Cutting In Line: Discontinuities in the Use of Large Numbers by Adults

Cutting In Line: Discontinuities in the Use of Large Numbers by Adults David Landy (dlandy@indiana.edu) Arthur Charlesworth (acharles@richmond.edu) Department of Psychological & Brain Sciences Indiana University, Bloomington IN Department of Mathematics and Computer Science University of Richmond, Richmond VA Erin Ottmar (erin.ottmar@richmond.edu) Department of Psychology University of Richmond, Richmond VA Abstract experiences of this many individual items are vanishingly rare, but small enough to play important roles in sciences such as geology, astronomy, and macroeconomics, and in political contexts such as budget discussions. There is little reason to think that evolution would have specially prepared us to deal with quantities of this magnitude. How do we come to access them, and take advantage of their properties? Perceptual tools such as telescopes allow the application of robust internal perceptual systems to apply beyond the range of their unadorned capacity. This paper explores how reasoning over culturally provided representations enables the perception of conceptually distant structures. In particular, this paper examines the behavior of typical adults estimating the position of large numbers (1 thousand to 1 billion) on a number line. Participants—even those who closely match linear placement—show discontinuities in placement in the immediate vicinity of 1 million. This pattern was predicted by a theoretical account in which linear behavior across many orders of magnitude is achieved through highly linear patterns of placement on smaller lines that are recycled and scaled to larger numerosities. Just as the telescope allows perception of the imperceptibly distant, reasoning processes over the natural numbers appear to allow intrinsically limited magnitude- perception systems to apply (with distortion) to much larger scales. Representations that Contribute to Quantity For small numbers, it appears that matching numbers to magnitudes requires the coordination of several systems. For instance, small exact set sizes may result from the active coordination of memorized count lists, pointing procedures, and a perceptual system that responds selectively to numerosity (Feigenson, Dehaene, & Spelke, 2004; Carey, 2009). Although exact numbers over 20 are typically not well linked to perceived set sizes (Izard & Dehaene, 2008; Sullivan & Barner, 2012) the linearity of the metric scale of the numbers extends out much further (Siegler & Opfer, 2003). Still, at some point even a log-based neural scale must run out. We simply cannot have lognormal response nodes that span the natural numbers, since our brains are finite in size, nor can we have created the entirety of the natural numbers through a completed infinity of recursions. Furthermore, typical environments do not require the individuation of sets as large as 10^9 (1 billion). How do we deal with these quantities when we do encounter them? One possibility is that we induce new tokens as needed throughout life, extending a process essentially identical to that used to induce numbers small enough to encounter frequently (Leslie, Gelman, and Gallistel, 2009; Gelman, 2011). We might call this the domain continuity hypothesis. Numbers in the range of 10^9 might be recursively generated through successorship. On this account, our cognitive systems do not represent a completed infinity, but represent the natural numbers through unbounded extensibility. A plausible but (to our knowledge) novel hypothesis is that a lognormal representation system could be extended to include large numbers, as needed, by the creation of new, log-normally responsive tokens with some spacing. Representing the entirety of numbers up to 10^12 this way requires less than 100 times the resources required to get from 2-3. While we cannot hope to construct a representation of a large Mersenne prime this way, we might well deal with large government deficits using the same log-normally distributed quantity representations that seem to be shared across many other species. Alternatively, Keywords: mathematical cognition, concepts, high-level reasoning, numerical cognition Introduction Much reasoning in mathematics involves taking structures well-defined in particular concrete domains and extending them to new less accessible contexts. For example, exponents are often presented initially to learners as repeated multiplications: x 2 =x*x. This idea is then extended to zero exponents, fractional and real exponents, and even complex-valued exponents. This paper focuses on a very elementary instance of the extension from the concrete to the abstract: the extension of the natural numbers beyond the feasibly countable range. Mathematics often deals with numbers far outside any normal experience. For instance, currently the largest known prime number (the 48 th Mersenne prime) would contain 17,425,170 digits if written as an Arabic numeral. It is easy enough to order these numbers from smallest to largest; it is much, much harder to have any sense of their actual size. Nor is their actual size of any importance, except for practical reasons of computation. Almost exclusively, when magnitude is important for very large numbers, it is in comparison to other numbers related by some thread of reasoning in the same problem. In this paper, we examine much more prosaic “big numbers”: those numbers just beyond the limit of practical countability—in the range of about 10^6-10^12. These numbers fall into an important boundary: so large that