From: Thomas Walker Lynch Date: Sun, 23 Aug 2026 14:07:32 +0000 (+0000) Subject: . X-Git-Url: https://git.reasoningtechnology.com/realizable_reverse.png?a=commitdiff_plain;h=fe083d4cd45c9d07528d38747f1a58e888f256a5;p=TM-2026 . --- diff --git a/document/book/TM-2026.html b/document/book/TM-2026.html index e87a696..11eaa36 100644 --- a/document/book/TM-2026.html +++ b/document/book/TM-2026.html @@ -24,7 +24,7 @@ @@ -37,7 +37,7 @@ Exordium -

My colleagues in computer arithmetic work tirelessly to confirm Schopenhauer’s thesis, so I am left to chose among only unfunded inconsequential problems that no one else wants.

+

My colleagues making computers do arithmetic work tirelessly to confirm Schopenhauer’s thesis, so I am left to chose among only unfunded inconsequential problems that no one else wants.

So perhaps we can discuss this one. Turing began with a Naturalist model, a clerk at a desk, yet the machine he described cannot be built. What follows completes the foundation he began. The subject of Computational Naturalism lies at the intersection of number theory, numerical analysis, computation theory, and computer architecture.

@@ -423,11 +423,11 @@ - Naturalism + Aristotlism -

Plato’s description of the Realm of Forms resembles religious teachings. The perfection of the circle hearkens to Buddhism. The Realm of Forms resembles religious teachings of heaven, and the allegory of the cave describes an earthly realm below it, or even of an underworld where people’s minds are kept in darkness. If not religious, it has a allure of mysticism and seems to require some element of belief rather than science.

+

Plato’s description of the Realm of Forms resembles religious teachings. The perfection of the circle hearkens to Buddhism. The Realm of Forms resembles religious teachings of heaven, and the allegory of the cave describes an earthly realm below it, or even of an underworld where people’s minds are kept in darkness. If not religious, it has the allure of mysticism and seems to require some element of belief.

-

Aristotle (384−322 BC) arrived at Plato’s Academy at seventeen and stayed for twenty years. He was the son of a physician, and he had from the beginning the disposition of a man who wants to see the thing itself. Where Plato was brilliant and intuitive, Aristotle was scholarly and methodical. +

Aristotle (384−322 BC) arrived at Plato’s Academy when he was seventeen and stayed for twenty years. He was the son of a physician, and he had from the beginning the disposition of a man who wants to see the thing itself. Where Plato was brilliant and intuitive, Aristotle was scholarly and methodical. Will Buckingham, Douglas Burnham, Peter J. King, John Marenbon, Clive Hill, and Marcus Weeks, The Philosophy Book, Big Ideas Simply Explained (New York: DK, 2011), page 58. The difference shows in what each took to be the starting point of an inquiry.

@@ -435,7 +435,7 @@ Aristotle, Metaphysics I.9 990b−991b, with the remark on poetical metaphors at 991a20−22, and again at XIII.4−5. Aristotle does use the word eidos, ordinarily translated as form, for the structure in virtue of which a thing is what it is. That structure is in the thing and nowhere else, and it is not Plato’s Form under another name. One English word is doing service for two concepts, and only the first is at issue here. Take away the Realm of Forms, and there can be no shadows, nor a cave to be let out of.

-

He then turned Plato upside down. For Plato, knowledge descends from on high: a person conceptualizes the Forms, and the senses report shadows that mislead more than they inform. For Aristotle, knowledge ascends: observation of particulars is where an inquiry starts, and many observations lead to the deduction of what they have in common. +

He then turned Plato upside down. For Plato, knowledge descends from on high: a person conceptualizes the Forms, and the senses report shadows that mislead as much as they inform. For Aristotle, knowledge ascends: observation of particulars is where an inquiry starts, and many observations lead to the deduction of what they have in common. Aristotle, Posterior Analytics II.19, for knowledge beginning in perception and rising by induction to the principles. The Greek term for the operation is epagōgē, rendered as induction.

@@ -443,142 +443,93 @@ Aristotle, Historia Animalium 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 Parts of Animals, 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., The Life and Letters of Charles Darwin (London: John Murray, 1887), vol. 3, 252. What he produced from it was a taxonomy: animals sorted by the features they share, the sorting answerable to the specimens and revised whenever a specimen refused its category.

-

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 Naturalism. In modern times term takes on two meanings, both that trace back to Aristotle, that of the biologist who studies nature, and that of the philosophy that the language of science and mathematics is the language of talking about nature.

+

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 Naturalism. In modern times the term takes on two meanings, both that trace back to Aristotle, that of the biologist who studies nature, and that of the philosophy that the language of science and mathematics is the language of talking about nature.

 - - The long transmission + + The argument since -

The Greek inheritance did not pass to the Enlightenment directly. It passed through Syriac Christian translators and then, from the eighth century, through the Arabic-speaking world, where Hunayn ibn Ishaq (809−873) and his circle rendered Aristotle, Galen, and Euclid into Arabic under Abbasid patronage. What arrived was not Aristotle alone but Aristotle layered with late-antique commentary, and with Neoplatonic works circulating under his name. - Notably the Theology of Aristotle, in fact an adaptation of Plotinus. See “Greek Sources in Arabic and Islamic Philosophy,” Stanford Encyclopedia of Philosophy. The mixing was consequential: a reader of that corpus met Plato and Aristotle already partly reconciled, and did not always know which he was reading. -

- -

Al-Kindi (c. 801−873) put the Greek apparatus to work describing a single creating First Cause, and so began the long project of fitting demonstration to revelation. Al-Farabi (c. 872−950) drew the distinction most useful here, between knowing a thing as it is and knowing it through a symbolic representation addressed to those who cannot follow a demonstration. - “al-Farabi’s Philosophy of Society and Religion,” Stanford Encyclopedia of Philosophy. The treatise On the Harmonization of the Opinions of the Two Sages, long ascribed to al-Farabi and directly concerned with reconciling Plato and Aristotle, is of disputed authorship; see Marwan Rashed, Arabic Sciences and Philosophy 19, no. 1 (2009): 43−82. - A representation, on that account, is neither the thing nor an arbitrary substitute for it, which is the reading section depends on.

- -

Ibn al-Haytham (c. 965−1040) went further than distinguishing knowledge from representation and insisted that a claim about the world be checked against the world, building apparatus to do it; his Book of Optics settles by experiment that light travels to the eye rather than from it. - Ibn al-Haytham, Kitāb al-Manāẓir, c. 1011−1021, translated into Latin as the De aspectibus and known to Bacon, Witelo, and Kepler. The dark chamber described there is the ancestor of the camera obscura. - He is a naturalist in both senses of the word this book uses, three centuries before the word existed in either.

- -

Ibn Sina (980−1037) rebuilt Aristotle into a system turning on the distinction between what exists necessarily and what exists contingently, and made metaphysics the science of being as such. Al-Ghazali (1058−1111) then attacked the philosophers where it counts for this chapter, denying that fire has any power to burn cotton and holding that what looks like a natural cause is God acting directly at each occasion. +

From the eighth century Greek texts reached Syriac Christian translators and then Arabic-speaking scholars rendered Aristotle, Euclid, and Galen into Arabic, and the problem those scholars faced was how to fit demonstration to faith. + Hunayn ibn Ishaq (809−873) and his circle were central to the translation movement under Abbasid patronage. What arrived was not Aristotle alone but Aristotle layered with late-antique commentary, and with Neoplatonic works circulating under his name, notably the Theology of Aristotle, in fact an adaptation of Plotinus. See “Greek Sources in Arabic and Islamic Philosophy,” Stanford Encyclopedia of Philosophy. + Al-Farabi (c. 872−950) distinguished knowing a thing as it is from knowing it through a representation addressed to those who cannot follow a demonstration, which is the reading section depends on. Ibn al-Haytham (c. 965−1040) built apparatus and settled by experiment that light travels to the eye rather than from it, making him a naturalist in both senses of the word three centuries before the word existed in either. Al-Ghazali (1058−1111) put the sharpest denial of Naturalism that anyone has written, holding that fire has no power to burn cotton and that what looks like a natural cause is God acting directly at each occasion. Al-Ghazali, Tahāfut al-Falāsifa, discussion 17. The position is called occasionalism, and it prefigures by six hundred years Hume’s argument that we observe succession and never necessity. - That is the sharpest denial of Naturalism anyone has written, since it does not dispute what is observed but denies that anything observed has any power of its own.

- -

Ibn Rushd (1126−1198) answered him, holding that a demonstrated conclusion cannot conflict with a revealed truth, and that where the two appear to conflict the reading of the text is what wants revision. His commentaries reached Latin Europe through the translators at Toledo, where Gerard of Cremona (c. 1114−1187) and others turned the Arabic corpus into Latin, and Aristotle re-entered a Europe that had lost him.

- -

What Europe then argued about for four hundred years was the same question in new dress, under the name of the problem of universals. Peter Abelard (1079−1142) held that what the many share is not a thing but what a word signifies. Thomas Aquinas (1225−1274) took the Aristotelian line, that the universal is in the particular and is separated only by the mind considering it. William of Ockham (c. 1287−1347) took the shortest road, holding that only particulars exist and a universal is a sign standing for many of them. - Ockham’s position is called nominalism, from nomen, a name. It is the third answer, and neither Greek had proposed it: the universal is neither in a realm nor in the thing but is a mark that stands for many things. - It is worth pausing on that third answer, since it locates what the many share in a symbol, and a symbol is something someone has to write down.

- + Ibn Rushd (1126−1198) answered that a demonstrated conclusion cannot conflict with a revealed truth, and that where the two appear to conflict it is the reading of the text that wants revision.

- - Age of enlightment - - - - Age of enlightment - -

The Aristotelian account was the working position of the schools for most of two thousand years. Thomas Hobbes (1588−1679) made geometry a science of bodies and of the motions that generate them. - Thomas Hobbes, De Corpore (London, 1655), Part II. He defended the position through a public quarrel with John Wallis that ran over twenty years; see Douglas M. Jesseph, Squaring the Circle: The War between Hobbes and Wallis (Chicago: University of Chicago Press, 1999). - Isaac Barrow (1630−1677) argued in his Cambridge lectures that mathematical magnitudes are abstracted from the magnitudes of physical bodies and have no other source. - Isaac Barrow, Lectiones Mathematicae, delivered 1664−1666 and published (London, 1683), Lectures I−VII. Barrow held the Lucasian chair before Newton, who succeeded him in 1669. +

His commentaries reached Latin Europe through the translators at Toledo, and Aristotle re-entered a continent that had lost him. What Europe then argued for four hundred years was the same question under a new name, the problem of universals, and William of Ockham (c. 1287−1347) gave it a third answer that neither Greek had proposed: only particulars exist, and a universal is a sign standing for many of them. + The position is called nominalism, from nomen, a name. It is worth pausing on, since it locates what the many share in a symbol, and a symbol is something somebody has to write down.

-

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. - A man who says that geometry is a branch of mechanics is saying that the forms are in the machinery. It is worth recalling that the vocabulary of the period made the claim easy to state. The study of the physical world was called natural philosophy, and Newton’s book of 1687 is the Mathematical Principles of Natural Philosophy. The word scientist was not coined until 1834. - William Whewell coined it in an unsigned review of Mary Somerville, On the Connexion of the Physical Sciences, Quarterly Review 51 (1834): 54−68, and put it into print under his own name in The Philosophy of the Inductive Sciences (London: Parker, 1840), vol. 1, cxiii. The older term survives in the chairs of natural philosophy at the Scottish universities. - The vocabulary changed before the claim was settled.

+

European Enlightenment philosophers anxiously took up the discussion.

-

The eighteenth century turned against the Platonic position from two directions at once. George Berkeley (1685−1753) denied that there are abstract general ideas at all, and then turned that denial on the calculus in The Analyst of 1734, where the vanishing increments of the new analysis are asked what they are and found to be neither finite quantities, nor nothing, but the ghosts of departed quantities. - George Berkeley, A Treatise Concerning the Principles of Human Knowledge (Dublin, 1710), Introduction §§7−25, against abstract ideas; The Analyst (London, 1734), §35 for the phrase. Berkeley’s target was the reasoning and not the results, and the objection stood unanswered until the rigorization of the following century. - The attack was fatal to complacency and did not restore the naturalist account, since Berkeley’s own position placed the objects in the mind rather than in the world. David Hume (1711−1776) then divided all inquiry into relations of ideas and matters of fact, and placed mathematics squarely among the former, where nothing observed bears on it. - David Hume, An Enquiry Concerning Human Understanding (London, 1748), §IV part 1. - An empiricist had thus removed mathematics from the reach of experience.

- - - - The turn inward +

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), who succeeded Barrow in the Lucasian chair, 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. + A man who says that geometry is a branch of mechanics is saying that the forms are in the machinery.

-

Immanuel Kant (1724−1804) settled the matter for the next century in the Critique of Pure Reason of 1781. Mathematical judgements are necessary, so they are not got from experience; they are also not empty, since seven and five taken together yield a twelve that no analysis of the concepts of seven, five, and sum will produce. They are therefore synthetic and a priori both, and what makes them possible is the pure intuition of space and time, which is a form contributed by the knowing subject rather than a feature found in the world. - Immanuel Kant, Kritik der reinen Vernunft (Riga: Hartknoch, 1781; second edition 1787), Introduction B14−B17 for the arithmetical example, and the Transcendental Aesthetic for space and time as forms of intuition. +

René Descartes (1596−1650) took the other road, holding that certain knowledge comes from reason reflecting on itself rather than from the senses, which deceive. His model for that certainty was mathematics, and the ideas he trusted most were those he took to be innate, present in the mind before any experience of the world, which is Plato’s recollection in modern dress. + 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).

-

It is worth being exact about what this does. Kant does not deny that mathematics is grounded. He relocates the ground, from the world to the structure of the mind that apprehends the world, and in doing so he answers the question the naturalist had been trying to answer, and answers it better than the naturalist then could. Geometry is necessary because space is our form of outer intuition, and it applies to everything we can experience because we can experience nothing outside that form. For a hundred and thirty years the position looked unassailable.

- -

What unmade it was not an argument but a discovery. Nikolai Lobachevsky (1792−1856) published a geometry in which the parallel postulate fails, in 1829; János Bolyai (1802−1860) published another in 1832; Carl Friedrich Gauss (1777−1855) had reached the same results earlier and withheld them. Bernhard Riemann (1826−1866) generalized the question in 1854, and Eugenio Beltrami (1835−1900), whose model is discussed in chapter , showed in 1868 that the new geometry is consistent if the old one is. - N. I. Lobachevsky, “On the Principles of Geometry,” Kazan Messenger, 1829−1830; János Bolyai, appendix to Farkas Bolyai, Tentamen (Maros-Vásárhely, 1832); Bernhard Riemann, “Über die Hypothesen, welche der Geometrie zu Grunde liegen,” delivered 1854, published Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen 13 (1868): 133−152; Eugenio Beltrami, “Saggio di interpretazione della geometria non-euclidea,” Giornale di Matematiche 6 (1868): 284−312. - If there are several consistent geometries and physical space satisfies at most one of them, then geometry is not the science of space, and the necessity Kant had explained is not there to be explained. The best example either party had of mathematics answering to the world had been taken off the table, and it was taken off by mathematicians rather than by philosophers.

- - - - The last stand and the demolition - -

The fullest naturalist account of number ever written appeared in the middle of this, and is therefore later than most readers expect. John Stuart Mill (1806−1873) argued in A System of Logic of 1843 that the truths of arithmetic are inductive generalizations from experience, of the same kind and standing as the generalizations of any other science. Two and one make three is a fact about collections of objects, learned the way facts about collections of objects are learned, and it is necessary only in the sense that nothing has ever contradicted it. - John Stuart Mill, A System of Logic, Ratiocinative and Inductive (London: Parker, 1843), Book II, chaps. 5−6, and Book III, chap. 24. +

While the philosophers argued, others built. 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. It added and subtracted, subtraction being done by complement, and carried automatically across digits. 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.

-

Gottlob Frege (1848−1925) destroyed the position in the Grundlagen der Arithmetik of 1884, and did so at leisure and by name. If a number is a property of a heap of things, then 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. If arithmetic rests on what has been observed, then its necessity is the necessity of a habit, and a sufficiently strange experience would revise it. And if numbers are ideas, then they are episodes in somebody’s head, and my two and your two are two different objects, and nothing has been said about number at all. This last charge Frege named psychologism, and it became the period’s term of abuse. - Gottlob Frege, Die Grundlagen der Arithmetik (Breslau: Koebner, 1884), Introduction and §§7−10 against Mill, §§21−25 against number as a property of external things, and §§26−27 against number as an idea. Frege pressed the charge again in his review of Husserl, Zeitschrift für Philosophie und philosophische Kritik 103 (1894): 313−332, with enough effect that Husserl abandoned the position. +

Baruch Spinoza (1632−1677) removed the distance between God and the world by identifying them, holding that there is one substance, called God or Nature indifferently, and that everything follows from it by necessity. He set the argument out in geometric order, with definitions, axioms, propositions, and demonstrations, so that mathematics supplied not only the model of certainty but the very form of the writing. + Baruch Spinoza, Ethica Ordine Geometrico Demonstrata (Amsterdam, 1677), Part I. The phrase Deus sive Natura, God or Nature, appears in Part IV, preface.

-

The demolition was thorough and it was, on its own terms, correct. Every one of those objections is answered in this book, and none of them is answered by disputing what Frege said about Mill. They are answered by giving numbers a source that Mill did not have available to him, which is the subject of chapter , and by an account of what a symbol is that makes my two and your two the same object, which is the subject of chapter . Mill was arguing from heaps of pebbles. There was no other mechanism then to argue from.

- -

After 1884 the field belonged to programmes that dispensed with the world. Logicism derived mathematics from logic; formalism treated it as the manipulation of marks under stated rules; intuitionism grounded it in mental construction. Richard Dedekind (1831−1916) wrote in 1888 that numbers are free creations of the human mind, and the remark drew no objection. - Richard Dedekind, Was sind und was sollen die Zahlen? (Braunschweig: Vieweg, 1888), preface. Leopold Kronecker’s better-known remark that God made the whole numbers and all else is the work of man is reported by Heinrich Weber, “Leopold Kronecker,” Jahresbericht der Deutschen Mathematiker-Vereinigung 2 (1893): 5−31, at 19. - Formalism in particular gained by not looking at the world, since a formal system is answerable to its own rules alone and is very much the easier to study for it. The crisis recounted in chapter was fought entirely among these three, and no party to it proposed consulting an apparatus.

+

John Locke (1632−1704) answered Descartes directly, holding that there are no innate ideas and that the mind begins empty, everything in it arriving through sensation and through reflection upon what sensation delivers. He did not thereby return mathematics to the world, holding instead that its certainty comes from the agreement among our own ideas rather than from anything observed. + 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. Note what it does here: knowledge is made to begin in the world, and mathematics is made the exception. +

-

It is worth recording what was available and went unused. In March of 1826 Babbage read a paper to the Royal Society describing a language he had invented for machines. - 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 called it 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. Babbage considered it one of his best inventions. - Charles Babbage, Passages from the Life of a Philosopher (London: Longman, Green, 1864), 104, where he adds that he doubts machinery of comparable complexity could be contrived without that language or an equivalent one. On its mature form, comprising labeled drawings, timing diagrams, and logic diagrams, see Anthony Hyman, Charles Babbage: Pioneer of the Computer (Oxford: Oxford University Press, 1982), 58. +

Gottfried Wilhelm Leibniz (1646−1716) saw one of Pascal’s machines 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 Stepped Reckoner, demonstrated to the Royal Society in 1673. “Explication de l’Arithmétique Binaire,” Mémoires de l’Académie Royale des Sciences (1703). The calculemus passage is from “The Art of Discovery” (1685). Leibniz also described unconscious perceptions lying below the threshold of notice, the petites perceptions, in the preface to the Nouveaux Essais (written 1704).

-

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 the drawing board in London while the foundational crisis was being fought in Germany. The two never met. Mathematics had, in that period, everything it needed to be grounded in a machine, and instead spent fifty years proving there was nothing to look at.

+

The same man supplied the standing objection to everything proposed in this book. Suppose a machine so contrived that it thinks, and suppose it enlarged until a man could walk about inside it as though in a mill. He would find only parts pushing against parts, and would never find a perception among them. + G. W. Leibniz, Monadologie (written 1714), §17. The argument is known as Leibniz’s Mill. + The objection is not answered here, and the reader is asked to hold it.

-

It is therefore worth noticing what the paper that closed the episode opens with. 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.

- +

George Berkeley (1685−1753) did not choose between the two accounts. He attempted instead to construct a position in which both were true at once. He did this by proposing that perception is reality itself, and not a report about some further thing standing behind it. When asked whether this meant that with no one in the room the chair and the desk would disappear, he explained that they were still there, because God perceived them. And thus he also tied in faith. + George Berkeley, A Treatise Concerning the Principles of Human Knowledge (Dublin, 1710), §§3 and 6 for the doctrine that to be is to be perceived, §§28−33 and §146 for the argument that reaches God, and §48 for the objects that persist unperceived by any man. The case is put again, and more accessibly, in Three Dialogues between Hylas and Philonous (London, 1713). His route to God is that a man does not choose what he experiences, the world presenting itself as it does whether he likes it or not, so the will producing his ideas of the world is not his own. See also Will Buckingham et al., The Philosophy Book (New York: DK, 2011), page. +

- - The return +

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 not an argument but a discovery, as 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.

-

Naturalism came back into philosophy in the second half of the twentieth century, chiefly through Quine, who held that epistemology is a chapter of natural science rather than a tribunal sitting above it, and that our commitment to mathematical objects stands or falls with the theories that cannot be stated without them. It did not come back into the foundations of mathematics in any form that touched practice. - W. V. Quine, “Epistemology Naturalized,” in Ontological Relativity and Other Essays (New York: Columbia University Press, 1969), 69−90. See also Imre Lakatos, Proofs and Refutations (Cambridge: Cambridge University Press, 1976), on mathematics as it is actually done, by conjecture, counterexample, and repair; Hilary Putnam, “What is Mathematical Truth?” Historia Mathematica 2 (1975): 529−543, which names the position quasi-empiricism; and Penelope Maddy, Naturalism in Mathematics (Oxford: Clarendon Press, 1997) and Second Philosophy: A Naturalistic Method (Oxford: Oxford University Press, 2007). -

+

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. Mill was arguing from heaps of pebbles, and there was no other mechanism then to argue from. + 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 closest of these to the present book is Philip Kitcher’s, which gives mathematical knowledge as knowledge of operations that an idealized agent performs, collecting, ordering, and matching, rather than of objects standing apart. - Philip Kitcher, The Nature of Mathematical Knowledge (New York: Oxford University Press, 1983). - The distance is still the whole of the distance. His agent is idealized, which is to say stipulated, and what it can do is settled by the philosopher who describes it. The machine in these pages is not stipulated. It is built, and what it does when it runs is a question with an answer that nobody supplies.

+

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.

 - - What the word means here + + Where this fits in -

Two things the word does not mean in this book. It does not mean natural computing, the field that draws its methods from living systems, from molecules, swarms, and neurons. The machine in these pages is a tape and a controller and nothing about it is borrowed from anything alive. And Natural does not qualify a number. What mathematics calls the natural numbers are Peano Numbers here, for the reason given in section , which leaves the adjective to the philosophy alone.

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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.

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What it does mean is the position reviewed above, held about mathematics, and pursued by the method the older sense of the word names. Computational Naturalism is the thesis that mathematics is a taxonomy of observations of a machine. The conditions such a machine must satisfy are set out in chapter , the machine itself is built in chapter , and the taxonomy occupies most of what follows that.

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With this language Babbage was describing a machine behavior in the way that a Naturalist would describe the natural world. At the same time he was describing hat computation would cause to happen. It was for the machine what Newton’s calculus was for moving bodies of mass.

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Consider a Turing Machine program that prints the character s, loops back, and prints it again without end. It cannot be run to completion. It can nonetheless be analyzed, discussed, and reasoned about, as we are doing at this moment. Here then is the situation in miniature: there are things the first order cannot reach, and a language standing above it in which those very things are said.

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Later came the rewrite rule interpretation of what a machine could do, where Mathematics was interpreted as a string of symbols being recognized and then replaced by a new string. This was encapsulated in recursive functions, the lambda calculus ..., and these are pure abstractions.

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Accordingly the Realm of Forms is not a separate realm. It is the tower of languages standing above the machine, each layer of it there to speak of the holes in the layer beneath. Those languages become separated from the machine that gave rise to them, so the base machine never need be run. What builds the tower is exhaustion. Each layer runs out of what it can say, and the running out is what makes the next layer necessary. The layers are the orders of analysis, defined in section .

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Then Alan Turing asked mathematicians to imagine what a clerk writing on paper could accomplish. And later Stephen Kleene opens his Introduction to Metamathematics with a flock of four sheep and a grove of four trees, and observes 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. However he then goes on to present an abstract symbol based computation.

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Both parties are therefore granted what they asked for. Aristotle is granted that the account begins in an object one can point at and that nothing enters from outside it. Plato is granted a realm above the particulars whose inhabitants are not to be found by running anything. What is denied is the separation, and only that. The tower has a floor, the floor is an apparatus, and the orders are how one climbs.

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Suppose for a moment, that mathematics itself spoke to what the slave boy was attempting to do when he attempted to double the size of a square, what Turing’s clerk was capable of doing, and what mathematicians do in general when they work. And all of these extend from observing the Realizable Machine presented in section [cross reference here], where said machine is a cousin of the Turing Machine, That is the thesis defended in this book, i.e. Computational Naturalism.

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Stephen Kleene opens his Introduction to Metamathematics with a flock of four sheep and a grove of four trees, and observes 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.

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Computational Naturalism does not declare a victor in the debate between the abstract down, or the bottom up, rather what comes out is a demonstration that the two are aspects of one thing. That of language and meta language.

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The sheep are then dismissed and the strings arrive. This is worth pausing over, because formalism was sold as the cure for Platonism. No abstract objects and no separate realm, only marks on paper and rules for replacing them. But consider the marks. A formal string never smudges, never exhausts the paper, and costs nothing to write. Two occurrences of a symbol are perfectly identical, on every page, in every century. Those are Forms. Formalism did not empty the Realm. It evicted the numbers and moved the symbols in.

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What this book proposes is that the appeal to the sheep be taken seriously rather than used and set aside, and that the flock be replaced by an apparatus whose behaviour can be established rather than assumed. The strings then cost something, the alphabet is finite because somebody had to build the decoder, and emptiness is a property of a cell rather than a character written in it. - The category error in treating emptiness as a symbol is taken up in section . The cost of an alphabet, measured in the size of the controller that must decode it, is taken up in section . -

- The search that led to the Turing Machine