# What's a number?

Discussion in 'Off-Topic' started by Souron, Apr 10, 2010.

1. ### warpusIn pork I trust

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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.

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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.

3. ### BirdjaguarHanafubukiRetired ModeratorSupporter

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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.

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5. ### BirdjaguarHanafubukiRetired ModeratorSupporter

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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.

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

7. ### BirdjaguarHanafubukiRetired ModeratorSupporter

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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".

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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".

9. ### BirdjaguarHanafubukiRetired ModeratorSupporter

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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.

10. ### SouronThe Dark Lord

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@Birdjaguar
What's a true number?

11. ### BirdjaguarHanafubukiRetired ModeratorSupporter

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

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

13. ### cardgameSensual Kitten

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then it is not a number

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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.

15. ### SouronThe Dark Lord

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So what's a true number?

16. ### warpusIn pork I trust

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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.

17. ### BirdjaguarHanafubukiRetired ModeratorSupporter

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

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

19. ### SouronThe Dark Lord

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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.

20. ### BirdjaguarHanafubukiRetired ModeratorSupporter

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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.