The One About Infinity

The infinite is one of the most confusing concepts to grasp. The truth is, in reality, things beyond a certain point of frequency or amount (even money) cease to hold any meaning for us. We just can’t imagine or visualize them, and thus they become meaningless. And yet, playing with this concept and understanding it a little better will help stretch your thinking and expand your mind. Taking a subject out of the ether and examining it in a reality-based and consequential way can help you extract a few lessons on extremely big, and extremely small, probabilities, and understand why a mathematical construct like 1/3 cannot truly be expressed as a decimal.

Any discussion of the infinite must begin with the thought experiment of Lucretius’ spear. If you contend that the universe if infinite, and then you walk right up to the edge (first of all, what would the edge even constitute), and then you throw a javelin, then what would happen? Would it bounce off something? Would it disappear? Would it just keep sailing because the universe is indeed infinite? What exactly would take place? Thinking through this abstraction makes clear that we have to suspend reality sometimes and redefine what we think is possible. You might even find that you have been asking wrong questions all along by not accounting for what you didn’t consider.

The Javelin Tells All

His argument is called “Lucretius’ Spear” and is what it sounds like: if, Lucretius argued, the universe is finite (or “bounded”), then what would happen if a man were to go right up to the edge of it all, and throw a javelin past the limit? What would happen? This is not a hypothetical question. If you insist that the universe does in fact have some point at which it ends, then you’d really be stumped to come up with an answer.
If the spear went hurtling onwards, it would mean that you weren’t actually at the limit. If the javelin bounced back, well, what was beyond the boundary? If there was something to stop the movement of the javelin, then that means he was not at the edge either.
Lucretius claimed, “For whatever bounds, that thing must itself be bounded likewise; and to this bounding thing there must be a bound again, and so on for ever and ever throughout all immensity.” You can see his point. Lucretius hoped, by this thought experiment, to show that the idea of a finite universe simply didn’t make sense. He also wanted to illustrate the concept of infinite and how it interacts with so-called boundaries.
Now, take a step back for a moment and survey your own thoughts. What’s going on, mentally? Has your mind run off with a potential counterargument? Are you imagining new and creative scenarios inspired by Lucretius’ javelin thrower? If so, congratulations; you are already experiencing the benefits of engaging with thought experiments.
The experiment might have indeed convinced you that the universe is infinite and not finite. But there’s more. After all, more modern thinking has suggested ways in which the universe can be thought of as both finite and infinite—for given definitions of both.
For example, imagine that all of existence is shaped like a doughnut, the surface of which we move about on. If you traveled on the surface of this doughnut in all directions, you would never technically come to an “edge” or boundary. However, the surface of the doughnut universe is finite in a concrete sense. It’s akin to saying that the javelin would race ahead and come flying from behind the javelin thrower, like a person traveling on a Mobius strip would eventually arrive at the same spot they left from.
A thought that might have occurred to you is to try and use the same argument, but for something you have more knowledge of. If someone claimed the earth was finite, someone else could make a javelin-style argument: What would happen if you went to the edge of the earth and threw a spear?
Because you already know that “edge of the earth” is in fact nothing more than a misunderstanding, you might be tempted to rethink the concept of “edge of the universe.” Here, you are using skillful induction and deduction, groping your way along new concepts, using what is already known to help understand what isn’t.
It’s likely that Lucretius’ early argument, and others like it, gave rise to the more sophisticated topological and mathematical models we have today for conceptualizing the size and scope of the universe. But what we do today with literal spacecraft, probes, mathematical models and theory, we were already attempting long ago with nothing more than our thoughts.
Einstein famously said that one cannot solve a problem at the same level of thinking that created it. In other words, all insight and illumination come from stepping outside of what you feel are the set facts of your world as you see it. This is where science and creativity amount to the same thing: imagination is something that allows you to think what has not been thought of before.
You cannot use the known to construct a truly novel vision—it can only ever serve as a crutch that takes you part of the way. Consider all the major leaps and bounds in mankind’s intellectual history, and you’ll see that they were all fundamental reimaginings of old rules. Evolution, Freud’s unconscious id, relativity… These are not simply improvements on previous models, but compete rethinks of the models themselves, i.e. not progress within a model, but between models, and beyond them.
Without ever having to go “out there” in the world to measure the actual length and breadth of the universe, thought experiments allow us to sharpen our mental tools and make conceptual space to conceive of the answer we are looking for. This may require completely dismantling old ideas by asking questions like: What does it actually mean to say something is “infinite”? Is this a quality that actually exists in the world, or is it more accurately a word that belies our own limited understanding, a linguistic vestige belonging to a level of thinking that posed the “paradox” in the first place?
The juiciest thought experiments have an uncanny way of making us think really hard about the questions we ask. The physicist Heisenberg claimed that, “What we see is not nature itself, but nature exposed to our method of questioning.”
You’re probably reminded of Schrodinger’s cat, but the quote reminds us that the quality of information we have about the world is very much determined by the manner in which we engage with it. Thought experiments are like questions, but in a way they allow us to look back on those questions themselves. What kind of information are we really looking for from the world? How can we get it? What are the best tools to do so, and what’s wrong with the ones we currently use?
By turning over Lucretius’ experiment in our minds, we invite ourselves to plunge a little deeper. We find ourselves in the company of some of the biggest of big ideas that have occupied man since the dawn of time. Similar arguments have been used to prove (or, curiously enough, disprove) the existence of God—i.e., if there is one meta-being who created everything, then who created the meta-being? What came before the Big Bang, exactly? Or, if you’d like to put a more Eastern spin on the same sort of contemplations, what did your face look like before you were born, and where does the candle flame go when you snuff it out?
In thinking about infinity, we are stretching our minds to the furthest reaches. It might not be overstatement to imagine that every question mankind has asked has been inspired in one way or another by this unfathomable mystery all around us.
Perhaps the greatest lesson this kind of mental exercise can teach us is to look beyond even our own conception of the problem, and the assumptions built into our very questions. In the mundane world we are often presented with what look like choices: do you believe in this thing or that thing? Pick a political candidate or opinion or laundry detergent. But what if there is a third option? What if you’re mistaken in thinking of it as a choice at all? What if you can choose both, or neither, or what if the entire question is where the problem lies?
Lucretius’ thought experiment teaches us to open up a window to other options. Not A or B, but a third variable, something new and as yet unseen. A space where marvelous and unthinkable things can happen. Psychologists , historians and philosophers can busy themselves endlessly with the question of human nature; a monk on the mountain might shrug and ask, who told you there was such a thing as human nature?