From: Thomas Walker Lynch Date: Mon, 24 Aug 2026 09:49:44 +0000 (+0000) Subject: . X-Git-Url: https://git.reasoningtechnology.com/?a=commitdiff_plain;h=2248867495075005a9001e4b796794ff55973bcb;p=TM-2026 . --- diff --git a/document/book/TM-2026.html b/document/book/TM-2026.html index 8afb12c..1a0739d 100644 --- a/document/book/TM-2026.html +++ b/document/book/TM-2026.html @@ -454,25 +454,21 @@

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-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-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.

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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.

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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.

+ 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.

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 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. - 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.

+ 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.

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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 (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, 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. +

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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. 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.