{\huge Why Is There Something?}

Or is there everything? \vspace{.3in}

Jim Holt is the author of the modestly titled book, Why Does The World Exist? He has---as his book's back cover says---written on string theory, time, infinity, numbers, truth, and even stercus taurinum, to use a learned translation of what the back cover actually says.

Today I thought it would be fun to ponder the main question he raises in his book.

The book is beautifully written, ponders deep questions, yet is fun to read. Holt is clearly a great writer who can make metaphysics seem accessible to those of us who are complete novices. The structure of the book is a tour, a literal tour, around the world; he talks to various great minds about his questions. This approach makes the book readable, accessible, and real. Grounding such a philosophical question in discussions with real people is a cool idea.

As you read the chapters, you journey around the world and get to hear from among others: David Deutsch, Andre Linde, Alex Vilenkin, Steven Weinberg, Roger Penrose, and the late John Updike.

The Question

Holt is most interested in the question: Why is there something rather than nothing? A pretty basic question. Perhaps more fundamental that our simple question like $latex {\mathsf{P}=\mathsf{NP}}&fg=000000$? I guess if there were nothing, then who would care whether $latex {\mathsf{P}}&fg=000000$ equals $latex {\mathsf{NP}}&fg=000000$ or not?

A Quick Proof

He gives a ``proof'' for those who are busy at the very front of the book: Suppose there were nothing. Then there would be no laws; for laws, after all, are something. If there were no laws, then everything would be permitted. If everything were permitted, then nothing would be forbidden. So if there were nothing, nothing would be forbidden. Thus nothing is self-forbidding.

Therefore, there must be something. Q.E.D.

Another Argument

One proof I found especially interesting is based on the subtraction argument. Apparently pioneered in the 1990's by British and American philosophers, this argument attempts to show that there could be a complete void---nothingness.

The arguments makes three assumptions. The world consists of a finite number of objects: people, tables, chairs, and so forth. It also supposes that each object is contingent. That is although the object does exist, it could actually have not existed. Holt says: ``This too seems plausible.'' Finally the argument assumes independence: the nonexistence of one object does not force the existence of another object.

The subtraction argument is now easy. Just start imaging that each of the objects does not exist. This is not meant in a literal sense, but one can perhaps imagine that each object in turn does not exist. Clearly, given the assumptions this eventually leads to a void of nothingness.

Pretty simple argument from a math viewpoint. Holt asks: But are the premises of the subtraction argument true?

He points out that the first two assumptions seem okay to him. But the independence one is as he says, ``more dubious.'' He worries that the removal of one object might force the existence of other objects. This clearly would destroy the argument.

But Wait $latex {\dots}&fg=000000$

What I like about the subtraction argument is that there is a repair if independence is weakened. Suppose that the removal of an object actually forces the existence of another object. This seems like a strong obstacle to the argument. But suppose that the situation is this: The world consists of a finite number of objects

$latex \displaystyle {\cal O}_{1}, \dots, {\cal O}_{N}. &fg=000000$

A simple concrete example is imagine that objects are divided into a finite number of types. The independence rule now says that the removal of an object $latex {\cal O}&fg=000000$ of type $latex {\tau}&fg=000000$ can add any finite number of objects of types less that $latex {\tau}&fg=000000$. Then it is easy to see that the subtraction process works provided you remove the highest types first.

This is nothing more than an argument about ordinals. Of course there are ways to order the objects of the world so that the subtraction always stops, but that this is hard to prove. See this note by Andrej Bauer on ``The hydra game'' for a discussion of such situations.

According to a paper by Aviv Hoffman, the ``Existential Subtraction'' (ES) premise is expressed via possible-worlds semantics as:

There is a finite world $latex {w_1}&fg=000000$ such that for every possible world $latex {w}&fg=000000$ that shares some objects with $latex {w_1}&fg=000000$, there is an object $latex {x}&fg=000000$ in both $latex {w_1}&fg=000000$ and $latex {w}&fg=000000$ such that there is a world $latex {w'}&fg=000000$ whose objects are those of $latex {w}&fg=000000$, minus $latex {x}&fg=000000$.

Granting this allows one to iterate from $latex {w = w_1}&fg=000000$ to nothing. With weakened independence, however, the premise only says there is an $latex {x}&fg=000000$ and a $latex {w'}&fg=000000$ which might have other objects besides $latex {x}&fg=000000$, such that the ordinal measure of $latex {w'}&fg=000000$ is less than that of $latex {w}&fg=000000$. The choice of $latex {x}&fg=000000$ itself might be to minimize this measure.

Ken's Takes

First, I (Ken) take the Platonist position that mathematical structures have objective existence, and witness the same laws throughout creation. This begins already with the empty set. I agree with this deduction by the philosopher Wesley Salmon (emphasis in original):

The fool saith in his heart that there is no empty set. But if that were so, then the set of all such sets would be empty, and hence it would be the empty set.

Creation then follows ex-\{nihilo\}, either by standard set-theoretic means, or the way Donald Knuth ascribed to ``J.H.W.H. Conway'' in his book \it Surreal Numbers.

Hence the question is really about whether ``mathematical something'' entails ``physical something.'' What I find missing from the above arguments, and even from the Stanford Encyclopedia of Philosophy's article on ``Nothingness'' (from which the Salmon quote is taken), is information. John Wheeler's dictum ``It From Bit'' doesn't necessitate physical presence from the fact that information exists mathematically, but I believe it channels the question.

Since the subtraction argument presumes a finite universe, it presumes one with a finite number $latex {m}&fg=000000$ of physically realized bits of information. That $latex {m}&fg=000000$ is well-defined follows from the principle conceded by Stephen Hawking in 2005, that the amount of physical information in the universe cannot be changed by any physical process. Note that this already rules out any idea that ``subtraction'' could be a process applied to our own world.

The All-or-Nothing Argument

It strikes me, then, that when the subtraction argument is made attentive to information, it goes this way:

1. Starting Premise: It is possible to have a finite world.
2. Hence it is not the case that for all $latex {m}&fg=000000$, at least $latex {m}&fg=000000$ bits of physically-realized information are entailed in any world.
3. Hence there exists an $latex {m}&fg=000000$ such that some world has $latex {m}&fg=000000$ bits of information.
4. Information Subtraction (IS) Premise: In any world with physically realized information, there is a physically realized bit whose removal leaves a possible world.

Granting both premises allows us to iterate from $latex {m}&fg=000000$ bits of information down to zero. We can give IS a formal statement like ES above, and also weaken it to say the bit is replaced by one or more other bits but reducing some ordinal measure. The weaker form still iterates down to zero.

The IS form helps one think of ways ``independence'' might be false. Perhaps every particle in a possible world is entangled with another particle, or is part of a ``virtual pair'' that disappears. These eventualities do not defeat the conceptualization in terms of information, however. Virtual pairs are actually the agents that conserve information in the resolution of the ``black-hole information paradox'' that Hawking accepted. An entangled pair of particles may represent just one bit of information, so removing that bit may mean removing both. A concrete example of a ``substrate'' in which physical elements may have less information than their number is the hashing scheme used by chess programs, as I described here. Thus it may be possible to answer objections to IS.

In my opinion, however, what the information view does best is raise a point of aesthetics that antecedes trying to make ES or IS work via points of contingency and possible worlds. I guess I should say ``humble opinion,'' but an argument from aesthetics about ultimate matters is the polar opposite of humble:

Denying the conclusion says there is an $latex {m \geq 1}&fg=000000$ such that every possible world must have at least $latex {m}&fg=000000$ bits of information. From the starting premise, there must be a finite $latex {m}&fg=000000$ that is the greatest such $latex {m}&fg=000000$, and then IS fails for some world of size $latex {m}&fg=000000$. But why should any particular $latex {m \geq 1}&fg=000000$ be analytically special in this way?

We can imagine $latex {m = 0}&fg=000000$ being special, or... we could have $latex {m = \infty}&fg=000000$. Thus I hold that the ``subtraction'' argument actually rebounds to speak about infinite addition:

Either the world is necessarily infinite, or it is possibly void.

Using the symbols from modal logic for `necessary' and `possible,' we can call this the $latex {\Box}&fg=000000$All-or-$latex {\lozenge}&fg=000000$Nothing Argument.

This leads in turn to what I (Ken) think are the three questions that really matter.

Something Versus Everything

The first question is technical, and some regard it as already answered yes.

In an infinite cosmos, must every possible finite configuration of matter occur infinitely often?

If quantum fluctuations can generate configurations on human-sized scales---which is a premise of the still-current argument over ``Boltzmann Brains''---then yes. Then the following main question becomes equivalent to whether the cosmos is finite or infinite.

Is there something rather than everything? And either way, why?

Unlike the original question of something-versus-nothing, this one is not counterfactual---and we honestly don't know the answer, let alone why. At least the infiniteness question may be answerable: even this month there have been claims of new evidence from the Planck project analysis of the cosmic microwave background. Note also that ``everything is permitted'' was already a step of Holt's ``proof'' forbidding nothing above.

The third question is a mash-up of a quotation by Hawking and one by Wheeler, both noted as-here by cosmologist Max Tegmark. I read it as presuming a negative answer to ``everything,'' but it can be asked also about our particular region of an infinite cosmos:

What `breathes fire' into one set of equations, and not another?

For myself, I am partial to ``something,'' in either a finite cosmos or an infinite one with the first question answered ``no.'' Else we have to face down the reality of infinitely many ``copies'' of ourselves---as blithely asserted in articles by Tegmark---and of Earths that are exactly like ours or differ in arbitrary ways. Such variety applies also to paths of histories: infinitely many worlds would be as up-to-now except that I actually go out and commit the unspeakable act I'm thinking of for the purpose of this sentence. The common ``I'' among my minions could still be ``preponderantly moral'' in a sense described mathematically by Deutsch in his book The Fabric of Reality, but such lack of parsimony strikes me as degrading rationality. However, I must concede that denying ``everything'' entails a limitation of scale for the principle that ``everything not forbidden is compulsory,'' which is observed to hold at quantum scales.

Open Problems

Why indeed is there something? Is there everything? Or as Christopher Hitchens says on the back cover of Holt's book: P.S. What makes you so sure that there's anything? Love, Hitch.