Is Infinity Real?

In the daytime, you are surrounded by trees, buildings and the all-too-familiar accoutrements of Nature, to which by evolution we were designed to appreciate and be familiar. But at night, we see an unimaginably different view: The dark, starry night sky, with no sense of perspective or depth. It is easy to understand how The Ancients thought it a celestial ceiling with pinpoint lights arrayed in noteworthy patterns. Many millennia of campfires were spent trying to figure it out.

We are stuck in the middle ground between two vast scales that stretch before us and within us. Both, we are told, lead to the infinitely-large and the infinitely-small. But is this really true?

Astronomically, we can detect objects that emerged from the Big Bang nearly 14 billion years ago, which means their light-travel distance from us is 14 billion light years or 13,000,000,000,000,000,000,000,000,000 centimeters. This is, admittedly, a big number but it is not infinitely-large.

In the microcosm, we have probed the structure of electrons to a scale of 0.000000000000000000001 centimeters and found no signs of any smaller distance yet. So again, there is no sign that we have reached anything like an infinitely-small limit to Nature either.

When it comes right down to it, the only evidence we have for the universe being infinitely large (or other aspects of it being infinitely small) is in the mathematics and geometry we use to describe it. Given that infinity is the largest number you can count to, it is pretty obvious that even the scale of our visible universe of 13,000,000,000,000,000,000,000,000,000 centimeters falls woefully short of being even a relatively stupendous number by comparison to infinity.

Infinity is as old as the Ancient Greeks. But even Aristotle (384 – 322 BCE) would only allow the integers (1,2,3,…) to be potentially infinite, but not actually infinite, in quantity. Since then, infinity or its cousin eternity, have become a part of our literary and religious vernacular when we mention something really, really, really….. big or old! Through literary and philosophical repetition, we have become comfortable with this idea in a way that is simply not justifiable.

Mathematics can define infinity very precisely, and even the mathematician Cantor (1845 – 1918) was able to classify ‘transfinite numbers’ as being either representing countable infinities or uncountable infinities. To the extent that mathematics is also used in physics, we inherit infinity as the limit to many of our calculations and models of the physical world. But the problem is that our world is only able to offer us the concept of something being very, very, very… big, like the example of the visible universe above.

If you take a sphere a foot across and place an ant on it, it crawls around and with a bit of surveying it can tell you the shape is a sphere with a finite closed surface. But now take this sphere and blow it up so that it is 1 million miles across. The ant now looks across its surface and sees something that looks like an infinite plane. Its geometry is as flat as a sheet of paper on a table.

In astronomy we have the same problem.

We make calculations and measurements within the 28 billion light years that spans our visible universe and conclude that the geometry of the universe is flat, and so geometrically it seems infinite, but the only thing the measurements can actually verify is that the universe is very, very, very large and LOOKS like its geometry is that of an infinite, flat, 3-dimensional space. But modern Big Bang cosmology also says that what we are seeing within our visible universe is only a portion of a larger thing that emerged from the Big Bang and ‘inflated’ to enormous size in the first microseconds.  If you identify our visible universe out to 14 billion light years as the size of the period at the end of this sentence, that larger thing predicted by inflation may be millions of miles across at the same scale. This is very, very big, but again it is not infinite!

Going the other way, the current best theoretical ideas about the structure of the physical world seems to suggest that at some point near a so-called Planck scale of 0.0000000000000000000000000000000015 centimeters we literally ‘run out of space’. This mathematical conclusion seems to be the result of combining the two great pillars of all physical science, quantum mechanics and general relativity, into a single ‘unified’ theory.  The mathematics suggests that, rather than being able to probe the nature of matter and space at still-smaller scales, the entire edifice of energy, space, time and matter undergoes a dramatic and final change into something vastly different than anything we have ever experienced: elements that are beyond space and time themselves.  These ideas are captured in theories such as Loop Quantum Gravity and String Theory, but frankly we are still at a very early stage in understanding what this all means. Even more challenging is that we have no obvious way to make any measurements that would directly test whether physical reality simply comes to an end at these scales or not.

So on the cosmological scene, we can convincingly say we have no evidence that anything as large as ‘infinity’ exists because it is literally beyond our 14 billion light-year horizon of detection. The universe is simply not old enough for us to sample such an imponderably large realm. Advances in Big Bang cosmology can only propose that we live in an incomprehensively alien ‘multiverse’ or that we inhabit one miniscule dot in a vastly larger cosmos, which our equations extrapolate as infinity. Meanwhile, the world of the quantum hints that no infinitely-small structures exist in the universe, not even what we like to call space itself can be indefinitely sub-divided below the Planck scale.

In the end, it seems that infinity is a purely  mathematical ideal that can be classified by Cantor’s transfinite numbers manipulated symbolically, and thought about philosophically, but is never actually found among the objects that inhabit our physical world.

Now let’s go back to the issue of space after the relativity revolution and try to make sense of where we stand now!

Check back here on Monday, December 19 for the next installment!

7 thoughts on “Is Infinity Real?”

  1. I believe it was Max Tegmark who said that “Any time scientists use the term infinity in a discussion, that just means that we don’t know what we are talking about.” We lay people appreciate the tip-off.

  2. Thanks for your comment, Gloria! Tegmark’s quip is perhaps a bit harsh. Infinity is a precise mathematical concept, and anyone that uses calculus and works with physics-based modeling has to make peace with this concept because we insist that we live in a universe where space is continuous at all scales. That is the basis for constructing an integral, and integrals over space and time are used in many different settings, especially in quantum calculations. But at least in astronomy, we recognize that out universe and its finite age sets a firm limit on the scale of space that we can ever hope to directly observe. Extrapolations beyond this 14-billion-light year horizon are driven by mathematics and the equations we use to describe spacetime in relativity, which require us to consider scales of space and time that tend towards the in finite in space and eternal in time. Not much we can do about that except be mindful of infinity as a mathematical idea not a physical one based on ‘observations’.

  3. There must be an infinitely number of universes. To travel into / to a specific infinitely small universe , we have to enter into matter… enter in molecules, enter atoms, go to neutrons, enter neutrons, got in quarks, see the cords etc etc … up to Plank dimension.
    So, there must be an infinity of infinitely small universes. Huge, probably for the creatures living in each of these.
    But for the many creatures living in the infinitely small universe (reachable from our “universe) , the reverse path would mean that they would arrive in our universe, and only ours ! Now we are probably in another infinite small ourselves…. that would open to another universe if we travel through space….. But we are probably too ‘small to be able to see it).

  4. Hi Pierre! I like your intellectual enthusiasm for this subject!!! Your idea of nested universes is not unlike the ideas that some science fiction writers were enthusiastic about back in the 1930s and 1940s. For example, He Who Shrank written by Henri Hasse and published in 1936 is a short story that draws a parallel between atoms and planetary systems. Quick summary: A Mad Scientist injects his assistant with a serum that will make him shrink indefinitely. The assistant shrinks to subatomic sizes, discovers that every atom is actually a solar system, lands on a planet, shrinks below the size of the atoms of that planet, and so on without end.

  5. Should the question not be : “Is our universe expanding in a finite or infinite space”.

    I think we should consider space and universe as different things.

    The universe is expanding, since the big bang, in space….. or am I wrong ? Space being the support for the universe.

  6. Hi Pierre! General relativity is our theory of gravity and spacetime that works real well, has been tested in non-cosmological phenomena and has demonstrated that it accurately describes gravitational phenomena. So for now we have to use general relativity as our basis for understanding cosmology. GR describes very accurately the Big Bang scenario, but in doing so it answers a fundamental question about the expansion itself. The universe is not expanding into a pre-existing space. That is a common-sense Newtonian idea that is in direct contradiction to general relativity itself. It is not possible for us to understand the concept of expansion, or the dilation of space, any more than we can understand quantum mechanics and the basic concept of particle-wave duality. General relativity explains gravity and spacetime together, and we can either elect to just ‘go with the math’, or we are left with dealing with our failing common sense. Given that common sense was never designed to give us insight to extreme physical phenomena, as an astronomer I choose to go with the math. That said, space(time) and universe are the same things, and space(time) does not pre-exist the universe.

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