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I am reading "Calculus on Manifolds" by Michael Spivak.
This book is a famous mathematics book.

In this book, the author wrote the following sentence:

R^1, R^2, R^3 are often called the line, the plane, and space, respectively.

I wonder why the author didn't write as follows:

R^1, R^2, R^3 are often called the line, the plane, and the space, respectively.

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The definite article, "the", refers to a specific instance of a grammatically countable type of object that both the listener and speaker are expected to know about.

  • There exist many lines, but R^1 is the line on which we do 1-dimensional math on.
  • There exist many planes, but R^2 is the plane on which we do 2-dimensional math on.
  • However, there are not many "spaces" in this way.

We do have a word for "spaces", meaning multiple open areas with a boundary, but as far as math is concerned, "space" has no boundary, it goes on infinitely in all directions. In this form, "space" is uncountable, because in our world and experience, there cannot be more than one, and it encompasses everything. As such, it does not receive the definite article, or any article at all. Just "space".

Now, astute mathematicians reading this may note that mathematically, we can talk about R^4, R^5, and so forth very easily. From an R^4 perspective it might make sense to discuss multiple infinite "spaces", of which R^3 would be the space we do math on. However, in this, math follows English, and English has no understanding or concept of dimensions higher than 3-- we don't really have words to describe such things, and mathematicians who need to do so generally use symbols or long form descriptions instead of English words. You can consider this a historical feature of the language, if you like.

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    The line and the plane don't have boundaries, either. Commented Jun 9, 2023 at 6:34
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    The line and the plane can both be viewed as subsets of 3-space, and therefore there is a boundary between what is part of the line and what isn't, although this boundary is colinear with the line, and coplanar with the plane (the plane can be viewed as having two boundaries, a top and a bottom, with different orientation but the same location).
    – Stuart F
    Commented Jun 9, 2023 at 11:04
  • Richard Winters, Thank you very much for your answer. @JackO'Flaherty Thank you very much for your comment. gotube, Thank you very much for your edit.
    – tchappy ha
    Commented Jun 9, 2023 at 11:29
  • @StuartF Thank you very much for your comment.
    – tchappy ha
    Commented Jun 9, 2023 at 11:29
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    @StuartF In that case, one can postulate the same sort of implied boundary between 3-space and other spaces. I think the reason for not saying "the space" is that it's unusual and uncomfortable. Commented Jun 9, 2023 at 17:11

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