From: Thomas Walker Lynch Date: Tue, 25 Aug 2026 19:39:42 +0000 (+0000) Subject: . X-Git-Url: https://git.reasoningtechnology.com/Realizable_Machine.svg?a=commitdiff_plain;h=ee8168e7530a1ef3870cd7b69b133cc7d20f8df5;p=TM-2026 . --- diff --git a/document/book/TM-2026.html b/document/book/TM-2026.html index 047265d..b8434ae 100644 --- a/document/book/TM-2026.html +++ b/document/book/TM-2026.html @@ -32,16 +32,18 @@ - Exordium

My well funded colleagues have been working tirelessly to confirm Schopenhauer’s thesis, which they were in an excellent position to do, leaving me with a fulfilling career making people happy by cleaning fish and stocking shelves. Still I have managed to do some writing.

-

Perhaps the most surprising part of the transition has been the paywall which publishers use to prevent access to scholarly knowledge. A man writing outside an institution pays forty dollars a paper or does without. If not for Libgen this book would have few citations.

+

One of the Ancient Athenian punishments was ostracism, ten years outside the citadel of culture. A man who has not paid tuition and holds no appointment lacks the institutional affiliation that reading a journal requires, and so pays forty dollars a paper or does without. The Athenians put a man outside one city. Institutional ostracism puts him outside the accumulated intellectual contributions of humankind, thus setting him back to no better off than the other apes. Libgen should be a basic human right.

+ +

This book would not exist without the encouragement of my young and beautiful wife. While cousins brought their husbands to family events, she sat by herself as I remained home in the company of computation theory. She is charming, and I count myself fortunate not to be wearing horns. When this book is done I plan to crack open Nathan’s book, invite Epicurus to dinner, and begin the process of making it up to her.

+ +

Hence, if you do not like the book, and want to prevent more from me in the future, support the publishers, and send a message to my wife suggesting she bring me along to more family events.

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But mostly, this book would not exist without the encouragement of my young and beautiful wife. While cousins brought their husbands to family events, she sat by herself as I remained home in the company of computation theory. She is charming, and I count myself fortunate not to be wearing horns. When this book is done I plan to crack open Nathan’s book, invite Epicurus to dinner, and begin the process of making it up to her.

 @@ -91,7 +93,7 @@ Conventio -

All communication starts from the knowledge held in common between the author and reader. This chapter is a first meeting and a search for that commonality. It is not the start of the thesis of the book, that being Computational Naturalism. That topic doesn’t even come up. A reader who skipped these pages would be more likely to misread what follows, but would not miss any of the argument. What follows is the notation this book is written in and the words it takes from mathematics and logic. There are some terms introduced by this book and mentioned here, and those are so marked.

+

All communication starts from the knowledge held in common between the author and reader. This chapter is a first meeting and a search for that commonality.

Notation @@ -154,7 +156,9 @@

The leading capital letter on the mathematical number types does not distinguish the classical notion from the one constructed herein, because this book denies that the two are distinct in the first place. A Real is a Real whether a mathematician defines it or a machine produces the definition.

-

What mathematics calls the natural numbers are called Peano Numbers in this book. A Peano Number is what a Peano Machine outputs. This convention frees the word ‘Natural’ to refer to the philosophy only, so a reader never has to work out which of the two is meant. Note that herein Peano Numbers are taken to begin at zero, while Counting Numbers begin at one, and the two are not the same.Peano’s own axioms began at one. Modern presentations of Peano arithmetic begin at zero, and that is the convention followed here. Giuseppe Peano, Arithmetices principia, nova methodo exposita (Turin: Bocca, 1889), §1. An Integer, a Rational, and a Real are likewise the outputs of machines, each constructed in its turn.

+

What mathematics calls the natural numbers are called Peano Numbers in this book. A Peano Number is what a Peano Machine outputs. This convention frees the word ‘Natural’ to refer to the philosophy only, so a reader never has to work out which of the two is meant. Note that herein Peano Numbers are taken to begin at zero, while Counting Numbers begin at one, and the two are not the same. + Peano’s original axioms began with the number one. Modern presentations of Peano arithmetic commonly begin with zero, and that is the convention followed here. The historical formulation is shared between Richard Dedekind and Giuseppe Peano: Dedekind described a “simply infinite system” in Was sind und was sollen die Zahlen? (Braunschweig: Vieweg, 1888), while Peano published a corresponding axiomatization in Arithmetices principia, nova methodo exposita (Turin: Bocca, 1889). The axioms are therefore also called the Dedekind−Peano axioms. “Dedekind number” is already the name of a different mathematical sequence, so it is not an alternative name for the objects defined here. Peano did not create arithmetic itself, but his axiomatic presentation and notation gave the successor-based arithmetic its familiar formal identity. This book therefore calls the machine’s outputs Peano Numbers: the Peano Machine is named for the Peano Numbers it produces. + An Integer, a Rational, and a Real are likewise the outputs of machines, each constructed in its turn.

 @@ -449,93 +453,167 @@

+ - The presumed dialectic + Those who walked around it + +

Our courtrooms, legislatures, and debating societies establish rules for adversarial contest. Those rules instill in our culture the expectation that progress is made by choosing a winner. Yet a dialectic requires a structure. + The pattern of a thesis met by its opposite and settled in a third position is commonly credited to Hegel, but it is not his. Fichte set out the triad in the Grundlage der gesammten Wissenschaftslehre (Leipzig, 1794), §§1−3, and Heinrich Moritz Chalybäus fixed the three names upon it in the Historische Entwicklung der speculativen Philosophie von Kant bis Hegel (Dresden, 1837), from which they passed into general use. Hegel’s own term is Aufhebung, in which a position is cancelled, preserved, and raised at once, and scholars of Hegel are quick to point out that the three-step formula is not in his vocabulary. + For example, when two people debate whether a room is too hot or too cold, their argument implies the existence of a concept we call temperature. Sometimes understanding the deeper structure dissolves the debate altogether. +

-

Our courtrooms, legislatures, and debating societies set rules for adversarial contest, and this instills our culture with the expectation that progress is made by picking winners. However each dialectic implies a deeper issue. - The pattern of a thesis met by its opposite and settled in a third position is commonly credited to Hegel, and it is not his. Fichte set out the triad in the Grundlage der gesammten Wissenschaftslehre (Leipzig, 1794), §§1−3, and Heinrich Moritz Chalybäus fixed the three names upon it in the Historische Entwicklung der speculativen Philosophie von Kant bis Hegel (Dresden, 1837), from which it passed into general use. Hegel’s own term is Aufhebung, in which a position is cancelled, preserved, and raised at once, and scholars of Hegel are quick to say that the three-step formula is not in his vocabulary. - Say two people debate if the room is too hot, or too cold, implied is the existence of temperature. Sometimes the deeper understanding resolves the debate by encompassing both sides. +

An old parable tells of blind men and an elephant. One grasps the trunk and reports a snake; another takes hold of the ear and reports a fan; a third embraces the leg and reports a pillar. If these men were to debate, whom would the judge declare the winner? Or should we regard their mutually contradictory accounts as pointing toward a deeper truth? Not a truth synthesized from their reports, but one discovered through them.

+ +

As Greek learning passed among late-antique, Syriac, and Arabic-speaking communities, the transmission of Greek philosophy became dispersed rather than simply preserved in one place. Western Europe retained fragments of Aristotle’s logic, while much of the philosophical and scientific corpus was translated and developed in the Syriac Christian and Arabic scholarly worlds. The Abbasid translation movement made Aristotle, Euclid, Galen, and other Greek authors available in Arabic, but the work was not merely translational: it also involved commentary, criticism, and the construction of new philosophical syntheses. + The Arabic translation movement flourished especially under the Abbasids. Syriac Christian scholars had already translated Greek philosophical and medical works into Syriac, and some of these scholars later participated in Arabic translations. The resulting Arabic tradition also incorporated Neoplatonic works and texts falsely attributed to Aristotle, notably the Theology of Aristotle, an adaptation of parts of Plotinus. See Cristina D’Ancona, “Greek Sources in Arabic and Islamic Philosophy,” Stanford Encyclopedia of Philosophy. +

+ +

+ One recurring problem was how philosophical demonstration could be reconciled with Islamic theology. Al-Farabi (c. 872−950) held that truth reaches the philosopher as a proof and everyone else as an image of it, with different nations holding different images of the one truth. Avicenna (980−1037) gave Aristotelian and Neoplatonic materials a systematic philosophical form. Ibn al-Haytham (c. 965−1040) argued, with the aid of experiments, that vision depends on light entering the eye rather than on rays emitted by the eye. Al-Ghazali (1058−1111) denied that fire possesses an independent and necessary power to burn cotton, arguing that what appears as natural causation depends at every occasion on God’s act. Ibn Rushd (1126−1198) answered that a demonstrated conclusion cannot conflict with a revealed truth, and that apparent conflicts call for a revision of the textual interpretation. + Al-Ghazali, Tahāfut al-Falāsifa, discussion 17. The position, termed occasionalism, denies necessary causal efficacy to created things; it need not deny the regular succession of events or the usefulness of ordinary causal descriptions. Its relationship to Hume’s later account of causation is one of resemblance and historical comparison, not simple identity.

-

There is an ancient story from India telling of blind men and an elephant. One takes hold of the trunk and reports a snake, one the ear and reports a fan, one the leg and reports a pillar. So then, if these men had a debate, who would the judge pick as the winner? Or should we accept mutually contradictory points of view as being indicative of a deeper truth? Perhaps not one synthesized, but one discovered.

+

Ibn Rushd’s commentaries, together with other Arabic philosophical works, entered Latin Europe through translators working in Toledo and elsewhere. The Arabic-Latin translation movements transformed several areas of Latin philosophy, including natural philosophy, psychology, metaphysics, logic, and ethics. + The Arabic-Latin translation movement was not a single return of Greek philosophy through one city or one author. Important translations were made in Toledo, Sicily, southern Italy, and elsewhere. The influence of Arabic philosophy on Latin Europe was especially strong in natural philosophy, psychology, and metaphysics, but it also reached logic and ethics. See Dag Nikolaus Hasse, “Influence of Arabic and Islamic Philosophy on the Latin West,” Stanford Encyclopedia of Philosophy. +

-

After the Athenian Academy declined, the transmission of Greek thought fractured. Western Europe retained fragments of Aristotle’s logic while the corpus migrated east, carried by Syriac Christians to eighth-century Arabic-speaking scholars who rendered Aristotle, Euclid, and Galen into Arabic. The problem those scholars faced was fitting philosophical demonstration to Islamic faith. - Hunayn ibn Ishaq (809−873) and his circle drove the translation movement under Abbasid patronage. The arriving texts contained Aristotle layered with late-antique commentary and Neoplatonic adaptations circulating under his name, notably the Theology of Aristotle (an adaptation of Plotinus). See “Greek Sources in Arabic and Islamic Philosophy,” Stanford Encyclopedia of Philosophy. - Al-Farabi (c. 872−950) held that a truth reaches the philosopher as a proof and everyone else as an image of it, and that different nations hold different images of the one truth. Ibn al-Haytham (c. 965−1040) built apparatus and settled by experiment that light travels to the eye rather than from it. Al-Ghazali (1058−1111) formulated the absolute denial of natural causes, arguing that fire possesses no inherent power to burn cotton, and that apparent natural causality is God acting directly at each occasion. - Al-Ghazali, Tahāfut al-Falāsifa, discussion 17. The position, termed occasionalism, prefigures by six hundred years Hume’s argument that human observers see succession rather than necessity. - Ibn Rushd (1126−1198) answered that a demonstrated conclusion cannot conflict with a revealed truth, and that apparent conflicts call for a revision of the textual reading. His commentaries returned the Greek philosophy to Latin Europe through the translators at Toledo.

+

The specifically medieval form of the question was shaped by Porphyry’s Isagoge, an introduction to Aristotle’s Categories. Porphyry famously set aside, for a deeper investigation, the questions of whether genera and species exist in reality or only in thought, whether they are bodies or incorporeal, and whether they exist separately or in sensible things. Boethius then carried these questions into Latin philosophy through two commentaries, and medieval authors spent centuries refining the relations among things, concepts, and the universal terms by which things are named. + Porphyry, Isagoge, in John Peter Anton and George L. Kustas, eds., Essays in Ancient Greek Philosophy, vol. 2 (Albany: State University of New York Press, 1971), 197−199. Boethius wrote two commentaries on Porphyry’s text, the first based on Marius Victorinus’s translation and the second on his own. +

-

Europe then argued the same question for four hundred years under a new name, the problem of universals. William of Ockham (c. 1287−1347) declined both of the answers on offer. What the many share is neither a Form standing above them nor an essence lodged within them, but a sign standing for many of them, and a sign is something somebody has to write down. - The position is called nominalism, from nomen, a name. - He is the first man in this account to refuse the pair outright and put a third thing in its place, and the third thing he put there was a representation.

+

William of Ockham (c. 1287−1347) gave one of the sharpest late-medieval attempts to dissolve this dialectic. He held that what the many instances of a thing have in common is neither a Form standing above them nor an essence lodged within them, but a sign that stands for all of them. + William of Ockham, Summa Logicae I.14−17 (c. 1323); in English as Ockham’s Theory of Terms: Part I of the Summa Logicae, trans. Michael J. Loux (Notre Dame: University of Notre Dame Press, 1974). The fuller critique is in the Ordinatio I d.2 qq.4−8. The position is called nominalism, from nomen, a name. Peter Abelard (1079−1142) reached a non-realist account two centuries earlier, while Ockham gave one of the sharpest and most influential late-medieval statements. Ockham’s sign is a concept in the soul, with spoken and written words subordinated to it, so the sign he means is nearer the abstraction of section than the representation is. + Plato places the explanatory model in an intelligible order distinct from sensible particulars, whereas Ockham denies that there is any independently existing universal of that kind to locate. In the language of section , Ockham keeps the representation and the instances below, and empties the layer above. +

-

René Descartes (1596−1650) argued that certain knowledge comes from reason reflecting on itself rather than from the senses, which deceive. The ideas he trusted most were those he took to be innate, present in the mind before any experience of the world, thus channeling Plato in modern terms. +

René Descartes (1596−1650) argued that certain knowledge comes from reason reflecting on itself, rather than from the senses which can deceive. The ideas he trusted most were those he regarded as arising from the mind’s own constitution rather than being derived from sensory experience, thus giving Plato a modern form. René Descartes, Meditationes de Prima Philosophia (Paris, 1641), Meditations III and V. The position is called rationalism. Descartes held the body to be a machine and animals to be machines entire, while reserving the mind from mechanism; the line he drew there was erased a century later by Julien Offray de La Mettrie, L’Homme Machine (Leyden, 1747). - John Locke (1632−1704) answered him by saying there are no innate ideas, the mind begins empty, and everything in it arrives through sensation and through reflection upon what sensation delivers. Yet Locke declined to include mathematics, which he conceded comes from the agreement among our own ideas. - John Locke, An Essay Concerning Human Understanding (London, 1690), Book I against innate ideas, Book II on the two sources, and Book IV chap. 4 on mathematical certainty. The position is called empiricism. + John Locke (1632−1704) answered that there are no innate ideas: the mind begins without ideas, and what is in it arrives through sensation and through reflection on what sensation delivers. Yet Locke treated mathematics as demonstrative knowledge of relations among ideas rather than as a direct report of sensation. + John Locke, An Essay Concerning Human Understanding (London, 1690), Book I against innate ideas, Book II on the two sources, and Book IV chap. 4 on mathematical certainty. The position is called empiricism. Locke’s denial concerns innate ideas, not the mind’s innate capacities for receiving and comparing ideas.

-

George Berkeley (1685−1753) attempted to dissolve the dialectic by proposing that perception is reality itself and not a report about some further thing standing behind it. Thus, the world as it is sensed, and the world as it is perceived, are one and the same. When asked what becomes of the furniture in a room when nobody is in it, he answered that God perceives it. +

George Berkeley (1685−1753) attempted to dissolve the dialectic by identifying sensible objects with ideas as perceived, rather than treating perception as a report about some further material thing standing behind them. The world as sensed is therefore a world of perceived ideas, but Berkeley does not reduce reality to a private person’s perceptions: the order of nature is secured by God’s perception and is not controlled by individual will. When asked what becomes of the furniture in a room when nobody is there, he answered that God perceives it. George Berkeley, A Treatise Concerning the Principles of Human Knowledge (Dublin, 1710), §§3 and 6 for the doctrine, §§28−33 and §146 for the argument reaching God, and §48 for the objects that persist unperceived by any man. Put again, and more accessibly, in Three Dialogues between Hylas and Philonous (London, 1713).

-

Thomas Hobbes (1588−1679) opened the Leviathan by asking why the heart is not a spring and the nerves so many strings, and made geometry a science of bodies and of the motions that generate them. - Thomas Hobbes, Leviathan (London, 1651), Introduction; De Corpore (London, 1655), Part II. The fuller statement of man as machine is Julien Offray de La Mettrie, L’Homme Machine (Leyden, 1747). - Isaac Barrow (1630−1677) argued that mathematical magnitudes are abstracted from the magnitudes of physical bodies and have no other source. And Newton (1642−1727) wrote in the preface to the Principia that geometry is founded in mechanical practice and is nothing other than the part of universal mechanics which measures accurately. - Isaac Newton, Philosophiæ Naturalis Principia Mathematica (London, 1687), Author’s Preface. The remark is made in passing, as something the reader will grant, which is itself evidence of how the matter then stood. The vocabulary of the period made the claim easy to state, the study of the physical world being called natural philosophy; the word scientist was not coined until 1834, by William Whewell. - Newton’s calculus is a language whose subject matter is a body in motion, its terms recording what was observed of such bodies in a form other men could take up and use.

+

Thomas Hobbes (1588−1679) pressed the naturalist argument into the human body. Against the separation of an immaterial mind from an extended body, he treated sensation, imagination, and reasoning as processes belonging to the natural order. Thought, on this account, was not a visitor from a higher realm but something that happened in a body. + Thomas Hobbes, Leviathan (London, 1651), Introduction and chaps. 1 and 5. In the Introduction Hobbes compares the body to an artificial machine, asking why the heart should not be a spring, the nerves strings, and the joints wheels. In chap. 1 he treats sense as a motion produced in the organ by an external body; in chap. 2 he describes imagination as decaying sense; and in chap. 5 he defines reasoning as a form of reckoning or computation. See also Hobbes, De Corpore (London, 1655), Part I, chap. 1, and Part II on geometry and motion. Hobbes’s materialism does not anticipate a modern computational theory in every detail, but it places bodily sensation, imagination, and reasoning within a natural order rather than assigning them to an immaterial realm. +

+ +

David Hume (1711−1776) divided all inquiry into relations of ideas and matters of fact. Hume placed mathematical certainty among the relations of ideas while noting that mathematical certainty belongs to intuition and demonstration rather than to empirical observation. + David Hume, An Enquiry Concerning Human Understanding (London, 1748), §IV part 1. +

+ +

In 1781 Immanuel Kant (1724−1804) placed mathematics within the conditions of human cognition. He argued that space and time are forms of intuition: space is the condition under which geometry is possible, and time is the condition under which arithmetic can be constructed through succession. + Immanuel Kant, Kritik der reinen Vernunft (Riga: Hartknoch, 1781), Introduction B14−B17 and the Transcendental Aesthetic. Kant does not merely say that mathematics is a private invention. He argues that space and time are forms of intuition, conditions under which objects can be given to us and mathematical cognition can arise. +

-

While the argument ran, machines were being built. Blaise Pascal (1623−1662) made a working calculating machine at nineteen, to relieve his father of the arithmetic of tax assessment, and some twenty of them were made. Gottfried Wilhelm Leibniz (1646−1716) saw one in Paris and built a better one, which multiplied and divided, and he later published an arithmetic in which every number is written with nothing but a nought and a one. He also proposed a notation in which reasoning itself would become calculation, so that two philosophers in disagreement might set aside the dispute, take up their pens, and say to one another, let us calculate. - The Pascaline, built from 1642, and the Stepped Reckoner, demonstrated to the Royal Society in 1673. Pascal is also the man who wrote that the heart has its reasons, which reason does not know; Pensées (Paris, 1670), §277 in the Brunschvicg numbering. Leibniz, “Explication de l’Arithmétique Binaire,” Mémoires de l’Académie Royale des Sciences (1703); the calculemus passage is from “The Art of Discovery” (1685). - One man thus had the machine, the number system it would eventually be built in, and the proposal that reasoning is what such a machine does. He had no way to join them, because the notation he wanted was a notation for reasoning, and what was missing was a notation for the apparatus. The machines went into cabinets, and no party to the argument thought to consult one. Section takes them up as specimens.

+

Kant’s position remained influential until nineteenth-century geometries challenged the identification of Euclidean space with the necessary structure of human intuition. In the 1820s and 1830s, Nikolai Lobachevsky and János Bolyai developed geometries in which Euclid’s parallel postulate fails. In 1868 Eugenio Beltrami constructed a model of Lobachevskian geometry within Euclidean geometry, establishing a relative-consistency result: if the Euclidean background is consistent, then so is the modeled geometry. + Nikolai Lobachevsky, “On the Principles of Geometry,” Kazan Messenger (1829−1830); János Bolyai, “Appendix Scientiam Spatii Absolute Veram Exhibens,” published with Farkas Bolyai’s Tentamen (Maros-Vásárhely, 1832); Eugenio Beltrami, “Saggio di interpretazione della geometria non-euclidea,” Giornale di Matematiche 6 (1868): 284−312. Gauss had reached similar results and published none of them. The historical point is not that non-Euclidean geometry simply refuted Kant, but that it forced philosophers to reconsider what Kant’s claims about mathematical intuition could mean. + This did not by itself refute Kant’s philosophy, but it made the location of mathematical necessity in human spatial intuition much more difficult to defend. +

-

David Hume (1711−1776) then divided all inquiry into relations of ideas and matters of fact and placed mathematics among the former, so that an empiricist had removed mathematics from the reach of experience. Immanuel Kant (1724−1804) settled the matter for the next century by relocating the ground once more, from the world to the mind: mathematical judgements are necessary and are not empty, and what makes them possible is the pure intuition of space and time, a form contributed by the knowing subject. - David Hume, An Enquiry Concerning Human Understanding (London, 1748), §IV part 1. Immanuel Kant, Kritik der reinen Vernunft (Riga: Hartknoch, 1781), Introduction B14−B17 and the Transcendental Aesthetic. Kant does not deny that mathematics is grounded; he answers the question the naturalist had been trying to answer, and answers it better than the naturalist then could. - What unmade that position was a discovery rather than an argument. Lobachevsky in 1829 and Bolyai in 1832 published geometries in which the parallel postulate fails, and Beltrami showed in 1868 that the new geometry is consistent if the old one is. If physical space satisfies at most one of several consistent geometries, then geometry is not the science of space, and the necessity Kant had explained is not there to be explained.

+

By the nineteenth century, the question of whether mathematics rose from observation or descended from the abstract had largely ceased to be the question mathematicians were asking. They turned instead to the construction of symbolic systems. A person might say that Plato had won. The Forms were no longer discussed as a separate realm; they had been recast as symbols, relations, and rules that could be manipulated within a formal system. George Boole put logical operations into an algebra. Georg Cantor made infinite collections a subject of mathematical investigation. Richard Dedekind sought to characterize the natural numbers through objects, systems, and mappings, and in Was sind und was sollen die Zahlen? (1888) described what he called a simply infinite system. Giuseppe Peano published a closely corresponding axiomatization in Arithmetices principia, nova methodo exposita (1889), acknowledging Dedekind’s earlier work. + George Boole, The Mathematical Analysis of Logic (Cambridge: Macmillan, Barclay & Macmillan, 1847), and An Investigation of the Laws of Thought (London: Walton and Maberly, 1854). Georg Cantor’s work on set theory and transfinite numbers expanded the mathematical study of infinity. Richard Dedekind, Was sind und was sollen die Zahlen? (Braunschweig: Vieweg, 1888); Giuseppe Peano, Arithmetices principia, nova methodo exposita (Turin: Fratres Bocca, 1889). Dedekind’s simply infinite system and Peano’s axiomatization are historically distinct formulations of the natural-number structure. Peano’s presentation is closely related to Dedekind’s earlier formulation, and modern historical accounts therefore often refer to the shared lineage as the Dedekind−Peano axioms. “Dedekind number” is already used for a different mathematical sequence, so it is not adopted here as an alternative name. +

-

The fullest naturalist account of number ever written came in the middle of this, later than most readers expect. John Stuart Mill (1806−1873) argued in 1843 that the truths of arithmetic are inductive generalizations from experience, on the same footing as those of any other science. Gottlob Frege (1848−1925) destroyed the position in 1884, at leisure and by name: a single pack of cards is one, and fifty-two, and four, depending on what one has chosen to count, so the number is not in the heap; and if numbers are ideas then my two and your two are different objects and nothing has been said about number at all. - John Stuart Mill, A System of Logic (London: Parker, 1843), Book II, chaps. 5−6. Gottlob Frege, Die Grundlagen der Arithmetik (Breslau: Koebner, 1884), §§7−10 against Mill, §§21−27 against number as a property of things and as an idea. The second charge Frege named psychologism, and it became the period’s term of abuse. - The demolition was thorough and was, on its own terms, correct. Notice what Mill was obliged to argue from. He had heaps of pebbles, and no other mechanism was then available, so the specimens he offered had no behavior in them to observe and nothing about them fixed what was being counted. A naturalist handed a heap has been handed the wrong animal.

+

Frege’s Begriffsschrift of 1879 supplied a formal language for logic, and his Grundlagen der Arithmetik of 1884 attacked the reduction of number to empirical aggregates or private ideas. Frege argued that numbers are objective and that arithmetic is not grounded merely in psychological observation. The question that had begun with Forms, bodies, concepts, and signs had now become a question about the logical conditions under which arithmetic itself could be constructed. + Gottlob Frege, Begriffsschrift (Halle: Louis Nebert, 1879), and Die Grundlagen der Arithmetik (Breslau: Wilhelm Koebner, 1884). Frege’s anti-psychologistic arguments distinguish objective mathematical content from subjective ideas and criticize John Stuart Mill’s account of number as a property of aggregates. +

-

After 1884 the field belonged to programmes that dispensed with the world altogether. Logicism derived mathematics from logic, formalism treated it as the manipulation of marks under stated rules, and intuitionism grounded it in mental construction. The crisis recounted in chapter was fought entirely among those three, and no party to it proposed consulting an apparatus.

+

The formal turn did not settle the question by defeating Naturalism. It changed the location at which the question was asked. Observation supplied the original problems, while logic and notation supplied structures in which the problems could be stated with increasing precision. The next section returns to the natural object and asks what would happen if such an object could think.

-

It is therefore worth noticing how the paper that closed the episode opens. Alan Turing, asked to settle a question in the foundations of logic, begins by describing a man sitting at a desk with paper and a pencil, and asks what such a man can be got to do. He had put a specimen on the table.

 + + + + A thinking machine dissolves the dialectic + +

There is an attractive way to dissolve the dialectic between the Realm of Forms and natural observation. Suppose that a natural object could think. Then thought would not be an activity belonging to a realm above nature, nor would observation be an activity available only to a mind standing outside the world. The thinking object could be observed while it thought. Its states could be recorded, its transitions could be compared, its inputs could be supplied, and its results could be checked. The thing that thinks and the thing that is observed would be one thing.

+

Many philosophers had tried to dissolve the dialectic by relocating one of its sides inside the mind. Descartes placed certainty in reason and in ideas belonging to the mind’s own constitution. Locke removed innate ideas and made the mind a place into which sensation and reflection write. Berkeley made the perceived idea the object itself, while God guaranteed the persistence and order of the world. Hume separated relations of ideas from matters of fact, placing mathematical necessity among the former. Kant then placed the conditions of mathematics in the forms of human intuition, space and time. Hobbes and, later, La Mettrie went further in the other direction, treating the body as a machine, but they still had to explain what, if anything, remained outside the machine. Each proposal moved the boundary. None removed the distinction between the knowing subject and the observed object.

+

A machine that thinks offers a different possibility. It would not choose between the Form above and the observation below. It would be a natural object whose behavior could be described by the very abstractions that the philosophers had placed in the mind. Its mathematics would be neither a report from a separate realm nor a private possession of the observer. It would be a language for recording what the object does.

+

This is the attraction of the thought experiment. If a machine can think, then every part of its thinking is available to Natural observation. A person can watch a state change, write down the symbols that caused it, record the rule that selected the next state, and compare the result with the machine’s behavior. The machine does not need to reveal a hidden Form, and the observer does not need to invent one. The dialectic disappears not because one side defeats the other, but because the two sides are recognized as descriptions of one process at different levels.

- - Where this fits in +

The story begins with counting pebbles. A pebble is a natural object, and a row of pebbles can represent a number when a person gives the row that use. The pebbles do not count themselves. They remain where they were placed, and nothing in one pebble compels the next pebble to appear. A person supplies the operation, and the heap supplies only the objects being operated upon. This is why a heap is such a weak foundation for a naturalist writing about number. It has objects but no behavior.

-

Mill lost for want of a specimen. The specimen existed. In March of 1826 Charles Babbage (1791−1871) read a paper to the Royal Society describing a language he had invented for describing machine behavior, which he called the mechanical notation. It gave each part of a machine a sign, classified that part as fixed or as moveable, recorded what drove it and what it drove in turn, and set down when each motion occurred relative to the others. - Charles Babbage, “On a Method of Expressing by Signs the Action of Machinery,” Philosophical Transactions of the Royal Society of London 116 (1826): 250−265, read 16 March 1826. He considered it one of his best inventions, and doubted that machinery of comparable complexity could be contrived without that language or an equivalent one; Passages from the Life of a Philosopher (London: Longman, Green, 1864), 104. - So a notation whose subject matter is an apparatus existed sixty years before Frege wrote, and the machine it was invented to describe was on a drawing board in London while the foundational crisis was being fought in Germany. The two never met.

+

The counting board and the abacus add a first mechanical distinction. A counter can be moved from one position to another, and its position can stand for a numerical state. The board does not merely contain a collection of things; it contains an arrangement that can change. Still, the person must supply the transitions. The hand moves the counter, the eye checks the position, and the mind carries the rule from one column to the next. The apparatus holds a state, but it does not yet carry the arithmetic through by itself.

-

Babbage had produced both halves of what a naturalist produces. The Analytical Engine was an object with behavior, which could be watched, and which would do the same thing again when set going again. The mechanical notation was a representation of that behavior, written in signs, in terms another engineer could take up and use to say what some further machine did. It was for the machine what Newton’s calculus was for moving bodies of mass, and what Aristotle’s sorted descriptions were for the animals. In each case a man observed, and then made a representation of what he had observed.

+

Hobbes did not produced a thinking machine, but he had made the body available for mechanical explanation. In the opening of the Leviathan, he asked why the heart should not be a spring and the nerves so many strings. In De Corpore, he treated geometry as a science of bodies and the motions that generate them. The claim is important here because it identifies mathematics with the description of mechanical behavior. But Hobbes’s machine was still a body described from outside. The next step would be an apparatus whose own states could carry out the description.

-

Recall the opening of chapter , that communication starts from the knowledge held in common between two parties, and the account in section , where a representation is the common knowledge that makes the communication possible. This is the word the whole argument turns on. Aristotle sorted the animals by the features they share and set the sorting down in terms a second man could use to report what he had found in a further specimen. The sorting answers to the specimens, the terms are held in common, and it is the holding in common that makes the report intelligible. Take away the representation and the observation dies with the observer.

+

Apparatus with more of the arithmetic in its own behavior had been built two centuries before Babbage. Blaise Pascal (1623−1662) constructed a working calculating machine at nineteen to relieve his father of the arithmetic of tax assessment, and some twenty of them were made. + The Pascaline, built from 1642. Étienne Pascal was a tax commissioner at Rouen, assessing in livres, sols, and deniers, twenty sols to the livre and twelve deniers to the sol, and the wheels of the machine were cut for those bases. It added and subtracted, subtraction being done by complement. Pascal is also the man who wrote that the heart has its reasons, which reason does not know; Pensées (Paris, 1670), §277 in the Brunschvicg numbering. + Turning the wheel that holds the units past nine lifts a weighted fork, which falls onto the wheel above and advances it by one. Should that wheel also stand at nine, it lifts the next fork in turn. A carry crossing a digit boundary is therefore a thing that happens in a room, at a time, and can be watched happening. The clerk supplies the power, but does not supply this part of the arithmetic, since once his hand has moved the carry the machine runs through a row of nines while he sits still. +

+ +

Here is what the heap of pebbles lacks. The machine has states, it passes between them under a rule, and the rule is realized in brass and can be examined. Ask a heap what follows nine and it is silent. Ask the machine and it answers. It answers the same way every time, and the answer can be checked against what the machinery was built to do. The number has become a state of an object, and addition has become a transition between states.

-

Notice that Plato needs the same middle term and puts it in the same place. Section set out his account as a Form above, a representation below it, and instances below that, and the geometers in the sand are arguing by way of the representation whatever else is true. Neither man disputes that a representation is what a person actually has in hand. What they dispute is what it is a representation of. For Plato it stands for a Form, and the shape of the shadow is fixed from above. For Aristotle it stands for what was observed, and is answerable to further observation and revised when a specimen refuses its category. That is the whole of the disagreement, and stating it this way puts it back on the tracks it belongs on.

+

Gottfried Wilhelm Leibniz (1646−1716) saw one of Pascal’s machines in Paris and built a more capable one, which multiplied and divided by means of a stepped drum and which he demonstrated to the Royal Society in 1673. Thirty years later he published an arithmetic in which every number is written with nothing but a nought and a one. + The Stepped Reckoner, or Staffelwalze. Leibniz, “Explication de l’Arithmétique Binaire,” Mémoires de l’Académie Royale des Sciences (1703). He held the binary notation to have theological significance, seeing in the generation of all numbers from nought and one an image of creation from nothing, and this was among his reasons for pursuing it. + The machine and the notation belong together. One is an apparatus whose states are changed by mechanical operations; the other is a compact way of writing the values those states can hold. +

-

So the word can be given its meaning for this book. Naturalism is the practice of forming knowledge by observing particulars and making a representation of what is observed. The making of the representation is not an afterthought to the observing, since an observation that cannot be stated is not yet knowledge of anything, and a representation is what makes the stating possible. This is Aristotle’s sense of the word and the older one. Present-day philosophy uses the term for the doctrine that nature is all there is, or for the position that inquiry proceeds by the methods of the natural sciences, and neither of those is what is meant in these pages. The two senses that do descend from Aristotle then hang together rather than merely sharing an ancestor. The biologist who studies nature is doing the observing, and the position that the language of science and mathematics is the language for talking about nature is a claim about the representation.

+

Leibniz then proposed something further. Let there be a notation in which every concept is written as a character, and let the rules for combining those characters be such that reasoning becomes calculation. Two philosophers in disagreement would then set aside the dispute, take up their pens, and say to one another: let us calculate. + The characteristica universalis and the calculus ratiocinator. The calculemus passage is from “The Art of Discovery” (1685). Neither the notation nor the calculus was completed, and much of what Leibniz wrote on the subject stayed unpublished in Hanover for two centuries. + Read that proposal against the machine on his desk and notice what it asks. A dispute is to be settled by carrying out a procedure rather than by argument, which is to say that the disputants are to consult an apparatus and abide by what it does. Leibniz had the machine, the number system it could eventually use, and the proposal that reasoning is what such a machine does. What he did not yet have was a language in which the apparatus itself could be described. +

-

Nothing in that definition says what is being observed, and this is the opening the book walks through. Point it at the animals and the representation is a taxonomy. Point it at bodies in motion and the representation is a calculus. Point it at a machine that computes and the representation is mathematics.

+

The difficulty is not simply that a machine holds numbers. It holds several numbers, each in a position, and changes them in an order determined by the connections among its parts. A language for the apparatus must therefore record a part, what that part is coupled to, the motion available to it, and what happens when the whole is set going. It must say not only what a number is, but what a wheel does with a number, when it does it, and what receives the result.

-

Consider what was done instead. The rewrite rule interpretation of what a machine could do took mathematics to be a string of symbols recognized and replaced by another string, and this was encapsulated in the general recursive functions and in the lambda calculus. - The formulations are those of chapter , where the history is given. - These have the form of Babbage’s notation, being rules for what mark follows what mark, with the apparatus the marks were an account of left out. A representation with nothing on the other side of it is no longer a representation of anything, and this is the condition mathematics was left in after 1884. The specimens had been burned and the catalog was being read as though it were the animals.

+

This is where the thesis of this book begins to take a precise form. Mathematics is a language for describing machine observations. Arithmetic describes discrete states such as the position of a digit wheel or the number held in a register. Logic describes the conditions under which one transition is selected rather than another. Geometry describes the arrangement of parts. Calculus describes quantities generated by motion and the rates at which those quantities change. The mathematics is abstract, but its subject matter can be the behavior of a natural object.

-

Then Alan Turing asked mathematicians to imagine what a clerk writing on paper could accomplish, and he put the specimen back. Later Stephen Kleene opened his Introduction to Metamathematics with a flock of four sheep and a grove of four trees, and observed that one can pair them off, a sheep to a tree, without counting either. - Stephen Cole Kleene, Introduction to Metamathematics (Amsterdam: North-Holland, 1952), 3. - The passage is there to introduce Cantor, and it does something else on the way. It appeals to Naturalism as a foundation, it leverages the reader’s intuition, and it makes the implicit point that mathematics has a purpose. Nobody objects to the sheep. Kleene then turns the page and presents an abstract symbol based computation, and the sheep are not heard from again.

+

Newton’s mathematics can be read in this way. I will call his calculus a rotational language, not because Newton used that phrase, and not because all calculus is a theory of gears, but because it gives a mathematical form to quantities generated by motion. A wheel makes the idea visible: its angular position is a state, its turning is a change of state, and its rate of turning is a rate of change. The same language can then be generalized from a wheel to a planet, from a planet to a body falling under gravity, and from a physical motion to an abstract variable.

-

Suppose for a moment that mathematics is the representation a Naturalist arrives at when the animal under observation is a machine that computes. Then the geometers arguing over a figure drawn in the sand, Turing’s clerk at his desk, and a mathematician at work today are all doing the one thing, which is observing such a machine and setting down what it does in terms held in common. - Plato’s strongest case for the opposing account is the slave boy of the Meno 82b−85b, who is brought to double the square under questioning alone and is said to be recollecting. The episode is the sharpest form of the question this book answers mechanically, and a reader who finds the present account thin is invited to set it against that passage. - The machine is the Realizable Machine of chapter , a cousin of the Turing Machine, and the conditions it has to meet to serve are set out in chapter . That is the thesis defended in this book, and it is what the title means by Computational Naturalism.

+

Newton himself placed geometry inside mechanics. In the preface to the Principia he wrote that “geometry is founded in mechanical practice, and is nothing but that part of universal mechanics which accurately proposes and demonstrates.” He also described the task of natural philosophy as proceeding “from the phenomena of motions to investigate the forces of nature, and then from these forces to demonstrate the other phenomena.” + Isaac Newton, Philosophiæ Naturalis Principia Mathematica (London, 1687), Author’s Preface. Newton’s distinction is between rational mechanics, which proceeds by demonstration, and practical mechanics, which belongs to the manual arts. The point used here is that the mathematical description is not detached from motion; it is the exact portion of mechanics that expresses and demonstrates what motion does. + In the language of this book, Newton’s calculus is not a descent from the Realm of Forms. It is a notation in which observations of changing natural objects can be recorded, transformed, and checked. +

-

Computational Naturalism declares no victor between the descent from above and the ascent from below. What comes out instead is that the two are aspects of one thing, that of a language and its metalanguage, and the demonstration of it is the business of the chapters that follow.

- +

Leibniz’s marks for the differential and integral remain among the most successful mathematical representations ever invented, but his proposed characteristica still took thought or quantity as its subject matter. What was missing was a notation for the apparatus: a way to describe the geometry of its parts, the time of their movements, and the paths by which force passed through them.

+ +

George Boole (1815−1864) supplied another part of the language. He gave the operations of logic an algebra, so that conjunction, disjunction, and negation could be written as marks and manipulated under stated rules. The truth of a compound could be computed from the truth values of its parts. + George Boole, The Mathematical Analysis of Logic (Cambridge: Macmillan, Barclay & Macmillan, 1847), and An Investigation of the Laws of Thought (London: Walton and Maberly, 1854). The direct connection between this algebra and switching circuits was not drawn until Claude Shannon, “A Symbolic Analysis of Relay and Switching Circuits,” Transactions of the American Institute of Electrical Engineers 57 (1938): 713−723. Boole’s work is included here as the formal language of conditional combination, not as an already completed theory of machines. + This is Leibniz’s calculus of reasoning delivered at last, but it arrived without an apparatus attached to it. The language could describe which result followed from which premises; it did not yet describe the physical mechanism that carried out the transitions. +

+ +

The separation of a procedure from the machine that carries it out came from weaving. Joseph Marie Jacquard (1752−1834) fitted a loom with a chain of punched cards, one card for each pass of the shuttle, the holes determining which warp threads were raised. The pattern was thereby held outside the loom and could be changed without altering the loom. + The Jacquard loom, from 1804, built on earlier card and cylinder mechanisms by Basile Bouchon (1725), Jean-Baptiste Falcon (1728), and Jacques de Vaucanson (1745). A portrait of Jacquard woven on such a loom from some twenty-four thousand cards hung in Babbage’s drawing room. + A loom so fitted is a machine whose behavior is written elsewhere, in a notation of holes. A person who wants to know what the loom will do reads the cards rather than the loom. The machine and the procedure have become separable, although the procedure still has to be embodied in a physical sequence of cards. +

+ +

Charles Babbage (1791−1871) took up the three unfinished ideas: arithmetic embodied in machinery, mathematical operations represented as procedures, and procedures separated from the machine that executes them. He began designing the Difference Engine in 1821 and published an account of it in 1822. The engine was to compute and print mathematical tables by finite differences. Its method was beautiful because it used repeated addition and thereby avoided the more difficult mechanical operations of multiplication and division. + Charles Babbage, “A Note on the Application of Machinery to the Computation of Astronomical and Mathematical Tables,” Memoirs of the Astronomical Society 1 (1822): 309−314. The Computer History Museum describes Difference Engines as specialized calculators based on finite differences, with a printer mechanically coupled to the calculating section. + The Difference Engine did not need to understand a polynomial. Its wheels held a value and successive differences, and the mechanism repeatedly added one row of differences to the next. That is the same behavior this book later describes as a function-extension machine. + The later computational-analysis chapter develops this connection explicitly: a tape holds a function value and its finite differences, and each call adds the first difference to the function value, the second difference to the first, and so on. See section and the difference-table sections that follow it. +

+ +

With the construction of the first Difference Engine stalled, Babbage conceived a more ambitious machine. Beginning in 1834, he designed the Analytical Engine as a general-purpose programmable computing engine. It would take instructions from punched cards, hold numbers and intermediate results in a Store, and perform arithmetic in a separate Mill. The separation is the one a modern reader recognizes as the separation of memory from arithmetic processing. + The Analytical Engine was never completed in Babbage’s lifetime. It was designed in several stages and included punched-card programming, a Store, a Mill, conditional behavior, repetition, and several forms of output. See the Computer History Museum, “The Engines.” + The machine was not merely a faster calculator. It was a proposal for an object whose states could represent data, whose cards could represent instructions, and whose mechanical transitions could transform one into the other. +

+ +

More than a decade before the Analytical Engine, Babbage had built the language needed to reason about machinery. In March 1826 he read to the Royal Society a paper entitled “On a Method of Expressing by Signs the Action of Machinery.” He gave the system the name Mechanical Notation. It recorded the actual shape and relative position of every piece, the time and duration of every motion, and the connection of each movable piece with every other on which it acted. The drawings, the times of action, and the trains for the transmission of force could all be represented in one language. + Charles Babbage, “On a Method of Expressing by Signs the Action of Machinery,” Philosophical Transactions of the Royal Society of London 116 (1826): 250−265, read 16 March 1826. Babbage’s later account in Passages from the Life of a Philosopher, chap. IX, enumerates the three kinds of information: shape and position, time and duration, and mechanical connection or “trains.” +

+ +

Babbage understood what he had made. In his later account he wrote: “I have called this system of signs the Mechanical Notation. By its application to geometrical drawing it has given us a new demonstrative science, namely, that of proving that any given machine can or cannot exist; and if it can exist, that it will accomplish its desired object.” The notation did not merely illustrate a finished machine. It allowed a machine to be reasoned about before it existed.

+ +

In another passage, Babbage says that three consequences of his work were of great importance. One was the conception of the Analytical Engine; another was the Mechanical Notation. He wrote that without this language he could not have invented the Analytical Engine, and that he did not believe machinery of equal complexity could be contrived without it or an equivalent language. He later called it “one of the most important additions I have made to human knowledge” and said that it had placed the construction of machinery in the rank of a demonstrative science. + Charles Babbage, Passages from the Life of a Philosopher (London: Longman, Green, Longman, Roberts, & Green, 1864), chap. V, for the claim that the language was necessary to the Analytical Engine; chap. IX, for the new demonstrative science; and chap. XXXIII, “The Author’s Contributions to Human Knowledge,” for the statement that he regarded it as “one of the most important additions I have made to human knowledge.” + The recollection that Babbage regarded the notation as one of his best inventions is therefore right in substance, but his own wording is more precise and more revealing. He did not merely rank it among his inventions. He claimed that it changed what could be demonstrated about a machine. +

+ +

In 1843 Ada Lovelace translated Luigi Menabrea’s account of the Analytical Engine and expanded it with notes, the last of which gave a table of the operations by which the proposed engine would compute the Bernoulli numbers. The table is more precisely an execution trace than a modern program, but it makes the central separation visible: instructions are written outside the machine, numbers are held inside it, and the machine carries out the instructed transformations. + L. F. Menabrea, “Notions sur la machine analytique de M. Charles Babbage,” Bibliothèque Universelle de Genève 82 (1842); translated and expanded by Ada Augusta, Countess of Lovelace, in Richard Taylor, ed., Scientific Memoirs, vol. 3 (London, 1843), 666−731, especially Note G. +

+ +

By the middle of the nineteenth century, mathematics had therefore acquired the beginnings of a language for describing machines. Arithmetic could describe the values held by their parts. Geometry could describe the parts and their positions. Newton’s calculus could describe quantities generated by motion, including the rotational changes through which a mechanical machine carries a value from one state to another. Boole’s algebra could describe conditional combinations. Babbage’s Mechanical Notation could describe the machine as a connected object whose parts acted at particular times and transmitted force along particular trains.

+ +

The thinking machine is attractive because it would make these descriptions meet. The machine is natural because it is made of matter and can be observed. It is formal because its behavior can be recorded as symbols and rules. It is mathematical because those symbols can be transformed into predictions of what the machine will do. The Realm of Forms has not been found above the world, and Naturalism has not been confined to heaps of passive objects. The abstract structure is the description of a natural process.

+ +

The following section turns from this historical line to the machines themselves. It will ask what kind of natural object a machine is, what it means for such an object to think, and how its behavior compares with the behavior of a living mechanism.

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Multiplication of synchronized difference vectors

- -

- Because the forward difference operator is linear, adding or subtracting two polynomials is achieved by the elementwise addition or subtraction of their initial difference vectors. Multiplication, however, requires a discrete convolution of the two vectors. -

- -

- Suppose a programmer has two initial difference vectors, A_0 representing function f(t) with an extent of \omega_a, and B_0 representing function g(t) with an extent of \omega_b. The goal is to compute the initial difference vector C_0 for the product function h(t) = f(t)g(t). -

- -

- Multiplying two polynomials of degrees \omega_a and \omega_b yields a polynomial of degree \omega_a + \omega_b. Therefore, the resulting vector C_0 will strictly have an extent of \omega_c = \omega_a + \omega_b, requiring a tape component count of \omega_a + \omega_b + 1. -

- -

- To determine the components of C_0 directly from A_0 and B_0 without evaluating the functions, we rely on the multiplication of their basis elements. In the calculus of finite differences, polynomials are expanded using binomial coefficients. The product of two binomial coefficients expands into a linear combination of higher binomial coefficients according to a known combinatorial identity: -

- -

- \binom{t}{i} \binom{t}{j} = \sum_{k=\max(i,j)}^{i+j} \binom{k}{i} \binom{i}{k-j} \binom{t}{k} -

- -

- By applying this identity across the summations of both input functions, the component k of the resulting vector C_0 can be computed algebraically. Each component C_{0, k} is the sum of the cross products of the input components, weighted by combinations of their indices: -

- -

- C_{0, k} = \sum_{i=0}^{\omega_a} \sum_{j=0}^{\omega_b} A_{0, i} B_{0, j} \binom{k}{i} \binom{i}{k-j} -

- -

- Thus, while a Turing Machine extending the function only requires a simple accumulator, a machine tasked with multiplying two initial tapes must perform a combinatorial cross multiplication to generate the expanded tape before the extension sequence can begin. -

- -

Division and the Reciprocal Difference Vector

- -

- If the multiplication of two polynomials in the finite difference domain is a discrete convolution, then division is a discrete deconvolution. By finding the reciprocal of a difference vector, a programmer can perform division using the same combinatorial architecture. -

- -

- Let A_0 be the initial difference vector for a polynomial f(t). We seek the reciprocal difference vector C_0, which represents the function h(t) = 1/f(t). Because the reciprocal of a polynomial is a rational function, its forward differences will never reduce to zero. Thus, C_0 is an infinite vector. -

- -

- Following the lazy evaluation strategy, C_0 is not written to a static tape. It is implemented as a generator machine. The main evaluator queries this generator for its values up to the required extent \omega only as they are demanded. -

- -

- By definition, f(t)h(t) = 1. In the difference domain, this means the convolution of A_0 and C_0 must equal the identity vector I_0, where I_{0, 0} = 1 and all subsequent components are exactly zero. -

- -

- Recall the convolution formula for component k of the product: -

- - - I_{0, k} = \sum_{i=0}^{k} \sum_{j=0}^{k} A_{0, i} C_{0, j} \binom{k}{i} \binom{i}{k-j} - - - -

- To perform the deconvolution, we isolate the unknown component C_{0, k}. This term occurs in the summation strictly when j = k. When j = k, the term k-j equals 0, making the binomial coefficient \binom{i}{0} = 1. Factoring C_{0, k} out of the sum yields: -

- - - C_{0, k} \sum_{i=0}^{k} A_{0, i} \binom{k}{i} - - -

- A person familiar with Newton's forward difference formula will recognize that the summation \sum_{i=0}^{k} A_{0, i} \binom{k}{i} is exactly the evaluation of the original function at step k, or f(k). -

- -

- We can now solve for C_{0, k} recursively. For the base case k = 0, where I_{0, 0} = 1: -

- - - C_{0, 0} = \frac{1}{A_{0, 0}} - - -

- For all subsequent components where k > 0 and I_{0, k} = 0, we subtract the previously known terms of the convolution and divide by f(k): -

- -

- C_{0, k} = \frac{-1}{f(k)} \sum_{i=0}^{k} \sum_{j=0}^{k-1} A_{0, i} C_{0, j} \binom{k}{i} \binom{i}{k-j} -

- -

- This reveals a strict recurrent structure. To generate component k of the reciprocal vector, the generator machine relies entirely on the static components of the input polynomial A_0 and the previously computed components of the reciprocal C_{0, 0} through C_{0, k-1}. By encapsulating this recurrence within a generator, a programmer can perform exact division while maintaining finite memory bounds, extending the reciprocal vector only when the execution demands it. Note, this is a reciprocal of a function, rather than that of a value. -

- - - ---------- - - in the original Turing machine Architecture, add three tapes, stdin, stdout, and stderr