Analog & Digital Entry Directory

by Corry Shores
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[Central Entry Directory]

[The following is a list of entries regarding the Analog/Digital debate in philosophy.]

Entry Directory for the topic of
Analog & Digital

Analog & Digital Aesthetic Communication (Deleuze's Analog & Digital)

Deleuze's Analog and Digital Communication; Isomorphism; and Aesthetic Analogy. Part 5 of Deleuze Posthumanism Paper, "Do Posthumanists Dream...?"

Analogical and "Digital" Language (Deleuze's Theory of Language)

Deleuze's Language Disorders: Speaking Schizophrenically – Writing Nonsensically

Analog & Digital Quantity Representation

Continuous & Discrete Forms – and Digital & Analog Representations – of Magnitude

Analog and Digital Synthesizer Modulation

Modulation of Sensation: Deleuze, Bacon, Cézanne, Wagner, and Richard Pinhas

Digital Logic (& Deleuze's Wild Digital Logic)

Deleuze's Wild Digital Computer Logic: A Formalization of Affirmative Synthetic Exclusive Disjunction's Deformities

Analog is Dense. Density Defined.

The Density of the Rational Numbers

The Essence of Digital (and Binary)

Pre-History and Early History of Digital Computation in Eves' An Introduction to the History of Mathematics

Leibniz' Representation and Computation of Binary Numerals

The Development of Digital Carry Technology

Digital Machinery & Their Programing

The Abstract Machines of Computation Theory. Marvin Minsky's Computation, Introduction and Finite-State Machines

Programing Abstract Machines: Turing Machines in Boolos & Jeffrey's Computability and Logic

Turing Computation of the Natural Numbers

Analog & Digital Neurocomputation

Deleuze's & Guattari's Neurophysiology and Neurocomputation

Analog & Entropy in Neural Computation

Analog Neural Potential Gradation in Marieb & Hoehn, Human Anatomy & Physiology

Quantity of Neurocomputation and Summation Types in Bear, Connors, & Michael, Neuroscience: Exploring the Brain

Arguments against Analog Computation (& My Defenses for It)

Nick Bostrom & Anders Sandberg: Whole Brain Emulation (WBE) (Mental Uploading / Mental Downloading) Part 3, Sect. 2, "Exotica" and Deleuzean Critique

Entry Directory, Schonbein, "Cognition and the Power of Continuous Dynamical Systems"

The Wheel is Come Full Pixel: Analog against Digital; Bostrom and Sandberg's Brain Emulation, Examined and Critiqued. Section 4

Beyond Human Numbers: Natural & Cosmic Computation

Pythagoras' Natural Computers: The Wild Cosmic Computation of Melodies and Flowers

Bateson, Natural Analog and Digital Computation, in Mind and Nature: A Necessary Unity

Humans as Thinking Machines

Summary of Alistair Welchman’s “Machinic Thinking”

The Abstract Machine of Love's Diagrammatic Functioning in Mille Plateaux / Thousand Plateaus §170

Becoming-Computer, Hayles, How We Became Posthuman: Virtual Bodies in Cybernetics, Literature, and Informatics

Andy Clark's Cyborgs and Deleuze's Restructive Element

External Sources

New Directions in Analog & Digital Philosophy at kvond's Frames/ sing:

Analog and Digital Intellect: Threshold Intensity, or Either/Or

Peaks and Troughs of Intensity: From Digital to Analog Embodiment

Turing Computation of the Natural Numbers

by Corry Shores
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Turing Computation

of the Natural Numbers

We will now build from our previous post on Turing machines. What we want is a mechanical computation of the natural numbers.

[I obtain the following program from Peter Bradley's wonderful animated site on Turing Machines.]

Our abstract machine will compute the natural numbers and display them in binary. So it is a "binary counter" machine.

This is its program, displayed as a "machine table." The S represents the number that the machine is reading. The subscript '0' means it would be a zero, and the subscript '1' means it is a one.

The q's stand for the two different instructions. L means to move left. R means to move right. There are two boxes for each q instruction, because the machine will see either of two symbols (1 or 0), and according to which one, it will perform some distinct task. We could articulate the instructions this way.

q1) If the scanned symbol is zero, write a zero overtop of it, move to the left, and begin instruction q1. If the scanned symbol is one, write a one overtop of it, move to the right, and begin instruction q2.

q2) If the scanned symbol is zero, write a one instead, move to the left, and begin instruction q1. If the scanned symbol is one, write a zero instead, move to the right, and begin instruction q2.

We will follow it through all the steps leading to its count of four.

It begins on a zero.

All the other boxes are zero except for one. At two spaces to the left of the starting point, there is a '1'. We do not read this with the number. Its function is to 'bump' the machine back to the right so it may continue its adding. All the boxes to the right of the bumper are the binary digit places that display each new counted number as it is added. However, this display shows them inversely. Hence we need to think of the ascending digit places moving to the right. So normally 100 equals four in binary. But here it would be displayed as 001.

So the machine finds itself at the second zero. We place a yellow arrow wherever the machine is found at the start of the given instruction. It always moves at the end of the instruction. We display its direction with black arrows.

We begin at the first instruction q1. Because the machine starts at a zero, we look to S0. It tells the machine to write a zero, and to move to the left. There is already a zero there, so nothing changes.

Likewise for the next step. Instruction q1 repeats.

But now the machine stands above a '1'.

It writes a '1' again, then it moves to right and begins instruction q2.

At this box there is a zero. So the machine replaces it with a '1', moves the left, and begins instruction q2. At this point, we have our first counted value: 1

The number it scans next is a '1', so it writes a '1' over it, moves to the right, and begins instruction q2.

Here the machine finds a '1'. So it replaces it with a zero, moves to the right to begin q2. But notice the similarity to when we clear the binary abacus.

At the next place it finds a zero. Instruction q2 tells it then to write a '1' instead and move to the left. This is equivalent to the carry procedure on the binary abacus.

So we see our first instance of clear and carry performed by the Turing Computer.

It now sees a zero. Instruction q1 tells it to write another zero, move to the left, and begin instruction q1.

It arrives upon the bumper. Instruction q1 tells it to write another '1' over it, move to the right, and begin instruction q2.

Now it sees a zero in the units place of our number display. Instruction q2 tells it to write a '1', move to the left, then begin instruction q1. In this way, we have added another number to get three.

It hits the bumper again, so it goes to the right and begins instruction q2.

Now it sees a '1' in the units place. Instruction q2 tells it to write a '0' instead, move to the right, and begin instruction q2. Here again is a clear procedure.

It sees another '1' in the next digit place. Instruction q2 tells it to write a zero, move to the right, and begin instruction q2. This is the second part of the clear/carry procedure.

Now it sees a '0'. Instruction q2 tells it to replace the '0' with a one, move the left, and begin instruction q1. This is the carry procedure that gets us to four. Now, the machine repeats the same clear/carry process as it had before.

Given an infinite amount of time, it will display the infinity of natural numbers.

See Peter Bradley's site on Turing Machines

Deleuze's Wild Digital Computer Logic: A Formalization of Affirmative Synthetic Exclusive Disjunction's Deformities

by Corry Shores
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(I dedicate this entry to Deleuze scholar Rocky Clancy. Thanks for your firestorm lightning-bolt mind.)

[I obtain the following from the sixth paragraph of the 24th chapter in Deleuze's Logic of Sense / Logique du Sens.]

When something becomes, there is change.

Wood becomes fire.

Fire burns wood.

So wood cannot be fire. Thus the wood becomes not-wood. So being becomes nothing. And, the fire is not wood. Fire comes from not-fire. Thus nothing becomes being.

All of logic is built on the principle of identity: A is A. Closely related to this law is the law of non-contradiction: A cannot be not-A. But in becoming, A becomes not-A. This defies logic.

Yet someone might clarify. A is A at time 1, and B (or not-A) at time 2. But A is never both itself and not itself at the same time.

However, change is a movement. And physicists can measure "instantaneous velocity." It is an instantaneous change of sorts. Leibniz says that the moving object occupies more spacial points then are in its actual length. And at the instant when the pumpkin hits the ground, it is both moving and smashed at the same time. It is virtually smashed, because it is fully smashed as a tendency. It has intensively stretched into non-being, while still maintaining its fullest form, all in an instant. In other words, the world around us is one mess of logical contradiction.

Deleuze's logic provides a solution.

His correction to formal logic is "affirmative synthetic exclusive disjunction." But by its nature, this logic cannot be formalized like other logical operators. However, by using the metaphor of electrical logic gates, we will provide a formalization of identity's deformity in the event of becoming.

Let's first review two logical operators, conjunction and exclusive disjunction.

A and B are different things. B is not-A, and A is not-B.

Consider we say, "It is Sunday, and it's cloudy." We could reduce both clauses to symbols: "A and B." For this to be true, both A and B must be true. Otherwise, the whole conjunction is false.

Or instead we say, "Either it is cloudy, or the sky is clear." It cannot be both, and it must be one of the two. This is exclusive disjunction: "A or B."

Deleuze's new logical operation is affirmative synthetic exclusive disjunction. It first presumes that A and B logically exclude each other. So both cannot be true; for example, the two propositions "there is wood," and "the fire consumed it" supposedly cannot both be true. But, says Deleuze, they are contracted together.

We strike a match. At one point it is wood. At another point it is fire. But consider that instant when the match is right at the limit of its still being wood. In the next instant it will be fire. But between its limit as wood, and its initiation as fire, there extends no time at all. In other words, at the limit of the match being wood, nothing stands between the wood and fire. So the wood and fire contract together. That does not mean that they both assimilate into the same thing. They maintain their differences. For, one lies at the limit of the other: it contracts upon it, but it is not placed within it. So the two maintain their differences. Because they are different, they are not the same, and hence there is a sort of distance that remains between them. But as we saw, there is no extent of time between them. So there is not an extensive difference between them. Rather, there is an intensive difference. When the match sits in its box, it decomposes slowly. When we strike it, it changes rapidly. We can measure degrees of change from one instant to the next. So at the limit of wood being wood, it has a profound internal tendency to become fire. For this reason, it is intensely fire, even while it is wood. At that limit, the wood is virtually fire. But, its being fire is no less real than its being wood, just as the pumpkin's being smashed is no less real than its still being whole, right as it hits the ground.

So, this is not a logical conjunction, Deleuze claims. For, the wood and the fire maintain their individuality. They are differences that are forced together and are thereby contracted as differences. This contraction that maintains their distances is the 'synthesis.' Imagine that we force together the north ends of two extraordinarily powerful magnets, despite their enormous repulsive force. Their being together does not make the one end any less repulsive to the other. The extensive magnitudes between them are contracted. But there is an incredible intensive force that is trying to push them apart. The force is there, because the contraction causes them to communicate their differences to one another, without them homogenizing into one thing.

Let's compare the truth tables for conjunction (AND) with exclusive disjunction (XOR).

Deleuze writes that synthetic disjunction cannot be reduced to conjunction. So we may not use the conjunction truth table. But he also says that the disjunction is affirmative. When A and B are contracted, they maintain their distance, which affirms their difference. Thus because it is affirmative, we cannot use the exclusive disjunction (XOR) truth table. For, we see it lists a negative truth value for the combination of true disjuncts.

Deleuze also says that the exclusive disjunction disperses its ramifications everywhere. Imagine that the magnets we force together have such incredible repulsive forces that they both explode and fly off away from each other in all directions. So the contraction that happens in this operation causes a multitude of divergent heterogeneous series to be disjunctively "coordinated" at once. And at the disjunction is the "aleatory point." How, where, and into what forms the series disperse is decided by both the competition of forces and pure chance.

Because we will be using a digital logic formalization, I need first to distinguish two types of chance. To do so, we imagine an electrical circuit. The circuit designer includes a random variable generator. At mathematically random times, the generator causes the current to randomly jump to another part of the circuit. Now consider instead that we overload the circuit. This causes it to "short circuit." The current jumps wildly around the circuitry (and maybe also outside the circuit system). I want to be very clear about these two types of chance: mathematically random vs wild. Random means that it goes somewhere that we cannot predict. But the places and the ways that it gets there we have predetermined. And the rest is left to chance.

When instead the circuit jumps wildly, it could go anywhere in any way. It could jump out of the circuit. It could jump to places that are impossible using artificial means. No one can predict where the lighting-bolt strikes. It's wild. It's the wilderness of chance. We will keep this sort of variable in mind when we design our Deleuzean Digital Logic Gate.

Let's first just look at how the conjunction and exclusive disjunction logic gates are symbolized.

[Thanks you image source: Tony R. Kuphaldt at cybermike.net]

I propose the following shape for Deleuze's Affirmative Synthetic Exclusive Disjunction Gate. Instead of XOR for exclusive disjunction, I name it SOR for synthetic exclusive disjunction.

I choose this shape because it bears the golden ratio, which is a ubiquitous proportion in nature and the cosmos. As an irrational number, it cannot be represented using digital numerals. It is undeniably a real number. But it will always defy digital logic.

Let's imagine now that there are two channels of electrical current that enter the SOR gate: A and B. Let's also imagine that the SOR gate can handle the electricity from either channel, just one at a time. But when both channels feed charge into the gate, then it overfills its carrying capacity. Still the logic of the gate forces the charges through. This contracts them together even though they are too large to both merge together in the gate.

Because the SOR gate is essentially an exclusive disjunction, we will follow the logic of exclusive disjunction up to the point where synthesis happens.

Consider first when no electricity flows through the channels. None should flow out of the gate.

But if it flows through just A, then the gate can handle the current, and it lets the charge through.

Likewise if only B's current is moving. Again, the gate can carry the charge and let it pass through.

However, we noted that the SOR gate cannot carry both charges at once. When they force themselves into the gate, they do not then become one same current flowing out the other end. Rather, they ramify wildly (at this aleatory point.) So they never converge. However, they contract, because they both entered the gate. Nonetheless, they shoot out wildly to any place and in any way whatsoever.

I have chosen a "w" to represent the logical value of this combination. It is not a 1 or a 0. We previously saw that it cannot be either of those. But it is still something. It is a variable. Yet not a mathematically random variable, but rather a wild variable. It can become anything anywhere depending on the battle of intensive forces that deform it in the instant of contraction.

I present the SOR gate as a possible formalization for Deleuze's logic. It has the advantage of metaphorically making use of real forces like electricity. A plain truth table will not suffice. I also think that even though it is a formalization, it is loyal to the way that intensities deform whatever they embody on account of their pushing-and-pulling it in many different directions at once. This we display as the wild dispersion of ramified current "series" and symbolize with the random variable w for its wild logical value.

Another advantage of this formalization will be when discussing a problem in Artificial Intelligence theory. Some are trying to replicate the human mind. And even a few think we can replicate someone's mind so well that the computer simulation is no different from the actual person herself. I do not argue that this is impossible. I will leave it for experiments to demonstrate. Rather, I am trying to use Deleuze's philosophy to suggest a way to replicate or at least explain human creativity. My hypothesis is that it is wild.



AND and XOR gate images from:
http://www.cybermike.net/reference/liec_book/Digital/DIGI_3.html
Thanks Tony R. Kuphaldt

The Development of Digital Carry Technology

by Corry Shores
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The Development of Digital Carry Technology

Later we will discuss the philosophical implications of the carry technique for positional numeral systems. Here we will follow its development from primitive techniques to modern technologies.

First an Illustration:

Carrying-Capacity and Carrying-Over

Water falls. A small bucket catches some.

Quickly it fills.

It can carry only one liter. When it reaches its carrying capacity, it tips-over and pours into a larger container below it

Then the little bucket flips back upward, and fills again.

When it reaches its carrying capacity again, it carries-over its contents another time to the larger container. Now the larger one has two liters in it.

The process repeats. Eventually this larger container reaches its carrying capacity, which is ten liters.

Then it carries over its contents to another container that is ten times as large. This one fills at one hundred liters. So after ten carry-overs of the middle container, the largest one fills. But to get to that point, one hundred of the smallest containers must carry over their contents.

We can imagine this series continuing. Each time, the next largest container is ten times the size of the previous one.

Hands: The First Digital Computers

Counting and adding most likely we performed using hand signs, before written symbols were invented.

Each of our fingers we call a 'digit.' As well, each discrete numerical value we call digits as well. These discrete quantities are the formal foundation of digital technology.

We have ten fingers, so we can have ten different digits (as well as zero, or the absence of a digit)

But when we get to ten fingers, we run out of digits (although some cultures went to their toes. They have a base twenty numeral system). Thus we "carry" the ten-value so we can then count together "one and ten." The English word eleven comes from "ein lifon" or "one left over." Twelve comes from "twe lif" or "two over ten." And thirteen "three and ten;" fourteen "four and ten." Twenty comes from "twe-tig," or "two tens." Also, hundred comes from "ten times," meaning, ten times ten. Then we add back again at the first decimal place, saying, "one hundred and one..." and so on. (Eves 12b) So we see that our hands can only contain ten digit values. Then they must carry-over to the tens place. When that containment category is filled, then we must carry-over to the hundreds place, and so on.

The Abacus' Token Digits

Until Arabic numerals had arrived in Europe, this carry procedure was aided by the abacus. We use rows of objects. They could be pebbles in a sand tray, beads on a rack, and so forth. Each row has a carrying-capacity of ten objects. But the objects' values for each row ascend according to powers of ten. So one bead in the single digit place is worth only one unit. But one token in the tens place is worth the value of ten smaller ones, and so on.

So let's look at an abacus with beads sliding along rods. We will set it so that all the beads are to one side.

This setting will be equal zero. Now to make it display the value of one, we take a bead from the bottom row, and slide it to the left.

So to make it read two, we slide over another bead.

We will slide seven more to display nine.

Now, when we slide over another to make ten, we have reached the carrying capacity for that row. So it carries that value over to the next one. But we only move one bead from the next row. We keep in mind, however, that every bead in this row symbolizes ten beads in the first row.

Hence the abacus now displays ten, even though only one bead is found to the left.

Let's add eighty-nine to the ten already shown. We then obtain ninety-nine.

We will add one. But that causes us to reach our carrying capacity for the first row. So we will carry the value over to the next row.

To make the carry, we first empty the first row by moving all the beads to the right. Then we add one more unit in the next row. This one bead is worth the same as the ten beads below it that carried over.

But now we have reached the carrying capacity for the tens row. So we empty it. And carry that value to the top row. Here, each bead is worth ten from the middle tens-row, and a hundred from the bottom units-row.

So we see how the abacus allows us to mechanically carry-over values to the next place.

Using our Hands to Manipulate Abstractions:

Symbolic Computation

From around 700 A.D. to 1500 A.D., the abacus was gradually replaced by Hindu-Arabic numerals:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

These allowed us to perform the carry operations by manipulating symbols rather than object-tokens.

Engineered Mathematics:

Mechanical Digital Calculators

As we see, the carry method using symbols is still very mechanical. And it would seem that a machine could do such a thing if it were correctly engineered. Philosophers Blaise Pascal and Gottfried Leibniz endeavored to create such machines. Leibniz said: "It is unworthy of excellent men to lose hours like slaves in the labour of calculation which could safely be relegated to anyone else if machines were used." [Smith Sourcebook 181] We will detail the mechanics of their machines later. What we will focus-on here is a simplification of the carry mechanism. Imagine we are looking at a box. It displays two numbers beside each other. There is a crank to turn the numbers on the right.

Now consider if the top panel were transparent.

In fact, we will merely look inside the box. The white squares around the bottom number just tells us which number is being displayed at that time.

We will presume that each number 'pops' into place as we turn the crank. In other words, we cannot set the gears to display a value in between 1 and 2 for example. This will keep the device digital and not analog [we later discuss analog computation machines.] Now we will turn the crank to display the value for one.

We turn it again to show two.

We turn it seven more times to display nine.

Now, we see that the single tooth on the units dial is currently touching tooth a of the gear above it. So far, we have moved up to nine, and the top gear has not turned at all. But now we see that if we turn the units gear one more time, its tooth will press against top gear's tooth a. This will cause it rotate. But that will then make the left gear turn one place.

So this single tooth triggered the carry mechanism, which is the top gear. It carries over the value to the next decimal place. Its value reads only "1." But it is worth ten of the units. So in this way, mechanical calculators are able to carry-over values automatically. Recall that with the abacus, the human operator had to make the decision to perform the carry operation. But here the machine performs that action automatically for us. So we do not need to "think" in order to perform the carry. The machine does this mathematical work.

Binary Digitalization

We will now look at the mechanisms involved in the digital computation of electronic circuitry.

But before we do that, we will examine the binary representation and computation of quantities, rather than the decimal system we have been using so far.

In binary, our containers' carrying-capacities progress according to powers of two, and not to powers of ten. So one liter pours into the first bucket.

But it reaches its carrying capacity at two liters.

Then at two liters it carries-over its contents to the next container.

We only have two liters so far. But because we carried at two, we now represent it as 10 [See the entry on Leibniz' explanation of the binary system.] Then the first container receives another liter, but it is now at half its carrying capacity.

So we do not yet carry from the units place. Hence the binary number for three is "11." Yet when the first container fills for the fourth time, it will carry-over to the next one. But then this one will reach its carrying capacity, so then it spills into an even larger container.

There is only one 'load' or 'spill' in this lowest container. But it is worth four liters. Yet, there were two carries, so the binary representation is "100," (a zero for each carry) even though the decimal form is "4." We can perhaps see this operation better on a binary abacus. Instead of each row having a capacity of ten beads, they carry-over when they reach two. So we set the abacus at zero.

Then we move one bead to the left to display one.

Then we bring-over the other one. But that fills the carrying capacity.

So we will 'empty' or clear the units row, and carry-over its value to the next row.

Now we want to add another value of one. So we move one bead over on the bottom units row.

The abacus now reads '11.' We will add another unit. But then we reach our carrying capacity again.

So we empty the units row, and carry one unit over in the next row above.

But then we reach the carrying capacity for this row. So we carry-over again.

So we can better see this carry operation. We could perform it using symbols this way.

The Logic of Discrete Flow:

Digital Electronic Computation

Now we will turn to the electronic digital circuits that perform calculations for us using binary values.

We need only picture the electric flow as water. It moves through pipes. Either it flows or it does not flow. When it reaches a gate, there are certain conditions that allow the water to pass through it. Our gates will take-in two channels of flow. They follow two basic logical operations, conjunction and exclusive disjunction.

Let's first review the logic of these gates.

Image a crime. Our boss was murdered in his office. The previous day we got in fight with him about whether or not we were allowed to take-off work the next day. The whole office of co-workers heard the commotion. No resolution was met. And we stormed out furious, saying out loud, "I could kill him! Just kill him!" So the next day when he was murdered, we were arrested as a suspect. Now we stand trial.

We give the prosecution our alibi: "I was away-from the scene of the crime, and I was on vacation." We have sworn-in, so any lie is perjury. Clearly if we were clocked-into work and thus near the crime scene, we have perjured ourselves.

Or consider this other possibility. We took-off work, but went there anyway to say 'hi' to co-workers. But we do not want to place ourselves at the scene. So if we say that we were on vacation, but we were not at the scene, then again we have perjured ourselves.

So even though one part of our statement was true, the whole claim is false. Or, maybe what really happened was we had snuck-out-of work while being clocked-in. We do not want to admit this fact, because then the jury would question our character. So in that case, we were actually away from the scene, but we lied by saying that we were on vacation.

So again, even though one-half of the whole defense is true, the conjunction of the claims is false. The only way that both our alibi's claims would be true is if we were both on vacation and we were away from the crime scene.

There is a logic behind these truth calculations. This is the logic of conjunction. Our adding-circuit will have two gates. Both will operate according to a logical operation. One of them will be conjunction or 'AND' like the one above. Instead of "true" or "T," we will use "1." And instead of "false" or "F," we will use "0."

So we will have two input channels of water (or electrical) current. They are A and B. Their pipes are connected directly to an AND gate. But if no water is flowing through either of them, then nothing will be flowing-out on the other side either.

And even if channel A is flowing, still that will not satisfy the conjunction condition.

Likewise if just B is flowing.

Only if both A and B are flowing will the AND gate allow the current to flow through it.

Now there is another level of logic calculus involved. The other gate is the exclusive disjunction or XOR gate.

B's current will travel there.

And A's current will also.

The current flowing out of the XOR gate will give us the sum of the two A and B values. If they reach the binary carrying-capacity of two, then the AND output will tell us that we have carried to the next binary place.

Now we need to grasp the logic of the XOR exclusive disjunction gate. We saw that for conjunction, the two logical values of two propositions were conjoined together by "and." Now for disjunction, two logical values will find themselves at a cross-roads. They are joined by an "either/or, but not both" relation. Only one can be true. Imagine we mistakenly told the prosecution at one point in our testimony that we were near our work place, but at another part of our examination, we said that we were definitely not near the crime scene. The prosecutor then says, "Well, either you were at the scene of the crime, or you were away from it." If it is one option or the other, we cannot say it was neither one.

However, there is no contradiction if we say that we were not at the crime scene, because we were away from it.

And vice versa, we also do not contradict ourselves if we say that were at the crime scene, because we were not away from it.

However, we will contradict ourselves if we say that we were both at the crime scene, and we were away from it. For, it must be one or the other. They cannot both be true.

Now let's return to our "half-adder" circuit.

A and B are different variables. So we do not take their values together as though they were two digits of the same number. So if A is '1' and B is '0,' that does not mean we are inputting '10,' which has the value of two in binary numerals. Rather, A is its own value, and B is its own value. After the circuit computes their additive value, then the output displays a binary number. So if the sum category shows '0' and the carry output shows '1,' then in fact we do have the binary numeral '10,' which means two, that is, the sum of one and one. We will now examine the logical calculus that the circuit performs so that it always gives us the correct addition.

First we will input zero plus zero. There is no current flowing through the channels. Given the conditions of our logic gates, none will flow out either.

So zero plus zero is 0. Now we will add one plus zero by sending current through the A channel.

A's flow alone will not trigger the AND gate to send current through the carry channel. However, it will trigger the XOR gate. So current flows through the sum channel, and our result of one plus zero is '1.' We will now add zero plus one by sending current through the B channel, but not through the A channel.

Again, B's current alone is not enough to trigger the AND gate, so the carry channel is not activated. However, B alone satisfies the conditions for the XOR gate, so it activates the sum channel. Like before, we obtain the output '1' for zero plus one. Now we will add one plus one by sending current through both the A and B channels.

We are sending current from both channels to both gates. But the XOR gate will not allow current through if it receives flow in both inputs. This is like saying "I was at the crime scene while I was away from it." So we send no current out through the sum channel. That is no matter, because we have satisfied the conditions to send current through the AND gate. Hence we have carried the value from the first digit place over to the second place. We obtain our sum for 'one and one' as the binary digit "10," which is equal to two or the decimal '2.'

So we see that there is still a mechanics of sorts involved in electronic digital computation. When we added two values to one row of the binary abacus, we first cleared the filled row. Then we carried-over the value of two into the second row, using only one token, which symbolized two single units.

We see now that the clearing operation in the digital circuit is performed by the exclusive disjunction (XOR) gate. When it receives both, it sends out no current, or a zero value. But, when the XOR gate receives both channels of flow, that means the conjunction (AND) gate is also receiving both flows. When it receives both, then it sends out a single channel of flow. So the AND gate performs the carry-over operation. Together, the two logic gates co-operate to perform the two hand-motions we use when operating the abacus, or when symbolically writing "0" in the sum column, then writing a '1' as carried-over in the next digit place column.

The gear mechanisms condect the clear operation by circularly placing the 9 next to the 0 on the first wheel. The carry operation that transfered the first dial's total value was performed by having a single tooth move the next wheel once every full revolution.

The development of the carry technique has been one of increasing automation of the mechanical procedure. Now with digital circuits, we do not need to

a) say "ten and one,"

b) manually clear one row of abacus tokens and carry the value over to the next row,

c) use pencils and paper to write the carry in the next written column, or

d) use physical force to turn extra gears so that the value can be carried to another wheel.

We now push single buttons, and logically-processed electrical currents perform both operations without us needing to apply extra mental or physical effort.

Eves, Howard. An Introduction to the History of Mathematics. London: Brooks/Cole - Thomson Learning, 1990.

Leibniz. "Machina arithmetica in qua non additio tantum et subtractio sed et multiplicatio nullo, divisio vero paene nullo animi labore peragantur" in David Eugene Smith A Soucebook in Mathematics Transl. Mark Kormes. Mineola, N.Y.: Dover Publications, 1959.

The hand sign image is from:

http://www.basiceducation.co.uk/sob/sob.htm

Two of the logic gate images are from:

http://www.opamp-electronics.com/tutorials/digital_theory_ch_003.htm