2019 recent trends in GPU price per FLOPS

Published 25 March, 2020
We estimate that in recent years, GPU prices have fallen at rates that would yield an order of magnitude over roughly:

17 years for single-precision FLOPS
10 years for half-precision FLOPS
5 years for half-precision fused multiply-add FLOPS

Details
GPUs (graphics processing units) are specialized electronic circuits originally used for computer graphics.1 In recent years, they have been popularly used for machine learning applications.2 One measure of GPU performance is FLOPS, the number of operations on floating-point numbers a GPU can perform in a second.3 This page looks at the trends in GPU price / FLOPS of theoretical peak performance over the past 13 years. It does not include the cost of operating the GPUs, and it does not consider GPUs rented through cloud computing.
Theoretical peak performance
‘Theoretical peak performance’ numbers appear to be determined by adding together the theoretical performances of the processing components of the GPU, which are calculated by multiplying the clock speed of the component by the number of instructions it can perform per cycle.4 These numbers are given by the developer and may not reflect actual performance on a given application.5
Metrics
We collected data on multiple slightly different measures of GPU price and FLOPS performance.
Price metrics
GPU prices are divided into release prices, which reflect the manufacturer suggested retail prices that GPUs are originally sold at, and active prices, which are the prices at which GPUs are actually sold at over time, often by resellers.
We expect that active prices better represent prices available to hardware users, but collect release prices also, as supporting evidence.
FLOPS performance metrics
Several varieties of ‘FLOPS’ can be distinguished based on the specifics of the operations they involve. Here we are interested in single-precision FLOPS, half-precision FLOPS, and half-precision fused-multiply add FLOPS.
‘Single-precision’ and ‘half-precision’ refer to the number of bits used to specify a floating point number.6 Using more bits to specify a number achieves greater precision at the cost of more computational steps per calculation. Our data suggests that GPUs have largely been improving in single-precision performance in recent decades,7 and half-precision performance appears to be increasingly popular because it is adequate for deep learning.8
Nvidia, the main provider of chips for machine learning applications,9 recently released a series of GPUs featuring Tensor Cores,10 which claim to deliver “groundbreaking AI performance”. Tensor Core performance is measured in FLOPS, but they perform exclusively certain kinds of floating-point operations known as fused multiply-adds (FMAs).11 Performance on these operations is important for certain kinds of deep learning performance,12 so we track ‘GPU price / FMA FLOPS’ as well as ‘GPU price / FLOPS’.
In addition to purely half-precision computations, Tensor Cores are capable of performing mixed-precision computations, where part of the computation is done in half-precision and part in single-precision.13 Since explicitly mixed-precision-optimized hardware is quite recent, we don’t look at the trend in mixed-precision price performance, and only look at the trend in half-precision price performance.
Precision tradeoffs
Any GPU that performs multiple kinds of computations (single-precision, half-precision, half-precision fused multiply add) trades off performance on one for performance on the other, because there is limited space on the chip, and transistors must be allocated to either one type of computation or the other.14 All current GPUs that perform half-precision or TensorCore fused-multiply-add computations also do single-precision computations, so they are splitting their transistor budget. For this reason, our impression is that half-precision FLOPS could be much cheaper now if entire GPUs were allocated to each one alone, rather than split between them.
Release date prices
We collected data on theoretical peak performance (FLOPS), release date, and price from several sources, including Wikipedia.15 (Data is available in this spreadsheet). We found GPUs by looking at Wikipedia’s existing large lists16 and by Googling “popular GPUs” and “popular deep learning GPUs”. We included any hardware that was labeled as a ‘GPU’. We adjusted prices for inflation based on the consumer price index.17
We were unable to find price and performance data for many popular GPUs and suspect that we are missing many from our list. In our search, we did not find any GPUs that beat our 2017 minimum of $0.03 (release price) / single-precision GFLOPS. We put out a $20 bounty on a popular Facebook group to find a cheaper GPU / FLOPS, and the bounty went unclaimed, so we are reasonably confident in this minimum.18
GPU price / single-precision FLOPS
Figure 1 shows our collected dataset for GPU price / single-precision FLOPS over time.19
Figure 1: Real GPU price / single-precision FLOPS over time. The vertical axis is log-scale. Price is measured in 2019 dollars.
To find a clear trend for the prices of the cheapest GPUs / FLOPS, we looked at the running minimum prices every 10 days.20
Figure 2: Ten-day minimums in real GPU price / single-precision FLOPS over time. The vertical axis is log-scale. Price is measured in 2019 dollars. The blue line shows the trendline ignoring data before late 2007. (We believe the apparent steep decline prior to late 2007 is an artefact of a lack of data for that time period.)
The cheapest GPU price / FLOPS hardware using release date pricing has not decreased since 2017. However there was a similar period of stagnation between early 2009 and 2011, so this may not represent a slowing of the trend in the long run.
Based on the figures above, the running minimums seem to follow a roughly exponential trend. If we do not include the initial point in 2007, (which we suspect is not in fact the cheapest hardware at the time), we get that the cheapest GPU price / single-precision FLOPS fell by around 17% per year, for a factor of ten in  ~12.5 years.21
GPU price / half-precision FLOPS
Figure 3 shows GPU price / half-precision FLOPS for all the GPUs in our search above for which we could find half-precision theoretical performance.22
Figure 3: Real GPU price / half-precision FLOPS over time. The vertical axis is log-scale. Price is measured in 2019 dollars.
Again, we looked at the running minimums of this graph every 10 days, shown in Figure 4 below.23
Figure 4: Minimums in real GPU price / half-precision FLOPS over time. The vertical axis is log-scale. Price is measured in 2019 dollars.
If we assume an exponential trend with noise,24 cheapest GPU price / half-precision FLOPS fell by around 26% per year, which would yield a factor of ten after ~8 years.25
GPU price / half-precision FMA FLOPS
Figure 5 shows GPU price / half-precision FMA FLOPS for all the GPUs in our search above for which we could find half-precision FMA theoretical performance.26 (Note that this includes all of our half-precision data above, since those FLOPS could be used for fused-multiply adds in particular). GPUs with TensorCores are marked in red.
Figure 5: Real GPU price / half-precision FMA FLOPS over time. Price is measured in 2019 dollars.
Figure 6 shows the running minimums of GPU price / HP FMA FLOPS.27
Figure 6: Minimums in real GPU price / half-precision FMA FLOPS over time. Price is measured in 2019 dollars.
GPU price / Half-Precision FMA FLOPS appears to be following an exponential trend over the last four years, falling by around 46% per year, for a factor of ten in ~4 years.28
Active Prices
GPU prices often go down from the time of release, and some popular GPUs are older ones that have gone down in price.29 Given this, it makes sense to look at active price data for the same GPU over time.
Data Sources
We collected data on peak theoretical performance in FLOPS from TechPowerUp30 and combined it with active GPU price data to get GPU price / FLOPS over time.31 Our primary source of historical pricing data was Passmark, though we also found a less trustworthy dataset on Kaggle which we used to check our analysis. We adjusted prices for inflation based on the consumer price index.32
Passmark
We scraped pricing data33 on GPUs between 2011 and early 2020 from Passmark.34 Where necessary, we renamed GPUs from Passmark to be consistent with TechPowerUp.35 The Passmark data consists of 38,138 price points for 352 GPUs. We guess that these represent most popular GPUs.
Looking at the ‘current prices’ listed on individual Passmark GPU pages, prices appear to be sourced from Amazon, Newegg, and Ebay. Passmark’s listed pricing data does not correspond to regular intervals. We don’t know if prices were pulled at irregular intervals, or if Passmark pulls prices regularly and then only lists major changes as price points. When we see a price point, we treat it as though the GPU is that price only at that time point, not indefinitely into the future.
The data contains several blips where a GPU is briefly sold very unusually cheaply. A random checking of some of these suggests to us that these correspond to single or small numbers of GPUs for sale, which we are not interested in tracking, because we are trying to predict AI progress, which presumably isn’t influenced by temporary discounts on tiny batches of GPUs.
Kaggle
This Kaggle dataset contains scraped data of GPU prices from price comparison sites PriceSpy.co.uk, PCPartPicker.com, Geizhals.eu from the years 2013 – 2018. The Kaggle dataset has 319,147 price points for 284 GPUs. Unfortunately, at least some of the data is clearly wrong, potentially because price comparison sites include pricing data from untrustworthy merchants.36 As such, we don’t use the Kaggle data directly in our analysis, but do use it as a check on our Passmark data. The data that we get from Passmark roughly appears to be a subset of the Kaggle data from 2013 – 2018,37 which is what we would expect if the price comparison engines picked up prices from the merchants Passmark looks at.
Limitations
There are a number of reasons why we think this analysis may in fact not reflect GPU price trends:

We effectively have just one source of pricing data, Passmark.
Passmark appears to only look at Amazon, Newegg, and Ebay for pricing data.
We are not sure, but we suspect that Passmark only looks at the U.S. versions of Amazon, Newegg, and Ebay, and pricing may be significantly different in other parts of the world (though we guess it wouldn’t be different enough to change the general trend much).
As mentioned above, we are not sure if Passmark pulls price data regularly and only lists major price changes, or pulls price data irregularly. If the former is true, our data may be overrepresenting periods where the price changes dramatically.
None of the price data we found includes quantities of GPUs which were available at that price, which means some prices may be for only a very limited number of GPUs.
We don’t know how much the prices from these datasets reflect the prices that a company pays when buying GPUs in bulk, which we may be more interested in tracking.

A better version of this analysis might start with more complete data from price comparison engines (along the lines of the Kaggle dataset) and then filter out clearly erroneous pricing information in some principled way.
Data
The original scraped datasets with cards renamed to match TechPowerUp can be found here. GPU price / FLOPS data is graphed on a log scale in the figures below. Price points for the same GPU are marked in the same color. We adjusted prices for inflation using the consumer price index.38 All points below are in 2019 dollars.
To try to filter out noisy prices that didn’t last or were only available in small numbers, we took out the lowest 5% of data in every several day period39 to get the 95th percentile cheapest hardware. We then found linear and exponential trendlines of best fit through the available hardware with the lowest GPU price / FLOPS every several days.40
GPU price / single-precision FLOPS
Figures 7-10 show the raw data, 95th percentile data, and trendlines for single-precision GPU price / FLOPS for the Passmark dataset. This folder contains plots of all our datasets, including the Kaggle dataset and combined Passmark + Kaggle dataset.41
Figure 7: GPU price / single-precision FLOPS over time, taken from our Passmark dataset.42 Price is measured in 2019 dollars. This picture shows that the Kaggle data does appear to be a superset of the Passmark data from 2013 – 2018, giving us some evidence that the Passmark data is correct. The vertical axis is log-scale.

Figure 8: The top 95% of data every 10 days for GPU price / single-precision FLOPS over time, taken from the Passmark dataset we plotted above. (Figure 7 with the cheapest 5% removed.) The vertical axis is log-scale.43

Figure 9: The same data as Figure 8, with the vertical axis zoomed-in.
Figure 10: The minimum data points from the top 95% of the Passmark dataset, taken every 10 days. We fit linear and exponential trendlines through the data. The vertical axis is log-scale.44
Analysis
The cheapest 95th percentile data every 10 days appears to fit relatively well to both a linear and exponential trendline. However we assume that progress will follow an exponential, because previous progress has followed an exponential.
In the Passmark dataset, the exponential trendline suggested that from 2011 to 2020, 95th-percentile GPU price / single-precision FLOPS fell by around 13% per year, for a factor of ten in ~17 years,45  bootstrap46 95% confidence interval 16.3 to 18.1 years.47 We believe the rise in price / FLOPS in 2017 corresponds to a rise in GPU prices due to increased demand from cryptocurrency miners.48 If we instead look at the trend from 2011 through 2016, before the cryptocurrency rise, we instead get that 95th-percentile GPU price / single-precision FLOPS price fell by around 13% per year, for a factor of ten in ~16 years.49
This is slower than the order of magnitude every ~12.5 years we found when looking at release prices. If we restrict the release price data to 2011 – 2019, we get an order of magnitude decrease every ~13.5 years instead,50 so part of the discrepancy can be explained because of the different start times of the datasets. To get some assurance that our active price data wasn’t erroneous, we spot checked the best active price at the start of 2011, which was somewhat lower than the best release price at the same time, and confirmed that its given price was consistent with surrounding pricing data.51 We think active prices are likely to be closer to the prices at which people actually bought GPUs, so we guess that ~17 years / order of magnitude decrease is a more accurate estimate of the trend we care about.
GPU price / half-precision FLOPS
Figures 11-14 show the raw data, 95th percentile data, and trendlines for half-precision GPU price / FLOPS for the Passmark dataset. This folder contains plots of the Kaggle dataset and combined Passmark + Kaggle dataset.
 Figure 11: GPU price / half-precision FLOPS over time, taken from our Passmark dataset. Price is measured in 2019 dollars.52  This picture shows that the Kaggle data does appear to be a superset of the Passmark data from 2013 – 2018, giving us some evidence that the Passmark data is reasonable. The vertical axis is log-scale.

Figure 12: The top 95% of data every 30 days for GPU price / half-precision FLOPS over time, taken from the Passmark dataset we plotted above. (Figure 11 with the cheapest 5% removed.) The vertical axis is log-scale.53

Figure 13: The same data as Figure 12, with the vertical axis zoomed-in.
Figure 14: The minimum data points from the top 95% of the Passmark dataset, taken every 30 days. We fit linear and exponential trendlines through the data. The vertical axis is log-scale.54
Analysis
If we assume the trend is exponential, the Passmark trend seems to suggest that from 2015 to 2020, 95th-percentile GPU price / half-precision FLOPS of GPUs has fallen by around 21% per year, for a factor of ten over ~10 years,55 bootstrap56 95% confidence interval 8.8 to 11 years.57 This is fairly close to the ~8 years / order of magnitude decrease we found when looking at release price data, but we treat active prices as a more accurate estimate of the actual prices at which people bought GPUs. As in our previous dataset, there is a noticeable rise in 2017, which we think is due to GPU prices increasing as a result of cryptocurrency miners. If we look at the trend from 2015 through 2016, before this rise, we get that 95th-percentile GPU price / half-precision FLOPS has fallen by around 14% per year, which would yield a factor of ten over ~8 years.58
GPU price / half-precision FMA FLOPS
Figures 15-18 show the raw data, 95th percentile data, and trendlines for half-precision GPU price / FMA FLOPS for the Passmark dataset. GPUs with Tensor Cores are marked in black. This folder contains plots of the Kaggle dataset and combined Passmark + Kaggle dataset.
Figure 15: GPU price / half-precision FMA FLOPS over time, taken from our Passmark dataset.59 price is measured in 2019 dollars. This picture shows that the Kaggle data does appear to be a superset of the Passmark data from 2013 – 2018, giving us some evidence that the Passmark data is correct. The vertical axis is log-scale.

Figure 16: The top 95% of data every 30 days for GPU price / half-precision FMA FLOPS over time, taken from the Passmark dataset we plotted above.60 (Figure 15 with the cheapest 5% removed.)

Figure 17: The same data as Figure 16, with the vertical axis zoomed-in.
Figure 18: The minimum data points from the top 95% of the Passmark dataset, taken every 30 days. We fit linear and exponential trendlines through the data.61
Analysis
If we assume the trend is exponential, the Passmark trend seems to suggest the 95th-percentile GPU price / half-precision FMA FLOPS of GPUs has fallen by around 40% per year, which would yield a factor of ten in ~4.5 years,62 with a bootstrap63 95% confidence interval 4 to 5.2 years.64 This is fairly close to the ~4 years / order of magnitude decrease we found when looking at release price data, but we think active prices are a more accurate estimate of the actual prices at which people bought GPUs.
The figures above suggest that certain GPUs with Tensor Cores were a significant (~half an order of magnitude) improvement over existing GPU price / half-precision FMA FLOPS.
Conclusion
We summarize our results in the table below.
Release Prices95th-percentile Active Prices95th-percentile Active Prices (pre-crypto price rise)11/2007 – 1/20203/2011 – 1/20203/2011 – 12/2016 $ / single-precision FLOPS12.517169/2014 – 1/20201/2015 – 1/20201/2015 – 12/2016 $ / half-precision FLOPS8108$ / half-precision FMA FLOPS44.5—
Release price data seems to generally support the trends we found in active prices, with the notable exception of trends in GPU price / single-precision FLOPS, which cannot be explained solely by the different start dates.65 We think the best estimate of the overall trend for prices at which people recently bought GPUs is the 95th-percentile active price data from 2011 – 2020, since release price data does not account for existing GPUs becoming cheaper over time. The pre-crypto trends are similar to the overall trends, suggesting that the trends we are seeing are not anomalous due to cryptocurrency.
Given that, we guess that GPU prices as a whole have fallen at rates that would yield an order of magnitude over roughly:

17 years for single-precision FLOPS
10 years for half-precision FLOPS
5 years for half-precision fused multiply-add FLOPS

Half-precision FLOPS seem to have become cheaper substantially faster than single-precision in recent years. This may be a “catching up” effect as more of the space on GPUs was allocated to half-precision computing, rather than reflecting more fundamental technological progress.
Primary author: Asya Bergal
Notes


Cortés, Pizarro, and Afonso as precedents for takeover

Daniel Kokotajlo, 29 February 2020
Epistemic status: I am not a historian, nor have I investigated these case studies in detail. I admit I am still uncertain about how the conquistadors were able to colonize so much of the world so quickly. I think my ignorance is excusable because this is just a blog post; I welcome corrections from people who know more. If it generates sufficient interest I might do a deeper investigation. Even if I’m right, this is just one set of historical case-studies; it doesn’t prove anything about AI, even if it is suggestive. Finally, in describing these conquistadors as “successful,” I simply mean that they achieved their goals, not that what they achieved was good.
Summary
In the span of a few years, some minor European explorers (later known as the conquistadors) encountered, conquered, and enslaved several huge regions of the world. That they were able to do this is surprising; their technological advantage was not huge. (This was before the scientific and industrial revolutions.) From these cases, I think we learn that it is occasionally possible for a small force to quickly conquer large parts of the world, despite:

Having only a minuscule fraction of the world’s resources and power
Having technology + diplomatic and strategic cunning that is better but not that much better
Having very little data about the world when the conquest begins
Being disunited

Which all suggests that it isn’t as implausible that a small AI takes over the world in mildly favorable circumstances as is sometimes thought. EDIT: In light of good pushback from people (e.g. Lucy.ea8 and Matthew Barnett) about the importance of disease, I think one should probably add a caveat to the above: “In times of chaos & disruption, at least.”  NEW EDIT: After reading three giant history books on the subject, I take back my previous edit. My original claims were correct.
Three shocking true stories
I highly recommend you read the wiki pages yourself; otherwise, here are my summaries:
Cortés: [wiki] [wiki]

April 1519: Hernán Cortés lands in Yucatan with ~500 men, 13 horses, and a few cannons. He destroys his ships so his men won’t be able to retreat. His goal is to conquer the Aztec empire of several million people.
He makes his way towards the imperial capital, Tenochtitlán. Along the way he encounters various local groups, fighting some and allying with some. He is constantly outnumbered but his technology gives him an advantage in fights. His force grows in size, because even though he loses Spaniards he gains local allies who resent Aztec rule.
Tenochtitlán is an island fortress (like Venice) with a population of over 200,000, making it one of the largest and richest cities in the world at the time. Cortés arrives in the city asking for an audience with the Emperor, who receives him warily.
Cortés takes the emperor hostage within his own palace, indirectly ruling Tenochtitlán through him.
Cortés learns that the Spanish governor has landed in Mexico with a force twice his size, intent on arresting him. (Cortés’ expedition was illegal!) Cortés leaves 200 men guarding the Emperor, marches to the coast with the rest, surprises and defeats the new Spaniards in battle, and incorporates the survivors into his army.
July 1520: Back at the capital, the locals are starting to rebel against his men. Cortés marches back to the capital, uniting his forces just in time to be besieged in the imperial palace. They murder the emperor and fight their way out of the city overnight, taking heavy losses.
They shelter in another city (Tlaxcala) that was thinking about rebelling against the Aztecs. Cortés allies with the Tlaxcalans and launches a general uprising against the Aztecs. Not everyone sides with him; many city-states remain loyal to Tenochtitlan. Some try to stay neutral. Some join him at first, and then abandon him later. Smallpox sweeps through the land, killing many on all sides and causing general chaos.
May 1521: The final assault on Tenochtitlán. By this point, Cortés has about 1,000 Spanish troops and 80,000 – 200,000 allied native warriors. He had 16 cannons and 13 boats. The Aztecs have 80,000 – 300,000 warriors and 400 boats. Cortés and his allies win.
Later, the Spanish would betray their native allies and assert hegemony over the entire region, in violation of the treaties they had signed.

Pizarro [wiki] [wiki]

1532: Francisco Pizarro arrives in Inca territory with 168 Spanish soldiers. His goal is to conquer the Inca empire, which was much bigger than the Aztec empire.
The Inca empire is in the middle of a civil war and a devastating plague.
Pizarro makes it to the Emperor right after the Emperor defeats his brother. Pizarro is allowed to approach because he promises that he comes in peace and will be able to provide useful information and gifts.
At the meeting, Pizarro ambushes the Emperor, killing his retinue with a volley of gunfire and taking him hostage. The remainder of the Emperor’s forces in the area back away, probably confused and scared by the novel weapons and hesitant to keep fighting for fear of risking the Emperor’s life.
Over the next months, Pizarro is able to leverage his control over the Emperor to stay alive and order the Incans around; eventually he murders the Emperor and makes an alliance with local forces (some of the Inca generals) to take over the capital city of Cuzco.
The Spanish continue to rule via puppets, primarily Manco Inca, who is their puppet ruler while they crush various rebellions and consolidate their control over the empire. Manco Inca escapes and launches a rebellion of his own, which is partly successful: He utterly wipes out four columns of Spanish reinforcements, but is unable to retake the capital. With the morale and loyalty of his followers dwindling, Manco Inca eventually gives up and retreats, leaving the Spanish still in control.
Then the Spanish ended up fighting each other for a while, while also putting down more local rebellions. After a few decades Spanish dominance of the region is complete. (1572).

Afonso [wiki] [wiki] [wiki]

1506: Afonso helps the Portuguese king come up with a shockingly ambitious plan. Eight years prior, the first Europeans had rounded the coast of Africa and made it to the Indian Ocean. The Indian Ocean contained most of the world’s trade at the time, since it linked up the world’s biggest and wealthiest regions. See this map of world population (timestamp 3:45). Remember, this is prior to the Industrial and Scientific Revolutions; Europe is just coming out of the Middle Ages and does not have an obvious technological advantage over India or China or the Middle East, and has an obvious economic disadvantage. And Portugal is a just tiny state on the edge of the Iberian peninsula.
The plan is: Not only will we go into the Indian Ocean and participate in the trading there — cutting out all the middlemen who are currently involved in the trade between that region and Europe — we will conquer strategic ports around the region so that no one else can trade there!
Long story short, Afonso goes on to complete this plan by 1513. (!!!)

Some comparisons and contrasts:

Afonso had more European soldiers at his disposal than Cortes or Pizarro, but not many more — usually he had about a thousand or so. He did have more reinforcements and support from home.
Like them, he was usually significantly outnumbered in battles. Like them, the empires he warred against were vastly wealthier and more populous than his forces.
Like them, Afonso was often able to exploit local conflicts to gain local allies, which were crucial to his success.
Unlike them, his goal wasn’t to conquer the empires entirely, just to get and hold strategic ports.
Unlike them, he was fighting empires that were technologically advanced; for example, in several battles his enemies had more cannons and gunpowder than he did.
That said, it does seem that Portuguese technology was qualitatively better in some respects (ships, armor, and cannons, I’d say.) Not dramatically better, though.
While Afonso’s was a naval campaign, he did fight many land battles, usually marine assaults on port cities, or defenses of said cities against counterattacks. So superior European naval technology is not by itself enough to explain his victory, though it certainly was important.
Plague and civil war were not involved in Afonso’s success.

What explains these devastating conquests?
Wrong answer: I cherry-picked my case studies.
History is full of incredibly successful conquerors: Alexander the Great, Ghenghis Khan, etc. Perhaps some people are just really good at it, or really lucky, or both.
However: Three incredibly successful conquerors from the same tiny region and time period, conquering three separate empires? Followed up by dozens of less successful but still very successful conquerors from the same region and time period? Surely this is not a coincidence. Moreover, it’s not like the conquistadors had many failed attempts and a few successes. The Aztec and Inca empires were the two biggest empires in the Americas, and there weren’t any other Indian Oceans for the Portuguese to fail at conquering.
Fun fact: I had not heard of Afonso before I started writing this post this morning. Following the Rule of Three, I needed a third example and I predicted on the basis of Cortes and Pizarro that there would be other, similar stories happening in the world at around that time. That’s how I found Afonso.
Right answer: Technology
However, I don’t think this is the whole explanation. The technological advantage of the conquistadors was not overwhelming.
Whatever technological advantage the conquistadors had over the existing empires, it was the sort of technological advantage that one could acquire before the Scientific and Industrial revolutions. Technology didn’t change very fast back then, yet Portugal managed to get a lead over the Ottomans, Egyptians, Mughals, etc. that was sufficient to bring them victory. On paper, the Aztecs and Spanish were pretty similar: Both were medieval, feudal civilizations. I don’t know for sure, but I’d bet there were at least a few techniques and technologies the Aztecs had that the Spanish didn’t. And of course the technological similarities between the Portuguese and their enemies were much stronger; the Ottomans even had access to European mercenaries! Even in cases in which the conquistadors had technology that was completely novel — like steel armor, horses, and gunpowder were to the Aztecs and Incas — it wasn’t god-like. The armored soldiers were still killable; the gunpowder was more effective than arrows but limited in supply, etc.
(Contrary to popular legend, neither Cortés nor Pizarro were regarded as gods by the people they conquered. The Incas concluded pretty early on that the Spanish were mere men, and while the idea did float around the Aztecs for a bit the modern historical consensus is that most of them didn’t take it seriously.)
Ask yourself: Suppose Cortés had found 500 local warriors, gave them all his equipment, trained them to use it expertly, and left. Would those local men have taken over all of Mexico? I doubt it. And this is despite the fact that they would have had much better local knowledge than Cortés did! Same goes for Pizarro and Afonso. Perhaps if he had found 500 local warriors led by an exceptional commander it would work. But the explanation for the conquistador’s success can’t just be that they were all exceptional commanders; that would be positing too much innate talent to occur in one small region of the globe at one time.
Right answer: Strategic and diplomatic cunning
This is my non-expert guess about the missing factor that joins with technology to explain this pattern of conquistador success.
They didn’t just have technology; they had effective strategy and they had effective diplomacy. They made long-term plans that worked despite being breathtakingly ambitious. (And their short-term plans were usually pretty effective too, read the stories in detail to see this.) Despite not knowing the local culture or history, these conquistadors made surprisingly savvy diplomatic decisions. They knew when they could get away with breaking their word and when they couldn’t; they knew which outrages the locals would tolerate and which they wouldn’t; they knew how to convince locals to ally with them; they knew how to use words to escape militarily impossible situations… The locals, by contrast, often badly misjudged the conquistadors, e.g. not thinking Pizarro had the will (or the ability?) to kidnap the emperor, and thinking the emperor would be safe as long as they played along.
This raises the question, how did they get that advantage? My answer: they had experience with this sort of thing, whereas locals didn’t. Presumably Pizarro learned from Cortés’ experience; his strategy was pretty similar. (See also: the prior conquest of the Canary Islands by the Spanish). In Afonso’s case, well, the Portuguese had been sailing around Africa, conquering ports and building forts for more than a hundred years.
Lessons I think we learn
I think we learn that:
It is occasionally possible for a small force to quickly conquer large parts of the world, despite:

Having only a minuscule fraction of the world’s resources and power
Having technology + diplomatic and strategic cunning that is better but not that much better
Having very little data about the world when the conquest begins
Being disunited

Which all suggests that it isn’t as implausible that a small AI takes over the world in mildly favorable circumstances as is sometimes thought.
 EDIT: In light of good pushback from people (e.g. Lucy.ea8 and  Matthew Barnett) about the importance of disease, I think one should probably add a caveat to the above: “In times of chaos & disruption, at least.”
Having only a minuscule fraction of the world’s resources and power
In all three examples, the conquest was more or less completed without support from home; while Spain/Portugal did send reinforcements, it wasn’t even close to the entire nation of Spain/Portugal fighting the war. So these conquests are examples of non-state entities conquering states, so to speak. (That said, their claim to represent a large state may have been crucial for Cortes and Pizarro getting audiences and respect initially.) Cortés landed with about a thousandth the troops of Tenochtitlan, which controlled a still larger empire of vassal states. Of course, his troops were better equipped, but on the other hand they were also cut off from resupply, whereas the Aztecs were in their home territory, able to draw on a large civilian population for new recruits and resupply.
The conquests succeeded in large part due to diplomacy. This has implications for AI takeover scenarios; rather than imagining a conflict of humans vs. robots, we could imagine humans vs. humans-with-AI-advisers, with the latter faction winning and somehow by the end of the conflict the AI advisers have managed to become de facto rulers, using the humans who obey them to put down rebellions by the humans who don’t.
Having technology + diplomatic and strategic skill that is better but not that much better
As previously mentioned, the conquistadors didn’t enjoy god-like technological superiority. In the case of Afonso the technology was pretty similar. Technology played an important role in their success, but it wasn’t enough on its own. Meanwhile, the conquistadors may have had more diplomatic and strategic cunning (or experience) than the enemies they conquered. But not that much more–they are only human, after all. And their enemies were pretty smart.
In the AI context, we don’t need to imagine god-like technology (e.g. swarms of self-replicating nanobots) to get an AI takeover. It might even be possible without any new physical technologies at all! Just superior software, e.g. piloting software for military drones, targeting software for anti-missile defenses, cyberwarfare capabilities, data analysis for military intelligence, and of course excellent propaganda and persuasion.
Nor do we need to imagine an AI so savvy and persuasive that it can persuade anyone of anything. We just need to imagine it about as cunning and experienced relative to its enemies as Cortés, Pizarro, and Afonso were relative to theirs. (Presumably no AI would be experienced with world takeover, but perhaps an intelligence advantage would give it the same benefits as an experience advantage.) And if I’m wrong about this explanation for the conquistador’s success–if they had no such advantage in cunning/experience–then the conclusion is even stronger.
Additionally, in a rapidly-changing world that is undergoing slow takeoff, where there are lesser AIs and AI-created technologies all over the place, most of which are successfully controlled by humans, AI takeover might still happen if one AI is better, but not that much better, than the others.
Having very little data about the world when the conquest begins
Cortés invaded Mexico knowing very little about it. After all, the Spanish had only realized the Americas existed two decades prior. He heard rumors of a big wealthy empire and he set out to conquer it, knowing little of the technology and tactics he would face. Two years later, he ruled the place.
Pizarro and Afonzo were in better epistemic positions, but still, they had to learn a lot of important details (like what the local power centers, norms, and conflicts were, and exactly what technology the locals had) on the fly. But they were good at learning these things and making it up as they went along, apparently.
We can expect superhuman AI to be good at learning. Even if it starts off knowing very little about the world — say, it figured out it was in a training environment and hacked its way out, having inferred a few general facts about its creators but not much else — if it is good at learning and reasoning, it might still be pretty dangerous.
Being disunited
Cortés invaded Mexico in defiance of his superiors and had to defeat the army they sent to arrest him. Pizarro ended up fighting a civil war against his fellow conquistadors in the middle of his conquest of Peru. Afonzo fought Greek mercenaries and some traitor Portuguese, conquered Malacca against the  orders of a rival conquistador in the area, and was ultimately demoted due to political maneuvers by rivals back home.
This astonishes me. Somehow these conquests were completed by people who were at the same time busy infighting and backstabbing each other!
Why was it that the conquistadors were able to split the locals into factions, ally with some to defeat the others, and end up on top? Why didn’t it happen the other way around: some ambitious local ruler talks to the conquistadors, exploits their internal divisions, allies with some to defeat the others, and ends up on top?
I think the answer is partly the “diplomatic and strategic cunning” mentioned earlier, but mostly other things. (The conquistadors were disunited, but presumably were united in the ways that mattered.) At any rate, I expect AIs to be pretty good at coordinating too; they should be able to conquer the world just fine even while competing fiercely with each other. For more on this idea, see this comment.
By Daniel Kokotajlo
Acknowledgements
Thanks to Katja Grace for feedback on a draft. All mistakes are my own, and should be pointed out to me via email at daniel@aiimpacts.org. Edit: Also, when I wrote this post I had forgotten that the basic idea for it probably came from this comment by JoshuaFox.
(Front page image from the Conquest of México series. Representing the 1521 Fall of Tenochtitlan, in the Spanish conquest of the Aztec Empire)


Incomplete case studies of discontinuous progress

Published 7 Feb 2020
This is a list of potential cases of discontinuous technological progress that we have investigated partially or not at all.
List
In the course of investigating cases of potentially discontinuous technological progress, we have collected around fifty suggested instances that we have not investigated fully. This is a list of them and what we know about them.
The Haber Process
This was previously listed as NSD, but that is tentatively revoked while we investigate a complication with the data.
Previous explanation
The Haber process was the first energy efficient method of producing ammonia, which is key to making fertilizer. The reason to expect that the Haber process might represent discontinuous technological progress is that previous processes were barely affordable, while the Haber process was hugely valuable—it is credited with fixing much of the nitrogen now in human bodies—and has been used on an industrial scale since 1913.
A likely place to look for discontinuities then is in the energy cost of fixing nitrogen. Table 4 in Grünewald’s Chemistry for the Future suggests that the invention of the Haber reduced the energy expense by around 60% per nitrogen bonded over a method developed eight years earlier. The previous step however appears to have represented at least a 50% improvement over the process of two years earlier (though the figure is hard to read). Later improvements to the Haber process appear to have been comparable. Thus it seems the Haber process was not an unusually large improvement in energy efficiency, but was probably instead the improvement that happened to take the process into the range of affordability.
Since it appears that energy was an important expense, and the Haber process was especially notable for being energy efficient, and yet did not represent a particular discontinuity in energy efficiency progress, it seems unlikely that the Haber process involved a discontinuity. Furthermore, it appears that the world moved to using the Haber process over other sources of fertilizer gradually, suggesting there was not a massive price differential, nor any sharp practical change as a result of the adoption of the process. In the 20’s the US imported much nitrogen from Chile. Alternative nitrogen source calcium cyanamide reached peak production in 1945, thirty years since the Haber process reached industrial scale production.
The amount of synthetic nitrogen fertilizer applied hasn’t abruptly changed since 1860 (see p24). Neither has the amount of food produced, for a few foods at least.
In sum, it seems the Haber process has had a large effect, but it was produced by a moderate change in efficiency, and manifest over a long period.

Aluminium
It is sometimes claimed that the discovery of the Hall–Héroult process in the 1880s brought the price of aluminium down precipitously. We found several pieces of quantitative data about this, but they seriously conflict. The most rigorous looking is a report from Patricia Plunkert at the US Geological Survey, from which we get the following data. However note that some of these figures may be off by orders of magnitude, according to other sources.
Plunkert provides a table of historic aluminium prices, according to which the nominal price fell from $8 per pound to $0.58 per pound sometime between 1887 and 1895 (during most of which time no records are available). This period probably captures the innovation of interest, as the Hall–Héroult process was patented in 1886 according to Plunkert, and the price only dropped by $1 per pound during the preceding fifteen years according to her table. Plunkert also says that the price was held artificially low to encourage consumers in the early 1900s, suggesting the same may have been true earlier, however this seems likely to be a small correction.
The sewing machine
Early sewing machines apparently brought the time to produce clothing down by an order of magnitude (from 14 hours to 75 minutes for a man’s dress shirt by one estimate). However it appears that the technology progressed more slowly, then was taken up by the public later – probably when it became cost-effective, at which time adoptees may have experienced a rapid reduction in sewing time (presumably at some expense). These impressions are from a very casual perusal of the evidence.
Video compression
Blogger John McGowan claims that video compression performance was constant at a ratio of around 250 for about seven years prior to 2003, then jumped to around 900.
Figure 1: video compression performance in the past two decades.
Information storage volume
According to the Performance Curves Database (PCDB), ‘information storage volume’ for both handwriting and printing has grown by a factor of three in recent years, after less than doubling in the hundred years previously. It is unclear what exactly is being measured here however.
Undersea cable price
The bandwidth per cable length available for a dollar apparently grew by more than 1000 times in around 1880.

Infrared detector sensitivity
We understand that infrared detector sensitivity is measured in terms of ‘Noise Equivalent Power’ (NEP), or the amount of power (energy per time) that needs to hit the sensor for the sensor’s output to have a signal:noise ratio of one. We investigated progress in infrared detection technology because according to Academic Press (1974), the helium-cooled germanium bolometer represented a four order of magnitude improvement in sensitivity over uncooled detectors.1 However our own investigation suggests there were other innovations between uncooled detectors and the bolometer in question, and thus no abrupt improvement.
We list advances we know of here, and summarize them in Figure 5. The 1947 point is uncooled. The 1969 point is nearly four orders of magnitude better. However we know of at least four other detectors with intermediate levels of sensitivity, and these are spread fairly evenly between the uncooled device and the most efficient cooled one listed.
We have not checked whether the progress between the uncooled detector and the first cooled detector was discontinuous, given previous rates. This is because we have no strong reason to suspect it is.
Figure 2: Sensitivity of infrared detectors during the transition to liquid-helium-cooled devices.
Genome sequencing – IIP
This appears to have seen at least a moderate discontinuity. An investigation is in progress.
It was suggested to us in particular that Next Generation Sequencing produced discontinuous progress in output per instrument run for DNA sequencing.

Aircraft shot down per shell fired

We’ve seen it claimed that the proximity fuse increased this metric by 2x or more. We don’t know what the trend was beforehand, however.

Time to produce clothing

The Sewing machine was proposed as discontinuous in this metric. We have not investigated.

Sensitivity of infrared detectors

Cryogenically cooled semiconductor sensors were proposed as discontinuous in this metric. We have not investigated.

Frames per Second

It was suggested that something in high sensitivity, high precision metrology, e.g. trillion frame-per-second camera from MIT, would be discontinuous in this metric. We have not investigated.

Access to Information

Smart phones were suggested as a discontinuity in this metric. We have not investigated.

Spread of minimally invasive surgery

Laparoscopic cholecystectomy was suggested as a discontinuity in this metric. We have not investigated.

Max. submerged endurance, submerged runs

Nuclear-powered submarines may be a discontinuity in this metric. We have not investigated.

Clothmaking efficiency

The Jacquard Loom and the Spinning Jenny were suggested as discontinuities in this metric. We have not investigated.

Personal armor protectiveness-to-weight ratio

Kevlar was suggested as discontinuities in this metric. We have not investigated.

Lumens per watt

High pressure sodium lamps were suggested as discontinuities in this metric. We have not investigated.

Linear programming

The Simplex algorithm was suggested as discontinuities in this metric. We have not investigated.

Fourier transform speed

Fast fourier transform was suggested as discontinuities in this metric. We have not investigated.

Polynomial identity testing efficiency

Probabilistic testing methods were suggested as discontinuities in this metric. We have not investigated.

Audio compression efficiency

The MP3 format was suggested as discontinuities in this metric. We have not investigated.

Crop yields

This amazing genetic modification, if it works as claimed, may well be a discontinuity in this metric. We have not investigated.
Thanks to Stephen Jordan, Bren Worth, Finan Adamson and others for suggesting potential discontinuities in this list.


Effect of AlexNet on historic trends in image recognition

AlexNet did not represent a greater than 10-year discontinuity in fraction of images labeled incorrectly, or log or inverse of this error rate, relative to progress in the past two years of competition data.
Details
This case study is part of AI Impacts’ discontinuous progress investigation.
Background
The annual ImageNet competition asks researchers to build programs to label images.1 It began in 2010, when every team labeled at least 25% of images wrong. The same was true in 2011, and would have been true in 2012, if not for AlexNet, a convolutional neural network that mislabeled only 16.4% of images.2
Trends
Percent of images mislabeled
Data
We collected data on the error rate (%) of the 2010 – 2012 ImageNet competitors from Table 6 of Russakovsky et al3 into this spreadsheet. See Figure 1 below.
Figure 1: Error rate (%) of ImageNet competitors from 2010 – 2012
Discontinuity measurement
The ImageNet competition had only been going for two years when AlexNet entered, so the past trend is very short. Given this, the shape of the curve prior to AlexNet is entirely ambiguous. We treat the trend as linear for simplicity, but given that, it is better to choose a transformation of the data that we expect to be linear, given our understanding of the situation.
Two plausible transformations are the log of the error, and the reciprocal of the error rate.4 These two transformations of the data are shown in Figures 2 and 3 below.
Figure 2: Log base 2 / error rate of ImageNet competitors from 2010 – 2012
Figure 3: 1 / error rate of ImageNet competitors from 2010 – 2012
The best 2012 AlexNet competitor gives us discontinuous jumps of 3 years of progress at previous rates for the raw error rate, 4 years of progress at previous rates for log base 2 of the error rate, or 6 years of progress at previous rates for 1 / the error rate.5 For the 6-year discontinuity, we tabulated a number of other potentially relevant metrics in the ‘Notable discontinuities under 10 years’ tab here.
Notes


Historic trends in transatlantic message speed

The speed of delivering a short message across the Atlantic Ocean saw at least three discontinuities of more than ten years before 1929, all of which also were more than one thousand years: a 1465-year discontinuity from Columbus’ second voyage in 1493, a 2085-year discontinuity from the first telegraph cable in 1858, and then a 1335-year discontinuity from the second telegraph cable in 1866.
Details
This case study is part of AI Impacts’ discontinuous progress investigation.
Background
Summary of historic developments
All communications between Europe and North America were carried on ships until 1858, when the first telegraph messages were transmitted over cable between the UK and US.1 That first cable only lasted six weeks, and took more than sixteen hours to send a message from the Queen.2
A permanent cable wasn’t laid until eight years later.3 Better telegraph cables were laid a further thirty and sixty years later. We do not investigate developments after 1929.
Figure 1: Undersea communications cables became common in the long run: map of undersea communications cables in 2007.4
Trends
Transatlantic message speed, 140 character message
We looked at historic times to send messages across the Atlantic Ocean.
Message speed can depend on the length of the message. Where this was relevant, we somewhat arbitrarily chose to investigate for a 140 character message. We measure fastest speeds of real historic systems that could send 140 character messages across the Atlantic Ocean. We do not require that a 140 character message was actually sent by the method in question.
We generally use whatever route was actually taken (or supposed in an estimate), and do not attempt to infer faster speeds possible had an optimal route been taken (though note that because we are measuring speed rather than time to cross the Ocean, route length is adjusted for to a first approximation).
We only investigated this metric from 1492-1493 and 1841-1928. We do not investigate 1493-1841 because our data is insufficiently complete to determine how continuous it was.5
Data
Our data for message speed came from a variety of online sources, and has not been thoroughly vetted. The full dataset with sources can be found here.6
Because message delivery coincided with passenger travel until the first telegraph, data until then coincides with that used in our investigation into historic trends in transatlantic passenger travel.
The resulting trend is shown in Figures 2-3.
 Figure 2: Average speed for message transmission across the Atlantic in recent centuries (see Figure 3 for longer term trend)
    Figure 3: Average speed for message transmission across the Atlantic.
Discontinuity Measurement
We measure discontinuities by comparing progress made at a particular time to the past trend. For this purpose, we treat the past trend at any given point as exponential or linear depending on apparent fit, and judge a new trend to have begun when the recent trend has diverged sufficiently from the longer term trend. See our spreadsheet, tab ‘Message’ to view the trends, and our methodology page for details on how to interpret our sheets and how we divide data into trends.
Given these judgments about past progress, there were three discontinuities of more than ten years, all of which were more than one thousand years: a 1465-year discontinuity from Columbus’ second voyage in 1493, then a 2085-year discontinuity from the first telegraph cable in 1858, and then a 1335-year discontinuity from the improved telegraph cable in 1866.7 In addition to the size of these discontinuity in years, we have tabulated a number of other potentially relevant metrics here.8.
Discussion of causes
 Transatlantic message speed is a narrower metric than overall message speed, precluding some technologies that can only deliver messages short distances or on land (e.g. the semaphore telegraph, which relied on a series of towers within line of sight). We expected this would make discontinuity more likely.
Notes


Historic trends in long-range military payload delivery

The speed at which a military payload could cross the Atlantic ocean contained six greater than 10-year discontinuities in 1493 and between 1841 and 1957:
DateMode of transportKnotsDiscontinuity size(years of progress at past rate)1493Columbus’ second voyage5.814651884Oregon18.6101919WWI Bomber (first non-stop  transatlantic flight)1063511938Focke-Wulf Fw 200 Condor174191945Lockheed Constellation288251957R-7 (ICBM)~10,000~500
Details
Background
The speed at which a weapons payload could be delivered to a target on the opposite side of the ocean appears to have been limited to the speed of a piloted vehicle (and so coincided with speed of passenger delivery) until the first long-range missiles became available in the late 1950s.1.
Trends
This case study is part of AI Impacts’ discontinuous progress investigation.
Transatlantic military payload delivery speed
We look at fastest speeds of real historic systems that could have delivered military payloads across the Atlantic Ocean. We do not require that any military payload was actually sent by the method in question.
We generally use whatever route was actually taken (or supposed in an estimate), and do not attempt to infer faster speeds possible had an optimal route been taken (though note that because we are measuring speed rather than time to cross the Ocean, route length is adjusted for to a first approximation).
We only investigated this metric from 1492-1493 and 1841-1957. We do not investigate 1493-1841 because our data is insufficiently complete to determine how continuous it was.2
Data
We collated records of historic potential times to cross the Atlantic Ocean for military payloads. These are available at the ‘Payload’ tab of this spreadsheet, and are displayed in Figure 1 and 2 below. We have not thoroughly verified this data.
Because military payload delivery coincided with passenger travel until the late 1950s, most of our data coincides with that used in our investigation into historic trends in transatlantic passenger travel.
The advent of ICBMs in 1957 probably increased the crossing speed to thousands of knots. We are fairly uncertain about how fast the first ICBMs were, but our impression is that they traveled at an average of least 5,000 knots and likely more like 10,000 knots.3 Uncertainty here makes little difference to measurement of discontinuities. To not be a discontinuity of more than a hundred years, the first ICBM would need to have traveled horizontally at less than 2314 knots, which seems unlikely, because that is insufficient speed to cross the ocean, assuming optimal angle of fire.4
We haven’t recorded the later trend, since our understanding is that modern ICBMs do not travel much faster than we think early ones may easily have done.5 so will not yield clear discontinuities, and we do not know of faster missiles than ICBMs.6
 Figure 1: Historic speeds of sending hypothetical military payloads across the Atlantic Ocean
 Figure 2: Historic speeds of sending hypothetical military payloads across the Atlantic Ocean since 1700 (close up of Figure 1)
Discontinuity measurement
Until 1957, discontinuities are the same as those for speed of transatlantic passenger travel, since the data coincides. This gives us five discontinuities.
We calculate the final development, the ICBM, to probably represent a discontinuity of around 500 years, but at least 1007. See this spreadsheet, tab ‘Payload’ for our calculation.8
This gives us six greater than 10-year discontinuities in total, including five shared with transatlantic passenger travel speed. Three of them represent more than one hundred years of past progress:
DateMode of transportKnotsDiscontinuity size(years of progress at past rate)1493Columbus’ second voyage5.814651884Oregon18.6101919WWI Bomber (first non-stop  transatlantic flight)1063511938Focke-Wulf Fw 200 Condor174191945Lockheed Constellation288251957R-7 (ICBM)~10,000~500
In addition to the sizes of these discontinuity in years, we have tabulated a number of other potentially relevant metrics here.9
Notes


Historic trends in bridge span length

We measure eight discontinuities of over ten years in the history of longest bridge spans, four of them of over one hundred years, five of them robust as to slight changes in trend extrapolation.
The annual average increase in bridge span length increased by over a factor of one hundred between the period before 1826 and the period after (0.25 feet/year to 35 feet/year), though there was not a clear turning point in it.
Details
This case study is part of AI Impacts’ discontinuous progress investigation.
Background
A bridge span is a section of bridge between supports.1 Bridges can have multiple spans, e.g. one for each arch.2 Bridges are often measured by their ‘main span’.
We investigated bridge span (rather than bridge length, mass, or carrying capacity) because it was suggested to us as discontinuous. We also expect it to be a good metric for seeing technological progress, rather than economic progress, because additional spending can probably add more spans to a structure more easily than it can make each span longer. Span length is also a less ambiguous metric than total length, since it is not always clear where a road ends and a bridge begins.
The Akashi Kaikyō Bridge, current record-holder for longest bridge span3

Trends
Longest bridge span length
Data
We gathered data for bridge span lengths from several Wikipedia lists of longest bridge spans over history for particular types of bridge, plus a few additional datapoints from elsewhere. Our data and citations are in this spreadsheet in the tab ‘five bridge types’.
Problems, ambiguities, and limitations of the data and our collection process:
Some span lengths are given as different lengths on different Wikipedia pages. We did not investigate this, and the one we used was arbitrary.We did not find a list of historic longest bridge spans for all bridge types, so used several pages about longest bridges for particular bridge types, for instance List of Longest Suspension Bridge Spans. It is quite possible we failed to find all such lists. In the data we have though, suspension bridges are usually longer than anything else, and the Wikipedia History of Longest Suspension Bridge Spans mentions in its list a few times when non-suspension bridges are the longest bridge span in the world, suggesting that the authors of that page at least believe that at all other times the suspension bridges are the longest. We had already found the other bridges they mention (all arch or cantilever bridges).We have not investigated the accuracy of the Wikipedia data.We are unsure what exact definition of ‘bridge’ is used in any of these pages. Our impression is that they need to allow foot vehicle traffic to cross independently (e.g. it looks like foot bridges are included, but not this cable car with a 2831m span, which it seems would hold the current record were it a bridge). We have not investigated more.We treated dates of N BC dates -N
Figure 1-3 show the length of the longest bridge span for five types of bridge over time. If we understand correctly, these include the longest bridges of any kind at least since around 500AD.
Figure 1: Entire history of longest bridge spans of five types, measured in feet. See text for further details.
Figure 2: Figure 1 only visible to 600 feet.
Figure 3: Figure 1 since 1800
Discontinuity measurement
To measure discontinuities relative to past progress, we treat past progress as linear, and belonging to five different periods (i.e. three times we consider the recent trend to be sufficiently different from the older trend that we base our extrapolation on a new period).4
Using this method, the length of the longest bridge span has seen a large number of discontinuities (see table below).
NameYear opened/became longest of typeMain span (feet)DiscontinuityChakzam Bridge*14304492230Menai Suspension Bridge1826577146Great Suspension Bridge*1834889403Wheeling Suspension Bridge1849101070Niagara Clifton Bridge*1869126014George Washington Bridge*19313501132Golden Gate Bridge1937420019Akashi-Kaikyo Bridge*1998653256*Entry was more robust to informal experimentation with  different linear extrapolations
Deciding what to treat as the previous trend at any point is hard in this dataset, because the shape of the trend isn’t close to being exponential or linear. The sizes of the discontinuities and even the particular bridges that count as notably discontinuous are not very robust to different choices. In a small amount of experimentation with different linear trends, five bridges were always discontinuities, marked with * in the above table. That the overall trend is marked by many discontinuities seems robust.
 In addition to the size of these discontinuity in years, we have tabulated a number of other potentially relevant metrics here.5
Change in rate of progress
The annual average increase in bridge span length increased by over a factor of one hundred between the period before 1826 and the period after (0.25 feet/year to 35 feet/year), though there was not a clear turning point in it. See spreadsheet for calculation (tab: ‘Five bridge types (longest)’)
Notes

Historic trends in bridge span length


Historic trends in light intensity

Maximum light intensity of artificial light sources has discontinuously increased once that we know of: argon flashes represented roughly 1000 years of progress at past rates.
Annual growth in light intensity increased from an average of roughly 0.4% per year between 424BC and 1943 to an average of roughly 190% per year between 1943 and the end of our data in 2008.
Details
This case study is part of AI Impacts’ discontinuous progress investigation.
Background
That which is uncited on this page is our understanding, given familiarity with the topic.1
Electromagnetic waves (also called electromagnetic radiation) are composed of oscillating electric and magnetic fields. They span in wavelength from gamma rays with wavelengths on the order of 10-20 meters to radio waves with wavelengths on the order of kilometers. The wavelengths from roughly 400 to 800 nanometers are visible to the human eye, and usually referred to as light waves, though the entire spectrum is sometimes referred to as light, especially in the context of physics. These waves carry energy and their usefulness and the effect that they have on matter is strongly affected by their intensity, or the amount of energy that they carry to a given area per time. Intensity is often measured in watts per square centimeter (W/cm2), and it can be increased either by increasing the power (energy per time, measured in watts) or focusing the light onto a smaller area.
Electromagnetic radiation is given off by all matter as thermal radiation, with the power and wavelength of the waves determined by the temperature and material properties of the matter. When the matter is hot enough to emit visible light, as is the case with the tungsten filament in a light bulb or the sun, the process is referred to as incandescence. Processes which produce light by other means are commonly referred to as luminescence. Common sources of luminescence are LEDs and fireflies.
The total power emitted by a source of incandescent source of light is given by the Stefan-Boltzman Law.2
Light intensity is relevant to applications such as starting fires with lenses, cutting with lasers, plasma physics, spectroscopy, and high-speed photography.
History of progress
Focused sunlight and magnesium
For much of history, our only practical sources of light have been the sun and burning various materials. In both cases, the light is incandescent (produced by a substance being hot), so light intensity depends on the temperature of the hot substance. It is difficult to make something as hot as the sun, so difficult to make something as bright as sunlight, even if it is very well focused. We do not know how close the best focused sunlight historically was to the practical limit, but focused sunlight was our most intense source of light for most of human history.
There is evidence that people have been using focused sunlight to start fires for a very long time.3 There is further evidence that more advanced lens technology has existed for over 1000 years4, so that humans have been able to focus sunlight to near the theoretical limit5 for a very long time. Nonetheless, it appears that nobody fully understood how lenses worked until the 17th century, and classical optics continued to advance well into the 19th and 20th century. So it seems likely that there were marginal improvements to be made in more recent times. In sum, we were probably slowly approaching an intensity limit for focusing sunlight for a very long time. There is no particular reason to think that there were any sudden jumps in progress during this time, but we have not investigated this.
Magnesium is the first combustible material that we found that we are confident burns substantially brighter than crudely focused sunlight, and for which we have an estimated date of first availability. It was first isolated in 18086, and burns with a temperature of 3370K7. Magnesium was bright enough and had a broad enough spectrum to be useful for early photography.
Mercury Arc Lamp
The first arc lamp was invented as part of the same series of experiments that isolated magnesium. Arc lamps generate light by using an electrical current to generate a plasma8, which emits light due to a combination of luminescence and incandescence. Although they seem to have been the first intense artificial light sources that do not rely on high combustion temperature9, they do not seem to have been brighter than a magnesium flame10 in the early stages of their development. Nonetheless, by the mid 1930s, mercury arc lamps, operated in glass tubes filled with particular gases, were the brightest sources available that we found. Our impression is that progress was incremental between their first demonstration around 1800 and their implementation as high intensity sources in the the 1930s, but we have not investigated this thoroughly.
Argon Flashes
Argon flashes were invented during the Manhattan project11 to enable the high speed photography that was needed for understanding plutonium implosions. They are created by surrounding a high explosive with argon gas. The shock from the explosive ionizes the argon, which then gives off a lot of UV light as it recombines. The UV light is absorbed by the argon, and because argon has a low heat capacity (that is, takes very little energy to become hot), it becomes extremely hot, emitting ~25000 Kelvin blackbody radiation. This was a large improvement in intensity of light from blackbody radiation. There does not seem to have been much improvement in blackbody sources in the 60 years since.
Lasers
Lasers work by storing energy in a material by promoting electrons into higher energy states, so that the energy can then be used to amplify light that passes through the material. Because lasers can amplify light in a very controlled way, they can be used to make extremely short, high energy pulses of light, which can be focused onto a very small area. Because lasers are not subject to the same thermodynamic limits as blackbody sources, it is possible to achieve much higher intensities, with the current state of the art lasers creating light 16 orders of magnitude more intense than the light from an argon flash.
Figure 1: Industrial laser12
Trends
Light intensity
We investigated the highest publicly recorded light intensities we could find, over time.13 Our estimates are for all light, not just the visible spectrum.
Data
One of our researchers, Rick Korzekwa, collected estimated light intensities produced by new technologies over time into this spreadsheet. Many sources lacked records of the intensity of light produced specifically, so the numbers are often inferred or estimated from available information. These inferences rely heavily on subject matter knowledge, so have not been checked by another researcher. Figures 2-3 illustrate this data.
Pre-1808 trend
We do not start looking for discontinuities until 1943, though we have data from beforehand, because our data is not sufficiently complete to distinguish discontinuous progress from continuous, only to suggest the rough shape of the longer term trend.
Together, focused sunlight and magnesium give us a rough trend for slow long term progress, from lenses focusing to the minimum intensity required to ignite plant material in ancient times to intensities similar to a camera flash over the course of at least two millenia. On average during that time, the brightest known lights increased in intensity by a factor of 1.0025 per year (though we do not know how this was distributed among the years).
Due to our uncertainty in the early development of optics for focusing sunlight, the trend from 424 BC to 1808 AD should be taken as the most rapid progress that we believe was likely to have occurred during that period. That is, we look at the earliest date for which we have strong verification that burning glasses were used, and assuming these burning glasses produced light that was just barely intense enough to start a fire. So progress may have been slower, if more intense light was available in 424 BC than we know about, however progress could only have been faster on average if burning glass (that could actually burn) didn’t exist in 424 BC, or if there were better things available in 1808 than we are aware, both of which seem less likely than that technology was better than that in 424 BC.

Figure 2: Estimated light intensity for some historic brightest artificial sources known to us. Note that the very earliest instances of a given type are not necessarily represented, for instance our understanding is that dimmer arc lamps existed in the early 1800s.
Figure 3: Close up of Figure 2, since 1800
Discontinuity measurement
We treat the rate of previous progress as an exponential between the burning glass in 424BC and the first argon candle in 1943. At that point progress has been far above that long term trend for two points in a row, so we assume a new faster trend and measure from the 1936 arc lamp. In 1961, after the trend again has been far surpassed for two points, we start again measuring from the first laser in 1960. See this project’s methodology page for more detail on what we treat as past progress.
Given these choices, we find one large discontinuity from the first argon candle in 1943  (~1000 years of progress in one step), and no other discontinuities of more than ten years since we begin searching in 1943.14
In addition to the size of these discontinuity in years, we have tabulated a number of other potentially relevant metrics here.15
Note on mercury arc lamp
The 1936 mercury arc lamp would be a large discontinuity if there were no progress since 1808. Our impression from various sources is that progress in arc lamp technology was incremental between their first invention at the beginning of the 19th century and the bright mercury lamps that were available in 1936. We did not thoroughly investigate the history and development of arc lamps however, so do not address the question of the first year that such lamps were available or whether such lamps represented a discontinuity.
Note on argon flash
The argon flash seems to have been the first light source available that is brighter than focused sunlight, after centuries of very slow progress, and represents a large discontinuity. As discussed above, because we are less certain about the earlier data, our methods imply a relatively high estimate on the prior rate of advancement, and thus a low estimate of the size of the discontinuity. So the real discontinuity is likely to be at least 996 years (unless for instance there was accelerating progress during that time that we did not find records of).
Change in rate of progress
Light intensity saw a large increase in the rate of progress, seemingly beginning somewhere between the arc lamps of the 30s and the lasers of the 60s. Between 424BC and 1943, light intensity improved by around 0.4% per year on average, optimistically. Between 1943 and 2008, light intensity grew by an average of around 190% per year.16
The first demonstrations of working lasers seems to have prompted a flurry of work. For the first fifteen years, maximum light intensity had an average doubling time of four months, and over roughly five decades following lasers, the average doubling time was a year.17
Discussion
Factors of potential relevance to causes of abrupt progress
Technological novelty
One might expect discontinuous progress to arise from particularly paradigm-shifting insights, where a very novel way is found to achieve an old goal. This has theoretical plausibility, and several discontinuities that we know of seem to be associated with fundamentally new methods (for instance, nuclear weapons came from a shift to a new type of energy, high temperature superconductors with a shift to a new class of materials for superconducting). So we are interested in whether discontinuities in light intensity are evidence for or against such a pattern.
The argon flash was a relatively novel method rather than a subtle refinement of previous technology, however it did not leverage any fundamentally new physics. Like previous light sources, it works by adding a lot of energy into a material to make it emit light in a relatively disorganized and isotropic manner. Achieving this by way of a shockwave from a high explosive was new.
It is unclear whether using an explosive shockwave in this way had not been done previously because nobody had thought of it, or because nobody wanted a shorter and brighter flash of light so much that they were willing to use explosives to get it.18
The advent of lasers did not produce a substantial discontinuity, but they did involve an entirely different mechanism for creating light to previous technologies. Older methods created more intense light by increasing the energy density of light generation (which mostly meant making the thing hotter), but lasers do it by creating light in a very organized way. Most high intensity lasers take in a huge amount of light, convert a small portion of it to laser light, and create a laser pulse that is many orders of magnitude more intense than the input light. This meant that lasers could scale to extremely high output power without becoming so hot that the output is that of a blackbody.
Effort directed at progress on the metric
There is a hypothesis that metrics which see a lot of effort directed at them will tend to be more continuous than those which are improved as a side-effect of other efforts. So we are interested in whether these discontinuities fit that pattern.
Though there was interest over the years in using intense light as a weapon19, and for early photographers, who wanted safe, convenient, short flashes that could be fired in quick succession, there seems to have been relatively little interest in increasing the peak intensity of a light source. The US military sought bright sources of light for illuminating aircraft or bombing targets at night during World War II. But most of the literature seems to focus on the duration, total quantity of light, or practical considerations, with peak intensity as a minor issue at most.
The argon flash appears to have been developed more as a high peak power device than as a high peak intensity device.20 It did not matter if the light could be focused to a small spot, so long as enough light was given off during the course of an experiment to take pictures. Still, you can only drive power output up so much before you start driving up intensity as well, and the argon flash was extremely high power.
Possibly argon flashes were developed largely because an application appeared which could make use of very bright lights even with the concomitant downsides.
There seems to have been a somewhat confusing lack of interest in lasers, even after they looked feasible, in part due to a lack of foresight into their usefulness. Charles Townes, one of the scientists responsible for the invention of the laser, remarked that it could have been invented as early as 193021, so it seems unlikely that it was held up by a lack of understanding of the fundamental physics (Einstein first proposed the basic mechanism in 191722). Furthermore, the first paper reporting successful operation of a laser was rejected in 1960, because the reviewers/editors did not understand how it was importantly different from previous work.23
Although it seems clear that the scientific community was not eagerly awaiting the advent of the laser, there did seem to be some understanding, at least among those doing the work, that lasers would be powerful. Townes recalled that, before they finished building their laser, they did expect to “at least get a lot of power”24, something which could be predicted with relatively straightforward calculations. Immediately after the first results were published, the general sentiment seems to have been that it was novel and interesting, but it was allegedly described as “a solution in search of a problem”.25 Similar to the argon flash, it would appear that intensity was not a priority in itself at the time the laser was invented, and neither were any of the other features of laser light that are now considered valuable, such as narrow spectrum, short pulse duration, and long coherence length.
Most of the work leading to the first lasers was focused on the associated atomic physics, which may help explain why the value of lasers for creating macroscopic quantities of light wasn’t noticed until after they had been built.
In sum, it seems the argon flash and the laser both caused large jumps in a metric that is relevant today but that was not a goal at the time of their development. Both could probably have been invented sooner, had there been interest.
Predictability
One reason to care about discontinuities is because they might be surprising, and so cause instability or problems that we are not prepared for. So we are interested in whether discontinuities were in fact surprising.
It is unclear how predictable the large jump from the argon flash was. Our impression is that without knowledge of the field, it would have been difficult to predict the huge progress from the argon flash ahead of time. High explosives, arc lamps, and flash tubes all produced temperatures of around 4,000K to 5,000K. Jumping straight from that to >25,000K would probably have seemed rather unlikely.
However as discussed above, it seems plausible that the technology allowing argon flashes was relatively mature earlier on, and therefore that they might have been predictable to someone familiar with the area.
Notes

Historic trends in light intensity


Historic trends in book production

The number of books produced in the previous hundred years, sampled every hundred or fifty years between 600AD to 1800AD contains five greater than 10-year discontinuities, four of them greater than 100 years.  The last two follow the invention of the printing press in 1492.
The real price of books dropped precipitously following the invention of the printing press, but the longer term trend is sufficiently ambiguous that this may not represent a substantial discontinuity.
The rate of progress of book production changed shortly after the invention of the printing press, from a doubling time of 104 years to 43 years.
Details
This case study is part of AI Impacts’ discontinuous progress investigation.
Background
Around 1439, Johannes Gutenburg invented a machine for making books commonly referred to as “the printing press”. The printing press was used to quickly copy pre-created sheets of letters of ink onto a print medium.1 Presses that stamped paper with carved blocks of wood covered in ink were already being used in Europe,2 but Gutenburg made several major improvements on existing methods, notably creating the hand mould, a device which allowed for quickly creating sheets of inked letters rather than carving them out of wood. The printing press allowed for the quick and cheap production of printed books like never before.3
Replica of the Gutenberg Printing Press4
Trends
We looked primarily at two different metrics– the rate of book production in Western Europe and the real price of books in England. We chose these two because they were some of the only printing-related data sources which had data that went back several centuries before the invention of the printing press.
Had the data been available, we would like to have looked at some metric correlated clearly with innovations in the writing / printing process — e.g. the number of pages produced per worker per hour. Then we could  check whether the printing press represented a discontinuity relative to earlier innovations (e.g., the pecia system for hand-copying manuscripts).5
Unfortunately, neither the rate of book production nor the price of book data we have correlate well with innovations in the writing / printing process. The authors of our rate of book production data claim that most of the variation in the pre-printing press numbers is explained by factors which are not innovation or close proxies to innovation.6 Our data on the price of books is similarly unhelpful, as the early price data is too sparse to be meaningful.
In addition to the two metrics described above, we looked cursorily at a few metrics with no early data which changed drastically as a result of the printing press– the number of unique titles printed per year, the variation of genres in books, the price of books in the Netherlands, and the total consumption of books.
Rate of book production in Western Europe
Data collection
Our data for the rate of book production come from estimates of Europe-only production generated in a 2009 paper by historians Eltjo Buringh and Jan Luiten Van Zanden.7 Rate data is represented as the number of books produced in the previous 100 years at various points in time.
When we use the term book, we mean it to refer to any copy of a written work, whether hand-copied manually or produced via some kind of printing technique. The paper separates book production into estimates of manuscript and printed book production, where the production of printed books starts only after the printing press is invented. We will also use the terms manuscript and printed book to talk about the data, but it’s unclear to us if the paper means manuscript to mean “any book not made using a Gutenburg-era printing press” or “any book transcribed by hand”. At one point the authors sum these two estimates into a single graph of production per capita,8 suggesting that the combination of manuscript and printed book data should cover all books.
The paper’s estimates for manuscript production are constructed by taking an existing sample of manuscripts and then attempting to correct for its geographical and temporal biases.9 Estimates for book production are constructed by counting new titles in library catalogues and multiplying by estimates for average prints per title at a given time.10
The estimates of manuscript production seem extremely non-robust given that large number of correction factors applied.11 The estimates of book production seem somewhat more robust, but should be taken as a lower bound as the authors did not correct for lost books and have estimated the average number of prints per title conservatively.12
Data
Figure 1a displays the raw data for rate of book production on a log scale, taken from the data in the paper described above and compiled in this spreadsheet. Each data point represents the total number of books produced in the previous 100 years.
Figure 1a: Book production in Western Europe
Figure 1b displays the same data as Figure 1a along with our interpretation.
Looking at the data, we assume an exponential trend up until 1500, and another one after that.13 The blue line is the average log rate of the rate of book production before the invention of the printing press (just manuscripts); the red line is the average log rate of the rate of book production after the invention of the printing press (manuscripts + printed books).
Figure 1b: Rate of book production in Western Europe. Blue and red lines are the average log rates of the rate of book production before and after the printing press. Grey points are projections of the average log rate before the printing press.
Grey points shown after 1500 reflect projected manuscript and therefore book production had the printing press not been invented. In practice, the actual number of manuscripts produced after 1500 were very small and not presented in the data.
Discontinuity measurement
If we just look at the trend of book production per past 100 years, measured once every 100 years before 1500, and then once every 50 years afterwards, we can calculate discontinuities of sizes 161 years in 900, 134 years in 1200, 23 years in 1300, 180 years in 1500, and 138 years in 1550.14 This is obviously a strange kind of trend–a discontinuity of one hundred years in a metric with datapoints every hundred years might mean nothing perceptible at the one-year scale. So in particular, these discontinuities do not tell us much about whether there would be discontinuities in a more natural metric, such as annual book production.
Changes in the speed of progress
There was a marked change in progress in the rate of book production with the invention of the printing press, corresponding to a change in the doubling time of the rate of book production from 104 years to 43 years.15
Interpreting this rate of change on the graph, before the invention of the printing press, the total rate of book production, which consists entirely of manuscripts, follows the exponential line shown in blue. The invention of the printing press in 1439 allows for mass production of printed books, causing the rate of book production to veer sharply off the existing exponential line, shown as the first point in red.  Note that our underlying data sources are non-robust, particularly for manuscript data pre-printing press, so the magnitude of this change in rate of progress may be under or overstated.
Discussion of causes
The change in the doubling time of the rate of book production caused by the printing press may reflect a large change in the factors that drove book production.
In their paper, Buringh and Van Zanden note that in the Middle Ages, 60% of the variation in book production is explained by the number of universities, the number of monasteries, and urbanization.16 They produce the following graph, correlating monastery numbers and early book production:
Figure 2: Buringh and Van Zanden’s figure of the relationship between book production and monestaries
By contrast, after the printing press was invented, Buringh and Van Zanden attribute a much more important role to individual book consumption and the forces of the market:
How to explain the significant increase in book production and consumption in the centuries following the invention of moveable type printing in the 1450s? The effect of the new technology (and import technological changes in the production of paper) was that from the 1470s on, book prices declined very rapidly. This had number of effects: consumption per literate individual increased, but it also became more desirable and less costly to become literate. Moreover, economies of scale in the printing industry led to further price reductions stimulating even more growth in book consumption.
It seems plausible that the move between exponential curves caused by the printing press was a shift from an exponential curve that reflected the growth of monasteries, universities, and cities to an exponential curve that reflected the growth of a complicated set of market forces.
Real price of books in England
Data collection
We took data from a 2004 paper written by economic historian Gregory Clark, who took data from historic records of price quotes,17 though data before 1450 is based on just 32 total price quotes, and Clark notes that “the prices vary a lot by decade since it is hard to control for the quality and size of the manuscript.”18 As such, we should not take much meaning out of the individual data points before 1450 or interpret the period between 1360 and 1500 as a rising trend.
Clark’s paper reports an index of the nominal price of books in Table 9 of his paper. To instead get an index of the real price of books,19 we divided each nominal price by Clark’s reported “cost of living” for each year (x 100), which was the amount of money paid by a relatively prosperous consumer for the same bundle of goods every year.20
Data
Figure 3 is a graph of (an index of) the real price of books in England, generated from the data described above and compiled in this spreadsheet. Each point represents the amount of money in each year needed to buy some bundle of books assuming you would pay 100 for that same bundle of books in 1860.
Economist Timothy Irwin, looking at this same dataset, claims that the drop in price around 1350 was due to knowledge about paper-making finally making its way to England: “As time passed, improvements in technology and the gradual spread of literacy reduced these obstacles to effective transparency. In particular, the diffusion of knowledge about paper-making (from around 1150) and then printing (from around 1450) dramatically reduced the price of books. (See Figure 1 for an estimate of the decline in England).”21
‘Figure 1’ in the quote above refers to a graph of the real price of books that uses the same data source and is identical to the one we generated. His claim seems plausible, but we are not confident about it given how sparse and noisy the early data is and given our lack of precise date information on how paper-making spread in England.
Discontinuity measurement
The graph contains two major drops– one as a result of the printing press, and one claimed to be the result of paper replacing parchment in England. Looking at just the set of blue data points between these two drops, we can see that they are clustered around some range of values. The data in this range is too noisy and sparse22 to generate a meaningful rate of progress, but it seems clear that for a wide variety of plausible rates, the printing press represented a discontinuity in the real price of books when compared to the trend in book prices after the spread of paper-making.
However, if you take the past trend to include the price of books before paper-making, then there is no clear discontinuity– the price of printed books could be part of an existing trend of dropping prices that started with paper-making. We also believe the data here is too poor to draw firm conclusions.
Figure 3: Real price of books in England. All prices are relative to a book in 1860 costing 100, so a real price of 1800 would be 18x as expensive as a book in 1860.
Discussion of causes
Whether or not it counts as a substantial discontinuity relative to the longer term trend, the printing press produced a sharp drop in the real price of books. This was because their price was largely driven by labor costs, which went down sharply (one author estimates by a factor of 341) when a laborer could use a machine to print massive numbers of books rather than manually transcribing each copy.23
Other noteworthy metrics
Many historians associate the invention of the printing press with other unsurprising book-related changes in the world, including:24
An increase in the productivity of book production, i.e. the ratio between the wage of a copy producer and the price of a standard book. In particular, one estimate measures a 20-fold increase in the productivity in the first 200 years after the invention.25 Another estimate guesses that there was a 340-fold decrease in the cost per book as a direct result of the printing press.26
A sharp decrease in the real price of books. One estimate of the real price of books in the Netherlands suggests a ~5-fold decrease between 1460 and 1550.27
An increase in genre-variety of books, and in particular a shift away from theological texts and an increase in the amount of fiction.28
An increase in the total consumption of books, likely as a result of their declining price and increased literacy levels.29
Most of these changes are gradual over at least a century, rather than involving a sharp change that might be a large discontinuity. Such gradual changes might suggest a sharper change in some underlying technology though, for instance. The fall in the price of Dutch books was relatively abrupt, but the data lacks a trend leading up to the printing press.
The increases in the number of genres and unique titles published suggest that there was a larger amount of information available in printed form. Decreased prices and increased consumption of books suggests that this information was easier to access than before. These things suggest there might have been an interesting change in availability of information in general, however we do not know enough about the past trend to say whether this was likely to be discontinuous.
Notes

Historic trends in book production