We have all heard at some point in our lives that mathematics is the language of science. The reason for this is that science deals with data, and data can be described by numbers that represent physical properties. Let's look at a simple situation where science tries to "model" with mathematics some properties of our world based on data.
Imagine a simple mathematical equation like y = mx + b, where m and b are two adjustable numbers, and x and y are a pair of interesting "unknowns" called variables. If you plug in an appropriate number for x and turn the crank, you get another number, y, as a prediction. Mathematically, x, y, b and m are pure numbers, but when you look around and collect data, data are not just pure numbers but represent quantities that have units of length, temperature, mass and so on. Relating the pure numbers to interesting physical quantities is the art of mathematical modeling. Let's look at a few applications of this mathematical model.
If x were the calendar year after 1980, m = 1.8 and b = -3229, then y would be the global average carbon dioxide concentration in parts per million for that year. If m = 2.2 and b = -4268, then y would be the sea level change in millimeters since 1940. So far, these two mathematical models just fit the accumulated data and do not explain why the values for m and b are what they are in each of the instances. The real power of physical science is that, through a knowledge of how temperature affects water, causing it to expand, and how greenhouse gases affect temperature, we can create a theory that predicts the values for m and b before you go out and observe their actual values by collecting data! This is what the current theory of global warming attempts to do by explaining how carbon dioxide and global warming increases ocean temperature, and ocean temperature causes water to expand and increase seal level.
Suppose we create a similar theory to account for how the universe formed out of a "big bang." We sift through our data and derive a number of mathematical relationships like the ones we just discussed for sea level rise. In the cosmological setting, the values for m and b determine, for example, how many helium atoms you will find for every hydrogen atom. In our universe, this primordial ratio is about 1 to 4. Just suppose, then, that the predicted mathematical model from Big Bang theory becomes y = 0.15x + 0.10.
One possibility for this Big Bang theory is that m and b are like the value for pi. These numbers are completely and self-consistently defined to be the unique values that we see. There is no further scientific discussion because m and b are logically derived within this theory from quantities we already know about. We call this a closed system, and it leads to the idea that there can be only one logically consistent universe, and we are living inside of it.
A second possibility is that our universe is part of a larger system called the multiverse in which all possible values for m and b can occur and lead to many -- in fact, an infinite number of -- separate, logically consistent universes. Most of those values lead to universes in which life does not exist, while others have randomly selected values for m and b that are within the very narrow range to allow life to eventually emerge. We observe the weird values for m and b because we are here to experience them. This is called the anthropic cosmological principle, and it represents the dilemma facing physicists today.
If we live in the first kind of universe, we expect that at some point we will eventually understand all there is to know about it and come up with a single, complete mathematical theory explaining all of its significant details. There will be no impediments to making all the necessary observations and creating the ultimate logical model with no unprovable assumptions to launch the mathematics. For example, all the fundamental constants in nature will be derivable from inside the mathematical theory and will no longer be quantities that we have to observe and measure.
Photo by Avenue (own work) via Wikimedia Commons
If we live in the second kind of universe, we have a problem. If the multiverse exists, we can never observe any of these other universes to prove or disprove their existence. The fundamental constants that make our universe look the way it does and make life possible are not unique and derivable from some grand supertheory but are randomly assigned and can in principle take on any value from 0 to infinity. We are simply lucky to be living in a life- and sentience-friendly universe where the constants we see and even the laws of nature make this possible.
For a long time, Big Bang theory could not help us decide between these two possibilities, but with the discovery of gravity B-modes in the cosmic background radiation, it looks like a small window has been opened to study this problem. These subtle fingerprints hidden in the polarization data seem to validate the idea that the cosmos rapidly inflated from a quantum speck to a vast spacetime soon after the Big Bang some 13.8 billion years ago. An important implication of "inflationary cosmology" is that this has happened many times, in fact an infinite number of times, in different parts of spacetime, creating a multiverse.
So although we may not be able to directly observe any of these other universes, their existence seems to be logically required if our model building is on the right track. The anthropic cosmological principle is not something derivable from inside Big Bang theory at all. It merely represents a selection effect that we apply to all the possible universes in the multiverse to find the one we actually live in!