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To me, it is not that hard to imagine an alternate universe where the fact that $H_0(GL_n(\mathbb{R})) = H_0(O_n(\mathbb{R})) = \mathbb{Z}^2$ is an unstable fact that holds for small $n$, but not in very high dimensions. In such a universe, determinant would only be defined for small square matrices, and $\bigwedge^n \mathbb{R}^n$ would be $0$-dimensional for large $n$.

By way of analogy, suppose we lived in a two dimensional universe. It would be completely intuitive to us that $\pi_1(SL_2)$ was $\mathbb{Z}$. Elementary school textbooks would say something like "every person, throughout their life, has made a certain number of full turns to the left, and a certain number of full turns to the right, and the difference between these numbers is called the 'winding number'." It would then be extremely surprising when we studied three dimensional geometry (that arcane, counterintuitive subject!) and learned that winding number is only defined modulo $2$.

To me, it is not that hard to imagine an alternate universe where the fact that $|\pi_0(GL_n(\mathbb{R}))| = |\pi_0(O_n(\mathbb{R}))| = 2$ is an unstable fact that holds for small $n$, but not in very high dimensions. In such a universe, determinant would only be defined for small square matrices, and $\bigwedge^n \mathbb{R}^n$ would be $0$-dimensional for large $n$.

By way of analogy, suppose we lived in a two dimensional universe. It would be completely intuitive to us that $\pi_1(SL_2)$ was $\mathbb{Z}$. Elementary school textbooks would say something like "every person, throughout their life, has made a certain number of full turns to the left, and a certain number of full turns to the right, and the difference between these numbers is called the 'winding number'." It would then be extremely surprising when we studied three dimensional geometry (that arcane, counterintuitive subject!) and learned that winding number is only defined modulo $2$.

To me, it is not that hard to imagine an alternate universe where the fact that $H_0(GL_n(\mathbb{R})) = H_0(O_n(\mathbb{R})) = \mathbb{Z}/2$ is an unstable fact that holds for small $n$, but not in very high dimensions. In such a universe, determinant would only be defined for small square matrices, and $\bigwedge^n \mathbb{R}^n$ would be $0$-dimensional for large $n$.

By way of analogy, suppose we lived in a two dimensional universe. It would be completely intuitive to us that $\pi_1(SL_2)$ was $\mathbb{Z}$. Elementary school textbooks would say something like "every person, throughout their life, has made a certain number of full turns to the left, and a certain number of full turns to the right, and the difference between these numbers is called the 'winding number'." It would then be extremely surprising when we studied three dimensional geometry (that arcane, counterintuitive subject!) and learned that winding number is only defined modulo $2$.

To me, it is not that hard to imagine an alternate universe where the fact that $H_0(GL_n(\mathbb{R})) = H_0(O_n(\mathbb{R})) = \mathbb{Z}^2$ is an unstable fact that holds for small $n$, but not in very high dimensions. In such a universe, determinant would only be defined for small square matrices, and $\bigwedge^n \mathbb{R}^n$ would be $0$-dimensional for large $n$.

By way of analogy, suppose we lived in a two dimensional universe. It would be completely intuitive to us that $\pi_1(SL_2)$ was $\mathbb{Z}$. Elementary school textbooks would say something like "every person, throughout their life, has made a certain number of full turns to the left, and a certain number of full turns to the right, and the difference between these numbers is called the 'winding number'." It would then be extremely surprising when we studied three dimensional geometry (that arcane, counterintuitive subject!) and learned that winding number is only defined modulo $2$.

To me, it is not that hard to imagine an alternate universe where the fact that $H_0(GL_n(\mathbb{R})) = H_0(O_n(\mathbb{R})) = \mathbb{Z}^2$ is an unstable fact that holds for small $n$, but not in very high dimensions. In such a universe, determinant would only be defined for small square matrices, and $\bigwedge^n \mathbb{R}^n$ would be $0$-dimensional for large $n$.

By way of analogy, suppose we lived in a two dimensional universe. It would be completely intuitive to us that $\pi_1(SL_2)$ was $\mathbb{Z}$. Elementary school textbooks would say something like "every person, throughout their life, has made a certain number of full turns to the left, and a certain number of full turns to the right, and the difference between these numbers is called the 'winding number'." It would then be extremely surprising when we studied three dimensional geometry (that arcane, counterintuitive subject!) and learned that winding number is only defined modulo $2$.

To me, it is not that hard to imagine an alternate universe where the fact that $H_0(GL_n(\mathbb{R})) = H_0(O_n(\mathbb{R})) = \mathbb{Z}/2$ is an unstable fact that holds for small $n$, but not in very high dimensions. In such a universe, determinant would only be defined for small square matrices, and $\bigwedge^n \mathbb{R}^n$ would be $0$-dimensional for large $n$.

By way of analogy, suppose we lived in a two dimensional universe. It would be completely intuitive to us that $\pi_1(SL_2)$ was $\mathbb{Z}$. Elementary school textbooks would say something like "every person, throughout their life, has made a certain number of full turns to the left, and a certain number of full turns to the right, and the difference between these numbers is called the 'winding number'." It would then be extremely surprising when we studied three dimensional geometry (that arcane, counterintuitive subject!) and learned that winding number is only defined modulo $2$.