Broadsheet

This week we’re going to look a specific piece of early Roman military equipment, the humble bronze pectoral, which it turns out is surprisingly tricky for us to confidently reconstruct, in part because the period of its use that most interests us (the run from c. 264 to c. 146 where Rome is winning its first big overseas wars) is a relative gap – fancy word, ‘lacuna‘ – in our evidence, making it really difficult to correlate what our literary source (Polybius) is telling us to the physical evidence we have (both preserved examples and artwork). This was, we are told (by Polybius) the armor of the common Roman soldier in the period of their greatest wars, yet on some level we do not really know what it looked like. Not with certainty, in any case.

In particular I am going to argue that the most common reconstruction of this armor, as a single bronze plate suspended usually by leather straps over the chest, is probably wrong and that the armor more likely existed as a complex harness, simplified in brief literary description down to just its core element. But as we’ll see, this is going to be a zone of what I term ‘real uncertainty’ – a situation where without new evidence coming out of the ground, we simply cannot know for sure.

So this is not just an exercise in working through how to reconstruct one specific kind of equipment, but also how historians engage in questions that exist in a zone of really low confidence.

But first, as always, affording a full panoply of heavy infantry equipment as is the duty of any propertied Roman citizen is expensive! If you want to help me ~~waste~~ spend my money on reproduction ancient military equipment, you can support this project over at Patreon. If you want updates whenever a new post appears or want to hear my more bite-sized musings on history, security affairs and current events, you can follow me on Bluesky (@bretdevereaux.bsky.social). I am also active on Threads (bretdevereaux) and maintain a de minimis presence on Twitter (@bretdevereaux).

Via Wikipedia, a rough map of cultural groups in pre-Roman Italy. Key:
Dark Blue: Ligures
Brown: Veneti
Pink: Etruscans
Light Blue: Piceni
Light Green: Umbrians
Dark Green: Oscans (including the Samnites, discussed below)
Orange: Messapii
Yellow: Greeks
Gold: Latins (including the Romans)

Polybius

Our first stop is Polybius. Polybius wrote in the mid-second century (that is, the 140s), but his history covers the period from 264 to 146 and his description of the pectoral is placed relatively early in the narrative, in 216, as part of a larger explanation of the Roman military system. There is thus immediately a question as to if the details Polybius is giving are correct for 216 or for the 140s when he wrote. In practice, the answer must be something of a mix: Polybius has sources that reach back and might give him details appropriate to the period (he seems to have the writings of a military tribune to use for this description of the dilectus), but it seems likely that his description of the pectoral comes from observing it. Consequently, while I suspect that Polybius’ description of who is required to wear what may be accurate for 216, he has clearly seen the pectoral and understands it to still be in use in his own day (indeed, at other points in this extended passage, he explicitly notes things that used to be one way but had changed by his own day).¹

That’s handy, because Polybius is the only source that describes this armor. Later historians – Livy, Plutarch, etc. – seem broadly unaware of it and it really does seem like the pectoral was in the process of going extinct when Polybius was writing (for reasons below). So we have one description of the armor, but at least it is by an eyewitness. Here it is (Polyb. 6.23.14-15, trans. mine):

The many [hoi polloi, “the common folk”] taking a bronze plate a span [c. 23cm] on all sides, which they place over their chests and call ‘heart protectors’ [καρδιοφύλαξ, very literally ‘heart protector’], finish their armaments. However those worth more than ten thousand drachmas [= the first class of Roman infantry], instead of the heart-protector wear mail coats [αλυσιδωτοί θώρακες, “hooked [or chain] cuirasses” which we know is the Greek way to say ‘mail coats.’]

And…that’s it. From later authors (Varro, De Ling. Lat. 5.116; Plin HN 34.18) we get the Latin name for this armor, pectorale (pectorale, pectoralis (n) for the Latin nerds), thus the English term ‘pectoral’ but no more details of its construction.²

Crucially, this armor doesn’t show up on any highly visible Roman military monuments. The reason is fairly simple: the earliest really visible Roman military monuments are the Pydna Monument (168) and the so-called Altar of Domitius Ahenobarbus (late second century) by which point the pectoral was already on the way out. Ancient artists tend to prefer high status equipment and so with the pectoral on the way out (though likely still very much in use in 168) and the poorer, lower status armor, they didn’t depict it, instead preferring to use mail armor to signal Roman soldiers (specifically, mail is used on the Pydna Monument to signal ‘these are Romans’ in contrast to Macedonians or Gauls).³

As a result, scholars initially didn’t have a lot to go on except Polybius’ description – the archaeology, as we’ll see, doesn’t really get sorted out until the last 40 years or so. So they reconstructed on that basis. A ‘span’ (σπιθαμή) is a ‘natural’ unit, the distance between the thumb and the little finger at full extension, which is conveniently more or less half of a cubit (the length of a forearm out to the end of the middle finger), which eventually becomes formalized in Attic measurements (which Polybius tends to use; other places might have slightly different measures for the same terms) as 23.1cm and 46.2cm respectively.

That leads to the most common thing we see in artistic reconstructions and reenactor kit: the pectoral is reconstructed as a brass or bronze plate, usually about 1-2mm thick (the normal thickness for breastplates), 23cm by 23cm square. Since obviously it needs to be attached to something it is often shown backed in leather, with leather straps around the waist and over the shoulders holding it in place. I am going to call this reconstruction – a single plate, 23cm square, on a leather harness – the ‘traditional’ reconstruction.

That size lets the pectoral cover most of the chest, but it does nothing for the belly, sides or shoulders. On that basis, I have very often heard scholars regard it as a very minimal, almost token defense, unlikely to do much at all to protect the men wearing it. And again, before there was much archaeology to work with (or before finds had been analyzed, arranged chronologically and had their development worked through), you can see how this is the most logical extrapolation of what Polybius is saying.

But I do want to note some things here. Polybius’ description of this armor is extremely brief. He does not even bother to explain what Roman mail armor is like at all – no description, for instance, of its length (to the knees) or shoulder-doubling or the front-closure mechanism. If it weren’t for period depictions of mail, we would probably reconstruct it without these elements. As for the pectoral, all he says is that it is a span square and the Romans have a funny name for it. Which is to say it is entirely possible that Polybius is leaving out some details here. Which brings us to:

The Development of the Italic Pectoral

This, of course, is the point at which we naturally turn to archaeology to provide us both physical examples of this kind of armor and also visual representations of it. And he we run into an immediate problem: the third and second century feature a near total lacuna of Italic armor, in both artwork and preserved examples. The problem is frustrating in its elegant simplicity: the Roman military system – terribly efficient and in its way, anti-aristocratic – coincides as it expands with the end of aristocratic ‘warrior burials’ wherever it goes. Thus as Rome during the fourth and early third century goes about consolidating control of Italy, the amount of nice tomb paintings with aristocratic warrior in procession or burials with arms and armor drop to basically nothing. The Roman army is removing the evidence we might have for the Roman army. Astoundingly frustrating.

The evidentiary record begins to pick up a bit in the second century with more artistic depictions of Roman soldiers as the Roman state engages in more monumental depictions of its soldiers (noted above), but by that point mail rather than the pectoral is the ‘national armor’ of Rome’s armies (even though the pectoral is likely in use) and pectorals never appear. The really strong archaeological record for armor will have to wait until the imperial period, when the permanent stationing of Rome’s armies on the frontier of the empire means they sit in one place long enough for us to recover bits of armor.⁴ Weapons show up more often than armor (pila more often than any other type of weapon, a testament to their disposability) and we get a lot of helmets (for reasons not entirely clear to me), but functionally no body armor from this period. The best we can do are tiny fragments of metal rings for mail and even those are rare.

Worse yet, as mentioned before, the pectoral was going extinct in this period. Notably, when our evidence improves massively in the first century BC and AD, the pectoral is nowhere to be found. No source mentions it as still in use in that period, no artist depicts it, no finds of it are recovered. Polybius is thus our last source for this armor, suggesting that by the start of the first century, it had been wholly replaced by mail. No shock, mail is awesome (if expensive). But that means we cannot look for later examples to help us understand what Polybius is saying.

But we can can look at earlier ones.⁵

The Italic pectoral seems to have arrived from the Middle East in the 8th or perhaps 7th centuries (sometime between c. 750 and c. 680). This form of armor, a more or less flat metal place (as opposed to an enclosing breastplate of the sort we see in Greece around this time) has Middle Eastern precedents (we see Assyrian soldiers in artwork wearing similar armor), though how exactly it made it to Italy is unclear – Phoenicians seems most probable, but uncertain. In either case, by the seventh century, these pectoral armors are quite common over all of Italy, including Latium (where Rome is) and Etruria. The armor at this point generally consists of two bronze plates (a front plate and a back plate), which might be rectangular or circular, about 20-25cm wide (or tall; sometimes these are even smaller than this) and which were connected by leather straps. We generally call these ‘rectangular’ and ‘single disc’ pectorals. When decorated (and they very frequently are), they usually feature either geometric designs (often rectangles within a rectangular cuirass) or animal designs, either punched into the plate or embossed.

Via the British Museum (1872,1008.1) an Italic kardiophylax (c. 700-600BC), 25.4cm wide.

And you can see how an archaeologist looking at these pectorals from the seventh century might be thinking, “ah, I see exactly what Polybius was talking about: a bronze plate a span square!” Except, of course, the sixth century is not the second century and these pectorals keep evolving.

Now, in significant parts of Italy, especially Etruria, these pectoral armors begin to be replaced in the late sixth century by Greek-style armor, especially for elite, high-status warriors. In particular, the Etruscans love the tube-and-yoke (linothorax) armor when it shows up and it swiftly becomes a marker of elite status, though pectorals do occasionally show up in Etruscan art, albeit less frequently, but they are certainly petering out. Annoyingly, at roughly this point the archaeological record for Rome specifically also dries up, so it isn’t clear exactly what armors are popular in Rome in the very early Republic (our literary sources assume Greek-style armors, which may be right, but they are guessing and deeply anachronistic in their assumptions).

However in central Italy, in the Apennines Mountains, the pectoral persists and undergoes some significant design changes. Around 600, we start to see changes to the strap mechanisms holding the armor together: one shoulder strap is replaced with a pair of bronze plates connected by a hinge. The resulting harness gets pretty complex, as you can see in the figure of the Capestrano Warrior (c. 550), where the harness that holds the pectoral also supplies a scabbard (suspended at the chest) for the sword and there is a clear contrast between the metal hinged plate (over the right shoulder) and the more reddish-colored leather straps (of which there are three, two wide and one narrow) holding the harness and scabbard together.

Via Wikipedia, the Capestrano Warrior (c. 550), found at Capestrano in Abruzzo (in Italy), depicting a warrior of the Piceni, a central Italic peoples on the Adriatic coast.

In the early fifth century, this design is both enhanced and greatly simplified with the emergence of the first ‘triple disc’ pectorals. These are so named because the front plate (and back plate) take the form of three discs in a triangular arrangement, though I must stress this is a single plate with three circular designs on it in a roughly triangular shape, not three individual circular plates. Indeed, earlier archaeologists supposed that the ‘triple disc’ cuirass must have evolved in two stages from the disc pectorals discussed above and posited a ‘double disc’ cuirass, which turns out not to have existed.

These triple-disc breast- and back-plates were joined together not by leather straps but by a simplified version of the hinged plate system used in the sixth century disc pectorals, except now there are four connecting plates: one each over the shoulders (each of them hinged) and one at the sides (without hinges). These plates also get a bit wider, providing relatively fuller coverage over the upper body and the armor is supplemented by a wide bronze belt worn around the waist which protects the lower abdomen. You can see the full armor clearly in artwork:

Via the British Museum, a fourth century squat lekythos showing a pectoral cuirass (in this case a ‘triple disk’ type) worn by a Campanian warrior

In the second-half of the fourth century (so 350 onwards), we see these triple-disc cuirasses joined by another type, particularly on the western coast of southern Italy (so the area south of Latium), the ‘rectangular anatomical cuirass.’ This takes the existing triple-disc harness structure, with its bronze belt and connecting side and shoulder plates, but instead of the triangular triple-disc cuirass, it substitutes rectangular breast- and back-plates, with the designs on these invariably mimicking the musculature on Greek muscle cuirasses, although – because these plates are smaller than Greek breastplates (which wrap around the body) – the muscles depicted are visibly smaller-than-lifelike. In short, the artistic form of the muscle cuirass is being copied, but this is not an effort to mimic the actual muscles of the man wearing the armor.

To give a sense of size, recovered triple disc cuirasses range from 27-32.5cm tall and 26-28cm at the widest, while the rectangular anatomical cuirasses range from 29.5 x 37cm tall to 25 x 30cm wide for the front plates.⁶ Combined with side and shoulder plates that tend to be 5-8cm wide and a wide bronze belt (7-12cm wide, 70-110cm long, ~1mm thick), these really do cover most of the upper body, albeit with gaps, and are something closer to an articulated breastplate than they are to the small ‘heart protector’ of the Capestrano Warrior.

And you may note that a rectangular plate over the chest of c. 30cm by c. 28cm is not very far from Polybius’ description of “a bronze plate a span on all sides” and better yet is far more likely to have actually be in use in the third and second centuries for Polybius to see.

Via Wikipedia, a triple-disc cuirass with its shoulder and side plates (but no bronze belt), in the Museo Archeologico Nazionale di Paestum.
This is, as an aside, a good example – particularly the triple-disc component – of how simple the decoration of these armors could get. The cuirass is cut out of sheet metal, has three simple discs hammered into its shape and is otherwise mostly unadorned. Assuming sufficient bronze, such cuirasses could likely be made relatively quickly and cheaply, compared to something like a muscle cuirass (or **certainly** compared to later mail armor).

Notably – and this is going to matter in a moment – these fifth and fourth century pectoral harnesses do not appear without bronze belts or connecting plates. You will find these pectorals in museums without those added elements, in many cases because when the first of these armors were excavated (and/or looted) it was done carelessly and so the smaller plates were missed. However, whenever we get these armors with secure provenance or see them depicted in artwork, as Michael Burns notes, without exception, we get the full harness with all seven elements (frontplate, backplate, 2 shoulder plates, 2 side plates, bronze belt). We never to my knowledge ever see them suspected in simple leather harnesses; it surely was possible to do so, but it is unclear that anyone ever did after the introduction of the four-plate harness.

What Michael Burns thinks is happening (revising earlier work by the late, great Peter Connolly), and I think he is right, is that Southern Italic peoples are responding to the increasing presence of Greek muscle cuirasses coming in through Greek colonies in Southern Italy. But rather than just copying the muscle cuirass, they seem to have innovated from their own single-disc pectorals (which didn’t always cover a whole lot of the chest) to the triple-disc to create a kind of ‘exploded’ muscle cuirass. Initially, they do this by taking their own armor form, the single-disc cuirass, and expanding it out into a full ‘exploded’ breastplate, but eventually, in the fourth century, there’s enough artistic crossover that designs that use a rectangular plate and intentionally mirror Greek artistic tropes appear alongside triple-disc styles (which do not go away). It is worth noting that some of these triple-disc and rectangular anatomical armors are wonderfully decorated with complex designs, but many of them are very minimally decorated, especially as we get into the fourth century, suggesting a demand for a cheaper, no-frills version of this protection.

And then in 290 the Romans win the Third Samnite War and take control of the non-Greek parts of Southern Italy. And as noted above, when the Romans incorporate a given part of Italy into their ‘alliance’ system, for reasons that are not entirely clear to us (but the pattern is very strong), warrior burials, ritual weapon depositions and aristocratic artwork of warriors stop. Which means right around the year 300, our evidence for the Italic pectoral tradition simply vanishes. Really, we basically have an expanding bubble of darkness, radiating out from Rome (which is also probably how the Roman conquest felt to the Samnites), blinding our ability to track the development of armor in Italy.

Via Wikipedia, the Ksour Essef Cuirass, a triple-disc cuirass found in a Punic tomb in Ksour Essef, Tunisia. This cuirass is now generally dated to the late fourth or early third century, before the First Punic War, so its presence suggests significant trade contacts between Carthage and Italy, such at that a local Punic elite might acquire a beautifully decorated piece of Italian armor.

So by the third century, we do not see any pectorals, because we don’t see much of anything (except helmets; we continue to see those) for quite some time.

Except…

The Weird Exception We Need To Dismiss

The one odd exception to this is a pectoral disc found in the siege camps at Numantia.⁷ It is 17cm wide and circular, with a pattern of concentric circles and a large central knob and for quite some time if you went looking for an actual Roman pectoral this is what you would find.

The problem is that it isn’t Roman, it is very obviously Spanish. This spent a century not getting noticed because archaeologists working on ancient arms and armor tend to be very geographically specialized, so folks working on Roman and Italic arms and armor are not likely to be very familiar with the arms and armor of the fifth century Celtiberian Meseta. But if you are familiar with that, it is very clear that this is not a Roman pectoral at all, but a Spanish one, despite it turning up in a Roman camp.

Left: The Numantia pectoral, as illustrated by M.C. Bishop in Bishop and Coulston, *Roman Military Equipment* (2006), image © M. C. Bishop
Right: Via the Museo Arqueologico Nacional, Madrid, a Celtiberian pectoral harness (MAN 1940/27/AA/314, main disc 18cm in diameter, late fifth to early fourth century), showing similar concentric circle motifs and punch-holes around the outer edge.

First, while Italy had single-disc circular pectorals these had been replaced in the archaeological and artistic record in the fifth century by the larger triple-disc pectorals discussed above. Moreover, those earlier single-disc Italic pectorals don’t feature raised concentric circles as part of their normal artistic motifs. The more often have animals on them, or punch-holed simple geometric designs. They were also flat and did not feature central knobs.

But you know who did have pectoral harnesses with circular central plates featuring raised concentric circle designs and prominent central knobs? The Celtiberians, who are the people who lived at Numantia, where these camps were. Now the tricky bit here is that these pectorals are also – as far as we can tell – long out of use in the Iberian Peninsula as well: they persist through the fifth century, but fade out at the beginning of the fourth.

But whereas it is a little difficult to imagine a second-century Roman soldier decided to bring a piece of armor with him to Spain that had been out of use in Italy for something like four centuries, it is a lot easier to imagine the same Roman soldier in Spain might have looted a temple or a tomb (or simply struck a burial while entrenching his camp) that contained a fifth or very early fourth century Celtiberian disc-harness and that this soldier then looted the shiny bronze plate, later to be (for whatever reason) discarded in the camp.

Reconstructing the Roman Pectoral

So that is the shape of our evidence: with the Numantia pectoral removed (because it is not Roman at all, but Celtiberian), we have no examples of this armor from the third or second centuries B.C. What we do have is a tradition of pectoral armors which lead to the emergence of the triple-disc and rectangular anatomical pectoral harnesses in the fourth century, which we lose sight of in the general lacuna for most non-helmet military equipment in the third and second century. When our evidence returns, they are gone but we have this report by Polybius that poorer-but-still-propertied Romans in the heavy infantry (so not the poorest Romans fighting, those are the velites or do not serve at all) wear a bronze pectoral plate about a span square over their chest.

That admittedly quite poor evidence base leaves us with really just two options, both of them somewhat unsatisfactory.

The first option, the one taken – so far as I can tell – by the great majority of modern artistic reconstructions, is to simply read Polybius and reconstruct exactly what he says. That gives these Roman soldiers a single metal plate, typically shown mounted on a leather backing with leather straps, about 23cm square. This is, in a sense, the philologically elegant solution: it assumes nothing not in our text. The problem, from an archaeological perspective, is that this effectively requires arguing one of two cases: either first that sixth century pectoral – with its simple leather suspension – somehow survived in Italy for four centuries to be observed in action on the battlefield by Polybius in the mid-second century without leaving any other evidence at all. Not one piece of artwork, not one surviving example in the intervening period, despite the fact that we have sixty-seven fifth and fourth century examples of the later pectoral types (45 triple-disc and 22 rectangular anatomical cuirass types). That could be right. But it is a heroic assumption.

Alternately, the argument would be that the Romans at some point developed their own version of the pectoral, probably based off of the rectangular anatomical type, which discarded with the wide bronze belt, the shoulder plates and the side plates and so consisted only of a breastplate and a backplate. The problem here is simple: as Michael Burns notes in his survey of Italic pectorals, that configuration never occurs in artwork or in archaeology where site and provenance are secure. We do not have a single example of those later Southern Italic pectorals – the types that emerge after the more complex harness structure discussed above – dispensing with those pieces. Could they have done? Of course. But as of 2005 (and so far as I know, to the present), we have no evidence that anyone ever did. This solution thus requires conjuring into existence an effectively unknown armor-type. That could be right, particularly given how bad our evidence for Roman arms and armor in the Early Republic is. You can even imagine, if we had evidence of it, how we’d explain it: the broadening participation in the Roman army leads to poorer Romans to take up the Samnite cuirasses (that is, triple-disc and rectangular anatomical cuirasses) they have seen, but to jettison the ‘extra bits’ to make it cheaper and more affordable, effectively reversing a few centuries of armor development to create a stripped down breast- and back-plate only version. That’s what we’d posit, if we had some, but we don’t have some and I would argue that it runs against the rules of evidence as practices in archaeology to conjure into existence an unattested variant of an object-class (which does not developmentally link to anything else you can see) simply because it would be convenient. That is not how we assess coins or pots, I do not see why we would do it with armor.

That leaves another option: Polybius is describing the Southern Italian pectoral harness we can see, but doing so incompletely. It is not hard to imagine how the Romans will have picked up this armor: they spent the period from 343 to 290 fighting the Samnites in Campania and the Samnites are the major users of the triple-disc cuirass and Campania is where we most often see them in artwork. If the Romans weren’t already using this armor (and remember, we have no evidence at all of what armor the Romans are using in c. 300), they could certainly pick it up.

Then Polybius comes along in the mid-second century, where this armor is already dying out, largely replaced by mail, but still hanging on here or there – perhaps as hand-me-downs used by poorer Romans. One advantage of the pectoral harness’ seven-part structure is that it is a sort of ‘one-size-fits-no-one’ set that would be reasonably easy to modify or pass down to new users (unlike a Greek-style muscle cuirass, which really needs to be fitted to the wearer). Polybius then, writing about the Roman army as it existed in the Second Punic War (218-201) and, as per Rawson, using perhaps the accounts of some military tribunes, is aware of this armor’s place in the military regulations of that time and so includes it but with only minimal description. As a Greek, Polybius is used to thinking about body armor as a single piece – a breastplate, a tube-and-yoke cuirass, a mail coat – rather than a harness, so looking at a rectangular anatomical cuirass that is, perhaps 30cm by 28cm for its front plate, he describe sit simply as ” bronze plate a span on all sides.” Just as he doesn’t include the details of Roman mail armor’s shoulder doubling, he feels no real need to include the shoulder and side plates of the harness and he may not even be aware that the wide bronze belt has any real armor value at all (early archaeologists made the same error, assessing it as purely decorative, but it would offer some protection).

I think these are the three options we are left with for the pectoral: surprising sixth-century survival in the mid-second century, otherwise un-evidenced recreation of an older form out of the fourth-century rectangular anatomical cuirass or simply that it is the rectangular anatomical cuirass, harness at all, that Polybius has described incompletely. My own instinct is that the latter is probably correct. One interesting thing is that compared to, say, muscle cuirasses, these pectoral cuirasses of both the triple-disc and rectangular anatomical types were probably produced from sheet metal (sheet bronze, in particular), rather than forged from an ingot, which would have made it relatively easier to produce larger numbers of armors – equally if one opted for a style with simple decorations, amply in evidence in the archaeological record. Meanwhile, as noted the design is fairly easy to adjust for size. Jeremy Armstrong and Nicholas Harrison suggest that this in part allowed for “the expansion of warfare in Italy seen in the fourth century and marked by Rome’s wars of conquest” and I think that is right.⁸

Now in the fourth century, that armor might still be restricted to the fairly well-off. But in the late third or early second century, it is not hard seeing how the introduction of an even better but also substantially more expensive armor – mail – might ‘push’ existing pectoral cuirasses (again, of both types) down the socioeconomic ladder as the Roman first census class was required – as Polybius tells us – to acquire mail. The spare armor might ‘flow downwards’ as it were, making the affluent man’s undecorated but still shiny bronze armor of c. 350 the poor man’s pectoral of c. 150. Indeed, there is no reason it couldn’t be the very same piece of armor.

I do not think the evidence allows us to answer this question with confidence, but I do think that simple inertia has led scholars to continue reproducing the ‘traditional’ pectoral reconstruction long after it stopped being the most likely one. Instead, the most likely solution is that the Romans had continued to use, in some form, the full triple-disc or rectangular anatomical cuirass, including metal connecting plates (and perhaps the wide bronze belt) and that what Polybius was seeing was not, in fact, the small decorative chest-plates of the sixth century but rather this armor.

I spend a lot of time reading about the nature of technological progress, and I’ve found that the literature on technology is somewhat uneven. If you want to learn about how some particular technology came into existence, there’s often very good resources available. Most major inventions, and many not-so-major ones, have a decent book written about them. Some of my favorites are Crystal Fire (about the invention of the transistor), Copies in Seconds (about the early history of Xerox), and High-Speed Dreams (about early efforts to build a supersonic airliner).

But if you’re trying to understand the nature of technological progress more generally, the range of good options narrows significantly. There’s probably not more than ten or twenty folks who have studied the nature of technological progress itself and whose work I think is worth reading.

One such researcher is Brian Arthur, an economist at the Santa Fe Institute.1 Arthur is the author of an extremely good book about the nature of technology (called, appropriately, “The Nature of Technology,”) which I often return to. He’s also the co-author, along with Wolfgang Polak, of an interesting 2006 paper, “The Evolution of Technology within a Simple Computer Model,” that I think is worth highlighting. In this paper Arthur evolves various boolean logic circuits (circuits that take ones and zeroes as inputs and give ones and zeroes as outputs) by starting with simple building blocks and gradually building up more and more complex functions (such as a circuit that can add two eight-bit numbers together).

Logic circuits invented by Arthur’s simulation.

I wanted to highlight this paper because I think it sheds some light on the nature of technological progress, but also because the paper does a somewhat poor job of articulating the most important takeaways. Some of what the paper focuses on — like the mechanics of how one technology gets replaced by a superior technology — I don’t actually think are particularly illuminating. By contrast, what I think is the most important aspect of the paper — how creating some new technology requires successfully navigating enormous search spaces — is only touched on vaguely and obliquely. But with a little additional work, we can flesh out and strengthen some of these ideas. And when we look a little closer, we find what the paper is really showing us is that finding some new technology is a question of efficiently acquiring information.

Outline of the paper

The basic design of the experiment is simple: run a simulation that randomly generates various boolean logic circuits and analyze the sort of circuits that the simulation generates. Boolean logic circuits are collections of various functions (such as AND, OR, NOT, EQUAL) that perform some particular operation on binary numbers. The logic circuit below, for instance, determines whether two 4-bit numbers are equal using four exclusive nor (XNOR) gates, which output a 1 if both inputs are identical, and a 4-way AND gate, which outputs a 1 if all inputs are 1. Boolean logic circuits are important because they’re how computers are built: a modern computer does its computation by way of billions and billions of transistors arranged in various logic circuits.

The simulation works by starting with three basic circuit elements that can be included in the randomly generated circuits: the Not And (NAND) gate (which outputs 0 if both inputs are 1, and 1 otherwise), and two CONST elements which always output either 1 or 0. The NAND gate is particularly important because NAND is functionally complete; any boolean logic circuit can be built through the proper arrangement of NAND gates.

Using these starting elements, the simulation tries to build up towards higher-level logical functions. Some of these goals, such as creating the OR, AND, and exclusive-or (XOR) functions, are simple, and can be completed with just a few starting elements. Others are extremely complex, and require dozens of starting elements to implement: an 8-bit adder, for instance, requires 68 properly arranged NAND gates.

To achieve these goals, during each iteration the simulation randomly combines several circuit elements — which at the beginning are just NAND, one, and zero. It randomly selects between two and 12 components, wires them together randomly, and looks to see if the outputs of the resulting circuit achieve any of its goals. If it has — if, by chance, the random combination of elements has created an AND function, or an XOR function, or any of its other goals — that goal is marked as fulfilled, and circuit that fulfills it gets “encapsulated,” added to the pool of possible circuit elements. Once the simulation finds an arrangement of NAND components that produces AND and OR, for instance, those AND and OR arrangements get added to the pool of circuit elements with NAND and the two CONSTS. Future iterations thus might accidentally stumble across XOR by combining AND, OR, and NAND.

An XOR gate made from a NAND, an OR, and an AND gate.

Because finding an exact match for a given goal might be hard, especially as goals get more complex, the simulation will also add a given circuit to the pool of usable components if it partially fulfills a goal, as long as it does a better job of meeting that goal than any existing circuit. Circuits that partially meet some goal (such as a 4-bit adder that gets just the last digit wrong) are similarly used as components that can be recombined with other elements. So the simulation might try wiring up our partly-correct 4-bit adder with other elements (NAND, OR, etc.) to see what it gets; maybe it finds another mini-circuit that can correct that last digit.

Over time, the pool of circuit elements that the simulation randomly draws from grows larger and larger, filled both with circuits that completely satisfy various goals and some partly-working circuits. A circuit can also get added to the pool if it’s less expensive — uses fewer components — than existing circuits for that goal. So if the simulation has a 2-bit adder made from 10 components, but stumbles across a 2-bit adder made from 8 components, the 8-component adder will replace the 10 component one.

When the simulation is run, it begins randomly combining components, which at the beginning are just NAND, one, and zero. At first only simple goals are fulfilled: OR, AND, NOT, etc. The circuits that meet these goals then become building blocks for more complex goals. Once a 4-way AND gate is found (which outputs 1 only if all its inputs are 1), that can be used to build a 5-way AND gate, which in turn can be used to build a 6-way AND gate. Over several hundred thousand iterations, surprisingly complex circuits can be generated: circuits which compare whether two 4-bit numbers are equal, circuits which add two 8-bit numbers together, and so on.

However, if the simpler goals aren’t met first, the simulation won’t find solutions to the more complex goals. If you remove a full-adder from the list of goals, the simulation will never find the more complex 2-bit adder. Per Arthur, this demonstrates the importance of using simpler technologies as “stepping stones” to more complex ones, and how technologies consist of hierarchical arrangements of sub-technologies (which is a major focus of his book).

We find that our artificial system can create complicated technologies (circuits), but only by first creating simpler ones as building blocks. Our results mirror Lenski et al.’s: that complex features can be created in biological evolution only if simpler functions are first favored and act as stepping stones.

Analyzing this paper

I don’t have access to the original simulation that Arthur ran, but thanks to modern AI tools it was relatively easy for me to recreate it and replicate many of these results. Running it for a million iterations, I was able to build up to several complex goals: 6-bit equal, a full-adder (which adds 3 1-bit inputs together), 7-bitwise-XOR, and even a 15-way AND circuit.

But I also found that not all of the simulation design elements from the original paper are load-bearing, at least in my recreated version. In particular, much of the simulation is devoted to the complex “partial fulfillment” mechanic, which adds circuits that only partially meet goals, and gradually replaces them as circuits that better meet those goals are found. The intent of this mechanic, I think, is to make it possible to gradually converge on a goal by building off of partly-working technologies, which is how real-world technologies come about. However, when I turn this mechanic off, forcing the simulation to discard any circuit that doesn’t 100% fulfill some goal, I get no real difference in how many goals get found: the partial fulfillment mechanic basically adds nothing (though this could be due to differences in how the simulations were implemented).

To me the most interesting aspect of this paper isn’t showing how new, better technologies supersede earlier ones, but how the search for a new technology requires navigating enormous search spaces. Finding complex functions like an 8-bit adder or a 6-bit equal requires successfully finding working functions amidst a vast ocean of non-working ones. Let me show you what I mean.

We can define a particular boolean logic function with a truth table – an enumeration of every possible combination of inputs and outputs. The truth table for an AND function, for instance, which outputs a 1 if both inputs are 1 and 0 otherwise, looks like this:

Every logic function will have a unique truth table, and for a given number of inputs and outputs there are only so many possible logic functions, so many possible truth tables. For instance, there are only four possible 1 input, 1 output functions.

However, the space of possible logic functions gets very very large, very very quickly. For a function with n inputs and m outputs, the number of possible truth tables is (2^m)^(2^n). So if you have 2 inputs and 1 output, there are 2^4 = 16 possible functions (AND, NAND, OR, NOR, XOR, XNOR, and 10 others). If you have 3 inputs and 2 outputs, that rises to 4^8 = 65,536 possible logic functions. If you have 16 inputs and 9 outputs, like an 8-bit adder does, you have a mind-boggling 10^177554 possible logic functions. By comparison, the number of atoms in the universe is estimated to be on the order of 10^80, and the number of milliseconds since the big bang is on the order of 4x10^20. Fulfilling some goal from circuit space means finding one particular function in a gargantuan sea of possibilities.

The question is, how is the simulation able to navigate this enormous search space? Arthur touches on the answer — proceeding to complex goals by way of simpler goals — but he doesn’t really look deeply at the combinatorics in the paper, or how this navigation happens specifically.2

The emergence of circuits such as 8-bit adders seems not difficult. But consider the combinatorics. If a component has n inputs and m outputs there are (2^m)^(2^n) possible phenotypes, each of which could be realized in a practical way by a large number of different circuits. For example, an 8-bit adder is one of over 10^177,554 phenotypes with 16 inputs and 9 outputs. The likelihood of such a circuit being discovered by random combinations in 250,000 steps is negligible. Our experiment— or algorithm—arrives at complicated circuits by first satisfying simpler needs and using the results as building blocks to bootstrap its way to satisfy more complex ones.

Navigating large search spaces

In his 1962 paper “The Architecture of Complexity,” Nobel Prize-winning economist Herbert Simon describes two hypothetical watchmakers, Hora and Tempus. Each makes watches with 1000 parts in them, and assembles them one part at a time. Tempus’ watches are built in such a way that if the watchmaker gets interrupted — if he has to put down the watch to, say, answer the phone — the assembly falls apart, and he has to start all over. Hora’s watches, on the other hand, are made from stable subassemblies. Ten parts get put together to form a level 1 assembly, ten level 1 assemblies get put together to form a level 2 assembly, and 10 level 2 assemblies get put together to form the final watch. If Hora is interrupted in the middle of a subassembly, it falls to pieces just like Tempus’ watches, but once a subassembly is complete it’s stable; he can put it down and move on to the next assembly.

It’s easy to see that Tempus will make far fewer watches than Hora. If both have a 1% chance of getting interrupted each time they put in a part, Tempus only has a 0.99 ^ 1000 = 0.0043% chance of assembling a completed watch; the vast majority of the time, the entire watch falls to pieces before he can finish. But when Hora gets interrupted, he doesn’t have to start completely over, just from the last stable subassembly. The result is that Hora makes completed watches about 4,000 times faster than Tempus.

Simon uses this model to illustrate how complex biological systems might have evolved; if a biological system is some assemblage of chemicals, it’s much more likely for those chemicals to come together by chance if some small subset of them can form a stable subassembly. But we can also use the Tempus/Hora model to describe the technological evolution being simulated in Arthur’s paper.

Consider a technology as some particular arrangement of 1,000 different parts, such as the NAND gates that are the basic building blocks of Arthur’s logic circuits. If you can find the proper arrangement of parts, you can build a working technology. Assume we try to build a technology by adding one part at a time, like Tempus and Hora build their watches, until all 1000 parts have been added. In this version, instead of having some small probability of being interrupted and needing to start over, we have a small probability (say 1%) of correctly guessing the next component. This mirrors Arthur’s simulation, where we had a small probability of randomly connecting a component correctly to fulfill some goal. Only by properly guessing the arrangement of each part, in order, can we create a working technology.

In Simon’s original model, assembling a watch was like flipping 1000 biased coins in a row. Each coin had a 99% chance of coming up heads, and only when 1000 heads were flipped was a watch successfully assembled. Our modified model is like flipping 1000 biased coins which have only a 1% chance of coming up heads. Creating a technology via the “Tempus” method is like flipping 1000 coins in a row and hoping for heads each time. The probability of producing a working technology is 0.01^1000, essentially zero. But if we create a technology via the “Hora” method of building it out of stable subassemblies, the combinatorics become much less punishing. Now instead of needing to flip 1000 heads in a row, we only need to flip 10 in a row. 10 successful coinflips — 10 parts successfully added — gives us a stable subassembly, letting us essentially “save our place.” Flipping a tails doesn’t send us all the way back to zero, just to the last stable subassembly. The odds are still low — for each subassembly, you only have a 0.01^10 chance of getting it right — but it’s enormously higher than the Tempus design. You’re much more likely to stumble across a working technology if that technology is composed of simpler stable components, and you can determine whether the individual components are correct.

Arthur’s circuit simulation is able to find complex technologies because it works like Hora, not Tempus: complex circuits are built up from simpler technologies, the way Hora’s watches are built from stable subassemblies. Going from nothing to an 8-bit adder is like Tempus trying to build an entire and very complex watch by getting every step perfect. Much easier to be like Hora, and be the one that only needs to get the next few steps to a stable subassembly correct: adding a few components to a 6-bit adder to get a 7-bit adder, then adding a few to that one to get an 8-bit adder, and so on.

We can illustrate this more clearly with a modified version of Arthur’s circuit search. In this version, rather than trying to fulfill a huge collection of goals, we’re just trying to find the design for a specific 8-bit adder made from 68 NAND gates. Rather than build this up from simpler sub-components (7-bit adders, 6-bit adders, full adders), in this simulation we simply go NAND gate by NAND gate. Each iteration we add a NAND gate, and randomly wire it to our existing set of NAND gates. If we get the wiring correct, we keep it, and go on to try adding the next gate. If it’s incorrect, we discard it and try again.

We can think of this as a sort of modular construction, akin to building a complex circuit up from simpler circuits; at each level, we’re just combining two components, our existing subassembly and one additional NAND gate. This loses verisimilitude, since each subassembly no longer implements some particular functionality (we essentially just dictate that the simulation knows when it stumbles upon the correct gate wiring). But we don’t lose that much: it is, notably, possible to build an 8-bit adder with a hierarchy that requires just two components at almost every level (a few steps require three components). And this simpler simulation has the benefit of making it very easy to calculate the combinatorics at each step.

Hierarchical 8-bit adder. FA is a full-adder (which adds 3 input values together), HA is a half-adder which adds 2 input values together). Full decomposition down to NAND gates not shown.

68 NAND gates can create around 2^852 possible wiring arrangements with 16 inputs and 9 outputs. This is much less than the 10^177,554 possibile 16 input and 9 output functions, but it’s still an outrageously enormous number. If we tried to find the right wiring arrangement by random guessing all 68 gates at once, we’d never succeed: even if every atom in the universe was a computer, each one trying a trillion guesses a second, we’d still be guessing for about 10,000,000,000,000…(140 more zeroes)..000 years.

But by going gate-by-gate, the correct arrangement can be found in 453,000 iterations, on average. Each time we add a gate, there’s only a few thousand possible ways that it can be connected, so after a few thousand iterations we guess it correctly, lock the answer in, and move on to the next gate. By determining whether each step is correct, instead of trying to guess the complete answer all at once, the search becomes feasible.

This is why Arthur’s original simulation couldn’t fulfill complex needs without fulfilling simpler needs first: if you try to take too many steps at once, the combinatorics become too punishing, and it becomes almost impossible to find the correct answer by random guessing. In our 68 NAND gate search, finding an 8-bit adder is relatively easy if we go one gate at a time, but if we change that to two gates at a time (randomly adding one gate, then another gate, then checking to see if we’re correct), the expected number of iterations rises from 453,000 to 1.75 billion: if the probability of guessing one gate correctly is 1/1,000, the probability of guessing two gates is 1/1,000,000. If we try to guess three gates at a time (1 in a billion odds of guessing correctly), the number of expected iterations to guess all 68 gates correctly rises to ~9.3 trillion.

The explosive combinatorics gives us a better understanding of some of the results that come out of Arthur’s simulation. For instance, Arthur notes that in each iteration the simulation combines up to twelve components, then checks to see if a working circuit has been found. But Arthur notes that you can vary the maximum value and it doesn’t impact the results of the simulation much, stating of the various simulation settings that “[e]xtensive experiments with different settings have shown that our results are not particularly sensitive to the choice of these parameters.” Indeed, if we re-run the simulation and only allow it to try a maximum of 4 components at once, it works basically just as well as with 12 components. The more random components you combine together, the more the combinatorial possibilities explode, and the lower the chance of finding something useful. The probabilities of finding a useful circuit amongst the various possibilities becomes so immensely low with larger numbers of components that you don’t lose much by not bothering with them at all. Similarly, this also explains another result in the paper, that it’s easier to find complex goals if you specify only a narrower subset of simpler goals related to them. Arthur notes that a complex 8-bit adder is found much more quickly if you only give the simulation a few goals related to building adders. With fewer goals specified, the pool of possible technology components will remain smaller, the number of possible random combinations becomes fewer, and the easier it becomes to find the complex goals.

In essence, using simpler components as stepping stones to more complex ones is a kind of hill-climbing. The simulation looks in various directions (possible combinations of building blocks), until it finds one that’s higher up the hill (finds a circuit that meets some simple goal), restarts the search at the new, higher point on the hill, until it reaches a peak (satisfies a complex goal). The simulation is able to satisfy complex goals because it specified a series of simpler ones that provide a path up the hill to the complex goals. Arthur notes that “[t]he algorithm works best in spaces where needs are ordered (achievable by repetitive pattern), so that complexity can bootstrap itself by exploiting regularities in constructing complicated objects from simpler ones.”

Trying to go to complex circuits directly, then, is akin to just testing random locations in the landscape and seeing if they’re a high point: this is obviously much worse than following the slope of the landscape to find the high points.

Technological search and information

We can sharpen these ideas even further by bringing in some concepts from information theory. Information theory was invented by Claude Shannon at Bell Labs in the late 1940s, and it provides a framework for quantifying your uncertainty, and how much a given event reduces that uncertainty.

I find the easiest way to understand information theory is with binary numbers. The normal math we use day to day uses base 10 numbers. When we count upward from zero, we go from 0 to 9, then reset the first digit to 0 and increment the next digit: 10. With binary, or base 2, we increment the next digit after we get to 1. So 1 in base 10 is 1 in binary, but 2 is 10, 3 is 11, 4 is 100, and so on.

Decimal (base 10) and binary (base 2) numbers.

In binary, each binary digit, or bit, doubles the potential size of the number we can represent. So with two digits, we can define 4 possible values (0, 1, 2, and 3 in base 10). With 3 digits, that doubles to 8 possible values (getting us from 0 through 7), with 4 digits that doubles again to 16 possible values, and so on. A 16 bit binary number can represent 2^16 = 65,536 possible values, which is why in computer programming the largest value that a 16 bit integer can represent is 65,535.

Say you have a string of bits, but don’t know whether they’re ones or zeroes. Because each bit doubles the number of possible values that can be represented, each unknown bit you fill in reduces the number of possible values by half. If you have 3 binary digits, there are 8 possible numbers that could be represented. Each time you learn what one of the bits is, you reduce the number of possible values by half.

With information theory, we generalize this concept somewhat. In information theory one bit of information reduces the space of possibilities by 50%; in other words, each bit reduces our uncertainty by half. Say you’re like me, and you often lose your phone in your jacket pockets. If you’re wearing a coat with 2 pockets and you know the phone is in one of them, specifying the location of your phone, narrowing it down from 2 possibilities to 1, takes one bit of information. If you’re wearing a coat with 4 pockets, you now need 2 bits of information: 1 bit to tell you whether it’s on the right or the left, and another bit to tell you whether it’s an upper or lower pocket. The first bit cuts the possibilities in half, leaving you with two possibilities, and the second bit cuts it in half again. If your jacket has 8 pockets, now you need 3 bits to specify its location, and so on. The more places that something could be, the more information it takes to specify its location.

Information theory is particularly useful for quantifying how much information we get from some particular outcome. Say someone flips a fair coin; how much information do I get when they reveal whether it was heads or tails? Well, before they reveal it, I knew it could be one of two options, heads or tails. Revealing it narrows the number of possibilities from two down to one. We’ve cut the number of possibilities in half, and thus gained 1 bit of information. More generally, the information provided by some outcome is equal to -log2(the probability of that outcome). So revealing how a fair coin was flipped gives us -log2(0.5) = 1 bit. If we’re dealt a single card from a deck face down, when we reveal that card we’ve reduced the number of possible cards from 52 down to 1, and gained -log2(1/52) = 5.7 bits of information.

For a repetitive process, we also want to know a related quantity: entropy. Entropy is determined by calculating the information received from each possible outcome, multiplying it by the probability of that outcome, then summing all those values together. It’s the expected quantity of information you’ll get by taking some particular action.

Say I’ve lost my phone in my jacket with eight pockets, and am looking for it by randomly trying pockets until I find it. A random guess has a 1/8 chance of successfully finding the phone, and a 7/8 chance of coming up empty. Guessing correctly will yield me -log2(1/8) = 3 bits of information, as expected: once I guess correctly, I know the phone’s location. But an incorrect guess will yield me only 0.19 bits of information: I already knew most of the pockets don’t have the phone, so failing to find the phone in one pocket doesn’t tell me much that I didn’t already know. The entropy of a guess is (1/8) * log2(1/8) + (7/8) * log2(7/8) = 0.54 bits. When I first check a pocket, I can expect to get a little more than half a bit of information. (If I rule out pockets that I’ve already checked, the expected amount of information I get will rise each time, though if you’re like me you might have to check the same pocket a few times before you find the phone.)

Because each bit of information we get cuts the number of possibilities remaining by 50%, it doesn’t take that much information to narrow down an enormously large search space. The 2^852 possible circuits that can be created by wiring up our 68 NAND gates requires only 852 bits — 852 times cutting the number of possibilities in half — to specify. (That’s approximately the same number of bits that it takes to specify each letter of this sentence.)

A key aspect of entropy is that we maximize how much information we get when each outcome is equally plausible. So the entropy of a fair coin, with a 50% chance of coming up heads, is 1 bit. But if the coin has a 90% chance of coming up heads, the entropy is now just 0.46 bits. If the coin has a 99% chance of coming up heads, the entropy falls to 0.08 bits. When one outcome is very likely, you learn much less on each attempt, because you mostly get the outcome you already knew was likely. This is why when playing the game “20 questions,” the most efficient strategy is to try and ask questions where the answer divides the number of possibilities in half. “Is it bigger than a breadbox?” is a good starting question because there are probably roughly similar numbers of items that are bigger and smaller than a breadbox. “Is it a 1997 Nissan Sentra?” is a bad starting question because most possibilities are not a 1997 Nissan Sentra, so we learn very little when the answer is “no.”

We can think of our 68 NAND gate search as flipping a series of very biased coins, each one with a ~ 1/(several thousand) probability of coming up heads (where “heads” is “guessing the right wiring combination for that particular NAND gate”). The entropy of this process — the expected amount of information that we get — is very low, around 0.003 bits per attempt. Each attempt we learn very little about the correct wiring diagram (“it wasn’t this arrangement, it wasn’t this arrangement either, or this one”) so we need a lot of attempts — around 453,000, on average — to accumulate the 852 bits needed to specify the correct wiring for our 8-bit adder.3

Trying to guess two gates at a time is like biasing the coin even further: now each one has a ~1/(several million) probability of coming up heads. We thus get vastly less information per attempt — less than 0.000001 bits per attempt on average — so it takes us many, many more attempts to accumulate the information needed.

A useful way of thinking about our 68 NAND gate search is that it’s like a huge, branching tree. At every step — every time we add a gate — there are thousands of branches, each one representing one possible way to wire up the gate. Each branch then splits into thousands more (representing all the possible ways of wiring up the next gate), which split into thousands more, which split into thousands more, until at the end we have 2^852 possible “leaves,” each one representing a unique way of wiring up all 68 gates. Trying to get all 68 gates right at once, and then checking to see whether or not you did, is like examining one single leaf, one path from the base of the tree all the way to the tip of a branch. Not only are you overwhelmingly likely to guess wrong, but you haven’t narrowed down your possibilities at all: all you know is that one single leaf wasn’t the right answer, leaving you with the rest of the 2^852 possible leaves to sift through.

Checking to see whether each gate we add is correct before we proceed to the next one, by contrast, massively narrows down the number of possibilities. Whenever we determine a gate isn’t in the right spot, it eliminates every possibility that branches off from that point. If there are 1000 possible ways to wire each gate, each time we guess correctly we’ve narrowed down the possibilities downstream of that choicepoint by 99.9%. Huge swaths of possibilities get eliminated at each correct guess, letting us converge on the correct answer much more quickly.

The same basic logic applies to Arthur’s simulation. (In fact, in another publication, Arthur uses a very similar metaphor, describing technological search as trying to find a working path up a mountain, which is full of various obstacles.) Building up complex functions without the aid of intermediate, simpler ones is like trying to find a single leaf on a tree the size of the universe. Building up to complex circuits gradually, using simpler components as building blocks, lets you screen off huge branches of the tree at once. Once you have a working 2-bit adder, every branch that has a non-working 2-bit adder in it gets screened off. Your iterations yield massively more information, and the search problem becomes tractable.

Conclusion

The logic of Arthur’s simulation, and our simpler simulation, also applies to creating new technologies more generally. Logic circuits are a useful model to explore, because they’re real technology that is very amenable to simulation (they have a well-defined, simple behavior), but technology in general can be thought of as a combination of simpler components or elements arranged in various ways to create more complex ones. As Arthur notes:

…in 1912 the amplifier circuit was constructed from the already existing triode vacuum tube in combination with other existing circuit components. The amplifier in turn made possible the oscillator (which could generate pure sine waves), and these with other components made possible the heterodyne mixer (which could shift signals’ frequencies). These two components in combination with other standard ones went on to make possible continuouswave radio transmitters and receivers. And these in conjunction with still other elements made possible radio broadcasting. In its collective sense, technology forms a network of elements in which novel elements are continually constructed from existing ones. Over time, this set bootstraps itself by combining simple elements to construct more complicated ones and by using few building-block elements to create many.

One takeaway from this paper, as Arthur notes and we explored more deeply, is that a hierarchical arrangement of components, where a complex technology is made of simpler components, which are in turn made from even simpler components, makes it much easier to create some new technology. But a more general takeaway is that successfully creating some new technology means getting new information as quickly as possible. Working from (or towards) a hierarchical, modular design for some technology, where each element has some specific job it must do, makes it easier to find new technologies in part because you learn vastly more from each attempt at building one of those subparts. Knowing whether some entire complex function works or not tells you much less than knowing which individual component is working right, and what specific functionality needs to be corrected to fix the problem.

In addition to Brian Arthur, some other folks who I think have done really good work on this are Bernard Carlson, Clay Christensen, Joel Mokyr, Hugh Aitken, Edward Constant, and various folks associated with the Trancik Lab. There’s also a few folks, such as Joan Bromberg and Lillian Hoddeson, who have produced multiple very good technological histories that I return to often.

Indeed, we find that if we just randomly combine dozens of NAND gates, we get a random truth table almost every time, and never solve even medium-complex functions with a few inputs and outputs.

Adding things up, you find that the search actually yields over 900 bits, rather than 852 bits. This is due to the information overhead of a sequential search: you end up getting “extra” information that you don’t need. In our 8-pocket jacket search, if we just guess randomly it will take us on average 8 tries to find the phone. 8 attempts * 0.54 bits per attempt yields 4.32 bits, more than the 3 bits we need to actually specify the phone’s location.