Machine as a projection of the human self

Faciamus machinam ad imaginem et similitudinem nostram.

Let us make the machine in our image and likeness.

Before machines became motors, engines and computers, the word already meant something broader:

an ingenious means by which something becomes possible.

Greek mythology gives us an extraordinary laboratory of artificial beings.

Athena represents intelligent making.

Hephaestus represents the workshop in which artifacts begin to act.

Artificial beings are not merely tools.

They become surfaces onto which humans project desire, identity and self-understanding.

Some artificial beings do not imitate us primarily as workers or companions.

They stand between inside and outside.

They decide who may approach, enter or survive.

A machine can do more than move.

It can contain knowledge unavailable to its operators.

The mythical Argo gives us an extraordinary example.

The Golem introduces another extraordinary structure:

matter + symbolic information → agent

Human — Can we manufacture ourselves?

Servant — Can we delegate labor?

Prosthesis — Can we extend ourselves?

Lover — Can we manufacture intimacy?

Protector — Can we delegate protection?

Weapon — Can artificial agency project violence?

Guardian — Can it decide who may pass?

Oracle — Can it know what we do not?

Mirror — What does it reveal about us?

Monster — What if it escapes our intentions?

We first imagined machines as versions of ourselves.

Then we built machines that extended parts of ourselves.

Eventually we built machines capable of modifying themselves through experience.

The artificial being is ancient.

The learning machine is the historical novelty.

“All is number.” (πάντα αριθμός) — attributed to Pythagoras, 6th century BCE.

For Pythagoras and his followers, numbers were not just a tool for counting. They were the hidden structure of reality itself — the ratios that govern music, geometry, even the harmony of the cosmos.

Today we trace that claim all the way to the silicon running inside your phone.

Here is the thread we will follow: a number can be written down in many different ways — and the way you write it changes what you can do with it.

That was true for Roman numerals. It is just as true for the floating-point formats inside a modern AI chip.

Rome counted like this: I, V, X, L, C, D, M.

No zero. No place value — the position of a symbol barely matters, only which symbols appear and in what combination.

Try multiplying MDCLXVI by XLIV in your head. Now try 1666 × 44. The second one is not just easier — it is a genuinely different kind of thinking.

Centuries before Pythagoras, the Baudhayana Sulba Sutra (India, c. 800 BCE) already states the same relationship between the sides of a right triangle.

The mathematics was not a single European discovery — it was discovered, independently and repeatedly, wherever people needed to measure, build, and trade.

Every numeral system so far shares the same gap: no symbol for nothing.

Without a zero, there is no clean way to write a number as a position-dependent code — and without position, arithmetic stays a craft for specialists, not a procedure anyone can follow.

Zero emerges independently more than once: among the Maya of Mesoamerica, and in India, where the mathematician Brahmagupta (7th century CE) is the first to give it formal rules of arithmetic.

It reaches the Islamic world through scholars like Muhammad ibn Musa al-Khwarizmi (9th century), whose own name — Latinized as algorismus — gives us the word algorithm.

A number system and a method for calculating with it arrive together. That is not a coincidence.

In 1202, Leonardo of Pisa — better known as Fibonacci — publishes Liber Abaci, introducing Hindu-Arabic numerals (zero included) to European merchants.

The pitch was practical: this notation makes bookkeeping and trade dramatically easier than the Roman system merchants were already using.

Here is the real leap: the same ten symbols, re-used, mean something different depending on where they sit.

106 and 160 use identical digits. Position alone carries the meaning.

This single idea — a small, finite alphabet whose meaning depends on position — is the one that modern computing was built on top of.

Gottfried Wilhelm Leibniz takes positional notation somewhere further: in 1703, he formally describes arithmetic in base 2 — binary.

He was reportedly delighted to find the same two-symbol logic echoed in the hexagrams of the I Ching, and saw in binary something close to a theological image: everything created from nothing (0) and unity (1).

Three centuries later, his once-mystical number system runs every computer on Earth.

Binary is just another positional system — base 2 instead of base 10. Only two symbols: 0 and 1.

That turns out to be the whole point. Building a reliable physical device with ten stable, distinguishable states is hard. Building one with two — voltage on, voltage off — is comparatively easy, cheap, and extremely reliable at enormous scale.

Every transistor in every chip is, at heart, a two-state switch pretending to be one of Leibniz's digits.

George Boole (1854) notices that 0 and 1 can represent more than quantity — they can represent truth: false and true.

Boolean algebra — AND, OR, NOT — becomes the logical skeleton underneath every circuit in every digital machine.

The same two symbols now carry both arithmetic and logic. That overlap is why a computer can calculate and also decide.

One binary digit is a bit. Group eight of them and you get a byte — 256 possible values.

Hexadecimal (base 16) exists purely as a convenience: one hex digit covers exactly 4 bits, so two hex digits describe a whole byte cleanly — far easier for a human to read than a string of eight zeros and ones.

You already used one today, probably without noticing: a colour code, a MAC address, a cryptographic key — all hexadecimal shorthand for binary underneath.

So far: how a culture chooses to write a number. Now: how a machine chooses to store one — and why, in a modern AI system, that choice is no longer a historical curiosity.

It is an engineering decision with real, measurable consequences.

Integer formats store whole numbers exactly. int8: one byte, 256 distinct values — typically -128 to 127.

Floating-point formats trade some of that exactness for enormous range: a sign bit, an exponent, and a mantissa (the significant digits) — the same idea as scientific notation, just in binary.

More bits generally buys you more precision, more range, or both. Fewer bits buys you speed and memory — at a cost.

Both fp16 and bf16 use 16 bits total — half the size of ordinary fp32 — but split those bits differently.

fp16 (IEEE half precision): 1 sign + 5 exponent + 10 mantissa bits. More precision, a narrower range — values that are too large or too small simply overflow to infinity or zero.

bf16 ("brain float", from Google): 1 sign + 8 exponent + 7 mantissa bits. The same exponent range as full 32-bit floats, less precision — far more forgiving during the unstable arithmetic of training a neural network.

Same bit budget. Different trade-off between "how exact" and "how big."

Models are usually trained in fp16, bf16, or fp32 — the precision that stability during training needs.

Once trained, they are often quantized down to int8 for inference (actually running the model): roughly a quarter of the memory of fp32, and hardware built for int8 can push through far more operations per second than the same silicon running floating point.

The accuracy cost is usually small. The speed and energy savings, at the scale a real deployment runs at, are not.

Take a real, current piece of edge-AI hardware: NVIDIA's Jetson AGX Orin (64GB module).

NVIDIA's own published figure: up to 275 TOPS (trillion operations per second) running INT8, with structured sparsity.

Switch the same chip to FP16 / BF16 and achievable throughput drops — roughly by half, since each floating-point operation occupies more of the tensor core's datapath than an 8-bit integer one.

Same transistors. Same chip. The number format you choose to compute in changes your real-world throughput by a factor of two or more.

Pythagoras claimed number was the hidden structure of reality.

Three thousand years later, the question "what exactly is this number, and how many bits does it cost to represent it" is no longer philosophy — it decides how fast your model runs, how much it costs to train, and how much energy it burns doing so.

The history of numerals is, in a very literal sense, the history of computation.

In which the reader is asked, quite simply, What is an Algorithm? — and finds the answer rather older, and rather stranger, than expected.

Being an account of cogs and looms and engines of brass; of Signor Pascal's calculating box and Mr. Babbage's unbuilt Engines; of a certain Lady Lovelace who saw further than most; and of sundry other contrivances got up by ingenious persons long before any machine was called a computer.

Wherein two short films are watched, in the company of a partner to be consulted not once but twice; wherein old Euclid's method is recalled, some two thousand years and more since first he wrote it down; and wherein, at the last, the curious Persian origin of a word now spoken a thousand times a day — one al-Khwarizmi's own — is at length discovered.

Question of the day: What is an algorithm?

You already use the word casually — "the algorithm" recommends your videos, "the algorithm" decides your feed. But what, precisely, is one?

By the end of today's session you'll have a real definition — and you'll know where the word itself came from.

Turn to the person next to you. Two minutes, no googling.

Together, try to answer: what is an algorithm? Give one example each, if you can.

Don't worry about being "right" — we'll come back to this twice more today, and compare notes.

Before the word, there were the machines. Crash Course Computer Science #1: Early Computing — from the abacus to Herman Hollerith's punch cards.

What surprised you?

Notice how many of these — the abacus, Pascal's calculator, the Jacquard loom, Babbage's engines — are not computers in our sense at all. They're physical machines for carrying out a fixed, repeatable procedure.

That procedure, independent of whatever metal or wood or silicon carries it out, is the thing we're actually interested in today.

Strip away the brass gears, the steam, the punch cards — what's left in each of those machines is the same abstract thing: a precise, repeatable sequence of steps that turns some input into some output.

That sequence of steps is an algorithm. The machine is just one possible body for it to live in.

An algorithm is a finite, well-defined sequence of steps that transforms a given input into a desired output.

Finite — it has to actually end, eventually.

Well-defined — each step is unambiguous; anyone, or anything, following it correctly gets the same result.

Input / output — it does something to something.

You run algorithms all the time without a computer in sight.

A recipe: fixed steps, specific ingredients (input), a finished dish (output).

Long division, tying your shoelaces, the instructions for assembling furniture — all algorithms, in the strict sense.

Computers didn't invent algorithms. They just got extremely good at running them fast.

Euclid described an algorithm for finding the greatest common divisor of two numbers around 300 BCE — still taught today, still called Euclid's algorithm.

Long division as you learned it in school is centuries old. Knitting patterns are algorithms. So is a court protocol, or a recipe passed down for generations.

The concept is ancient. Only the name is comparatively recent — and it has a surprisingly specific origin.

Back to your partner. Compare: how close was your first guess to the definition we just built together?

This time, try to name an algorithm from your own daily life that has nothing to do with computers.

"Algorithm" doesn't come from Greek, like most technical vocabulary in mathematics. It comes from a name.

Whose?

A short answer, in under three minutes — BBC Ideas: Why algorithms are called algorithms.

Muhammad ibn Musa al-Khwarizmi, a 9th-century Persian scholar working in Baghdad's House of Wisdom, wrote a treatise on systematic methods for solving equations.

His name, Latinized centuries later as "algorithmi," became attached to the step-by-step methods he described — and eventually to the general concept itself.

(His other book's title gave us a second word: al-jabr — "algebra.")

Rewind two centuries. Long before Babbage, long before steam, the same ancient appetite for mechanizing the mind was already stirring — in the Baroque.

In 1642, at nineteen, Blaise Pascal built the Pascaline to help his father with tax calculations — a gear-driven adding machine, among the first of its kind ever built and sold.

A generation later, Gottfried Wilhelm Leibniz — one of the last true polymaths, equally at home in philosophy, law, and mathematics — went further still, designing the Stepped Reckoner: a machine that could multiply and divide, not merely add and subtract.

Leibniz wanted more than a calculator. He dreamed of a calculus ratiocinator — a universal symbolic method by which any dispute, of any kind, could in principle be settled by calculation alone: "Let us calculate."

Cogs and steam would eventually belong to Babbage's century. But the dream — that reasoning itself might be mechanized — is, properly speaking, Baroque.

By the early 1800s, the Industrial Revolution had already proven something remarkable: steam and gearwork could replace human muscle at scale.

Joseph-Marie Jacquard's loom (1801) went one step further — it replaced not just labor, but a kind of decision-making. A chain of punched cards, read mechanically, told the loom exactly which threads to raise for every single pass of the shuttle.

Change the cards, change the pattern. The procedure — the algorithm — was now separate from the machine that executed it, encoded on something you could hold in your hands.

Babbage noticed. He planned to use these very same punched cards to feed instructions into his own Analytical Engine.

Charles Babbage (1791–1871) — mathematician, philosopher, mechanical engineer, and by most accounts a thoroughly restless Victorian gentleman.

His Difference Engine was a mechanical calculator, designed to compute polynomial values automatically — and to print the results, removing human error from mathematical tables.

His far more ambitious Analytical Engine was something else entirely: a general-purpose, programmable computer — a century before electronics made one practical. It had a "mill" (processor), a "store" (memory), and was to be fed instructions on Jacquard-style punched cards.

He never finished building it. But he had, in effect, designed a computer.

Augusta Ada King, Countess of Lovelace (1815–1852), daughter of Lord Byron, was trained rigorously in mathematics from childhood — and became Babbage's closest intellectual collaborator.

In 1843, translating an Italian paper on the Analytical Engine, she added her own notes — three times longer than the original text. Among them: a step-by-step method for the Engine to compute Bernoulli numbers.

That method is widely considered the first published algorithm written specifically for a machine — making Lovelace, by most accounts, the first computer programmer, over a century before there was a computer to run it on.

She also saw further than Babbage himself: the Engine, she wrote, might one day manipulate not just numbers, but music and symbols of any kind. Others called her the Enchantress of Numbers.

One more figure worth knowing, even briefly: George Boole (1815–1864), who in 1854 published The Laws of Thought — reducing logical reasoning itself to a kind of algebra, with only two values: true and false.

Boolean algebra sat unused for decades as a philosophical curiosity. In the 20th century it became the literal switching logic inside every digital circuit ever built.

Babbage's engine, Lovelace's program, Boole's logic — three separate threads in the 1800s, waiting nearly a hundred years for electronics to braid them together. We'll pick that thread up again later in this course.

Today: a definition, a history, a name.

Next week: Alan and his Enigma — the story of Alan Turing, his machine(s), and his test(s).

We'll meet the Turing machine that finally gave "algorithm" a single, rigorous, universal meaning — and ask what happens when the body running one never gets tired, never gets bored, and never stops.