Science states that space is endless, and ever expanding. But, if we are inside the planet earth, the planet earth is inside the galaxy, the galaxy is inside space, then what is space inside? What is it expanding in? And if space is endless, how can it expand?

Space is not expanding "in" anything else. The distances between points in space are increasing, but not because they are moving through some "superspace" that contains space.

Mathematicians distinguish between two different approaches to defining geometric properties of a space: the extrinsic approach and the intrinsic approach. The extrinsic approach involves relating the space to some larger space that it sits inside; the intrinsic approach makes use of only the space itself, and not some larger space that it sits inside.

For example, suppose we want to study the curvature of the surface of the earth. One way to see that the surface of the earth is curved is to image a flat plane tangent to the surface of the earth at some point. We can detect and measure the curvature of the surface of the earth by noting that the surface deviates from the tangent plane, and measuring the size of this deviation. But this deviation takes place within the 3-dimensional space that the surface of the earth is embedded in, so this is an extrinsic measure of the curvature. The curvature can also be detected by making measurements that take place entirely on the surface of the earth. For example, if you lay out a large triangle on the surface of the earth and measure the angles of the triangle, you will find that they add up to more than 180 degrees. This measurement makes no reference to a larger space containing the earth's surface, so it is an intrinsic measure of the curvature of the surface.

Cosmologists use only the intrinsic approach when discussing the geometry of spacetime. Thus, none of this discussion involves any reference to a larger space that spacetime sits inside. Although they may use words that seem to suggest such a larger space, such as "expansion" or "curvature", those words are always being used to refer to some intrinsic property of spacetime itself, and not some relationship between spacetime and a larger space.

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