Robin Hanson
Curt Adams
Bryan Caplan
Joel Cohen
Roger Congleton
Tyler Cowen
Mark Crain
Eirikur Hallgrimsson
Carl Feynman
Hal Finney
Brad De Long
Perrin Meyer
Carlos Ramirez
John Schroeter

Long-Term Growth As A Sequence of Exponential Modes

Introduction Many have argued that current U.S. stock prices are much higher than can be justified by reasonable profit forecasts based on empirical models derived from the recorded history of the U.S. economy  (Shiller, 2000) . While it might be theoretically possible for the economy to suddenly switch to a fundamentally new mode with much higher corporate profits, our experience with the U.S. economy offers little evidence that a mode switch of this magnitude has happened anytime in the last century or so  (Barro & Sala-i-Martin, 1995) , though we may be able to understand smaller changes in stock prices in terms of milder fluctuations in profit trends  (Barsky & De Long, 1993) . The arguments against high current U.S. stock prices are thus largely empirical, in contrast with the theoretical nature of the (typically informal) arguments offered in support of investor optimism. These empirical arguments, however, are based primarily on the last century or so of economic experience. Could empirical forecasts based on a longer term history of economic growth offer more support for the rationality of high stock prices? To address this question, this paper will take an empirical approach to modeling very long term data on economic growth, with an eye toward inferring the chances for very large changes in economic growth modes. In order to empirically model very long term economic history, we need one or more very long term time series, and we need to choose one or more functional forms to use in modeling those time series. Most attempts to formally model long term human history have focused on population time series, as population estimates have been offered going back many thousands of years, and such time series have been considered to be the most reliable long term series available. Also, for most of human history population seems to have reasonably reflected most fundamental changes, including areas colonized, capital accumulated, and technologies adopted. This seems to be because new productive abilities have mainly resulted in larger populations under Malthusian conditions of near-subsistence per capita consumption. Nearly constant per capita consumption, however, implies that reasonable estimates of total consumption can be obtained from reasonable estimates of population. We thus have reasonable estimates of world product up until about 1500, when per capita product began to rise. Economic historians have also provided us with reasonable estimates for world product over the past century or so  (Maddison, 1995) . So all that has stood in the way of a full long term world product time series was to fill in the gap from roughly 1500 to 1850. Recently economic historian Brad De Long has cleverly used a simple empirical Malthusian relation to fill this gap, and has thus created a world product time series ranging from one million B.C. to today  (De Long, 1998) . His estimates for before 10,000 B.C., however, are based on forty year old speculations  (Deevey, 1960) . This paper will therefore replace those old speculations with recent published estimates going back to two million B.C.  (Hawks, Hunley, Lee, & Wolpoff, 2000) . This paper will then attempt to empirically model that time series with a few simple functional forms. After considering models of gradually accelerating growth, which have been the most popular in formal models of long term population growth, we will turn to attempting to formalize the more common informal descriptions of history as a sequence of growth modes. And since exponential growth is the most common formal description of short term growth, we will consider modeling long term world product history as a sequence of exponential growth modes. The simplest sequence of exponentials is a sum of exponentials, and we will find that a sum of four exponential modes fits the data with less than a 2% average error, which is less than the average uncertainty in the estimates. One of these modes, however, appears to be there just to allow a more gradual transition between two other modes than a simple sum of exponentials will allow. So we will then try a constant elasticity of substitution (CES) form for combining exponentials, which can allow a transition to be smooth or sudden by varying a single parameter. We will find that a CES combination of three exponential modes also fits the data with less than a 2% average error. After describing these models, and discussing their interpretation, we will then use them to consider when we might see a transition to yet another mode of economic growth, and how fast growth might then be. It seems quite unreasonable to expect an infinite series of faster and faster growth modes, since that leads to an infinite world product in finite time. But it may be reasonable to wonder if a world that has so far seen at least three distinct growth modes might see yet one more growth mode.
 The Data Economic historians have reasonable estimates for world product over the last century or so, while other historians have reasonable estimates for world population over the last ten thousand years, and cruder estimates for the two million years before that. Until a few centuries ago, per-capita consumption was typically near subsistence levels, making it easy to infer world product from world population. Thus the obstacle to having a full long term world product time series has been to get reasonable estimates for the intermediate period, from roughly 1500 to 1850. Recently, economic historian Brad De Long found a simple linear relation between world population growth rates and per capita product, a relation which seems valid at the low values of per capita product observed from roughly 1850 to 1950  (De Long, 1998) . This linear relation has allowed him to created world product estimates for the missing intermediate period, and thus to create a world product time series ranging from one million B.C. to today. De Long developed several time series based on alternative assumptions, and this paper uses his preferred world product series. The only change made here is to substitute more recent estimates for population prior to 10,000 B.C. De Long continued the use by Kremer  (Kremer, 1993)   Deevey's estimates, however, were never published in a peer-reviewed journal, cite no other sources, and are now forty years old. The data set used here excises Deevey's forty year old estimates and instead substitutes two more recent population estimates. Hawks and coauthors  (Hawks et al., 2000)  estimate a "population bottleneck" of about ten thousand at about two million B.C., and that we "cannot reject exponential growth from" then until ten thousand B.C. Consistent with that conclusion, they also accept a Weiss  (Weiss, 1984)   The resulting world product time series has 54 data points, is graphed in Figure  1 , and is listed in the appendix. In Figure  1 , world product is described in units of an equivalent number of humans at a subsistence consumption level. 2 Time is described by the number of years before the somewhat arbitrary date of 2050. This data is also graphed later in Figure  2 , which shows world product growth rates versus levels, displaying each step between data points as a horizontal line. This graph looks noisier, because growth rates are noisier, but it has the virtue of avoiding the arbitrary choice of a reference date like 2050. 3
 Choosing a Functional Form What functional form should we use to model our world product time series? Simple exponential growth is probably the most popular general model for describing positive quantities that grow in time by many orders of magnitude, as both population and world product have. It is, for example, widely used to model the last half century of economic growth  (Barro & Sala-i-Martin, 1995)  and 1 These estimates are based on a "multi-regional" model of human evolution  (Relethford, 1998 ). An alternative "out of Africa" model posits a similar population bottleneck at roughly 200 thousand B.C.  (Rogers & Jorde, 1995; Ingman, Kaessmann, Paabo, & Gyllensten, 2000) . The multi-regional model is based more heavily on non-genetic fossil evidence. Genetic evidence may well estimate the number of our direct ancestors alive on a given date, but nongenetic fossils seem better estimates of the size of the economy or ecological niche occupied by our direct ancestors and their very close relatives on that date. Further footnotes will, however, mention how the alternative model changes this paper's estimates. 2 Specifically, "subsistence" is $80 1990 "International" dollars, which is the ratio of De Long's preferred world product and population estimates for one million B.C. This per capita product has been assigned to my substitute dates of two million B.C. and 750 thousand B.C., and retained De Long's estimates for all other dates. 3 In the analysis which follows, we will occasionally compare world product growth since two million B.C. to the earlier exponential growth of maximum brain size. Maximum animal mass and relative brain size have both grown roughly exponentially since the Cambrian explosion about 550 million years ago, when animals first appeared in large numbers in the fossil record. The mass of the currently largest animal doubled roughly every 70 million years, while maximum brain volume relative to body volume (raised to the 2/3 power) has doubled roughly every 50 million years  (Russell, 1983) . (More details on the details of brain size history are given by Jerison  (Jerison, 1991) .) Together these estimates suggest that maximum animal brain size has doubled roughly every 34 million years over the last 550 million years (for a total of about 16 doublings). For comparison, the exponential decay rate of Marine genera over this period was a half-life of about 23 million years  (Valen, 1973; Newman & Sibani, 1999) . the last few centuries of growth in scientific literature  (de Solla Price, 1963) . Simple variations on exponential growth, such as linear trends in growth rates, have been particularly popular in making short term population forecasts  (Lee, 1990) , but seem clearly inadequate for longer term forecasts. Variations on logistic growth have also long been popular in estimating various population trends  (Meyer & Ausubel, 1999) , but also seem clearly inadequate for modeling very long term population trends  (Cohen, 1995) . Many researchers have had more empirical success explicitly modeling long term population histories with functions that describe accelerating change. The first such efforts used simple hyperbolics  (von Foerster, Mora, & Amiot, 1960) , and soon became infamous warnings against taking one's model too literally, since they predicted an infinite human population in the early twenty first century. More recent related efforts  (Kremer, 1993; Kapitza, 1996; Johansen & Sornette, 2000)  have all included corrections which could describe a recent or upcoming slowdown and other complexities, mostly to allow growth to today be approaching fundamental limits, and hence a final end to accelerating growth. Some authors have also informally summarized world history as continually accelerating change, but the more common informal summary is of human history as a sequence of specific growth modes. History has, for example, been described as the slow expansion of hunter-gatherers, followed by faster growth following the domestication of plants and animals, followed by even faster growth with commerce, science, and industry  (Cipolla, 1967) . Simple exponential growth is a very common model for describing particular historical periods. It is, for example, widely used to model the last half century of world economic growth  (Barro & Sala-i-Martin, 1995) . Since historical periods tend to be described as exponential growth, formal models of transitions between periods tend to be models of transitions between exponential growth asymptotes. Many have modeled the industrial revolution this way  (Marvin Goodfriend, 1995; Hansen & Prescott, 1998; Jones, 1999; Steinmann, Prskawetz, & Feichtinger, 1998; Galor & Weil, 2000; Lucas, 2000) . At least one earlier transition, to cities, has also been modeled this way  (Marvin Goodfriend, 1995) . And possible current or upcoming transitions, such as due to computers or the internet, have also been modeled this way  (Helpman, 1998) . Given all this, it seems natural to formally model the long term history of humanity as a sequence of exponential growth modes. Yet to my knowledge no one has done this. Some authors seem to have come close, by drawing exponential curves suggestively next to data on log-log graphs of population growth  (Deevey, 1960; Kates, 1996; Livi-Bacci, 1997) . (Some of these authors, however, do not seem to have realized that it could be simple exponentials they were drawing, and instead of "exponential growth followed by a period of approximate stability"  (Kates, 1996) .) One author did more explicitly consider and reject a sum of exponentials model for world population history, though without any explicit model fitting or test  (Cohen, 1995) . (Another author considered a sum of two exponentials, one of which was decreasing  (Lee, 1988) .) In this paper we thus take on the neglected task of more formally modeling long term human history as a sequence of exponential growth modes. In particular, we consider such a model of world product history. Once we have chosen to model history as a sequence of exponential modes, there still remains the question of how we should mathematically model transitions between exponential growth modes. We will first try a simple sum of exponentials, and then try to replace this sum with a CES (constant elasticity of substitution) form. Such a form allows each transition to be sudden or gradual, depending on a single free parameter per transition. We will end up with a model describing the three classic economic growth modes, which we might call hunting, farming, and industry.
 A Sum of Exponentials Model Apparently, no one has yet tried to formally model long-term growth as a sequence of exponentially growing modes. This paper thus attempts to do so for world product. One very simple model of a sequence of growth modes is a sum of exponentials, which for n modes we will write as Y (t) = Y n (t), where Y 0 (t) = 0, and for i from 1 to n, Y i (t) = Y i−1 (t) + M i (t), M i (t) = c i 2 t/τ i . The parameter τ i is the doubling time of mode i. If data consists of pairs (t j , Y j ), a min-square-error of logs method searches for model parameters of a function Y (t) that minimize j (log(Y (t j )) − log(Y j )) 2 . Applying this method to my data, the best fit sum of four exponential terms gives an eyeballpleasing fit, and a (root mean square) average percentage error of 1.8%. This is about the same magnitude as the smallest uncertainty or "indifference range" attributed to the population estimates that this data is based in part on, which is about 2%  (McEvedy & Jones, 1978) . Figure  1  compares this model with the data, and indicates that the main errors seem to be due to unmodeled fluctuations with a period of roughly 500-1500 years. Figure  1     M i (d i ) = M i−1 (d i ). (The questionable doubling time factor for the hunting mode comes from comparing it to the previous growth rate of animal brains, as discussed in the footnotes. 6 ) Table  1  also gives a world product factor, defined as Y i (d i+1 )/Y i−1 (d i ), which says how much world product increased due to each term.
 CES-Combined Exponentials While most of the modes in the above model seem to have a familiar interpretation, the second to last mode, the one with a 58 year doubling time, seems more difficult to interpret. We might consider it to represent a commercial revolution, distinct from the later industrial revolution. But visibly, it seems to just be there to allow the model to describe a slower and smoother transition than a simple sum of exponentials will allow between the modes before and after it. To more directly model the possibility of a slow versus fast transitions between modes, we can combine terms not with a sum, but instead with a constant elasticity of substitution (CES) form  (Barro & Sala-i-Martin, 1995) , as in Y i (t) = (Y i−1 (t) a i + M i (t) a i ) 1/a i . When the CES power is a i = 1, this is x + y, the previous case. When a i = 0, this is proportional to (Y i−1 M i ), which for exponential terms essentially makes an infinitely slow transition. When If we now fit such a CES combination of three exponential terms to our data, again using the min square error of logs method, we get an eyeball-pleasing fit, and an average percentage error of 1.7%. This model is described 7 in Table  2 , and compared to the data in Figures  1 and 2 . This model has the same number of free parameters as the four-term sum of exponentials model, and a modestly smaller total squared error of logs. a i is positive infinity, this is max(Y i−1 , M i ), Note that the industry mode will not begin to dominate until 2020, has a doubling time which is half of what we have observed over the last half century, and is already responsible for a larger world product factor than either of the previous two modes. The very low transition power allows a new mode to have a strong influence long before it officially dominates, and predicts a mid-transition growth rate of about half of the new mode growth rate.
 Interpreting the Modes By describing world product history as either a sum or CES-combination of exponentials, one seems to essentially be saying that among the thousands of large and important changes that the world economy has seen, only a handful are fundamentally big news. Other changes are either "small," at least in the big picture, or such that a change of broadly similar economic magnitude and timing was largely predetermined by the fundamental economic forces at play. Typically, the economy is dominated by one particular mode of economic growth, which produces a roughly constant growth rate. While there are often economic processes which grow exponentially at a rate much faster than that of the economy as a whole, such processes almost always slow down as they become limited by the size of the total economy. Very rarely, however, a faster process reforms the economy so fundamentally that overall economic growth rate accelerates to track this new process. The economy might then be thought of as composed of an old sector and a new sector, a new sector which continues to grow at its same speed even when it comes to dominate the economy. Following this line of interpretation, the simple sum of exponentials model seems to assume that during each transition, the "new" and "old" economies co-exist but do not influence each other. In contrast, the CES-combination form seems to describe interactions between modes. Modes might complement each other (0 < a < 1), making for a more gradual transition, or substitute for each other (a > 1), making for a sharper transition. The large transition centered around 5000 B.C. (but surely begin thousands of years earlier) is naturally interpreted as the transition to from "hunting" to "farming" (both broadly construed). Populations relying on food from the domestication of plants and animals seem to have grown exponentially for several millennia before this transition date, nicely fitting a simple interpretation of co-existing new and old economies  (Diamond, 1997) . In the CES model, the power estimate for the farming transition suggests that farming mostly substituted for hunting, rather than complementing it. Because the data is poor, this suggestion can only be weak. But it does make sense, given that hunting and farming economic modes competed for the use of both land and labor. 8 The recent "industrial revolution" transition has a CES power describing a much more gradual transition, suggesting less substitution and more complementarity between the old and new economic growth modes. If we interpreted the old and new modes relatively literally as "farming" and "industry", this makes some sense, in that improved farming technology has freed workers for industry while improved industry technology has often transferred to farming technology. Such a literal interpretation, however, is argued against by the CES model estimate that we will not "officially" transition to the new mode until 2020, in the sense of having the new contribution to the world economy exceed the old part. For this to make sense, much of our non-farming economy needs to be interpreted as still dominated by the "old" growth mode. The form of our model suggests that the new growth mode might be found in small quantities well before it has a noticeable effect on total growth rates, and that what distinguishes this new economy inside the old one is that it is growing rapidly, at a rate closely related to what will become the new total economic growth rate. We might thus better understand what distinguishes the current "new" and "old" growth modes by looking for processes which doubled before the transition at a rate similar to the modeled new rates of 6 or 15 years. Some interesting candidates are found in measures of scientific progress. The number of scientific journals has doubled steadily about every 15 years since about 1750, even though the world product doubling time in 1750 was around a century. (From about 1650 to 1750 both journals and scientific societies doubled roughly every 30 years.) Narrower measures of progress grow slower, while broader measures grow faster. For example, the number of "important" discoveries has doubled every 20 years, while the number of U.S. engineers has doubled every 10 years  (de Solla Price, 1963) . This rough coincidence in timing and doubling times suggests, though only weakly, that a new "scientific" evolution and diffusion of knowledge via a subject-specific network of articulate specialists is fundamental to the new industry growth mode. Perhaps the diffusion of seeds and artifacts via a simpler more spatial network of less specialized people is fundamental to the farming mode. If so, the new 15 year doubling time of the sum of exponentials model suggests that published papers are close to the fundamental unit of growth, while the 6 year doubling time of the CES model suggests that a very broad measure of "scientific"-like activity is closer to the fundamental unit of growth. How can we make sense of the basic idea of the world economy repeatedly switching to new growth modes? That is, if the economy is capable of faster growth modes, what prevented them from happening earlier? Among the many models mentioned before of transitions between historical economic growth modes, perhaps the simplest postulate minimum scale effects, so that a new growth mode will not occur in economies below a certain size, density, or technology level. A stochastic variation on this is to make the number of attempts or chances to discover and transition to a new growth mode be proportional to the current scale of the economy. This variation also predicts that a transition will occur at roughly the time when the economy reaches some predetermined scale. Such models could interpret the roughly stair-step shape of Figure  2  somewhat literally. The lip of each step would describe a new mode which would not be realized until world product, moving along the x-axis, reached the base of that step.
 Could It Happen Again? Current U.S. stock prices seem to be much higher than can be justified by the expectation that the U.S. economy will continue much as it has for the last century  (Shiller, 2000) . If speculators are rational, it seems that they must be assigning a substantial probability to the possibility of a large change in the economy, such as a fall in the risk-premium investors demand, a fall in the real risk of corporate profits, a rise in the share of income taken by corporate profits, or a rise in real economic growth rates. It is this last possibility, of a large rise in economic growth rates, that seems to be offered some support by the empirical analysis of very long term economic growth. If one takes seriously the idea of long-term economic history as a sequence of exponential growth modes, one might naturally wonder if a world economy that seems to have so far seen at least three dramatic transitions to much faster growth modes will see yet another such transition. One pair of authors, Ausubel and Meyer, did informally "speculate on a fourth [Deevey-like] 'information' pulse starting now that would enable another order-of-magnitude rise" in population. But they took the idea no further  (Ausubel & Meyer, 1994) . Are there ways an economy can grow fast that will not be possible until the world economy reaches a threshold in size, density, or the price of some key technology? While it is certainly possible that the economy is approaching fundamental limits to economic growth rates or levels, so that no faster modes are possible, we should also consider the other possibility. To use our models to estimate how soon the world economy might jump again to a faster growth mode, and how much faster that mode might be, we must make some assumptions about what it is that is similar across past and future modes and mode transitions. The apparent regularities in Figure  2  suggest one approach. They suggest that we treat multiples of world product and growth rates as similar across growth modes. More precisely, if we assume that at each transition new values of the doubling time factor and world product factor are drawn again from the same constant-in-time distributions for these parameters, then previous values of these parameters offer "sample" estimates of future parameter values. In the sum of four exponentials model, if the current mode were to last through as large a world product factor as one of the previous three growth modes, it would last until one of the sample dates of 1963, 2032, or 2066. If the next doubling time factor were the same as one of the last three values, the next doubling time would be either would be .05, 1, or 4 years. 9 The world product method expects that the world product factor, as previously defined, of the current mode will be similar to that of the previous modes. The doubling time method instead expects to see a similar value for (d i+1 − d i )/τ i , which is the number of doubling times during the period when a mode is dominant. Since the "current" mode does not even start to dominate until 2020, the doubling time method naturally gives later date estimates. The estimates of the new doubling time offer what seems to be a remarkably precise estimate of an amazingly fast growth rate. One is tempted to immediately reject such rapid growth as too preposterous to consider, until one remembers how preposterous it would have seemed to forecast the previous two transitions that did in fact occur. The robustness of these forecasts can be examined by considering what these types of models would have predicted using only the data up to 1900. From 1900 to 2000, world product increased by a factor of 37. The best fit four term sum of exponentials model forecasts that a factor of 35 increase, while the best fit three term CES model forecasts a factor of 22 increase. The main reason for the CES model failure here is estimating a transition date of about 2060, instead of the 2020 date suggested by the larger dataset. Other parameter estimates were in much closer agreement. 10 Another row corresponding to a brain growth mode would give sample dates of 2072 and 2120. Using the out of Africa population estimate would change the new doubling time estimates from  14.9?,9.3,15.8 to 1.49?,93,15.8    3  and Figure  4  show how a new transition might look, if the new mode had a 15 day doubling time and the same strong complementarity with our current mode that our current mode had with the previous mode, and if the new mode first started to have a noticeable effect on growth rates in 2041. (Transitions starting at any other date would look very similar.) Table  4  makes it clear that in this scenario, the new mode has an enormous effect on economic growth rates very soon after it has any noticeable effect on growth rates. If this growth mode did not hit limits to slow it down, then by 2047, i.e., within six years of becoming noticeable, the economy would then grow by a larger factor than it had from two million B.C. until 2040. This is thus not at all a reasonable model for describing the recent small but noticeable increase in U.S. economic growth rates that some now attribute to a "new industrial revolution." Thus if stock prices are now high in anticipation of a transition to a new growth mode, it must be in anticipation of a much smaller change in growth rates, of a change that has not yet detectably changed growth rates, or of a much more gradual transition between growth modes. In summary, if one takes seriously the model of economic growth as a series of exponential growth modes, and if relative change parameters of a new transition are likely to be similar to such parameters describing old transitions, then it seems hard to escape the conclusion that the world economy could see a very dramatic change within the next century, to a new economic growth mode with a doubling time of roughly two weeks or less. While it is hard to see in much detail how the world economy could possibly grow that fast, it is at least suggestive that computer hardware costs have in fact been consistently falling for many decades at a doubling time of one to two years  (Moravec, 1998; Kurzweil, 1999) . Several observers have predicted a transition to a dramatically new economic mode where computers substitute for most human labor, perhaps not long after the raw computational ability of a desktop computer matches the computational ability of a human brain, around 2025  (Moravec, 1998; Kurzweil, 1999) . Simple formal models have even been developed suggesting that adding robots to very standard growth models can allow growth rates to naturally increase by a factor of ten, a hundred, or even more  (Hanson, 1998) . Others have speculated that even faster growth rates are possible with mature nanotechnology. If the next mode had a "slow" doubling time of two years, and if it lasted through twenty doubling times, longer than any mode seen so far, it would still last only forty years. After that, it is not clear how many more even faster growth modes are possible before hitting fundamental limits. But it is hard to see how such fundamental limits would not be reached within a few decades at most.
 Conclusion Many people have tried to make sense of very long term time series estimating human population. Some have explicitly modeled this history as steadily accelerating growth, while others have more informally described it as a sequence of growth modes. And exponential growth is by far the most common formal model of shorter term growth. Meanwhile, Brad De Long has recently constructed long term world product estimates from recent world product estimates and older population estimates. This paper has thus attempted the neglected task of more formally describing long term world product growth as a sequence of exponential growth modes. We have found that a time series of world product over the last two million years can be comfortably described as either as a sum of four exponentials, or as a CES-combination of three exponentials. But the CES model seems preferable on theoretical grounds, which can be thought of as describing the three modes of "hunt-ing," "farming," and "industry," broadly construed. (An earlier period of exponential growth in brain sizes may be the relevant previous growth mode.) In the CES-combination model, there is a sharp transition between hunting and farming, and a smooth transition between farming and industry. These can be interpreted as due to the interaction between hunting and farming being more one of substitution, relative to a more complementary relation between farming and industry. The rough timing and doubling time coincidence between industry and measures of scientific progress weakly suggests that scientific-like creation and diffusion of knowledge might be a key to the current growth mode. Since there seem to be some rough regularities regarding how much the economy grows between transitions, and how much faster each new growth mode is, this paper has also considered what these regularities suggest about when we might see yet another transition to a much faster mode, if such faster modes are possible. The suggestion is fantastic, namely of a transition to a doubling time of two weeks or less sometime within roughly the next century. One might think this suggestion too fantastic to consider, were it not for the fact that similar predictions before previous transitions would have seemed similarly fantastic. We should also keep in mind that investors in U.S. stocks seem to be betting that a large change in the nature of the economy is likely soon. They may well be wrong, but this paper shows that the empirical case against such expectations is not as clear as one might expect from empirical analyzes looking at the last century or so. From a purely empirical point of view, very large changes are actually to be expected within the next century. of population estimates from McEvedy and Jones (McEvedy & Jones, 1978) going back to 10 thousand B.C., and of three population estimates from Deevey (Deevey, 1960) for 25 thousand B.C., 300 thousand B.C. and one million B.C.
 population estimate of about half a million between one million B.C. and 500 thousand B.C., which I've coded as at 750 thousand B.C. 1
 also shows the best fit hyperbolic model, 4Y (t) = c(T − t) −α ,which has a power of α = 1.46, goes to infinity at T = 2004, and has a much inferior average percentage error of 9.6%. (Using three terms instead of four in the sum of exponentials model also gives a visibly much worse fit.)For each term,
 with an infinitely fast transition. Other positive values of a i allow for other intermediate speeds of transition.

 Table 1 : 1 Table 1 lists a mode description, doubling time 5 , the factor by which the doubling Sum of Exponentials Growth Modes time accelerated between terms, and a "date began to dominate," defined as the d i that solves Growth Doubling Date Began Doubling World Product Mode Time (years) To Dominate Time Factor Factor Hunting 230,000 2,000,000 B.C. 149? 406 Farming 860 4700 B.C. 267 178 ?? 58 1730 A.D. 14.9 8 .9 Industry 15 1903 A.D. 3.8 > 78
 Table 2 : 2 CES-combined Exponentials Growth Modes Growth Doubling Date Began Doubling World Product Transition Mode Time (year) To Dominate Time Factor Factor CES Power Hunting 224,000 2,000,000 B.C. 153? 480 ? Farming 909 4860 B.C. 247 190 2.4 Industry 6.3 2020 A.D. 145 > 590 0.094
 Table 3 : 3 Table 3 describes sample forecasts based on the CES model (which the author prefers). It9 If the next mode were like the brain growth mode, it would start in 2162 and have a doubling time of 0.1 years. Using the CES Model to Forecast The New Mode includes sample forecasts of the next economic doubling time, and two kinds of estimates of the next transition date. 10 Growth Doubling Time New Doubling World Product New Date New Date Mode (DT) Factor Time (days) (WP) Factor from WP from DT Hunting 153? 14.9? 480 1996 2075 Farming 247 9.3 190 1976 2067 Industry 145 15.8 > 590
 Table 4 : 4 . New Transition Scenario, Instantaneous Annual Growth Rates For example, the up-to-1900 CES estimate of the new doubling time is 6.4 years, very close to the up-to-2000 estimate of 6.3 years. Timing estimates thus seem less reliable than growth rate estimates. 2039 2040 2041 2042 2043 2044 2045 2046 2047 6.1% 6.1% 6.6% 8.0% 14% 41% 147% 476% 1023%
 Table 4 , 4 Figure
			 The best fit "log-periodic model," of the formY (t) = c(T − t) −α (1 + b cos(ω ln(T − t) + φ)) (Johansen & Sornette, 2000) , has a power of α = 1.48, goes to infinity at T = 2005, and has an average percentage error of 7.4%.5 Under the "out of Africa" theory, the hunting mode began ten times more recently, at 200 thousand B.C. Substituting this estimate for the two multi-regional estimates would reduce the hunting doubling time, increase the hunting doubling time factor, and reduce the farming doubling time factor, all by a factor of ten. All other entries would remain unchanged.
			 For reference, an additional row of Table1describing the earlier brain size growth mode would have entries: 34,000,000, 550,000,000 B.C., ??, and "67,000." The last entry is the total increase of brain size, which isn't directly comparable to the other rows since it is not an increase in world product.
			 An additional row describing the earlier brain size growth mode would have the same entries as before, and the "out of Africa" theory would change the same entries by a factor of ten.
			 Maximum brain size does not translate very directly into world product, but the long roughly steady growth in maximum brain size has been mentioned in this paper because it may be the relevant growth mode prior to the early spread of the human population. Product growth in an economy of human-like creatures seems to have awaited the evolution of a land-based animal with hands, a large enough brain, and perhaps other unknown prerequisites.

Abstract: A world product time series covering two million years is well fit by either a sum of four exponentials, or a constant elasticity of substitution (CES) combination of three exponential growth modes: "hunting," "farming," and "industry." The CES parameters suggest that farming substituted for hunting, while industry complemented farming, making the industrial revolution a smoother transition. Each mode grew world product by a factor of a few hundred, and grew a hundred times faster than its predecessor. This weakly suggests that within the next century a new mode might appear with a doubling time measured in days, not years.


Artificial Interdisciplinarity: Artificial Intelligence for Research on Complex Societal Problems

Introduction This paper explores the question of how artificial intelligence (AI) can be of value to interdisciplinary research (IDR) aimed at addressing major societal problems such as public health, environmental protection, and emerging technology governance. These problems are highly complex and multifaceted and do not fit neatly into traditional academic disciplines. For example, the study of global warming integrates environmental science, social science, policy, energy engineering, and more. This extreme breadth makes IDR a very difficult cognitive challenge for even the most capable human researchers. If AI can help meet this challenge, that could be of considerable value for addressing many major societal problems. This paper is especially interested in the possibilities for near-to intermediate-term AI systems that use existing AI techniques or relatively straightforward extensions of them. These would ideally be systems that computer scientists could work toward right now, and that would also be well short of strong AI, i.e., human-level or super-humanlevel artificial general intelligence (AGI). A central question of this paper is: Can we imagine tractable, real-world AI system designs that would make significant contributions to understanding and addressing complex and important societal problems, but without posing the major risks and ethical issues associated with strong AI? I humbly acknowledge that I do not know the answer to this question, but I do know that it is a worthy question to ask, and I can lay out some relevant considerations. That is the aim of this paper. To streamline the discussion, let us introduce the following term: Artificial interdisciplinarity (A-ID): Artificial intelligence that performs interdisciplinary research or supports other agents in the performance of interdisciplinary research. Some emphasis is warranted on the clause "or supports other agents in the performance of." Per this definition, an AI system that human researchers use to perform IDR counts as A-ID. Given this wider definition, it follows that there is A-ID in active use right now. Likewise, ideas for new forms of A-ID can leverage human collaboration. To make a valuable contribution to addressing societal problems, A-ID only needs to help with some aspect of IDR; it does not need to be able to complete IDR projects with no human collaboration. Indeed, it is plausible that an A-ID system that could succeed at the full range of IDR tasks would need to be strong AI. As a scholarly contribution, this paper sits at the intersection of literatures on IDR and both near-term and long-term AI, as well as the nascent concept of intermediateterm AI. Regarding IDR, there is an active line of research on the cognitive challenges faced by human interdisciplinary researchers  (Bracken and Oughton 2006; Keestra 2017; MacLeod 2018) . To the best of my knowledge, this literature has not yet considered the current or potential future role of AI. Regarding near-term AI, an extensive literature applies existing AI tools to support the research process. Most of this is oriented toward narrow disciplinary research, but some studies involve IDR  (Nunez-Mir et al. 2017; Tuhkala et al. 2018) . There is also a more general, extensive, and longstanding use of computers in IDR, especially in the physical sciences  (Crease 2017) , though that is beyond the scope of this paper. Regarding long-term AI, some studies consider the prospect of superintelligent AI designed as "oracles" to answer a wide range of questions that humans might have, some of which are in the domain of IDR  (Armstrong et al. 2012; Yampolskiy 2012) . Each of these lines of research is discussed in greater detail later in the paper. Finally, recent research has argued that intermediate-term AI has gone overlooked relative to near-term and long-term AI  (Parson et al. 2019a (Parson et al. , 2019b Baum 2020) . This paper provides dedicated attention to intermediate-term AI and concurs that this time period merits attention. The paper also seeks to encourage AI research to orient itself toward addressing important societal problems and to provide some direction for this. At present, much of AI research is instead oriented toward more intellectual goals of expanding the capacity of AI systems or toward profitable activities of technology corporations. As is wellknown in a lot of IDR, intellectual progress and profit do not necessarily bring progress on societal problems and in some cases can make the problems worse, for example, intellectual progress on dangerous technologies and profitable activity that pollutes the environment. IDR is also not necessarily good for society and can be oriented toward a wide range of ends including intellectual progress and profit. Furthermore, even when IDR is focused on addressing societal problems, it is still not a panacea for them-the existence of literature addressing societal problems does not on its own ensure that the problems actually get addressed. That said, IDR is vital for addressing complex societal problems, and this is the primary focus of IDR today. The paper therefore encourages A-ID research to join forces with the extensive communities of IDR focused on addressing societal problems. I approach the topic of AI and IDR as a veteran interdisciplinary researcher with some knowledge of both AI and the meta-study of IDR. I am not a computer scientist, and so I can speak with less confidence on what the AI options for IDR may be. Therefore, this paper seeks to lay out some general considerations and to hopefully serve as a prompt for an interdisciplinary conversation among computer scientists, scholars of IDR, and others with relevant perspectives. As this paper explains, this sort of interdisciplinary conversation can be difficult, but it is vital for making progress on a wide range of cross-cutting topics. In keeping with much of the literature on IDR, this paper is written in terms of "societal problems" instead of the concept of "social good" that frames this special issue. These two terms are broadly compatible, in that addressing societal problems is one major way to advance the social good. IDR is commonly used to make progress on specific problems or issues, and is perhaps less commonly used to understand or advance the good in a more abstract or general sense. The challenge of understanding the complex and distinctive features of many specific societal problems is a primary focus of IDR. As the first dedicated discussion of A-ID, the paper provides a broad discussion. Section 2 introduces IDR and outlines its cognitive challenges. Sections 3-5 discuss A-ID over the near-term, intermediate-term, and long-term. Section 6 discusses issues raised by A-ID. Section 7 concludes with general thoughts on the role of AI in addressing societal problems. Sections 3-5 are organized around time scales because each time scale poses distinct issues for A-ID. Near-term AI can be defined as AI that currently exists or could exist in the near future via relatively straightforward extensions of current AI systems. Attention to near-term A-ID includes description of existing A-ID systems and prospects for extending them. Intermediate-term AI goes beyond what is already feasible or now in development, but it stops short of the more extreme forms of AI that could exist over the long term. This paper defines long-term AI as AI with capabilities that equal or exceed human cognition; terms for this include strong AI, human-level AI, artificial general intelligence, ultraintelligence, and superintelligence. Discussions of AI are often divided between the near term and the long term, with some debate over which is more important  (Baum 2018; Cave and Ó hÉigeartaigh 2019; Prunkl and Whittlestone 2020) . Less attention has been paid to AI over intermediate time scales  (Parson et al. 2019a (Parson et al. , 2019b , though this paper finds that A-ID at all time scales can be important.
 Interdisciplinary Research and its Cognitive Challenges IDR has been defined in a variety of ways  (Klein 2017)  Interdisciplinarity is often compared with related concepts, especially multidisciplinarity and transdisciplinarity. To generalize, multidisciplinary research is commonly understood to be the closest to disciplinary research, consisting of research that has input from multiple disciplines without integrating them; IDR takes an additional step away from disciplinary research by also integrating insights from across disciplines; transdisciplinary research is the furthest from disciplinary research by transcending disciplinary boundaries altogether, as if they did not exist, and often also transcending boundaries between academic and nonacademic sources of knowledge. These distinctions are fuzzy and contested  (Klein 2017)  but suffice for purposes of this paper. Note that some of the research described in this paper may be more precisely classified as transdisciplinary, though the distinction between interdisciplinarity and transdisciplinarity is not crucial for this paper and the paper will use the two terms more or less interchangeably. IDR has traditionally had two somewhat distinct motivations. See  Bernstein (2015)  for an intellectual history. One motivation is intellectual, seeking the understanding of cross-cutting topics and the synthesis of knowledge accumulated in (and beyond) the disciplines. An especially ambitious form of this motivation, associated mainly with transdisciplinarity, is for unifying all of the various strands of human knowledge and intellectual activity. The concept of consilience as developed by  Wilson (1998)  is one work in this direction. While intellectually motivated IDR is commonplace, IDR is perhaps more commonly motivated by practical societal problems. Environmental problems have been a longstanding focus, given their societal importance and multifaceted nature, involving both social and ecological systems. More generally, IDR often-but not always-addresses social and policy issues associated with science and technology; this includes social and policy issues raised by AI. A different type of motivation for IDR is the complexity of many important subjects of study. Complexity can be defined in a variety of ways; one definition proposed in the context of IDR is that complexity is an attribute of systems whose components-which may themselves be systems-interact in predominantly nonlinear ways  (Newell 2001) . I would add that while this definition's emphasis on nonlinear systems has a mathematical tone, the interconnections found in IDR are often best understood in more qualitative terms. Regardless, the definition points to the distinction between multidisciplinarity and interdisciplinarity. Many research subjects do not fit neatly within any one academic discipline, but research on these subjects does not necessarily require integration across disciplines. For example, oceanography and battery technology are both relevant to the study of global warming, but oceanographers and battery engineers typically do not need to consider each others' work when doing their own. (This example is from a critique of IDR by  Jacobs 2013, p.130-131.)  This disconnected work would classify as multidisciplinary but not interdisciplinarity. As a contrasting example, global warming policy may affect industrial activity, which affects greenhouse gas emissions, which affect climate patterns, which affect natural hazards, which affect human welfare. Interdisciplinary global warming policy research may consider these complex interconnections to assess the overall merits of a policy idea. All research, whether interdisciplinary or disciplinary, has social as well as intellectual dimensions. Indeed, one argument in favor of disciplines is that they provide a useful structure for organizing populations of human researchers within universities  (Jacobs 2013) . Interdisciplinary research centers serve a similar social purpose. Likewise, some challenges for IDR are of a social and not intellectual nature. For example, the discipline-based department system at most universities often incentivizes disciplinary research over IDR. These social dimensions may be less relevant to an A-IDR system. AI may be more skilled at IDR than humans because AI lacks social reasons to cluster into disciplines. However, if an A-IDR system derives its knowledge from the corpus of human scholarship, then it may learn and perpetuate disciplinary divisions, just as current AI systems learn and perpetuate human biases obtained from other human-produced datasets. The remainder of this section surveys some cognitive challenges of IDR. All forms of research can be cognitively difficult, but IDR poses some distinct challenges that make it especially challenging for human researchers. Note that these challenges also apply to transdisciplinary research and to a lesser degree to multidisciplinary research. For more general discussion of interdisciplinarity, multidisciplinarity, and transdisciplinarity, see e.g.,  Hoffmann et al. (2013 ), McGregor (2014 ,  Bernstein (2015) ,  Lawrence (2015) ,  Scholz and Steiner (2015) ,  Menken and Keestra (2016), and Frodeman (2017) ; for an alternative and more critical perspective, see Jacobs (2013).
 Disciplinary Divides Different academic disciplines commonly approach the same topic from different perspectives, using different conceptual paradigms and different language, featuring different intellectual traditions and standards, and often covering different aspects of the same topic. This creates the cognitive challenge of integrating the disparate input into a unified understanding of a topic  (Bracken and Oughton 2006; Keestra 2017; MacLeod 2018) . As a consequence, IDR requires a laborious and often frustrating process of translation between different disciplines. It is a translation of language as well as of intellectual norms and epistemic perspectives. Much of the challenge comes from the fact that human expertise builds from years of focused study and practice. The human mind can achieve high levels of performance on specific tasks by forming complex representations of the task and its surrounding context-for example, chess masters "chunking"  (Chase and Simon 1973)  together complex patterns of chess pieces. The sheer difficulty of developing expertise precludes individual researchers from being expert on the full range of subjects that are relevant to complex societal problems. Furthermore, the very nature of human expertise can make it harder for experts in one discipline to understand other disciplines. Indeed, the term "discipline" implies a certain disciplining of the human mind to think in certain ways. A human mind disciplined to think in one way can struggle to think in another way. Compounding the problem is the fact that academia is traditionally divided into thematic disciplines that do not map to societal problems. A scholar is typically trained and employed as, for example, a computer scientist or a social scientist, not as an expert on the social implications of computers. Academia has made some effort to restructure toward IDR, but the traditional disciplines still dominate. As a consequence, disciplinary divides are even greater than they in principle need to be.
 Massive Literatures Even if a researcher or research team is able to parse the disparate disciplinary contributions to a particular topic, there remains the challenge of reading the relevant literature. Even within a single discipline, literature review can be a highly laborious task. For complex interdisciplinary topics, it quickly becomes overwhelming. For example, the Intergovernmental Panel on Climate Change (IPCC) is a body of top global warming researchers from around the world. It produces periodic syntheses of the global warming literature for high-level policy audiences. IPCC reviews are massive exercises, involving hundreds of scientists each putting in hundreds of hours over periods of up to 5 years  (Victor 2015) . Despite this massive scale, the literature is now at a point where the IPCC's teams of researchers are struggling to keep up  (Minx et al. 2017) . Climate change is an especially complex issue with a large literature even by IDR standards, but the IPCC is also an unusually large IDR project. Other IDR projects with narrower scopes and smaller teams face similar literature challenges. The IPCC is distinctive in its effort to synthesize the entire peer-reviewed literature on a complex interdisciplinary topic. Researchers who merely wish to contribute to the literature do not need to read so much. However, gaining a basic understanding of global warming in all its facets still requires reading at least some literature across numerous subjects in natural science, social science, engineering, the humanities, policy, etc. This is an extensive undertaking, and it is compounded by the added challenge of translating across the disciplinary divides.
 Peer Review IDR poses distinct peer-review challenges. IDR commonly works at the interface of multiple traditional academic disciplines. In many cases, it is uncommon to have expertise in each of the disciplines. Indeed, the person or team performing the IDR may be the only one in the world working across that particular set of topics. For example, this paper works at the intersection of AI and IDR; I am not aware of any other research that does this. The eclectic mixes of topics found in IDR make it unusually difficult to peer review  (Laudel 2006; Pautasso and Pautasso 2010; Holbrook 2017 ). If no other researchers have expertise across the range of topics covered in the work, then it may be necessary to form an interdisciplinary team of reviewers, but then this team must overcome its own disciplinary divides, which is a substantial task just to complete a single review. Or, the task of working across the disciplinary divides may fall to the people handling the submissions-journal editors, grant program managers, etc.-but then this just adds to their burden. It has been suggested that the difficulty of reviewing IDR, combined with the tendency of researchers to favor work from their own specialty, biases reviewers against IDR  (Laudel 2006) . One study has found that IDR grant proposals are indeed funded at a lower rate  (Bromham et al. 2016) . This is another way in which the current academy may be systematically biased against IDR, limiting its contribution to addressing important societal problems.
 Transfer Some research aims to transfer insights from the study of one societal issue to another. Transfer is seldom discussed in literature about IDR, which tends to focus on one issue at a time (but see  Krohn 2017) . Transfer is not a panacea, especially given the distinct attributes that each societal issue has, which limits the extent to which insights transfer from one issue to another. Nonetheless, it can be a powerful way to study societal issues. Transfer can be especially helpful for newly emerging issues that have not yet built up a robust literature, such as issues involving emerging technologies. For example, Altmann and Sauer (2017) transfer insights from strategic studies of nuclear weapons to the study of newly emerging autonomous weapon systems.  Baum (2017a)  transfers insights from the psychology of promoting environmentally beneficial behaviors among the lay public to the study of how to encourage AI researchers to pursue socially beneficial AI designs. The psychology of transfer finds that people tend to be more successful at it when there is relatively little "distance" between the domain transferred from and the domain transferred to  (Perkins and Salomon 1992) . For example, it may be easier to transfer insight from one weapon system to another (as in Altmann and Sauer 2017) than it is to transfer insight from lay public environmental behavior to expert behavior on technology development (as in Baum 2017a). Unfortunately, much of the insight available to be transferred lies at a considerable conceptual distance. A major challenge is simply recognizing the cross-issue similarity. Indeed, another finding from the psychology of transfer is that people often struggle to apply lessons to "problem isomorphs," i.e., to another domain that is functionally equivalent but different in appearance  (Simon and Hayes 1976) . People even struggle with problem isomorphs when both domains are presented to them. IDR transfer is more difficult because researchers typically must identify the two domains for themselves. On top of that, IDR transfer can require bridging three sets of epistemic divides: the divide between the two issues and the divides within each of the two issues. This can make for an especially difficult cognitive task.
 Near-Term A-ID This section discusses A-ID that currently exists or could exist with relatively straightforward extensions of current technology. Note that this section takes a relatively broad view of what qualifies as AI. By narrower standards, some of what is presented here may not qualify as AI. The exact definition and scope of AI is complex and controversial and beyond the scope of this paper; for further discussion, see, e.g.,  Legg and Hutter (2007) on definitions and McCorduck (2004)  on how the scope of what is considered to classify as AI has gotten narrower over time.
 Search Engines Perhaps the most ubiquitous form of A-ID is the search engine for Internet and scholarly databases, including Dimensions.ai, Euretos, Google Scholar, IRIS.AI, Microsoft Academic, Omnity, Semantic Scholar, SourceData, and Web of Science. Search engines are valuable for many forms of research, but they are especially valuable for IDR. In narrow disciplinary fields, researchers need to follow a relatively small body of literature. They may even be able to keep up with the relevant literature without the use of search engines, instead identifying literature by reading certain journals and sharing relevant literature within peer communities. In contrast, IDR frequently pushes researchers outside their areas of familiarity and requires them to identify relevant literature from within extremely large bodies of work. Search engines can help with each of the IDR challenges described in Section 2. They can provide some help to the challenge of overcoming disciplinary divides by providing a tool to explore the literatures of other disciplines, though this leaves open the challenge of understanding publications in unfamiliar disciplines. They can be used to navigate the massive literatures on interdisciplinary topics, such as by searching for multiple keywords to find literature on specific themes for a societal problem-for example, "global warming" and "cost-benefit analysis." They can be used to identify peer reviewers-for example, the authors of papers on global warming and cost-benefit analysis. And, with some creativity, they can support the transfer of insights across societal problem-for example, one could search for literature on global warming and cost-benefit analysis to identify insights that can be transferred to cost-benefit analysis of other global environmental issues. Search engines could be improved. To my eyes, an important area for improvement is in the handling of synonyms and, more generally, the identification of terminology. When searching for literature on an unfamiliar topic, identifying the right keywords to search for can be a major impediment. Once one knows the right keywords, relevant literature often flows abundantly. It would be especially valuable to have tools that could identify keyword synonyms used by other disciplines. Handcrafted resources, such as Wikipedia, can be helpful, but such resources are not available for all keywords. Automated tools for this could be quite helpful. Synonym identification is an active subject of AI research. One approach is to analyze patterns in the text surrounding a word, known as "distributional word vectors" or "word embeddings," on grounds that synonyms can be used in the same way and therefore tend to be surrounded by similar text  (Leeuwenberg et al. 2016; Mohammed 2020 ). However, this work uses large linguistic datasets (i.e., large collections of text). Much less text is available for identifying synonyms in academic literature, especially for the many specialized concepts and small subfields. Indeed, the rarity of specialist terms can be a distinguishing characteristic of academic publications. One academic search engine, Omnity (https://www.omnity.io), uses the rarest words of a document to identify other documents with a similar mix of rare words on grounds that documents with the same rare words are likely to be related  (Perkel 2017 ). Omnity's approach may work well within a discipline, but it may be less well suited for search across disciplines that use different terms for the same concepts. Further research may be needed to develop techniques for synonym identification in small academic literatures.
 Recommendation Engines At least two forms of recommendation engines are currently available for research. The first provides recommendations of publications. For example, Google Scholar provides publication recommendations that are customized for individual user profiles, and Elsevier provides publication recommendations that are customized for specific publications on its ScienceDirect website. A more specialized example is the project http://xrisk.net that provides recommendations of publications in the field of catastrophic risk  (Shackelford et al. 2020) . These recommendations provide an additional way for researchers to identify relevant literature on the topics they are studying. Note that the recommendations cover topics that researchers have already identified, via their user profiles or the publications they are already looking at, so they are less valuable for the IDR challenge of exploring unfamiliar lines of research. Therefore, in terms of the challenges described in Section 2, these recommendation engines can be especially valuable for handling massive literatures, and they may be of relatively limited value for overcoming disciplinary divides. A second form of research recommendation engine recommends potential peer reviewers to journal editors. This is used, for example, in the Elsevier Evise system, which was launched in 2015. Insofar as Evise provides good recommendations of peer reviewers, it can lighten the burden on journal editors, as noted, for example, by the editor of the interdisciplinary Elsevier journal Ecological Economics (Hukkinen 2017). Another example is the Toronto Paper Matching System developed for matching papers to reviewers for computer science conferences  (Charlin and Zemel 2013) . While the Toronto Paper Matching System is used predominantly for computer science, it potentially could be expanded to more IDR domains. The value of recommendation engines for IDR could be especially large due to the difficulty of identifying appropriate reviewers for IDR. One way to improve interdisciplinary recommendation engines would be to streamline the process of building specialized recommendation engines. The recommendation engine presented by  Shackelford et al. (2020)  uses a custom artificial neural network trained with a hand-coded dataset of articles found to be relevant to the field of catastrophic risk by an open ("crowdsourced") team of researchers in the field. Other fields could apply the same approach. Hand-coding the training dataset is laborious but is vital for aligning recommendation engine results with the interests of people in the field. The labor required for this may be large, but it does not require any skills other than knowledge of the field. In contrast, developing the artificial neural network does require more specialized skills. Fields that lack people with these skills may struggle to develop their own recommendation engines. Therefore, the process of building fieldspecific recommendation engines could be streamlined by developing an artificial neural network that could be trained with data from any field and in particular by developing a more accessible user interface for it.
 Automated Content Analysis A final application of AI to research is the use of machine learning for analyzing the content of the literature on a given topic, known as "automated content analysis"  (Nunez-Mir et al. 2017 ). Essentially, it treats academic literature as an instance of "big data", sometimes referred to as "big literature"  (Nunez-Mir et al. 2016) . Automated content analysis analyzes clusters of key words and phrases to produce statistical trends in literatures. This can be of value for learning what a literature has tended to find, where gaps may lie, etc. It can also be used for identifying relevant literature on a topic and in this capacity may be considered a form of search engine. It is of particular interest in the context of systematic reviews, in which the aim is to provide a complete and unbiased account of the literature on a particular topic. In terms of the challenges described in Section 2, this can be of high value for handling the massive literatures of IDR. While automated content analysis has been used mainly for narrow disciplinary research, it has been used for IDR studies of forest ecology (Nunez-Mir et al. 2017) and participatory design  (Tuhkala et al. 2018 ). The Nunez-Mir et al. (  2017 ) study is illustrative of the possibilities. It analyzes the text of 14,855 abstracts in 7 forestry journals. It uses a mix of supervised and unsupervised seeding, meaning that central concepts are obtained via a mix of human input and algorithmic text mining. Subsequent algorithmic analysis assesses the preponderance of the concepts in the abstract database. A primary finding of the study is that most of the abstracts do not show an interdisciplinary approach and very few considered social dimensions of forest ecology. In consideration of this finding, Nunez-Mir et al. (  2017 ) call for more IDR on forestry, especially its social dimensions. Note that the Nunez-Mir et al. (  2017 ) study does not reveal anything about the nature of forests or their interdisciplinary attributes beyond the list of central concepts. The output is a map of the literature, not an interpretation of what the literature means and what its societal implications are. This is a general limitation of current automated content analysis. Thus, as  Sutherland and Wordley (2018, p.366)  write: Advances in artificial intelligence and machine learning could make it easier to perform tasks such as locating papers for defined topics using search terms, categorizing papers as relevant for further consideration, and producing systematic maps. But for all fields, assessing the quality of individual studies, writing up summaries and so on will continue to require skilled humans, at least for the foreseeable future. The above remark about "the foreseeable future" suggests that the AI needed for interpreting literature and its societal implications is significantly beyond the capacity of current AI. My own judgment is that this is probably correct. Therefore, interpretation is considered in the following section on intermediate-term A-ID.
 Intermediate-Term A-ID Intermediate-term A-ID systems would go significantly beyond current capabilities but not so far beyond that they would constitute "strong" A-ID with human-level or superhuman-level capabilities. Intermediate-term is of interest because it could be of greater value to IDR than near-term A-ID while avoiding the downsides associated with strong AI (see Section 5). Ideally, designs for intermediate-term A-ID would derive at least in part from existing AI techniques such that current AI researchers could work toward their development. There can also be value to envisioning intermediate-term A-ID that requires new techniques but would stop short of strong AI. Of course, the sooner new A-ID capabilities become available, the sooner they can contribute to addressing important societal problems. As stated in the introduction, I am not a computer scientist, and so I am less confident in my thoughts on the design of intermediate-term A-ID systems. Nonetheless, a few basic thoughts on computer science factors are offered as a contribution to the topic, alongside some social factors that I can comment on more confidently.
 Interpretation One direction to extend current A-ID capabilities would be via the interpretation of research publications. As  Sutherland and Wordley (2018)  explain, this is a task currently left for human researchers (see Section 3.3). However, progress in the field of machine reading (a.k.a. natural-language understanding/interpretation) could change this. For example, one project to automate interpretation is the Elsevier-sponsored ScienceIE  (Augenstein et al. 2017) . The current focus of ScienceIE is on identifying phrases and relations between phrases within narrow disciplinary literatures. Its aims are ambitious: "Say you have a question about a paper: A machine learning model reads the paper and answers your question" (Augenstein as interviewed by  Stockton 2017) . Automated interpretation of research publications would be of considerable value for many forms of research, but it may be of particularly large value for IDR, due to the large and diverse literatures involved in IDR. An integrated system for automated content analysis and interpretation would be especially valuable for synthesizing the insights contained in vast and diverse IDR literatures. Such capability could drastically reduce the burden of human researchers in IDR synthesis projects like the IPCC and could likewise enable similar synthesis projects across a wider range of societal problems. In terms of the IDR challenges described in Section 2, interpretation would be especially valuable for handling disciplinary divides and massive literatures, especially if combined with automated content analysis. Publications can be interpreted in many different ways, some of which may be easier for A-IDR. For example, interpreting whether a publication has presented results that are statistically significant may be easier for A-IDR than interpreting whether the publication has presented results that could help address some societal problem. It may be the case that current AI paradigms based on neural networks will struggle with the interpretation of complex interdisciplinary texts because, as discussed by  Marcus (2018) , neural networks are skilled at finding statistical relationships in complex datasets but struggle to handle causal relationships, hierarchies, and open-ended environments, all of which are important attributes of interdisciplinary texts. However, the extent to which this is indeed the case is left as a question for future research on A-ID.
 Translation Given the difficulty of translating across the epistemic divides that exist between academic disciplines (Section 2.1), it could be of considerable value to have A-ID systems that could assist with the translation. A-ID translation would convert between the jargons of different disciplines or between that of one jargon and a more widely accessible plain English. This would reduce the linguistic burden of IDR, thereby facilitating researchers to work across a wider range of disciplines, either alone or in teams. It would also be of value for peer reviewers tasked with reviewing interdisciplinary work that includes topics outside the reviewers' own areas of expertise. Potentially, A-ID translation could build off existing work on machine translation between languages (e.g., English to Spanish). An active area of research is for unsupervised machine translation (e.g.,  Lample et al. 2018 ; summarized in plain English by  Ranzato et al. 2018) , in which translation does not require large datasets of pre-translated text. Unsupervised machine translation would be important for interdisciplinary translation because large datasets generally do not exist and would be expensive to produce. A-ID translation may require techniques that go beyond cross-language machine translation because the text of different disciplines commonly varies by conceptual basis and not just by language. In other words, much of the challenge of reading text from other disciplines is that readers do not know the concepts that the words are describing or at least are not accustomed to thinking in terms of these concepts. Interdisciplinary translation is about explaining unfamiliar concepts just as much as it is about explaining unfamiliar terminology. In contrast, cross-language machine translation generally involves concepts that are familiar in both languages. For example,  Ranzato et al. (2018)  presents the translation of the phrase "cats are lazy" from English to Urdu; presumably Urdu speakers are already familiar with the concept of a lazy cat. Therefore, successful A-ID translation may need to include a capacity for concept identification and translation that goes substantially beyond cross-language machine translation.
 Transfer The challenge of interdisciplinary transfer (Section 2.4) maps neatly to the AI topic of transfer learning  (Pan and Yang 2009) . Both involve the same essential process of applying knowledge gained in one domain to another domain with similar features, especially to avoid the resource-intensive process of building complex knowledge sets for multiple domains. Complex societal problems often have some similarities with each other. Human researchers can barely scratch the surface of the potential of interdisciplinary transfer due to its cognitive difficulty and the very large number of combinations of similar societal problems. If AI transfer learning could be applied to IDR, it could open up vast possibilities for learning about societal problems. Current AI transfer learning involves tasks that are rather far removed from IDR. For example, one common AI transfer learning task is image recognition  (Zamir et al. 2018) . Whereas images are readily expressed in statistical terms (e.g., via the binary representation of a digital image file), societal problems are not. To be sure, there are ways to express societal problems in some quantitative terms, such as via cost-benefit analysis for the evaluation of solutions. But much of the insight involved in interdisciplinary transfer is qualitative. A-ID transfer learning would need reliable ways to quantify this insight. It would also need some degree of translation, given the differing terminology and other linguistic constructs that can exist across societal problems. There is some reason to believe that A-ID transfer learning could be quite difficult to achieve. Computers are very capable at churning through large numbers of combinations of predefined concepts, just as they are very capable at churning through large numbers in general. However, they may struggle at identifying the sorts of connections between concepts that humans find meaningful. In a discussion of creativity in humans and computers,  Boden (2009)  argues that humans are relatively skilled and computers relatively unskilled, at "combinatorial creativity," meaning creativity in making new associations between seemingly unrelated concepts. Humans have rich mental models of the world built up throughout lifetimes of experience and observation, which enable us to devise analogies and imagery and other forms of combinatorial creativity. Endowing computers with anything similar is very difficult, and computers likewise struggle at combinatorial creativity. Interdisciplinary transfer is, at least in part, a form of combinatorial creativity, and so A-ID systems may struggle with it.
 Long-Term A-ID As an abstract matter, it is not hard to imagine some future AI that is capable of performing the full range of cognitive tasks involved in IDR. A strong AI/human-level AGI would presumably be able to do IDR at least as well as humans could. Whereas "narrow" AI is only intelligent for a narrow range of cognitive tasks, AGI is general in the sense of being able to perform a wide range of cognitive tasks. Human-level AGI could perform the same breadth of cognitive tasks as human minds and with the same skill as human minds. Such an AI could do any other cognitive task that humans can do, so it would presumably also be capable of IDR, potentially even so capable as to effectively solve all of society's problems. The AI may not be operating in complete isolation from humans-studies of human problems would likely require some human participation-but the AI may be able to play the entire suite of cognitive roles currently played by human interdisciplinary researchers. The prospect of solving societal problems is one common motivation for developing AGI. This is seen in the stated goals of active AGI research and development projects. For example, the AGI project AIDEUS observes that the "limitation of intellectual possibilities is fully experienced by scientists" and that this "concerns all problems of peoplefrom traffic jams to economic crises and wars." That is very much in the spirit of the present paper. AIDEUS states this as a motivation for building strong AI.  1  As another example, DeepMind envisions AI for "helping humanity tackle some of its greatest challenges, from climate change to delivering advanced healthcare." 2 Finally, Jeff Hawkins of Numenta calls for AI to help humanity "face challenges related to disease, climate, and energy"  (Hawkins 2017) . Note that these and other examples can be obtained from  Baum (2017b) . As valuable as AGI could be for IDR and for addressing complex societal problems, it faces several important downsides. First is the technical difficulty of building AGI. Despite the numerous projects working on AGI, the current state of the art remains quite far removed from it. It is not clear if or when AGI would be built; projections of decades or longer are common  (Baum et al. 2011; Grace et al. 2018) . That is a long time to wait for solutions to major societal problems. For some problems, such as nuclear war or extreme pandemics, the effects could be so severe that civilization would fail before it has the chance to build AGI. The desire to use AGI for societal problems could be additionally problematic by putting a dangerous time pressure on AGI development. Herein lies a dilemma. On one hand, earlier AGI would help with other societal problems. On the other hand, a rushed AGI could skimp on built-in safety measures and ethical design, increasing the risk of adverse or even catastrophic harm caused by the AGI. To the extent that major societal problems can be addressed without AGI, this buys AGI developers more time to get it done right. As an aside, note that some scholarship on IDR may be of value for the safe development of AGI, noting that the project of building AGI is itself an interdisciplinary task. For example,  Keestra (2017)  documents how a failure of interdisciplinary communication contributed to the 1986 space shuttle Challenger disaster and analyzes how future interdisciplinary communication could go better. Communication among those developing and launching AGI may benefit from this analysis. Careful AGI design may be needed even for AGI systems that exist only to provide input on societal problems. This is a theme of the literature on the prospect of superintelligent "oracle" AIs that are designed to only answer questions that humans pose to it  (Armstrong et al. 2012; Yampolskiy 2012) . The oracle design is sometimes conceived as a way to make AGI safe. The hope is that by restricting the AI to only answering questions, humanity can prevent the AI from taking dangerous physical actions to alter the world. However, studies of superintelligent oracles have raised the concern that they could still be dangerous, for example, by manipulating the people who interact with it (ibid.). The prospect of strong A-ID being dangerous in this way is another reason to explore near-and intermediate-term A-ID that can help address societal problems without raising the concerns associated with strong AI. Finally, it should be noted that the development of AGI is itself an interdisciplinary issue that would benefit from improved capacity for IDR. The development of AGI involves computer science, ethics, risk management, and more. Important questions include how best to design the AGI, how best to manage the human teams that are working on the design, and when to launch the AGI given the potential benefits and harms that could come from it. Insofar as near-and intermediate-term A-ID could improve the available answers to these sorts of questions, it could improve the outcomes from AGI development. This gives another reason to pursue near-and intermediate-term A-ID, including the concepts outlined in this paper.
 Discussion The limitations of near-term A-ID and the difficulties and risks of long-term A-ID suggest an important role for intermediate-term A-ID. A-ID interpretation, translation, and transfer could offer powerful capabilities for IDR and, more generally, for understanding and addressing complex societal problems. They could combine with existing A-ID systems (search engines, recommendation engines, and automated content analysis) to provide support across all of the cognitive challenges of IDR. This would not necessarily automate the full suite of IDR tasks-it could still leave important roles for human researchers-but it could nonetheless be of considerable value to IDR. On the other hand, some of these new capabilities could push uncomfortably toward AGI.  Yampolskiy (2013)  argues that the understanding of natural language is "AIcomplete," meaning that any AI system that is capable of understanding natural language would necessarily also be an AGI with a wide range of other capabilities. Similarly,  Armstrong et al. (2012, p.305)  postulate that "the task of advancing any field involving human-centred issues-such as economics, marketing, politics, language understanding, or similar-is likely to be  AI-complete," and Stockton (2017)  suggests that an AI capable of peer-reviewing research papers would require AGI. Exactly what, if anything, is AI-complete may be controversial and is beyond the scope of this paper, but it is nonetheless worth considering the implications of these arguments. These arguments imply that significant portions of A-ID interpretation, translation, and transfer may be impossible without AGI. Interpretation inherently involves understanding natural language, and so there may be very little A-ID interpretation that can be done without AGI. The translation of concepts may require understanding natural language, and so, without AGI, A-ID translation may be limited to the translation of terminology. Finally, transfer may also require understanding natural language and therefore AGI unless the societal problems of IDR can be reduced to a simpler quantitative form. The extent to which A-ID interpretation, translation, and transfer can be done without AGI is an important open question. Another concern is that intermediate-term A-ID could introduce certain biases into the research. This could occur if the A-ID interprets or evaluates research differently than human interdisciplinary researchers. For example,  Hukkinen (2017)  expresses concern that AI may succeed at assessing the intellectual rigor of research publications but struggle at assessing their significance for societal problems. Indeed, the ScienceIE project is explicitly framed in terms of scientific publications, but the interpretation of publications in other (non-science) areas that are essential to IDR for societal problems (such as ethics and public policy) require different ways of thinking and potentially different AI techniques. An A-ID system that can perform well on scientific rigor but poorly on ethics and policy could lead to weak or even harmful insight on how to address societal problems. More generally, intermediate-term A-ID systems must be carefully evaluated to ensure that they are producing constructive assistance to IDR. This will likely benefit from close collaboration between computer science developers of A-ID and human interdisciplinary researchers and potentially also people with other backgrounds, including stakeholders from outside academia. The inclusion of non-academic stakeholders is especially important with an eye toward actually addressing societal problems instead of just advancing the academic study of them. The existence of IDR literature does not on its own imply that the problems actually get addressed. Absent close engagement from non-academic stakeholders, there is a risk of producing solutions that work well in theory but not in practice. Additionally-and importantly-engagement with nonacademic stakeholders can result in different formulations of what the problems are in the first place. For these reasons, IDR-and especially transdisciplinary research-often calls for stakeholder participation in the research process. Readers should be aware that IDR often involves controversy, especially when it reaches out beyond the academy. Complex and important societal problems often involve heated disagreements between different stakeholders: between environmentalists and the fossil fuel industry over global warming, between rival geopolitical powers over nuclear weapons, etc. In some cases, there can be clever technical solutions that please everyone or at least do not upset anyone. However, more typically, solutions involve difficult tradeoffs between competing factions and fundamental moral values. Readers are encouraged to consider contributing to A-ID, but they should do so with an awareness of all that this entails. In the extreme case, A-ID could even be misused so as to make societal problems worse. For example, some of the same IDR that can help environmentalists refine their messages can also do the same for the fossil fuel industry, a dynamic that may be compounded by the extensive financial resources that the industry has at its disposal. Existing interdisciplinary research communities have often handled this by engaging with the ethical dimensions of societal problems and striving for ethically sound IDR practice. Work on A-ID should do the same. Finally, it should be noted that A-ID could help with all IDR, not just IDR on societal problems. As discussed in Section 2, IDR is perhaps most commonly performed for societal problems, but some IDR is oriented toward scientific and intellectual progress without any particular regard for societal problems. Such A-ID may be significant from the perspective of philosophy of science; this matter is left for future research.
 Conclusion This paper has articulated the concept of A-ID, outlined its importance for helping human researches cope with and overcome the considerable cognitive challenges of IDR, and explained how this can help address complex societal problems. A-ID exists today in the form of search engines, recommendation engines, and automated content analysis, but this leaves human researchers with the bulk of the work. Insofar as A-ID systems could contribute more to IDR, this could be a major contribution to addressing major societal problems. AI researchers interested in applying AI to societal problems may find this a worthy area of application. This paper describes some potential A-ID research directions; readers with more background in computer science than myself may have further suggestions. However, there are some difficult issues involved in A-ID that researchers should be cautious about, especially regarding the controversies inherent to complex societal issues, the potential for misuse of A-ID, and the potential relation of A-ID to strong AI/AGI. For their part, interdisciplinary researchers can contribute by using A-ID tools in their research and by contributing to efforts to improve the tools. Advancing A-ID is itself an interdisciplinary task that will benefit from contributions from people with a variety of backgrounds. Essential IDR cognitive tasks may be AI-complete, meaning that A-ID capable of these tasks would need to be strong AI. These tasks could include the interpretation of research publications, the translation of research concepts across disciplinary divides, and the transfer of IDR insights to new societal problems. If weak/narrow A-ID is unable to handle these tasks, or other essential IDR tasks, then human interdisciplinary researchers may be largely on their own until the advent of strong AI. This would mean that, despite all the disparate ongoing accomplishments of narrow AI, it would make relatively little contribution to addressing society's most complex and important problems. Alternatively, if there could be progress on A-ID that would not require AGI, then this could be a major contribution and would be a worthy focus for the field of AI. . One notable definition is provided in the US National Academies report Facilitating Interdisciplinary Research: Interdisciplinary research (IDR) is a mode of research by teams or individuals that integrates information, data, techniques, tools, perspectives, concepts, and/or theories from two or more disciplines or bodies of specialized knowledge to advance fundamental understanding or to solve problems whose solutions are beyond the scope of a single discipline or area of research practice (Committee on Facilitating Interdisciplinary Research and Committee on Science, Engineering, and Public Policy 2005, p.2).
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			 http://aideus.com/community/community.html 2 https://deepmind.com/blog/learning-through-human-feedback Artificial Interdisciplinarity: Artificial Intelligence for...

Abstract: This paper considers the question: In what ways can artificial intelligence assist with interdisciplinary research for addressing complex societal problems and advancing the social good? Problems such as environmental protection, public health, and emerging technology governance do not fit neatly within traditional academic disciplines and therefore require an interdisciplinary approach. However, interdisciplinary research poses large cognitive challenges for human researchers that go beyond the substantial challenges of narrow disciplinary research. The challenges include epistemic divides between disciplines, the massive bodies of relevant literature, the peer review of work that integrates an eclectic mix of topics, and the transfer of interdisciplinary research insights from one problem to another. Artificial interdisciplinarity already helps with these challenges via search engines, recommendation engines, and automated content analysis. Future "strong artificial interdisciplinarity" based on human-level artificial general intelligence could excel at interdisciplinary research, but it may take a long time to develop and could pose major safety and ethical issues. Therefore, there is an important role for intermediate-term artificial interdisciplinarity systems that could make major contributions to addressing societal problems without the concerns associated with artificial general intelligence.