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\u00a9 Richard Elwes\/Basic Books <\/p>\n
Gayoung Lee, Gizmodo<\/strong>: The title of the books is Huge Numbers. What does that mean? What makes a number big?<\/p>\n Richard Elwes<\/strong>: Well, that\u2019s a fairly important question in the book. Quite early in the process, I had to basically answer the question you\u2019ve just asked me. What is a big number? And you can give a contextual answer. If you\u2019re trying to balance golf balls on top of each other, then 5 is a very big number. The consequence is that any number can either be very big or very small, just depending on the context.<\/p>\n What I came down to was how human beings think about numbers, use numbers, and work with numbers. We do that through various Intellectual tools or systems that we\u2019ve built. And I started to think, well, what I mean by a \u201cbig number\u201d is a number that starts to challenge some of those systems.<\/p>\n Gizmodo<\/strong>: The subtitle mentions 4\u00bd and Fish 7. How do these numbers fit this approach?<\/p>\n Elwes<\/strong>: The most basic tool we have for working with numbers is what cognitive scientists call subitizing\u2014basically instant recognition. If I put three marbles on the table, and said, \u201cGayoung, how many marbles are there?\u201d You\u2019d see immediately there are three. You don\u2019t have to stop. You don\u2019t have to count. You\u2019re not going to get it wrong.<\/p>\n If I put nine marbles on the table, you\u2019d probably get it wrong. If you want to get it right, you have to use something more sophisticated: stop and count. According to William Stanley Jevons, who did the experiment first<\/a>, 4\u00bd was the limit. Why four and a half? Well, he always got it right when it was four. And he almost always got it right when it was five. So he said, \u201cOkay, the limit\u2019s somewhere in between, so four and a half.\u201d<\/p>\n Gizmodo<\/strong>: So in this context, to our brains, something beyond 4\u00bd is a \u201cbig number.\u201d<\/p>\n Elwes<\/strong>: Exactly. If what you\u2019re thinking about is this inbuilt, subitizing process for instantly recognizing numbers, then anything beyond 4\u00bd is, in that sense, a big number.<\/p>\n Gizmodo<\/strong>: And Fish 7?<\/p>\n Elwes<\/strong>: So Fish 7<\/a> is the biggest number that appears in the book. There\u2019s a Japanese googologist\u2014someone who\u2019s interested in big numbers\u2014who writes under the pseudonym \u201cFish<\/a>.\u201d And it\u2019s \u201c7\u201d because it\u2019s his seventh number. It\u2019s his attempt to just write down the biggest number he possibly could, really. It is certainly one of the biggest numbers anyone has ever described. He did that by arming himself with a powerful language in modern mathematical logic.<\/p>\n In between those are lots of interesting systems humans have developed. Quite a lot of the book lies in between the two extremes: about numbers big enough to break, or at least challenge, some human system.<\/p>\n Gizmodo<\/strong>: At the very beginning, you say that the story is about numbers that are big from the perspective of a human user. That implies numbers are a construct, even as we believe they constitute our objective reality. What do numbers reveal about the human consciousness?<\/p>\n Elwes<\/strong>: That\u2019s an interesting question. As you say, numbers are just the language of science. I think many people have said that. It\u2019s interesting that we are the only life form that we know of that can handle numbers precisely beyond that subitizing limit of 4\u00bd or whatever the limit may be for some other species. But we don\u2019t do that instinctively, or innately. That\u2019s not something we are born with.<\/p>\n And indeed, there are plenty of people whose culture has not developed precise language for large numbers. There are people who spoke unwritten languages where the number system just runs out at some point. That\u2019s actually quite striking for people who live in more technologically sophisticated societies. We would think of being able to count up to whatever number you like as just something everyone can do, but that\u2019s not the case.<\/p>\n Gizmodo<\/strong>: Why do you think society, generally speaking, gravitated toward a very defined number system?<\/p>\n Elwes<\/strong>: Going back in history, a really pivotal cause was the beginning of cities. Because with cities, we started getting, well, money. If you want money, you\u2019ve got to have numbers. And you also have bigger numbers of people in one place. You want to pay taxes. You want to make sure you\u2019ve got enough food for the community. Whatever it is, you\u2019re going to have to count things.<\/p>\n A lot of early number systems wouldn\u2019t work for the globalist, high-tech age that we\u2019re in now. An easy example is Roman numerals. Imagine trying to run the modern world on Roman numerals\u2014it\u2019s just not going to work! Roman numerals also just run out of numbers at some point. That was true for a lot of older, traditional numeral systems.<\/p>\n The system that we\u2019re now using originated in India. There\u2019s no point where it runs out. It doesn\u2019t have a \u201cbiggest number.\u201d What does eventually happen is that it starts to become a bit unwieldy. If you want to do science, you need to talk about numbers in the billions, trillions, and beyond for the number of cells in the body or the number of stars in the galaxy. We don\u2019t need a completely new system, but a slight modification: modern scientific notation. Instead of writing 1 followed by 12 zeros, you write 10 with a little superscript, 12, which means 10\u00b9\u00b2, or 10 multiplied by itself 12 times.<\/p>\n And that makes a big difference. On the one hand, it\u2019s a ridiculously big number. On the other hand, we can express it with a very short sequence of symbols, and none of them are complicated. That\u2019s how powerful this system is. And it\u2019s become absolutely essential to how humans look up to the universe and try and describe our home.<\/p>\n \u201cNumbers make you think about things on a certain scale. Your mind is being taken to this enormous scale that you can\u2019t really comprehend. There\u2019s something sort of a bit dazzling about it\u2026 It\u2019s a bit scary.\u201d <\/p>\n Gizmodo<\/strong>: And if you just add one little dash, a minus symbol, to the superscript, now you\u2019re describing impossibly small things.<\/p>\n Elwes<\/strong>: Exactly! Isn\u2019t that amazing? Many people throughout history haven\u2019t had access to that. So relative to them, our horizons are much broader, because we can easily just ask questions and discuss things on these really, really extreme scales\u2014tiny and huge.<\/p>\n Gizmodo<\/strong>: The history of number systems really highlights how they emerged due to necessity\u2014as in, there\u2019s a practical need for these scales. But as I understand it, there are folks who are chasing large numbers, well, just because. Is there a sophisticated reason behind this curiosity? Where\u2019s the necessity behind these ventures?<\/p>\n Elwes<\/strong>: There have been people throughout different points in history that just went way beyond this system. The classical Maya who lived in Central America certainly engraved on monuments numbers far bigger than any practical need would ever require them to do. But they also had a really streamlined, effective numerical system that could easily extend beyond what they needed. So it looks like, you know, they took it and ran with it.<\/p>\n Gizmodo<\/strong>: This fascination with incomprehensibly large numbers was a long-running thing for humans.<\/p>\n Elwes<\/strong>: As soon as you mention any number\u2014whether it\u2019s big or small or tiny or enormous\u2014you\u2019re thinking about a certain scale. If you\u2019re talking about hundreds of people, that gives you an image in your mind. Then imagine thousands of people, millions of people, billions of people. Numbers make you think about things on a certain scale.<\/p>\n Then you say some unimaginably enormous number, like a quintillion. Your mind is being taken to this enormous scale that you can\u2019t really comprehend. There\u2019s something a bit dazzling about it. People react with a sense of vertigo. It\u2019s a bit scary. It causes some sort of emotional response, I think, because the mind tries to tackle them, and it can\u2019t. Because we just don\u2019t have intuition for things on those scales. And that causes maybe some discomfort or some wonder.<\/p>\n Gizmodo<\/strong>: You mention asking your mathematician colleagues of the largest number they encountered. Any memorable responses?<\/p>\n Elwes<\/strong>: There was one which I wasn\u2019t expecting at all, which comes out of the mathematics of music. So it turns out that the numbers 353 and 284\u00a0 are very close together. It was discovered 2,000 years ago and is a way of generating a 53-note scale. Just within pure mathematics, there are a number of enormous numbers that arise. The branch of mathematics that generates the largest numbers is mathematical logic. The reason for that is that it\u2019s all about studying different computational systems or different languages in a sort of formal, mathematical, logical sense. The absolute biggest numbers in the book come out of using set theory, which is a technical branch of mathematical logic.<\/p>\n Gizmodo<\/strong>: As a mathmatician yourself, how has investigating huge numbers influenced the way you approach your work?<\/p>\n Elwes<\/strong>: I emphasize in the book that it\u2019s a human story. The tools we develop to describe numbers\u2014or to access very big numbers in describing the universe on whatever scale\u2014are all [human-made] tools. I think most mathematicians are, at least for practical purposes, what you might call Platonists. That is to say, we just think this stuff exists, and we\u2019re out there studying it. I think, you know\u2014for practical purposes, I don\u2019t want to philosophically commit to it\u2014I work in that in the same way.<\/p>\n It doesn\u2019t directly affect the way I teach or research math, but the realization feels like a healthy thing to have as background awareness. We are the products of history, and we are humans using human technology.<\/p>\n