Table of Contents

God invented and gave us sight to the end that we might behold the courses of intelligence in the heaven, and apply them to the courses of our own intelligence which are akin to them, the unperturbed to the perturbed; and that we, learning them and partaking of the natural truth of reason, might imitate the absolutely unerring courses of God and regulate our own vagaries. The same may be affirmed of speech and hearing: they have been given by the gods to the same end and for a like reason. For this is the principal end of speech, whereto it most contributes. By ratiocination, I mean computation. Prologue An encouraging development in the education of programmers in recent years has been a renewed interest in the relationship between programming, logic and abstract algebra. Recent enthusiasm over functional programming is no doubt the main motivation behind this happy development. However, this interest is at times accompanied by a kind of awed appreciation of this relationship, perhaps among practitioner enthusiasts, who consider it almost miraculous, more than among academics. Some researchers notable for popularizing the interest in this relationship encourage this view with talk of “computational trinitarianism” and statements like, “this is further proof that mathematics is discovered rather than invented,” one going so far as to call the relationship “a manifestation of the divine” and to say that “all three have ontological force; they codify what is, not how to describe what is already given to us. In this sense they are foundational; if we suppose that they are merely descriptive, we would be left with the question of where these previously given concepts arise, leading us back again to foundations”. Another has said, “Have you heard of the Curry-Howard-Lambek isomorphism? … These three different theories … were developed separately — constructive logic, typed lambda calculus, and category theory — and the isomorphism says that these are actually identical. Why do these three completely separate theories have the same structure? Because all three of them are about composability.” While he is not too far off the mark when he later says, “Maybe we are just discovering the way our brains work. … [I]f mathematics is all about composition, then composition is something that our brains came up with in order to deal with complexity,” the truth is far simpler yet more interesting, even if less mysterious. Awed wonder is a powerful marketing tool, but it mystifies rather than clarifies, and fosters a kind of magical thinking in a field that strives to be most at odds with it (although, as we’ll see in part 2, it sometimes fails). The goal of mathematics is to simplify, not to wow. This attitude towards that — quite obvious, as we’ll see — relationship also distracts from the actual, less obvious, discoveries made. The existence of this relationship was not among them; it was not the denouement — it was the setup. Confusing the two muddles the very essence of any story. I also believe that it is this very attitude that puts the focus on the chosen mathematical abstractions rather than on the things they describe creates an atmosphere where there can only be one True interpretation — even one with three personas. I call this absolutist view that confuses the peculiarities of a method of observation with those of the observed system and so denies alternative interpretations the “Protestant” view, and it is a hindrance to understanding logic, where truth is, almost by definition, always relative to a specific system. As Alonzo Church wrote in the very paper he first introduced the λ-calculus: We do not attach any character of uniqueness or absolute truth to any particular system of logic. The entities of formal logic are abstractions, invented because of their use in describing and systematizing facts of experience or observation, and their properties, determined in rough outline by this intended use, depend for their exact character on the arbitrary choice of the inventor… [T]here exist, undoubtedly, more than one formal system whose use as a logic is feasible, and of these systems one may be more pleasing or more convenient than another, but it cannot be said that one is right and the other wrong. The following text will attempt to demistify this relationship between computation, logic and abstract algebra. Category theory is a generalization of the last, but programming languages are sometimes interchangeably assigned to both the second, logic, and the first, computation, thus adding to the confusion. In my view, programming languages (at least when studied in a theoretical context) strictly belong to the second persona of logic, and not at all to the first, computation, other than by virtue of the ordinary relationship between the three; programming languages are no more related to computation than formal logic is. I have previously written about the distinction between programming and computation and the misunderstanding caused by confusing the two, and I hope that this text will clarify the matter further. While I will not opine on whether or not mathematical objects have “an ontological force”, as the Platonic ontology of mathematics is a matter of debate in the philosophy of mathematics — a fascinating field, but not a focus of this discussion — I will show that they are not only very much descriptive, but that what they attempt to describe, and therefore how they arise, has been explicitly stated by the originators of those ideas. There is nothing incidental, miraculous, surprising or “divine” in the relationship between computation, logic and algebra; rather, nothing is more creatively human than the process that led to their correlation, and while some things in mathematics are perhaps discovered, logic and algebra were invented, and rather than carrying an independent “foundational ontological force”, they were devised to describe a physical phenomenon (although I do not claim that the mathematical systems invented may not also correspond to some Truth). In particular, I will show that computation, logic and algebra were studied as a single subject for most of their relevant history, related in the following way: human reason works by computation, logic is a description of the process of reasoning, and algebra is the mathematical modeling of logic. The three began to separate into different disciplines only towards the end of the nineteenth century and the beginning of the twentieth. Logic was separated from algebra by Gottlob Frege, immediately triggering the criticism that the correspondence between the two is an argument for having only one. Frege responded that there would obviously be a correspondence between the two because he had invented one to describe the same thing as the other, but that one can translate between two modes of expression does not mean that one of them is redundant. Computation separated from logic some decades later as a more foundational or primitive concept, while exposing the fact that the basic principles of how logic supposedly described human reason — which Gottfried Leibniz called the “Art of Combination” — hadn’t changed for literally millennia, and while logicians/algebraists declared themselves to be studying the computation in the human mind, they were actually just modeling a very particular, and ancient, view of human reason that they had taken on unquestionable faith. The systems were in such harmonious correspondence not just because they described the same physical phenomenon, but because they all re-described the same ancient description of that phenomenon. While researching the topic — not being a logician or a historian, but rather a programmer and an amateur interested in the history of mathematics — I was surprised how explicit this connection was made from the very beginning of the disciplines. Quite the opposite of accidental or miraculous, calling this relationship obvious is an understatement. That there is some relationship between a house cat and a leopard can be said to be obvious; the one between the original Star Wars trilogy and its many prequels and sequels is much more than just obvious. If you think that it is in any way accidental or miraculous, or that it is proof that Darth Vader was discovered rather than invented, you are clearly missing something very important in understanding the meta-theory of stories — meaning, how they come to be. I delight at the opportunity to use the history of science — or of logic and mathematics in this case — to clarify the development of mathematical concepts by tracing their evolution and the creative process of their originators. This book-length text — and not a short book — is published as a three-part blog post series, but I personally wrote very little of it. While the text is not different in content or conclusions from books about the history of logic and computation (or the wonderful recent Atlantic article, How Aristotle Created the Computer), it is certainly different in style, being comprised mostly of original sources. I chose to compose the text in this way for several reasons, listed here in no particular order: 1. I am not a professional historian and do not have sufficient knowledge to provide insight on historical trends. 2. I find reading (carefully selected) primary sources a lot more fun than sweeping descriptions. 3. As most mathematicians and programmers do not make good historians, a lot of inaccuracies and downright mistakes are perpetrated because few have seen the actual sources. 4. As my goal is to demystify logic and mathematics, I think it is especially enlightening to read how the originators of the ideas describe their process of arriving at their insight. My text is also different in focus; it attempts to emphasize the evolution of the connection between computation, logic and algebra, and is concerned with each of the three subjects mostly inasmuch as they relate to the other two. The themes you will want to pay close attention to in the primary sources are the analysis of the human mind, and the role of language in reasoning. While the thesis regarding nature of that connection is easily established — in fact, it is satisfactorily established by picking at random almost any single one of the original sources I’ll cover — we must take care not to fall into the trap of an overarching, coherent, Gladwellian narrative, and remember that there is a wide gap between an early, vague notion of ideas — even if it employs modern terms, as that is often a later homage to early sparks — and a rigorous treatment of them; the two must not be considered the same or one risks falling into the trap of anachronism. Of course, the thesis is only a guiding question and an excuse and to cover what a fascinating subject in the history of science and ideas. So I allowed myself to quote longer excerpts when I found them interesting or amusing, even if they are tangential to the main thrust of the narrative. I’ve also skipped most of the overly technical portions of the texts, quoting only the basics, as this is not a mathematical textbook. Nevertheless, I do assume some familiarity with the relevant mathematical terms in parts of the discussion, so those who are familiar with the subjects will gain more from the text. Like most histories, this one is incomplete. I’ve only chosen those works that have had a strong and direct influence on the mainline development of the subject in the West — thus ignoring the development of the subject in the East, especially in India, and only briefly mentioned imported influences — and even then had to restrict the selection. For essential context, I rely on secondary sources; these are just a few notable ones: In part 1, on the classical and embryonic period of logic, I relied heavily the book Mechanization of Reasoning in a Historical Perspective by Witold Marciszewski and Roman Murawski (review here). For part 2, about the algebraic period, I used Daniel J. Cohen’s, Equations from God: Pure Mathematics and Victorian Faith. For part 3, about the logicist/mathematical/computational period I relied on Robin Gandy’s paper The Confluence of Ideas in 1936 and Andrew Hodges’s biography of Alan Turing. There were others as well, all referenced in the text and bibliography.

I. The Language of Thought

Simplex Apprehensio

No idea is born, spontaneously, from nothingness; every invention grows in a time and a place, and is produced by a context in addition to ingenuity. But a story must begin somewhere, and a good place to start the story of logic is with Aristotle of Stagira (384-322 BC), who was interested in the question of how we know what is true, and how we demonstrate the truth of a claim to others. Unfortunately, Aristotle’s works — especially on logic — are rather dull and abstract; To be fair to Aristotle, his surviving works were likely not meant for publication as written, but rather served as private lecture notes. quoting long passages from them would put an end to my attempt at writing a coherent text composed mostly of primary sources before it begins in earnest. Nevertheless, we cannot do without him completely:

⊕ Prior Analytics, c. 350 BCE We must first state the subject of our inquiry and the faculty to which it belongs: its subject is demonstration and the faculty that carries it out demonstrative science. We must next define a premiss, a term, and a syllogism, …; and after that, the inclusion or noninclusion of one term in another as in a whole, and what we mean by predicating one term of all, or none, of another. A premiss then is a sentence affirming or denying one thing of another. This is either universal or particular or indefinite. By universal I mean the statement that something belongs to all or none of something else; by particular that it belongs to some or not to some or not to all; by indefinite that it does or does not belong, without any mark to show whether it is universal or particular, e.g. ‘contraries are subjects of the same science’, or ‘pleasure is not good’. … Therefore a syllogistic premiss without qualification will be an affirmation or denial of something concerning something else in the way we have described; it will be demonstrative, if it is true and obtained through the first principles of its science. ⊕

Plato (left) who believed in the divine world of ideals and Aristotle (right) who believed in the power of earthly empiricism, depicted here in a Vatican fresco by Raphael as they’re debating Haskell vs. ML among Go programmers … A syllogism is discourse in which, certain things being stated, something other than what is stated follows of necessity from their being so. I mean by the last phrase that they produce the consequence, and by this, that no further term is required from without in order to make the consequence necessary. … Every premiss states that something either is or must be or may be the attribute of something else; of premisses of these three kinds some are affirmative, others negative, in respect of each of the three modes of attribution; again some affirmative and negative premisses are universal, others particular, others indefinite. It is necessary then that in universal attribution the terms of the negative premiss should be convertible, e.g. if no pleasure is good, then no good will be pleasure; the terms of the affirmative must be convertible, not however, universally, but in part, e.g. if every pleasure, is good, some good must be pleasure; the particular affirmative must convert in part (for if some pleasure is good, then some good will be pleasure); but the particular negative need not convert, for if some animal is not man, it does not follow that some man is not animal. First then take a universal negative with the terms A and B. If no B is A, neither can any A be B. For if some A (say C) were B, it would not be true that no B is A; for C is a B. But if every B is A then some A is B. For if no A were B, then no B could be A. But we assumed that every B is A. Similarly too, if the premiss is particular. For if some B is A, then some of the As must be B. For if none were, then no B would be A. But if some B is not A, there is no necessity that some of the As should not be B; e.g. let B stand for animal and A for man. Not every animal is a man; but every man is an animal.

Aristotle’s main contribution, which has dominated most work in logic to this day, was in pointing out that if our premises are stated in that particular syntactic form, such as all A is B and all B is C, then, using the syntactic formal inference rule of the syllogism, we may deduce that all A is C, regardless of what A, B and C mean. As the nineteenth century logician Augustus De Morgan put it, ⊕ Formal Logic, p. 1 “[t]his logical truth depends upon the structure of the sentence and not upon the particular matters spoken of.”

But another significant contribution, though less essential, served as the basis of all work in formal logic until Turing and has to do less with the formal proof theory of the syllogism, but with how logical sentenses are formed, namely as a composition of terms denoting (having the meaning, or semantics, of) attributes, concepts, or classes that stand for some collection of objects, such as man, animal or virtue. The composition takes the general form [quantifier] A is B, and there are four kinds of propositions: positive universal propositions, for example, all men are animals, positive particular (or existential in modern terms) propositions, namely, some men are virtuous, and their negation This logic resembles monadic first-order logic in the modern nomenclature. .

It should be noted that while Aristotle’s logic deals with some form of quantification (all A is B or some C is D), it did not include the propositional connectives of conjunction and disjunction (“and” and “or”) that allow forming compound predicates. This was likely deliberate Robin Smith, Aristotle’s Logic . The connectives and compound predicates were developed later in Ancient Greece and during the Middle Ages Kevin Klement, Propositional Logic .

Aristotle’s general framework of reasoning — elaborated upon and expanded somewhat during the Middle Ages — can be summarized as the mechanization of reasoning through the manipulation of language, or syntax, that expresses meaning through a combination of terms denoting concepts. This framework could be called combinatory logic, but as the name is nowadays used for something much more technical and specific, I will call it the Aristotelian framework.

In the book Mechanization Of Reasoning In A Historical Perspective by Witold Marciszewski and Roman Murawski, Marciszewski writes that this Aristotelian framework,

⊕ Mechanization Of Reasoning, Kindle Locations 828-875 started from singling out three hierarchically arranged operations of the mind. Fundamental among those three operations was that of grasping things by concepts. In a sense it could be called the simplest one, and so it was called by scholastic logicians…, when they used the term ‘simple apprehension’ (simplex apprehensio). It is the simplest one in the sense that in that part of the act of grasping things which occurs at the level of consciousness, and so is accessible to introspection, we do not perceive any components that could be clearly isolated. For instance, some simple apprehension leads one to the concept of natural number, whose content includes, inter alia, the fact that every natural number has its successor. …The second operation of the mind, second in the sense that it assumes the existence of concepts and is more complex than conceptualization, is the formation of judgements. … [I]t consists in the combination of concepts into a judgement (judicium)… A judgement was treated as an invariably tripartite structure consisting of the subject, the copula, and the predicate, the copula expressing either affirmation (‘is’) or negation (‘is not’). This is how a judgement (protasis) was understood by Aristotle. … The third operation of the mind, the most complex one in the sense that it assumes the two preceding ones and brings the most complex product, consists in the proof, construed in the Aristotelian logic as a syllogism.

We shall now skip some two thousand years to the seventeenth century, and so must turn to a secondary source, Marciszewski again, for some background of the developments that occurred during that long period and the context in which they occured:

⊕ Mechanization Of Reasoning, Kindle Locations 898-927 That context involved two philosophical tendencies, namely finitism and formalism. … In the Greek philosophy of mathematics finitism established its place for good owing to the paradoxes of Zeno of Elea (490-430 B.C.), which showed how formidably perplex are the problems resulting from the concept of (actual) infinity. That was reinforced by the authority of Aristotle who in his Physics and Metaphysics advanced arguments, to be later repeated for centuries, against the existence of actually infinite domains. … Likewise, a kind of formalism was at that time [of the 17th century] nothing extraordinary. Its lively presence can be seen in both the intensification of the nominalistic trends from the 12th century to the late Middle Ages and the 17th century, and also in other elements of medieval culture. It was not without importance that human minds were at that time imbued with biblical ideas, which included the faith in the perfection of the concepts to be found in the Scripture and the adequacy of the words which rendered those concepts. For instance, since Adam in Paradise gave names to all animals, that is, came in the possession of their concepts, nothing more was left in that respect to be done. No new intuition was necessary, it just sufficed to make use of the words assigned to the concepts. Likewise astronomy was a complete and closed system, which was moreover perfectly synthetized with the theological one (which we can finely seen, for instance, in Dante’s Divine Comedy). The ideas of the potential infinity of human cognition, of the limits of verbalization, of the approximative nature of scientific theories and the like, have become familiar to the modern mind only recently. As long as it was believed that concepts and judgements had adequate mappings in language, on the one-to-one basis, there were reasons to believe that thoughts could be fully replaced by words, and these, being material objects, could be processed mechanically. This picture of medieval mentality applies to only some of its trends (it would be naive to treat it as a monolith). For instance, Augustinism opposed Aristotelianism both by its infinitism and its doctrine of illumination, which stressed the intuitive, non-mechanizable elements of cognition; that tendency even more manifested itself in the gnostic movements. But as for the problems with which we are concerned here the essential point is that both the finitistic and the formalistic trend were firmly rooted in the medieval thought.

While the Aristotelian framework served as the basis for virtually all work on logic and the mechanization of reasoning up to the twentieth century, not deviated from until the major philosophical breakthrough by Alan Turing, it inhabited an intellectual context very different from ours. Marciszewski writes that Mechanization Of Reasoning, Kindle Locations 828-910 “[p]aradoxically enough the problem of a certain formalization of arguments was for centuries discussed in terms of the logic of discovery, and hence in terms of processes which we now treat as typically creative and not subject to mechanical procedures was due to a combinatory interpretation of the process of discovery, which involved finitism and formalism. … [F]or the representatives of the Platonic-Aristotelian views (which came to be opposed by modern empiricism) truth in the strict sense of the term was identical with necessary truth or (approximately) analytic truth, and the set of such truths was supposed to be decidable. … The view which implied the finiteness of both the domain of individuals and the set of concepts provided people with reasons to postulate the decidability of the system of human knowledge. True statements would be deducible from a finite set of first principles (as Aristotle claimed), and false ones would be refutable by demonstrating that they contradict those principles.”

There were dissenting views, some anti-formalistic, and some, like Francis Bacon opposed this deductive program altogether, pointing out that deduction can never create new knowledge, only expose, or demonstrate, truths already present in the assumed premises, and wished to create a logic based on inductive reasoning. However, Mechanization Of Reasoning, Kindle Locations 1669-1673 “[t]he greatness which logic was destined to attain and which it did attain in our century, was reached by the old Aristotelian road when it merged with the path along which mathematics was developing. Those two paths came closer to one another for the first time in the 17th century, and the process was due to Leibniz. Thus it was he and not Bacon who became the forerunner of future logic. And those thinkers who, while underestimating Aristotle, did not commit the mistake of underestimating deduction, also proved to have come closer to what was ahead.”

The Clockwork Mind

⊕



While the intellectual climate changed considerably with the scientific revolution and the rise of empiricism, which no longer saw human knowledge as a closed system, the Aristotelian framework was able to both survive a migration to the different climate, while at the same time gaining more rigor in the 17th century, due to the work of Gottfried Wilhelm Leibniz (1646-1716), who became, according to Marciszewski, “the first logician of our epoch.”

Leibniz reestablished the Aristotelian framework on scientific grounds by reconstructing it on top of two new foundations: 1. that the universe is a machine that behaves according to fixed laws, and 2. that the highest of God’s creation, man’s reason, is no different, itself being a sophisticated machine. Leibniz compared the universe (including the human mind) to an elaborate clock, ticking without the constant intervention of the clockmaker (an idea that Leibniz’s contemporary, Isaac Newton, rejected as marginalizing the role of God in His creation). No longer a rhetorical tool used to make an argument or “demonstrate”, as Aristotle presented it and as it had been perceived in the Middle Ages, logic was to Leibniz a description of the workings of the mind.

For a full treatment of the intellectual climate in Leibniz’s time, and how both empiricism and Platonism seved as influence, I will refer the interested reader to Marciszewski, while we move on to examine the primary sources.

Leibniz wrote that he had been inspired by his recent predecessor Thomas Hobbes (1588-1679) (who, to put him in historical context, was a contemporary of William Shakespeare) to consider even the human mind as an automaton. Hobbes opens his masterwork, Leviathan, thus:

⊕ Introduction to Leviathan (or The Matter, Forme, & Power of a Common-Wealth Ecclesiastical and Civill), 1651 Nature (the art whereby God hath made and governes the world) is by the art of man, as in many other things, so in this also imitated, that it can make an Artificial Animal. For seeing life is but a motion of Limbs, the begining whereof is in some principall part within; why may we not say, that all Automata (Engines that move themselves by springs and wheeles as doth a watch) have an artificial life? For what is the Heart, but a Spring; and the Nerves, but so many Strings; and the Joynts, but so many Wheeles, giving motion to the whole Body, such as was intended by the Artificer? Art goes yet further, imitating that Rationall and most excellent worke of Nature, Man.

Man-made machines and the natural creations, including the mind, differ not in kind, Leibniz believed, but merely in degree of complexity, as natural beings are machines “all-the-way-down”:

⊕ The Monadology, 1714, in Loemker, Philosophical Papers, p.644 [E]ach organic body belonging to a living being is a kind of divine machine or natural automaton infinitely surpassing all artificial automata. For a machine made by human art is not a machine in each of its parts; for example, the tooth of a brass wheel has parts or fragments which are not artificial so far as we are concerned, and which do not have the character of a machine, in that they fit the use for which the wheel was intended. But the machines of nature, living bodies, are still machines in their smallest parts, into infinity. It is this that makes the difference between nature and art, that is, between the divine art and ours.

The idea that reason itself can be mechanized by formalization, and that it resembles algebra, also came from Hobbes, who in his work, Computation or Logique wrote:

⊕ Computation or Logique, in Elements of Philosophy, pp. 2-4, 1656 By RATIOCINATION, I mean computation. Now to compute, is either to collect the sum of many things that are added together, or to know what remains when one thing is taken out of another. Ratiocination therefore is the same with Addition and Substraction; and if any man adde Multiplication and Division, I will not be against it, seeing Multiplication is nothing but Addition of equals one to another, and Division nothing but a Substraction of equals one from another, as often as is possible. So that all Ratiocination is comprehended in these two operations of the minde, Addition and Substraction. But how by the ratiocination of our mind, we add and substract in our silent thoughts, without the use of words, it will be necessary for me to make intelligible by an example or two. If therefore a man see something afar off and obscurely, although no appellation had yet been given to anything, he will, notwithstanding, have the same idea of that thing for which now, by imposing a name on it, we call it body. Again, when, by coming nearer, he sees the same thing thus and thus, now in one place and now in another, he will have a new idea thereof, namely, that for which we now call such a thing animated. Thirdly, when standing nearer, he perceives the figure, hears the voice, and sees other things which are signs of a rational mind, he has a third idea, though it have yet no appellation, namely, that for which we now call anything rational. Lastly, when, by looking fully and distinctly upon it, he conceives all that he has seen as one thing, the idea he has now is compounded of his former ideas, which are put together in the mind in the same order in which these three single names, body, animated, rational, are in speech compounded into this one name, body-animated-rational or man. In like manner, of the several conceptions of four sides, equality of sides, and right angles, is compounded the conception of a square. For the mind may conceive a figure of four sides without any conception of their equality, and of that equality without conceiving a right angle ; and may join together all these single conceptions into one conception or one idea of a square. And thus we see how the conceptions of the mind are compounded. … We must not therefore thinke that Computation, that is, Ratiocination, has place onely in numbers; as if man were distinguished from other living Creatures (which is said to have been the opinion of Pythagoras) by nothing but the faculty of numbring; for Magnitude, Body, Motion, Time, Degrees of Quality, Action, Conception, Proportion, Speech and Names (in which all the kinds of Philosophy consist) are capable of Addition and Substraction.

Note how Hobbes equates computation with the combination of concepts (moreover, he describes a process of abstraction, i.e., naming a general concept, and refinement, i.e., restricting that concept to a narrower contained concepts).

Leibniz’s Instrument

Leibniz set out to give a more rigorous, mathematical, treatment to Hobbes’s ideas. To understand the more technical state of affairs at the time we must know that that the use of letters to stand for arbitrary concepts was pioneered by Aristotle, but over the middle ages and early-modern period, algebra was developed in the Arab world and in Europe, and letters were also taken to mean an arbitrary number, a class of numbers, or an unknown number — meaning, the variable came to be.

In 1666, as part of his admission process to the philosophical faculty at Leipzig, 20-year-old Leibniz wrote the Dissertation on the Art of Combinations. After dispensing with the obligatory proof of God’s existence in the introduction, he roughly outlines the idea of expressing things universally with numbers, he writes,

⊕ Dissertation On the Art of Combination, part 3, Thomas Hobbes, everywhere a profound examiner of principles, rightly stated that everything done by our mind is a computation, by which is to be understood either the addition of a sum or the subtraction of a difference … So just as there are two primary signs of algebra and analytics, + and −, in the same way there are as it were two copulas, ‘is’ and ‘is not’”.

He came up with an ambitious plan:

⊕ On Universal Synthesis and Analysis, or The Art of Discovery and Judgment, l679 in Loemker, Philosophical Papers, pp. 229-233 Seeing that there are categories for the simple terms by which concepts are ordered, why should there not also be categories for complex terms, by which truths may be ordered? … It seemed to me, however, that this could be achieved universally if we first had the true categories for simple terms and if, to obtain these, we set up something new in the nature of an alphabet of thoughts, or a catalogue of the highest genera or of those we assume to be highest, such as a, b, c, d, e, f, out of whose combination inferior concepts may be formed. … All derivative concepts, moreover, arise from a combination of primitive ones, and the more composite concepts from the combination of less composite ones. [But one must take care that the combinations do not become useless through the joining-together of incompatible concepts. This can be avoided only by experience or by resolving them into distinct single concepts. One must be especially careful, in setting up real definitions, to establish their possibility, that is, to show that the concepts from which they are formed are compatible with each other.] ⊕

Frontispiece of Leibniz’s Dissertatio de arte combinatoria, printed in 1690 (source: Wikipedia) … Thus any truth whatever can be justified, for the connection of the predicate with the subject is either evident in itself as in identities, or can be explained by an analysis of the terms. This is the only, and the highest, criterion of truth in abstract things, that is, things which do not depend on experience — that it must either be an identity or be reducible to identities. … the art of combinations in particular, as I take it (it can also be called a general characteristic or algebra), is that science in which are treated the forms or formulas of things in general, that is, quality in general or similarity and dissimilarity; in the same way that ever new formulas arise from the elements a, b, c themselves when combined with each other, whether these elements represent quantities or something else. This art is distinct from common algebra, which deals with formulas applied to quantity only or to equality and inequality. This algebra is thus subordinate to the art of combinations and constantly uses its rules. But these rules of combination are far more general and find application not only in algebra but in the art of deciphering, in various games, in geometry itself when it is treated linearly in the manner of the ancients, and finally, in all matters involving relations of similarity.

Here we can already see the idea of a hierarchy, or an order, of propositions or categories (sometimes also called by Leibniz and most future logicians, concepts, notions or classes), as well as the idea that complex concepts are built from simpler ones by applying combinations, which are themselves finer (or coarser) concepts with semantics that can be manipulated computationally. We also see Leibniz making a connection between logic and algebra — both manipulate formulas and symbols — even though algebra is ultimately concerned with quantities and logic with more abstract concepts. What matters, according to Leibniz, is the general idea of combination of symbols.

But to take advantage of algebra for the actual study of logic, another step had to be made. While algebra as the art of manipulating symbols had already been established by Leibniz’s time, algebraic variables could only stand for numbers for their combinations to make sense. Leibniz, however, recognized this was not a limitation, as numbers can be used to encode, or characterize, anything:

⊕ On the General Characteristic, Ca. 1679 in Loemker, Philosophical Papers, p. 221-227 There is an old saying that God created everything according to weight, measure, and number. But there are things which cannot be weighed, those namely which have no force or power. There are also things which have no parts and hence admit of no measure. But there is nothing which is not subordinate to number. Number is thus a basic metaphysical figure, as it were, and arithmetic is a kind of statics of the universe by which the powers of things are discovered. Men have been convinced ever since Pythagoras that the deepest mysteries lie concealed in numbers. It is possible that Pythagoras brought over this opinion, like many others, from the Orient to Greece. But, because the true key to the mystery was unknown, more inquisitive minds fell into futilities and superstitions, from which there finally arose a kind of popular Cabbala, far removed from the true one, and that multitude of follies which is falsely called a kind of magic and with which books have been filled. Meanwhile there remained deep-rooted in men the propensity to believe that marvels can be discovered by means of numbers, characters, and a certain new language, which some called the Adamic language, by numbers; Jacob Böhme called it the Natursprache. But perhaps no mortal has yet seen into the true basis upon which everything can be assigned its characteristic number. For the most scholarly men have admitted that they did not understand what I said when I incidentally mentioned something of the sort to them. And although learned men have long since thought of some kind of language or universal characteristic by which all concepts and things can be put into beautiful order, and with whose help different nations might communicate their thoughts and each read in his own language what another has written in his, yet no one has attempted a language or characteristic which includes at once both the arts of discovery and of judgment, that is, one whose signs or characters serve the same purpose that arithmetical signs serve for numbers, and algebraic signs for quantities taken abstractly. Yet it does seem that since God has bestowed these two sciences on mankind, he has sought to notify us that a far greater secret lies hidden in our understanding, of which these are but the shadows. Some unknown fate has brought it about, however, that when I was a mere boy I became involved in these considerations, and as first inclinations usually do, they have remained strongly flxed in my mind ever since. Two things which are otherwise of doubtful merit and are harmful to many people, proved wonderfully useful to me: first, I was self-taught, and second, I looked for something new in every science when I first studied it, often before I even understood its already established content. But so I gained a double reward: first, I did not fill my head with empty and cumbersome teachings accepted on the authority of the teacher instead of sound arguments; second, I did not rest until I had traced back the tissues and roots of every teaching and had penetrated to its principles. By such training I was enabled to discover by my own effort everything with which I was concerned. ⊕

Leibniz mansplaining monads to his friend and disciple, Queen Sophia Charlotte of Hanover, queen consort of Prussia, in front of the Charlottenburg Palace in Berlin (source: Leibnitiana) When I turned, therefore, from the reading of history, which had delighted me from my earliest youth, and from the cultivation of style, which I carried out with such ease both in prose and in more restricted forms that my teachers feared that I might remain stuck in such frivolities, and took up logic and philosophy and had barely begun to understand something about these fields, what a multitude of fancies came to birth in my brain and were scratched down on paper to be laid before my astonished teachers. Among other things I once raised a doubt concerning the categories. I said that just as we have categories or classes of simple concepts, we ought also to have a new class of categories in which propositions or complex terms themselves may be arranged in their natural order. For I had not even dreamed of demonstrations at that time and did not know that the geometricians do exactly what I was seeking when they arrange propositions in an order such that one is demonstrated from the other. My question was thus superfluous, but when my teachers failed to answer it, I pursued these ideas for the sake of their novelty, attempting to establish such categories for complex terms or propositions. Upon making the effort to study this more intently, I necessarily arrived at this remarkable thought, namely that a kind of alphabet of human thoughts can be worked out and that everything can be discovered and judged by a comparison of the letters of this alphabet and an analysis of the words made from them. This discovery gave me great joy though it was childish of course, for I had not grasped the true importance of the matter. But later, the more progress I made in my thinking about these things, the more confirmed I was in my decision to carry the problem further. … For this is what I finally discovered after most intent thought. Nothing more is necessary to establish the characteristic which I am attempting, at least to a point sufficient to build the grammar of this wonderful language and a dictionary for the most frequent cases, or what amounts to the same thing, nothing more is necessary to set up the characteristic numbers for all ideas than to develop a philosophical and mathematical ‘course of studies’, as it is called, based on a certain new method which I can set forth, and containing nothing more difficult than other courses of study, or more remote from use and understanding, or more alien to the usual way of writing. … Once the characteristic numbers for most concepts have been set up, however, the human race will have a new kind of instrument which will increase the power of the mind much more than optical lenses strengthen the eyes and which will be as far superior to microscopes or telescopes as reason is superior to sight. The magnetic needle has brought no more help to sailors than this lodestar will bring to those who navigate the sea of experiments. What other consequences will eventually follow from it must be left to the decree of the fates; however, they cannot be the great and good. For men can be debased by all other gifts; only right reason can be nothing but wholesome. But reason will be right beyond all doubt only when it is everywhere as clear and certain as only arithmetic has been until now. Then there will be an end to that burdensome raising of objections by which one person now usually plagues another and which turns so many away from the desire to reason. When one person argues, namely, his opponent, instead of examining his argument, answers generally, thus, ‘How do you know that your reason is any truer than mine? What criterion of truth have you?’ And if the first person persists in his argument, his hearers lack the patience to examine it. For usually many other problems have to be investigated first, and this would be the work of several weeks, following the laws of thought accepted until now. And so after much agitation, the emotions usually win out instead of reason, and we end the controversy by cutting the Gordian knot rather than untying it. This happens especially in deliberations pertaining to life, where a decision must be made; here it is given to few people to weigh the factors of expediency and inexpediency, which are often numerous on both sides, as in a balance. The more strongly we are able to present to ourselves, now one circumstance and now another, in order to balance the varying inclinations of our own minds, and the more eloquently and effectively we can adorn and point them out for others, the more firmly we shall act and carry the minds of other men with us, especially if we make wise use of their emotions. There is hardly anyone who could work out the entire table of pros and cons in any deliberation, that is, who could not only enumerate the expedient and inexpedient aspects but also weigh them rightly. Thus two disputants seem to me almost like two merchants who are in debt to each other for various items, but who are never willing to strike a balance; instead, each one advances his own various claims against the other, exaggerating the truth and magnitude of certain particular items. Their quarrel will never end on this basis. And we need not be surprised that this is what has happened until now in most controversies in which the matter is not clear, that is, is not reduced to numbers. ⊕

A replica of Leibniz’s stepped reckoner, the first mechanical calculator to perform all four arithmetic operations. The device was discovered in 1879 in an attic at the University of Göttingen by workers fixing a leak in the roof (source: History of Computers) ⊕

Details of the mechanism of the stepped reckoner. An illustration in Leupold, Jacob, Theatrum arithmetico-geometricum, das ist…, 1727 (source: Library of Congress) Now, however, our characteristic will reduce the whole to numbers, so that reasons can also be weighed, as if by a kind of statics. For probabilities, too, will be treated in this calculation and demonstration, since one can always estimate which of the given circumstances will more probably occur. Finally, anyone who is certainly convinced of the truth of religion and its consequences, and so embraces others in love that he desires the conversion of mankind, will surely admit, if he understands these matters, that nothing will be more influential than this discovery for the propagation of the faith, unless it be miracles, the holiness of an apostle, or the victories of a great monarch. Where this language can once be introduced by missionaries, the true religion, which is in complete agreement with reason, will be established, and apostasy will no more be feared in the future than would an apostasy of men from the arithmetic or geometry which they have once learned. So I repeat what I have often said: that no man who is not a prophet or a prince can ever undertake anything of greater good to mankind or more fitting for the divine glory. But we must go further than words! Since the admirable connection of things makes it most difficult to give the characteristic numbers of a few things separated from others, I have thought of an elegant device, if I am not mistaken, by which to show that ratiocination can be proved through numbers. Thus I imagine that these most remarkable characteristic numbers are already given, and, having observed a certain general property to be true of them, I set up such numbers as are somehow consistent with this property, and applying these, I at once demonstrate through numbers, in wonderful order, all the rules of logic and show how we can know whether certain arguments are in good form. But the material soundness or truth of an argument can be judged without much mental effort and danger of error only when we have the true characteristic numbers of things themselves. … One cannot go to infinity in his proofs, however, and therefore some things must be assumed without proof — not silently and by stealth, indeed, dissimulating our own laziness as philosophers customarily do, but keeping clearly in mind what we have used as first assertions, after the example of geometricians who, to show their good faith, acknowledge at the very start the assumed axioms they are to use, so that they may be sure that all the conclusions are proved at least hypothetically from these assumptions. First of all, I assume that every judgment (i.e., aflirmation or negation) is either true or false and that if the affirmation is true the negation is false, and if the negation is true the affirmation is false; that what is denied to be true -— truly, of course — is false, and what is denied to be false is true; that what is denied to be affirmed, or affirmed to be denied, is to be denied; and what is affirmed to be affirmed and denied to be denied is to be affirmed. Similarly, that it is false that what is false should be true or that what is true should be false; that it is true that what is true is true, and what is false, false. All these are usually included in one designation, the principle of contradiction. … In general, every true proposition which is not identical or true in itself can be proved a priori with the help of axioms or propositions that are true in themselves and with the help of definitions or ideas. For no matter how often a predicate is truly affirmed of a subject, there must be some real connection between subject and predicate, such that in every proposition whatever, such as A is B (or B is truly predicated of A), it is true that B is contained in A, or its concept is in some way contained in the concept of A itself. … Such truth could itself be deduced from the analysis of concepts, if this were always within human power, and will certainly not escape the analysis of an omniscient substance who sees everything a priori from ideas themselves and from his decrees. It is certain, therefore, that all truths, even highly contingent ones, have a proof a priori or some reason why they are rather than are not. And this is what is commonly asserted: that nothing happens without a cause, or these is nothing without a reason. ⊕ Leibniz succinctly states something that had also been discussed by Aristotle: the axiom of causality, probably the most important one in metaphysics and in all of the philosophy of science, that explains some of the “surprising effectiveness” of mathematics and seemingly surprising similarities between different theories, that, at their core, are just descriptions of this fundamental assumption. Most importantly, it explains why so many mathematical theories have a notion of transitivity, as causality is transitive: if A necessarily (or possibly) causes B, and B necessarily (or possibly) causes C, then A necessarily (or possibly) causes C. … This axiom, however, that there is nothing without a reason, must be considered one of the greatest and most fruitful of all human knowledge, for upon it is built a great part of metaphysics, physics, and moral science; without it, indeed, the existence of God cannot be proved from his creatures, nor can an argument be carried fom causes to effects or from effects to causes, nor any conclusions be drawn in civil matters. So true is this that whatever is not of mathematical necessity, as for instance are logical forms and numerical truths, must be sought here entirely.

Thus was born the dream of the general characteristic, or characteristica universalis, the universal language of all human thought. Note that Leibniz’s characteristica universalis was intended to describe not only acts of deduction but also of scientific discovery (“both the arts of discovery and of judgment”).

His vision of a formal system that puts an end to dispute by answering all questions is nowhere given a better expression than in this famous passage:

⊕ Die philosophischen Schriften, vol. 7, p. 200 Translation: Wikiquote When this is done, if controversies were to arise, there would be no more need of disputation between two philosophers than between two calculators. For it would suffice for them to take their pencils in their hands and to sit down at the abacus, and say to each other (and if they so wish also to a friend called to help): Let us calculate [calculemus].

Leibniz justified the feasibility of a universal characteristic in the inherent mechanical nature of the mind:

⊕ A New System of the Nature and the Communication of Substances, as Well as the Union Between the Soul and the Body, Journal des savants, June 27, 1695, in Loemker, Philosophical Papers, p. 458 For why should God be unable to give to substance in the beginning a nature or internal force which enables it to produce in regular order — as in an automaton that is spiritual or formal but free in the case of that substance which has a share of reason — everything which is to happen to it, that is, all the appearances or expressions which it is to have, and this without the help of any created being? Especially since the nature of substance necessarily demands and essentially involves progress or change and would have no force of action without it. And since it is the nature of the soul to represent the universe in a very exact way, though with relative degrees of distinctness, the sequence of representations which the soul produces will correspond naturally to the sequence of changes in the universe itself. So the body, in turn, has also been adapted to the soul to fit those situations in which the soul is thought of as acting externally. This is all the more reasonable inasmuch as bodies are made solely for the spirits themselves, who are capable of entering into a society with God and of extolling his glory. Thus as soon as one sees the possibility of this hypothesis of agreement, one sees also that it is the most reasonable one and that it gives a wonderful idea of the harmony of the universe and of the perfection of the works of God.

Leibniz explained how, if the mind is a machine, it is capable of exhibiting elaborate behavior:

⊕ Clarification of the Difficulties Which Mr. Bayle Has Found in the New System of the Union of Soul and Body, Histoire des ouvrages des savants, July, 1698, in Loemker, Philosophical Papers, p. 495 I have compared the soul with a clock only with regard to the regulated precision of its changes, which is only imperfect; even in the best clocks, but which is perfect in the works of God. And one can say that the soul is a most exact immaterial automaton. When it is said that a simple being will always act uniformly, a distinction needs to be made. If to act uniformly is to follow perpetually the same law of order or of succession, as in a certain scale or series of numbers, I agree that in this sense every simple being and even every composite being acts uniformly. But if uniformly means similarly, I do not agree. … We must also take into consideration that the soul, however simple it may be, always has a feeling [sentiment] composed of many perceptions at once, a fact which serves our purpose as well as if it were composed of parts like a machine. For each preceding perception influences those which follow in conformity with a law of order which is found in perceptions as well as in movements. For many centuries, too, most philosophers, who ascribe thoughts to souls and to angels, whom they believe to be without any bodies (not to speak of the intellects of Aristotle), have admitted spontaneous change in simple beings. I add that the perceptions which are found together in one soul at the same time include a veritably infinite multitude of little indistinguishable feelings, which the subsequent series must develop, so that we need not be astonished at the infinite variety of what must result from it in time. All this is only a consequence of the representative nature of the soul, which must express what happens, and even what will happen in its body and in some way in all other bodies, through the connection or correspondence between all the parts of the world. Perhaps it would have sufficed to say that God, having made material automatons, could also make immaterial ones which represent the former ones, but I believed that it would be well to elaborate my views a little more fully.

We therefore see that Leibniz, in the 1600s, believed that the mind was a “natural automaton” and reason a “computation”, and seeing the similarity between Aristotle’s logic, the description of correct reasoning in a semi-symbolic language, and algebra, the science of symbol combination, sought to invent an algebra-like calculus that to provide a “determined procedure” for reasoning.

The centrality of language in a mechanical description of thought was explained by Leibniz thus:

⊕ New Essays, p. 134 If we didn’t want to make ourselves understood we indeed wouldn’t ever have created language… But once it has been created it serves also for purposes other than communication; for it also enables man to reason to himself, both because words provide the means for remembering abstract thoughts and also because symbols and ‘blind thoughts’ are useful in reasoning, as it would take too long to lay everything out and always replace terms by definitions.

He explains what he means by “blind thought”, i.e., thought expressible as symbolic, formal or mechanical computation:

⊕ New Essays, p. 82 [O]ur thoughts are for the most part what I call ‘blind thoughts’. I mean that they are empty of perception and sensibility, and consist in the wholly unaided use of symbols—like people doing algebraic geometry and mostly not attending to the geometrical figures that are being dealt with. Usually words are in this respect like the symbols of arithmetic and algebra. We often reason in words, with the object itself virtually absent from our mind.

And elsewhere:

⊕ Dialogue, August 1677, in Loemker, Philosophical Papers, pp. 183-184 B. What of it? Thoughts can occur without words. A. But not without some other signs. Try, I pray, whether you can begin any arithmetical calculation without numerical signs. … B. … Yet I notice that, if characters can be used for ratiocination, there is in them a kind of complex mutual relation [situs] or order which fits the things; if not in the single words at least in their combination and inflection, although it is even better if found in the single words themselves. Though it varies, this order somehow corresponds in all languages. This fact gives me hope of escaping the difficulty. For although characters are arbitrary, their use and connection have something which is not arbitrary, namely a definite analogy between characters and things, and the relations which different characters expressing the same thing have to each other. This analogy or relation is the basis of truth. For the result is that whether we apply one set of characters or another, the products will be the same or equivalent or correspond analogously. But perhaps certain characters are always necessary for thinking.

In the margin he scribbled, “When God calculates and exercises his thought, the world is made.”

This emphasis on language cannot be understated, and it would come to dominate all work in logic in the subsequent ceturies, at least until Turing. But note that language requires not only a syntax of characters, but also a semantics. Despite the “blind” computation, “[t]his analogy or relation” — between characters and things, i.e., semantics — “is the basis of truth.” So while reasoning can be carried out “blindly”, or mechanically, it is predicated on an a priori semantics.

Some of Leibniz’s contemporaries and immediate predecessors, most notably René Descartes, rejected the formalistic view of logic, and believed that it should be a direct description of the work of the mind for which the use of language is inessential. Leibniz, however, believed that thought — or, at least, much of it — is or can be mediated by the mechanical manipulation of language. Indeed, “blind thought”, he claimed, is essential. For example, when multiplying 3 by 4 one can have a mental image of the operation, but multiplication of larger numbers requires mechanical, symbolic operations on numerals. Leibniz makes the point about the importance of algorithmic thought — although he doesn’t use the term algorithm, but rather calculus — in a letter to Ehrenfried Walther von Tschirnhaus:

⊕ Letter to Walter von Tschirnhaus, May, 1678, in Loemker, Philosophical Papers, p. 194 You are entirely of my opinion when you say that in very composite matters a calculus is necessary. For this is the same as if you had said that characters are necessary, for a calculus is nothing but operation through characters, and this has its place not only in matters of quantity but in all other reasoning as well. Meanwhile I have a very high regard for such problems as can be solved by mental powers alone insofar as this is possible, without a prolonged calculation, that is, without paper and pen. For such problems depend as little as possible on external circumstances, being within the power even of a captive who is denied a pen and whose hands are tied. Therefore we ought to practice both in calculating and in meditating, and when we have reached certain results by calculation, we ought to try afterward to demonstrate them by meditation alone, which has in my experience often been successful.

Here Leibniz contrasts calculation employing a pen and paper with meditation, that can be done entirely in the mind and involves the direct manipulation of concepts rather than symbols. A similar analysis of the work of a person thinking while scribbling would serve as the starting point for Turing’s work as well, only he would come up with a very different conclusion, one that removes the distinction between meditation and calculation.

According to Marciszewski, all three ideas comprising Leibniz’s program — symbolic or algorithmic process, composition and separation of concepts, and a universal language of philosophy — were popular subjects of inquiry in the 17th century, but Leibniz was the first to combine them all. For example, the idea of a computation by a systematized application of small steps gained popularity in Europe after arriving from the Arab world in the work of al-Khwarizmi, who, in turn, relied on Hindu methods, in particular, on the decimal system, required for symbolic computation on numbers.

Leibniz went further in his development of a calculus ratiocinator — his “elegant device… by which to show that ratiocination can be proved through numbers” — a calculus of reasoning inspired by algebra to mechanically, i.e., through a “determined procedure”, to deduce propositions in the characteristica universalis:

⊕ Studies in a Geometry of Situation with a Letter to Christian Huygens, 1679, in Loemker, Philosophical Papers, p. 249-250 I have discovered certain elements of a new characteristic which is entirely different from algebra and which will have great advantages in representing to the mind, exactly and in a way faithful to its nature, even without figures, everything which depends on sense perception. Algebra is the characteristic for undetermined numbers or magnitudes only, but it does not express situation, angles, and motion directly. Hence it is often difficult to analyze the properties of a figure by calculation, and still more difficult to find very convenient geometrical demonstrations and constructions, even when the algebraic calculation is completed. But this new characteristic, which follows the visual figures, cannot fail to give the solution, the construction, and the geometric demonstration all at the same time, and in a natural way and in one analysis, that is, through determined procedure. … If it were completed in the way in which I think of it, one could carry out the description of a machine, no matter how complicated, in characters which would be merely the letters of the alphabet, and so provide the mind with a method of knowing the machine and all its parts, their motion and use, distinctly and easily without the use of any figures or models and without the need of imagination. Yet the figure would inevitably be present to the mind whenever one wishes to interpret the characters. One could also give exact descriptions of natural things by means of it, such, for example, as the structure of plants and animals. With its aid people who find it hard to draw figures could explain a matter perfectly, provided they have it present before them or in their mind, and could transmit their thoughts and experiences to posterity — a thing which cannot be done today because the words of our languages are not sufliciently fixed or well enough fitted for good explanations without figures. This is the least useful aspect of this characteristic, however, for if only description were involved, it would be better — assuming that we can and are willing to bear the expense — to have figures and even models or, better still, the original things themselves. But its chief value lies in the reasoning which can be done and the conclusions which can be drawn by operations with its characters, which could not be expressed in figures, and still less in models, without multiplying these too greatly or without confusing them with too many points and lines in the course of the many futile attempts one is forced to make. This method, by contrast, will guide us surely and without effort. I believe that by this method one could treat mechanics almost like geometry, and one could even test the qualities of materials, because this ordinarily depends on certain figures in their sensible parts.

I believe that it would not be completely anachronistic to attribute to Leibniz a belief in something close to what is now called universal computation (although perhaps not one that extends to all functions of the mind), as he believed that a precise and formal language could be used to describe the workings of both man-made machines, “no matter how complicated”, and “natural things,” like “the structure of plants and animals”. Indeed, Alan Turing, who, as we’ll see, was the first to try to describe the “immaterial automaton” of the mind by formal means that are not Aristotelian or linguistic (yet still “not without some other signs”), soon after set out to precisely describe the very “structure of plants and animals” using universal computation.

Leibniz began considering concrete mechanisms for the manipulating concepts and their combinations using numbers and algebra, stating with the assumption that every primitive concept is assigned a characteristic number. The following description of the calculus ratiocinator is probably the very first attempt at the algebraization, or mathematization of logic in general, and Aristotle’s framework in particular:

⊕ Two Studies in the Logical Calculus, 1679, in Loemker, Philosophical Papers, p. 235-244 A term is the subject or predicate of a categorical proposition. Thus I include neither the sign nor the copula among the terms. So when it is said, ‘The wise man believes’, the term is not believes, but a believer, just as if I say, ‘The wise man is a believer’. … To every term whatever may be assigned its characteristic number, which we may use in calculating, as we use the term itself in reasoning. I choose numbers in writing; in time I shall adapt other signs both to numbers and to speech itself. For the present numbers are the most useful because of their accuracy and the ease with which they are handled and because it is thus clear to the eye that all of the relations of concepts are certain and determined after the likeness of numbers. The rule for discovering fitting characteristic numbers is this one only: when the concept of a given term is composed directly out of the concepts of two or more other terms, then the characteristic number of the given term is to be produced by multiplying the characteristic numbers of the terms composing it. For example, since man is a rational animal, if the number of animal is a, for instance, 2 and the number of rational is r, for instance 3, the number of man, or h, will be $2 \times 3$ or 6. We shall introduce letters (such as a, r, and h here) when the numbers are not given or at least need not be considered specifically but are dealt with in general, as it is proper for us to do here in dealing with the elements. Like the practice in symbolic algebra or the arithmetic of figures, this is a way of avoiding the effort to try to do for each individual case what can be shown at one and the same time for an infinite number of instances. I shall explain the manner of using these letters below. … ⊕

Leibniz entertaining the guests at a soirée at Sophie Charlotte’s (source: Museum Schloss Herrenhausen) To make clear the use of characteristic numbers in propositions, the following must be kept in mind. Every true categorical proposition, affirmative and universal, signifies nothing but a certain connection between the predicate and the subject — in the direct case, that is, of which I am always speaking here. This connection is such that the predicate is said to be in the subject, or to be contained in it, and this either absolutely and viewed in itself, or in some particular case. Or in the same way, the subject is said to contain the predicate; that is, the concept of the subject, either in itself or with some addition, involves the concept of the predicate. And therefore the subject and predicate are mutually related to each other either as whole and part, or as whole and coinciding whole, or as part to whole. In the first two cases the proposition is universal affirmative. So when I say, ‘All gold is a metal’, I mean by this only that the notion of metal is contained in the notion of gold in a direct sense, for gold is the heaviest metal. And when I say, ‘All pious people are happy’, I mean only that the connection between piety and happiness is such that whoever understands the nature of piety perfectly will see that the nature of happiness is involved in it in the direct sense. But in every case, whether the subject or the predicate is a part or a whole, a particular affirmative proposition always holds. … But a distinction between the subjects of a universal and a particular proposition is found in the manner of this inclusion. For the subject of a universal proposition, viewed in itself and taken absolutely, must contain the predicate; so that the concept of gold, viewed in itself and taken absolutely, involves the concept of metal, since the notion of gold is that of the heaviest metal. But in an affirmative particular proposition it suffices that the inclusion is successful when something is added to the subject. The concept of metal, viewed absolutely and in itself, does not involve the concept of gold; something must be added to involve it, namely, the sign of particularity. For it is some certain metal which contains the concept of gold. In the future, however, when we say that a term is contained in another or a concept in another concept, we understand this to mean simply and in itself. … The schools speak otherwise, because they are considering not concepts but instances subsumed under universal concepts. Thus they say that metal is wider than gold, since it contains more species than does gold. If we were to count the individuals made of gold on the one hand, and those made of metal on the other, there would certainly be more of the latter than of the former, and hence the former would be contained in the latter as part in a whole. In fact, by applying this observation and using fitting characters, we could demonstrate all the rules of logic by another kind of calculus than the one developed here, merely by an inversion of our own calculus. But I prefer to consider universal concepts or ideas and their composition, for these do not depend on the existence of individuals. So I say that gold is greater than metal, because more constituents are required for the concept of gold than for that of metal, and more is needed to produce gold than to produce just a metal. Thus our phrases here and the Scholastic phrases do not contradict each other but must nevertheless be carefully distinguished. It will be clear to the careful student that I make no innovations in my way of speaking which do not have a definite reason and application. … Everything that we have so far said about terms that contain or do not contain each other in various ways, we may now transfer to their characteristic numbers. This is easy because, as we said in Article 4, when a term helps to constitute another term, that is, when the concept of one term is contained in that of another, then the characteristic number of the one enters by multiplication into the characteristic number assumed for the term so constituted. Or what amounts to the same thing, the characteristic number of the term to be constituted (or that which contains the other) is divisible by the characteristic number of the constituting term (or that which is in the other). For example, the concept of animal enters into the formation of the concept of man, and so the characteristic number of animal, a (for example, 2), combines with some other number r (such as 3), to produce the number ar or It by multiplication ($2 \times 3$); that is, the characteristic number of man. Hence number ar or h (or 6) must necessarily be divisible by a (or by 2). ⊕

From Candide by Voltaire, a novella satirizing Leibniz’s claim that ours is the best of all possible worlds. Dr. Pangloss, Candide’s mentor and ‘the greatest philosopher of the Holy Roman Empire’ — who probably represents Leibniz — is reduced to panhandling, and, after being cured from syphilis, is shown in this 18th century illustration as he is about to be tortured and hanged in an auto-de-fé. (source: Literature Wikia)… Hence we can also determine through characteristic numbers which term does not contain another. One has merely to test whether the number of one term can be divided exactly by the number of the other. For example, if the characteristic number of man is found to be 6, and that of the ape is 10, it is obvious that the concept of ape does not include that of man, nor that of man the ape, since 10 cannot be divided evenly by 6, nor 6 by 10. So if you wish to know whether the concept of wisdom is contained in that of a just being, that is, whether nothing more is required for wisdom than what is already contained in justice, you need merely to examine whether the characteristic number of just can be divided exactly by the characteristic number of wise. If the division is impossible, it will be clear that something more is required for wisdom than what is in justice, namely, a knowledge of reasons. For one can be just by custom or habit, even though he cannot give a reason for what he does. I will show later how that minimum which is necessary or must be added for the purpose can be discovered by characteristic numbers. So we can learn in this way whether any universal affirmative proposition is true. For in such a proposition the concept of the subject, taken absolutely and indefinitely and in general viewed in itself, always contains the concept of the predicate. For example, all gold is a metal, that is, the concept of metal is contained in the concept of gold generally and viewed in itself… Thus if we wish to know whether all gold is a metal … we merely see whether the defiinition of metal is contained in it; that is, by a very simple procedure when characteristic numbers are introduced, we see whether the characteristic number of gold can be divided by the characteristic number of metal. … II. SPECIMEN OF UNIVERSAL CALCULUS … If one thing can be substituted anywhere in place of another without destroying truth, the other thing can be substituted conversely in place of the first… For assuming two terms a and b, such that b can be substituted anywhere for a, then I say, a can be substituted anywhere in place of b. This I prove as follows… … If a is b, and d is c, then ad is bc. This is an admirable theorem, which can be demonstrated in this way. a is b, therefore ad is bd, by the above. d is c, therefore bd is bc, also by the above. ad is bd, and bd is bc, therefore ad is bc, which was to be demonstrated. … If a is b and b is a, then a and b are said to be the same. Thus every pious man is happy, and every happy man is pious. Therefore pious and happy are the same. … Propositions true in themselves: (1) a is a. Animal is animal. (2) ab is a. Rational animal is animal. (3) a is not non-a. Animal is not nonanimal. (4) Non-a is not a. Nonanimal is not animal. (5) What is not a is non-a. What is not an animal is nonanimal. (6) What is not non-a is a. What is not a nonanimal is an animal. From these many others can be derived. Consequences true in themselves: a is b, and b is c, therefore a is c. God is wise, wise is just; therefore God is just. This chain can be continued further. For example, God is wise, wise is just, just is austere; therefore God is austere. Principles of the calculus: (1) Whatever is concluded in certain indefinite letters must be understood to be concluded in whatever other letters have the same relation. Thus, since it is true that ab is a, it is also true that be is b… (2) The transposition of letters in the same term changes nothing; thus ab coincides with ba, or rational animal with animal reasoner. (3) The repetition of the same letter in the same term is useless; thus b is aa, or bb is a; man is an animal animal, or man man is an animal. It suffices to say that a is b, or man is an animal.

This should be considered remarkable by anyone familiar with the modern notions of formal logic and algebras of logic considering that it was written almost 350 years ago. It is not hard to guess how Frege, when attempting to continue, in his own words, Leibniz’s work on the characteristica exactly two hundred years later, formalized Leibniz’s (or, indeed, Aristotle’s) construct of a “concept” or a “notion” — and we will revisit that development. But let’s examine Leibniz’s algebra: he assigns every concept a number — in a way that should remind us of Gödel numbering — and the conjunction or intersection of two concepts as the multiplication of their characteristic numbers. Implication or inclusion of concepts is represented by division, but note that as the numbers grow with conjunction or intersection, A divides B corresponds to B implies A or B is contained in A, and therefore the number 1 would represent true (i.e., as it divides everything, it corresponds to the concept that is implied by anything) and 0 would represent false. Usually, when speaking about divisibility as an order relation we have it the other way around, namely 1 is the bottom element and 0 is the top, and, indeed Leibniz recognizes the possible inversion in section 12 above when referring to the schools, and notes that “we could demonstrate all the rules of logic by another kind of calculus than the one developed here, merely by an inversion of our own calculus.”

Leibniz’s calculus would run into a small problem. His axioms state that “the repetition of the same letter in the same term is useless”, but mere multiplication does not satisfy this. Thus, the multiplication of, “big man” with “green man” does not divide “big green man” because of the repetition of the factors of the characteristic number corresponding to “big”. To correct this, Leibniz would have had to use the least common multiple (i.e., the smallest number dividing a and b) instead of multiplication, and then he would have been surprisingly close to a very decent algebraization of logic. In fact, combining the axiom that repetition of the same concept, such as animal animal is equal to just one occurrence of the term (yielding a least-common-multiplier approach), with the observation that the algebra can be inverted (to use a greatest-common-divisor), and that the order and its dual can be used together to form a distributive lattice, would have given Leibniz the modern Boolean algebra (as opposed to Boole’s original algebra, as we’ll see), at least for finite domains — a nice algebra of Aristotelian logic — with the difference that Leibniz’s algebra is concrete, namely, terms signify natural numbers as opposed to abstract concetps. However, Leibniz specifically alludes to the possibility of thinking of terms more abstractly, when he writes, “I choose numbers in writing; in time I shall adapt other signs both to numbers and to speech itself. For the present numbers are the most useful because of their accuracy and the ease with which they are handled and because it is thus clear to the eye that all of the relations of concepts are certain and determined after the likeness of numbers.” So what is important is not that the terms represent actual numbers, but that the relations between terms bears the likeness of numbers, where concrete numbers are just used for demonstration purposes and for the gaining of intuition.

We may wonder if this is some mathematical miracle, but it isn’t. A distributive lattice arises directly from the Aristotelian logic’s combination of concepts, where concepts are collections of elements. Indeed, all Boolean algebras are isomorphic to fields of sets, and distributive lattices are isomorphic to set operations, but in order to get that structure for Leibniz’s representation of concepts as integers those modern theorems aren’t necessary, as integers ordered with the divisibility relation are just a straightforward encoding of finite sets of their prime factors, by the fundamental theorem of arithmetic (known since antiquity) and so can nicely serve as a direct representation of a combinatorial logic of concepts, as long as those concepts comprise a finite number of elements. That such an encoding should exist in the first place isn’t surprising to Leibniz, who intuits that a universal description of man-made and natural systems exists, and that numbers could encode it. Turing would give a more satisfying explanation.

Leibniz’s attempts to algebrize logic continued with his followers, who were all part of a historical development that Marciszewski describes thus: Mechanization Of Reasoning, Kindle Locations 2109-2114 “We are here to do with a case in which the intersection, at a certain point of time, of two mutually independent processes opens a new stage in history. Such intersection may also appear in the fact that a certain group of scholars is engaged with similar intensity in both processes. This was just in the case under consideration: people strongly rooted in the tradition of scholastic logic came to be interested in algebra, and some of them proved creative in that newly born discipline called then logistica speciosa universalis, that is (in a free translation), the general theory of calculating with variables (i.e., sign denoting species of objects, e.g. numbers, instead of individual objects).”

While not an essential part of our main theme, a further curious connection between Leibniz and modern computer science can be observed in his interest in binary arithmetic:

⊕ Leibniz, Explanation of Binary Arithmetic, Which Uses Only the Characters 0 and 1, with Some Remarks on Its Usefulness, and on the Light It Throws on the Ancient Chinese Figures of Fuxi, 1703, in Die philosophischen Schriften, vol 7., p. 223, translation by Lloyd Strickland, 2007 The ordinary reckoning of arithmetic is done according to the progression of tens. Ten characters are used, which are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, which signify zero, one, and the successive numbers up to nine inclusively. And then, when reaching ten, one starts again, writing ten by “10”, ten times ten, or a hundred, by “100”, ten times a hundred, or a thousand, by “1000”, ten times a thousand by “10000”, and so on. But instead of the progression of tens, I have for many years used the simplest progression of all, which proceeds by twos, having found that it is useful for the perfection of the science of numbers. Thus I use no other characters in it bar 0 and 1, and when reaching two, I start again. This is why two is here expressed by “10”, and two times two, or four, by “100”, two times four, or eight, by “1000”, two times eight, or sixteen, by “10000”, and so on. Here is the Table of Numbers of this way, which may be extended as far as is desired. ⊕

Examples by Leibniz of binary addition, subtraction, multiplication and division. (Source)… Establishing this expression of numbers enables us to very easily make all sorts of operations. And all these operations are so easy that there would never be any need to guess or try out anything, as has to be done in ordinary division. There would no longer be any need to learn anything by heart, as has to be done in ordinary reckoning, where one has to know, for example, that 6 and 7 taken together make 13. … However I am not in any way recommending this way of counting in order to introduce it in place of the ordinary practice of counting by ten. For, aside from the fact that we are accustomed to this, we have no need to learn what we have already learned by heart. … But reckoning by twos, that is, by 0 and 1, as compensation for its length, is the most fundamental way of reckoning for science, and offers up new discoveries, which are then found to be useful, even for the practice of numbers and especially for geometry. The reason for this is that, as numbers are reduced to the simplest principles, like 0 and 1, a wonderful order is apparent throughout.

The Algebra of Thought

Others continued Leibniz’s quest for the algebraization of logic in the 17th and 18th centuries, like Gottfried Ploucquet (1716-1790), Johann Heinrich Lambert (1728-1777), Johann Andreas Segner (1704-1777), or like the brothers Johann (1667-1748) and Jacob Bernoulli (1655-1705), who frequently corresponded with Leibniz and are considered his disciples, who in 1685 wrote a book called, Parallelismus ratiocinii logici et algebraici (the parallelism between logical reasoning and algebra), that again showed the similarity between the two disciplines, both concerned with the combination of symbols, or like Daniel Bernoulli, Johann Bernoulli’s son. But I will focus on Leonhard Euler (1707-1783), Johann Bernoulli’s student and Daniel’s close friend, mostly because his writings are particularly clear.

The passages I will quote are from a 1795 English translation by the Scottish minister Henry Hunter, of a series of letters Euler wrote between 1760 and 1762 to Friederike Charlotte of Brandenburg-Schwedt and her sister.

In the following passage Euler beautifully explains a central notion in mathematics — the one that both justifies its suitability for the description of natural phenomena and explains its universality — namely abstraction. While not uniquely related to logic and algebra, abstraction, Euler explains, is a prerequisite to any formal treatment of logic, as it is how the primitives of logic — the notion or concepts — are formed:

⊕ Letters to a German Princess, pp. 385-389 Letter 100. Of the Abstraction of Notion The senses represent objects only which exist externally; and sensible ideas all refer to them; but of these sensible ideas, the soul forms to itself a variety of other ideas, which are indeed derived from these, but which no longer represent objects really existing. ⊕

The German Princess When, for example, I look at the full moon, and fix my attention only on its contour, I form the idea of roundness; but I cannot affirm that roundness exists of itself. The moon is round, but the round figure does not exist separately out of the moon…. The ideas of numbers have the same origin. Having seen two or three persons, the soul forms the idea of two or three, without attaching it any longer to the persons. Having already acquired the idea of three, the soul is able to proceed, and to form the ideas of greater numbers, … without ever having precisely seen so many things together. A single instance, therefore, in which we have seen two or three objects, may carry the soul forward to the formation of the ideas of other numbers, be they ever so great. … Here the soul exerts a new faculty, which is called the power of abstraction; this takes place when the soul fixes its attention on only one quantity or quality of the object, and considers it separately, as if it were no longer attached to the object. When, for instance, I put my hand on a heated stone, and confine my attention to the heat only, I form from it the idea of heat, which is no longer attached to the stone. This idea of heat is formed by abstraction, as it is separated from the stone, and the soul might have derived the same idea from touching a piece of wood heated, or by plunging the hand into hot water. Thus, by means of abstraction, the soul forms a thousand other ideas of the quantities and properties of objects, by separating them afterwards from the objects themselves… These ideas, acquired by abstraction, are denominated notions, to distinguish them from sensible ideas, which represent to us objects really existing. There is still farther a species of notions, likewise formed by abstraction… When I see a pear-tree, a cherry-tree, an apple-tree, an oak, a fir, &c. all these ideas are different; I, nevertheless, remark in them several things which they have in common; as the truck, the branches, and the root; I stop short only at those things which the different ideas have in common, and the object, in which all such qualities meet, I call a tree. Thus the idea of tree, which I have formed in this manner, is a general notion, and comprehends the sensible ideas of the pear-tree, the apple-tree, and, in general, of every tree that exists. … This manner of forming general ideas is, therefore, likewise performed by abstraction, and it is here, chiefly, that the soul exerts the activity and performs the operations from which all our knowledge is derived. Without these general notions, we should differ nothing from the brutes. 7th February, 1761

In the next letter, Euler explains how language denotes abstract notions, and how it is essential to reasoning:

⊕ Letters to a German Princess, pp. 389-393 Letter 101. Of Language; its Nature, Advantages, and Necessity, in order to the Communication of Thought, and the Cultivation of Knowledge Whatever aptitude a man may have to exercise the power of abstraction, and to furnish himself with general ideas, he can make no considerable progress without the aid of language, spoken or written. Both the one and the other contains a variety of words, which are only certain signs, corresponding to our ideas, and whose signification is settled by custom, or the tacit consent of several men who live together. It would appear, from this, that the only purpose of language to mankind is mutually to communicate their sentiments, and that a solitary man might do very well without it; but a little reflection only is necessary to be convinced, that men stand in need of language, as much to pursue and cultivate their own thoughts, as to keep up a communication with others. … The essence of language consists, rather, in its containing words to denote general notions; as that of tree corresponds to a prodigious number of individual beings. These words serve not only to convey to others, who understand the same language, the same idea which I affix to the words; but they are, likewise, a great assistance to me, in representing this idea to myself. Without the word tree, which represents to me the general notion of a tree, I must imagine to myself at once a cherry-tree, a pear-tree, an apple-tree, a fir, &c. This would necessarily oppress the mind, and speedily involve it in the greatest perplexity. … You may easily conceive how many abstractions it was necessary to make, in order to arrive at the notion of virtue. The actions of men were first to be considered; they were, then, to be compared with the duties imposed on them; in consequence of this, we give the name of virtue to the disposition which a man has to regulate his actions conformably to his duties. But, on hearing the word virtue rapidly pronounced in conversation, do we always connect with it this complex notion? And what idea is excited in the mind, on hearing the particle and or also pronounced? … You are now enabled to judge of what advantage language is to direct our thoughts; and that, without language, we should hardly be in a condition to think at all. 10th February, 1761

After establishing, or, rather, asserting, the importance of language, Euler introduces the Aristotelian logical form of propositions and judgments:

⊕ Letters to a German Princess, pp. 393-397 Letter CII. Of the Perfections of a Language. Judgments and Nature of Propositions, affirmative negative; universal, or particular. I have been endeavouring to shew you, how necessary language is to man, not only for the mutual communication of sentiment and thought, but, likewise, for the improvement of the mind, and the extension of knowledge. These signs, or words, represent, then, general notions, each of which is applicable to an infinite number of objects: as, for instance, the idea of hot, and of heat, to every individual object which is hot; and the idea, or general notion of tree, is applicable to every individual tree in a garden, or a forest, whether cherries, oaks or firs, &c. ⊕ A passage from the translator’s preface to the second edition of Euler’s Letters of Euler On Different Subjects in Physics and Philosophy Adressed to A German Princess, 1802, by Henry Hunter, D.D. The words may sound cryptic to the modern reader, especially ones educated in the arts of logic and programming, but in the 18th century people claimed that women are naturally less inclined for intellectual pursuits to justify not encouraging them to learn:



I was mortified to reflect that the specious and seductive productions of a Rousseau, and the poisonous effusions of a Voltaire, should be in the hands of so many young men, not to say young women, to the perversion of the understanding, and the corruption of the moral principle, while the simple and useful instructions of the virtuous Euler were hardly mentioned… I considered myself as rendering a meritorious service to my country.



The subjects of these Letters, and the Author’s method of treating them, seem to me much adapted to this purpose. With the assistance of a very moderate apparatus, they might conduct youth of both sexes, with equal delight and emolument, to a very competent knowledge of natural philosophy. Euler wrote these Letters for the instruction of a young and sensible female, and in the same view that they were written, they are translated, namely, the improvement of the female mind; and object of what importance to the world! I rejoice to think I have lived to see female education conducted on a more liberal and enlarged plan. I am old enough to remember the time when well-born young women, even of the north, could spell their own language but very indifferently, and some hardly read it with common decency… While the boys of the family were conversing with Virgil, perhaps with old Homer himself, the poor girls were condemned to cross-stitch, on a piece of gauze-canvass, and to record their own age at the bottom of a sampler. They are now treated as rational beings, and society is already the better for it. And wherefore should the terms female and philosophy seem a ridiculous combination? Wherefore preclude to a woman any source of knowledge to which her capacity, and condition in life, entitle her to apply? It is cruel and ungenerous to expose the frivolity of the sex, after reducing it to the necessity of being silly and frivolous. Cultivate a young woman’s understanding, and her person will become, even to herself, only a secondary concern; let her time be filled up in the acquisition of attainable and useful knowledge, and then she will cease to be a burden to herself and to every body about her; make her acquainted with the world of nature, and the world of art will delude her no longer. Hence you must be sensible how one language may be more perfect than another. A language always is so, in proposition as it is in a condition to express a greater number of general notions, formed by abstraction. It is with respect to these notions that we must estimate the perfection of a language. … These general notions, formed by abstraction, are the source of all of our judgments and all our reasonings. A judgment is nothing else but the affirmation, or negation, that a notion is applicable, or inapplicable; and when such judgment is expressed in words, we call it a proposition. To give an example: All men are mortal, is a proposition which contains two notions; the first, that of men in general; and the second, that of mortality, which comprehends whatever is mortal. The judgment consists in pronouncing and affirming, that the notion of mortality is applicable to all men. This is a judgment, and, being expressed in words, it is a proposition; and, because it affirms, we call it an affirmative proposition. If it denied, we would call it negative, such as this, no man is righteous. These two propositions, which I have introduced as examples, are universal, because the one affirms of all men, that they are mortal, and the other denies that they are righteous. There are likewise particular propositions, both negative and affirmative; as, some men are learned, and some men are not wise. What is here affirmed, and denied, is not applicable to all men, but to some of them. Hence we derive four species of propositions. The first is that of affirmative and universal propositions, the form of which in general is: Every A is B. The second species contains negative and universal propositions, the form of which in general is: No A is B. The third is, that of affirmative propositions, but particular, contained in this form: Some A is B. And, finally, the fourth is that of negative and particular propositions, of which the form is: Some A is not B. All these propositions contain, essentially, two notions, A and B, which are called the terms of the proposition: the first of which affirms or deines some things; and this we call the subject; and the second, which we say is applicable, or inapplicable, to the first, is the attribute. Thus, in the proposition All men are mortal, the word man, or men, is the subject, and the word mortal the attribute; these words are much used in logic, which teaches the rules of just reasoning. These four species of propositions may likewise be represented by figures, so as to exhibit their nature to the eye. This must be a great assistance toward comprehending more distinctly wherein the accuracy of a chain of reasoning consists. As a general notion contains an infinite number of individual objects, we may consider it a space in which they are all contained. Thus, for the notion of man we form a space [fig 1] in which we conceive all men to be comprehended. For the notion of mortal we form another (fig 2) in which we conceive every thing mortal to be comprehended. And when I affirm, all men are mortal, is is the same thing with affirming, the first figure is contained within the second…. 14th February, 1761

Euler clearly explains that the notions or concepts formed by abstraction