Insight Editions · No. 276
The Computer and the Brain
The 1956 Silliman Lectures, with Forewords by Ray Kurzweil and Paul and Patricia Churchland
John von Neumann
John von Neumann's last, unfinished work turns the vocabulary of the stored-program computer on the brain, comparing neurons with vacuum tubes and asking what kind of language the nervous system computes in.
A study guide to The Computer and the Brain by John von Neumann. Shared free for personal reading; not for sale and not endorsed by the authors or publishers. Please read and buy the original.
The Book on One Page
In his unfinished 1956 Silliman Lectures, John von Neumann describes the computer in exact terms, then measures the brain against it. The neuron looks digital, but it is slow, imprecise, tiny and frugal, so the brain must be massively parallel and logically shallow. His conclusion is that the brain computes with a statistical notation, trading precision for reliability, and that its native language is not the mathematics we consciously use.
The argument in eight steps
- Two ways to represent a number. An analog machine lets a physical quantity be the number; a digital machine writes digits held by markers and reduces arithmetic to logic (sum is odd parity, carry is majority).
- Control is the program. From fixed wiring through plugboards and tapes, control evolved into memory-stored orders: instructions are numbers with addresses, so a machine can compute on and modify its own program.
- Why 12 digits. Real data rarely justify 3 digits, but long chains of operations superpose errors (as √N) and amplify them, so each step needs 10 to 12 decimals. Precision is bought with depth.
- The memory ladder. Fast memory is costly, so machines provide a little at high speed and more at each slower level: the hierarchic principle that still shapes every computer.
- The neuron is prima facie digital. It fires a standard pulse or nothing, depending on combinations of inputs: an AND or OR organ with a threshold. Against tubes and transistors it is 10⁴ to 10⁵ times slower but 10⁸ to 10⁹ times smaller and thriftier, so nature builds many slow organs in parallel.
- But more than a gate. Summation time, synaptic geometry, fatigue and change-detecting receptors mean the true count of active organs is the synapse count, and the system may be partly analog.
- A memory nobody can find. A brain needs a large memory (his estimate: 2.8 × 10²⁰ bits if nothing is forgotten), but its location and physics are unknown, and it need not use the neuron's componentry.
- The statistical language. Nerves send numbers as pulse frequencies (50 to 200 per second): 2 to 3 digits, but robust to lost pulses. The brain keeps logical depth short (three synapses in the retina) and trades arithmetical precision for logical reliability. Our mathematics may be a short code running on a primary language we have not yet described.
Three threads run under the eight steps. First, component physics shapes logical organization: cheap, slow parts push toward breadth; expensive, fast parts toward depth plus memory. Second, von Neumann keeps finding the brain's tricks already latent in his machine vocabulary: the pulse density code of Section 3 is the nerve's frequency code, and Turing's short code of Section 8 becomes his model for mathematics itself. Third, he is scrupulous about ignorance: the memory's physical basis and the brain's "language" are declared unknown rather than guessed.
The three prefatory pieces frame the fragment. Klara von Neumann (1957) explains why it is unfinished: he wrote it at night as an AEC commissioner dying of bone cancer, and died on 8 February 1957. The Churchlands (2000) read it as the founding text of parallel, vector-matrix computation in the brain and call him the "Newton of the mind". Ray Kurzweil (2012) places it among five ideas behind the information age (Shannon, Turing, the von Neumann machine, machine intelligence, the singularity) and uses it to argue that a digital machine can simulate the brain, though the brain cannot simulate a computer. Both forewords contain factual slips flagged in their chapters.
Chapter map
| Ch | Title | One-line payload |
|---|---|---|
| Fwd3 | Foreword to the Third Edition (Kurzweil, 2012) | Five ideas built the information age; von Neumann's universality argument implies machines can simulate the brain |
| Fwd2 | Foreword to the Second Edition (Churchlands, 2000) | The architect of the serial computer saw that the brain is a parallel, vector-matrix machine |
| Pref | Preface (Klara von Neumann, 1957) | Written at night by a dying AEC commissioner; the lectures were never delivered |
| Intro | Introduction | A mathematician's speculations, aimed at how brain study could change logic and mathematics |
| S1 | The Analog and the Digital Procedure | Numbers as physical quantities or as digits; digital arithmetic is pure logic |
| S2 | Logical Control | From fixed wiring and plugboards to stored orders with addresses, conditional transfers and self-modification |
| S3 | Mixed Procedures and Precision | Pulse-density hybrids; error superposition and amplification force 10 to 12 digits per step |
| S4 | Characteristics of Modern Machines | 1956 speeds and counts; access time, the memory hierarchy, direct addressing |
| S5 | The Nature of the Nerve Impulse | Neurons are digital, 10⁴ to 10⁵ slower and 10⁸ to 10⁹ thriftier; hence parallelism |
| S6 | Stimulation Criteria | Thresholds, summation and geometry make the synapse the real active organ |
| S7 | The Problem of Memory | 2.8 × 10²⁰ bits postulated; physical basis unknown and possibly unlike the neuron |
| S8 | Mixed Processes and Codes | Digital and analog stages alternate; Turing's short code lets one machine imitate another |
| S9 | Not Digital but Statistical | Rate codes give 2 to 3 digits but high reliability; the brain's language is shallow and statistical |
Foreword to the Third Edition (Ray Kurzweil, 2012)
Ray Kurzweil reads von Neumann's last manuscript as the first serious bridge between computer science and neuroscience, and sets it inside five ideas that built the information age, three of them von Neumann's own. His reading: von Neumann used the universality of computation to argue that a digital machine can simulate whatever the brain does, even though the brain is built nothing like a computer. The foreword is a confident, sometimes careless piece of advocacy for Kurzweil's own programme of reverse engineering the brain.
Kurzweil's frame: reverse engineering the brain
Kurzweil opens from his familiar premise: every information technology improves exponentially in price-performance and capacity, so the information age keeps widening its reach. The most important information process to understand is human intelligence itself, and he calls this book perhaps the earliest serious study of the link between thinking and the computer, written by the man who designed the computer's basic architecture.
- The project. Science is making accelerating gains in reverse engineering how humans think, and these biologically inspired methods are being used to build ever smarter machines.
- The forecast. AI built this way will eventually "soar past" unenhanced human thinking.
- The value claim. The point is not to displace humans but to extend what is already a human-machine civilization, which Kurzweil calls the thing that makes our species unique.
He then counts five key ideas under the information age. Von Neumann is "largely responsible" for three and made a basic contribution to a fourth; Claude Shannon solved reliability and Alan Turing defined universality (influenced, Kurzweil says, by an early von Neumann lecture). In this book von Neumann sets out his model of computation, then argues for the essential equivalence of brain and computer: he admits the deep structural differences but uses Turing's equivalence of all computation to plan a route to understanding, re-creating and finally extending the brain's methods.
- Reliable information (Shannon, 1948). Redundancy and error-correcting codes make communication, memory and logic arbitrarily accurate even over noisy channels.
- Universality of computation (Turing, 1936). A minimal machine reading one bit at a time can compute anything any machine can compute.
- The von Neumann machine (1945). CPU, random access memory holding both program and data, program counter, input/output: the working architecture of nearly every computer since.
- Machine intelligence (Turing 1950; von Neumann 1956). Learn the brain's methods and use them to build smarter machines.
- The singularity (von Neumann via Ulam, 1958). Accelerating technology approaches an "essential singularity" beyond which human affairs cannot continue unchanged.
Idea 1: Shannon makes digital computing possible
The problem everyone assumed was fatal
Around 1940 the word "computer" meant an analog computer: numbers held as voltage levels, with special components for adding and multiplying. Accuracy was about one part in a hundred, and because each operation passed its error on to the next, errors accumulated; beyond a handful of steps the answer meant nothing. Kurzweil's analogy is copying analog tape: each generation is noisier, and by the tenth copy it is almost all noise.
Engineers expected digital machines to suffer the same decay, and the arithmetic of a noisy channel seemed to prove it. If each bit gets through with probability p, a message of n bits arrives intact with probability pⁿ, which collapses as messages grow.
- p = 0.9 per bit. One bit arrives intact 90% of the time; two bits 0.9² = 81%; one byte (8 bits) 43%; five bytes about 1%.
- One error per million bits. A file of half a million bytes (4 million bits) survives intact with probability (1 − 10⁻⁶)^(4×10⁶) ≈ e⁻⁴, under 2%.
- Consequence. A single flipped bit can wreck a program, so no improvement in raw channel quality fixes the problem: error risk grows with message size regardless.
Analog machines coped by graceful degradation: confine them to short, constrained calculations and they are usable. Digital computers cannot do that, because communication is pervasive at every level: between computers, between memory and the central processing unit, from register to register, back and forth to the arithmetic unit, and even between bit registers inside it. With error rates compounding over all that traffic, digital computation looked doomed.
Shannon's answer: redundancy
That consensus held until Shannon's "A Mathematical Theory of Communication" (Bell System Technical Journal, July and October 1948). His noisy-channel coding theorem says that over any channel with an error rate other than exactly 50% per bit (which would be pure noise), messages can be sent with an error rate as low as desired: one bad bit in n, for any n. Even a channel right only 51% of the time can deliver one error per million, per trillion, or per trillion trillion bits.
- Mechanism. The route is redundancy, obvious now but not then. The simplest scheme sends each bit three times and takes a majority vote; if that is not enough, repeat more.
- Worked check (not in Kurzweil). At p = 0.9, a triple-sent bit is decoded wrongly only if two or three copies fail: 3(0.1²)(0.9) + 0.1³ = 0.028, cutting the error rate from 10% to 2.8% at three times the cost.
- Efficiency. Plain repetition is the easiest illustration but not the most efficient code. Shannon's paper founded information theory and gave optimal methods of error detection and correction for any non-random channel.
Kurzweil's everyday proofs: telephone modems sent accurate data over hissing, popping analog lines, and CDs, DVDs and program disks still read correctly after being scratched. The same logic covers all three ingredients of computation, which he lists as communication, memory and logic gates: each can be made arbitrarily reliable with codes. Without Shannon there would be no large, complex digital data or algorithms, and no buildable von Neumann machine.
Idea 2: Turing's universal machine
The machine as thought experiment
Turing's 1936 machine is not hardware but a thought experiment, deliberately as simple as possible ("but no simpler", Kurzweil adds, paraphrasing Einstein).
- Tape. Infinitely long, one 0 or 1 per square; input is written on it and output is left on it.
- Head. Reads one square at a time.
- Rule table. In effect a stored program: numbered states, each giving one action if the current square is 0 and another if it is 1.
- Actions. Write a 0 or 1, move the tape one square left or right, or halt; each rule also names the next state.
- Finiteness. Any program that halts uses only a finite stretch of tape, so a finite-memory version still solves a useful class of problems.
With his former professor Alonzo Church, Turing arrived at the Church-Turing thesis: a problem that a Turing machine cannot solve cannot be solved by any machine obeying natural law. Despite a handful of commands and one-bit steps, the machine can compute anything any computer can. "Strong" readings extend this to minds: the brain obeys natural law, so its information processing cannot exceed a machine's, and hence a Turing machine's. This is the premise the whole foreword leans on.
Influence, undecidability and decisions
- Mutual influence. Kurzweil claims Turing was shaped by a 1935 Cambridge lecture by von Neumann on the stored-program concept, and that von Neumann in turn made Turing's 1936 paper required reading for colleagues in the late 1930s and early 1940s (the first claim is doubtful; see the critical lens).
- Unsolvable problems. The same paper showed that some well-defined problems with provably unique answers can never be computed by any machine, overturning the nineteenth-century faith that whatever can be defined can eventually be solved. Kurt Gödel's 1931 incompleteness theorem reached a kindred result.
- Why simplicity is enough. Because a Turing machine can choose its next action from results already computed, it can make decisions and model arbitrarily complex hierarchies of information.
Kurzweil adds that Turing "designed and completed" Colossus by December 1943 to break Enigma, a single-purpose machine that could not be reprogrammed, and credits it with helping Britain overcome the Luftwaffe's 3:1 advantage in the Battle of Britain. Almost every element of that story is wrong (see below).
Idea 3: the von Neumann architecture
In "First Draft of a Report on the EDVAC", dated 30 June 1945, von Neumann set out the concepts that Kurzweil says have ruled computing "for the past sixty-six years", from the washing-machine microcontroller to the largest supercomputer. Meant as an internal project paper, it became the designers' bible of the 1940s and 1950s.
- Central processing unit. Performs the arithmetical and logical operations.
- Memory unit. Holds both program and data.
- Mass storage. Bulk storage beyond working memory.
- Program counter. Tracks which instruction comes next.
- Input/output channels. Connect the machine to the world.
| Feature | Turing machine | von Neumann machine |
|---|---|---|
| Purpose | Map what is computable; efficiency irrelevant | A practical computing machine |
| Unit of work | One bit per step | Multi-bit words, usually a multiple of 8 bits |
| Memory | Sequential tape; programs waste time shuttling it | Random access: any item retrieved at once |
| Program | Rule table fixed in the machine | Stored in the same memory as data; reprogrammable, even self-modifying |
| Instruction | State transition | Operation code + operand address, the shape of machine language today |
- Stored program. Kurzweil says von Neumann introduced it a decade earlier. Storing code like data lets one machine be reprogrammed for any task and, if the program store is writable, permits self-modifying code, which he calls a powerful form of recursion. Before this, virtually every computer, Colossus included, was built for one task; the stored program made Turing's universality practical.
- Credit and chronology. The EDVAC design came out of work with J. Presper Eckert and John Mauchly. The EDVAC itself ran only in 1951, by which time the Manchester Small-Scale Experimental Machine, EDSAC, BINAC and a stored-program version of ENIAC existed, all shaped by the report and (Kurzweil says) involving Eckert and Mauchly. Von Neumann contributed directly to several, including the later stored-program ENIAC.
Precursors and the one "true" ancestor
1837 Charles Babbage describes the Analytical Engine: program on punched cards borrowed from the Jacquard loom, a memory of 1,000 words of 50 decimal digits (about 21 KB), opcode plus operand, conditional branching and looping. Kurzweil's "one true precursor"; never built.
1843 Ada Lovelace translates Luigi Menabrea's article on the Engine, adds long notes and programs she must debug in her head ("table checking"), and denies the Engine any power to "originate anything".
1941 Konrad Zuse's Z-3 reads its program from punched film; no conditional branch. Used by the German Aircraft Research Institute for wing flutter, but his request to replace relays with vacuum tubes is refused: computing is "not war important".
1944 Howard Aiken's Mark I executes instructions straight from punched paper tape; no stored program, no conditional branch, so not a von Neumann machine.
1945 The EDVAC report becomes the template for every later design.
Lovelace's line that the Engine "weaves algebraic patterns" as the loom weaves flowers and leaves frames her as the first to speculate on machine intelligence, and the first to deny it. Kurzweil judges Babbage's design "miraculous" for its era but lost by the mid-twentieth century: it is unclear whether von Neumann or the other pioneers knew of it, so the credit for articulating the modern computer stays with von Neumann. He closes the section by tying ideas 1 and 3 together: a machine that ships data constantly between and within its units could not be built without Shannon's methods.
Idea 4: machine intelligence, and Kurzweil's verdict on the manuscript
The fourth idea is to give computers intelligence, going past Lovelace's denial. Turing's 1950 paper "Computing Machinery and Intelligence" set the goal and proposed the Turing test. Von Neumann's book, after laying out his architecture, turns to the best example of an intelligent system we have, the human brain, so that its methods can inspire better machines. Kurzweil calls it the earliest serious look at the brain by a mathematician and computer pioneer: before it, computer science and neuroscience were "two islands with no bridge". The irony he notes is that this was von Neumann's last work: written for Yale, never delivered and never finished because of cancer.
What von Neumann got right
Kurzweil finds the 1955-56 text "remarkably accurate" on the details that matter for the comparison, given neuroscience's primitive tools.
- Digital output, analog input. The axon fires or does not, which was not obvious then (the output might have been analog). Processing in the dendrites and soma is analog: a weighted sum of inputs against a threshold.
- Connectionism. That neuron model fed connectionism, systems built on it in hardware and software; the first was Frank Rosenblatt's program on an IBM 704 at Cornell in 1957. Modern neuron models are richer, but analog processing of dendritic inputs via neurotransmitter concentrations has held up.
- One-way simulation. By universality, a von Neumann machine can simulate a brain; the converse fails because the brain has no stored program: its algorithms are implicit in its structure.
- Learning. Neurons learn patterns from their inputs (now known to be coded in neurotransmitter concentrations); what von Neumann could not know is that learning also happens by creating and destroying connections.
- Slow but parallel. Neurons manage about 100 calculations per second, compensated by massive parallelism: all 10¹⁰ neurons work at once (Kurzweil gives modern estimates of 10¹⁰ to 10¹¹), and in fact each of roughly 10³ connections per neuron computes simultaneously.
- Shallow algorithms. Fast decisions with slow neurons rule out long sequential algorithms. A third baseman choosing to throw to first rather than second has time for only a handful of neural cycles; reverse engineering of the visual cortex shows sophisticated visual judgments in three or four cycles.
Digital computation can emulate analog values to any precision, and the brain's analog precision is low; computers' serial speed advantage (vastly larger now than in 1956) plus parallel von Neumann machines, the design of every supercomputer, covers the brain's parallelism. The brain is plastic, but a computer is far more plastic: it can rewrite its methods by changing software.
The one major dissent: memory
Von Neumann. Nothing is truly forgotten. 10¹⁰ neurons × about 14 inputs per second × 60 years (about 2 × 10⁹ s) gives roughly 10²⁰ bits of lifetime storage. Kurzweil adds that the 14-per-second rate is itself low by at least three orders of magnitude.
Kurzweil. We keep only a tiny fraction of experience, stored not as low-level bit patterns (like video) but as sequences of high-level patterns from a cortical hierarchy of pattern recognizers; recall is reconstruction, not playback.
Kurzweil's hierarchy runs upward: recognizers for topological forms (the crossbar of a capital A, its lower concavity), then printed letters, then words such as "Apple"; parallel regions recognize the object apple and the spoken word; near the top a recognizer concludes "That was funny." His test: recall your last walk. Who was the fifth person you met, was there a baby carriage, a mailbox, what was in the second shop window? Most people can only reconstruct from a few cues; machines, by contrast, recall perfectly, one of AI's advantages.
Beyond memory he finds very little at "significant odds" with current knowledge: no 1956 book could describe the brain perfectly, but each mechanism von Neumann describes is paired with how a computer could do the same thing. On capacity, the brain outclassed the few computers of 1956; Kurzweil writes that the first supercomputers are now reaching the conservative estimate of 10¹⁶ operations per second needed to simulate brain function, and predicts that this hardware will cost $1,000 early in the 2020s. Von Neumann, he argues, wrote these lectures because he was confident both the hardware and software of intelligence would eventually arrive.
Idea 5: the singularity
The last idea is acceleration. A year after von Neumann's death in 1957, Stanislaw Ulam (1958) recalled him saying that the ever accelerating progress of technology gives the appearance of approaching "some essential singularity" in human history, beyond which human affairs as we know them could not continue. Kurzweil calls this the first known use of "singularity" for human history, which makes von Neumann a forebear of Kurzweil's own best-known thesis.
- Colossus. Turing did not design it (Tommy Flowers did); it attacked the German Lorenz teleprinter cipher, not Enigma; it ran from early 1944, so it could not have helped win the 1940 Battle of Britain.
- The 1935 lecture. No record shows von Neumann lecturing in Cambridge on a stored-program concept in 1935; his visit concerned almost periodic functions, and the stored program dates from the mid-1940s (so "a decade earlier" is also wrong).
- Unsolvable problems. Turing did not show there are "as many" unsolvable as solvable problems; computable functions are countable, so almost all problems are uncomputable.
- Connectionism's origin. The threshold neuron is McCulloch and Pitts (1943), a paper von Neumann cited in the EDVAC report; Rosenblatt's perceptron report dates from 1957 and its IBM 704 demonstration from 1958.
- Babbage as a "true von Neumann machine". The Engine's program lived on cards outside its store, so it was not a stored-program design in von Neumann's sense.
- Further readings. "A Mathematical Theory of Communication" is listed under von Neumann; it is Shannon's.
- "14 inputs per neuron". Von Neumann's 14 per second is per receptor, and the neuron count has since settled at about 86 billion (Azevedo et al., 2009).
- Consumer hardware. Nvidia's RTX 5090 (January 2025, $1,999) is rated at 3,352 "AI TOPS", about 3 × 10¹⁵ low-precision (FP4, sparse) operations per second; the $549 RTX 5070 offers 988 TOPS. At $1,000 that is roughly 10¹⁵, an order of magnitude short of 10¹⁶, and only at 4-bit precision.
- Top end. Nvidia's Vera Rubin NVL72 rack (production 2026) claims 3.6 × 10¹⁸ FP4 operations per second; the fastest TOP500 machine in June 2026, China's LineShine, became the first to pass 2 exaflops on HPL (FP64).
- Kurzweil himself restated his timelines in The Singularity Is Nearer (2024), keeping human-level AI around 2029.
Kurzweil's foreword is the optimist's reading: universality guarantees brain simulation, so the remaining questions are speed and cost. The Churchlands' 2000 foreword, next, is the neuroscientist's reading, more interested in what von Neumann got wrong about the brain's architecture. Read both before the lectures themselves.
Foreword to the Second Edition (Paul M. and Patricia S. Churchland, 2000)
The philosophers Paul and Patricia Churchland call the book "the eye of a hurricane": a calm, balanced 1956 assessment at the center of the fight over how brains compute. Their thesis is that von Neumann, architect of the serial digital computer, saw past his own design and recognized the brain as a massively parallel analog machine. They end by naming him the "Newton of the mind", already come and gone.
The hurricane and its calm center
The Churchlands frame the book as a "locus of clarity and calm" inside a vortex of competing research programmes about mind and machine. What makes it singular is its date: written at the very start of the electronic computer explosion, it attempts a balanced assessment of what the brain might be computing, seen through the computational theory, computer technology and neuroscience of 1956. One would expect such an assessment to be hopelessly dated; they argue the opposite on each of three fronts.
- Computational theory. The theory of generating any computable function, founded by Church and Turing and "to a degree" von Neumann, has proved as sturdy and fertile as its founders hoped: the lens "was well turned at the start".
- Computer technology. Every office and home machine is an instance of the von Neumann architecture; he would recognize all of them as his own design.
- Empirical neuroscience. The most complicated front: a half-century of work has produced a substantially new science, yet it has developed apart from computing.
The machines have not left him behind
The von Neumann architecture is a functional organization in which a sequential "program" held in modifiable "memory" dictates the nature and order of the steps carried out by a "central processor". Von Neumann's own words in the book outline its rationale; only the vocabulary and the clock speeds have changed.
| His term (1956) | Our term (2000) |
|---|---|
| Code | Program |
| Complete code | Machine-language program |
| Short code | High-level programming language |
Every machine in sight in 2000, "from PalmPilot organizers to supercomputers", whether running poker games or galactic-genesis simulations, is a further instance of his vision. In no relevant sense, they conclude, has he been left behind.
Neuroscience became a new science
Neuroanatomy, neurophysiology, developmental neurobiology and cognitive neurobiology all made "monumental advances", driven by new techniques. Together they show the brain's filamentary microstructure, the electrochemical behaviour of its parts and its global activity during conscious cognition. The brain still holds many mysteries but is no longer a black box.
- Microstructure. Electron and confocal microscopy.
- Single cells. Patch clamping of membranes and channels.
- Whole-brain electrical activity. Electro- and magneto-encephalography (EEG, MEG).
- Imaging. CAT, PET and MRI scans.
Two sister sciences in isolation
The puzzle the Churchlands pose is that the science of artificial cognition and the science of natural cognition ran in "parallel streams" from the 1950s to 2000, each a roaring success, each teaching the other almost nothing.
Computer scientists. Learned little or nothing about the biological brain; spent careers writing programs, designing languages and building ever-better microchip hardware, none of which touched neuroscience.
Neuroscientists. Learned little or nothing of computational theory, automata theory, formal logic, binary arithmetic or transistor electronics; spent their time staining brain slices or putting microelectrodes into living neurons. If they used computers, it was as a lab tool, like a voltmeter, calculator or filing cabinet.
The recurring explanation is that the brain has a physical organization and computational strategy very different from the von Neumann machine, so for nearly fifty years the two fields studied different subject matters. The Churchlands concede this answer is "fiercely disputed" and may be mistaken, but it sits at the heart of debate on how brains work and how to build AI. Their questions frame the choice:
- Path one. Push past biology's limits (mainly speed and reliability) and exploit electronics, which even on a von Neumann design can simulate any computation.
- Path two. Mimic the organization of the brains of insects, fishes, birds and mammals, for whatever reasons.
- The open question. What is that organization, and does it differ in any important way from what our machines do?
Von Neumann's argument, step by step
Von Neumann's answer, they say, is "prescient, powerful, and decidedly nonclassical". He spends the first half of the book on the classical conceptions he is famous for, then, turning to the brain, tentatively calls its functioning "prima facie digital". The Churchlands pun that this first take is also "prima facie procrustean": it forces the data into the wrong frame, as von Neumann himself immediately admits and then explores at length.
Four points of comparison
- Wiring. Logic gates have "two lines in and one line out". Each neuron sends out one axon, as the classical picture requires, but receives more than a hundred, even several thousand inputs. Not decisive (there are multivalent logics), but it gives him pause.
- Size. Comparing "basic active organs" (neurons against logic gates), neurons are at least 10² times smaller. Exactly right in 1956; photo-etched microchips then erased the advantage for two-dimensional sheets: "we can forgive von Neumann this one".
- Speed. Neurons are perhaps 10⁵ times slower per basic logical operation. If anything he understated it: at a neuron "clock" of about 10² Hz against desktop clocks near 1,000 MHz (10⁹ operations per second), the gap is closer to 10⁷. A von Neumann brain would be a "computational tortoise".
- Accuracy. Computers handle 8, 10 or 12 decimal places; the neuron's presumed code, the spike-train frequency, gives at most two, about ±1% of a maximum near 100 Hz. Over many steps small early errors accumulate, and in many important computations they are amplified exponentially. A digital brain with two-place accuracy would be a "computational dunce".
From depth to breadth
Speed and accuracy together force the conclusion that the brain's regime must involve a minimum of "logical depth": it cannot be running thousands of sequential steps like a processor's high-frequency, recursive cycle. Too slow to finish any but trivial computations, it would be incompetent even with enough time. The poignancy, the Churchlands note, is that the brain plainly does sophisticated computing "in the twinkling of an eye", and nothing is wrong with his arguments.
His resolution is extraordinary logical breadth. Natural automata, he wrote, will tend to be "highly parallel", artificial ones "serial", picking up and processing as many items as possible simultaneously. That obliges us to count the synapses, not just the neurons, among the brain's basic active organs.
- Synapses. About 10¹⁴, each modulating the arriving signal before passing it on.
- Fan-in. Up to 10,000 synapses onto a single cell, whose job is to sum or integrate them and fire in turn.
- Rate. Each synapse active about 100 times per second (typical spiking near 100 Hz).
- Total. 10² × 10¹⁴ = 10¹⁶ operations per second, all simultaneous, against 10⁹ for a cutting-edge 2000 desktop.
The brain is neither tortoise nor dunce, because it was never a serial digital machine at all: it is a massively parallel analog machine. Von Neumann suggests this, and modern neuroscience and network modelling, they say, strongly confirm it.
A population's simultaneous spike frequencies form a very large input vector; the synaptic junctions connecting it to the next population form an even larger matrix of coefficients; their product is an output vector, a new pattern of spiking in the receiving population. The acquired configuration of trillions of synapses both stores the brain's knowledge and skills and performs the transformation, giving the speed and freedom from recursively amplified error von Neumann required.
Optimism, then caution
The insight does not dim digital and serial technology or hopes for AI: the reverse. Engineers can build electronic synapses and, dropping the classical architecture, assemble vast parallel networks of artificial neurons with the brain's shallow-but-broad regime.
Electronic components would make such networks roughly 10⁶ times faster than their biochemical namesakes. A synapse-for-synapse duplicate of your brain could then run, in thirty seconds, a train of thought that takes your own brain a year, and in half an hour an intellectual life of "three-score-and-ten" years. Intelligence, they write drily, "has an interesting future". (Check: 70 years ≈ 3.7 × 10⁷ minutes; divided by 10⁶ ≈ 37 minutes.)
Then the caution. Small artificial neural networks already exist, built on three assumptions: synapses are tiny multipliers, neurons are tiny summators with a sigmoid output, and information is coded solely in spike frequency. After lengthy training many show striking "cognitive" talent, yet they capture little of the subtlety and variety of real synapses and neurons. First-pass models are crude approximations and may prove as mistaken as the old guess that the brain is primarily digital.
- More than one way to code information in an axon's behaviour.
- More than one way to modulate it at a synapse.
- More than one way to integrate it within a neuron.
The lesson is humility before empirical facts, as von Neumann showed.
The Newton of the mind
They place von Neumann's computer revolution alongside Isaac Newton's mechanics and James Maxwell's electromagnetics in long-term impact, and praise his character in seeing past his own architecture to a possibly more powerful explanatory paradigm. Commentators often call for a "Newton of the mind"; the Churchlands reply that the case is strong he has already come and gone, and his name is John von Neumann.
- "William Church" should be Alonzo Church, co-author of the Church-Turing thesis.
- Size claim. Von Neumann's own text puts the linear size advantage at about 10³ (it is internally inconsistent); the Churchlands' 10² is the lower figure.
- "Analog machine". Von Neumann himself calls the system mixed: pulses are all-or-none, and only the frequency code and chemical processes are analog-like. The Churchlands' label sharpens his tentative conclusion into a firm one.
- The 10⁶ speed-up assumes a copy can be built synapse for synapse with electronic timing; no such emulation of even an insect brain existed in 2000.
- "Newton of the mind" is generous: the book is a sketch of an unfinished lecture series and offers no laws, only a well-argued set of constraints.
- Deep learning made the Churchlands' vector-matrix description the dominant form of AI; GPUs and TPUs are essentially matrix-multiply engines. An Nvidia Rubin GPU (2026) holds 336 billion transistors and is rated at 50 petaflops at 4-bit precision.
- Speed. Desktop CPUs reach about 5-6 GHz (2026), only 5 to 6 times the 2000 clock: the gains since came from parallelism, just as von Neumann predicted for large automata.
- Connectomics began measuring the matrix itself: the FlyWire fly-brain wiring diagram (Nature, October 2024) maps about 140,000 neurons and over 50 million synapses.
- Their caution held too. Transformers still use the multiply-and-sum neuron the Churchlands called crude, while neuroscience keeps adding dendritic computation and spike-timing codes the models ignore.
Preface (Klara von Neumann, 1957)
Klara von Neumann, the author's widow, explains why the book is a fragment: von Neumann wrote it at night while serving full time on the Atomic Energy Commission and dying of bone cancer, and never finished it. The Silliman Lectures were the one engagement he refused to cancel, and they were never delivered. Her preface doubles as the book's only biography of its author.
The invitation and the bargain
The Silliman Lectures at Yale are among the oldest and most distinguished lecture series in the US. The custom is a run of talks over about two weeks, then a book shaped from them and published by Yale. In early 1955 Yale invited von Neumann to lecture in the spring term of 1956, in late March or early April; his chosen theme, "The Computer and the Brain", had interested him for a long time.
- The one condition. He accepted only if the lectures were cut to one week, because President Eisenhower had just appointed him to the Atomic Energy Commission (AEC), a full-time Washington job; the written manuscript would still cover the topic in full.
- How he meant to write it. As he always had, at home, at night or at dawn: his capacity for work was "practically unlimited" when a subject held him, and the unexplored possibilities of automata did.
- Yale's response. It agreed at once, the first of several accommodations Klara credits to the university, and he began at the AEC planning to continue his automata theory on the side ("a little en cache").
- The move. In spring 1955 the couple left Princeton for Washington; he took leave from the Institute for Advanced Study (IAS), where he had been professor of mathematics since 1933.
A compressed life
1903 Born in Budapest; an almost photographic memory shows itself early.
1920s Studies chemistry, then mathematics, in Berlin, Zurich (Technische Hochschule) and Budapest.
1927 Privatdozent at the University of Berlin, probably one of the youngest in German universities in decades; later teaches at Hamburg.
1930 First Atlantic crossing, as a one-year guest lecturer at Princeton.
1931 Joins the Princeton faculty and makes the US his permanent home.
1933 Professor in the School of Mathematics at the IAS.
Late 1930s Turns to theoretical hydrodynamics and the difficulty of solving partial differential equations analytically.
1943 Joins the Manhattan Project, one of the scientists who "disappeared into the West", shuttling between Washington and Los Alamos.
1952 Joins the AEC's General Advisory Committee.
15 Mar 1955 Sworn in as an AEC commissioner; household moves to Washington in early May.
Aug 1955 Severe pain in the left shoulder; after surgery, bone cancer is diagnosed.
Late Nov 1955 Lesions found on his spine; walking becomes difficult.
Jan 1956 Confined to a wheelchair, still attending meetings and writing.
Early Apr 1956 Admitted to Walter Reed Hospital, which he never leaves.
8 Feb 1957 Dies; the manuscript is still unfinished.
From pure mathematics to machines
Klara traces a clear causal path from abstraction to computing to the brain.
- Pure work. In the 1920s and 1930s he published on quantum theory, mathematical logic, ergodic theory, continuous geometry, rings of operators and much else.
- Applied turn. Shock-wave interaction, an important defence problem, needed more calculation than any known analytic method could deliver, which drove him to high-speed machines.
- ENIAC. Built in Philadelphia for the Ballistic Research Laboratories of Army Ordnance, it was his first contact with automatic computation; he helped modify parts of its mathematical-logical design and stayed fascinated by automata "until his last conscious hours".
- Los Alamos. There he became convinced, and tried to convince others in many fields, that fast electronic computing would open many unsolved scientific problems.
- The IAS machine. After the war he and a small team built an experimental electronic computer at the IAS (Klara calls it the "JONIAC"), which became the pilot model for similar machines across the country; some of its principles were still in use in 1957.
- The brain link. In designing it, the team tried to imitate known operations of the living brain. That led him to study neurology, seek out neurologists and psychiatrists, attend their meetings, and lecture on copying an extremely simplified brain model in machines, the ideas the Silliman Lectures were meant to develop.
- Postwar spread. He worked on meteorology, where numerical calculation opened "entirely new vistas", and on nuclear-physics calculations for the AEC's laboratories.
The machine and the brain were linked for von Neumann from the start: the IAS computer was designed partly by analogy with the brain, and these lectures were meant to run the analogy back the other way, using machine concepts to understand neural tissue.
Writing against the illness
The manuscript was written almost entirely in the months of uncertainty after the August 1955 diagnosis. By day he worked in his office and travelled for the Commission; by night he wrote, turning systematically to the Silliman text. Hopes rose when the shoulder lesion seemed isolated and fell with each new pain.
After the spinal lesions of November 1955, every trip and speaking engagement was cancelled except the Silliman Lectures. The hope was that X-ray treatment would strengthen his spine enough to reach New Haven by late March; Yale was asked to cut the series to one or two lectures. By March all hope of travel was gone. Yale still did not cancel, offering to have someone read the manuscript in his place, but he could not finish it in time, and never did. He took it to Walter Reed in April 1956 and made a few more attempts before, in Klara's words, even "Johnny's exceptional mind" could not overcome the body's weariness.
She closes, dated Washington, September 1957, by thanking the Silliman Lecture Committee, Yale University and Yale University Press for admitting the unfinished, fragmentary manuscript to the Silliman series. That status matters for every chapter that follows: the text stops mid-argument, and its figures are working estimates rather than revised conclusions.
- John von Neumann. Mathematician, IAS professor, AEC commissioner; author of the lectures.
- Klara (Klári) Dan von Neumann. His wife and the preface's author; herself one of the first programmers, she coded Monte Carlo simulations on the ENIAC in 1948, a fact her modest preface omits.
- Dwight D. Eisenhower. Appointed him to the AEC in 1955, the reason the lectures were shortened.
- Yale University and its Silliman Committee. Host and publisher, credited with patience throughout.
Klara calls the IAS computer the "JONIAC"; it is normally called the IAS machine. JOHNNIAC was a separate copy built at RAND and named in his honour. Her preface is also a memoir of devotion, so it is silent on his hawkish nuclear politics, including his strong support for the hydrogen bomb, and on the scientific collaborators of the IAS project (such as engineer Julian Bigelow).
Introduction
Von Neumann opens with a disclaimer and an ambition. He offers only a "somewhat systematized set of speculations" about how a mathematician should approach the nervous system. His real aim is the reverse direction: a mathematical study of the brain may change how we understand mathematics and logic themselves.
Two qualifications
Von Neumann, "neither a neurologist nor a psychiatrist" but a mathematician, describes the work as an approach toward understanding the nervous system from the mathematician's point of view, then qualifies both halves of that phrase at once.
- Not an approach, a guess about approaches. From the "hazy distance" at which most lines of attack are seen, he tries to sort the mathematically guided ones that look promising a priori from those that do not, and offers rationalizations for each guess.
- A particular kind of mathematician. Beyond general mathematical technique, logic and statistics are in the foreground, treated mainly (not exclusively) as tools of information theory.
- Practical experience as data. Much of that information theory is the experience gathered in planning, evaluating and coding large logical and mathematical automata, the most typical being the large electronic computing machines.
He regrets that no theory of automata yet exists. What he can appeal to is only an "imperfectly articulated and hardly formalized body of experience": a frank statement that the computer half of the comparison rests on engineering know-how, not settled theory.
Treat the computer and the brain as two species of automaton, compare them on the same terms (representation, control, precision, speed, size, memory, code), and let the mismatches reveal what kind of logic the brain must be running.
The real aim
His main purpose, he says, lies elsewhere. A deeper mathematical study of the nervous system, "mathematical" in the logical and statistical sense just defined, will affect our understanding of the parts of mathematics involved, and may change how we view mathematics and logic proper. He promises reasons later; they arrive in the closing sections, where he argues that the brain's language is not the language of mathematics and that our mathematics is a secondary language built on top of it.
How the argument is organized
The introduction ends by opening Part 1, The Computer, whose first move is to split existing machines into two classes, analog and digital, according to how they represent numbers. That classification becomes the first yardstick later applied to neurons.
| Part | Sections | What it establishes |
|---|---|---|
| Part 1: The Computer | Analog and digital procedures; logical control; mixed procedures; precision; machine characteristics | The vocabulary: representation, control, memory hierarchy, why digital machines need 10 to 12 decimal digits |
| Part 2: The Brain | Neuron and nerve impulse; stimulation criteria; memory; codes; logical structure; statistical notation; language of the brain | The neuron is prima facie digital, but slow, imprecise and parallel; hence a shallow, statistical logic unlike our mathematics |
- Warren McCulloch and Walter Pitts. Their 1943 paper modelled neurons as threshold logic units; von Neumann used their notation in the 1945 EDVAC report.
- Claude Shannon. His 1948 theory supplies the "information theory" von Neumann names as his toolkit.
- Von Neumann himself. He had lectured on the "General and Logical Theory of Automata" (Hixon Symposium, 1948) and on building reliable organisms from unreliable components (Caltech, 1952; published 1956).
It may alter the way in which we look on mathematics and logics proper.
The modesty is real, not rhetorical: the manuscript is unfinished, its numbers are order-of-magnitude guesses, and he never reached the promised full account of the brain's mathematics. The Silliman charter adds an irony he never mentions: the series was endowed to show "the presence and providence of God" through natural science, and it ended up hosting a mathematician's comparison of the brain with a machine.
The Computer: The Analog and the Digital Procedure
Part 1 opens by splitting all computing machines into two classes according to how they represent numbers. An analog machine lets a physical quantity be the number; a digital machine writes the number as digits and manipulates them by strict logical rules. Everything later said about precision, control and the brain builds on this distinction.
The analog procedure
In an analog machine each number is a physical quantity whose value, measured in a pre-assigned unit, equals the number: the angle a disk has turned, the strength of a current, a relative voltage. To compute, the machine needs organs (components) that perform the basic operations of mathematics directly on these quantities, following a predetermined plan.
The conventional basic operations
The usual set is the "four species of arithmetic": x + y, x − y, xy, x/y.
- Addition and subtraction. Easy with currents: merge them in parallel or antiparallel.
- Multiplication and division. Harder, but various electrical components can do them.
- The unit problem. For multiplication and division, though not for addition and subtraction, the unit of measurement matters: doubling the unit halves every represented number, which leaves a sum consistent but distorts a product.
Unusual basic operations: the differential analyzer
Some analog machines are built around other basic operations altogether, a feature von Neumann flags as important for later. The classical differential analyzer represents numbers as rotation angles of disks and works with two primitives.
- Half-sum and half-difference. Instead of x + y and x − y it offers (x ± y)/2, because a cheap standard part produces exactly that: the differential gear on a car's back axle.
- Integration instead of multiplication. All quantities are functions of time, and an organ called the integrator forms the Stieltjes integral z(t) = ∫ x(t) dy(t) for two inputs x(t), y(t).
- Coverage. In suitable combinations the two primitives reproduce addition, subtraction and multiplication.
- Feedback. With feedback tricks they also give division. Feedback looks like a device for solving implicit relations but is really an elegant short-circuited iteration and successive approximation.
- Economy, the true justification. Programming a machine means replacing (or approximating to any prescribed degree) a complex operation by combinations of the machine's basic operations. For a given class of problems one basic set needs simpler, shorter combinations than another; for systems of total differential equations, which the analyzer was built for, (x ± y)/2 plus integration beats the four species.
Von Neumann's point is general: which operations count as "basic" is a matter of efficiency for a problem class, not a law of computation. He returns to this when asking what the nervous system's basic operations might be.
The digital procedure
Markers and their combinations
A decimal digital machine writes each number as in ordinary print, a sequence of decimal digits, and each digit is carried by a system of markers.
- Ten-valued marker. One marker with ten forms suffices, for example a pulse appearing on one of ten pre-assigned lines.
- Two-valued markers. A pulse present or absent on a line, or a pulse of positive or negative polarity; these must be used in groups per digit.
- The minimum group. Three two-valued markers give 2³ = 8 combinations (too few); four give 2⁴ = 16 (more than enough). So at least four binary markers per decimal digit, with any 10 of the 16 patterns stipulated as the digits; larger groups may be used for other reasons.
- Redundancy. The ten-line marker is really a group of ten two-valued markers, highly redundant compared with the four-line minimum.
- Gating. Markers are usually electrical pulses (or voltages and currents held as long as their indication is valid) and must be controlled by electrical gating devices.
- 2³ = 8 patterns: cannot cover 10 digits.
- 2⁴ = 16 patterns: covers 10, leaving 6 unused (modern binary-coded decimal uses exactly this scheme).
- 10 lines for a one-of-ten code versus 4 for the minimum: 2.5 times the wiring.
- 12 markers or groups per channel for a machine handling twelve-place numbers.
Machine types and components
The successive generations of digital machines are defined by their switching parts: electromechanical relays, vacuum tubes, crystal diodes, ferromagnetic cores and transistors, used alone or in combination, some preferably in the memory organs and others in the "active" organs outside memory.
Parallel and serial schemes
The digits of a number can appear in parallel (simultaneously, in different organs) or in series (one after another in a single organ). A machine built for twelve-place decimals, say six places on each side of the point, needs twelve markers or marker groups in every information channel that passes numbers. Almost all machines also make the decimal point position adjustable, a refinement he sets aside.
The four operations as logic
Digital machines have so far always used the four species of arithmetic, but carried out by strict logical rules rather than physical processes.
- Addition. Rules say how to form digit sums, when to carry, and how to repeat and combine. In binary the two-digit table (0+0=00, 0+1=01, 1+1=10) means: sum = 1 if the digits differ; carry = 1 if both are 1. Because a carry may arrive, a three-input table is really needed, and it reads: sum digit = 1 if the number of 1s is odd (1 or 3); carry = 1 if the 1s form a majority (2 or 3).
- Subtraction. Logically similar to addition, and usually reduced to it by "complementing" the subtrahend.
- Multiplication. More plainly logical and more involved: products of the multiplicand with each multiplier digit are formed (often pre-formed for all ten digits) and added with shifts. In binary it is transparent: for a multiplier digit 0 the partial product is omitted, for 1 it is the multiplicand itself. Signed factors add rules for the four sign cases.
- Division. Like multiplication but with iterated trial-and-error subtraction, with rules for forming quotient digits in each alternative case, run as a serial, repetitive scheme.
Add 6 (0110) and 3 (0011). Rightmost column: 0, 1, no carry gives one 1, odd, so sum 1, carry 0. Next: 1, 1, carry 0 gives two 1s, even and a majority, so sum 0, carry 1. Next: 1, 0, carry 1 gives two 1s again, sum 0, carry 1. Last: 0, 0, carry 1 gives sum 1, carry 0. Result 1001 = 9: four applications of the same parity-and-majority rule.
All four operations are "patterns of alternative actions", organized in highly repetitive sequences and governed by strict, logical rules, radically unlike the physical processes of analog machines. Multiplication and division in particular are logically complex; our long, almost instinctive familiarity with them hides this until one tries to state the rules in full.
| Feature | Analog machine | Digital machine |
|---|---|---|
| Number is | A physical quantity (angle, current, voltage) | A string of digits held by markers |
| Arithmetic by | Physical processes (merging currents, gears, integrators) | Logical rules (sum, carry, complement, shift) |
| Basic operations | Can be tailored, e.g. (x ± y)/2 and integration | So far always the four species of arithmetic |
| Organs per operation | As many as the problem needs | One per operation (see Section 2) |
| Precision | Physically limited (Section 3) | Add digits as needed |
The analog option von Neumann describes did not die. Because neural networks tolerate low precision, several groups now compute matrix products physically: IBM Research's analog in-memory chip (Nature, 2023) used about 35 million phase-change memory devices to run speech-recognition models, letting Ohm's and Kirchhoff's laws do the multiply-and-add, exactly the "physical quantity as number" principle of this section.
Logical Control
A machine must not only execute basic operations but run them in the logical pattern that produces the solution. Von Neumann traces control from fixed wiring, through plugboards and tapes, to the scheme that won. In memory-stored control the orders are numbers held in memory, so the machine can compute on its own program and modify it as it runs. Control is the half of the computer comparison that later lets him ask what plays the role of "orders" in the nervous system.
Control in analog machines
Fixed connections: wiring as program
Executing operations one at a time is not enough; the machine must perform them in the sequence, or rather the logical pattern, that generates the answer. In the traditional differential analyzer this is done physically.
- Enough organs up front. The machine must contain, a priori, as many differential gears and integrators as the calculation needs operations of each kind, (x ± y)/2 and ∫ x(t) dy(t).
- Connections form a replica. Their input and output shafts are linked, by cogwheels in early models and by electrical follower arrangements ("selsyns") later, so that the assembly is a physical replica of the calculation.
- Setting up is programming. The connection pattern can be set at will, and this is how the user's intention is impressed on the machine: mechanically in cogwheel machines, by plugging in electrical ones.
- The limitation. In every type the setting is fixed for the entire duration of a problem.
Plugged control and logical tape control
Some of the very last analog machines added a trick. Their plugged connections were actually made by electromechanical relays, so they could be changed by electrically energizing the relay magnets. Those stimuli were driven by punched paper tapes, and the tapes could be started, stopped and restarted by electrical signals from the calculation itself.
Such a signal means some numerical organ has reached a pre-assigned condition: a number's sign has turned negative, or one number has been exceeded by another. Sensing is simple in each medium.
- Voltages or currents. The sign is read by rectifier arrangements.
- Rotating disks. The sign shows whether the disk has passed its zero position moving right or left.
- Comparisons. "a exceeds b" is detected when the sign of the difference b − a turns negative.
The result is a "logical" tape control, better called "state of calculation combined with tape" control, superposed on the basic fixed-connection control. This is conditional behaviour, the germ of branching, achieved in analog hardware.
Principles of digital machines
Only one organ for each basic operation
Digital machines have uniformly one organ per basic operation: one adder, one multiplier, and so on. Most analog machines instead need as many organs of each kind as the problem demands. Von Neumann stresses that this is a historical fact, not an intrinsic requirement.
- Analog could follow suit. Electrically connected analog machines could in principle be built with one organ per operation and a digital-style logical control; the latest relay-and-tape analog control is already a step in that direction.
- Digital exceptions are superficial. Machines that seem to break the rule can be brought back to it by simple reinterpretation, sometimes just by treating them as a duplex or multiplex machine with intercommunication.
Hence the need for memory
With one organ per operation, the results of partial, intermediate calculations must be parked somewhere while the single adder or multiplier does other work. This forces a large number of passive storage organs.
- Memory register. An organ that stores a number (erasing what it held before), accepting it from whatever organ it is then connected to, and repeats it on "questioning", emitting it to another organ.
- Memory. The totality of the registers.
- Capacity. The number of registers in the memory.
Two basic methods of digital control
Von Neumann describes two pure types, with simplifications and idealizations, then their combinations.
Method 1: control sequence points
The machine contains logical control organs called control sequence points, numbering several hundred in some newer machines.
- Simplest mode. Each point is wired to the operation organ it actuates, to the memory registers supplying inputs and to the one receiving the output.
- Handing on. After a definite delay long enough for the operation, or on receipt of a "performed" signal (needed when the operation's duration is variable or its maximum unacceptably long, at the price of an extra connection), it actuates its successor, which does the same according to its own wiring.
- Result. On its own this gives an unconditioned, repetitionless calculation.
Branching points add sophistication. Such a point has two successors and two states, A and B: A sends the process to the first successor, B to the second. Normally in state A, it is connected to two memory registers; say a negative sign in the first flips it from A to B, and a negative sign in the second flips it back (registers store a sign as well as digits, for which one two-valued marker suffices).
- Disjoint branches. The two successors can be entirely separate branches chosen by numerical criteria, with "B to A" used to reset for a new computation.
- Reuniting branches. The branches may merge again in a common later successor.
- Loops. If the A branch leads back to the branching point, the procedure repeats until the criterion that commands "A to B" is met: the basic iterative process.
- Combination. All these tricks can be combined and superposed.
The totality of electrical connections is still the set-up of the problem, the expression of the user's intention, so this is again plugged control: changeable between problems but, in the simplest form, fixed during one. Refinements include points that trigger several operations at once, and pluggings made by relays set from tapes that move on signals from the calculation.
Method 2: memory-stored control
The second method, which has gone "quite far toward displacing the first", replaces control sequence points by orders. An order is, in most designs, physically the same thing as a number: in a decimal machine a sequence of decimal digits, for example the 12 digits of a standard word (sometimes several orders share one word).
- Addresses. All memory registers are numbered serially; a register's number is its address. Basic operations are numbered too.
- Content of an order. The operation number plus the addresses of the input and output registers, as digits in a fixed order.
- Variants. One order may control several operations; it may direct that its addresses be modified before execution, normally by adding the contents of a specified register to them (which von Neumann calls practically the most important variant); or these functions may sit in special orders.
- Operation number. Which basic operation to perform.
- Input addresses. Where the operands come from.
- Output address. Where the result goes.
- Successor. By default the order at X + 1; a transfer order names address Y instead, or every order may carry an explicit successor address.
- Conditional transfer. Go to X or Y depending on whether the number at Z is negative; the order holds a type number in the place of the operation number, plus X, Y and Z.
- Address modification. Add a specified register's contents to the addresses before execution: the later index register.
In plugged control the control points were real, physical objects and their wiring expressed the problem. Now the orders are "ideal entities" stored in memory, and it is the contents of a segment of memory that express the problem: hence the name memory-stored control.
How memory-stored control operates
Because the orders sit in memory, the machine reaches more flexibility than any earlier scheme. Its normal work is to fetch numbers (or orders) from memory, process them as numbers, and write them back. So it can change the very orders that control it.
- Self-modifying order systems. Programs that keep modifying themselves, and with them the computations they control, make possible processes more complex than iteration.
- Practice, not fantasy. Von Neumann notes that, far-fetched as it sounds, this is widely used and very important in contemporary computation planning.
- Loading. The order system, which is the problem and the user's intention, is put into the machine by loading it into memory, usually from a previously prepared tape.
To sum the numbers in registers 100 to 199, with a counter in register I starting at 0: the order at X adds the register at address (100 + contents of I) to an accumulator, using address modification; X + 1 adds 1 to I; X + 2 stores I − 100 in a test register Z; X + 3 is a conditional transfer: if Z is negative go back to X, otherwise continue at X + 4. Without address modification the program would have to rewrite the address inside order X on every pass, exactly the self-modification von Neumann describes.
| Scheme | Where the problem lives | Changeable during a run? | Conditional behaviour |
|---|---|---|---|
| Fixed connections (analog) | Gears, shafts, selsyns: a physical replica | No | None |
| Plugged, relay-and-tape (analog) | Plugboard driven by punched tape | Yes, by tape | Tape starts and stops on sign or comparison signals |
| Control sequence points (digital) | Plugged wiring among several hundred points | Simplest form: no | Branching points with states A and B; loops |
| Memory-stored control (digital) | Orders held as numbers at addresses | Yes, the machine rewrites them | Conditional transfer on sign of number at Z |
Mixed forms of control
The two pure modes combine. Take a plugged-control machine that also has a memory of the stored-control type.
- Plugging as data. Its complete plugging state can be written as a sequence of digits, occupying several consecutive registers; the first one's address serves as the scheme's address, and several plugging schemes can be stored at once.
- Order 1. Reset the whole plugged set-up from the digit sequence stored at a given address.
- Order 2. A system of orders that change single items of plugging. Both require the plugging to be made by electrically controllable devices: relays, vacuum tubes or ferromagnetic cores.
- Order 3. Hand control from the memory-stored regime to the plugged regime.
- The return path. The plugging scheme must in turn be able to name the memory-stored control, at a specified address, as the successor (or one branch successor) of a control sequence point.
Von Neumann writes as if memory-stored control were an impersonal development; he does not mention the EDVAC dispute with Eckert and Mauchly over credit for the stored-program idea, nor Turing's 1936 universal machine, whose tape already held instructions as symbols. His "several hundred" control points and 12-digit decimal words describe the early-1950s machines he knew, not later binary designs.
- Still the norm. Virtually every CPU and GPU core still fetches stored instructions with addresses, branches and index registers.
- Self-modifying code fell out of favour; index registers (first built as the Manchester "B-line", 1949) did the address arithmetic instead, and modern systems mark code pages read-only for security.
- The bottleneck. John Backus named the "von Neumann bottleneck" in his 1977 Turing Award lecture: the single channel between processor and memory. AI accelerators now attack it with on-chip memory and compute-in-memory designs.
- Mixed control lives on. FPGAs are the modern plugged machine: their wiring is itself stored as a digit sequence (a bitstream) that a processor can load, much like von Neumann's "plugging state held in memory".
Mixed Numerical Procedures and Precision
Von Neumann describes hybrid machines, including a "pulse density" code that quietly foreshadows the nervous system, then asks why digital machines carry 12 decimal digits when real data rarely justify 3. The answer is depth: long chains of operations accumulate and, worse, amplify errors, so each step needs far more precision than the result. That argument becomes the yardstick for judging how the brain, with two or three digits, can compute at all.
Mixed numerical procedures
Analog and digital parts in one machine
The first mixed type joins an analog part and a digital part that exchange numerical material under a common control; alternatively each part has its own control and the two controls exchange logical material. Either way the machine needs converters in both directions.
- Digital to analog. Build a continuous quantity from its digital expression.
- Analog to digital. Measure a continuous quantity and express the result as digits.
- Feasibility. Components for both tasks are well known, including fast electrical ones.
Mixed representation: the pulse density system
A second, more intimate mix combines analog and digital principles in every computing step (though not in the logical procedure), most simply by representing each number in a part-analog, part-digital way. Von Neumann describes one such scheme, used occasionally in components, planning and communications, though no large-scale machine was ever built on it.
- Representation. A number is carried by a sequence of electrical pulses on a single line; the sequence's length is irrelevant, and its average density in time is the number.
- Averaging window. Two intervals t₁ and t₂ (t₂ much larger than t₁) are fixed, and densities are averaged over durations between them; a unit must also be fixed.
- Monotone variant. The density may instead equal a fixed monotone function of the number, such as its logarithm, giving fine resolution where it is needed (small numbers) and coarse where it is acceptable (large ones), with continuous shadings between.
- Arithmetic. If density equals the number, addition is just merging the two pulse streams; the other operations are trickier but have adequate procedures; negative numbers are handled by "suitable tricks".
- Precision and change. Each sequence needs many pulses within each t₁; a number may change during the computation only slowly compared with t₂.
- Control. Sensing numerical conditions directly is awkward, so densities are converted to analog quantities: pulses each delivering a standard charge through a resistance into a slowly leaking condenser hold it at a nearly constant voltage and leakage current, both usable for logical control.
Von Neumann presents pulse density as an engineering curiosity, but it is exactly the frequency code he later attributes to nerve fibres (Section 9): all-or-none pulses whose rate carries a magnitude, read out by something like a leaky integrator. The precision limits he states here are the limits he will impose on the brain.
Precision
Analog against digital
Setting other factors aside, precision is the main limit of analog machines.
- 1:10³. Electrical analog machines rarely do better.
- 1:10⁴ to 1:10⁵. Mechanical differential analyzers at best. Going from 10³ to 10⁴ is relatively simple, 10⁴ to 10⁵ is about the best current technology can do, 10⁶ is impossible with present means.
- 1:10¹². A twelve-decimal digital machine, a typical modern level; one more digit costs about 1/12 = 8.3% more equipment and a similar loss of speed (not everywhere), which is not serious.
- 1:10². Pulse density, intrinsically worse than analog: it needs about 10² pulses per t₁, cutting speed 100-fold by that alone; such losses are hard to accept and larger ones prohibitive.
| Representation | Typical precision | Cost of one more digit | Verdict |
|---|---|---|---|
| Electrical analog | 1:10³ | Hard engineering | Limited |
| Mechanical analog (differential analyzer) | 1:10⁴ to 1:10⁵ | 10⁶ impossible | Near its ceiling |
| Pulse density | ~1:10² | 10× more pulses, 10× slower | Intrinsically low |
| Digital (12 places) | 1:10¹² | ~8.3% more equipment and time | Any precision desired |
Why demand such extreme precision
The puzzle: data in applied mathematics and engineering are seldom better than 1:10³ or 1:10⁴, often not even 1:10², and answers are neither needed nor meaningful beyond that; chemistry, biology and economics are less exacting still. Yet the uniform experience of high-speed computing is that 1:10⁵ is inadequate for many important problems and 1:10¹⁰ to 1:10¹² is fully justified. Von Neumann traces this to the structure of our present mathematical and numerical procedures.
- Procedures are long. Every problem worth a fast machine, anything of at least medium complexity, breaks down into very long sequences.
- Why long. Current methods analyse all functions into combinations of basic operations, usually the four species of arithmetic; most functions can only be approximated that way, by long, often iteratively defined chains.
- Arithmetical depth. The number of basic operations in sequence is therefore great.
- Logical depth. Greater still, by a large factor, since each arithmetic operation is itself a long logical chain; he needs only arithmetical depth for the argument.
Two ways error grows
Superposition. Errors from N operations add, but since they are largely (not entirely) random, the total grows like √N, not N. That alone cannot explain the requirement: to turn a stepwise 1:10¹² into an overall 1:10³ would need √N of about 10⁹, far beyond any real run. Even an extreme case, one operation every 20 microseconds for 48 hours on one problem, gives only N ≈ 10¹⁰ (172,800 s ÷ 20 μs ≈ 8.6 × 10⁹).
Amplification. The decisive factor is that later operations can magnify errors introduced earlier, which "can cover any numerical gulf very quickly". The gap between 1:10³ and 1:10¹² is a factor of 10⁹, yet 425 successive operations that each enlarge an error by only 5% produce it (1.05⁴²⁵ ≈ 10⁹).
High stepwise precision is insurance against depth: the more steps in sequence, the more digits each step must carry. Von Neumann declines a detailed estimate, noting that much of the art of computing consists of keeping amplification down, but experience shows the high levels are justified for any reasonably complicated problem. A system that cannot afford the digits must instead keep its calculations shallow, which is precisely his later inference about the brain.
He closes the computer discussion of method here and turns to speeds, sizes and the like, the hardware parameters he needs before comparing them with neurons.
- 10¹⁶ or 10¹⁸. The superposition step implies N ≈ 10¹⁸ (since √N must reach 10⁹); this edition's text gives 10¹⁶, apparently a slip. The conclusion stands either way, since real N was about 10¹⁰.
- Randomness assumed. The √N law holds only for independent, unbiased errors; systematic rounding bias grows linearly in N, which strengthens rather than weakens his case.
- The √ sign is lost in some extractions of this passage ("about times"); the intended law is √N.
- Science still runs in FP64 (about 16 decimal digits).
- AI went brain-ward: training and inference moved to FP8 and FP4 (one or two significant decimal digits), and Microsoft's BitNet b1.58 (2025) restricts weights to -1, 0, +1. Neural networks are shallow, statistical and error-tolerant, which is exactly von Neumann's description of the brain (Section 9 onward).
- Analog returns. The IBM analog in-memory chip noted in Section 1 (Nature, 2023) reached 12.4 TOPS/W on speech tasks, trading digits for energy exactly as this section predicts.
- Pulse density returns. Spiking neuromorphic systems carry values as pulse timing and rate: Intel's Hala Point (April 2024) links 1,152 Loihi 2 chips simulating about 1.15 billion neurons.
Characteristics of Modern Analog and Digital Machines
Von Neumann closes Part 1 with the hardware numbers of 1956: component speeds, counts, memory access times and the hierarchic principle that organizes memory. Fast memory is expensive, so a machine needs only a little of it at the fastest access time and progressively more at slower ones, a design rule every computer still follows. These are the figures he will set against neurons in Part 2.
Modern analog machines
- Size. The largest analog machines have one or two hundred basic-operation organs, recently all electrical or electromechanical (the mechanical stage kept for extra precision).
- Control hardware. An elaborate logical control adds typical digital organs, relays or vacuum tubes (not driven at extreme speed), up to a few thousand.
- Cost. In extreme cases the investment reaches about $1,000,000.
Modern digital machines: the active organs
Large digital machines are more complex. Von Neumann splits them into active organs and memory organs, counting input and output among the memory organs, which he admits is unusual. The classification matters later: the outside world becomes the last level of the memory hierarchy.
- Logic. Sense coincidences, combine stimuli, and possibly sense anticoincidences. Nothing more is necessary (more complex logical organs are sometimes provided), and these are the elements from which arithmetic is built.
- Regeneration. Restore a pulse's gradually attrited energy, or lift it from one part's energy level to a higher one (both called amplification), and restore a standardized pulse shape and timing within tolerances.
Speed of the active components
The same functions have been performed, in succession, by relays, vacuum tubes, crystal diodes, ferromagnetic cores and transistors, or small circuits combining them.
- 10⁻² s per elementary logical action for relays.
- 10⁻⁵ to 10⁻⁶ s for vacuum tubes, in extreme cases a half or a quarter of 10⁻⁶.
- 10⁻⁶ s (sometimes a small multiple) for the new solid-state devices, likely to reach 10⁻⁷ s or better.
- 10⁻⁸ to 10⁻⁹ s expected from other devices within a decade.
- 3,000 to 30,000 active organs in a large machine.
- 300 to 2,000 of them in the arithmetical organ, one more or less merged group of subassemblies.
- 200 to 2,000 aggregated to perform memory functions (the registers below).
- 300 to 2,000 ancillary organs to service the second memory level; across all memory, ancillary circuits can reach 50% of the machine.
Memory: access time
Memory organs are classified by access time, which has two parts.
- "In" time. Time to store a number already elsewhere in the machine (usually in an active-organ register), erasing what was there.
- "Out" time. Time to repeat the stored number on "questioning" to another part that can accept it.
- One figure. One may keep the two apart, take the larger, or the average; von Neumann uses a single access time for simplicity.
- Random access. If access time does not depend on the address, the memory is random access. If it varies, use the maximum or the average, which may depend on the statistics of the problems being run.
Registers built from active organs
Registers made from active organs have the shortest access time and are the most expensive.
- Cost. At least four vacuum tubes (or not many fewer solid-state devices) per binary digit or sign, so at least four times that per decimal digit. A 12-decimal-digit word plus sign needs 12 × 4 = 48 binary digits plus 1 sign = 49 positions, × 4 tubes = a 196-tube register.
- Speed. Access in one or two elementary reaction times, very fast compared with the alternatives.
- Why they are still needed. Several can be integrated with economies; they are required as "in" and "out" access organs for other memories; one to three are part of the arithmetic organ. In moderate numbers they are cheaper than expected and necessary.
- Why not bulk memory. They cannot supply the large capacities nearly every big machine needs. (In the earlier relay epoch, relay registers were the main memory; his analysis covers vacuum-tube machines and later.)
The hierarchic principle
A machine may need N words at access time t, but providing all N at t is technically hard or, what amounts to the same, very expensive, and usually unnecessary.
- Step 1. Perhaps only a much smaller N′ words are needed at t.
- Step 2. Once N′ are provided at t, the full N may be needed only at a slower t″.
- Step 3. Intermediate capacities at intermediate access times may be cheapest of all.
- General scheme. A ladder of k levels with capacities N₁ < N₂ < … < Nₖ and access times t₁ < t₂ < … < tₖ, so that Nᵢ words are available at tᵢ; capacities grow more demanding and access times less demanding as one descends. (N₁ = N′, t₁ = t; Nₖ = N, tₖ = t″.)
The levels in a 1956 machine
A full count in a large high-speed machine shows at least three, possibly four or five levels.
- Registers. N₁ at least 3, sometimes more; up to 20 has been proposed. Access t₁ is the basic switching time, or twice it.
- Specific memory organs. Devices distinct from the switching organs: electrostatic tubes and magnetic core arrays, cores "definitely on the ascendant" (ferro-electric and other techniques may enter). N₂ from a few thousand to a few tens of thousands of words (the larger sizes still in design); t₂ five to ten times t₁.
- Later levels. Mostly magnetic drums and tapes; magnetic discs suggested and occasionally explored. Capacity rises by about 10× per level, access time even faster, with special rules intervening.
- The outside world. The part of the world the machine communicates with directly: punched paper tape or cards in, printed paper out; sometimes magnetic tape is the final medium, transcribed to paper off the machine.
Complications of access time
The slower devices have special access rules. A drum presents all its parts successively and cyclically; a tape has practically unlimited capacity but presents its parts in fixed linear order, which can be stopped and reversed. Any of these can be combined with arrangements that synchronize the machine's operation with the memory's fixed sequence, so "access time" is no longer one number.
- 5 to 15 µs for ferromagnetic core memories.
- 8 to 20 µs for electrostatic memories.
- 2,500 to 20,000 rpm for drums, one revolution per 24 to 3 ms, during which 1 to 2,000 words may be fed.
- 70,000 lines per second for tape, one line per 14 µs; a word takes 5 to 15 lines (so roughly 70 to 210 µs per word).
The principle of direct addressing
Every existing machine uses direct addressing: each word in the whole memory (all hierarchic levels together) has its own numerical address, which identifies it and its position uniquely and is always specified explicitly when the word is read or written.
- Partial availability. Not all parts of memory may be accessible at once; there may be multiple memories with access priorities, so access depends on the general state of the machine at the moment.
- No ambiguity. Even so, there is never any doubt about the address or the place it designates.
Direct addressing is the assumption von Neumann never manages to transfer to the nervous system: his memory section (Part 2) cannot even say where the brain's memory sits, let alone how an item is found. Later writers (and today's AI) supplied the alternative: content-addressable, associative recall, where a fragment of a pattern retrieves the rest.
- Hierarchy notation. The printed text garbles the ordering (it shows N(k+1) and an "&" for a primed t); the intended rule is simply N₁ < … < Nₖ and t₁ < … < tₖ.
- Counting I/O as memory is his own convention, chosen with the brain in mind (sense organs as an interface with an "outside" store), not standard practice then or now.
- Speed. His "10⁻⁸ to 10⁻⁹ s within a decade" was about right: 1960s supercomputers had ~100 ns cycles with nanosecond logic. Transistors on TSMC's 2 nm N2 process (volume production from late 2025, its first gate-all-around nanosheets) switch in picoseconds.
- The hierarchy is permanent. Caches, DRAM and flash are his N, t ladder; the memory wall (processors outrunning memory) is the main constraint on AI hardware today, which is why GPUs stack HBM beside the die.
- Discs, merely "suggested" in his text, arrived in 1956 with IBM's RAMAC 350 (5 million characters).
| Hierarchy level | 1956 (von Neumann) | 2026 |
|---|---|---|
| Registers | 3 to 20 words, one switching time | Hundreds of registers per core, sub-nanosecond |
| Fast memory | Cores, electrostatic tubes; thousands of words; 5-20 µs | SRAM caches (MBs) in ns; HBM4, 288 GB per Nvidia Rubin GPU at 22 TB/s |
| Bulk storage | Drums (3-24 ms per turn), tape | NAND flash SSDs (µs), hard discs (ms) |
| Outside world | Punched cards, paper tape, print | Networks, cloud object storage |
| Active organs | 3,000 to 30,000 tubes or transistors | 336 billion transistors in one Rubin GPU |
The Brain: The Nature of the Nerve Impulse
Part 2 turns to the other "automaton". Von Neumann calls the nervous system "prima facie digital", because a neuron either emits a standard pulse or stays silent, then sets the neuron against the vacuum tube and transistor on speed, size and energy. Natural components are about 10⁴ to 10⁵ times slower but 10⁸ to 10⁹ times more economical, so an efficient brain must be massively parallel while an efficient computer is serial. The three conclusions drawn here organize the rest of the book.
The plan of Part 2
Part 1 supplied the vocabulary; Part 2 applies it to the human nervous system, treated as a second kind of automaton. Von Neumann sorts the comparison into three kinds of point.
- Similarities. Well-known territory, covered quickly.
- Superficial dissimilarities. Obvious ones of size and speed, which still need careful numbers.
- Deep dissimilarities. Differences in principles of functioning and control and in overall organization: his primary aim, which can only be judged properly against the other two.
Why the nervous system looks digital
The neuron and its impulse
The first observation is that the nervous system's functioning is "prima facie digital". The basic component is the nerve cell or neuron, whose normal function is to generate and propagate a nerve impulse. That impulse is complex, with electrical, chemical and mechanical aspects, yet it is nearly the same under all conditions: an essentially reproducible, unitary response to a wide variety of stimuli.
A neuron is a cell body from which one or more branches, the axons, originate. The impulse is a continuous change that travels along each axon, usually at a fixed speed that may depend on the cell.
- Electrical. Most often described this way: a potential of about 50 millivolts lasting about one millisecond.
- Chemical. Where the pulse passes, the ionic composition of the fluid inside the axon changes, and so do the conductivity and permeability of the axon wall, the membrane. At the axon's endings the chemistry is plainest: specific substances appear when the pulse arrives.
- Mechanical. Very probably the permeability changes require reorientation of membrane molecules, a change in their relative positions.
- Reversibility. All these changes reverse: once the impulse passes, every condition along the axon and every constituent returns to its original state.
- Why the aspects merge. The membrane is a few tenth-microns thick (of order 10⁻⁵ cm), a molecular dimension for the large organic molecules involved. At that scale every chemical change is driven by changed intramolecular forces (so is mechanically induced), and every mechanical change alters electrical properties. Distinctions that are sharp at the macroscopic scale blur at the membrane, so the impulse can be described in any of the three vocabularies.
Stimulation: success or failure
Since the developed impulse is the same however it was produced, its cause can be described electrically or chemically. Inside the nervous system it is mostly triggered by one or more other nerve impulses, and the outcome is binary.
- Failure. A passing disturbance arises but dies out within a few milliseconds; nothing propagates.
- Success. The disturbance quickly takes a nearly standard form and spreads along the axon, its appearance largely independent of how it was induced.
- Where. Stimulation normally occurs in or near the cell body; propagation runs along the axon.
Pulses as binary markers
This is the core of the digital reading. A pulse is a two-valued marker in the sense of Part 1: absence = 0, presence = 1, interpreted as an event on a specific axon (all the axons of a given neuron) and possibly in a specific time relation to other events, so that each marker plays a definite logical role.
- Conditional stimulation. A neuron's output pulse is normally triggered by pulses arriving at its body, but only certain combinations and synchronisms of incoming pulses succeed; all others fail.
- The neuron as active organ. It accepts and emits definite physical entities, pulses; the rules saying which input groups make it fire are the rules governing it as an active organ.
- Conclusion. That is exactly how an organ of a digital machine is described, which justifies the "prima facie digital" claim.
The description contains idealizations and simplifications (taken up in Sections 6 and 9). Once they are allowed for, the digital character no longer stands out "so clearly and unequivocally". Von Neumann still starts here because these are the most conspicuous traits: the honest order is digital first, then the qualifications.
Speed: synapse, fatigue, recovery
Before the qualifications come orienting figures on size, energy and speed, stated against the main artificial competitors, the vacuum tube and (more recently) the transistor.
Anatomy needed for the timing
- Stimulation along an axon. A suitable electrical potential or chemical stimulant applied mid-axon starts a disturbance that becomes a standard pulse travelling both up and down the axon.
- Dendrites. The "usual" stimulation lands on short branches near the cell body that are essentially small axons; these stimulation receptors are the dendrites, from which the pulse spreads to the body and on to the regular axons.
- Synapse. Stimulation by another neuron's pulse comes from a special ending of that neuron's axon, the synapse.
- Side question. Whether a pulse travelling along an axon can directly excite an exceptionally close-lying neighbour axon, bypassing synapses, is left open; appearances favour such a "short-circuited" process.
Two definitions of reaction time
- A few × 10⁻⁴ s. Trans-synaptic stimulation time: from a pulse's arrival at a synapse to the stimulated pulse appearing on the nearest point of the next neuron's axon.
- 1.5 × 10⁻² s. Minimum interval after which a second stimulation can again produce a standard response: the more significant measure for machine economy.
- 1-2% of that interval is actual synaptic transmission; the rest is recovery from fatigue.
- 0.5 × 10⁻² s. Partial recovery: the neuron can already fire a standard pulse, but only to a significantly stronger stimulus (a point he promises to revisit, Section 6).
- 10⁻⁶ to 10⁻⁷ s. Reaction times of tubes and transistors in large logical machines, full recovery included.
- 10⁴ to 10⁵. The artifacts' speed advantage.
Immediately after firing, the neuron has not returned to its pre-stimulation state: it is fatigued and cannot respond normally to another pulse. So the neuron's reaction time lies anywhere from 10⁻⁴ to 10⁻² s depending on definition, and von Neumann takes the longer, recovery-inclusive figure as the meaningful one, the same convention he applies to tubes.
Size: three measures, one answer
Linear size of the active part
A neuron's linear size varies enormously: some sit in tight aggregates with very short axons, others carry pulses between distant parts of the body and are comparable in length to the body itself. So he compares only the logically active part of each organ.
- Neuron. The cell membrane, a few times 10⁻⁵ cm.
- Vacuum tube. The grid-to-cathode distance, 10⁻¹ to a few × 10⁻² cm.
- Transistor. The distance between the "whisker electrodes" (the non-ohmic emitter and control electrode), multiplied by about 3 to include the active surroundings: somewhat under 10⁻² cm.
- Result. Natural components lead by about 10³ in linear size.
Volume and packing density
Packing density is the better measure on both sides, since what matters is how many organs fit in a given space.
- Brain. The central nervous system fills about a litre (10³ cm³) and holds about 10¹⁰ neurons, "or somewhat higher": about 10⁻⁷ cm³ per neuron.
- Machines. A few thousand vacuum tubes occupy several times 10 ft³; the same number of transistors fit in one or a few ft³. Taking the best figure: about 10⁵ cm³ for a few × 10³ organs, so 10 to 10² cm³ per active organ.
- Result. Natural components lead by 10⁸ to 10⁹ in volume.
- Cross-check. The cube root of 10⁸ to 10⁹ is 0.5 to 1 × 10³, which he says agrees with the direct linear estimate.
Energy dissipation
A logical organ does no work: its output pulse need carry no more energy than its prorated share of the input pulses, and there is no necessary link between the two. So virtually all energy consumed is dissipated as heat, and "consumption" and "dissipation" are the same thing.
- Brain. About 10 watts across 10¹⁰ neurons: 10⁻⁹ W per neuron.
- Vacuum tube. About 5 to 10 W.
- Transistor. As little as 10⁻¹ W.
- Result. Again 10⁸ to 10⁹ in nature's favour, matching the volume figure.
| Measure | Neuron | Tube or transistor | Factor |
|---|---|---|---|
| Linear size of active part | Membrane, a few × 10⁻⁵ cm | Grid-cathode gap 10⁻¹ to a few × 10⁻² cm; transistor whiskers (×3) under 10⁻² cm | ~10³ for the neuron |
| Volume per organ | 10³ cm³ brain / 10¹⁰ neurons = 10⁻⁷ cm³ | A few thousand transistors in about 1 ft³: 10 to 10² cm³ | 10⁸ to 10⁹ for the neuron |
| Power per organ | 10 W / 10¹⁰ = 10⁻⁹ W | Tube 5-10 W; transistor 10⁻¹ W | 10⁸ to 10⁹ for the neuron |
| Reaction time | 10⁻² s (10⁻⁴ s synaptic) | 10⁻⁶ to 10⁻⁷ s | 10⁴ to 10⁵ for the artifact |
Three conclusions
Summing up: the size factor is about 10⁸ to 10⁹ for nature (from the cube of the linear factor, from volume and from dissipation), against 10⁴ to 10⁵ for the artifacts on speed. Von Neumann stresses that the discussion is still "very near to the surface" and the conclusions open to revision, but states them anyway.
- Figure of merit. In actions performed per unit time by organs of the same total size (volume or dissipation), nature leads by about 10⁴: the quotient of 10⁸ to 10⁹ by 10⁴ to 10⁵.
- Parallel versus serial. Nature favours more, slower organs; artifacts favour fewer, faster ones. An efficient large natural automaton will pick up and process as many items as possible simultaneously (highly parallel); an efficient artificial one will do things one at a time, or nearly (serial).
- Not freely interchangeable. Conclusion 1 assumes parallel and serial operation can be swapped, which is false. Some operations need earlier results and cannot run alongside them, so moving from serial to parallel may be impossible or may require a change in logical approach. Serializing a parallel procedure almost always creates new memory requirements, since early results must be stored while later steps run. Hence the logical structures of natural and artificial automata should differ widely, and artificial ones should need systematically more memory.
The size-over-speed "figure of merit" is arithmetic; the real claim is that component physics shapes logical organization. Cheap, slow parts push a designer toward breadth; expensive, fast parts toward depth plus memory. Everything later about the brain's shallow, statistical computation follows from this.
- Membrane thickness. A cell membrane is about 5-10 nm (5-10 × 10⁻⁷ cm), roughly thirty to fifty times thinner than "a few tenth-microns", so the neuron's linear lead was larger than he computed.
- Internal slip. He says the cube-root result agrees with "the 10² arrived at above", but his direct estimate was 10³; the Churchlands quote the lower figure (Foreword, 2000).
- Recovery and amplitude. The absolute refractory period is about 1 ms, and many neurons fire at hundreds of Hz; his 15 ms describes typical firing intervals, not a limit. A spike swings about 100 mV (roughly -70 to +30 mV), not 50.
- Neurons are not the only parts. The brain's 10 W also powers glia, roughly as numerous as neurons, so "10⁻⁹ W per neuron" overstates a neuron's share slightly.
- Counts. The human brain has about 86 billion neurons (Azevedo et al., 2009) and 10¹⁴ to 10¹⁵ synapses; it runs on about 20 W, twice his 10 W.
- Silicon now wins per component on size, speed and switching energy: an Nvidia Rubin GPU (in production 2026) packs 336 billion transistors, switching in picoseconds.
- The brain still wins as a system. One Rubin GPU draws 1.8-2.3 kW, a Vera Rubin NVL72 rack 190-230 kW. Intel's neuromorphic Hala Point (April 2024) simulates 1.15 billion neurons and 128 billion synapses, about 1% of a human brain, at up to 2,600 W.
- Conclusion 2 came true. Clocks stalled near 5 GHz in the mid-2000s, so computing grew by parallelism, the strategy he assigned to nature.
- Conclusion 3 too. Serial-to-parallel conversion is limited by dependent steps (Amdahl's law, 1967), and today's accelerators are bound more by memory traffic than arithmetic, the "new memory requirements" he predicted.
Stimulation Criteria
Von Neumann now takes back part of the "prima facie digital" picture. A neuron with the right threshold is an AND or an OR gate, but real neurons sum many inputs over time, may respond to the geometry of their synapses, and some react to change rather than level. Whatever these complexities do, they mean a nerve cell is more than one basic active organ: the meaningful count is of synapses, perhaps 10 to 100 times the neuron count.
Three kinds of stimulus
The neuron's normal output is the standard nerve pulse, and it can be triggered in three ways. This section examines the idealizations hidden in the earlier digital description, which he warns are "not at all trivial to evaluate".
- Other neurons. The arrival of one or more nerve pulses from other cells, the case treated first.
- The outside world. Phenomena to which a particular neuron is specifically sensitive: light, sound, pressure, temperature.
- The inside of the organism. Physical and chemical changes at the neuron's own location.
The simplest criterion: elementary logic
Take a neuron contacted, via synapses, by the axons of two other neurons.
- Threshold of two simultaneous pulses. It fires only when both stimulators are active: an "and" organ performing conjunction.
- Threshold of one pulse. It fires when either is active: an "or" organ performing disjunction.
- Completeness. "And" and "or", together with "no" (negation), form a complete set: every logical operation, however complex, can be built from them. He skips how neurons might produce "no", or what tricks avoid needing it.
- Conclusion. Viewed this way, neurons are the basic logical, and hence the basic digital, organs.
A single parameter, the minimum number of simultaneous inputs, turns one kind of cell into different logical operations. This is the bridge between neurophysiology and Boolean logic on which the whole "prima facie digital" reading rests, and the rest of the section shows how quickly it strains.
More complicated stimulation criteria
The two-input neuron is an idealization; real neurons are "not so simply organized".
- A few synapses. Some neurons do carry only one, two or a few easily counted synapses.
- Many synapses. More often a neuron's body carries synapses from the axons of many other neurons, and occasionally several axons from the same neuron synapse on one cell.
- Counting rule. With many synapses the simplest behaviour is to fire when a minimum number of simultaneous pulses (or more) arrive.
- Geometry rule. Plausibly, things are more complex: with hundreds of synapses on one cell, the effective input combinations may be defined by which special regions of the body or dendrite system they cover, by the spatial relations between those regions, and by still more complicated quantitative and geometrical relationships.
The threshold
When the criterion is the simple one, a minimum number of simultaneous pulses, that minimum is the neuron's threshold. Physiologists customarily describe stimulation requirements this way, but von Neumann insists it is by no means established that the requirement has this simple form rather than the geometric one above.
The summation time
Further complexities lie outside the simple stimulus-response relation.
- Period of grace. "Simultaneous" never means exactly simultaneous: two pulses within a finite summation time act as if simultaneous.
- Fuzzy window. The summation time may not be sharp: a slightly earlier pulse may still be summed with gradually decreasing, partial effect.
- Trains. Pulse sequences spaced farther apart than the summation time may, by their length, produce more than their individual effect.
- Abnormal states. Superposed fatigue and recovery can leave a neuron with response characteristics different from the standard ones.
- Verdict. Incomplete bodies of observation all suggest the neuron can be, in special situations, much more complicated than the "dogmatic" stimulus-response description by elementary logic can express.
- Elementary logic. Two inputs, threshold 2 (AND) or 1 (OR).
- Threshold. A minimum number of simultaneous pulses among many synapses.
- Spatial pattern. Which synapses fire, where on the cell, and how those regions relate geometrically.
- Summation time. "Simultaneous" means within a finite, possibly fuzzy period of grace; trains add extra effect; fatigue and recovery create abnormal states.
- Derivative criteria (receptors). Response to a change of stimulus, not its level.
Stimulation criteria for receptors
Von Neumann proposes calling all neurons that respond to stimuli other than nerve pulses receptors, divided into external (responding to the organism's surface: light, sound, pressure, temperature) and internal (responding to local chemical or physical change). The same question arises: under what condition does a pulse fire?
| Receptor type | Threshold criterion (minimum intensity of…) |
|---|---|
| External: light | Illumination |
| External: sound | Sonic energy within a given frequency band |
| External: pressure | Overpressure |
| External: temperature | Rise in temperature |
| Internal: chemical | Change in concentration of the critical agent |
| Internal: physical | Change in the relevant physical parameter |
Threshold is not the only criterion. In the optic nerve, many fibres respond not to a level of illumination but to its change: some to the passage from darkness to light, others from light to darkness. The stimulus is the size of the derivative, not the level itself. These responses might come not from single neurons but from the output of more complicated neuron systems, a question he leaves aside; either way, receptors too use criteria beyond threshold.
What the complexities might mean
Von Neumann weighs three readings of all these complications.
- No function. They may play no useful role at all.
- A mixed system. More interestingly, in an essentially digitally organized system they might play an analog or "mixed" role. Some authors have suggested that general electrical potentials influence functioning, so that the system responds to solutions of potential-theory problems "in toto", less elementary than digital stimulation criteria. Since the system is probably primarily digital, such effects would interact with digital ones, giving a mixed rather than genuinely analog system. He refers the reader to the literature and does not pursue it.
- A bigger count. Whatever their function, every such complication means a nerve cell is more than one basic active organ. If certain combinations of synapses fire a cell and others do not, the significant count is of synapses, not cells, raising the number of basic active organs by a factor like 10 to 100; mixed phenomena make counting harder still.
- 10¹⁰ neurons in his estimate becomes 10¹¹ to 10¹² basic active organs at 10 to 100 synapses each.
- Each such organ is still slow (~10⁻² s cycle), so the parallelism argument of Section 5 gets stronger, not weaker.
- The size advantage of nature (10⁸ to 10⁹ per neuron) is shared among more organs, so the per-organ lead shrinks by the same factor.
- Unnamed source. The AND/OR neuron is the McCulloch-Pitts model (1943), which von Neumann had used in his 1945 EDVAC report; he does not cite it here.
- Inhibition skipped. Negation in real circuits comes from inhibitory synapses, which he leaves aside, although inhibition was well established by Sherrington's work decades earlier.
- Factor too small. A cortical neuron receives thousands of synapses, so a synapse count exceeds the neuron count by roughly 10³ to 10⁴, not 10 to 100.
- The derivative fibres he describes were reported by H. K. Hartline (1938) in "on", "off" and "on-off" retinal ganglion cells; they are indeed outputs of retinal circuits, not single receptors, as he suspected.
- Dendritic logic. Gidon et al. (Science, 2020) found calcium-mediated spikes in human cortical dendrites that compute an XOR-like function, which a single threshold unit cannot: his "spatial relations" criterion made concrete.
- A neuron as a deep network. Beniaguev, Segev and London (Neuron, 2021) needed a temporally convolutional network of 5 to 8 layers to reproduce one cortical pyramidal neuron's input-output behaviour.
- Field effects are real but small. Ephaptic coupling, neurons influencing each other through the extracellular electric field, has been measured in cortex and hippocampus, a modest version of his "general electrical potentials".
- AI kept level 2. Artificial neural networks still use a weighted sum plus a nonlinearity, essentially his threshold neuron with graded weights.
The Problem of Memory within the Nervous System
Every computer built so far has needed a memory, so von Neumann postulates one in the nervous system while admitting nobody knows where or what it is. His lifetime estimate of 2.8 × 10²⁰ bits rests on the assumption that nothing is ever truly forgotten; his firmer point is that memory need not be built from the same parts as the active organs.
Why postulate a memory
So far the comparison has ignored one component whose presence in the nervous system is "highly plausible, if not certain". Every artificial computing machine has needed a memory, which suggests its role is a matter of principle, not accident. The case is inference from machine experience, not observation.
- One or several. There may be several memories, not one.
- All physical claims are guesses. Its nature, embodiment and location are equally hypothetical: a separate organ or parts of known organs, possibly a large system of specific nerves, possibly linked to the body's genetic mechanism.
- The Greek comparison. We are as ignorant as the Greeks who placed the mind in the diaphragm.
- The two certainties. It must have large capacity, and it is hard to see how an automaton as complicated as the nervous system could do without one.
How to measure capacity
Von Neumann borrows the standard bookkeeping of computing machines and of information theory (Shannon "and others"): any information converts into binary digits (bits), and the capacity of a mixed memory is the sum of its parts.
- Decimal digit. log₂ 10 ≈ 3.32 bits. Check: 2¹⁰ = 1,024 ≈ 10³, so three decimal digits ≈ 10 bits and one digit ≈ 10/3 ≈ 3.33 bits.
- 1,000 eight-place numbers. 1,000 × 8 × 3.32 = 2.66 × 10⁴ bits.
- Typed character. An 88-way choice: 2 cases × 26 letters + 35 punctuation marks, numerals and spaces; worth log₂ 88 ≈ 6.45 bits.
- 1,000 letters. 6,450 = 6.45 × 10³ bits.
- Shapes and colours. Geometric shapes and colour nuances, given a stated precision and resolution, also reduce to bits.
The lifetime estimate
A modern machine's memory is about 10⁵ to 10⁶ bits. The brain, a far larger automaton, should need much more; how much is hard to say, but a rough orienting estimate is possible.
- Receptor rate. A standard receptor accepts about 14 distinct digital impressions per second, counted as 14 bits/s.
- Every cell a receptor. Allow 10¹⁰ nerve cells and assume each is, under suitable conditions, essentially an inner or outer receptor: total input 14 × 10¹⁰ bits/s.
- No true forgetting. Assume (with "some evidence") that impressions may leave the centre of attention but are never truly erased.
- Lifetime. Take 60 years ≈ 2 × 10⁹ s.
- Product. 14 × 10¹⁰ × 2 × 10⁹ = 2.8 × 10²⁰ bits.
- Sanity check. This exceeds the machine figure of 10⁵ to 10⁶ by a lot, but not unreasonably more than the brain's excess in basic active organs found earlier.
Each step is chosen to maximize the figure: every neuron as receptor, every impression kept, every second counted. The number tells us what total recall would cost, not what the brain stores. Von Neumann's real use of it is comparative: the brain's memory surplus over machines is of the same order as its surplus in active organs.
Possible physical embodiments
Various authors have proposed mechanisms; von Neumann lists them without choosing, noting that many more could be offered "with equal or even greater plausibility".
- Variable stimulation criteria. A cell's threshold, or more broadly its stimulation criteria, changes with its history: frequent use might lower its threshold. Memory would then reside in the variability of the criteria. A possibility he declines to discuss.
- Variable connections. A more drastic version: the distribution of conducting axons itself changes. Persistent disuse might make an axon ineffective; unusually frequent use might facilitate (lower the threshold of) that path. Parts of the system would vary with history and so be a memory.
- Genetic memory. "Obviously present": the chromosomes and genes are memory elements whose states affect, and partly determine, the whole system's functioning.
- Self-perpetuating chemistry. Some traits of chemical composition in certain areas might maintain themselves, since the self-perpetuating properties of genes can apparently also reside outside the genes, elsewhere in the cell.
- Reverberating circuits. Systems of nerve cells that stimulate each other cyclically are memories built from active elements, like the vacuum-tube "flip-flop" (two tubes gating each other), the first kind of machine memory and still used, in flip-flop-like form, in transistor and other high-speed technologies.
| Candidate (1956) | Von Neumann's description | Status in 2026 |
|---|---|---|
| Variable thresholds | Frequent use lowers a cell's threshold | Real (intrinsic plasticity), but secondary |
| Changing connections | Disuse makes an axon ineffective; heavy use facilitates a path | Leading account: synaptic plasticity and long-term potentiation (Bliss and Lømo, 1973) |
| Genetic memory | Genes store states that shape the system | Species memory, not episodes; epigenetic marks aid consolidation |
| Self-perpetuating chemistry | Self-maintaining chemical traits outside genes | Persistent protein complexes such as KIBRA-PKMζ sustain potentiated synapses (2024) |
| Reverberating circuits | Neurons stimulating each other in cycles, like tube flip-flops | Persistent activity supports working memory, not long-term storage |
Memory need not resemble the active organs
Von Neumann thinks it a priori unlikely that the nervous system relies mainly on reverberating circuits, because memories made of basic active organs are in every sense that matters extremely expensive. Machine history shows why.
The first large vacuum-tube machine, ENIAC, used flip-flops only for its primary (fastest, most directly available) memory. It had 22,000 tubes yet held only a few dozen ten-digit decimal numbers: a few hundred bits, under 10³. Machines of 1956 strike a balance of about 10⁴ basic active elements to 10⁵ to 10⁶ bits of memory, reached by using memory technology entirely different from the active organs.
- Different media. A tube or transistor machine may keep its memory in an electrostatic system (a cathode-ray tube) or large arrays of ferromagnetic cores.
- Hard to classify. Other important types, acoustic delay, ferroelectric and magnetostrictive delay lines among them, do not fit neat categories; he declines a full classification.
- The point. The componentry of memory may be entirely different from that of the active organs.
He ends on three statements of knowledge and ignorance: we know the brain's basic active organs (nerve cells); there is every reason to believe a very large memory goes with them; and we "most emphatically" do not know what physical entities make up that memory.
- Misprint. The text credits information theory to "G. E. Shannon"; it is Claude E. Shannon.
- Inflated estimate. Treating all 10¹⁰ neurons as receptors and assuming nothing is forgotten are both unsupported. Landauer (1986) estimated about 10⁹ bits retained over a lifetime, eleven orders of magnitude lower; Kurzweil also rejects total recall (Foreword).
- Addressing left out. He never asks how stored items would be retrieved; the machine model of Part 1 assumes explicit addresses, while later work (Hopfield, 1982) showed how networks can recall by content.
- Bits per synapse. Bartol et al. (eLife, 2015) found about 26 distinguishable synapse sizes in rat hippocampus, roughly 4.7 bits each; across 10¹⁴ to 10¹⁵ synapses that suggests a petabit-to-petabyte scale.
- Engrams. Tonegawa's group (2012) revived a fear memory by stimulating its engram cells with light.
- Mapping it. One cubic millimetre of human cortex (Shapson-Coe et al., Science, 2024) contains about 57,000 cells and 150 million synapses and took 1.4 petabytes of images; the MICrONS mouse cubic millimetre (Nature, 2025) has over 500 million synapses.
Mixed Processes and Codes
Von Neumann adds two pieces to his model. Processes in the body switch from digital to analog and back many times, and the genes are the clearest example. He then defines codes and, following Turing, the short code. A short code makes one machine imitate another, and this idea becomes the key to his final claim that our mathematics may be a secondary language running on the brain's primary one.
Digital and analog parts in the nervous system
Before leaving memory, which he has called a deep and wide-open problem, von Neumann adds a "minor aspect" in a brief and admittedly incomplete note: how the digital and analog ("mixed") parts of the nervous system relate.
Alternation along a process
A process passing through the nervous system can change character repeatedly.
- Digital stage. Nerve pulses control a particular step, such as the contraction of a specific muscle or the secretion of a specific chemical.
- Analog stage. That contraction or secretion is itself an analog phenomenon: a continuous chemical change or mechanical displacement.
- Back to digital. Sensed by suitable inner receptors, the analog event becomes the origin of a new train of nerve pulses.
- Repetition. These switches may alternate several times, so any given process can have a mixed character.
The genetic mechanism as the typical case
The genes are "clearly" parts of a digital system of components, but their effects consist of stimulating the formation of definite enzymes characteristic of each gene, which belong to the analog domain. Gene expression is thus one specific instance of the general digital-analog alternation.
A system that keeps converting between digital and analog stages cannot be described by logic alone or continuous mathematics alone. This prepares the next section's claim that the nervous system's notation is neither purely digital nor analog but statistical.
Codes and their role in control
Von Neumann now moves beyond memory to principles of organizing logical orders, important in any complicated automaton.
- Code. A system of logical instructions an automaton can carry out, causing it to perform some organized task.
- Logical orders. In the nervous system, things like nerve pulses on the appropriate axons; in general, anything that makes a digital logical system function in a reproducible, purposive way.
Complete code. Specifies everything: in nerve terms, the sequence of impulses and the axons they appear on, which fully defines one behaviour. In computers it is a set of orders with all specifications; any problem solved by calculation needs one, and using a machine means formulating complete codes.
Short code. A system of instructions that makes one machine behave as if it were another, specified machine. Shown possible by A. M. Turing ("in 1937") and since realized by computing experts in various particular ways.
Why another machine's orders are nonsense
A machine is controlled by strings of symbols, usually bits, and its instructions must define which strings are orders and what each does. For two machines, the meaningful strings need not be the same, and their effects may differ entirely. Fed orders meant for machine B, machine A meets, at least in part, nonsense:
- Invalid strings. Bit strings outside A's family of meaningful orders.
- Purposeless actions. Strings A can "obey" but whose effects form no part of the organized plan for solving the intended problem.
What a short code must contain
- Inspect. Instructions, in terms the first machine understands, to examine every order it receives and decide whether it has the structure of an order of the second machine.
- Execute. Enough orders in the first machine's own system to carry out the actions the second machine would have taken for that order.
- Compound. Each imitated order may stand for several operations, iterative processes or multiple actions of any kind, yet is handled as if it were an elementary, uncompounded order.
Turing's important result: in this way the first machine can imitate the behaviour of any other machine. The imitated order structure may differ entirely from the real one and may be much richer: anything the first machine can do, in any length of time and under any order system, can be packaged so that it looks like a primitive action.
Why "short"
The name is historical. Short codes were developed as an aid to coding: to write programs more briefly than the machine's natural order system allowed, by treating it as a different machine with a more convenient, fuller order system that permits simpler, more straightforward coding.
Suppose a machine's only arithmetic orders are add, subtract and conditional transfer. A short code can define a "multiply" order: on meeting it, the code runs a loop of shifted additions (Section 1) and returns the product, so the programmer writes one line instead of dozens. The cost is time: each imitated order takes many real steps. Von Neumann's next section uses exactly this trade-off to ask whether logic and mathematics are a short code running on the brain's native machinery.
- Date. Turing's paper appeared in the Proceedings of the London Mathematical Society in 1936 (volume dated 1936-37).
- Two senses. "Short Code" was also the name of John Mauchly's 1949 interpreted language for BINAC and UNIVAC; von Neumann uses the term more broadly, for any imitating code.
- Genes as digital switches. Calling genes purely digital is a simplification: expression levels are graded and regulated, so the gene-enzyme link is itself partly analog.
Software is layer upon layer of short codes: interpreters (Python bytecode), virtual machines (the JVM), emulators such as Apple's Rosetta 2 (x86 on Arm since 2020), and GPU code compiled through Nvidia's PTX virtual instruction set. The Churchlands' gloss holds: complete code is machine language, short code a high-level language.
Not Digital but Statistical: The Language of the Brain
Because the nervous system regulates quantities, it must have an arithmetical part, and computing experience says such a part needs 10 to 12 decimal digits. Nerves deliver 2 to 3. Von Neumann's resolution: the brain trades arithmetical precision for logical reliability through a statistical notation and very shallow logical depth, so its primary language cannot be the mathematics we consciously use. These are the manuscript's last pages; the argument stops where his illness stopped it.
The logical structure of the nervous system
The final topic is unconnected to memory or to codes: the respective roles of logic and arithmetic in any complicated automaton, and specifically in the nervous system.
Why any controller has an arithmetical part
- Human-designed automata. Any automaton built for human use, especially for controlling complicated processes, normally has a purely logical part (no arithmetic involved) and an arithmetical part.
- The reason. With our habits of thought it is very hard to express a truly complicated situation without formulae and numbers.
- Examples. Keeping the body's temperature, certain pressures or its chemical isostasy constant would, if a human designer formulated the task, be defined by numerical equalities and inequalities.
- The logical remainder. Other parts can be stated qualitatively: which combinations of circumstances should produce a physiological response and which should not.
So the nervous system, viewed as an automaton, has both parts, and its arithmetic matters as much as its logic. It is therefore "a computing machine in the proper sense", open to analysis in computing-machine terms.
The expected precision, and its implausibility
The natural question is then: with what precision does the arithmetical part work?
- Machine experience. A machine facing tasks as complicated as the nervous system's must provide high precision, because calculations are long: errors add up, and early errors are amplified by later steps (Section 3), so far more precision is needed than the physical problem suggests.
- Expected answer. Ten- or twelve-decimal precision "would not be an exaggeration".
- His comment. The conclusion was worth working out "because of, rather than in spite of, its absolute implausibility": a deliberate reductio, setting up the contrast with what nerves actually do.
Not digital but statistical
How nerves transmit numbers
Numerical data travel as periodic or nearly periodic pulse trains, and the mechanism follows from Section 5's fatigue and recovery.
- Strong stimulus. A receptor fires again as soon as its absolute refractoriness has passed.
- Weaker stimulus. It must also wait out a period of relative refractoriness before each response, so it fires periodically but at a lower frequency.
- Result. Intensity is rendered as frequency, a monotone function of stimulus intensity: a kind of frequency-modulated signalling, directly observed in fibres of the optic nerve and in nerves reporting important pressures.
- Scale effects. Because frequency is a monotone function of intensity rather than equal to it, the system can build in scale effects and express precision in terms convenient to the scales that arise (compare the logarithmic pulse density of Section 3).
- 50 to 200 pulses per second: the usual frequency range.
- 2 to 3 decimal digits: the precision this permits.
- 10 to 12 digits: what a machine of comparable task complexity would need.
- No known computer can operate reliably and significantly at the nervous system's precision, a fact he says "must be emphasized again and again".
Low precision, high reliability
The same notation brings a high level of reliability.
Digital (machines). Meaning lies in the position, presence or absence of every marker. One missing pulse can cause "absolute perversion of meaning", that is, nonsense. Precise, deep, fragile.
Statistical (nerves). Meaning lies in statistical properties: pulse frequencies, possibly correlations between trains on a nerve's many fibres. Losing one or several pulses, or inserting spurious ones, distorts the meaning only "inessentially". Imprecise, shallow, robust.
Resolving the conflict
How can a system need arithmetic yet have only 2 to 3 digits? Anyone who has studied deterioration of precision in long calculations knows the answer.
- Source of deterioration. Error superposition and, more, amplification of early errors by later steps: in short, the large number of operations in series, the arithmetical depth.
- Depth is logical too. Many operations in series is equally a feature of the logical structure, so all deterioration phenomena are due to great logical depth.
- The statistical message. What counts in nerves is not the exact position of markers but the statistical characteristics of their occurrence.
- The trade. This notation, radically different from ordinary arithmetic and mathematics, gives lower arithmetical precision but higher logical reliability: a deterioration in arithmetic has been traded for an improvement in logic.
Other statistical carriers
So far only one statistical trait carries information: the frequency (pulses per second) of a single, nearly periodic train. But every relevant nerve has many fibres, each carrying numerous trains, so it is "perfectly plausible" that statistical relationships between trains also carry information; correlation coefficients and the like are the natural candidates.
Frequency is a single-fibre statistic; correlation is a population statistic. Von Neumann's hint that information may live in relations among many parallel trains anticipates population and synchrony codes, and it also answers the precision worry: averaging across many fibres buys back precision that one fibre cannot provide.
The language of the brain is not the language of mathematics
Two kinds of communication
The nervous system rests on two kinds of communication: those without arithmetic formalism (communication of orders, logical) and those with it (communication of numbers, arithmetical). The first may be called language proper, the second mathematics.
Logic and mathematics as historical accidents
- Languages are accidents. Human languages are transmitted to us historically, and their multiplicity proves nothing about them is absolute or necessary; Greek and Sanskrit are historical facts, not logical necessities.
- So, perhaps, are logic and mathematics. It is reasonable to assume they too are historical, accidental forms of expression, with possible essential variants.
- The nervous system says so. Its nature and message systems indicate "positively" that this is the case: whatever language it uses has less logical and arithmetical depth than we are used to.
The retina performs a considerable reorganization of the visual image, and does it, at the point where the optic nerve begins, through only three successive synapses: three consecutive logical steps. Combined with the statistical character and low precision of the message system, which mean precision cannot degenerate far, this shows the brain uses different logical structures from ordinary logic and mathematics, with far less depth "under otherwise similar circumstances".
Mathematics as a short code
The language involved may correspond to a short code rather than a complete code (Section 8). When we talk mathematics we may be using a secondary language, built on the primary language the central nervous system truly uses. Hence:
- Outward forms mislead. The visible form of our mathematics is not decisive evidence of what the brain's mathematical or logical language really is.
- The firm conclusion. The remarks on reliability and on logical and arithmetical depth prove that whatever that language is, it must differ considerably from what we consciously and explicitly regard as mathematics.
- Need. The nervous system controls quantities, so it has an arithmetical part.
- Expectation. Machine experience says such a part needs 10 to 12 digits.
- Fact. Rate-coded nerves give 2 to 3 digits, with high reliability.
- Resolution. Precision is preserved only with shallow logical depth; the brain swaps arithmetic precision for logical reliability through a statistical notation.
- Consequence. The brain's primary language is shallow and statistical; conscious mathematics is a secondary, short-code language on top of it.
- Retina. The direct path (photoreceptor, bipolar, ganglion cell) has two synapses, with lateral horizontal and amacrine circuits; the point about shallowness stands.
- Precision. Averaging over many fibres raises a population's precision above one fibre's 2-3 digits, which von Neumann's own correlation remark implies.
- Which language is the short code? The text can be read two ways: the brain's own language as a short code, or our mathematics as a short code running on it. The surrounding sentences favour the second.
- Unfinished. The argument stops here: the lectures were never delivered, and von Neumann died in February 1957. The promised account of how this would alter mathematics never came.
- Beyond rate. Population codes (Georgopoulos's motor population vector, 1986), spike timing (images categorized in about 150 ms, Thorpe et al., 1996) and spike correlations are all established, confirming his hunch.
- AI went statistical. Networks run at FP8 and FP4 with noise-tolerant distributed representations, while depth stays modest: tens to about a hundred layers. Statistical machines reach formal mathematics: in July 2025 systems from Google DeepMind and OpenAI scored at gold-medal level at the International Mathematical Olympiad, a statistical primary layer carrying mathematics as a secondary language, the arrangement von Neumann proposed for the brain.
Master Takeaways
Von Neumann's fragment reduces to one comparison: computers are fast, precise and serial, brains are slow, imprecise and parallel, and each design follows from its components. The brain survives low precision by keeping its computations shallow and its notation statistical, which is the recipe modern AI hardware has rediscovered.
- The brain trades precision for reliability. Pulse-frequency codes carry only 2 to 3 digits but survive lost pulses, so the brain computes shallowly and statistically.
- Slow parts force parallelism. Neurons are 10⁴ to 10⁵ times slower than tubes but 10⁸ to 10⁹ times thriftier, so nature uses many slow organs at once.
- Depth costs precision. Errors superpose and amplify along long chains: 425 steps at 5% each magnify an error 10⁹-fold, so serial machines need 10 to 12 digits.
- The synapse is the unit. Thresholds, summation and geometry make each neuron many active organs; count synapses, not cells.
- Fast memory must be rationed. The hierarchic principle (a little fast storage, more slow) still organizes every computer from cache to cloud.
- Our maths may be a short code. The brain's primary language differs from conscious mathematics, which may run as a secondary language on top of it.
- Representation defines the machine. An analog machine lets a physical quantity be the number; a digital one writes digits with markers, at least four binary markers per decimal digit.
- Basic operations are a design choice. The differential analyzer's (x ± y)/2 and integration beat the four species of arithmetic for differential equations.
- Digital arithmetic is logic. In binary, the sum digit is parity and the carry is majority; every operation is a pattern of alternative actions under strict rules.
- One organ per operation forces a memory. With a single adder and multiplier, intermediate results must wait in registers, and capacity becomes a design parameter.
- Stored orders made programs data. Once instructions are numbers with addresses, a machine can compute on and modify its own program, the basis of all software.
- Branching is the heart of control. From relay tapes to branching points to the conditional transfer on the sign of a number at Z, decisions on computed results make iteration possible.
- Precision is cheap in digital, dear in analog. One extra digit costs a digital machine about 8.3%; an analog machine cannot reach 1:10⁶ at all.
- Pulse density foreshadows the nerve. Numbers as average pulse rates, read by a leaky condenser, are the frequency code he later finds in nerves, at the same low precision.
- "Prima facie digital" is the right starting point. The all-or-none spike makes a neuron a binary marker, but the idealization hides analog summation.
- Measure reaction time with recovery. Synaptic delay is 10⁻⁴ s, but the meaningful cycle, recovery included, is about 10⁻² s.
- Three measures, one size factor. Volume, power and the cube of linear size all give nature a 10⁸ to 10⁹ lead per organ.
- Parallel and serial are not interchangeable. Dependent steps resist parallelization, and serializing parallel work creates memory demands, so the two automata's logics differ.
- Thresholds make gates. A neuron needing two simultaneous inputs is AND, one input OR; with negation these form a complete logic.
- Some receptors read derivatives. Optic fibres that fire on light-to-dark or dark-to-light transitions respond to change, not level.
- Capacity can be counted in bits. One decimal digit is 3.32 bits, a typed character 6.45 bits; any memory's capacity is the sum.
- Total recall is an upper bound. 14 bits/s × 10¹⁰ cells × 2 × 10⁹ s = 2.8 × 10²⁰ bits only if every cell is a receptor and nothing is forgotten.
- Memory need not look like logic. ENIAC's 22,000 tubes held under 10³ bits; machines moved to cores and drums, and the brain's memory may be non-neural too.
- Biology mixes digital and analog. Pulses trigger contractions and secretions that are sensed and re-encoded; genes (digital) make enzymes (analog).
- Turing universality cuts one way. A digital machine can imitate any other through a short code, which is why Kurzweil argues computers can simulate brains, though not the reverse.
- Meaning can live in statistics. Frequencies and correlations between pulse trains can carry information; modern neuroscience confirmed rate, timing and population codes.
- The retina proves shallowness. Major image reorganization happens in about three synaptic steps, impossible for a deep, sequential program.
- Shannon made digital buildable. Redundancy lets any channel better than 50% error reach any reliability, which is why pervasive internal communication does not doom computers.
- Precise humility. Von Neumann marks the memory question "most emphatically" unanswered rather than guess; the book is a model of stating what is unknown.
- A fragment. The text stops where the argument becomes interesting; the "language of the brain" is announced, never described.
- Dated physiology. Membrane thickness, recovery time and spike amplitude are off; neuron counts were low (86 billion, not 10¹⁰) and the synapse factor far above his 10 to 100.
- Inflated memory. The 2.8 × 10²⁰-bit estimate assumes total recall and all neurons as receptors; later estimates of retained memory are far lower.
- Thin citations. McCulloch and Pitts go unnamed, Shannon is misprinted, and Turing's paper is misdated to 1937.
- Arithmetic slips. The √N step implies 10¹⁸ operations, not the printed 10¹⁶, and the size cross-check cites "10²" where his own figure was 10³.
- Forewords with errors. Kurzweil's history of Colossus and his "1935 lecture" are wrong; the Churchlands write "William Church".
- What held up. Parallelism, low precision, the synapse as the unit and the gap between brain logic and serial code are now textbook, and drive modern AI hardware.
Frameworks & mental models
| Name | What it says | Use it when |
|---|---|---|
| Analog vs digital representation | A number is a physical quantity, or a string of digits manipulated by logic | Choosing hardware: analog for cheap low precision, digital for scalable precision |
| Efficient basic operations | The best primitive set depends on the problem class | Designing instruction sets, accelerators or domain-specific languages |
| Memory-stored control | Orders are numbers in memory with addresses, branches and modification | Explaining why software is data, and why the memory channel bottlenecks |
| Mixed control | Plugging states stored as data; control handed between regimes | Reasoning about FPGAs, microcode and reconfigurable hardware |
| Error superposition and amplification | Random errors grow as √N; amplification grows geometrically with depth | Setting numeric precision for long computations or quantized AI models |
| Hierarchic principle | Capacities N₁ < N₂ < … at access times t₁ < t₂ < … | Designing storage, caches and data pipelines |
| Size-speed figure of merit | Size factor ÷ speed factor gives actions per volume per time | Comparing computing substrates: neurons, GPUs, neuromorphic chips |
| Logical depth vs breadth | Low-precision, slow parts favour shallow, wide computation | Judging whether a task suits brains, neural networks or serial code |
| Ladder of stimulation criteria | Logic, threshold, geometry, summation time, derivative | Deciding how much neuron detail a model needs |
| Complete vs short code | Full specification vs a code that makes one machine imitate another | Thinking about interpreters, emulators and layered languages |
| Statistical notation | Meaning lies in frequencies and correlations, not marker positions | Designing robust, noise-tolerant signalling and representations |
| Kurzweil's five ideas | Reliable bits, universality, stored program, machine intelligence, singularity | Placing any computing advance in the history of the field |
Who's who
| Name | Role | Why they matter |
|---|---|---|
| John von Neumann | Mathematician, IAS, AEC commissioner | Designed the stored-program architecture; wrote these lectures while dying |
| Klara von Neumann | His widow; early ENIAC programmer | Wrote the preface and saw the fragment into print |
| Alan Turing | English logician | His universal machine underwrites the short code and simulation arguments |
| Claude Shannon | Information theorist, Bell Labs | Supplied bits as the unit of memory and reliable coding |
| Paul and Patricia Churchland | Philosophers of mind | Second-edition foreword: the brain as vector-matrix parallel processor |
| Ray Kurzweil | Inventor and futurist | Third-edition foreword: five ideas, simulation of the brain, singularity |
| Warren McCulloch and Walter Pitts | Neurophysiologist and logician | 1943 threshold-logic neuron behind von Neumann's AND/OR analysis |
| Eckert and Mauchly | ENIAC and EDVAC engineers | Co-designers of the first stored-program machines |
| Stanislaw Ulam | Mathematician | Recorded von Neumann's "essential singularity" remark (1958) |
Connect it
- AI hardware. Modern AI accelerators converge on what von Neumann described for the brain: massive parallelism, low precision (FP8, FP4) and memory placed beside compute. His size-speed figure of merit remains the right lens for comparing GPUs with neuromorphic chips.
- Neural networks. Deep learning is his statistical notation in silicon: distributed, noise-tolerant, shallow relative to programs, and judged by robustness more than exactness.
- Enterprise architecture. The hierarchic principle and the stored-program bottleneck still govern system design: tier data by access time and move compute to the data.
- Banking systems. Precision depends on depth: ledgers need exact decimal arithmetic through long chains, while fraud and risk models tolerate low-precision statistical inference.
About this book
The Computer and the Brain: The 1956 Silliman Lectures, with Forewords by Ray Kurzweil and Paul and Patricia Churchland by John von Neumann, 1958; third edition 2012. This is a chapter-by-chapter study of the book, with summaries, maps, tables and diagrams; short quotations keep their page references. It is shared free for personal study, is not for sale, and is not endorsed by the author or publisher. Please read and buy the original.
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