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The Borsuk–Ulam theorem proofs that on earth, there will always be at least two points that have exactly the same temperature at once. This can be mathematically proven to be true and it indeed has been proven.

Is this knowledge a priori or a posteriori?

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    Borsuk-Ulam theorem proves nothing a priori about Earth, it proves something about continuous maps of a sphere. That Earth is a sphere (actually, not quite), or that temperature can be modeled by such a map (actually, strictly speaking, it can't be, it is not even defined at every "point") is certainly not a priori. "As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality", Einstein. You can replace "certain" with "a priori".
    – Conifold
    Aug 12 '20 at 11:05
  • @Conifold Why not just post your comment as an answer? Aug 12 '20 at 16:07
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Conifold has provided a very good answer, so he/she should be credited for it. I'll just add the following clarification. Your question contains two parts:

  1. Every continuous function from an n-sphere into Euclidean n-space maps some pair of antipodal points to the same point (Borsuk-Ulam theorem).
  2. The Borsuk-Ulam theorem is applicable to the earth and its temperature.

The first statement can be considered to be a priori knowledge as it does not depend on empirical investigation to determine its truth. The second statement does require empirical research (e.g. is the earth a sphere, in euclidean space,...), hence, if true, would be a posteriori (empirical) knowledge.

In general, mathematics, without observation, does not say anything about the world.

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