From: Thomas Walker Lynch Date: Sun, 23 Aug 2026 12:19:38 +0000 (+0000) Subject: . X-Git-Url: https://git.reasoningtechnology.com/realizable_reverse.png?a=commitdiff_plain;h=7c05371f24602da2231c5b1a30a17f9e2412238f;p=TM-2026 . --- diff --git a/document/book/TM-2026.html b/document/book/TM-2026.html index 154be16..e87a696 100644 --- a/document/book/TM-2026.html +++ b/document/book/TM-2026.html @@ -389,8 +389,8 @@ - - Naturalism + + Why the title of this book Platonism @@ -409,15 +409,15 @@

When a geometer erases his drawing the theorem is untouched. He draws it again badly and the theorem is untouched. Every square that has ever been drawn could be wiped away and the theorem would still hold, and it held before any of them were drawn. Such is the non-temporal nature of Forms.

- The definition of a symbol given in section is Platonic in its structure. It begins with an unnamed math object that has required properties. The definition then states that we do not write the symbol directly, as it remains abstract. A representation convention is chosen for example, a Greek letter say, and what appears on a page is an instance of that representation. Here the symbol is the Form of a symbol, the representation is the shape drawn in the sand, and there can of course be many instances drawn. + The definition of a symbol given in section is Platonic in its structure. It begins with an unnamed math object that has required properties. The definition then states that we do not write the symbol directly, as it remains abstract. A representation convention is chosen for example, a Greek letter say, and what appears on a page is an instance of that representation. Hence the Form is the symbol, the shape of the shadow is the representation, and the shadow itself is an instance.

- If we were to rephrase the explanation of the geometers drawing squares in the sane, we would say there exists a math object called a square and that it is described by properties, but we can not write it down directly, so it remains as an abstraction. What the geometers are using as a guide an agreed upon representation of the square that they all can recognize instances of. What is scratched into the sand and is under discussion is an such an instance. Furthermore, whether the geometers decide to instead do something else on that day, and no instances of squares are drawn in the sand, the math object known as a square remains unchanged. + 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.

- There is subtle difference between a Symbol, being a form, and the symbol described in the section , namely that symbol of section is said to be a math object, whereas the Platonic Symbol is a Form. This might be significant, as the section is being separated out of a defined set. + A subtle difference remains between the Symbol as a Form and the symbol of section . The structure is the same in both: an unwritten thing above, a representation, and many instances below. What differs is where the thing above is found. A Form is found in the Realm of Forms. The symbol of section is found among math objects. A math object, being anything a mathematician cares to name, includes representations and instances, the only requirements are that the symbols be distinct and can be represented.

@@ -425,30 +425,62 @@ Naturalism +

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.

+

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. 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 to double the number of things standing in need of explanation while leaving the original question where it was, and to say that a drawn circle participates in the Circle is, in his words, to speak in poetical metaphors. +

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 nothing of the account survives. There is no elsewhere for the ideal circle to be, so there is no original for the drawn one to fall short of, so there are no shadows, so there is no cave to be led out of. The drawn circle is a circle.

+ 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: one conceptualizes the Forms, and the senses report shadows that mislead more than they inform. For Aristotle, knowledge ascends: perception of particulars is where an inquiry starts, many perceptions leave what is common to them, and the general principle is arrived at from below. +

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. 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.

-

Upon leaving the Academy Aristotle then did what his position obliges a man to do, and went and looked. 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. +

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. 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. So Aristotle is a naturalist in both of the senses this book uses. He held that an account is owed in terms of the world, and he spent his life observing and sorting. The two are not separate facts about him. The second is what the first commits a man to.

+ 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.

+ + + + + The long transmission + +

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. +

-

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. Where the position and the method are distinguished, the first is called metaphysical naturalism and the second methodological naturalism, and a person can hold the second while remaining quiet about the first. The title of this book follows from both. Computation theory is the result of observing and categorizing what the Realizable Machine of chapter does and can be made to do.

+

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. + 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.

+ + + + Age of enlightment - Number as a natural object + Age of enlightment -

The Aristotelian account was the working position of the schools for most of two thousand years, and it was still the working position when the mathematics that mattered began to change. In the seventeenth century the naturalist reading was held plainly, and held by the people doing the work. Thomas Hobbes (1588−1679) made geometry a science of bodies and of the motions that generate them. +

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. @@ -460,7 +492,7 @@ 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.

-

The eighteenth century turned against the 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. +

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.