The atomic human, p.16
The Atomic Human, page 16
In October 1911, just as the Principia was finished, one of Bertrand Russell’s tea parties was interrupted by an Austrian engineer. He turned up unannounced to Russell’s rooms at Trinity, carried along by the wave of interest in logic. He declared his desire to change his field of study and to work on mathematical logic with Russell.
The would-be student attended Russell’s lectures and followed him around Cambridge. He even tagged along when Russell headed back to his rooms. He was constantly challenging Russell’s assertions. Russell was initially unsure whether he was a crank or a genius, but after a month he decided on the latter. The engineer’s name was Ludwig Wittgenstein.
Wittgenstein went on to write some of the most important philosophical works of the twentieth century, but the only one that was published in his lifetime was a logical theory of language. In Tractatus Logico-Philosophicus he described an idea that helps explain how logic can be mapped on to the computer. It’s called the truth table. Perhaps the term ‘truth table’ conjures up images of Romany tents and a wizened lady hovering over a crystal ball – the truth table in mathematics is less exotic, but much more useful.
The truth table represents the different possible inputs and outputs to a given logical operation. Let’s assume we have a proposition we’d like to discuss. It could be that ‘Sherlock Holmes is a fictional character.’ We might assert that this is True. To represent our proposition in Boolean algebra, we use a letter. Let’s use the letter P to represent the idea that Holmes is a fictional character.
Logical operations are similar to the arithmetic operations we learn at school. They are equivalent to addition, multiplication, division and subtraction. But they operate on logical predicates. For example, the logical operation Not reverses an assertion. So, if we say P is True, then Not P would be False. This logical operation feels like negation in arithmetic. The negation operation in arithmetic would turn P to −P and, just like negation, if you apply it twice, you recover the original answer. So −−P = P and Not Not P = P.
Wittgenstein’s idea was to represent these logical operations in a table. So for this operation on P, in the left-hand column is the original input value of P; in the right is the output value after modification with Not:
Input
Output
P
Not P
False
True
True
False
The Not operation is the most basic operation of Boolean algebra. The truth table gives us the output value of the operation Not for all the possible inputs. With this simple truth table we seem to have gone to a lot of effort for very little gain, but the Boolean algebra also has more complex operations. One of these is called And, which takes two inputs. Its output is True if both the inputs are True. So, if we think of proposition P as ‘Sherlock Holmes is a fictional character’ and then we think of proposition Q as ‘Dr Watson is a fictional character’, we can define P And Q in a truth table:
Input
Output
P
Q
P And Q
False
False
False
False
True
False
True
False
False
True
True
True
If both ‘Sherlock Holmes is a fictional character’ and ‘Dr Watson is a fictional character’ are True, then the output of P And Q is True. The truth table lists in two columns on the left all the possible combinations of inputs for P and Q. The right-hand column indicates the output for each combination.
OK, so the And operation may not seem a lot more complex than the Not operation, and it still feels like a long-winded way of capturing the very natural concept of the word ‘and’. But the beauty of algebras arises when we start to bring in lots of different rules and the relationships become hard to remember in our heads. In mathematics this is known as composition. Mathematical composition is the process of feeding the output of one mathematical function into the input of another. It’s the opposite of Babbage’s process of decomposing a complex task into simpler components. When Babbage was reviewing Victorian approaches to manufacturing, he suggested that division of labour should separate the manufacturing process into simpler tasks, each of which could be done by a specialist. In computer software, the same approach leads to separation of concerns. A more complex system is separated into simpler parts, and each part of the system has a role that is easier to understand. This is how Bletchley Park decrypted German codes and how Facebook built its social network. Mathematical composition is the opposite process: in mathematical composition we bring together the simpler components to form a more complex whole. This is how Shannon modelled a telephone exchange with logic, and also how Hodgkin and Huxley modelled the squid’s giant axon. They composed simpler mathematical operations to form a more complex system.
You can think of composition as feeding the output of one process into the input of another. The Albion fly-ball mechanism is the composition of a fly-ball and a series of linkages with an actuator at the regulator. We can build a mathematical model of each of these components in the form of a function and then look at the system as a whole by considering the composition of those mathematical models: we feed the outputs of one function into the inputs of another.
We can see composition occurring across the systems we’ve looked at so far. In the Supreme Allied Command, Eisenhower’s orders were received by his subordinates as inputs, combined with those subordinates’ domains of expertise, then relayed through the information topography, eventually reaching my grandfather in his dressing gown and pyjamas. In Bletchley Park, the Enigma coded messages went through different processes: first they were transliterated from the radio signal, then cross-referenced for possible ‘cribs’, and then processed on the bombes to decrypt the message. Processes were being composed to form the final outcome.
We already have two operations for our truth tables. So, let’s try composing them. We will take the output of the And operation and put it into the Not operation. To write this mathematically we need to invent a notation. In mathematics, it’s common to use brackets to compose operations. So, we will write Not (P) and And (P,Q) to represent our operations. Now, if want to apply Not after applying And, we can write Not (And (P,Q)). This gives us a new operation, which we call Nand.14 Unlike Not and And, Nand doesn’t have an equivalent word to represent the idea in the English language, but we can use the rules of Boolean algebra to write down its truth table:
Input
Output
P
Q
Not (And (P,Q))
False
False
True
False
True
True
True
False
True
True
True
False
These Boolean operations are all simple mathematical functions. They are so common that when we use them in practice we use symbols to represent them. So, the And operation P And Q is normally written with a little upside down ‘v’, so P ∧ Q, just like ‘multiply’ is normally denoted with a small ×. Not is normally written like a minus sign with a bent end, ¬.
When explaining the Enigma machine in Bletchley Park I mentioned how it could be seen as implementing a mathematical function. Tommy Flowers was given a job that was the other way around: to build a machine that would implement a mathematical function. He used electronic valves to implement the different operations, and that was the basis of the electronic computer. When we implement these logical operations in computers, we call them logic gates. The gate the team wanted Tommy Flowers to build for the Heath Robinson machine was called the Xor gate (the Xor gate is an exclusive Or gate. It is like the Or gate, except it outputs a 0 instead of a 1 when the inputs are both 1); the Lorenz cipher also used this gate to encode the German High Command’s messages. The Lorenz machine implemented a Vernam cipher in which the message was initially encoded as a string of 1s and 0s, equivalent to True and False in our Boolean algebra. It was encrypted by taking the key generated by the Lorenz machine and applying a Xor operation to the message alongside the key. The Xor gate gave the encrypted message as the output. Bill Tutte had reverse-engineered how the Lorenz machine was producing that key and in doing so he revealed a weakness in the encryption that Tommy Flowers’s Colossus machine exploited.
But the excitement Turing and others felt about Colossus wasn’t just down to this Xor gate. Colossus was also programmable. Programming in Colossus consisted of connecting a set of logic gates together to form a composition. Modern computers store their programs electronically, but programming Colossus was done by sticking together the little electronic machines Flowers had built to represent the logic gates. Flowers provided switches to enable the programmers to change the connections in the machine. He commented on the result in an interview with Brian Randell, a professor of computer science from Newcastle:
It just changed the whole picture… they wanted programmable logic – and we provided them with a big panel with a lot of keys on it and by throwing the keys they could – the mathematicians – could program the machine – the keys did the ‘and’ and ‘or’ functions – and we didn’t do any multiplying; we didn’t need a multiply but we added ‘and’ and ‘or’ functions and put them in series and parallel and so forth, and they were quite happy. In fact they were like a lot of schoolboys with a new toy when we first gave it to them; they thought it so wonderful they were playing with it for ages just to see what you could do with it.15
The Colossus machine allowed Turing, Good and the others to program a composition of logic gates. That is what makes it the first electronic computer. The logic gates were implemented in valves, but they could be composed together, just like one neuron feeds into another, or like the different processes flowing together in an assembly line.
There are three magical things about Boolean algebra. The first is the composition idea. It turns out that by composing different gates we can create any truth table you can imagine. So, in other words, for any inputs on the truth table, we can construct logical gates that will produce any set of outputs we choose. We merely have to choose the right composition of the gates. We need to decide which gates should feed into which others.
The second magical thing about Boolean algebra is what happens when we feed back signals. Norbert Wiener believed that feedback was at the heart of intelligence, naming the entire field of cybernetics after Watt’s governor. He was inspired by its innovation of feeding back the speed of the engine to the regulator. What happens if you feed back the output of a logic gate to its input? If you do it right, you obtain memory. Let’s just repeat that, again with emphasis: memory. At which point were we going to mention memory? Shouldn’t memory be at the heart of intelligence? Aren’t many intelligence tests based on the ability to remember? Is Braitenberg’s squid, with its simple nervous system, really intelligent if it can’t remember things? Is Shannon’s communications network an intelligence if it cannot remember? Is the Colossus machine really intelligent without this ability?
Our notion of directed intelligence didn’t say anything about memory being a necessary condition for intelligence. It only said that you would achieve your goal more efficiently through information. But we’ve seen the influence of the printed word on the propagation of ideas. Writing allows words to be stored, and printing allows those words to be rapidly reproduced and shared. DNA allows the instructions that build our cells to be stored. Memory is the storage of information, and it is a way of communicating across time, a way of passing information from the past to the future.
Most of our perspective on the atomic human has been driven by our ability to communicate, by the way we’ve moved information from one place to another. Memory is a special kind of communication. Its ability to communicate with the past means that anything we place in memory is something we are choosing to share with the future, regardless of whether it’s a filing system for storing German cribs or DNA’s system of storing protein instructions as a chemical string. If we have access to memory, we can review the state of the world as it was rather than merely as it is. This can be of tremendous value, particularly if you’re trying to find your car keys.
A simple feedback system involving two Nand gates is enough to provide memory. It’s endearingly called a flip-flop. The flip-flop combines two features we’ve already explored: cross wiring, like we saw in one variant of Braitenberg’s vehicles, and feedback, like we saw in Watt’s governor.
In the flip-flop, the output of each Nand gate is fed back to one input of the other. This causes the gate to remember inputs. You can no longer predict the output of this system with a simple truth table. The output of the system will depend on what it’s seen in the past. The flip-flop has the ability to communicate with the past. From a technical perspective, we say that the system has internal state. What we mean by that is it can make decisions not just on the basis of inputs from the external world, like Braitenberg’s squid, but also on the basis of previous experience. So, feedback allows our systems to hold internal state, what we call memory, and that feedback also gives our systems a basic form of self-awareness. Perhaps you’re beginning to understand why Wiener thought feedback was so important.
You might now use your own memory to recall that I suggested that there are three magical things about Boolean algebra and, so far, I’ve only described two of them. The third magical thing is the notion of the universal gate. Composition, memory and now the universal gate. Perhaps the term ‘universal gate’ makes this third thing sound the most exciting of the three, and the universal gate is a very cool concept. But it is not a universal gate that allows us to travel across the universe via wormholes, although that would be pretty cool too. A universal gate in Boolean algebra is one that provides a fundamental form from which all other representations can be developed.
Let’s repeat that in a different way. A universal gate is like the ultimate Lego brick for truth tables. A universal gate is the only logical gate we need to create any truth table we need. If I told you that you can only have access to one logical gate but you can have as many of them as you like, you would be wise to choose a universal gate. Because you can always compose universal gates to form any other gate. With a universal gate, you can create any logical mapping.
The Nand gate is an example of a universal gate. Using just Nand gates, we can combine the three magical ideas of Boolean algebra. We can compose Nand gates to create any truth table we can imagine, and we can also use feedback with Nand gates to create memory. So as long as we have enough Nand gates we can create any logical machine we like, and we can give it memory. These were also the requirements for Turing’s universal computer, the imaginary machine he had conceived of before the war that could compute anything. No wonder Flowers’s creation made the mathematicians at Bletchley Park behave like schoolchildren. He’d just given them a glimpse of the digital future.
The printing press triggered the scientific Enlightenment. The easy sharing of ideas in books led to a revolution in the way we think about the world around us. After Newton’s Principia a new view of our universe based on fundamental scientific truths emerged. It suggested that by understanding all the underlying phenomena by which the universe operates, we can make predictions about those phenomena. This enticing view of intelligence is captured in Laplace’s demon. Echoes of Laplace’s demon are found in the machines used to decrypt German messages at Bletchley Park. Those machines exhaustively explored different solutions to decode German messages, but, when we review decision-making systems we find in the real world, we see simpler systems based on fly-balls, levers, sensors and muscles.
Braitenberg’s book described the behaviours of simply wired vehicles as an exercise in synthetic psychology. He then discussed how evolution or, as we introduced it, selection bias, determines which of those behaviours are sustained. Even the simplest of Braitenberg’s vehicles can exhibit complex behaviour. This causes the problems we experience with Braitenberg’s law: the ‘law of uphill analysis and downhill invention’. Where does this complexity come from in these simply wired entities? The answer recurs when examining the atomic human: it comes from the outside world. The squid we described responded to light in its environment; any complexity in its behaviour emerges from the changing patterns of light in its world. These apparently simpler systems react quickly to the world around them with very little computation involved. As a result, they are not locked in in the manner of our reflective intelligence, and yet systems like this are also an integral part of the atomic human.
The atomic human is a composition of fast-reacting reflexive decisions and slow-reacting reflective decisions. Naturally, we are much more aware of our reflective decisions than the reflexive ones, just as Eisenhower was more aware of his orders than of the actions of an individual soldier like my grandfather. But the Normandy landings were as much about the experiences of individual soldiers as about those of their supreme commander.
Eisenhower’s decision to attack was informed by critical intelligence decrypts from Bletchley Park, but it was also informed by his experience leading amphibious assaults on North Africa and Sicily. After he made his decision, Eisenhower, along with Winston Churchill, went to inspect the troops. In the film of this inspection of the US 101st Airborne Division, Eisenhower is next to Churchill in his formal military attire,16 and together they walk behind the gliders that will carry the troops across the Channel. They shake hands with Brigadier General Don Pratt, who will lead the assault, then inspect the ordered ranks of soldiers. There’s another film in which Eisenhower strides across a temporary fence into a wider camp filled with rows of tents. Eisenhower talks and laughs with troops who are not ready for inspection but are ready for battle. Their faces are painted and their uniforms augmented with camouflage. Eisenhower walks casually among the men, and their smiles and casual stances are reflected in his smiles, his nods and his easy manner. Because Eisenhower’s decision was informed by the fact that he was a human, in that inspection we see him sharing his humanity with his troops. Afterwards he headed home and wrote the following note:
