 I was going to do a visual gag about Graham's number, but I've only eaten a Googleplex of these things and I don't know if I can go through with it. There's a branch of mathematics known as Ramsey theory, which is concerned with finding the limits on how many things you absolutely need to satisfy some criteria. For example, if you arrange a number of dots into a circle and then connect them all with either red or blue lines, how many dots do you need minimum so that no matter how you end up choosing line colors, you can't avoid drawing either a red or a blue triangle? Questions like these piqued the interest of mathematician Ron Graham, who tried his hand at a slightly different problem. Rather than circles of dots and red or blue triangles, he was thinking about dimensions of cubes and red or blue this thing. How many dimensions does a cube need to be before no matter what, you can't avoid drawing this figure in all the connections between its corners, either in all red or all blue. In a two-dimensional cube, which is just a square, it's pretty easy to avoid. Just use a different color for one of the edges. Duh. In a three-dimensional cube, which is just a cube, it's still pretty easy. You've got double the number of corners and a bunch more edges to worry about, but with a little bit of thought, you're still probably fine. Now, if you're anything like me, you're probably not used to thinking in four dimensions. But moving from 3D to 4D is very much like moving from the square to the cube. You double the number of corners again, so 16 now, and then you connect everything to everything else. We're now looking at 120 edges, any one of which could be colored red or blue, which means that we're looking at two to the power of 120 different colorings of this figure. That's a number that's 36 digits long. But computers and mathematicians are pretty good at this sort of thing. And it turns out that in four dimensions, we can still avoid drawing this figure. We can avoid it in five and six dimensions all the way up to twelve. So can we just keep on going forever? Well, no. Graham didn't find an answer for how many dimensions we absolutely need before we have to draw this figure somewhere inside the cube. But he did find an upper limit, a number that's definitely bigger than the one that we're looking for. In fact, it's definitely bigger than the vast majority of numbers that we use. It actually held a record for the largest finite number used in a productive mathematical proof, and it's not hard to understand why. Mathematicians have invented some notational tools to represent stupidly large numbers in a way that won't take all day to write out. This up arrow is used in Graham's proof, and it denotes a sort of recursive power of exponents. x up arrow up arrow y means raise x to the x-th power y times. So two up arrow up arrow three means that you get a stack of three twos, two squared squared, which is just two to the fourth power, or 16. Now, if you're crazy enough to add a third arrow, you're saying raise x to the x-th power x double arrow y times. So two triple arrow three means make a stack of sixteen twos. If you've got a second, pause the video, find a calculator, and try squaring two fifteen times. That's a pretty big number. Let's say it's just much larger than the number of particles in the observable universe. Graham defined a number, G1, as three quadruple arrow three. That's a three with three triple arrow three exponents on it. If you were able to divide the entire observable universe into Planck lengths, the smallest meaningful distance in quantum mechanics, and then somehow fill each of those locations with a single binary digit, essentially making a universe-sized hard drive, you'd have around 10 to the 180th power digits to work with, and you still wouldn't even come close to being able to write down G1. But that's not Graham's number. That's just where we're starting from. G2 is three G1 arrows three. G3 is three G2 arrows three. So on and so forth up to G64 Graham's number, which is the upper limit to this problem that he discovered. So the answer to the original question is a finite number somewhere between 13 and Graham's number. It used to be between six and Graham's number, but mathematicians have managed to close the gap a little bit. As I've mentioned in previous videos, there are all sorts of clever ways that we can dissect large numbers to try and interpret them meaningfully. Like an aircraft carrier is about three football fields long. It has a crew about the size of a small town, that sort of stuff. But G64 is too freaking big for us to pare down so that we can handle it in any way. I mean, there is that limitation of the size of the universe, but perhaps more appropriately, there are limitations of psychology too. In that episode, I cited some evidence that we used different neurological systems to crunch numbers and to estimate answers. The system that we use for doing math, for pushing symbols around counting and following mathematical rules, is very closely associated with the parts of the brain that are normally used for language, while rough estimation of answers and figuring out what those symbols might practically mean, is more closely associated with the parts of the brain that are used for visual memory. It turns out that isn't the only split that happens in our heads when it comes to numbers. Our estimation apparatus has a couple of different gears that it uses for cognitive representation of quantity, and you don't need to get to Graham's number for them to start doing something weird. Number sense is the name for our ability to intuitively grasp numbers of things, and their relationships to other numbers of things, without resorting to counting or using math, is what allows us to look at a room full of hungry kids and the single cheese pizza that we bought and think, oh crap. A lot of research has gone into trying to understand the details of how number sense works. Most researchers agree that it's composed of two distinct systems, one that's good for small exact numbers and the other that's good for moderately large, inexact numbers. They both have some very interesting and unique properties unto themselves. Fascinatingly, this isn't just a human thing. There are all sorts of diverse species, which exhibit some sort of number sense. Guppies, monkeys, pigeons, rats, ravens, dogs, and cuttlefish all exhibit a similar pair of psychological mechanisms at work with similar strengths and weaknesses. First, the parallel individuation system is concerned with quantities of one to four things, which it can track with ruthless efficiency. Whether you catch a brief flash of some assortment of stuff, or have someone poke a few fingertips, so long as we're talking about fewer than four whatever's, parallel individuation is on the job. It's called parallel because it allows for discrete tracking of each one of those objects at the same time. You don't need to count to know exactly how many cookies are on that plate. You can just look and know that there are three. 2. Koko? Of course, Graham's number is larger than four, so no help there. How about this other system? Well, the approximate number system works by grouping a quantity of stuff into a single mental symbol, like about 300. While the parallel individuation system can tell if one thing goes missing every single time, the approximate system has a bit of a buffer zone where it doesn't really register that anything's different. 260, 340, meh, who cares? It's about 300. And weirdly, the amount of change necessary for the system to register that something's different is logarithmic. It tends to exaggerate the differences between smaller numbers and kind of fall asleep for larger ones. Like going from $10 to $15 for a movie ticket, that feels huge, but hearing that someone's going from a $40,000 to a $41,000 a year salary, unless I sit down and really think about how much money that is, how many movie tickets, I barely register a blip on my mental radar. That logarithmic error scale does improve some with training. Children are more susceptible to this phenomenon at lower numbers than adults are, possibly because adults have some experience working with larger ones. But it's still present even in highly trained adults. The larger the numbers, the less we care about the same amount of change. Now, Graham's number is just so far outside anything that we might realistically encounter in the world that NumberSense just doesn't work on it. The best we can do is a numb feeling of incomprehensible bigness, which, although accurate, isn't really informative. But there are large numbers that we do encounter that NumberSense is nonetheless ill-equipped to handle. A 1 in 14 million chance of winning the lottery? A microchip that can perform 20 trillion operations a second? Even a $200,000 mortgage is just so big that we can't bring the same intuitive faculties to bear that we would use to decide between an $8 medium or a $12 large popcorn. The only recourse that we have to interpret these numbers in any meaningful way is to dissect them with mathematics, to break them down into quantities that the approximate number system can work with reasonably well without reporting back that the difference of $1,000 isn't really a big deal. And speaking of math, in 2014, a team of mathematicians came up with a smaller upper bound on the problem that Graham was working on. Two triple arrow six. That's just two to the power of four trillion or so. Chump change, right? Best I can do is two double arrow six. How's your NumberSense of Graham's number? Please leave a comment below and let me know what you think. Thank you very much for watching. Don't forget to blah, blah, subscribe, blah, share, and don't stop thunking.