<!--
+higher order machines and Russel and Turing
+
+
VOLUME BOUNDARY, provisional.
Deferred to a later volume, from the first edition:
</RT·section>
<!--------------------------------------------------------------------------------->
- <RT·section counter="Front Matter,Front Matter Section" id="Section·Preface">
- <RT·name>Preface</RT·name>
+ <RT·section counter="Front Matter,Front Matter Section" id="Section·Prefatio">
+ <RT·name>Praefatio</RT·name>
<p>The IEEE Std 754 floating-point standard equips the numerical analyst with the tools to design platform independent algorithms with predictable numerical behavior. Though the principal founder of the standard did commit one sin. He allowed for the expanded precision of the intermediate calculations on the Intel 8087 as a matter of backwards compatibility. He apparently succumbed to that final temptation because he had been part of that project. Consequently, the standard does not provide universal platform independence.</p>
<RT·section counter="Front Matter,Front Matter Section" id="Section·Conventio">
<RT·name>Conventio</RT·name>
- <p>All communication starts from the knowledge held in common between the author and reader. This chapter is a first meeting and a search for that commonality.</p>
-
- <RT·section id="Section·Conventio·Notation">
- <RT·name>Notation</RT·name>
+ <RT·section id="Section·Conventio·Notatio">
+ <RT·name>Notatio</RT·name>
<p>In math spans and blocks, each container, such as a sequence or a set, is represented with a single capital letter, even when in Greek. Each object that is not a container is represented with a single lower case letter. Character pairs or full symbol names can also be used to represent math objects, in which case the same convention is followed for the first letter of the symbol, while the remaining letters are lower case. The context will make it clear if a non-letter Unicode character represents a container or a non-container.</p>
<p>The Unicode middle dot, ·, is used as an ad hoc namespace operator in identifiers. Hence <RT·code>N·x</RT·code> would be the variable <RT·code>x</RT·code> from the <RT·code>N</RT·code> namespace. This is a typographic symbol with a long history of use as a separator. It appears in dictionaries where words are broken into syllables, e.g. ‘op·er·a·tor’. It is a letter of the Catalan alphabet, the <em>punt volat</em>, which distinguishes the geminate <RT·code>l·l</RT·code> of ‘col·legi’ from the digraph <RT·code>ll</RT·code>. And Japanese uses it, as the <em>nakaguro</em>, to separate the components of a transliterated foreign name. It is accepted in identifiers by modern C compilers,<RT·endnote>C11 permits the character anywhere in an identifier, including as the first character: it falls in the 00B7−00BA range of Annex D.1, and appears in none of the ranges disallowed initially in D.2. C23 restates the rule in terms of Unicode UAX #31, where the character belongs to Other_ID_Continue and so remains permitted, though no longer in first position. ISO/IEC 9899:2011, Annex D; ISO/IEC 9899:2024, §6.4.2.</RT·endnote> and it is part of the RT·gcc compiler mods as an ad hoc namespace operator.</p>
</RT·section>
- <RT·section id="Section·Conventio·Usage">
- <RT·name>Usage</RT·name>
+ <RT·section id="Section·Conventio·Usus">
+ <RT·name>Usus</RT·name>
<p>In the RT coding culture, we say that argument values are <em>given</em> to a function. Functions do not <em>take</em> arguments, as it is the caller that has authority over making the call, and thus over sending the arguments. A function is only said to <em>accept</em> or <em>reject</em> a value in the context of guard code evaluating argument validity, something it has authority to do. Similarly, machines, factories, and software <em>make</em> objects by assembling components they are given; they do not <em>create</em> them, since bringing components into being is not within an engineer’s authority. That leaves ‘creation’ to imply a more fundamental synthesis outside the scope of engineering, perhaps done by artists or supernatural beings.</p>
<p>Dates and times are given in the ISO 8601 format. The components run from the most-significant to the least-significant, year, then month, then day, in the same order the digits of a Hindu-Arabic numeral run. Two consequences follow from that ordering, and both are the reason for choosing it. A lexicographic comparison of two such dates agrees with their chronological comparison, so a list of them sorts correctly with no collation rules particular to dates. And the format is unambiguous, whereas a date such as 03/04/2026 is read as the third of April by some readers and the fourth of March by others. A time of day, when one is given, follows the date, and a trailing <RT·code>Z</RT·code> marks the time as UTC. Hence <RT·code>2026-06-01 08:28:00Z</RT·code>. ISO 8601 itself separates the date from the time with a <RT·code>T</RT·code>, which is difficult to read. RFC 3339 anticipates the objection and permits a space in its place, and that is the option taken here.<RT·endnote>Graham Klyne and Chris Newman, <em>Date and Time on the Internet: Timestamps</em>, RFC 3339, July 2002, §5.6.</RT·endnote> Where a date appears without a time, the date alone is intended, e.g. <RT·code>2026-06-01</RT·code>. As per the standard, if the day is not to be specified, it is simply dropped, e.g. <RT·code>2026-06</RT·code>.</p>
</RT·section>
+ </RT·section>
+
+ <!--------------------------------------------------------------------------------->
+ <RT·section id="Section·Received">
+ <RT·name>Common ground</RT·name>
+
+ <p>All communication starts from the knowledge held in common between those who are communicating, here me the author and you the reader. This chapter is our first meeting, and a search for this commonality.</p>
+
<RT·section id="Section·Conventio·Number_types">
<RT·name>Number types</RT·name>
</RT·section>
-
-
- <RT·section id="Section·Conventio·Fields_of_study">
+ <RT·section id="Section·Received·Fields_of_study">
<RT·name>Fields of study</RT·name>
<p>Number Theory consists of the analysis of structures built from <RT·term>Counting Number</RT·term>s, admitting zero where a structure requires it, as modular arithmetic does for its residues. It gives properties to these structures, e.g. the familiar ‘odd’ and ‘even’. It gives properties to the relationship between numbers and operations on those numbers, e.g. proving that an odd number squared is odd. It deals with the solution of mixed integer equations, and systems of such equations, and sets the rules for modulus arithmetic. Groups, rings, and fields belong to it, and so does the whole of algebraic structure. For example, encryption algorithms belong to Number Theory.</p>
<p>Computer Architecture is the design discipline concerned with the structure of a machine: what its parts are, what each is responsible for, and how the boundaries between them are drawn. It is distinct from the engineering of any particular part, and its characteristic question is where a boundary should fall rather than how a component should be built. The levels treated in <RT·Counter·read snapshot="Section·Design_stack" key="counter count"></RT·Counter·read> are its subject matter.</p>
- <p>Communication Theory studies the conveyance of a message from a sender to a receiver over a channel, and in particular what must be agreed in advance for the message to be recoverable at the far end. This book borrows from it because a tape written by one machine and read by another is a channel, and the questions that arise there — how the receiver is to know where the message ends, and how control is to be told apart from data — are ones communication engineers have already had to answer.</p>
+ <p>Communication Theory studies the conveyance of a message from a sender to a receiver over a channel, and in particular what must be agreed in advance for the message to be recoverable at the far end. This book borrows from it because a tape written by one machine and read by another is a channel, and the questions that arise there, such as how the receiver is to know where the message ends, and how control is to be told apart from data, are ones communication engineers have already had to answer.</p>
</RT·section>
- <RT·section id="Section·Conventio·Symbol">
+ <RT·section id="Section·Received·Symbol">
<RT·name>Symbol</RT·name>
<p>A <em>math object</em> is anything mathematics is willing to name: a number, a set, a function, a formula, or a symbol. Nothing further is meant by it. A <RT·term>value</RT·term>, where the term is used without qualification, is synonymous with math object.</p>
</RT·section>
- <RT·section id="Section·Conventio·Logic">
+ <RT·section id="Section·Received·Logic">
<RT·name>Logic</RT·name>
<p>Frege and Russell and Whitehead began from logic because they took it to be the foundation. In this book the machine is the foundation, and logic is among its phenomena. This chapter has as a purpose to establish language for discussion, rather than being a starting point of the work itself.</p>
- <RT·section id="Section·Conventio·Logic·Propositional_calculus">
+ <RT·section id="Section·Received·Logic·Propositional_calculus">
<RT·name>Propositional calculus</RT·name>
<p>A <RT·term>proposition</RT·term> is a statement that is given no arguments and carries a <RT·term>truth value</RT·term>, that being one of exactly two things, written <RT·term-em>true</RT·term-em> and <RT·term-em>false</RT·term-em>. A proposition is not predicated upon anything, so settling it requires nothing to be looked up and no value to be supplied. "The moon is made of green cheese" is a proposition, and it is false. "Seven is prime" is a proposition, and it is true. In either case the statement stands complete as it is written. Nothing in the calculus that follows looks inside a proposition or has any means of altering one. A proposition enters the calculus carrying its truth value, and the calculus makes use of nothing else about it.</p>
<p>The <RT·term>connective</RT·term>s are symbols, and instances of their representations are these: <RT·math>∧</RT·math> (and), <RT·math>∨</RT·math> (or), <RT·math>¬</RT·math> (not), <RT·math>→</RT·math> (implies), and <RT·math>↔</RT·math> (if and only if). Alongside them stand the parentheses, <RT·math>(</RT·math> and <RT·math>)</RT·math>, which group. Each connective is fixed, in that what it does is settled once for the whole calculus and nothing later on is at liberty to change it. What each one does is combine truth values to give a truth value, so <RT·math>∧</RT·math> gives true exactly when both of the truth values given to it are true, <RT·math>¬</RT·math> gives the truth value it was not given, and so on for the rest.</p>
- <p>Beside the connectives stand the <RT·term>proposition symbol</RT·term>s, whose representations are commonly <RT·math>p</RT·math>, <RT·math>q</RT·math>, and <RT·math>r</RT·math>. These are variables. A proposition symbol has no meaning of its own, and marks a place where a proposition is later bound to it. It is the only kind of symbol in this calculus that is not fixed.<RT·endnote>The letters follow the notation convention of <RT·Counter·read snapshot="Section·Conventio·Notation" key="counter count"></RT·Counter·read>, applied to whatever the symbol marks a place for. A proposition is not a container, so the letter is lower case. A formula is a string and therefore a container, so the letters standing for formulas below are capitals.</RT·endnote></p>
+ <p>Beside the connectives stand the <RT·term>proposition symbol</RT·term>s, whose representations are commonly <RT·math>p</RT·math>, <RT·math>q</RT·math>, and <RT·math>r</RT·math>. These are variables. A proposition symbol has no meaning of its own, and marks a place where a proposition is later bound to it. It is the only kind of symbol in this calculus that is not fixed.<RT·endnote>The letters follow the notation convention of <RT·Counter·read snapshot="Section·Conventio·Notatio" key="counter count"></RT·Counter·read>, applied to whatever the symbol marks a place for. A proposition is not a container, so the letter is lower case. A formula is a string and therefore a container, so the letters standing for formulas below are capitals.</RT·endnote></p>
<p>Symbol instances are strung together according to the <RT·term>formation rules</RT·term>, and what the rules build is called a <RT·term>formula</RT·term>. The rules are three, and the third is what closes the definition. The Greek letters <RT·math>Φ</RT·math> and <RT·math>Ψ</RT·math> stand for whole formulas below, and <RT·math>∘</RT·math> stands for any one of the binary connectives. All three belong to this discussion rather than to the calculus being discussed.</p>
</RT·section>
- <RT·section id="Section·Conventio·Logic·First-order_predicate_calculus">
+ <RT·section id="Section·Received·Logic·First-order_predicate_calculus">
<RT·name>First-order predicate calculus</RT·name>
<p>Now suppose that a proposition could be predicated on something else. A <RT·term>predicate</RT·term> is a logic function. It is given values for its arguments, and it then evaluates to true or to false. Give a predicate its arguments and what results is a proposition, so the predicate is the general case and the proposition is the settled one. <RT·math>\text{prime}(x)</RT·math> is a predicate of one argument, and <RT·math>\text{prime}(7)</RT·math> is true while <RT·math>\text{prime}(8)</RT·math> is false. <RT·math><(x, y)</RT·math> is a predicate of two arguments, which is how a relation is written. Thus <RT·math><(3, 5)</RT·math> is true, while <RT·math><(5, 3)</RT·math> is false. Predicate arguments are drawn from a <RT·term>domain</RT·term>, a set holding at least one member, and that set is whatever mathematics cares to name: the <RT·neologism>Counting Number</RT·neologism>s, the points of a plane, the strings over an alphabet. Nothing requires the arguments to be truth values, and nothing forbids it. What is required is that the result be a truth value.</p>
<RT·Counter·read snapshot="List·first-order-predicate-logic_stack" key="counter count step"></RT·Counter·read>
</RT·Counter·step>
</RT·section>
- <RT·section id="Section·Conventio·Logic·Second-order_predicate_calculus">
+ <RT·section id="Section·Received·Logic·Second-order_predicate_calculus">
<RT·name>Second-order predicate calculus</RT·name>
<p>In the first-order calculus a quantifier claims a domain variable and nothing else. Each predicate symbol is bound by an interpretation, so a person supplies a particular predicate for <RT·math>F</RT·math> and asks what follows, and no formula of that calculus ranges over the predicates themselves. The <RT·term>second-order predicate calculus</RT·term> lifts that restriction by admitting a second kind of claimable variable, the <RT·term>predicate variable</RT·term>, which carries a fixed number of argument places. The two fates are as before, and now apply to this second kind of symbol as well. In <RT·math>∀F \, Φ</RT·math> the quantifier has claimed <RT·math>F</RT·math>, so no interpretation binds a predicate to it, and the quantifier runs it over the properties of the domain in the same way that <RT·math>∀x</RT·math> runs a domain variable over the domain itself.</p>
</RT·section>
- <RT·section id="Section·Conventio·Logic·More_logic_terminology">
+ <RT·section id="Section·Received·Logic·More_logic_terminology">
<RT·name>More logic terminology</RT·name>
<p>A formula that holds under every interpretation is <RT·term>valid</RT·term>, and in the propositional calculus such a formula is also called a <RT·term-em>tautology</RT·term-em>. A formula that holds under at least one interpretation is <RT·term>satisfiable</RT·term>. So <RT·math>((p ∧ p) ↔ p)</RT·math> is valid, <RT·math>(p ∧ q)</RT·math> is satisfiable and not valid, and <RT·math>(p ∧ ¬p)</RT·math> is neither. The two are duals: a formula is valid exactly when its negation is unsatisfiable.</p>
</RT·section>
</RT·section>
+
<!--------------------------------------------------------------------------------->
<RT·section id="Section·Naturalism">
- <RT·name>Establishing the book title</RT·name>
+ <RT·name>The Dialectic</RT·name>
<RT·section id="Section·Naturalism·The_dialectic">
- <RT·name>The dialectic</RT·name>
+ <RT·name>Origin</RT·name>
<RT·section id="Section·Naturalism·Platonism">
<RT·name>Platonism</RT·name>
<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 <RT·Counter·read snapshot="Section·Conventio·Symbol" key="counter count"></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 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.
+ The definition of a symbol given in <RT·Counter·read snapshot="Section·Received·Symbol" key="counter count"></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 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.
</p>
<p>
</p>
<p>
- A subtle difference remains between the Symbol as a Form and the symbol of <RT·Counter·read snapshot="Section·Conventio·Symbol" key="counter count"></RT·Counter·read>. 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 <RT·Counter·read snapshot="Section·Conventio·Symbol" key="counter count"></RT·Counter·read> 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.
+ A subtle difference remains between the Symbol as a Form and the symbol of <RT·Counter·read snapshot="Section·Received·Symbol" key="counter count"></RT·Counter·read>. 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 <RT·Counter·read snapshot="Section·Received·Symbol" key="counter count"></RT·Counter·read> 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.
</p>
</RT·section>
</RT·section>
<RT·section id="Section·Naturalism·The_argument_since">
- <RT·name>Metamorphosis of the dielectic</RT·name>
+ <RT·name>Metamorphosis</RT·name>
- <p>Debate over the dialectic continued for two millennia between the decline of the Athenian Academy and the 19th century. There is a brief review of this interim period in the appendix of <RT·Counter·read snapshot="Section·Appendix_dialectic_to_Kant" key="counter count"></RT·Counter·read>. In the 19th century there were two revolutionary developments that changed the form of the dialectic. The symbolic logic foundation of mathematics, and the advent of thinking machines.</p>
+ <p>Debate over the Dialectic continued for two millennia from the decline of the Athenian Academy up to the 19th century. There is a brief review of this interim period in <RT·Counter·read snapshot="Section·Appendix·dialectic_to_Kant" key="counter count"></RT·Counter·read>. In the 19th century there were two revolutionary developments that changed the form of the Dialectic. The symbolic logic foundation of mathematics, and the advent of thinking machines.</p>
+
+ <RT·section id="Section·Naturalism·Symbolic_logic">
+ <RT·name>Symbolic logic</RT·name>
<p>A person might say that Plato had won over the mathematicians. The Forms were no longer discussed as a separate realm; they had been recast as symbols, relations, and rules that could be manipulated within a formal system. George Boole put logical operations into an algebra. Georg Cantor made infinite collections a subject of mathematical investigation. Richard Dedekind sought to characterize the natural numbers through objects, systems, and mappings, and in <em>Was sind und was sollen die Zahlen?</em> (1888) described what he called a <em>simply infinite system</em>. Giuseppe Peano published a closely corresponding axiomatization in <em>Arithmetices principia, nova methodo exposita</em> (1889), acknowledging Dedekind’s earlier work.
<RT·endnote>George Boole, <em>The Mathematical Analysis of Logic</em> (Cambridge: Macmillan, Barclay & Macmillan, 1847), and <em>An Investigation of the Laws of Thought</em> (London: Walton and Maberly, 1854). Georg Cantor’s work on set theory and transfinite numbers expanded the mathematical study of infinity. Richard Dedekind, <em>Was sind und was sollen die Zahlen?</em> (Braunschweig: Vieweg, 1888); Giuseppe Peano, <em>Arithmetices principia, nova methodo exposita</em> (Turin: Fratres Bocca, 1889). Dedekind’s simply infinite system and Peano’s axiomatization are historically distinct formulations of the natural-number structure. Peano’s presentation is closely related to Dedekind’s earlier formulation, and modern historical accounts therefore often refer to the shared lineage as the <em>Dedekind−Peano axioms</em>. “Dedekind number” is already used for a different mathematical sequence, so it is not adopted here as an alternative name.</RT·endnote>
<p>Alan Turing provided a notable exception in 1936. When he evaluated the foundations of computation, he began his argument by describing a person sitting at a desk with paper and a pencil. The question he posed was not what a mathematician knows, but what such a person can be made to do. Only after establishing this naturalistic observation did the formulas follow.</p>
- <p>Alonzo Church went the other way. His lambda calculus is a purely symbolic system, a calculus of function abstraction and application, and it makes no appeal to anything observable. His doctoral student Stephen Cole Kleene developed it further.
- <RT·endnote>Alonzo Church, “A set of postulates for the foundation of logic,” <em>Annals of Mathematics</em>, second series, vol. 33 (1932): 346−366, and vol. 34 (1933): 839−864. Church offered the system as a foundation for logic. Stephen Cole Kleene and J. Barkley Rosser proved it inconsistent in 1935, and what survived was the calculus of function abstraction and application itself, later shown coextensive with Turing computability. The foundational ambition failed; the notation did not.</RT·endnote>
+ <p>Kleene opens "Metamathematics" with brief but beautiful statement before discussing transfinite numbers, "A flock of four sheep and a grove of four trees are related to each other in a way in which neither is related to a pile of three stones or a grove of seven trees. ... Without counting the sheep or the trees, one can pair them with each other, for example by tethering the sheep to the trees, so that each sheep and each tree belongs to exactly one of the pairs."
+ <RT·endnote>Stephen Cole Kleene, <em>Introduction to Metamathematics</em> (Amsterdam: North-Holland, 1952), 3.</RT·endnote>
+ Later Kleene proposes that symbols be thought of concretely as marks on paper, or more precisely as abstracted from our experience with marks on paper, and then observes in a parenthesis that proof theory has to remain abstract to some degree because it supposes arbitrarily long symbol sequences to be constructible, while the paper and ink in the world are finite.
+ <RT·endnote>Kleene 1952, 62. The parenthesis is the concession this book is built on: the idealization is named as an idealization, and the reason given for it is a physical bound.</RT·endnote></li>
+ At the definition of the machine Kleene borrows the Natural example from Turing, of a person computing a function. The subsequent defense of Turing’s thesis is conducted entirely over such a person, who ticks figures off in pencil to avoid counting them twice, and who may read a symbol on paper by eye, another in braille by hand, and take a signal by ear.
+ <RT·endnote>Kleene 1952, 356 for the derivation, 377−381 for the defense.</RT·endnote></li>
</p>
- <p>Yet Kleene is not the counterexample he first appears to be. His 1952 <em>Introduction to Metamathematics</em> returns to the natural setting at each of the three places where the formalism has to be tied to something outside itself:</p>
+ </RT·Section>
- <RT·Counter·step counter="List" id="List·Kleene_natural-settings">
- <RT·name>Where Kleene reaches for a natural setting</RT·name>
- <RT·Counter·snapshot counter="List" snapshot="List·Kleene_natural-settings"></RT·Counter·snapshot>
- <ol>
- <li>At the opening of the book, to ground one-to-one correspondence before cardinality is defined. Kleene writes, "A flock of four sheep and a grove of four trees are related to each other in a way in which neither is related to a pile of three stones or a grove of seven trees. ... Without counting the sheep or the trees, one can pair them with each other, for example by tethering the sheep to the trees, so that each sheep and each tree belongs to exactly one of the pairs."<RT·endnote>Stephen Cole Kleene, <em>Introduction to Metamathematics</em> (Amsterdam: North-Holland, 1952), 3.</RT·endnote></li>
- <li>At the introduction of the formal system, to say what a symbol is. Kleene proposes that the symbols be thought of concretely as marks on paper, or more precisely as abstracted from our experience with marks on paper, and then observes in a parenthesis that proof theory has to remain abstract to some degree because it supposes arbitrarily long symbol sequences to be constructible, while the paper and ink in the world are finite.<RT·endnote>Kleene 1952, 62. The parenthesis is the concession this book is built on: the idealization is named as an idealization, and the reason given for it is a physical bound.</RT·endnote></li>
- <li>At the definition of the machine, and again in its defense. Kleene does not open the chapter on computable functions with a formalism. He opens with a person computing a function by following pre-assigned instructions, using finitely many tokens, able to observe only finitely many at once and to remember only finitely many more, and derives the atomic acts of the machine from what that person can do. The subsequent defense of Turing’s thesis is conducted entirely over such a person, who ticks figures off in pencil to avoid counting them twice, and who may read a symbol on paper by eye, another in braille by hand, and take a signal by ear.<RT·endnote>Kleene 1952, 356 for the derivation, 377−381 for the defense.</RT·endnote></li>
- </ol>
- <RT·Counter·read snapshot="List·Kleene_natural-settings" key="counter count step"></RT·Counter·read>
- </RT·Counter·step>
- <p>At the same time that the dialectic appeared to have been resolved in favor of the abstract, it reappeared incarnate. The resolution had been to say that mathematics is the manipulation of symbols under rules within a formal system. But the manipulation of symbols under rules is something a device can be built to do. In declaring itself formal, mathematics had declared itself mechanizable, and a mechanism is an object with a mass and a temperature that a person can put on a bench and turn a crank on. The Forms had been driven out of the heavens and into the notation, and the notation could be cut into brass. Whatever else that is, it is a natural object, and it can be observed.</p>
- <p>Such objects were not new. The Antikythera mechanism carries a model of the heavens in bronze gearing, and it predates the argument it settles by two thousand years.
- <RT·endnote>The shipwreck is dated to about 60 BC, which fixes the latest possible construction. Estimates of construction itself range from about 205 BC, taken from the epoch of the eclipse dials, to Derek de Solla Price’s earlier figure of about 87 BC, with the inscriptions favouring roughly 150 to 100 BC. No consensus exists on whether the device was built shortly before the wreck or well before it. The field remains active: a 2024 re-analysis of the calendar ring hole positions put the count near 354 or 355, which argues for a lunar rather than a solar ring.</RT·endnote>
- In 1642 Blaise Pascal built a machine that added, and in doing so discovered that a carry is not free. His sautoir lifted a weight and dropped it into the next wheel, and the number of wheels a single carry could propagate through was a matter of how much force the mechanism could raise. Leibniz took up the problem thirty years later with a stepped drum that multiplied, and his carriage never worked reliably across the full width of the register. He wanted more than arithmetic from it. He wanted a <em>calculus ratiocinator</em>, a reckoning that would settle disputes by computation, which is Hobbes’s claim that reasoning is a kind of reckoning taken seriously enough to be machined. Charles Babbage then built a machine that evaluated polynomials by the method of forward differences, the same construction taken up in <RT·Counter·read snapshot="Section·Computational_Analysis" key="counter count"></RT·Counter·read>, and went on to design the Analytical Engine, treated in <RT·Counter·read snapshot="Section·Prescience_and_credit" key="counter count"></RT·Counter·read>.
- <RT·endnote>Blaise Pascal began the machine in 1642 and some fifty were made. Gottfried Wilhelm Leibniz demonstrated a stepped reckoner to the Royal Society in 1673 and had a working instrument by 1694; the carry across the full register was never dependable. On the <em>calculus ratiocinator</em>, see the 1666 <em>Dissertatio de arte combinatoria</em> and the later sketches for a universal characteristic.</RT·endnote>
+ <!--=============================== revision ===============================-->
+ <RT·section id="Section·Naturalism·Thinking_machines">
+ <RT·name>Thinking machines</RT·name>
+
+ <p>Mathematics became the manipulation of symbols under rules within a formal system, and the Dialectic appeared to have been resolved. At the same moment it reappeared incarnate. In declaring itself formal, mathematics had declared itself mechanizable, and a mechanism is an object with a mass and a temperature that a person can put on a bench and turn a crank on. The Forms had been driven out of the heavens and into the notation, and then the notation was cut into brass. They had become a Natural object that could be observed.</p>
+
+ <p>A series of machines that modeled what could be seen in the heavens started in Greece about a hundred years after Aristotle’s death. These inventions followed the Greek philosophy into the Middle East, and then back to Europe. A brief history of them is given in <RT·Counter·read snapshot="Section·Appendix·Ancient_computing_machines" key="counter count"></RT·Counter·read>.</p>
+
+ <p>Such machines are the Dialectic incarnate. They are manifestly physical, yet they predict the events that occur in the heavens. An observer watching their mechanisms sees the same inertia and same couplings of rhythms in miniature as to what is happening in the greater cosmos. At the time they were invented, celestial mechanics was abstract. However, what we do not see in these machines is discrete symbol manipulation.</p>
+
+ <p>The model comes first. Newton did not begin with a notation and look for a world to fit it. He described the motions, and the symbols were written afterward to record the description. On their own the symbols do nothing. Cut the model into brass and the gears turn as the notation says they will, and they go on doing so for as long as the machine is kept oiled. The notation is exact about the machine. It is not exact about the sky, and since relativity we have known that it is not. So the orrery does not model the heavens. It models the model, and it is the one place where the description is perfectly true. A notation can be sound about a mechanism and wrong about the cosmos the mechanism was built to imitate.</p>
+
+ <p>This is also why the heliocentric account displaced the other. It is the simpler machine. It has fewer parts, it needs no epicycle laid upon an epicycle, and a man who sets out to build both will discover which is which with his hands before he discovers it with an argument.</p>
+
+ <p>In 1642 Blaise Pascal built a machine that added, and in doing so discovered that a carry is not free. The unusual thing about Pascal’s calculator was that it was digital, tracking discrete integer states across its dials, though not all of its registers were base-10; some were adapted for the non-decimal French currency of the era. The internal <RT·term>sautoir</RT·term> mechanism slowly lifted a weight that was released upon a carry to push the next digit. When successive digits carried, the weights would fall as dominoes.<RT·endnote>Blaise Pascal began the machine in 1642 and some fifty were made. Michael R. Williams, <em>A History of Computing Technology</em>, 126−130.</RT·endnote></p>
+
+ <p>Imagine a person who does not know how to add. That person could run experiments and observe this machine. If he were clever he would develop a notation for writing down his experiments and their results. The person could then learn how to add numbers from the observations. His notes could be shared with others to communicate what he had learned. Thus the Forms do have a location, as they live in the mechanism.</p>
+
+ <p>Perhaps someone would challenge this interpretation by saying that the mechanism in motion is not thinking, rather it is merely another form of writing. Pascal made the machine, so the machine is thus a message from Pascal explaining how to add numbers. It is even better than an animated video, a 4D instruction manual as a message, and no special glasses are required to see it. However, if this is the case, then Pascal himself is a message. When Pascal adds numbers he is not thinking, rather he is re-enacting the lessons from his grade school teacher who had taught him arithmetic. Learning the motions is no different than learning to dance.</p>
+
+ <p>Leibniz took up the problem thirty years later with a stepped drum that multiplied, though his carriage never worked reliably across the full width of the register. Of profound philosophical importance, he wanted more than arithmetic from it. He wanted a <em>calculus ratiocinator</em>, a reckoning that would settle human disputes by computation.<RT·endnote>Gottfried Wilhelm Leibniz demonstrated a stepped reckoner to the Royal Society in 1673 and had a working instrument by 1694. On the <em>calculus ratiocinator</em>, see the 1666 <em>Dissertatio de arte combinatoria</em> and the later sketches for a universal characteristic.</RT·endnote></p>
+
+ <p>Charles Babbage increased the complexity again when he designed and built a machine to compute polynomials by the method of forward differences, the construction taken up in <RT·Counter·read snapshot="Section·Computational_Analysis" key="counter count"></RT·Counter·read>. As Pascal had imagined relieving the accountant, Babbage imagined relieving the human computers who produced survey tables. As it turned out they correctly viewed the machine as more of a threat to their livelihoods than as a calculation aid.</p>
+
+ <p>Then Babbage crossed the threshold. He designed the Analytical Engine, which read its program from cards and could branch on a quantity it had just computed. I.e. he invented the first modern digital computer architecture, treated in <RT·Counter·read snapshot="Section·Prescience_and_credit" key="counter count"></RT·Counter·read>.</p>
+
+ <p>The machine as a messenger rather than a thinker now becomes harder to maintain. There is no longer anything to point at. The mechanism will not serve, because it is general purpose and what it does depends upon the cards it is given. So the message must be the program, and the sender the programmer. Consider then what the programmer knows. A man who cannot do sums is not thereby excluded from thinking; he is said to be thinking precisely when he works one out. Set the machine the same sum and the same must be granted to it. Give it a long division and more must be granted, for the procedure is longer and the result is further from anything the programmer held in his head. Give it a polynomial and more again. Give it an unsolved problem, and the programmer sends a message whose content he does not know, and cannot know without waiting for his own machine to tell him. At each rung the sender knows less of what he sent, and the machine supplies more of it.</p>
+
+ <p>Now ask whether the program will halt. No one knows, and no one can be made to know in general. This is not the next rung. It is where the ladder ends. The programmer is not merely ignorant of what he sent, in the way a man is ignorant of a sum he has not yet done. There is no procedure that would let him read his own message, and none can be built. The bound is on knowledge itself, and it falls on the man and the machine alike.</p>
+
+ <p>That is the bound as it stands at the first order of analysis. It is not the last word on the matter. <RT·Counter·read snapshot="Section·Consequentiality·Orders_of_analysis" key="counter count"></RT·Counter·read> takes the machine itself as the object of observation, and Turing’s example is answered there.</p>
+
+ <p>So the machine is observable, and what is observable is Natural. It performs abstraction, and what performs abstraction thinks. Neither description defeats the other, and there is no need to choose. The blind men are still holding the elephant. Mind and body have been put in one place, on a bench, where a person may turn a crank on them.</p>
+
+ <p>In any given state the machine is going to do something that leads to its next state, and it does so whatever we may think of the result. So what is it that the machine does that causes us to say it made an <em>error</em>? Nothing. Error is not among the things the mechanism has. It is a relation between what the machine did and an intention held somewhere off the bench, which is to say that error exists only if there is a message. The messenger account cannot be given up, and by the argument above it cannot be made to carry the whole weight either. Both are needed, and this is the Dialectic incarnate.</p>
+
+ <p>Babbage created a mechanical notation to write his observations about the consequences of the motions of the various parts of his machine upon the other parts. If Newton’s notation annotates the motion of the parts of a celestial analog because those parts carry his model, then what do Babbage’s notations carry? There is no cosmos behind the Analytical Engine for them to be about. They are about brass, and the brass is about whatever the cards say it is about today.</p>
+
+ </RT·section>
+ <!--============================= end revision =============================-->
+
+
+ <RT·section id="Section·Naturalism·Thinking_machihnes">
+ <RT·name>Thinking machines</RT·name>
+
+ <p>At the same time that the Dialectic appeared to have been resolved as mathematics because the manipulation of symbols under rules within a formal system, it reappeared incarnate. In declaring itself formal, mathematics had declared itself mechanizable, and a mechanism is an object with a mass and a temperature that a person can put on a bench and turn a crank on. The Forms had been driven out of the heavens and into the notation, and then the notation was cut into brass. It became a Natural object that could be observed.</p>
+
+ <p>A series of machines that modeled what could be seen in the heavens started in Greece about a hundred years after Aristotle’s death. These inventions followed the Greek philosophy into the Middle East, and then back to Europe. A brief history of them is given in <RT·Counter·read snapshot="Section·Appendix·Ancient_computing_machines" key="counter count"></RT·Counter·read>.
</p>
- <p>Notice what the metal adds. A carry in arithmetic is instantaneous and costs nothing; a carry in Pascal’s machine lifts a weight, takes time, and can jam. Precision in analysis is unbounded; precision in gearing is whatever the shop could cut. The abstraction says nothing about any of this, and the object cannot avoid it. Everything this book has to say about tape length, about the cost of a step, and about what happens at the end cases is already present in the difference between a carry on paper and a carry in brass.</p>
+ <p>Such machines are the Dialectic incarnate. They are manifestly physical, yet they predict the events that occur in the heavens. An observer watching their mechanisms sees the same inertia and same couplings of rhythms in miniature as to what is happening in the greater cosmos. At the time they were invented, celestial mechanics was abstract. However, what we do not see in these machines is discrete symbol manipulation.</p>
+
+ <p>Newton created a notation for predicting the motion of the gears in a celestial machine. To the extent that machine was an accurate model, those annotations also predicted the motion of celestial objects.</p>
+
+ <p>In 1642 Blaise Pascal built a machine that added, and in doing so discovered that a carry is not free. The unusual thing about Pascal’s calculator was that it was digital, tracking discrete integer states across its dials, though not all of its registers were base-10; some were adapted for the non-decimal French currency of the era. The internal <RT·term>sautoir</RT·term> mechanism slowly lifted a weight that was released upon a carry to push the next digit. When successive digits carried, the weights would fall as dominoes. <RT·endnote>Blaise Pascal began the machine in 1642 and some fifty were made. Michael R. Williams, <em>A History of Computing Technology</em>, 126-130.</RT·endnote></p>
+
+ <p>Imagine a person who does not know how to add. That person could run experiments and observe this machine. If he were clever he would develop a notation for writing down his experiments and their results. The person could then learn how to add numbers from the observations. His notes could be shared with other to communicate what he had learned. Thus the Forms do have a location, as they live in the mechanism.</p>
+
+ <p>Perhaps someone would challenge this interpretations by saying that the mechanism in motion is not thinking, rather it is merely another form of writing. Pascal made the machine, so the machine is thus a message from Pascal explaining how to add numbers. It is even better than an animated video, a 4D instruction manual as a message, and no special glasses are required to see it. However, if this is the case, then Pascal himself is a message. When Pascal adds numbers he is not thinking, rather he is re-enacting the lessons from his grade school teacher who had taught him arithmetic. Learning the motions are no different than learning to dance.</p>
+
+ <p>Leibniz took up the problem thirty years later with a stepped drum that multiplied, though his carriage never worked reliably across the full width of the register. Of profound philosophical importance, he wanted more than arithmetic from it. He wanted a <em>calculus ratiocinator</em>, a reckoning that would settle human disputes by computation.<RT·endnote>Gottfried Wilhelm Leibniz demonstrated a stepped reckoner to the Royal Society in 1673 and had a working instrument by 1694. On the <em>calculus ratiocinator</em>, see the 1666 <em>Dissertatio de arte combinatoria</em> and the later sketches for a universal characteristic.</RT·endnote> Charles Babbage then built a machine that evaluated polynomials by the method of forward differences, the same construction taken up in <RT·Counter·read snapshot="Section·Computational_Analysis" key="counter count"></RT·Counter·read>, and went on to design the Analytical Engine, treated in <RT·Counter·read snapshot="Section·Prescience_and_credit" key="counter count"></RT·Counter·read>.</p>
+
+ <p>Charles Babbage increased the complexity yet again when he designed and built a machine to compute polynomials. Simila to how Pascal imagined helping accounting and survey computations be less error prone, Babbage had imagined it being used by the human computers who created survey tables. As it turned out they correctly viewed the machine as more of a threat to their livelihoods than as a calculation aid.</p>
+
+ <p>Then Babbage crossed the threshold. He designed the Analytical engine that was capable of reading a stored program and performing conditional branches. I.e. he invented the first modern digital computer architecture.</p>
+
+ <p>The machine as a messenger rather than thinker at the Thinkery now gets more complex. The machine itself can’t be pointed at to say that it is a message, as the machine is general purpose, what it does depends on the program it is given. Then the question becomes one of whether the program can be considered to be a message from the programmer that is filtered through the machine. The answer to this question is, "not always", because sometimes programs are written to find solutions to problems for which the programmer does not know the answer to. We are left to say that the computer is thinking while working on such a problem.</p>
+
+ <p>In any given state, the machine is going to do something that leads to its next state. So then when observing the machine, what is it that the machine does that causes us to say it made an <em>error</em>?</p>
- <p>So is a machine an animal? Set a trilobite and a calculating engine side by side on the bench. Both are objects a person can hold, measure, and sort by the features they share, the sorting answerable to the specimens and revised whenever a specimen refuses its category. The trilobite came by its features through descent and the engine through a decision in Dorset Street, but the naturalist does not need to know which in order to begin. He needs only that the features are there to be looked at. Pick up the glass and start the notebook.</p>
+ <p>Babbage created a mechanical notation to write his observations about the consequences of the motions of the various parts in his machine upon other parts in the machine. If Newton”s observations about motion can be said to annotate the motion of the parts in the celestial analog machine, then what are Babbage’s notations about the motion of the components of his machine saying?</p>
</RT·section>
<RT·section id="Section·Naturalism·Machine_anatomy">
- <RT·name>The anatomy of a machine</RT·name>
+ <RT·name>Animal and Machine</RT·name>
<p>A naturalist does not begin with a theory. He begins with two specimens and a question about how they differ. The theory, if it comes, comes out of the comparison and is answerable to it. So take the two specimens seriously and describe them.</p>
</RT·section>
<RT·section id="Section·Definition_Computational_Naturalism">
- <RT·name>The definition of Computational Naturalism</RT·name>
+ <RT·name>Computational Naturalism</RT·name>
<p>Our courtrooms, legislatures, and debating societies establish rules for adversarial contest. Those rules instill in our culture the expectation that progress is made by choosing a winner. Yet a dialectic requires a structure.
<RT·endnote>The pattern of a thesis met by its opposite and settled in a third position is commonly credited to Hegel, but it is not his. Fichte set out the triad in the <em>Grundlage der gesammten Wissenschaftslehre</em> (Leipzig, 1794), §§1−3, and Heinrich Moritz Chalybäus fixed the three names upon it in the <em>Historische Entwicklung der speculativen Philosophie von Kant bis Hegel</em> (Dresden, 1837), from which they passed into general use. Hegel’s own term is <em>Aufhebung</em>, in which a position is cancelled, preserved, and raised at once, and scholars of Hegel are quick to point out that the three-step formula is not in his vocabulary.</RT·endnote>
</p>
<p>
- The main diagonal contains strictly non-zero factorials, ensuring the matrix is invertible. By inverting this matrix, a person replaces the cascading back substitution with a direct, closed form equation to recover any constant <RT·math>a_i</RT·math>. The inversion utilizes the signed Stirling numbers of the first kind, denoted <RT·math>s(n,i)</RT·math> (see the Appendix on Stirling numbers, <RT·Counter·read snapshot="Section·Appendix_Stirling_Numbers" key="counter count"></RT·Counter·read>).
+ The main diagonal contains strictly non-zero factorials, ensuring the matrix is invertible. By inverting this matrix, a person replaces the cascading back substitution with a direct, closed form equation to recover any constant <RT·math>a_i</RT·math>. The inversion utilizes the signed Stirling numbers of the first kind, denoted <RT·math>s(n,i)</RT·math> (see the Appendix on Stirling numbers, <RT·Counter·read snapshot="Section·Appendix·Stirling_Numbers" key="counter count"></RT·Counter·read>).
</p>
<RT·Counter·step counter="Equation" id="Equation·polynomial-constant_from_differences">
</RT·section>
+ <RT·section id="Section·Appendix·Ancient_computing_machines">
+ <RT·name>Ancient computing machines</RT·name>
+
+ <p>The architectural progression of computing machinery began by modeling physical celestial mechanics through geared linkages. The earliest recorded precursor is Archimedes’ Planetarium (circa 250 BC), a gear-driven sphere built in Syracuse to model the kinematic motions of the sun, moon, and five known planets.<RT·endnote>Cicero, <em>De re publica</em>, 1.14.21–22.</RT·endnote> This established the epicyclic gearing architecture later found in the Antikythera mechanism (circa 150 BC), which functioned as a complex analog computer dedicated to calculating the Metonic cycle, lunar motions, and eclipses.</p>
+
+ <p>The Antikythera mechanism is mysterious. It was discovered in 1901, and its function was not taken seriously until 1959.<RT·endnote>Derek J. de Solla Price, "An Ancient Greek Computer", <em>Scientific American</em>, Vol. 200, No. 6, June 1959, 60–67.</RT·endnote> It is highly probable that none of the philosophers of the early modern period knew such devices had existed in antiquity.<RT·endnote>The shipwreck is dated to about 60 BC, which fixes the latest possible construction. Estimates of construction itself range from about 205 BC, taken from the epoch of the eclipse dials, to Derek de Solla Price’s earlier figure of about 87 BC, with the inscriptions favouring roughly 150 to 100 BC. No consensus exists on whether the device was built shortly before the wreck or well before it. The field remains active: a 2024 re-analysis of the calendar ring hole positions put the count near 354 or 355, which argues for a lunar rather than a solar ring.</RT·endnote> Because the artifact was lost to the sea, the mechanical computation timeline was structurally severed.</p>
+
+ <p>However, the differential gearing architecture of the Hellenistic period survived into late antiquity and the Middle Ages. The Byzantine Geared Sundial (circa 500 AD) utilized a gear train to align a lunar dial and solar pointer.<RT·endnote>J. V. Field and M. T. Wright, "Early Gearing: Geared Mechanisms in the Ancient and Mediaeval World", <em>Science Museum</em>, 1985.</RT·endnote> Islamic engineers automated calendar tracking by embedding gear sequences inside astrolabes, notably those engineered by Al-Biruni (circa 1000 AD) and Abu Bakr of Isfahan (1221 AD).<RT·endnote>Donald Routledge Hill, <em>Islamic Science and Engineering</em>, Edinburgh University Press, 1993.</RT·endnote> Medieval European engineers applied this architecture to complex astronomical modeling. Richard of Wallingford engineered the Albion (1326 AD), an analog computing equatorium for finding planetary positions without manual trigonometric calculation.<RT·endnote>John D. North, <em>Richard of Wallingford: An Edition of His Writings</em>, Oxford University Press, 1976.</RT·endnote> The apex of this simulation paradigm was the Astrarium of Giovanni de’ Dondi (1348–1364 AD), which utilized over 107 discrete gears in epicyclic configurations to model the non-circular Ptolemaic orbits.<RT·endnote>Silvio A. Bedini and Francis R. Maddison, "Mechanical Universe: The Astrarium of Giovanni de’ Dondi", <em>Transactions of the American Philosophical Society</em>, Vol. 56, No. 5, 1966.</RT·endnote></p>
+
+ <p>The decisive structural pivot from astronomical simulation to pure arithmetic processing occurred in the 17th century, shifting the mechanisms from analog representations to digital states. Wilhelm Schickard constructed the Calculating Clock (1623 AD), a discrete mechanical calculator utilizing a geared carry mechanism for addition and subtraction, while integrating rotating cylinders based on Napier’s bones for multiplication.<RT·endnote>Michael R. Williams, <em>A History of Computing Technology</em>, IEEE Computer Society Press, 1997, 122-125.</RT·endnote></p>
+
+ </RT·section>
+
<RT·section id="Section·Appendix·formal_methods·Formal_methods_in_design">
<RT·name>Do formal methods belong in computer design?</RT·name>
</RT·section>
<RT·section counter="Appendix,Appendix Section" id="Section·Appendix·dialectic_to_Kant">
- <RT·name>The dialectic from the Athenian decline to Kant</RT·name>
+ <RT·name>The Dialectic from the Athenian decline to Kant</RT·name>
<p><RT·Counter·read snapshot="Section·Naturalism·The_argument_since" key="counter count"></RT·Counter·read> gives the short account of what became of the argument between Plato and Aristotle. This appendix gives the long one. It is arranged by date, with one exception noted where it occurs. A reader who accepts the summary loses nothing by skipping it.</p>
</p>
<p>William of Ockham (c. 1287−1347) gave one of the sharpest late-medieval attempts to dissolve this dialectic. He held that what the many instances of a thing have in common is neither a Form standing above them nor an essence lodged within them, but a sign that stands for all of them.
- <RT·endnote>William of Ockham, <em>Summa Logicae</em> I.14−17 (c. 1323); in English as <em>Ockham’s Theory of Terms: Part I of the Summa Logicae</em>, trans. Michael J. Loux (Notre Dame: University of Notre Dame Press, 1974). The fuller critique is in the <em>Ordinatio</em> I d.2 qq.4−8. The position is called nominalism, from <em>nomen</em>, a name. Peter Abelard (1079−1142) reached a non-realist account two centuries earlier, while Ockham gave one of the sharpest and most influential late-medieval statements. Ockham’s sign is a concept in the soul, with spoken and written words subordinated to it, so the sign he means is nearer the abstraction of <RT·Counter·read snapshot="Section·Conventio·Symbol" key="counter count"></RT·Counter·read> than the representation is.</RT·endnote>
- Plato places the explanatory model in an intelligible order distinct from sensible particulars, whereas Ockham denies that there is any independently existing universal of that kind to locate. In the language of <RT·Counter·read snapshot="Section·Conventio·Symbol" key="counter count"></RT·Counter·read>, Ockham keeps the representation and the instances below, and empties the layer above.
+ <RT·endnote>William of Ockham, <em>Summa Logicae</em> I.14−17 (c. 1323); in English as <em>Ockham’s Theory of Terms: Part I of the Summa Logicae</em>, trans. Michael J. Loux (Notre Dame: University of Notre Dame Press, 1974). The fuller critique is in the <em>Ordinatio</em> I d.2 qq.4−8. The position is called nominalism, from <em>nomen</em>, a name. Peter Abelard (1079−1142) reached a non-realist account two centuries earlier, while Ockham gave one of the sharpest and most influential late-medieval statements. Ockham’s sign is a concept in the soul, with spoken and written words subordinated to it, so the sign he means is nearer the abstraction of <RT·Counter·read snapshot="Section·Received·Symbol" key="counter count"></RT·Counter·read> than the representation is.</RT·endnote>
+ Plato places the explanatory model in an intelligible order distinct from sensible particulars, whereas Ockham denies that there is any independently existing universal of that kind to locate. In the language of <RT·Counter·read snapshot="Section·Received·Symbol" key="counter count"></RT·Counter·read>, Ockham keeps the representation and the instances below, and empties the layer above.
</p>
<p>Thomas Hobbes (1588−1679) pressed the naturalist argument into the human body. Against the separation of an immaterial mind from an extended body, he treated sensation, imagination, and reasoning as processes belonging to the natural order. Thought, on this account, was not a visitor from a higher realm but something that happened in a body.
<RT·endnote>John Locke, <em>An Essay Concerning Human Understanding</em> (London, 1690), Book I against innate ideas, Book II on the two sources, and Book IV chap. 4 on mathematical certainty. The position is called <RT·term>empiricism</RT·term>. Locke’s denial concerns innate ideas, not the mind’s innate capacities for receiving and comparing ideas.</RT·endnote>
</p>
- <p>George Berkeley (1685−1753) attempted to dissolve the dialectic by identifying sensible objects with ideas as perceived, rather than treating perception as a report about some further material thing standing behind them. The world as sensed is therefore a world of perceived ideas, but Berkeley does not reduce reality to a private person’s perceptions: the order of nature is secured by God’s perception and is not controlled by individual will. When asked what becomes of the furniture in a room when nobody is there, he answered that God perceives it.
+ <p>George Berkeley (1685−1753) attempted to dissolve the Dialectic by identifying sensible objects with ideas as perceived, rather than treating perception as a report about some further material thing standing behind them. The world as sensed is therefore a world of perceived ideas, but Berkeley does not reduce reality to a private person’s perceptions: the order of nature is secured by God’s perception and is not controlled by individual will. When asked what becomes of the furniture in a room when nobody is there, he answered that God perceives it.
<RT·endnote>George Berkeley, <em>A Treatise Concerning the Principles of Human Knowledge</em> (Dublin, 1710), §§3 and 6 for the doctrine, §§28−33 and §146 for the argument reaching God, and §48 for the objects that persist unperceived by any man. Put again, and more accessibly, in <em>Three Dialogues between Hylas and Philonous</em> (London, 1713).</RT·endnote>
</p>
<!-- LocalWords: Hopcroft Ullman endnote unformatted dereference
LocalWords: computable cardinality uniquifier namespace strived
<!-- LocalWords: Entscheidungsproblem metamathematics geometers
-<!-- LocalWords: metamathematical Aristotelianism
+<!-- LocalWords: metamathematical Aristotelianism Praefatio
-->
-->
-->
projects / TM-2026 / commitdiff
