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Source: from the TTC video course High School Algebra I with Monica Neagoy

Listen: short audio clip

Transcript:

So, that's numeration. Language pushed forward the evolution of numbers. But in these first two phases which are very old, what's very important to realize is that the numbers were not independent of the objects counted. So, when there were five objects, for example, there was no concept of the number five. They were just the five objects—five sheep, five people, five whatever. Then came number. The concept of number was a big step forward. It's not clear from history because we don't have enough information and research has not been able to determine when number first came into being—the concept of number. But it was clearly present in many of the great civilizations.

As you can see, there is no article, neither definite nor indefinite, in front of the word number in those two cases that I highlighted for you. How do you think that can be explained? I'm not asking you to explain this from a grammatical perspective per se, but rather from the perspective of why in this case it feels, to you as native speakers, like the right choice of words.

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    I think it is mentioning "number" as a concept, not actual numbers or numbering systems. Something like "Then came numbering.". Does sound odd though. – user3169 Oct 18 '15 at 21:55
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    I guess it could also be seen as an example of personification - perhaps in the spirit of historic peoples to have a personification of various abstract things like luck (Fortuna), the narrator wants to sound more interesting this way. Your transcript should use a capital N to reflect this - perhaps then it becomes more clear to you. – Sanchises Oct 18 '15 at 22:03
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    If I were to transcribe it, I may transcribe it as: "... Language pushed forward the evolution of numbers. But ..., what's very important to realize is that the numbers were not independent of the objects counted. So, when there were five objects, for example, there was no concept of the number five. They were just the five objects ..., five whatever. Then came number. The concept of number was a big step forward. It's not clear from history because we don't have enough information and research has not been able to determine when number first came into being the concept of number." – Damkerng T. Oct 18 '15 at 22:15
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As has already been noted in the comments the speaker here is using 'number' to refer to a concept - specifically to very basic set theory.

Cardinal numbers (1,2,3 etc) are the way we describe the size of sets, the sets of objects she refers to in the presentation

Swap 'number' for cardinality and it makes sense (but doesn't read as well).

While the use is uncommon it does rate a mention in the full OED:

The faculty of reckoning or calculating

Supported by a quote from none other than Noam Chomsky:

Birds may have certain limited capacities to match arrays of not too many items, but that has nothing to do with the faculty of number.

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