to say that all of their properties will have such parallels.

 

[ParadigmShifter]

There's only a few properties that we insist on though.

a + b = b + a
a * b = b * a
a * ( b + c ) = a*b + a*c
a + 0 = 0 + a = a
a * 1 = 1 * a = a
(a+b)+c = (a+b)+c
(a*b)*c = a*(b*c)

Those can be shown using pictures for positive integers and those properties are extrapolated for larger sets of numbers. Pictures aren't rigorous enough though so they form the basic axioms of arithmetic.

 

[Birdjaguar]

Numbers and abstract counting were invented 5000 years ago in Sumeria when the pictograph for "one jar of oil" became two: "1 [and] jar of oil". From that everything else mathematical flowed.

 

[ParadigmShifter]

They have found notches carved on bones that predate that though.

EDIT:

http://en.wikipedia.org/wiki/Ishango_bone

 

[Birdjaguar]

They have found notches carved on bones that predate that though.

Tally sticks date back 10s of thousands of years. They are not numbers though. They are a precursor to concrete counting which appeared about 10,000 years ago.

 

[ParadigmShifter]

In what way are they not numbers? They represent numbers do they not?

 

[Birdjaguar]

No, they represent a one-to one correspondence between the mark and some unknown event. Kill an animal, make a mark; kill an animal, make a mark; etc. There is no actual counting involved. They are records of "events".

 

[ParadigmShifter]

How can you be sure of that? It's prehistoric after all.

A one-to-one correspondence to the natural numbers is exactly how we define something being "countable".

 

[Birdjaguar]

The progression from tally sticks to concrete counting and then to abstract counting is clear in the archaeological record in the Mideast. The changes seem to follow the increased complexity of community life and the need for record keeping. If tallies actually represented abstract counting, why does the archaeological/cultural record show 5000 years of concrete counting before the first true numbers appear?

I am not at home at the moment, but tonight, I'll post a link to some more info on all this.

 

[Souron]

@Birdjaguar
What's a true number?

 

[Birdjaguar]

@Birdjaguar
What's a true number?

I don't understand your question.

"true number" doesn't mean anything to me.

 

[Perfection]

There's only a few properties that we insist on though.

a + b = b + a
a * b = b * a
a * ( b + c ) = a*b + a*c
a + 0 = 0 + a = a
a * 1 = 1 * a = a
(a+b)+c = (a+b)+c
(a*b)*c = a*(b*c)

Those can be shown using pictures for positive integers and those properties are extrapolated for larger sets of numbers. Pictures aren't rigorous enough though so they form the basic axioms of arithmetic.

Not all numbers follow these properties. For example, quaternions are non-commutative in multiplication.

 

[cardgame]

then it is not a number

 

[ParadigmShifter]

Quaternions are numbers in a way.They just don't form a a field under addition and multiplication (they form a division ring). The can be represented by 2x2 matrices with entries in the complex field though. (Complex numbers are isomorphic to a subset of 2x2 real matrices).

Octonians aren't even associative

The axioms I described are just the properties of the integers that apply up to complex numbers.

 

[Souron]

I don't understand your question.

"true number" doesn't mean anything to me.

The progression from tally sticks to concrete counting and then to abstract counting is clear in the archaeological record in the Mideast. The changes seem to follow the increased complexity of community life and the need for record keeping. If tallies actually represented abstract counting, why does the archaeological/cultural record show 5000 years of concrete counting before the first true numbers appear?

I am not at home at the moment, but tonight, I'll post a link to some more info on all this.

So what's a true number?

 

[warpus]

Quaternions are numbers in a way.They just don't form a a field under addition and multiplication (they form a division ring). The can be represented by 2x2 matrices with entries in the complex field though. (Complex numbers are isomorphic to a subset of 2x2 real matrices).

Octonians aren't even associative

The axioms I described are just the properties of the integers that apply up to complex numbers.

Which I think is another indicator that while there are parallels between numbers and nature, there isn't really anything deep to the relationship that some are suggesting.

 

[Birdjaguar]

So what's a true number?

I looked to see if I had used the phrase but didn't see that post. In my post it is a non technical term that refers to the introduction of abstract counting though the use of "symbols" that represent the concept of "twoness" or "fiveness" etc. It It marks the transition from concrete counting to abstract counting: the ties between the quantity and object have been severed. So "true" in this context means the abstract concept with which we are familiar today. Perhaps this would have been a clearer post:

The progression from tally sticks to concrete counting and then to abstract counting is clear in the archaeological record in the Mideast. The changes seem to follow the increased complexity of community life and the need for record keeping. If tallies actually represented abstract counting, why does the archaeological/cultural record show 5000 years of concrete counting between the use of tally sticks and the first appearance of numerals?

The use of numerals by the Sumerians in 3100 BCE are our first evidence of abstract counting.

 

[ParadigmShifter]

Numerals are just a shorthand for numbers. A tally represents a number just as well.

 

[Souron]

Tallies are numbers, but the problem with them is that they get unwieldy with large numbers. So naturally, numerals were invented only when there was a need to precisely count large quantities.

Also, I recently read an article about how humans naturally count logarithmically, not linearly. So the use of tally marks is a revolution itself. Particularly if they are use to count past 4.

 

[Birdjaguar]

Numerals are just a shorthand for numbers. A tally represents a number just as well.

I don't think so. A tally represents an event: one mark every time something happens. We count them because we use abstract counting as a part of our everyday life.

How do X notches on a stick or bone lead you to believe that the maker understood abstract counting? Are there paleolithic tally sticks with totals? Your familiarity with numbers is leading you to believe that a tally stick maker counted 1,2,3... as marks were made. There is no evidence of this at all. In fact the evidence tends to point to only the use of concrete counting prior to 3100 BCE.

People appear to be born to compute. The numerical skills of children develop so early and so inexorably that it is easy to imagine an internal clock of mathematical maturity guiding their growth. Not long after learning to walk and talk, they can set the table with impressive accuracy--one plate, one knife, one spoon, one fork, for each of the five chairs. Soon they are capable of noting that they have placed five knives, spoons, and forks on the table--and, a bit later, that this amounts to fifteen pieces of silverware. Having thus mastered addition, they move on to subtraction. It seems almost reasonable to expect that if a child were secluded on a desert island at birth and retrieved seven years later, he could enter a second-grade mathematics class without any serious problems of intellectual adjustment.

Of course, the truth is not so simple. This century, the work of cognitive psychologists, notably Jean Piaget, has illuminated the subtle forms of daily learning on which intellectual progress depends. Piaget observed children at play as they slowly grasped--or, as the case might be, bumped into---concepts that adults take for granted, as they refused, for instance, to concede that quantity is unchanged as water pours from a short stout glass into a tall thin one. Psychologists have since demonstrated that young children, asked to count the pencils in a pile, readily report the number of blue or red pencils, but must be coaxed into finding the total. Such studies have suggested not only that the rudiments of mathematics are mastered gradually, and with effort, but that the very concept of abstract numbers--the idea of a oneness, a twoness, a threeness that applies to any class of objects and is a prerequisite for doing anything more mathematically demanding than setting a table--is itself far from innate.

This observation draws support from linguistics and anthropology, in particular from the study of cultures that have evolved in isolation from modern society. Anthropologists have found that when a Vedda tribsman, of Sri Lanka, wanted to count coconuts, he would collect a heap of sticks and assign one to each coconut. Every time he added a new stick he said, "That is one". But if asked how many coconuts he possessed, he could only point to the pile of sticks and say, "That many", for the Vedda had no words devoted to expressing quantities. Thus, while capable of a kind of counting--counting in one-to-one correspondence, rather like the child setting the table-- the Vedda apparently had no conception of numbers that exist independently of sticks and coconuts and can be applied to either without reference to the other.

---Denise Schmandt-Besserat