From: Thomas Walker Lynch Date: Tue, 25 Aug 2026 08:41:22 +0000 (+0000) Subject: . X-Git-Url: https://git.reasoningtechnology.com/realizable_reverse.png?a=commitdiff_plain;h=4ad33c3bee9b939890a062d010e24c3b9770c74c;p=TM-2026 . --- diff --git a/document/book/TM-2026.html b/document/book/TM-2026.html index 1a0739d..7011047 100644 --- a/document/book/TM-2026.html +++ b/document/book/TM-2026.html @@ -391,8 +391,6 @@ 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 @@ -426,7 +424,7 @@ Aristotelianism -

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.

+

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 depends upon the reputation of the man who came back with the testimony. 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 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. @@ -440,49 +438,56 @@ 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.

-

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. +

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. 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 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.

+

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 carries two meanings, that of the biologist who studies nature, and that of the philosophy holding that nothing exists outside nature. This latter meaning is generally taken to deny the Realm of Forms, and with it any abstraction standing outside nature and apart from anything that could be observed. This book takes up the first meaning: observing the machine as a natural phenomenon, writing down what it does in a useful notation, and holding each conclusion answerable to further observation. While doing this we will build a structure, which defines the term abstract concept, and thus integrates Plato’s Realm of Forms. Perhaps in so doing, Plato is pulled into Naturalism, and the two are not distinct. If the argument is successful, many philosophers who have debated them as a dialectic over the centuries would surely be disappointed. + Aristotle is ancestor to the first meaning and only distantly to the second, holding as he did that the first mover is no part of nature; Metaphysics XII.7. The metaphysical doctrine is a nineteenth-century development. +

+ The dialectic without a specimen - The machine as an animal to be observed +

The pattern of a thesis met by its opposite and settled in a third position is commonly credited to Hegel, and it is not his. Fichte set it out, and Heinrich Moritz Chalybäus fixed the three names upon it in a popular history of 1837, from which it passed into the schoolroom and has stayed there. + Johann Gottlieb Fichte, Grundlage der gesammten Wissenschaftslehre (Leipzig, 1794), §§1−3. Heinrich Moritz Chalybäus, Historische Entwicklung der speculativen Philosophie von Kant bis Hegel (Dresden, 1837). Hegel’s own term is Aufhebung, in which a position is cancelled, preserved, and raised at once. The three-step formula is not in his vocabulary, and scholars of Hegel are quick to say so. + Wherever it came from, it trains a reader to sort positions into opposed camps and to wait for the resolution to arrive from somewhere above them. The sorting is the easy part and it is habit-forming.

+ +

There is an older story, of blind men and an elephant. One takes hold of the trunk and reports a snake, one the ear and reports a fan, one the leg and reports a pillar. It is usually told against them, as a lesson about partial views. It can be told the other way. Every report was accurate, every man had hold of something real, and what was wanting was not better observation but the recognition that the reports were of one animal. Plato and Aristotle have been handled as an opposed pair for a long while, and this chapter follows what came of that. It is a survey and it is selected, the question put to each man being whether he went and looked at something.

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. Ibn al-Haytham (c. 965−1040) built apparatus and settled by experiment that light travels to the eye rather than from it. Al-Ghazali (1058−1111) formulated the absolute denial of natural causes, arguing that fire possesses no inherent power to burn cotton, and that apparent natural causality is God acting directly at each occasion. + Al-Farabi (c. 872−950) held that a truth reaches the philosopher as a proof and everyone else as an image of it, and that different nations hold different images of the one truth. Ibn al-Haytham (c. 965−1040) built apparatus and settled by experiment that light travels to the eye rather than from it. Al-Ghazali (1058−1111) formulated the absolute denial of natural causes, arguing that fire possesses no inherent power to burn cotton, and that apparent natural causality is God acting directly at each occasion. Al-Ghazali, Tahāfut al-Falāsifa, discussion 17. The position, termed occasionalism, prefigures by six hundred years Hume’s argument that human observers see succession rather than necessity. - Ibn Rushd (1126−1198) 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 returned the Greek philosophy to Latin Europe through the translators at Toledo.

+ Ibn Rushd (1126−1198) answered that a demonstrated conclusion cannot conflict with a revealed truth, and that apparent conflicts call for a revision of the textual reading. His commentaries returned the Greek philosophy to Latin Europe through the translators at Toledo.

-

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

+

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

-

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

René Descartes (1596−1650) argued that certain knowledge comes from reason reflecting on itself rather than from the senses, which deceive. The ideas he trusted most were those he took to be innate, present in the mind before any experience of the world, thus channeling Plato in modern terms. René Descartes, 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 Descartes by saying there are no innate ideas, the mind begins empty, and everything in it arrives through sensation, and through reflection upon what sensation delivers. Yet Locke declined to include mathematics, which he conceded comes from the agreement among our own ideas. - John Locke, An Essay Concerning Human Understanding (London, 1690), Book I against innate ideas, Book II on the two sources, and Book IV chap. 4 on mathematical certainty. The position is called empiricism. 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. + John Locke (1632−1704) answered him by saying there are no innate ideas, the mind begins empty, and everything in it arrives through sensation and through reflection upon what sensation delivers. Yet Locke declined to include mathematics, which he conceded comes from the agreement among our own ideas. + John Locke, An Essay Concerning Human Understanding (London, 1690), Book I against innate ideas, Book II on the two sources, and Book IV chap. 4 on mathematical certainty. The position is called empiricism. 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, 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. - 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.

+

George Berkeley (1685−1753) attempted what is attempted here, which is to dissolve the opposition rather than to settle it. He held that perception is reality itself and not a report about some further thing standing behind it, so that the world as it is met and the world as it is are the one world, with nothing left for them to disagree about. Asked what becomes of the furniture in a room when nobody is in it, he answered that God perceives it. + George Berkeley, A Treatise Concerning the Principles of Human Knowledge (Dublin, 1710), §§3 and 6 for the doctrine, §§28−33 and §146 for the argument reaching God, and §48 for the objects that persist unperceived by any man. Put again, and more accessibly, in Three Dialogues between Hylas and Philonous (London, 1713). + The unification held. The price was an entity outside the system, brought in to keep a truth available when no instance of it was to be had. A reader is entitled to ask what this book pays for the same purchase, and to hold the question open until it is answered.

-

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.

+

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

-

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. - 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.

+

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

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. @@ -494,10 +499,12 @@

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.

+

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

 + + Where this fits in