<RT·endnote>Aristotle, <em>Posterior Analytics</em> II.19, for knowledge beginning in perception and rising by induction to the principles. The Greek term for the operation is <em>epagōgē</em>, rendered as <RT·term>induction</RT·term>.</RT·endnote>
</p>
- <p>After having persisted twenty years in challenging Plato over the Realm of Forms, when Plato passed away, Aristotle was not chosen to take his place to head the Academy. Plato’s estate, including the Academy, was inherited by Plato’s nephew Speusippus, a name few people are familiar with today. Upon leaving Aristotle dedicated his time to studying the natural world. He dissected, he collected, and he questioned fishermen and beekeepers about what they had seen; his account of the developing chick came from opening eggs on successive days.
+ <p>After years of challenging Plato over the Realm of Forms, the debate ended when Plato passed away. Aristotle was not chosen to take his place to head the Academy. The chair went to Plato’s nephew Speusippus, a name few people are familiar with today, and the estate to family.
+ <RT·endnote>Speusippus, son of Plato’s sister Potone, headed the school for eight years; Diogenes Laertius, <em>Lives</em> IV.1. The will leaves the estate at Iphistiadae to a boy named Adeimantus, taken to be a young kinsman since Plato had a brother of that name, and lists a second estate at Eiresidae; <em>Lives</em> III.41−42. Headship of the school and title to the land were separate things, and neither came to Aristotle. The Academy grove itself was a public gymnasium, not Plato’s to bequeath, and Aristotle, a Stagirite and therefore a metic at Athens, could not have held Attic land in any case.</RT·endnote>
+ Upon leaving Aristotle dedicated his time to studying the natural world. He dissected, he collected, and he questioned fishermen and beekeepers about what they had seen; his account of the developing chick came from opening eggs on successive days.
<RT·endnote>Aristotle, <em>Historia Animalium</em> VI.3 561a for the chick; V.12 541b for the modified arm of the male octopus, which was thought a fable until confirmed in 1857. He was wrong about a great deal, holding the heart to be the seat of thought and the brain an organ for cooling the blood. Darwin, thanking William Ogle in 1882 for a translation of the <em>Parts of Animals</em>, wrote that Linnaeus and Cuvier had been his two gods, but that they were mere schoolboys compared to old Aristotle. Charles Darwin to William Ogle, 22 February 1882, in Francis Darwin, ed., <em>The Life and Letters of Charles Darwin</em> (London: John Murray, 1887), vol. 3, 252.</RT·endnote>
What he produced from it was a <RT·term>taxonomy</RT·term>: animals sorted by the features they share, the sorting answerable to the specimens and revised whenever a specimen refused its category.</p>
- <p>This perspective, that knowledge comes from the categorization of observations of nature rather than, as Plato taught, from the conceptualization of Forms, is the pillar of the philosophy of <RT·term>Naturalism</RT·term>.</p>
+ <p>This perspective, that knowledge comes from the categorization of observations of nature rather than, as Plato taught, from the conceptualization of Forms, is the essence of <RT·term>Naturalism</RT·term>. There are two modern definitions for Naturalism. One definition is that of a philosophy that nothing exists outside of nature. This philosophy is generally taken to deny the existence of any abstraction standing apart from what could be observed, including that of the Realm of Forms. The other definition is that of a study of nature, that of a wildlife biologist.</p>
- <p>There are two modern definitions for Naturalism. One definition is that of a philosophy that nothing exists outside of nature. This philosophy is generally taken to deny the existence of any abstraction standing apart from what could be observed, including that of the Realm of Forms. The other definition is that of a study of nature, that of a wildlife biologist. Computational Naturalism holds that a machine is a natural phenomenon and can be observed, and that computation theory is a useful notation for cataloging these observations. However, within the study of Computational Naturalism there arises a metamathematics and layers of abstraction that do the work the Realm of Forms was posited to do. Plato described that realm and never defined it, and Computational Naturalism supplies a definition for at least some of what he described. Thus Computational Naturalism does not debate the question as to which view is correct, but instead holds that they are two aspects of the same thing.
- <RT·endnote>Aristotle is ancestor to the biologist’s meaning and only distantly to the philosophical one, holding as he did that the first mover is no part of nature; <em>Metaphysics</em> XII.7. The metaphysical doctrine is a nineteenth-century development.</RT·endnote>
- </p>
</RT·section>
-
<RT·section id="Section·Naturalism·The_argument_since">
- <RT·name>Those who walked around it</RT·name>
-
- <p>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.
- <RT·endnote>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 <em>Grundlage der gesammten Wissenschaftslehre</em> (Leipzig, 1794), §§1−3, and Heinrich Moritz Chalybäus fixed the three names upon it in the <em>Historische Entwicklung der speculativen Philosophie von Kant bis Hegel</em> (Dresden, 1837), from which they passed into general use. Hegel’s own term is <em>Aufhebung</em>, 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.</RT·endnote>
- 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 <em>temperature</em>. Sometimes understanding the deeper structure dissolves the debate altogether.
- </p>
-
- <p>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.</p>
-
- <p>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.
- <RT·endnote>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 <em>Theology of Aristotle</em>, an adaptation of parts of Plotinus. See Cristina D’Ancona, “Greek Sources in Arabic and Islamic Philosophy,” <em>Stanford Encyclopedia of Philosophy</em>.</RT·endnote>
- </p>
-
- <p>
- 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.
- <RT·endnote>Al-Ghazali, <em>Tahāfut al-Falāsifa</em>, 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.</RT·endnote>
- </p>
+ <RT·name>Metamorphosis of the dielectic</RT·name>
- <p>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.
- <RT·endnote>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,” <em>Stanford Encyclopedia of Philosophy</em>.</RT·endnote>
- </p>
+ <p>Debate over the dialectic continued for two millennia between the decline of the Athenian Academy and the 19th century. There is a brief review of this interim period in the appendix of chapter <RT·Counter·read snapshot="Section·Appendix_dialectic_to_Kant"></RT·Counter·read>. In the 19th century there were two revolutionary developments that changed the form of the dialectic. The symbolic logic foundation of mathematics, and the advent of thinking machines.</p>
- <p>The specifically medieval form of the question was shaped by Porphyry’s <em>Isagoge</em>, an introduction to Aristotle’s <em>Categories</em>. 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.
- <RT·endnote>Porphyry, <em>Isagoge</em>, in John Peter Anton and George L. Kustas, eds., <em>Essays in Ancient Greek Philosophy</em>, 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.</RT·endnote>
+ <p>A person might say that Plato had won over the mathematicians. 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 <em>Was sind und was sollen die Zahlen?</em> (1888) described what he called a <em>simply infinite system</em>. Giuseppe Peano published a closely corresponding axiomatization in <em>Arithmetices principia, nova methodo exposita</em> (1889), acknowledging Dedekind’s earlier work.
+ <RT·endnote>George Boole, <em>The Mathematical Analysis of Logic</em> (Cambridge: Macmillan, Barclay & Macmillan, 1847), and <em>An Investigation of the Laws of Thought</em> (London: Walton and Maberly, 1854). Georg Cantor’s work on set theory and transfinite numbers expanded the mathematical study of infinity. Richard Dedekind, <em>Was sind und was sollen die Zahlen?</em> (Braunschweig: Vieweg, 1888); Giuseppe Peano, <em>Arithmetices principia, nova methodo exposita</em> (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 <em>Dedekind−Peano axioms</em>. “Dedekind number” is already used for a different mathematical sequence, so it is not adopted here as an alternative name.</RT·endnote>
</p>
- <p>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.
- <RT·endnote>William of Ockham, <em>Summa Logicae</em> I.14−17 (c. 1323); in English as <em>Ockham’s Theory of Terms: Part I of the Summa Logicae</em>, trans. Michael J. Loux (Notre Dame: University of Notre Dame Press, 1974). The fuller critique is in the <em>Ordinatio</em> I d.2 qq.4−8. The position is called nominalism, from <em>nomen</em>, 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 <RT·Counter·read snapshot="Section·Conventio·Symbol"></RT·Counter·read> than the representation is.</RT·endnote>
- 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 <RT·Counter·read snapshot="Section·Conventio·Symbol"></RT·Counter·read>, Ockham keeps the representation and the instances below, and empties the layer above.
+ <p>Frege’s <em>Begriffsschrift</em> of 1879 supplied a formal language for logic, and his <em>Grundlagen der Arithmetik</em> 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.
+ <RT·endnote>Gottlob Frege, <em>Begriffsschrift</em> (Halle: Louis Nebert, 1879), and <em>Die Grundlagen der Arithmetik</em> (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.</RT·endnote>
</p>
- <p>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.
- <RT·endnote>René Descartes, <em>Meditationes de Prima Philosophia</em> (Paris, 1641), Meditations III and V. The position is called <RT·term>rationalism</RT·term>. 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, <em>L’Homme Machine</em> (Leyden, 1747).</RT·endnote>
- 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.
- <RT·endnote>John Locke, <em>An Essay Concerning Human Understanding</em> (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 <RT·term>empiricism</RT·term>. Locke’s denial concerns innate ideas, not the mind’s innate capacities for receiving and comparing ideas.</RT·endnote>
- </p>
+ <p>When symbolic logic and formulas dominated the debate, Naturalism was sidelined. Mathematicians and logicians conducted their arguments entirely in the syntax of formal systems, and physical observations were reduced to anecdotal framing devices.</p>
- <p>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.
- <RT·endnote>George Berkeley, <em>A Treatise Concerning the Principles of Human Knowledge</em> (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 <em>Three Dialogues between Hylas and Philonous</em> (London, 1713).</RT·endnote>
- </p>
+ <p>Alan Turing provided a notable exception in 1936. When he evaluated the foundations of computation, he began his argument by describing a person sitting at a desk with paper and a pencil. The question he posed was not what a mathematician knows, but what such a person can be made to do. Only after establishing this naturalistic observation did the formulas follow.</p>
- <p>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.
- <RT·endnote>Thomas Hobbes, <em>Leviathan</em> (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, <em>De Corpore</em> (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.</RT·endnote>
+ <p>Alonzo Church went the other way. His lambda calculus is a purely symbolic system, a calculus of function abstraction and application, and it makes no appeal to anything observable. His doctoral student Stephen Cole Kleene developed it further.
+ <RT·endnote>Alonzo Church, “A set of postulates for the foundation of logic,” <em>Annals of Mathematics</em>, second series, vol. 33 (1932): 346−366, and vol. 34 (1933): 839−864. Church offered the system as a foundation for logic. Stephen Cole Kleene and J. Barkley Rosser proved it inconsistent in 1935, and what survived was the calculus of function abstraction and application itself, later shown coextensive with Turing computability. The foundational ambition failed; the notation did not.</RT·endnote>
</p>
- <p>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.
- <RT·endnote>David Hume, <em>An Enquiry Concerning Human Understanding</em> (London, 1748), §IV part 1.</RT·endnote>
- </p>
+ <p>Yet Kleene is not the counterexample he first appears to be. His 1952 <em>Introduction to Metamathematics</em> returns to the natural setting at each of the three places where the formalism has to be tied to something outside itself:</p>
- <p>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.
- <RT·endnote>Immanuel Kant, <em>Kritik der reinen Vernunft</em> (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.</RT·endnote>
- </p>
+ <RT·Counter·step counter="RT·List·counter" id="List·Kleene_natural-settings">
+ <RT·Counter·snapshot counter="RT·List·counter" snapshot="List·Kleene_natural-settings"></RT·Counter·snapshot>
+ <ol>
+ <li>At the opening of the book, to ground one-to-one correspondence before cardinality is defined. Kleene writes, "A flock of four sheep and a grove of four trees are related to each other in a way in which neither is related to a pile of three stones or a grove of seven trees. ... Without counting the sheep or the trees, one can pair them with each other, for example by tethering the sheep to the trees, so that each sheep and each tree belongs to exactly one of the pairs."<RT·endnote>Stephen Cole Kleene, <em>Introduction to Metamathematics</em> (Amsterdam: North-Holland, 1952), 3.</RT·endnote></li>
+ <li>At the introduction of the formal system, to say what a symbol is. Kleene proposes that the symbols be thought of concretely as marks on paper, or more precisely as abstracted from our experience with marks on paper, and then observes in a parenthesis that proof theory has to remain abstract to some degree because it supposes arbitrarily long symbol sequences to be constructible, while the paper and ink in the world are finite.<RT·endnote>Kleene 1952, 62. The parenthesis is the concession this book is built on: the idealization is named as an idealization, and the reason given for it is a physical bound.</RT·endnote></li>
+ <li>At the definition of the machine, and again in its defense. Kleene does not open the chapter on computable functions with a formalism. He opens with a person computing a function by following pre-assigned instructions, using finitely many tokens, able to observe only finitely many at once and to remember only finitely many more, and derives the atomic acts of the machine from what that person can do. The subsequent defense of Turing’s thesis is conducted entirely over such a person, who ticks figures off in pencil to avoid counting them twice, and who may read a symbol on paper by eye, another in braille by hand, and take a signal by ear.<RT·endnote>Kleene 1952, 356 for the derivation, 377−381 for the defense.</RT·endnote></li>
+ </ol>
+ List <RT·Counter·read snapshot="List·Kleene_natural-settings"></RT·Counter·read>. <RT·name>Where Kleene reaches for a natural setting</RT·name>
+ </RT·Counter·step>
- <p>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.
- <RT·endnote>Nikolai Lobachevsky, “On the Principles of Geometry,” <em>Kazan Messenger</em> (1829−1830); János Bolyai, “Appendix Scientiam Spatii Absolute Veram Exhibens,” published with Farkas Bolyai’s <em>Tentamen</em> (Maros-Vásárhely, 1832); Eugenio Beltrami, “Saggio di interpretazione della geometria non-euclidea,” <em>Giornale di Matematiche</em> 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.</RT·endnote>
- 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.
- </p>
+ <p>At the same time that the dialectic appeared to have been resolved in favor of the abstract, it reappeared incarnate. The resolution had been to say that mathematics is the manipulation of symbols under rules within a formal system. But the manipulation of symbols under rules is something a device can be built to do. In declaring itself formal, mathematics had declared itself mechanizable, and a mechanism is an object with a mass and a temperature that a person can put on a bench and turn a crank on. The Forms had been driven out of the heavens and into the notation, and the notation could be cut into brass. Whatever else that is, it is a natural object, and it can be observed.</p>
- <p>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 <em>Was sind und was sollen die Zahlen?</em> (1888) described what he called a <em>simply infinite system</em>. Giuseppe Peano published a closely corresponding axiomatization in <em>Arithmetices principia, nova methodo exposita</em> (1889), acknowledging Dedekind’s earlier work.
- <RT·endnote>George Boole, <em>The Mathematical Analysis of Logic</em> (Cambridge: Macmillan, Barclay & Macmillan, 1847), and <em>An Investigation of the Laws of Thought</em> (London: Walton and Maberly, 1854). Georg Cantor’s work on set theory and transfinite numbers expanded the mathematical study of infinity. Richard Dedekind, <em>Was sind und was sollen die Zahlen?</em> (Braunschweig: Vieweg, 1888); Giuseppe Peano, <em>Arithmetices principia, nova methodo exposita</em> (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 <em>Dedekind−Peano axioms</em>. “Dedekind number” is already used for a different mathematical sequence, so it is not adopted here as an alternative name.</RT·endnote>
+ <p>Such objects were not new. The Antikythera mechanism carries a model of the heavens in bronze gearing, and it predates the argument it settles by two thousand years.
+ <RT·endnote>The shipwreck is dated to about 60 BC, which fixes the latest possible construction. Estimates of construction itself range from about 205 BC, taken from the epoch of the eclipse dials, to Derek de Solla Price’s earlier figure of about 87 BC, with the inscriptions favouring roughly 150 to 100 BC. No consensus exists on whether the device was built shortly before the wreck or well before it. The field remains active: a 2024 re-analysis of the calendar ring hole positions put the count near 354 or 355, which argues for a lunar rather than a solar ring.</RT·endnote>
+ In 1642 Blaise Pascal built a machine that added, and in doing so discovered that a carry is not free. His sautoir lifted a weight and dropped it into the next wheel, and the number of wheels a single carry could propagate through was a matter of how much force the mechanism could raise. Leibniz took up the problem thirty years later with a stepped drum that multiplied, and his carriage never worked reliably across the full width of the register. He wanted more than arithmetic from it. He wanted a <em>calculus ratiocinator</em>, a reckoning that would settle disputes by computation, which is Hobbes’s claim that reasoning is a kind of reckoning taken seriously enough to be machined. Charles Babbage then built a machine that evaluated polynomials by the method of forward differences, the same construction taken up in section <RT·Counter·read snapshot="Section·Computational_Analysis"></RT·Counter·read>, and went on to design the Analytical Engine, treated in section <RT·Counter·read snapshot="Section·Prescience_and_credit"></RT·Counter·read>.
+ <RT·endnote>Blaise Pascal began the machine in 1642 and some fifty were made. Gottfried Wilhelm Leibniz demonstrated a stepped reckoner to the Royal Society in 1673 and had a working instrument by 1694; the carry across the full register was never dependable. On the <em>calculus ratiocinator</em>, see the 1666 <em>Dissertatio de arte combinatoria</em> and the later sketches for a universal characteristic.</RT·endnote>
</p>
- <p>Frege’s <em>Begriffsschrift</em> of 1879 supplied a formal language for logic, and his <em>Grundlagen der Arithmetik</em> 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.
- <RT·endnote>Gottlob Frege, <em>Begriffsschrift</em> (Halle: Louis Nebert, 1879), and <em>Die Grundlagen der Arithmetik</em> (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.</RT·endnote>
- </p>
+ <p>Notice what the metal adds. A carry in arithmetic is instantaneous and costs nothing; a carry in Pascal’s machine lifts a weight, takes time, and can jam. Precision in analysis is unbounded; precision in gearing is whatever the shop could cut. The abstraction says nothing about any of this, and the object cannot avoid it. Everything this book has to say about tape length, about the cost of a step, and about what happens at the end cases is already present in the difference between a carry on paper and a carry in brass.</p>
- <p>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.</p>
+ <p>So is a machine an animal? Set a trilobite and a calculating engine side by side on the bench. Both are objects a person can hold, measure, and sort by the features they share, the sorting answerable to the specimens and revised whenever a specimen refuses its category. The trilobite came by its features through descent and the engine through a decision in Dorset Street, but the naturalist does not need to know which in order to begin. He needs only that the features are there to be looked at. Pick up the glass and start the notebook.</p>
</RT·section>
- </RT·section>
-
- <RT·section id="Section·Naturalism·A_thinking_machine_dissolves_the_dialectic">
- <RT·name>A thinking machine dissolves the dialectic</RT·name>
-
- <p>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.</p>
-
- <p>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.</p>
- <p>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.</p>
+ <RT·section id="Section·Naturalism·Machine_anatomy">
+ <RT·name>The anatomy of a machine</RT·name>
- <p>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.</p>
+ <p>A naturalist does not begin with a theory. He begins with two specimens and a question about how they differ. The theory, if it comes, comes out of the comparison and is answerable to it. So take the two specimens seriously and describe them.</p>
- <p>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.</p>
+ <p>The trilobite is built from a repeated part. A head shield at the front, a tail plate at the back, and between them a run of thoracic segments, each one made to the same plan and hinged to its neighbors so that the animal could roll into a ball. Down its length run three lobes, one along the axis and one to either side, which is what the name records. The eyes are compound and their lenses are calcite, so that the optics survive in the rock along with the shell. The animal moulted, which means a single individual could leave several specimens behind. The class ran from the early Cambrian to the end of the Permian and left something over twenty thousand described species, which is a very large number of variations on one body plan.</p>
- <p>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.</p>
-
- <p> Hobbes did not produced a thinking machine, but he had made the body available for mechanical explanation. In the opening of the <em>Leviathan</em>, he asked why the heart should not be a spring and the nerves so many strings. In <em>De Corpore</em>, 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.</p>
+ <p>Difference Engine No. 2 is also built from a repeated part. Two hundred and forty-eight figure wheels, every one of them made to the same plan, stacked into eight columns of thirty-one wheels each. One column holds the value of the function and the other seven hold its differences. A calculating cycle adds the seventh difference into the sixth, the sixth into the fifth, and so on down the line, the odd columns moving on one half of the cycle and the even columns on the other. A wheel passing from nine to zero trips a nib that latches a warning, and the carriage that acts on the warning is the part Babbage worked hardest on. The whole is some eight thousand parts of bronze, cast iron, and steel, five tons of it, eleven feet long, and it computes a seventh-order polynomial to thirty-one places.
+ <RT·endnote>Doron Swade, <em>Charles Babbage’s Difference Engine No. 2: Technical Description</em> (London: Science Museum, 1996 and later online editions), and the accompanying User Manual, for the figure wheel count, the column arrangement, the odd and even difference cycle, and the warning and carriage mechanism. On the anticipating carriage, A. G. Bromley, “Charles Babbage’s Analytical Engine, 1838,” <em>Annals of the History of Computing</em> 4, no. 3 (1982). The engine was completed to Babbage’s drawings at the Science Museum in 2002 and a second was built for the Computer History Museum in 2008.</RT·endnote>
+ </p>
- <p>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.
- <RT·endnote>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; <em>Pensées</em> (Paris, 1670), §277 in the Brunschvicg numbering.</RT·endnote>
- 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.
- </p>
+ <p>Two specimens, and the same observation about each. There is a unit, the unit is repeated, and the repetitions are differentiated by where they sit. In the animal, a segment near the head does something a segment near the tail does not. In the engine, the column holding the seventh difference does something the column holding the function value does not. Neither difference is a difference in the plan of the part. It is a difference in position, and position is doing the work.</p>
- <p>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.</p>
+ <p>This observation has a name, and it was given one in the same city in the same decade the engine was being drawn. Richard Owen, lecturing at the Royal College of Surgeons in 1843, separated two kinds of resemblance that had been running together. An <RT·term>analogue</RT·term> he defined as a part in one animal that has the same function as a part in another. A <RT·term>homologue</RT·term> he defined as the same organ in different animals under every variety of form and function. The repetition of a part within one organism he called <RT·term>serial homology</RT·term>.
+ <RT·endnote>Richard Owen, <em>Lectures on the Comparative Anatomy and Physiology of the Invertebrate Animals</em> (London: Longman, Brown, Green & Longmans, 1843), 374 and 379 for the two definitions. Owen distinguished special homology, holding between species, from serial homology, holding between the repeated parts of one organism, and from general homology, relating a part to the archetype. His criteria for deciding a homology were position, development, and composition. The account is pre-evolutionary: the archetype is a plan, not an ancestor. Darwin supplied the ancestor in 1859 and the vocabulary survived the substitution intact, which is the point being borrowed here.</RT·endnote>
+ </p>
- <p>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.
- <RT·endnote>The Stepped Reckoner, or <em>Staffelwalze</em>. Leibniz, “Explication de l’Arithmétique Binaire,” <em>Mémoires de l’Académie Royale des Sciences</em> (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.</RT·endnote>
- 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.
- </p>
+ <p>Notice that Owen’s distinction is doing work in this book already, under other names. Two architectures that compute the same class of functions are analogues: same function, different parts. That is what <RT·term>Turing Complete</RT·term> asserts, and it asserts nothing about how either machine is built. A transform that carries one machine to another while leaving every computation theoretic question with the same answer is asserting a homology: the same organ, under a variety of form. Section <RT·Counter·read snapshot="Section·Three_conditions"></RT·Counter·read> observes that the second requirement is the stronger of the two, and Owen would have said the same of his pair. The wing of a bat and the wing of a bee are analogues and tell you nothing about each other’s construction. The wing of a bat and the arm of a man are homologues, and either one can be read off the other bone by bone.</p>
- <p>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.
- <RT·endnote>The <em>characteristica universalis</em> and the <em>calculus ratiocinator</em>. The <em>calculemus</em> 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.</RT·endnote>
- 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.
- </p>
+ <p>That is why the comparison is worth making rather than merely amusing. A naturalist who has only analogy has a list of things that do the same job. A naturalist who has homology has a method: he can carry a finding from one specimen to another and know what he is entitled to carry. The transforms in the chapters that follow are homologies, and the argument that they are inconsequential is the argument that the carrying is legitimate.</p>
- <p>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.</p>
+ <p>Where does the comparison break? In two places, and both are instructive. The trilobite’s plan arrived by descent and the engine’s by a decision, so the engine has a designer to interrogate and the animal does not. Babbage left drawings; the Cambrian left none. But this is a difference in how the specimen came to be, not in what can be observed about it, and Owen was doing homology sixteen years before anyone had an account of descent to hang it on.</p>
- <p>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.</p>
+ <p>The second break runs the other way, and it favors the machine. A trilobite is a cast in rock. Its anatomy can be read and its behavior can only be inferred, so the naturalist argues about whether the enrollment was for defense and cannot settle it. The engine still turns. A person can set the initial differences, crank it, and watch the carry propagate through a column of nines, and if he wants to know what it will do he does not have to guess. As a subject for a naturalist, a working machine is a better animal than an animal.</p>
- <p>Newton’s mathematics can be read in this way. I will call his calculus a <em>rotational language</em>, 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.</p>
+ <p>And here the naturalist runs out of notebook. Morphology can record that a part exists and where it sits. It cannot record what the part does when the crank turns, in what order, at what time, and to what effect. Describing a running machine takes more than a description of its pieces. It takes a language for the values its parts hold, for the arrangement of the parts, for the conditions under which one motion is selected rather than another, and for the quantities that motion generates. Each of those languages had been built, by people who did not think they were building an instrument for observing machines.</p>
- <p>Newton himself placed geometry inside mechanics. In the preface to the <em>Principia</em> 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.”
- <RT·endnote>Isaac Newton, <em>Philosophiæ Naturalis Principia Mathematica</em> (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.</RT·endnote>
- 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.
- </p>
+ <p>Arithmetic describes the discrete states: the position of a digit wheel, the number standing on a column. Geometry describes the arrangement of the parts. Boole’s algebra, published while the engine was on the drawing board, describes conditional combination, so that which transition is selected can be written and computed rather than merely observed.
+ <RT·endnote>George Boole, <em>The Mathematical Analysis of Logic</em> (Cambridge: Macmillan, Barclay & Macmillan, 1847), and <em>An Investigation of the Laws of Thought</em> (London: Walton and Maberly, 1854). The connection between this algebra and switching circuits was not drawn until Claude Shannon, “A Symbolic Analysis of Relay and Switching Circuits,” <em>Transactions of the American Institute of Electrical Engineers</em> 57 (1938): 713−723. Boole is included here as the language of conditional combination, not as a theory of machines.</RT·endnote>
+ </p>
- <p>Leibniz’s marks for the differential and integral remain among the most successful mathematical representations ever invented, but his proposed <em>characteristica</em> 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.</p>
+ <p>Newton supplied the language for the motion itself. I will call his calculus a rotational language, not because he used the phrase and not because all calculus is a theory of gears, but because it gives mathematical form to quantities generated by motion, and a wheel is where that idea is easiest to see. The angular position of a wheel is a state. Its turning is a change of state. Its rate of turning is a rate of change. The same notation carries from a wheel to a planet, from a planet to a falling body, and from a physical motion to an abstract variable, and it carries in that direction because it started at the physical end. Newton placed geometry inside mechanics in the preface to the <em>Principia</em>, calling it the part of universal mechanics that states and demonstrates accurately, and described natural philosophy as reasoning from the phenomena of motion to the forces, and from the forces back to the phenomena.
+ <RT·endnote>Isaac Newton, <em>Philosophiæ Naturalis Principia Mathematica</em> (London, 1687), Author’s Preface; in English, Andrew Motte’s translation of 1729. Newton’s own distinction is between rational mechanics, which proceeds by demonstration, and practical mechanics, which belongs to the manual arts. The wording of the familiar sentence about geometry being founded in mechanical practice belongs to a particular translation and should be quoted only with the edition named. The point taken here is narrower than the sentence: the mathematical description is not detached from motion, it is the part of mechanics that expresses what motion does.</RT·endnote>
+ </p>
- <p>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.
- <RT·endnote>George Boole, <em>The Mathematical Analysis of Logic</em> (Cambridge: Macmillan, Barclay & Macmillan, 1847), and <em>An Investigation of the Laws of Thought</em> (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,” <em>Transactions of the American Institute of Electrical Engineers</em> 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.</RT·endnote>
- 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.
- </p>
+ <p>The last of the languages was built by the man who built the engine, and built because the engine could not be designed without it. In March 1826 Babbage read a paper to the Royal Society on expressing the action of machinery by signs. He called the system his Mechanical Notation, and it recorded three things that no drawing records together: the shape and relative position of every piece, the time and duration of every motion, and the connection of each moving piece to every other it acts upon, which he called the trains. He later wrote that it had given a new demonstrative science, namely that of proving whether a given machine can exist and, if it can, that it will accomplish what it was meant to. He said he could not have invented the Analytical Engine without it, and ranked it among the most important additions he had made to human knowledge.
+ <RT·endnote>Charles Babbage, “On a Method of Expressing by Signs the Action of Machinery,” <em>Philosophical Transactions of the Royal Society of London</em> 116 (1826): 250−265, read 16 March 1826. The three kinds of information are enumerated in <em>Passages from the Life of a Philosopher</em> (London: Longman, Green, Longman, Roberts, & Green, 1864), chap. IX, which is also where the claim about a new demonstrative science appears; chap. V for the dependence of the Analytical Engine upon it, and chap. XXXIII for his own ranking of it. Babbage’s claim is stronger than it is usually reported to be. He does not say the notation described his machines. He says it changed what could be demonstrated about a machine that did not yet exist.</RT·endnote>
+ </p>
- <p>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.
- <RT·endnote>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.</RT·endnote>
- 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.
- </p>
+ <p>Read the list again and notice what it is. Arithmetic for the states, geometry for the arrangement, logic for the selection of transitions, calculus for the motion, and a mechanical notation for the parts, their times, and the paths by which force passes between them. Nobody set out to assemble an instrument for observing machinery. What was assembled is an instrument for observing machinery.</p>
- <p>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.
- <RT·endnote>Charles Babbage, “A Note on the Application of Machinery to the Computation of Astronomical and Mathematical Tables,” <em>Memoirs of the Astronomical Society</em> 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.</RT·endnote>
- 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.
- <RT·endnote>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 <RT·Counter·read snapshot="Section·Computational_Analysis·Evaluating_and_Extending"></RT·Counter·read> and the difference-table sections that follow it.</RT·endnote>
- </p>
+ <p>This is where the thesis of this book takes its precise form. Mathematics is the language in which observations of machines are recorded. It is abstract, as a language must be, and its subject matter is the behavior of a natural object. The Realm of Forms was never found above the world, and Naturalism was never confined to heaps of passive objects. What the philosophers placed in a separate realm, and what their opponents tried to reduce to sensation, turns out to be the notation in which a person writes down what the thing on the bench will do when he turns the crank.</p>
- <p>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.
- <RT·endnote>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.”</RT·endnote>
- 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.
- </p>
+ </RT·section>
- <p>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 <em>Mechanical Notation</em>. 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.
- <RT·endnote>Charles Babbage, “On a Method of Expressing by Signs the Action of Machinery,” <em>Philosophical Transactions of the Royal Society of London</em> 116 (1826): 250−265, read 16 March 1826. Babbage’s later account in <em>Passages from the Life of a Philosopher</em>, chap. IX, enumerates the three kinds of information: shape and position, time and duration, and mechanical connection or “trains.”</RT·endnote>
- </p>
+ <RT·section>
+ <RT·name>The definition of Computational Naturalism</RT·name>
- <p>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.</p>
+ <p>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.
+ <RT·endnote>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 <em>Grundlage der gesammten Wissenschaftslehre</em> (Leipzig, 1794), §§1−3, and Heinrich Moritz Chalybäus fixed the three names upon it in the <em>Historische Entwicklung der speculativen Philosophie von Kant bis Hegel</em> (Dresden, 1837), from which they passed into general use. Hegel’s own term is <em>Aufhebung</em>, 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.</RT·endnote>
+ 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 <em>temperature</em>. Sometimes understanding the deeper structure dissolves the debate altogether.
+ </p>
- <p>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.
- <RT·endnote>Charles Babbage, <em>Passages from the Life of a Philosopher</em> (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.”</RT·endnote>
- 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.
- </p>
+ <p>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.</p>
- <p>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.
- <RT·endnote>L. F. Menabrea, “Notions sur la machine analytique de M. Charles Babbage,” <em>Bibliothèque Universelle de Genève</em> 82 (1842); translated and expanded by Ada Augusta, Countess of Lovelace, in Richard Taylor, ed., <em>Scientific Memoirs</em>, vol. 3 (London, 1843), 666−731, especially Note G.</RT·endnote>
- </p>
- <p>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.</p>
+ <p> Computational Naturalism holds that a machine is a natural phenomenon and can be observed, and that computation theory is a useful notation for cataloging these observations. However, within the study of Computational Naturalism there arises a metamathematics and layers of abstraction that do the work the Realm of Forms was posited to do. Plato described that realm and never defined it, and Computational Naturalism supplies a definition for at least some of what he described. Thus Computational Naturalism does not debate the question as to which view is correct, but instead holds that they are two aspects of the same thing.
+ <RT·endnote>Aristotle is ancestor to the biologist’s meaning and only distantly to the philosophical one, holding as he did that the first mover is no part of nature; <em>Metaphysics</em> XII.7. The metaphysical doctrine is a nineteenth-century development.</RT·endnote>
+ </p>
- <p>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.</p>
- <p>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.</p>
+ </RT·section>
</RT·section>
<RT·Counter·snapshot counter="RT·Code·counter" snapshot="Code·five_AST"></RT·Counter·snapshot>
<RT·code>
(defparameter *five-ast*
- '(progn
+ ’(progn
(write s) (step)
(write s) (step)
(write s) (step)
<RT·Counter·snapshot counter="RT·Code·counter" snapshot="Code·three_AST"></RT·Counter·snapshot>
<RT·code>
(defparameter *three-ast*
- '(progn
+ ’(progn
(write s) (step)
(write s) (step)
(write s) (step)
(
(and
(cdr cmds)
- (equal (first cmds) '(write s))
- (equal (second cmds) '(step)))
+ (equal (first cmds) ’(write s))
+ (equal (second cmds) ’(step)))
(append
- '((left) (write □))
+ ’((left) (write □))
(invert-direction (cddr cmds))
))
(
)
;; *primitive-five-minus-3-ast* evaluates to:
- ;; '(progn
+ ;; ’(progn
;; (write s) (step)
;; (write s) (step)
;; (write s) (step)
(
(and
(cdr cmds)
- (equal (first cmds) '(step))
- (equal (second cmds) '(left)))
+ (equal (first cmds) ’(step))
+ (equal (second cmds) ’(left)))
(remove-annihilations (cddr cmds)))
(
(and
(cdr cmds)
- (equal (first cmds) '(write s))
- (equal (second cmds) '(write □)))
+ (equal (first cmds) ’(write s))
+ (equal (second cmds) ’(write □)))
(remove-annihilations (cddr cmds)))
(
t
)
;; *simplified-five-minus-3-ast* evaluates to:
- ;; '(progn
+ ;; ’(progn
;; (write s) (step)
;; (write s) (step))
</RT·code>
</RT·section>
+<!--------------------------------------------------------------------------------->
+ <RT·section id="Section·Appendix_dialectic_to_Kant">
+ <RT·name>Appendix: The dialectic from the Athenian decline to Kant</RT·name>
+
+ <p>Section <RT·Counter·read snapshot="Section·Naturalism·The_argument_since"></RT·Counter·read> gives the short account of what became of the argument between Plato and Aristotle. This appendix gives the long one. It is arranged by date, with one exception noted where it occurs. A reader who accepts the summary loses nothing by skipping it.</p>
+
+ <p>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.
+ <RT·endnote>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 <em>Theology of Aristotle</em>, an adaptation of parts of Plotinus. See Cristina D’Ancona, “Greek Sources in Arabic and Islamic Philosophy,” <em>Stanford Encyclopedia of Philosophy</em>.</RT·endnote>
+ </p>
+
+ <p>
+ 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.
+ <RT·endnote>Al-Ghazali, <em>Tahāfut al-Falāsifa</em>, 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.</RT·endnote>
+ </p>
+
+ <p>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.
+ <RT·endnote>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,” <em>Stanford Encyclopedia of Philosophy</em>.</RT·endnote>
+ </p>
+
+ <p>Here the arrangement by date is set aside once, to introduce the question the schoolmen were answering. The specifically medieval form of it was shaped by Porphyry’s <em>Isagoge</em>, an introduction to Aristotle’s <em>Categories</em>. 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.
+ <RT·endnote>Porphyry, <em>Isagoge</em>, in John Peter Anton and George L. Kustas, eds., <em>Essays in Ancient Greek Philosophy</em>, 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.</RT·endnote>
+ </p>
+
+ <p>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.
+ <RT·endnote>William of Ockham, <em>Summa Logicae</em> I.14−17 (c. 1323); in English as <em>Ockham’s Theory of Terms: Part I of the Summa Logicae</em>, trans. Michael J. Loux (Notre Dame: University of Notre Dame Press, 1974). The fuller critique is in the <em>Ordinatio</em> I d.2 qq.4−8. The position is called nominalism, from <em>nomen</em>, 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 <RT·Counter·read snapshot="Section·Conventio·Symbol"></RT·Counter·read> than the representation is.</RT·endnote>
+ 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 <RT·Counter·read snapshot="Section·Conventio·Symbol"></RT·Counter·read>, Ockham keeps the representation and the instances below, and empties the layer above.
+ </p>
+
+ <p>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.
+ <RT·endnote>Thomas Hobbes, <em>Leviathan</em> (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, <em>De Corpore</em> (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.</RT·endnote>
+ </p>
+
+ <p>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.
+ <RT·endnote>René Descartes, <em>Meditationes de Prima Philosophia</em> (Paris, 1641), Meditations III and V. The position is called <RT·term>rationalism</RT·term>. 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, <em>L’Homme Machine</em> (Leyden, 1747).</RT·endnote>
+ 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.
+ <RT·endnote>John Locke, <em>An Essay Concerning Human Understanding</em> (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 <RT·term>empiricism</RT·term>. Locke’s denial concerns innate ideas, not the mind’s innate capacities for receiving and comparing ideas.</RT·endnote>
+ </p>
+
+ <p>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.
+ <RT·endnote>George Berkeley, <em>A Treatise Concerning the Principles of Human Knowledge</em> (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 <em>Three Dialogues between Hylas and Philonous</em> (London, 1713).</RT·endnote>
+ </p>
+
+ <p>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.
+ <RT·endnote>David Hume, <em>An Enquiry Concerning Human Understanding</em> (London, 1748), §IV part 1.</RT·endnote>
+ </p>
+
+ <p>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.
+ <RT·endnote>Immanuel Kant, <em>Kritik der reinen Vernunft</em> (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.</RT·endnote>
+ </p>
+
+ <p>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.
+ <RT·endnote>Nikolai Lobachevsky, “On the Principles of Geometry,” <em>Kazan Messenger</em> (1829−1830); János Bolyai, “Appendix Scientiam Spatii Absolute Veram Exhibens,” published with Farkas Bolyai’s <em>Tentamen</em> (Maros-Vásárhely, 1832); Eugenio Beltrami, “Saggio di interpretazione della geometria non-euclidea,” <em>Giornale di Matematiche</em> 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.</RT·endnote>
+ 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.
+ </p>
+
+ </RT·section>
+
<!--------------------------------------------------------------------------------->
<RT·section id="Section·Appendix_Stirling_Numbers">
<RT·name>Appendix: Stirling Numbers</RT·name>
projects / TM-2026 / commitdiff
