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Assuming that the Earth is what everything revolves cold be philosophical, but it's not Real. Why privilege the Earth (as some might say)? Why not Mars or Jupiter?

This seems to be more of a screed about Jesuits and other Catholics than about Astronomy.

If one is a mathematician, he can put his origin anywhere, but if one is an astronomer, the Sun get s the most useful origin.

I think that in the time period you're talking about (Copernicus &c), it was not scientism [did that begin with the Enlightenment] but religionism that prevailed (Xism: the idea that X has All the answers, no point looking elsewhere).

Extrapolating a bit further, are we supposed to believe this nonsense that the Earth is [almost] spherical when our very senses tell us that it's flat?

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The 'Enlighteners' claimed and still claim that if you criticse Copernicus you are ill-informed and probably religious - the supposed conflict between the 'rationalists' and the religious. Criticising Copernicus or Galileo has little to do with a religious viewpoint. The Jesuits (amongst many other religious) have been prominent in developing all the current strands of 'Science' and Scientism, so the claim is rather bizarre and that is why I included it. Copernican theory itself is very religious - Platonic in fact. Indeed as you said, you can create a model where Jupiter is the barycentre of the solar system and universe and develop maths and elliptical quants to support that view. You could base this on philosophy or say a Roman cult. What has always been missing for Copernican theory is the mechanical, visual proofs. Given Einstein's ultimate Scientism of Relativity the Science-narrative has conveniently removed that from being a serious consideration.

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Proofs? Which proofs are stronger: Sun-centered or something else? We manage to scoot around the Solar System and beyond based on the >>>current model<<< of the Solar System.

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oof. Copernicus doesn't care if you're convinced.

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