Babbage, Lovelace and the Analytical Engine: Symbols vs Numbers

 

Ada Lovelace (often called “the first computer programmer”) had an important insight when she contemplated Charles Babbage’s difference engine. She believed that it should move beyond mere calculation. It should, instead, tackle computation more generally. In other words, it should move from numerical operations to “symbolic manipulations”. This raises the philosophical issue (at least in this essay) as to what numbers are: Are they abstract objects or mere “marks on paper”? Moreover, how do this question relate to our view of computers generally?

Charles Babbage’s analytical engine, via Wikimedia Commons. Source here.)

General Purpose?

When the words “general purpose computer” are used it’s often the case that they aren’t really explained. Indeed, some laypeople may even interpret such words literally. However, the “general” in “general purpose” is far less… general than you may think.

In very simple terms, the words “general purpose” refer to a machine that’s Turing-complete. Moreover, that term only refers to a machine’s relation to other machines, not to being able to do the washing up or take the dog for a walk.

A machine (or something purely abstract) is said to be Turing-complete if it can be used to simulate any Turing machine. Thus, this refers to a machine understanding, decoding or simply recognising other machines — and, more importantly, working on — or with — their rules, algorithms, data, etc. This is technically expressed in terms of Turing equivalence. Thus:

Two computers P and Q are called equivalent if P can simulate Q and Q can simulate P.

Charles Babbage

Charles Babbage. Source, Wikimedia Commons here.

The interest in general-purpose computers dates back to the work and designs of Charles Babbage. So it’s here that Babbage’s “difference engine” and “analytical engine” need to be discussed.

The analytical engine was a proposed computer (or engine) designed by English mathematician Charles Babbage. He first described it in 1837 as the successor to his own difference engine, which was a design (i.e., not a concrete thing) for a mechanical calculator.

The historian of computing, Doron Swade, states that the analytical engine

“is a general-purpose computational engine [which] embodies [ ] almost every single significant logical feature of the modern digital computer”.

[See note.]

The difference engine, on the other hand, dealt with “something specific that has a fixed set of functions”.

The analytical engine was general purpose because “it was meant to be programmable, and it would automatically execute multiplication, division, subtraction and addition”.

In terms of the title of this essay: it can be seen here that we’re still dealing with numbers when it comes to the difference engine. Or, more accurately, we’re dealing with arithmetic. Clearly, a machine that deals exclusively with numbers (or arithmetical operations) cannot be “all purpose”. So the advance here is to use numbers to deal with… all purposes. Or, more widely, to use symbols to deal with all purposes. (Numbers and symbols as they are instantiated within “engines”, machines or computers.)

Ada Lovelace

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Ada Lovelace, by Alfred Edward Chalon. Public domain, via Wikimedia Commons here:

An historical account of the difference between the difference engine and the analytical engine may be of help here. The following paragraph is Doran Swade on that subject as it relates to Ada Lovelace:

“[ ] For those years [in the late 1930 and early 1940s], John Fuegi suggested, there is no evidence that the first machines were moving beyond the difference machine in the way Lovelace’s notes move beyond. They were still continuing with calculation, as though they had gone back to the difference engine rather than the analytical engine. After that it is hard to define historically when computers moved from calculating, say, the simple trajectories of artillery shells and mathematics to general-purpose computing.”

The important word in the long quote above is “calculation”. Simply put, calculation alone couldn’t have been classed as general purpose. So it can be asked here how the move from calculation to “general-purpose computing” came about.

Swade tells us that Ada Lovelace

“saw that these machines were not bound exclusively by numbers, making the essential transition to a number representing something other than quantity”.

So numbers weren’t actually transcended here. Instead, the idea of numbers as exclusively representing “quantity” was transcended. This is what flummoxes many laypersons. They see numbers, and then they assume either that such numbers have a relation to other numbers, or that the numbers must refer exclusively to quantities of some kind.

These distinctions were captured by Lovelace herself. She made the distinction by stating that calculation and computation aren’t the same thing. This also shows us that calculation is a subbranch of computing, rather than computing being a subbranch of calculation.

In terms of the analytical engine, it “could represent something other than quantity, such as notes of music or letters of the alphabet”. Indeed, if you randomly throw a pack of cards onto the floor, the many relations between the cards can still be represented numerically — even if they don’t instantiate symmetries or patterns. (There is little point in actually doing this.) On the other hand, numbers can represent the physical nature of a quark or the tensile strength of a bridge over a river.

Lovelace also captured something that later became important in late-19th-century logic. As Swade puts it, Lovelace “made the transition from arithmetic to symbolic manipulation”. Indeed, what the manipulation of numbers and symbols share is that this is carried out “according to rules”. (It can now even be said that rules are fundamental: numbers or symbols are secondary.)

[All this can be seen in Ada Lovelace’s “notes”, as found in ‘Sketch of of the Analytical Engine Invented by Charles Babbage’. These notes can also be found here.]

Gödel Numbering, Numbers and Symbols

Now take the specific and well-known case of Gödel numbering some eighty years after Lovelace’s death.

Kurt Gödel assigned numbers to things which aren’t quantities, such as mathematical and logical statements, proofs, etc. In this limited sense, then, numbers are convenient tools for representing things which aren’t themselves numbers.

However, what about using numbers to symbolise literally anything at all?

This is why non-mathematicians are often intimidated by the use of numbers and other mathematical devices. In other words, many people don’t realise that there can be a numerical account of almost anything! (For example, window open = 1. Window shut = 0.)

Numbers are as useful or convenient as hammers, nails, or whatever. This idea is a seemingly non-Platonic view of numbers. And even in physics a physicist can take a non-Pythagorean (rather than non-Platonic) position on numbers as they relate to the world, and on their use in physical theories.

Symbols can represent things other than numbers or quantities. Added to that, if one is a non-Platonist (rather than a non-Pythagorean), then, say, the number 2 is a symbol even if there are no quotation marks around it. In other words, 2 (not just ‘2’) is a (both metaphorically and literally) “mark on paper”. This stance is directly relevant to the use of numbers in computers and other machines.

Plato. Wikimedia Commons here.

Platonists and many others, on the other hand, believe that the number 2 is an abstract object. (They may never express their position in that precise way.) This basically means that 2 is an abstract Platonic form in Platonic heaven — alongside 3, 1001, Truth, Justice, Man, etc.

Of course, even non-Platonists deny that numbers are purely symbols. Take this categorical statement:

“Numbers are not symbols, but they do have a meaning that allows them to be added, multiplied, compared, and so on.”

If a number is only a mark on paper (or purely syntactic in nature), then it doesn’t offer us information or a meaning. Arguably, if 2 isn’t a mark, then it it must have, say, a “meaning” or even a “referent”.

This debate parallels one which occurred within modern logic.

Take this remark which states that modern logic is

“fundamentally a calculus whose rules of operation are determined only by the shape and not by the meaning of the symbols it employs, as in mathematics [ ]”.

It’s also often said that logic concerns itself with the form of arguments, not with their contents.

So now we can say that arithmetic (i.e., not all of mathematics) concerns itself with the forms of numbers, not with their contents.

This huge and historical debate needn’t be discussed here, save to say that it has relevance to the use of numbers and symbols in computers. In this case at least, the “shape” of numbers is, arguably, all that’s required.

Note:

Those outside computer nerdery may now be wondering how the word “logical” (as in the earlier clause “almost every logical feature of the modern digital computer”) is being used here. More precisely, some readers may wonder how “every logical feature” can be captured or represented by the actions of a computer.

For example, how do we move from either/or (logically symbolised as ∨) to 1 or 0, and then to an “on or off” operation (as found in logic gates) in the computer’s circuitry? Moreover, what about every other logical feature (or operation) human beings have created and been interested in?

“Metaphysical Lunatics” vs Common Sense

 

The Scottish philosopher Thomas Reid (1710–1796) used the words “metaphysical lunatics” about those philosophers who argued against “common sense principles” and beliefs. This essay will also discuss Reid’s influence on Ludwig Wittgenstein, as well as Herbert Marcuse’s attack on both “common sense philosophy” and Wittgenstein himself.

Paul Austin Murphy

9 min read

·

Apr 25, 2025

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A “metaphysical lunatic” and Thomas Reid.

The common-sense beliefs that Thomas Reid had in mind included such things as “you exist and have a mind”, “this table is real”, “I exist”, and “I can have reason to trust my senses and testimony generally”. [See note 1.]

Most readers will accept that these examples are deemed to be common-sense beliefs by most people — even if only when pressed or asked questions about them.

All the examples above have been extensively dissected by various “metaphysical lunatics”.

A Philosophy You Can Live By

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Thomas Reid and the commonsensicalists believed that philosophy should be unified with (as it’s often put) “the real world”.

According to the Scottish philosopher Alexander Broadie:

“As far as Reid is concerned, what this demonstrates is that there is something desperately wrong with Hume’s philosophy, because it is not a philosophy that you can live.”

It’s not immediately obvious why philosophy — Hume’s or anyone else’s — should have anything at all to do with what “you can live”. This position seems to erase truth itself from philosophy, and turn it into some vehicle for personal development, the advancement of religious/moral/political causes/goals, or pragmatic convenience. Thus, if the belief that “God loves [or hates] you” (or “The universe is conscious”) is something you can live by, then perhaps it doesn’t matter if it’s true or false.

Oddly enough, Hume’s own example shows this.

Hume was the first to admit that his philosophical conclusions had little or no impact on what he lived by. After all, he indulged in sceptical scenarios in the armchair, and then forgot about all of them when he went to play backgammon or billiards.

However, Hume might still have argued in the following manner:

Who cares if I ignore my sceptical conclusions when I play billiards or make love to a librarian?

Alternatively, Hume might have argued:

I haven’t, in fact, ignored my sceptical conclusions in my everyday affairs.

… Or at least Hume might have argued that he didn’t contradict his sceptical conclusions when he lived his life.

This way of looking at Humean scepticism was characterised by Melissa Lane in this way:

“There is a wonderful nineteenth-century Edinburgh story that said that really what happened was that Reid was shouting out very loudly, ‘We must believe in an external world,’ but whispering, ‘We have no reason for our beliefs.’ And then Hume saying very loudly, ‘We have no reason for our beliefs,’ but then whispering, ‘But we must believe in an external world.’ There is a sense in which, one can argue, are they in some ways saying the same thing?”

In this is a correct interpretation, then readers are just as warranted in arguing that Thomas Reid himself didn’t have a philosophy he could live by. After all, if it’s correct that he believed that “We have no reason for our beliefs” (yet we must believe in an “external world” all the same), then all he was doing was inverting Hume’s own position. Indeed, it’s an inversion which itself doesn’t make the slightest difference to what we live by. Of course, whether or not Reid did believe that “We have no reason for our beliefs” may just be speculation. After all, this is only part of a “nineteenth-century Edinburgh story”. That said, there may be an argument that Reid must have believed this — even if he never articulated it. In other words, Reid never demonstrated (let alone “proved”) that we do have reason for our beliefs. And, deep down, perhaps he knew that.

What’s more, if both Reid and Hume knew that the sceptical scenarios couldn’t be disproved (yet, nonetheless, we must ignore them outside the study), then what A.C. Grayling says about Hume adds meat to all this when he stated the following words:

“It may then be that Hume was not at all a sceptic about the external world, but about the potentials of reason to prove it.”

This makes sense. It does so because Hume didn’t have a problem with the existence of the external world: he had a problem with the grand claims of the rationalists.

Thomas Reid’s Philosophy

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Let Thomas Reid himself articulate the basic principle of Common Sense Realism:

“If there are certain principles, as I think there are, which the constitution of our nature leads us to believe, and which we are under a necessity to take for granted in the common concerns of life, without being able to give a reason for them — these are what we call the principles of common sense; and what is manifestly contrary to them, is what we call absurd.”

In terms of phrasing alone, if “common sense” beliefs and values are those we needn’t “give a reason for”, then that may strike some readers as being very convenient indeed. At its most extreme, if I say that “Killing Welsh people is a common-sense belief”, then I needn’t reason — or argue — for that belief. So perhaps this extreme example is too particular to be a common-sense “principle”. After all, it’s not a belief about the existence of a chair, or other people, or time. That said, Reid did apply his common-sense philosophy to moral beliefs and values. [See here.] Indeed, Reid’s “first principles” grounded his own moral and religious beliefs. Thus, Reid’s common-sense philosophy went way beyond metaphysics and epistemology.

In any case, the section about common-sense principles being those “which we are under a necessity to take for granted in the common concerns of life, without being able to give a reason for them” squares very well with Ludwig Wittgenstein’s doubts about doubt, and the “hinges” on which all our reasonings depend.

In On Certainty, Wittgenstein put it this way:

“The questions that we raise and our doubts depend on the fact that some propositions are exempt from doubt, are as it were like hinges on which those [doubts] turn.

“That is to say, it belongs to the logic of our scientific investigations that certain things are in deed not doubted…

“My life consists in my being content to accept many things.”

The main difference here is that Reid talked in terms of being under a necessity to accept certain principles, whereas Wittgenstein believed that (to put it simply) we can’t get the ball rolling unless we “exempt [certain propositions] from doubt”.

Not that an adherence to common sense was unthinking. It was a reasoned position. It also needs to be said that it wasn’t simply that, for example, Reid observed a table, and then argued that a belief in the existence of tables is “common sense”. His position on tables, etc. was based on “reflection” too. What’s more, Reid believed that our “faculties are all fallible”. In Reid’s own words:

“I can likewise conceive an individual object that really exists, such as St. Paul’s Church in London. I have an idea of it; I conceive it. The immediate object of this conception is 400 miles distant; and I have no reason to think it acts upon me or that I act on it.”

To state the obvious. Reid didn’t believe that St. Paul’s Church ceased to exist when he wasn’t observing it. Yet he might have concluded that Bishop Berkeley — and even some of the other empiricists - must have believed that…

Yes! Reid believed that these philosophers were metaphysical lunatics.

(It’s worth noting that this interpretation puts empiricists in exactly the same position as idealists.)

Finally, was the philosophy of common sense itself grounded on common sense?

Is Common-Sense Philosophy Conservative?

The danger of stressing “common sense” is that it can be used as a simple defence of the status quo. [See note 2.] The American philosopher Edward S. Reed picked up on this when he wrote the following words:

“[Whereas] Thomas Reid wished to use common sense to develop philosophical wisdom, much of this school simply wanted to use common sense to attack any form of intellectual change.”

So is it that the adherents of common sense don’t like “intellectual change”? This isn’t to say that intellectual change is always a good thing. And, politically, common sense may have a lot going for it… However, firstly it needs to be established what common sense is, and what beliefs and views fall under it.

On a mundane or anecdotal level, many people often simply mean my beliefs by their words “common-sense beliefs”. This has the consequence that “The earth is flat” and “The earth isn’t flat” can both be deemed to be common-sense beliefs by different people.

All that said, such everyday and mundane conceptions of common sense are certainly not identical to anything believed by Reid. Or at least it doesn’t seem so on the surface.

Reid’s philosophical — and indeed technical — account is that men (all men?) have an “innate ability to perceive common ideas”, and that such common ideas also exist alongside “judgement”. On the surface, this is a philosophical position on what common sense is. However, it can also be squared with conservative beliefs and views, and Reid’s philosophy may be taken as being mere (philosophical) gloss on conservativism (i.e., with a small ‘c’).

Despite all that, readers mustn’t automatically assume that common-sense philosophy was conservative… at least not in the 18th century. After all, the Scottish school of philosophers provided scientific accounts of historical events. They also believed that education should be free from religious and political dogma. Indeed, common-sense philosophy was part of the Scottish Enlightenment.

Marcuse on Common Sense and Wittgenstein

One philosopher who most certainly did believe that common-sense philosophy (or even just plain common sense) is conservative (or “reactionary”) was the German-American philosopher and political theorist Herbert Marcuse.

Marcuse argued that common sense is a weapon of ideology. Or, at the very least, he argued that it could be — and often has been — such a weapon. Moreover, an adherence to common sense actually “obscures reality”, and, in so doing, upholds existing power structures.

In terms of its affect on individual people. An adherence to common sense leads to a lack of critical thinking, intellectual sterility, and, more importantly, a blind acceptance of the status quo.

Earlier, Wittgenstein was tied to Thomas Reid, and the former was a particular target of Marcuse.

Take Wittgenstein’s claim (which will be very familiar to all those who’ve read his work) that philosophy “leaves everything as it is”. Marcuse believed (as found in his book One-Dimensional Man) that Wittgenstein’s statement is one of “academic sadomasochism [and] self humiliation”. He also believed that it was a “self denunciation of the intellectual whose labour does not issue in scientific, technical or like achievements”.

Marcuse interpreted the words “leaves everything as it is” as Wittgenstein stating that “the existing structure of society is all right as it is”, whereas other philosophers interpret Wittgenstein as arguing that only “language is all right as it is”.

Earlier it was said that common-sense philosophers believed that philosophy should be something to live by, and then Wittgenstein was tied to Reid, yet Marcuse believed that Wittgenstein, J.L. Austin and other linguistic philosophers had (in the words of Colin Lyas) “detached philosophy from its traditional concern with the large issues of life and rendered it trivial”.

Notes:

(1) The reference to “testimony” seems like the odd man out here. It can be doubted that many people would argue that a belief (or trust) in testimony has much to do with common sense. [See the ‘philosophy of testimony’.]

(2) The Conservative Party of the United Kingdom, right up until the 1990s and beyond, often talked about “common sense”. [See here.] And, as can be guessed by some readers, Marxist/Leftist commentators took a Marcusian position on such utterances. It’s certainly the case that Conservatives have often argued that they don’t indulge in what they called “theory”. Yet this is clearly false: they simply don’t like certain types of theory. As for Conservative leaders and politicians, many of their ideas and policies are both based on theory and are world’s away from being common sense. Then again, the essay above has partially shown that this is a problematic notion anyway, and is only on secure ground when it involves body counting what various populations believe.