From: Thomas Walker Lynch Date: Mon, 24 Aug 2026 09:17:26 +0000 (+0000) Subject: . X-Git-Url: https://git.reasoningtechnology.com/realizable_reverse.png?a=commitdiff_plain;h=3ac86f86c09c001f992405d73db09fe03f36e33c;p=TM-2026 . --- diff --git a/document/book/TM-2026.html b/document/book/TM-2026.html index 11eaa36..8afb12c 100644 --- a/document/book/TM-2026.html +++ b/document/book/TM-2026.html @@ -37,12 +37,11 @@ Exordium -

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.

+

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

-

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.

- -

This book would not exist without the encouragement of my wife. While cousins brought their husbands to family events, she sat by herself as I remained home in the company of computation theory. She is a beautiful, charming woman. I count myself fortunate not to be wearing horns.

+

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

+

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

 @@ -392,6 +391,8 @@ Why the title of this book +

This chapter is a survey not intended to be a philosophy tutorial. The biographical information is incomplete even relative to what is known today, and serves only to make the prose flow.

+ Platonism @@ -413,7 +414,7 @@

- If we were to rephrase the explanation of the geometers drawing squares in the sand in the language of this book, we would say there exists a math object distinct from other math objects, one that we are naming a square only so as to have something to point at, and that goes unspoken in everything the geometers do. It is not written down directly, and remains an abstraction. The shape the geometers recognize as a square is its representation. The knowledge of abstract concepts and their representations forms the common knowledge that is required for communication. The instance scratched into the sand is recognized as the representation and then facilitates the discussion of the concepts. + If we were to rephrase the explanation of the geometers drawing squares in the sand in the language of this book, we would say there exists a math object distinct from other math objects. It is not written down directly, thus remains an abstraction. The shape the geometers recognize as a square is its representation. The knowledge of abstract concepts and their representations forms the common knowledge that is required for communication. The instance scratched into the sand is recognized as the representation and then facilitates the discussion of the concepts.

@@ -423,111 +424,116 @@ - Aristotlism + Aristotelianism -

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.

+

The Realm of Forms is not anywhere, nothing done in this world bears upon what is true of it, and the man who has been up into the light returns with a report that can be checked against no other thing. Whether the report is believed turns on the standing of the man who carries it. That is the arrangement the priestess at Delphi works under, and it is the arrangement upon which every claim about an unseen realm hinges. This type of reasoning attracts those who believe in mysticism, and it repels those who seek grounding.

-

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. +

Aristotle (384−322 BC) arrived at Plato’s Academy when he was seventeen and Plato was sixty. Aristotle was the son of a physician, and had a pragmatic view. Where Plato was brilliant and intuitive, Aristotle was scholarly and methodical. The difference shows in what each took to be the starting point of an inquiry. Due to the passing of the centuries, not much is known about their personal dynamic at the Academy, though that Aristotle remained there for twenty years suggests that it was rich. 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.

+

-

Aristotle rejected the Realm of Forms entirely. His objection was that it explains nothing: to account for the many circles in the world by positing one Circle elsewhere is not to clarify the situation, but rather it is to introduce yet another thing in need of explanation. To say that a drawn circle participates in the Circle is, in his words, to speak in poetical metaphors. +

Aristotle and Plato both rejected relativism, they both disagreed with Democritus’s theory of the atom, and they both held that the universe appeared to have some basic guiding principles. However, Aristotle rejected the Realm of Forms entirely. His objection was that it explains nothing: to account for the many circles in the world by positing one Circle elsewhere is not to clarify the situation, but rather it is to introduce yet another thing in need of explanation. To say that a drawn circle participates in the Circle is, in his words, to speak in poetical metaphors. 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 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. +

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.

-

When Plato passed, Aristotle was not chosen to take his place as many had suspected would happen. 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 was got by opening eggs on successive days. +

After spending twenty years 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. 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 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.

+

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 of which 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. If computing machines are taken to be a genre of animals to be observed, then this book makes use of both meanings.

 - The argument since -

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. - 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.

+ The machine as an animal to be observed -

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. -

+

After the Athenian Academy declined, the transmission of Greek thought fractured. Western Europe retained fragments of Aristotle’s logic while the corpus migrated east, carried by Syriac Christians to eighth-century Arabic-speaking scholars who rendered Aristotle, Euclid, and Galen into Arabic. The problem those scholars faced was fitting philosophical demonstration to Islamic faith. + Hunayn ibn Ishaq (809−873) and his circle drove the translation movement under Abbasid patronage. The arriving texts contained Aristotle layered with late-antique commentary and Neoplatonic adaptations circulating under his name, notably the Theology of Aristotle (an adaptation of Plotinus). See “Greek Sources in Arabic and Islamic Philosophy,” Stanford Encyclopedia of Philosophy. + Al-Farabi (c. 872−950) distinguished knowing a thing exactly as it is from knowing it through a representation addressed to those unable to follow a demonstration, which is the distinction 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, three centuries before there was a word for what he was doing. Al-Ghazali (1058−1111) formulated the absolute denial of natural causes, arguing that fire possesses no inherent power to burn cotton, and that apparent natural causality is God acting directly at each occasion. + Al-Ghazali, Tahāfut al-Falāsifa, discussion 17. The position, termed occasionalism, prefigures by six hundred years Hume’s argument that human observers see succession rather than necessity. + Ibn Rushd (1126−1198) resolved this by stating that a demonstrated conclusion cannot conflict with a revealed truth, and that apparent conflicts call for a revision of the textual reading.

+ +

Ibn Rushd’s 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 locates what the many share in a symbol, and a symbol is something somebody has to write down. + The position is called nominalism, from nomen, a name. + Aristotle held that what the many share is in the things and is recovered by observing them. Ockham keeps the observing and grants the categories no standing beyond the representation they are recorded in. This book arrives near that position, with one addition Ockham had no means to supply, which is a specimen that writes the representation and can be watched while it does so.

-

European Enlightenment philosophers anxiously took up the discussion.

+

What follows is three threads running together for two centuries. One asks where the certainty of mathematics comes from and answers by turning inward, to the mind. One answers by turning outward, to bodies and their motions. The third is on a workbench, and consists of machines that carry out arithmetic while knowing nothing, together with the languages invented to say what those machines do. The third is the one this book follows, and for most of the period nobody took it for a contribution to the question at all.

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.

+ A man who says that geometry is a branch of mechanics is saying that the forms are in the machinery. Newton’s calculus is a language whose subject matter is a body in motion, its terms recording what was observed of such bodies in a form other men could take up and use. Aristotle’s sorted account of the animals and Newton’s calculus of the moving bodies are the same kind of product, which is a representation of what was observed.

-

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 (1596−1650) took the road inward, holding that certain knowledge comes from reason reflecting on itself rather than from the senses, which deceive. The ideas he trusted most were those he took to be innate, present in the mind before any experience of the world, 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). -

+ John Locke (1632−1704) answered him directly: there are no innate ideas, and the mind begins empty, everything in it arriving through sensation and through reflection upon what sensation delivers. Yet Locke declined to return mathematics to the world along with the rest, holding 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. George Berkeley (1685−1753) took the doctrine further, holding that perception is reality itself rather than a report about some further thing standing behind it, and reaching God as the perceiver who keeps the furniture in an empty room in being; A Treatise Concerning the Principles of Human Knowledge (Dublin, 1710), §§3, 6, 28−33, 48, and 146. + This is the pattern to watch through the whole period. Knowledge is made to begin in the world, and mathematics is made the exception. Every man who takes the outward road arrives at the same door and declines to open it, and the reason is always to hand: he can point at what a mathematician does and see no apparatus.

-

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. +

While the philosophers argued, others built, and what they built is the third thread. 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. -

- -

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. -

- -

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. -

+ A carry crossing a digit boundary in that machine is a thing that happens, in a room, at a time, and can be watched happening. Nothing in it recollects a Form and nothing in it consults an intuition. Here was the animal, and for two hundred years no party to the argument over the foundation of arithmetic thought to examine one.

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). -

+ One man thus supplied the machine, the number system it would eventually be built in, and the proposal that reasoning is what such a machine does. The three together are the programme of this book. He had all three in hand and no way to join them, because the notation he wanted was a notation for reasoning, and what was missing was a notation for the apparatus.

-

The 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. +

The same man supplied the standing objection to everything proposed here. 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.

- -

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. -

+ It is the objection that a dissection never turns up the thing the dissector went looking for. The objection is not answered here, and the reader is asked to hold it.

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.

+ What unmade that position was a discovery rather than an argument. Lobachevsky in 1829 and Bolyai in 1832 published geometries in which the parallel postulate fails, and Beltrami showed in 1868 that the new geometry is consistent if the old one is. If physical space satisfies at most one of several consistent geometries, then geometry is not the science of space, and the necessity Kant had explained is not there to be explained.

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.

+ John Stuart Mill, A System of Logic (London: Parker, 1843), Book II, chaps. 5−6. Gottlob Frege, Die Grundlagen der Arithmetik (Breslau: Koebner, 1884), §§7−10 against Mill, §§21−27 against number as a property of things and as an idea. The second charge Frege named psychologism, and it became the period’s term of abuse. + The demolition was thorough and was, on its own terms, correct. Notice what Mill was obliged to argue from. He had heaps of pebbles, and no other mechanism was then available, so the specimens he offered had no behavior in them to observe and nothing about them fixed what was being counted. A naturalist handed a heap has been handed the wrong animal.

-

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.

+

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.

+ +

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

 + Where this fits in -

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. +

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

-

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.

+

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

-

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.

+

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

-

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.

+

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

+ +

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

-

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.

+

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

-

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.

+

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

+

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

+

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

+ +

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

 + @@ -5263,7 +5269,7 @@ --> -->