<RT·section id="Section·Naturalism·Platonism">
<RT·name>Platonism</RT·name>
- <p>Plato (427−347 BC) spoke of an entity with an independent existence he called a Form. A Form is immutable, non-spatial, and non-temporal, and exists in a realm that has no location, called the Realm of Forms. Examples of Forms include mathematical properties (Equality, Circularity, Squareness, Unity, Doubleness), moral concepts (Justice, Courage, Piety, Temperance), aesthetic ideals (Beauty), physical archetypes (Tableness, Bedness, Treeness), and above them all the Good, which Plato held to be what makes the other Forms knowable. Plato then explains that <em>knowledge</em> is the conceptualization of Forms.
- </p>
+ <p>Plato (427−347 BC) spoke of an entity with an independent existence that he called a Form. A Form is immutable, non-spatial, and non-temporal, and exists in a realm that has no location, called the Realm of Forms. Examples of Forms include mathematical properties (Equality, Circularity, Squareness, Unity, Doubleness), moral concepts (Justice, Courage, Piety, Temperance), aesthetic ideals (Beauty), physical archetypes (Tableness, Bedness, Treeness), and above them all the Good, which Plato held to be what makes the other Forms knowable. Plato then explains that <em>knowledge</em> is the conceptualization of Forms.</p>
<p>
- As an illustration of the independent existence of Forms, consider what Plato says about geometers in the <em>Republic</em>. They draw a square and a diagonal, and then they argue about it. But they are not arguing about the figure they drew, and everyone participating in the discussion knows this to be the case. They are not asking whether that line, the one in the sand, is straight, nor whether that corner is square, for it plainly is not, and no conclusion they reach depends upon it. The drawn figure is used as an image of the Square itself, and what they say holds of the Square itself whether or not the drawing was made well, or made at all.<RT·endnote>Plato, <em>Republic</em> VI 510c−511a. The passage places mathematics on the third segment of the divided line, above opinion and below the account that would justify its hypotheses. Aristotle grants the same observation about mathematical practice and gives it the opposite explanation, at <em>Metaphysics</em> XIII 1078a17−31.</RT·endnote>
+ When people sense the material world, what they are sensing are the shadows from the Realm of Forms. Plato explains this with an allegory about a world where people are chained in a cave for their entire lives in a manner they only face a wall. Behind them a fire throws the shadows of objects onto the wall. Having never seen anything else, they mistake the shadows for reality itself. One of them is freed, is brought up into the light, and sees the objects themselves; and when he returns to tell the others, they do not believe him.
+ <RT·endnote>Plato, <em>Republic</em> VII 514a−520a.</RT·endnote>
+ </p>
+
+ <p>As an illustration of the independent existence of Forms, consider what Plato says about geometers in the <em>Republic</em>. They draw a square and a diagonal, and then they argue about it. But they are not arguing about the figure they drew, and everyone participating in the discussion knows this to be the case. They are not asking whether that line, the one in the sand, is straight, nor whether that corner is square, for it plainly is not, and no conclusion they reach depends upon it. The drawn figure is used as an image of the Square itself, and what they say holds of the Square itself whether or not the drawing was made well, or made at all. The lines in the sand are merely the shadow of the Square being discussed.
+ <RT·endnote>Plato, <em>Republic</em> VI 510c−511a. The passage places mathematics on the third segment of the divided line, above opinion and below the account that would justify its hypotheses. Aristotle grants the same observation about mathematical practice and gives it the opposite explanation, at <em>Metaphysics</em> XIII 1078a17−31.</RT·endnote>
+ </p>
+
+ <p>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.</p>
+
+ <p>
+ The definition of a symbol given in section <RT·Counter·read snapshot="Section·Conventio·Symbol"></RT·Counter·read> 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 down a symbol directly. 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.
</p>
<p>
- 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.
+ 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, what the geometers are attempting to draw in the sane is a representation of the square that they all agree on, and what is scratched into the sand is an instance of that representation. Furthermore, whether the geometers decide to instead do something else on that day, and now instances of squares are drawn in the sane, the math object remains unchanged.
</p>
<p>
- The definition of a symbol given in section <RT·Counter·read snapshot="Section·Conventio·Symbol"></RT·Counter·read> is Platonic, and was written that way before the matter had been thought through. It began with a symbol, which is an abstraction; it required a representation, such as a Greek letter; and it noted that what appears on the written page is an instance of that representation, of which there can be many. Set that beside the geometers. There is the Square, there is the manner in which a square is to be presented, four sides drawn equal and joined square, and there are the many figures scratched in the sand. The symbol is the Form, the representation is the manner in which the Form is met with, and the instance is the shadow. Three levels in each case, and the middle one is the reason section <RT·Counter·read snapshot="Section·Conventio·Symbol"></RT·Counter·read> declines the philosophers’ pair of type and token.
+ There is subtle difference between a Symbol, being a form, and the symbol described in the section <RT·Counter·read snapshot="Section·Conventio·Symbol"></RT·Counter·read>, namely that symbol of section <RT·Counter·read snapshot="Section·Conventio·Symbol"></RT·Counter·read> is a math object, and the Platonic Symbol has no such limitation. This might be significant, as the section <RT·Counter·read snapshot="Section·Conventio·Symbol"></RT·Counter·read> is being separated out of a defined set, where as a Platonic Symbol consists only of the required properties placed on a Symbol.
</p>
+
</RT·section>
<RT·section id="Section·Naturalism·Naturalism">
<RT·endnote>Aristotle, <em>Posterior Analytics</em> II.19, for knowledge beginning in perception and rising by induction to the principles. The Greek term for the operation is <em>epagōgē</em>, rendered as <RT·term>induction</RT·term>.</RT·endnote>
</p>
- <p>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.
+ <p>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.
<RT·endnote>Aristotle, <em>Historia Animalium</em> 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 <em>Parts of Animals</em>, 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., <em>The Life and Letters of Charles Darwin</em> (London: John Murray, 1887), vol. 3, 252.</RT·endnote>
What he produced from it was a <RT·term>taxonomy</RT·term>: 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.</p>
- <p>
- 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 <RT·term>Naturalism</RT·term>. The title of this book follows from it. Computation theory is the result of observing and categorizing what the Realizable Machine of chapter <RT·Counter·read snapshot="Section·Realizable_computation_theoretic"></RT·Counter·read> does and can be made to do.
- </p>
-
- </RT·section>
-
-
- <RT·section id="Section·Naturalism·The_word">
- <RT·name>The word and its practitioners</RT·name>
-
- <p>
- <RT·term>Naturalism</RT·term> is the position that whatever there is, is natural, and that an account of anything is owed in terms of the world rather than in terms of something standing outside it. Its method follows from the position: an inquiry begins in observation of what is there. Where the two are separated, the first is called metaphysical naturalism and the second methodological naturalism, and a person can hold the second while remaining quiet about the first.
- </p>
-
- <p>
- A <RT·term>naturalist</RT·term> as a noun is one who observes and sorts. John Ray (1627−1705), Carl Linnaeus (1707−1778), the Comte de Buffon (1707−1788), and Charles Darwin (1809−1882) were naturalists in this sense, and what a naturalist produces is a <RT·term>taxonomy</RT·term>: a classification answerable to the specimens, revised whenever a specimen refuses one of its categories. Nothing in that method requires the categories to have existed before the observing. The two uses of the word are related and are not identical, and both are wanted in these pages.
- </p>
+ <p>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 <RT·term>Naturalism</RT·term>. 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 <RT·Counter·read snapshot="Section·Realizable_computation_theoretic"></RT·Counter·read> does and can be made to do.</p>
- <p>
- It is worth recalling how recently the vocabulary shifted. The study of the physical world was called <RT·term-em>natural philosophy</RT·term-em> until the nineteenth century, and Newton’s book of 1687 is the <em>Mathematical Principles of Natural Philosophy</em>. The word <em>scientist</em> was coined in 1834.<RT·endnote>William Whewell coined it in an unsigned review of Mary Somerville, <em>On the Connexion of the Physical Sciences</em>, <em>Quarterly Review</em> 51 (1834): 54−68, and put it into print under his own name in <em>The Philosophy of the Inductive Sciences</em> (London: Parker, 1840), vol. 1, cxiii. The older term survives in the chairs of natural philosophy at the Scottish universities.</RT·endnote> A seventeenth century author who called mathematics a part of natural philosophy was making a claim that his vocabulary made easy, and the vocabulary changed before the claim was settled.
- </p>
</RT·section>
<RT·section id="Section·Naturalism·Number_as_a_natural_object">
<RT·name>Number as a natural object</RT·name>
- <p>
- 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. The sixteenth century argued at length over whether mathematical demonstrations are demonstrations in the full sense, the <em>quaestio de certitudine mathematicarum</em>, and the argument turned on whether a mathematical object has a cause, which an Aristotelian object must.<RT·endnote>Opened by Alessandro Piccolomini, <em>Commentarium de certitudine mathematicarum disciplinarum</em> (Rome, 1547), and continued by Pereira, Barozzi, and Biancani into the seventeenth century. See Paolo Mancosu, <em>Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century</em> (Oxford: Oxford University Press, 1996), chap. 1.</RT·endnote>
+ <p>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.
+ <RT·endnote>Thomas Hobbes, <em>De Corpore</em> (London, 1655), Part II. He defended the position through a public quarrel with John Wallis that ran over twenty years; see Douglas M. Jesseph, <em>Squaring the Circle: The War between Hobbes and Wallis</em> (Chicago: University of Chicago Press, 1999).</RT·endnote>
+ 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.
+ <RT·endnote>Isaac Barrow, <em>Lectiones Mathematicae</em>, delivered 1664−1666 and published (London, 1683), Lectures I−VII. Barrow held the Lucasian chair before Newton, who succeeded him in 1669.</RT·endnote>
</p>
- <p>
- In the seventeenth century the naturalist reading was held plainly and by the people doing the work. Thomas Hobbes (1588−1679) made geometry a science of bodies and of the motions that generate them, and defended the position through a public quarrel with John Wallis that ran for over twenty years.<RT·endnote>Thomas Hobbes, <em>De Corpore</em> (London, 1655), Part II. On the dispute with Wallis, see Douglas M. Jesseph, <em>Squaring the Circle: The War between Hobbes and Wallis</em> (Chicago: University of Chicago Press, 1999).</RT·endnote> 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.<RT·endnote>Isaac Barrow, <em>Lectiones Mathematicae</em>, delivered 1664−1666 and published (London, 1683), Lectures I−VII. Barrow held the Lucasian chair before Newton, who succeeded him in 1669.</RT·endnote> And Newton (1642−1727), in the preface to the <em>Principia</em>, wrote that geometry is founded in mechanical practice and is nothing other than the part of universal mechanics which measures accurately.<RT·endnote>Isaac Newton, <em>Philosophiæ Naturalis Principia Mathematica</em> (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.</RT·endnote> A man who says that geometry is a branch of mechanics is saying that the forms are in the machinery.
- </p>
+ <p>Newton (1642−1727) wrote in the preface to the <em>Principia</em> that geometry is founded in mechanical practice and is nothing other than the part of universal mechanics which measures accurately.
+ <RT·endnote>Isaac Newton, <em>Philosophiæ Naturalis Principia Mathematica</em> (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.</RT·endnote>
+ 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 <RT·term-em>natural philosophy</RT·term-em>, and Newton’s book of 1687 is the <em>Mathematical Principles of Natural Philosophy</em>. The word <em>scientist</em> was not coined until 1834.
+ <RT·endnote>William Whewell coined it in an unsigned review of Mary Somerville, <em>On the Connexion of the Physical Sciences</em>, <em>Quarterly Review</em> 51 (1834): 54−68, and put it into print under his own name in <em>The Philosophy of the Inductive Sciences</em> (London: Parker, 1840), vol. 1, cxiii. The older term survives in the chairs of natural philosophy at the Scottish universities.</RT·endnote>
+ The vocabulary changed before the claim was settled.</p>
- <p>
- 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 <em>The Analyst</em> 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.<RT·endnote>George Berkeley, <em>A Treatise Concerning the Principles of Human Knowledge</em> (Dublin, 1710), Introduction §§7−25, against abstract ideas; <em>The Analyst</em> (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.</RT·endnote> 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.<RT·endnote>David Hume, <em>An Enquiry Concerning Human Understanding</em> (London, 1748), §IV part 1. The division is anticipated in <em>A Treatise of Human Nature</em> (London, 1739), Book I.</RT·endnote> An empiricist had thus removed mathematics from the reach of experience.
- </p>
+ <p>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 <em>The Analyst</em> 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.
+ <RT·endnote>George Berkeley, <em>A Treatise Concerning the Principles of Human Knowledge</em> (Dublin, 1710), Introduction §§7−25, against abstract ideas; <em>The Analyst</em> (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.</RT·endnote>
+ 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.
+ <RT·endnote>David Hume, <em>An Enquiry Concerning Human Understanding</em> (London, 1748), §IV part 1.</RT·endnote>
+ An empiricist had thus removed mathematics from the reach of experience.</p>
</RT·section>
<RT·section id="Section·Naturalism·The_turn_inward">
<RT·name>The turn inward</RT·name>
- <p>
- Immanuel Kant (1724−1804) settled the matter for the next century in the <em>Critique of Pure Reason</em> 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.<RT·endnote>Immanuel Kant, <em>Kritik der reinen Vernunft</em> (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.</RT·endnote>
+ <p>Immanuel Kant (1724−1804) settled the matter for the next century in the <em>Critique of Pure Reason</em> 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.
+ <RT·endnote>Immanuel Kant, <em>Kritik der reinen Vernunft</em> (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.</RT·endnote>
</p>
- <p>
- This is the decisive turn, and it is worth being exact about what it 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.
- </p>
+ <p>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.</p>
- <p>
- 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 his habilitation lecture of 1854, and Eugenio Beltrami (1835−1900), whose model is discussed in chapter <RT·Counter·read snapshot="Section·Search_for_Turing_Machine"></RT·Counter·read>, showed in 1868 that the new geometry is consistent if the old one is.<RT·endnote>N. I. Lobachevsky, "On the Principles of Geometry," <em>Kazan Messenger</em>, 1829−1830; János Bolyai, appendix to Farkas Bolyai, <em>Tentamen</em> (Maros-Vásárhely, 1832); Bernhard Riemann, "Über die Hypothesen, welche der Geometrie zu Grunde liegen," delivered 1854, published <em>Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen</em> 13 (1868): 133−152; Eugenio Beltrami, "Saggio di interpretazione della geometria non-euclidea," <em>Giornale di Matematiche</em> 6 (1868): 284−312.</RT·endnote> 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.
- </p>
+ <p>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 <RT·Counter·read snapshot="Section·Search_for_Turing_Machine"></RT·Counter·read>, showed in 1868 that the new geometry is consistent if the old one is.
+ <RT·endnote>N. I. Lobachevsky, “On the Principles of Geometry,” <em>Kazan Messenger</em>, 1829−1830; János Bolyai, appendix to Farkas Bolyai, <em>Tentamen</em> (Maros-Vásárhely, 1832); Bernhard Riemann, “Über die Hypothesen, welche der Geometrie zu Grunde liegen,” delivered 1854, published <em>Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen</em> 13 (1868): 133−152; Eugenio Beltrami, “Saggio di interpretazione della geometria non-euclidea,” <em>Giornale di Matematiche</em> 6 (1868): 284−312.</RT·endnote>
+ 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.</p>
</RT·section>
<RT·section id="Section·Naturalism·The_last_stand_and_the_demolition">
<RT·name>The last stand and the demolition</RT·name>
- <p>
- 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 <em>A System of Logic</em> 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.<RT·endnote>John Stuart Mill, <em>A System of Logic, Ratiocinative and Inductive</em> (London: Parker, 1843), Book II, chaps. 5−6, and Book III, chap. 24.</RT·endnote>
+ <p>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 <em>A System of Logic</em> 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.
+ <RT·endnote>John Stuart Mill, <em>A System of Logic, Ratiocinative and Inductive</em> (London: Parker, 1843), Book II, chaps. 5−6, and Book III, chap. 24.</RT·endnote>
</p>
- <p>
- Gottlob Frege (1848−1925) destroyed the position in the <em>Grundlagen der Arithmetik</em> 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 <RT·term>psychologism</RT·term>, and it became the period's term of abuse.<RT·endnote>Gottlob Frege, <em>Die Grundlagen der Arithmetik</em> (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, <em>Zeitschrift für Philosophie und philosophische Kritik</em> 103 (1894): 313−332, with enough effect that Husserl abandoned the position.</RT·endnote>
+ <p>Gottlob Frege (1848−1925) destroyed the position in the <em>Grundlagen der Arithmetik</em> 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 <RT·term>psychologism</RT·term>, and it became the period’s term of abuse.
+ <RT·endnote>Gottlob Frege, <em>Die Grundlagen der Arithmetik</em> (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, <em>Zeitschrift für Philosophie und philosophische Kritik</em> 103 (1894): 313−332, with enough effect that Husserl abandoned the position.</RT·endnote>
</p>
- <p>
- 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 <RT·Counter·read snapshot="Section·Peano_Number"></RT·Counter·read>, 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 <RT·Counter·read snapshot="Section·Symbol"></RT·Counter·read>. Mill was arguing from heaps of pebbles. There was no other mechanism then to argue from.
- </p>
+ <p>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 <RT·Counter·read snapshot="Section·Peano_Number"></RT·Counter·read>, 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 <RT·Counter·read snapshot="Section·Symbol"></RT·Counter·read>. Mill was arguing from heaps of pebbles. There was no other mechanism then to argue from.</p>
- <p>
- 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.<RT·endnote>Richard Dedekind, <em>Was sind und was sollen die Zahlen?</em> (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," <em>Jahresbericht der Deutschen Mathematiker-Vereinigung</em> 2 (1893): 5−31, at 19.</RT·endnote> 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 <RT·Counter·read snapshot="Section·Search_for_Turing_Machine"></RT·Counter·read> was fought entirely among these three, and no party to it proposed consulting an apparatus.
- </p>
+ <p>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.
+ <RT·endnote>Richard Dedekind, <em>Was sind und was sollen die Zahlen?</em> (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,” <em>Jahresbericht der Deutschen Mathematiker-Vereinigung</em> 2 (1893): 5−31, at 19.</RT·endnote>
+ 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 <RT·Counter·read snapshot="Section·Search_for_Turing_Machine"></RT·Counter·read> was fought entirely among these three, and no party to it proposed consulting an apparatus.</p>
- <p>
- 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.
+ <p>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.
+ <RT·endnote>Charles Babbage, “On a Method of Expressing by Signs the Action of Machinery,” <em>Philosophical Transactions of the Royal Society of London</em> 116 (1826): 250−265, read 16 March 1826.</RT·endnote>
+ He called it the <RT·term-em>mechanical notation</RT·term-em>. 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.
+ <RT·endnote>Charles Babbage, <em>Passages from the Life of a Philosopher</em> (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, <em>Charles Babbage: Pioneer of the Computer</em> (Oxford: Oxford University Press, 1982), 58.</RT·endnote>
</p>
+
+ <p>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.</p>
+
+ <p>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.</p>
</RT·section>
<RT·section id="Section·Naturalism·The_return">
<RT·name>The return</RT·name>
- <p>
- Naturalism came back into philosophy in the second half of the twentieth century, though not into the foundations of mathematics in any form that touched practice. Willard Van Orman Quine (1908−2000) 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.<RT·endnote>W. V. Quine, "Epistemology Naturalized," in <em>Ontological Relativity and Other Essays</em> (New York: Columbia University Press, 1969), 69−90; the indispensability argument is developed in <em>Word and Object</em> (Cambridge: MIT Press, 1960) and taken up by Hilary Putnam, <em>Philosophy of Logic</em> (New York: Harper and Row, 1971).</RT·endnote> Imre Lakatos (1922−1974) described mathematics as it is actually done, by conjecture, counterexample, and repair, which is a naturalism about the practice if not about the objects.<RT·endnote>Imre Lakatos, <em>Proofs and Refutations: The Logic of Mathematical Discovery</em>, ed. John Worrall and Elie Zahar (Cambridge: Cambridge University Press, 1976), from papers of 1963−1964.</RT·endnote> Hilary Putnam (1926−2016) gave the name <RT·term>quasi-empiricism</RT·term> to the view that mathematical statements are confirmed and overturned in something like the way scientific ones are.<RT·endnote>Hilary Putnam, "What is Mathematical Truth?" <em>Historia Mathematica</em> 2 (1975): 529−543.</RT·endnote> Philip Kitcher gave an account of mathematical knowledge as knowledge of operations that an idealized agent performs, collecting, ordering, and matching, rather than of objects standing apart.<RT·endnote>Philip Kitcher, <em>The Nature of Mathematical Knowledge</em> (New York: Oxford University Press, 1983).</RT·endnote> Penelope Maddy has argued the case at book length under the name second philosophy, the position that there is no vantage point above the sciences from which to correct them.<RT·endnote>Penelope Maddy, <em>Realism in Mathematics</em> (Oxford: Clarendon Press, 1990); <em>Naturalism in Mathematics</em> (Oxford: Clarendon Press, 1997); <em>Second Philosophy: A Naturalistic Method</em> (Oxford: Oxford University Press, 2007).</RT·endnote>
+ <p>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.
+ <RT·endnote>W. V. Quine, “Epistemology Naturalized,” in <em>Ontological Relativity and Other Essays</em> (New York: Columbia University Press, 1969), 69−90. See also Imre Lakatos, <em>Proofs and Refutations</em> (Cambridge: Cambridge University Press, 1976), on mathematics as it is actually done, by conjecture, counterexample, and repair; Hilary Putnam, “What is Mathematical Truth?” <em>Historia Mathematica</em> 2 (1975): 529−543, which names the position quasi-empiricism; and Penelope Maddy, <em>Naturalism in Mathematics</em> (Oxford: Clarendon Press, 1997) and <em>Second Philosophy: A Naturalistic Method</em> (Oxford: Oxford University Press, 2007).</RT·endnote>
</p>
- <p>
- Kitcher’s agent is the closest of these to the present book, and 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.
- </p>
+ <p>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.
+ <RT·endnote>Philip Kitcher, <em>The Nature of Mathematical Knowledge</em> (New York: Oxford University Press, 1983).</RT·endnote>
+ 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.</p>
</RT·section>
<RT·section id="Section·Naturalism·What_the_word_means_here">
<RT·name>What the word means here</RT·name>
- <p>
- Two things the word does not mean in this book. It does not mean <em>natural computing</em>, 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 <RT·term>Natural</RT·term> does not qualify a number. What mathematics calls the natural numbers are <RT·neologism>Peano Number</RT·neologism>s here, for the reason given in section <RT·Counter·read snapshot="Section·Conventio·Number_types"></RT·Counter·read>, which leaves the adjective to the philosophy alone.
- </p>
+ <p>Two things the word does not mean in this book. It does not mean <em>natural computing</em>, 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 <RT·term>Natural</RT·term> does not qualify a number. What mathematics calls the natural numbers are <RT·neologism>Peano Number</RT·neologism>s here, for the reason given in section <RT·Counter·read snapshot="Section·Conventio·Number_types"></RT·Counter·read>, which leaves the adjective to the philosophy alone.</p>
- <p>
- What it does mean is the position reviewed above, held about mathematics, and pursued by the method the older sense of the word names. <RT·neologism>Computational Naturalism</RT·neologism> 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 <RT·Counter·read snapshot="Section·Three_conditions"></RT·Counter·read>, the machine itself is built in chapter <RT·Counter·read snapshot="Section·Realizable_computation_theoretic"></RT·Counter·read>, and the taxonomy occupies most of what follows that.
- </p>
+ <p>What it does mean is the position reviewed above, held about mathematics, and pursued by the method the older sense of the word names. <RT·neologism>Computational Naturalism</RT·neologism> 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 <RT·Counter·read snapshot="Section·Three_conditions"></RT·Counter·read>, the machine itself is built in chapter <RT·Counter·read snapshot="Section·Realizable_computation_theoretic"></RT·Counter·read>, and the taxonomy occupies most of what follows that.</p>
- <p>
- A reader who knows the literature will by now expect the thesis to be set against Plato, since that is how the opposition has been drawn since Aristotle drew it, and since a naturalist is supposed to be someone who denies that there is anywhere else for a number to be. This book does not draw it that way, and the remainder of this section says why, in advance of the machinery that makes the point properly.
- </p>
+ <p>Consider a Turing Machine program that prints the character <RT·code>s</RT·code>, 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.</p>
- <p>
- Consider a Turing Machine program that prints the character <RT·code>s</RT·code>, 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.
- </p>
+ <p>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 <RT·term>orders of analysis</RT·term>, defined in section <RT·Counter·read snapshot="Section·Consequentiality·Orders_of_analysis"></RT·Counter·read>.</p>
- <p>
- 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 <RT·term>orders of analysis</RT·term>, defined in section <RT·Counter·read snapshot="Section·Consequentiality·Orders_of_analysis"></RT·Counter·read>.
- </p>
+ <p>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.</p>
- <p>
- 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; the first order is the machine running, and it is where every term in this book is finally cashed. Plato is granted a realm of forms that is genuinely above the particulars, whose inhabitants are not to be found by running anything, and which is reached by reasoning rather than by looking. What is denied is only the separation. The tower has a floor, the floor is an apparatus, and the orders are how one climbs. Neither party is the foil of the other; they are describing the same structure from opposite ends of it.
- </p>
+ <p>Stephen Kleene opens his <em>Introduction to Metamathematics</em> 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.
+ <RT·endnote>Stephen Cole Kleene, <em>Introduction to Metamathematics</em> (Amsterdam: North-Holland, 1952), 3.</RT·endnote>
+ 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.</p>
- <p>
- Stephen Kleene opens his <em>Introduction to Metamathematics</em> 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.<RT·endnote>Stephen Cole Kleene, <em>Introduction to Metamathematics</em> (Amsterdam: North-Holland, 1952), 3.</RT·endnote> 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. What this book proposes is that the appeal 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.
+ <p>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.</p>
+
+ <p>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.
+ <RT·endnote>The category error in treating emptiness as a symbol is taken up in section <RT·Counter·read snapshot="Section·Conventional_Turing_Machine·Memory_of_emptiness"></RT·Counter·read>. The cost of an alphabet, measured in the size of the controller that must decode it, is taken up in section <RT·Counter·read snapshot="Section·Conventional_Turing_Machine·Analysis"></RT·Counter·read>.</RT·endnote>
</p>
</RT·section>
</RT·section>
</RT·section>
</RT·section>
- <!--------------------------------------------------------------------------------->
- <RT·section id="Section·Naturalism">
- <RT·name>Naturalism</RT·name>
-
- <RT·section id="Section·Naturalism·Platonism">
- <RT·name>Platonism</RT·name>
-
- <p>Plato (427−347 BC) spoke of an entity with an independent existence he called a Form. A Form is immutable, non-spatial, and non-temporal, and exists in a realm that has no location, called the Realm of Forms. Examples of Forms include mathematical properties (Equality, Circularity, Squareness, Unity, Doubleness), moral concepts (Justice, Courage, Piety, Temperance), aesthetic ideals (Beauty), physical archetypes (Tableness, Bedness, Treeness), and above them all the Good, which Plato held to be what makes the other Forms knowable. Plato then explains that <em>knowledge</em> is the conceptualization of Forms.</p>
-
- <p>Plato pressed the point with an allegory. Men are chained in a cave facing a wall, and behind them a fire throws the shadows of objects onto the wall before them. Having seen nothing else, they take the shadows for the whole of what there is. One of them is freed, is brought up into the light, and sees the objects themselves; and when he returns to tell the others, they do not believe him.
- <RT·endnote>Plato, <em>Republic</em> VII 514a−520a.</RT·endnote>
- </p>
-
- <p>As an illustration of the independent existence of Forms, consider what Plato says about geometers in the <em>Republic</em>. They draw a square and a diagonal, and then they argue about it. But they are not arguing about the figure they drew, and everyone participating in the discussion knows this to be the case. They are not asking whether that line, the one in the sand, is straight, nor whether that corner is square, for it plainly is not, and no conclusion they reach depends upon it. The drawn figure is used as an image of the Square itself, and what they say holds of the Square itself whether or not the drawing was made well, or made at all.
- <RT·endnote>Plato, <em>Republic</em> VI 510c−511a. The passage places mathematics on the third segment of the divided line, above opinion and below the account that would justify its hypotheses. Aristotle grants the same observation about mathematical practice and gives it the opposite explanation, at <em>Metaphysics</em> XIII 1078a17−31.</RT·endnote>
- </p>
-
- <p>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.</p>
-
- <p>The definition of a symbol given in section <RT·Counter·read snapshot="Section·Conventio·Symbol"></RT·Counter·read> is Platonic, and was written that way before the matter had been thought through. It began with a symbol, which is an abstraction; it required a representation, such as a Greek letter; and it noted that what appears on the written page is an instance of that representation, of which there can be many. Set that beside the geometers. There is the Square, there is the manner in which a square is to be presented, four sides drawn equal and joined square, and there are the many figures scratched in the sand. The symbol is the Form, the representation is the manner in which the Form is met with, and the instance is the shadow. Three levels in each case, and the middle one is the reason section <RT·Counter·read snapshot="Section·Conventio·Symbol"></RT·Counter·read> declines the philosophers’ pair of type and token.</p>
-
- </RT·section>
-
- <RT·section id="Section·Naturalism·Naturalism">
- <RT·name>Naturalism</RT·name>
-
- <p>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.
- <RT·endnote>Will Buckingham, Douglas Burnham, Peter J. King, John Marenbon, Clive Hill, and Marcus Weeks, <em>The Philosophy Book</em>, Big Ideas Simply Explained (New York: DK, 2011), page 58.</RT·endnote>
- The difference shows in what each took to be the starting point of an inquiry.</p>
-
- <p>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 <em>participates</em> in the Circle is, in his words, to speak in poetical metaphors.
- <RT·endnote>Aristotle, <em>Metaphysics</em> I.9 990b−991b, with the remark on poetical metaphors at 991a20−22, and again at XIII.4−5. Aristotle does use the word <em>eidos</em>, 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.</RT·endnote>
- 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.</p>
-
- <p>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.
- <RT·endnote>Aristotle, <em>Posterior Analytics</em> II.19, for knowledge beginning in perception and rising by induction to the principles. The Greek term for the operation is <em>epagōgē</em>, rendered as <RT·term>induction</RT·term>.</RT·endnote>
- </p>
-
- <p>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.
- <RT·endnote>Aristotle, <em>Historia Animalium</em> 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 <em>Parts of Animals</em>, 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., <em>The Life and Letters of Charles Darwin</em> (London: John Murray, 1887), vol. 3, 252.</RT·endnote>
- What he produced from it was a <RT·term>taxonomy</RT·term>: 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.</p>
-
- <p>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 <RT·term>Naturalism</RT·term>. 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 <RT·Counter·read snapshot="Section·Realizable_computation_theoretic"></RT·Counter·read> does and can be made to do.</p>
-
- </RT·section>
-
- <RT·section id="Section·Naturalism·Number_as_a_natural_object">
- <RT·name>Number as a natural object</RT·name>
-
- <p>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.
- <RT·endnote>Thomas Hobbes, <em>De Corpore</em> (London, 1655), Part II. He defended the position through a public quarrel with John Wallis that ran over twenty years; see Douglas M. Jesseph, <em>Squaring the Circle: The War between Hobbes and Wallis</em> (Chicago: University of Chicago Press, 1999).</RT·endnote>
- 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.
- <RT·endnote>Isaac Barrow, <em>Lectiones Mathematicae</em>, delivered 1664−1666 and published (London, 1683), Lectures I−VII. Barrow held the Lucasian chair before Newton, who succeeded him in 1669.</RT·endnote>
- </p>
-
- <p>Newton (1642−1727) wrote in the preface to the <em>Principia</em> that geometry is founded in mechanical practice and is nothing other than the part of universal mechanics which measures accurately.
- <RT·endnote>Isaac Newton, <em>Philosophiæ Naturalis Principia Mathematica</em> (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.</RT·endnote>
- 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 <RT·term-em>natural philosophy</RT·term-em>, and Newton’s book of 1687 is the <em>Mathematical Principles of Natural Philosophy</em>. The word <em>scientist</em> was not coined until 1834.
- <RT·endnote>William Whewell coined it in an unsigned review of Mary Somerville, <em>On the Connexion of the Physical Sciences</em>, <em>Quarterly Review</em> 51 (1834): 54−68, and put it into print under his own name in <em>The Philosophy of the Inductive Sciences</em> (London: Parker, 1840), vol. 1, cxiii. The older term survives in the chairs of natural philosophy at the Scottish universities.</RT·endnote>
- The vocabulary changed before the claim was settled.</p>
-
- <p>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 <em>The Analyst</em> 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.
- <RT·endnote>George Berkeley, <em>A Treatise Concerning the Principles of Human Knowledge</em> (Dublin, 1710), Introduction §§7−25, against abstract ideas; <em>The Analyst</em> (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.</RT·endnote>
- 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.
- <RT·endnote>David Hume, <em>An Enquiry Concerning Human Understanding</em> (London, 1748), §IV part 1.</RT·endnote>
- An empiricist had thus removed mathematics from the reach of experience.</p>
- </RT·section>
-
- <RT·section id="Section·Naturalism·The_turn_inward">
- <RT·name>The turn inward</RT·name>
-
- <p>Immanuel Kant (1724−1804) settled the matter for the next century in the <em>Critique of Pure Reason</em> 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.
- <RT·endnote>Immanuel Kant, <em>Kritik der reinen Vernunft</em> (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.</RT·endnote>
- </p>
-
- <p>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.</p>
-
- <p>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 <RT·Counter·read snapshot="Section·Search_for_Turing_Machine"></RT·Counter·read>, showed in 1868 that the new geometry is consistent if the old one is.
- <RT·endnote>N. I. Lobachevsky, “On the Principles of Geometry,” <em>Kazan Messenger</em>, 1829−1830; János Bolyai, appendix to Farkas Bolyai, <em>Tentamen</em> (Maros-Vásárhely, 1832); Bernhard Riemann, “Über die Hypothesen, welche der Geometrie zu Grunde liegen,” delivered 1854, published <em>Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen</em> 13 (1868): 133−152; Eugenio Beltrami, “Saggio di interpretazione della geometria non-euclidea,” <em>Giornale di Matematiche</em> 6 (1868): 284−312.</RT·endnote>
- 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.</p>
- </RT·section>
-
- <RT·section id="Section·Naturalism·The_last_stand_and_the_demolition">
- <RT·name>The last stand and the demolition</RT·name>
-
- <p>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 <em>A System of Logic</em> 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.
- <RT·endnote>John Stuart Mill, <em>A System of Logic, Ratiocinative and Inductive</em> (London: Parker, 1843), Book II, chaps. 5−6, and Book III, chap. 24.</RT·endnote>
- </p>
-
- <p>Gottlob Frege (1848−1925) destroyed the position in the <em>Grundlagen der Arithmetik</em> 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 <RT·term>psychologism</RT·term>, and it became the period’s term of abuse.
- <RT·endnote>Gottlob Frege, <em>Die Grundlagen der Arithmetik</em> (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, <em>Zeitschrift für Philosophie und philosophische Kritik</em> 103 (1894): 313−332, with enough effect that Husserl abandoned the position.</RT·endnote>
- </p>
-
- <p>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 <RT·Counter·read snapshot="Section·Peano_Number"></RT·Counter·read>, 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 <RT·Counter·read snapshot="Section·Symbol"></RT·Counter·read>. Mill was arguing from heaps of pebbles. There was no other mechanism then to argue from.</p>
-
- <p>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.
- <RT·endnote>Richard Dedekind, <em>Was sind und was sollen die Zahlen?</em> (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,” <em>Jahresbericht der Deutschen Mathematiker-Vereinigung</em> 2 (1893): 5−31, at 19.</RT·endnote>
- 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 <RT·Counter·read snapshot="Section·Search_for_Turing_Machine"></RT·Counter·read> was fought entirely among these three, and no party to it proposed consulting an apparatus.</p>
-
- <p>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.
- <RT·endnote>Charles Babbage, “On a Method of Expressing by Signs the Action of Machinery,” <em>Philosophical Transactions of the Royal Society of London</em> 116 (1826): 250−265, read 16 March 1826.</RT·endnote>
- He called it the <RT·term-em>mechanical notation</RT·term-em>. 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.
- <RT·endnote>Charles Babbage, <em>Passages from the Life of a Philosopher</em> (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, <em>Charles Babbage: Pioneer of the Computer</em> (Oxford: Oxford University Press, 1982), 58.</RT·endnote>
- </p>
-
- <p>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.</p>
-
- <p>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.</p>
- </RT·section>
-
- <RT·section id="Section·Naturalism·The_return">
- <RT·name>The return</RT·name>
-
- <p>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.
- <RT·endnote>W. V. Quine, “Epistemology Naturalized,” in <em>Ontological Relativity and Other Essays</em> (New York: Columbia University Press, 1969), 69−90. See also Imre Lakatos, <em>Proofs and Refutations</em> (Cambridge: Cambridge University Press, 1976), on mathematics as it is actually done, by conjecture, counterexample, and repair; Hilary Putnam, “What is Mathematical Truth?” <em>Historia Mathematica</em> 2 (1975): 529−543, which names the position quasi-empiricism; and Penelope Maddy, <em>Naturalism in Mathematics</em> (Oxford: Clarendon Press, 1997) and <em>Second Philosophy: A Naturalistic Method</em> (Oxford: Oxford University Press, 2007).</RT·endnote>
- </p>
-
- <p>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.
- <RT·endnote>Philip Kitcher, <em>The Nature of Mathematical Knowledge</em> (New York: Oxford University Press, 1983).</RT·endnote>
- 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.</p>
- </RT·section>
-
- <RT·section id="Section·Naturalism·What_the_word_means_here">
- <RT·name>What the word means here</RT·name>
-
- <p>Two things the word does not mean in this book. It does not mean <em>natural computing</em>, 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 <RT·term>Natural</RT·term> does not qualify a number. What mathematics calls the natural numbers are <RT·neologism>Peano Number</RT·neologism>s here, for the reason given in section <RT·Counter·read snapshot="Section·Conventio·Number_types"></RT·Counter·read>, which leaves the adjective to the philosophy alone.</p>
-
- <p>What it does mean is the position reviewed above, held about mathematics, and pursued by the method the older sense of the word names. <RT·neologism>Computational Naturalism</RT·neologism> 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 <RT·Counter·read snapshot="Section·Three_conditions"></RT·Counter·read>, the machine itself is built in chapter <RT·Counter·read snapshot="Section·Realizable_computation_theoretic"></RT·Counter·read>, and the taxonomy occupies most of what follows that.</p>
-
- <p>Consider a Turing Machine program that prints the character <RT·code>s</RT·code>, 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.</p>
-
- <p>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 <RT·term>orders of analysis</RT·term>, defined in section <RT·Counter·read snapshot="Section·Consequentiality·Orders_of_analysis"></RT·Counter·read>.</p>
-
- <p>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.</p>
-
- <p>Stephen Kleene opens his <em>Introduction to Metamathematics</em> 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.
- <RT·endnote>Stephen Cole Kleene, <em>Introduction to Metamathematics</em> (Amsterdam: North-Holland, 1952), 3.</RT·endnote>
- 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.</p>
-
- <p>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.</p>
-
- <p>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.
- <RT·endnote>The category error in treating emptiness as a symbol is taken up in section <RT·Counter·read snapshot="Section·Conventional_Turing_Machine·Memory_of_emptiness"></RT·Counter·read>. The cost of an alphabet, measured in the size of the controller that must decode it, is taken up in section <RT·Counter·read snapshot="Section·Conventional_Turing_Machine·Analysis"></RT·Counter·read>.</RT·endnote>
- </p>
- </RT·section>
- </RT·section>
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