HPS 0410 | Einstein for Everyone |

Back to main course page

or

What does it all mean, again?

Cosmos and Infinity

John
D. Norton

Department of History and Philosophy of Science

University of Pittsburgh

Here are two further morals, both concerned with matters on cosmic scales.

The King James version of the bible begins in Genesis with the following memorable account of the origin of the world:

In
the beginning God created the heaven and the earth. And the earth was without form and void, and darkness was upon the face of the deep. And the Spirit of God moved upon the face of the waters. And God said, “Let there be light”; and there was light. And God saw the light, that it was good; and God divided the light from the darkness. |

It is tempting to see in big bang cosmology a vindication of this Genesis account of creation. For the big bang, loosely speaking, has the universe issuing out of a conflagration of great heat and energy. The match, of course, is imperfect. Genesis supposes dark waters prior to the moment of light; and there is no early water in big bang cosmology. Still, we might imagine some vindication through these questions:

**Question:** Is Genesis' moment of the
creation of light the same as the moment of the big bang in big bang
cosmology? ("Let there be light.")

**Question:** The creation of the universe is a miraculous
breakdown of physical laws. Is this miraculous moment of creation the same
as the singular breakdown of physical laws in big bang cosmology at the
big bang?

These questions lead us into the tangled relationship of science and religion. I do not wish to enter into the broader issues they raise. My concern here, however, will be a very narrow one. Can we find in the science itself a positive answer to these two questions? That is, is there a moment in bang bang cosmology at which light is created? Is there a miraculous breakdown of physical laws at this moment?

The answer to both questions is no.

For the point made repeatedly in an earlier
chapter is that there is no moment of the big
bang. In big bang cosmology, for every moment in time, there is
always an earlier one, but there is no earliest, first moment. As the
chapter explains, this is quite compatible with our universe extending
only finitely many years into the past.

Nonetheless, it is tempting to imagine God at some moment immediately prior to the moments of time of big bang cosmology. At this moment of creation, God creates light. The difficulty is that this tempting image is outside the science. For the moment supposed is not a moment in spacetime. It is a moment, outside the time of big bang cosmology. The cosmology can tell us nothing about moments outside of itself, for the cosmology does not apply there. Big bang cosmology provides no notion of "earlier" to make sense of how this moment of creation can come before the moments of time of big bang cosmology.

To posit a moment outside the moments of big bang cosmology is to posit a moment outside of the spacetime of physics. One does not need big bang cosmology to make such a posit. It can be made in any cosmology, including those that extend for infinite times into the past. In such a universe we could also imagine a moment of creation prior to all the moments of the cosmology. The situation is the same in both big bang and infinite past time cosmology: neither supplies a notion of "earlier" that locates the moment of creation.

We are often told that the big bang originates in a singularity, which is portrayed as a place where the laws of physics break down. Once again it is tempting to imagine this is where God miraculously suspends the laws of physics to create the world. Once again, we find no comfort for this idea in big bang cosmology. For the cosmology itself provides no definite moment of time where a singularity happens.

Talk of a singularity as existing in any positive sense is an oversimplification that is usually benign. Here, however, it proves misleading. Speaking more precisely, there is no place where the laws of big bang cosmology break down. Rather the very notion of a breakdown occurs within the scope of a reductio argument. We posit such a candidate place in order to show that, according to big bang cosmology, there is no such place. That is, we ask can there be a first moment in big bang cosmology? We project back and find that such a place would have properties that contradict the physical theory. For that theory asserts that the curvature of spacetime is everywhere finite and well-defined. At this first moment--if there were such a thing--the curvature would be infinite and thus ill-defined. We complete the reductio by concluding that there is no such place.

One of the most intriguing results of metamathematics is that there are certain computational tasks that cannot be completed as a matter of principle, even though they appear quite achievable. One of the simplest is the "halting problem." The task is to write a computer program that can check the operation of other programs. Call a computer running the program "CHECK." In operation, CHECK is fed as input a candidate computer program of interest and some nominated input to the program. CHECK would then tell us whether that candidate program would come to a halt and return a result on that input; or whether it would continue to compute indefinitely and return no result. In more familiar language, will it "crash"?

The surprising and, it turns out, easily proved result is that there can be no CHECK program. The halting problem is insoluble. This is the most familiar of many further "uncomputable" tasks.

What is the nature of this insolubility? It is easy to imagine that it is a simple matter of logic. These tasks are uncomputable in the same way that there is no whole number that is the square root of ten. That is not so. The proofs of uncomputability all depend on what is actually a physical statement:

It is impossible to complete a computation that requires infinitely many steps.

Of course in the world of ordinary experience, this is an impossibility. No one (I hope) is expecting us to be able to build a computer that will complete an infinite number of computational steps. It is easy to conflate that practical impossibility with a logical impossibility. Here is an argument that tries to make it a logical impossibility:

If a computation requires infinitely many steps, then it cannot be completed, since there is no last step that would complete the computation.

Plausible as this last argument seems, it
is a fallacy. To complete an computation with infinitely many
steps requires only that we do all the steps. If "do all" can be achieved,
then the computation is completed. There is nothing more to ask of it.
Adding the extra demand "... including the last one" is an unnecessary,
additional requirement that is the cause of the logical difficulty. It
supposes that there has to be a last step where there can be none by
supposition.

The impossibility of completing infinitely many steps in a computation is not a logical restriction. It is a physical restriction. If we could circumvent this physical restriction and complete infinitely many actions in a finite time, then traditionally uncomputable tasks would become soluble. For example, it is possible to devise a computer that can implement any program. This is a "universal Turing machine." It would solve the halting problem. We would simply ask the machine to implement the operation of the chosen program on the input of interest. We would then only need to observe the operation of the machine to see if it halted on that input.

https://commons.wikimedia.org/wiki/File:Gluehlampe_01_KMJ.png

For a good introduction to this literature see
John Manchak and Bryan W. Roberts, "Supertasks", The Stanford
Encyclopedia of Philosophy <https://plato.stanford.edu/archives/win2016/entries/spacetime-supertasks/>. |
The literature on "supertasks" explores the possibility of systems that complete an infinity of actions in finite time. Needless to say, these systems are highly idealized and fictional. The goal, however, is not to establish how we could make such systems in practice. It is to show that the completion of infinitely many actions is not a contradiction in logic. |

Within this literature, general relativity provides us with an intriguing scenario in which this completion becomes possible. For it provides us with spacetimes in which the infinite lifetime of one entity can be fully contained within the past light cone of another.

In such a spacetime, the first entity is an idealized calculator who is programmed to compute for an infinity of the calculator's proper time. For example, the calculator may check Goldbach's conjecture that every even number is the sum of two primes. The calculator simply goes through the infinite list 2, 4, 6, 8, ... checking each even number in turn. If a counterexample is found, the calculator sends out a signal that will be received by the second entity, the observer. The calculator will never know at any finite time in its life whether a counterexample is found. However the observer will know if one is found, since the observer can survey the entire, infinite life of the calculator. This calculator-observer combination can now compute things that are normally regarded as uncomputable.

This case is represented in the spacetime diagram below. The calculator "C's" worldline has infinite length in proper time and is fully contained within the past light cone shown. The observer O lies outside that light cone, such that the entirety of C's world line is within observer O's past light. As C checks each even number, C sends a signal to O indicating the result. Since O will receive all the signals in a finite time along O's world line, O will know whether there is a counterexample to Goldbach's conjecture; or whether it holds for all even numbers.

Of course this system of calculator and observer involves some extreme idealizations. We assume a calculator who can continue calculating for infinite time. We assume an observer who can disentangle infinitely many signals arriving in finite time. There will be other pathologies in the structure of a spacetime that allows this arrangement in the first place. That we need such idealizations impinges on the question of practical achievability. However it does not compromise the main point: there is no logical impossibility in completing an infinity of actions. Here we see a spacetime hospitable to such completions.

For an account of these
spacetimes, see my paper with John Earman, "Forever
is a Day: Supertasks in Pitowsky and Malament-Hogarth Spacetimes," *Philosophy
of Science*, *60* (1993), pp. 22-42.

Copyright John D. Norton.
February 23, 2013. November 14, 2019.