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Dinesh

Dinesh club

Posted: 02 Dec 2023

Taken: 06 Mar 2019

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Poincare Conjecture •
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Infinity

Infinity

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 Dinesh
Dinesh club
Although the Greeks believed that the earth was spherical, many of them felt the universe extended infinitely. Here, for example is Archytas, en.wikipedia.org/wiki/Archytas the very accomplished Pythagorean mathematician and friend of Plato, on the subject:

“If I were at the outside, say at the heaven of the fixed stars, could I stretch my hand or my stick outward or not? To suppose that I could not is absurd: and if I can stretch it out, that which is outside must be either body or space (it makes no difference which it is as we shall see). We may then in the same way get to the outside of that again, and so on, asking on arrival at each new limit the same question; and if there is always a new place to which the stick may be held out, this clearly involves extension without limit. If now what so extends is body, the proposition is proved; but even if it is space, then, since space is that in which body is or can be, and in the case of eternal things we must treat that which potentially is as being, it follows equally that there must be body and space extending without limit.”

Archytas is arguing that there is no boundary to the universe – wherever we are in the universe, he argues, when we look in the sky, apart from whatever bodies it contains, it will look roughly the same. We won’t see an edge – wherever we are, we can stick a hand or stick out. Since there is no boundary, he concludes erroneously that the universe must be infinite. To see why his argument is flawed, note that we could repeat the same argument on the face of the Earth. When we stand outdoors, any place on our planet, we can stretch out a hand or stick horizontally and no barrier prevents us from extending our own body in this fashion. There is no boundary, no edge. If Archytas’s conclusion were correct, the Earth would extend infinitely in all directions. But it does not. It is a sphere. (It could even have been a torus.) To say that the universe has no boundary is not to say that it goes on infinitely, just as to say that the Earth has no edge is not to say that it goes on forever.

It may be that the Universe goes on forever, but it seems very unlikely. Space and matter are intimately related, and the assertion that the universe has an infinite amount of matter causes serious theoretical problems. The universe could also have a boundary of some kind, but this is bit like assuming that the world is a disk with an edge that one could fall off. Few scientists with mathematical training seriously believe this either.
5 months ago.
 Dinesh
Dinesh club
Poincare Conjecture
12 days ago.

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