# What is wrong with this argument?

#### 0bydefinition

This is an excerpt from a manuscript by William James Sidis published in 1920 called "The Animate and the Inanimate":

But there is one outstanding objection to this theory that the stellar universe is infinite. There may be supposed to be no reason why the average brightness of stars should be any different in one part of space from what is in any other part; multiplying this average brightness by the average number of stars per unit volume (the average star-density that we suppose for the infinite universe), we will get the average amount of light issuing from a unit volume anywhere in space; let us call this product L. Now, as the apparent brightness of any source of light is inversely proportional to the square of the distance between that source and the observer, then, if we call that distance d, the average apparent brightness of a unit of volume at distance d from the observer could be represented as L/d². If we divide space into an infinite number of concentric spherical shells, with the observer at the center, each with equal thickness, let us say the unit distance divided by 4n, then, especially when the sphere is very large, the volume of each shell is approximately d². Multiplying the average apparent brightness of a unit volume at distance d by the volume of the shell of distance d, we find that the volume of each such shell is a constant, L. Since the stellar universe consists of an infinite number of such shells, each of which has the same apparent brightness, it follows that the brightness of the sky, or indeed of the smallest part of it, must be altogether infinite. The consequence of the theory of an infinite universe is obviously contradicted by facts.

Now, there is obviously some fallacy in this argument, but I can't find it. Any thoughts would be appreciated.

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#### mathman

One problem with this is that the universe is 13.7 billion years old, so that no light can reach us from more than 13.7 billion light years away.

#### 0bydefinition

One problem with this is that the universe is 13.7 billion years old, so that no light can reach us from more than 13.7 billion light years away.
Yes, of course. The "observable universe". But that doesn't readily refute Sidis' argument.

By the way, I think there are two typos. The first one is in the sentence "let us say the unit distance divided by 4n". The "n" isn't supposed to be there. And the second when he says "we find that the volume of each such shell is a constant"; what he means is apparent brightness not volume.

The volume of each shell behaves like d^2 - d, so when d is very large, the apparent brightness of each shell is indeed approximately L. Since lim {L + L/d} as d-> Inf equals L.
Is there a perfectly obvious reason why the whole argument is fallacious?

Staff Emeritus
Yes, of course. The "observable universe". But that doesn't readily refute Sidis' argument.
Of course it does. If I replace 13 billion light years with 13 feet, doesn't that make it obvious?

#### DaveC426913

Gold Member
Now, there is obviously some fallacy in this argument, but I can't find it. Any thoughts would be appreciated.
Why does there have to be some fallacy in his argument? Sidi is arguing against an infinite universe. He is merely putting numbers to Olbers' Paradox.

#### vanesch

Staff Emeritus
Gold Member
Another version of Olber's paradox, right ?

edit: oops, Dave said this already.

#### 0bydefinition

Of course it does. If I replace 13 billion light years with 13 feet, doesn't that make it obvious?
I don't understand why it would refute the argument. If nothing outside the observable universe can reach us, then wouldn't that be only in agreement with his observation that the brightness of the sky isn't infinite?

#### 0bydefinition

Why does there have to be some fallacy in his argument? Sidi is arguing against an infinite universe. He is merely putting numbers to Olbers' Paradox.
Precisely because he is arguing against an infinite universe makes me doubt the validity of his argument.

And, yes, now I understand it. The argument is equivalent to Olbers' paradox.

#### A.T.

I don't understand why it would refute the argument.
Does he argue that the entire universe cannot be infinite, or that just the observable universe cannot be infinite? Or doesn't he care because he assumes a static universe? An evolving & expanding universe can be infinite without a full bright sky.

#### vanesch

Staff Emeritus
Gold Member
The reasoning is: from (premise) follows (conclusion).

The reasoning is correct, that is from the premise does follow the conclusion.

The premise is a static, infinite universe with a constant density (even small) of active, static stars.

The conclusion is: the night sky must be bright as hell.

Now, the conclusion is false. Hence, given the correctness of the reasoning, the premise must be false.

And that's true, the premise is false: the universe isn't infinite AND static AND filled with a finite constant density of static stars.

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