1890-93: Herman Hollerith (MIT and US Census Bureau)
was hired by the U.S. Census Bureau in 1880 as a statistician
after graduating from Columbia School of Mines in New York in 1879.
The 1880 census took 7 years to complete.
Processor:
After failing with data encoded on (easy ripping) paper strips,
he considered punched cards.
At MIT: Processor for tabulating data on punch cards
[monotype/Differential Analyzer].
The holes close circuits triggering mechanical counters and sorter bins.
Won the competition for the delivery of data processing equipment
for the 1890 US Census with his Punch Card Tabulating Machine,
mechanical processors (programmable with wires and plugs)
to read, count, tally and sort data on punch cards.
saved $5 million and 7 1/2 years' time.
[No multiplication? -- 1931 ``first'' IBM multiplying machine].
C.f. National Geographic on the 1900 Census.
[from here]:
«It is important to note that the typical card processing applications from the 1890's to the 1950's did
not require the use of computers! A deck of cards from a retail application, for example, could be
sorted by the category field on a card sorter, and then each category could be run through a
tabulating machine to sum the price fields of all cards in that category or similar accounting
functions.»
A description of machines for sorting and for comparing cards (by their information content).
Cards: A character of data was stored by punching holes into some of 12 rows
with a column on a card [from here]:
«The original code used for punched card data recording
had only 240 distinct punch positions per card [implying 20 columns],
but in the early 1900's, a new standard card format was introduced with 45 columns of round
holes per card and 12 punch positions in each column (540 punch positions).»
(12 rows × 45 columns)
«In 1928, Hollerith's
company, now renamed IBM, introduced the rectangular hole 80 column format, almost doubling
the amount of data that could be recorded on a card».
This one IBM could (later) patent and make most money.
(«Babbage's proposed use of cards played a crucial role in later years,
providing a precident that prevented Hollerith's company
from claiming patent rights on the very idea of storing data on punched cards»).
From this card derives the 80 columns text screen.
Encod.:
- 3 zones:
1 optional punch in first 2 rows (1:1 selection of 3 ``zone'')
× 1 optional punch in remaining 10 rows (1:1 selection)
= 33 combinations (10 digits + 22 letters + space)
- 4 zones:
optional punch in first 3 rows (4 ``zones'')
× optional punch in remaining 9 rows
= 40 combinations (all digits and letters + `*' `-' `.' space)
- Full BCD allows additional combinations:
(#1) 1st or 2nd row together with 3rd row if rest is empty, and
(#2) additional punch in 11th row if no punch in 4th or 12th row
=> 66 combinations.
- EBCD disallows BCD extension #1 => 64 combinations
(8-bit tape codes based on EBCD)
- binary zones allow any combination of punches in first 3 rows (8 zones),
and BCD extension #2 => 256 combinations.
Commerce, 1896: Founded company to sell his machines to companies
for processing buisness data
(1914, merged with two other companies;
1924, renamed to IBM).
|
- Calculator, 1893: Otto Shweiger's (Swiss engineer) Millionaire, the first efficient four-function calculator
[Burroughs'/Fahrdiagraph],
- Notat. for logic & relations, 1895:
Schröder introduces
a;b for relation composition (DeMorgan, Peirce: ab),
(reintroduces DeMorgan's???) ab for AND (Peirce: a,b).
He writes a00 for transitive closure of a,
a0 for transitive reflexive closure,
and a11 and a1 for their DeMorgan duals.
- Technology, 1897: Karl Ferdinand Braun's cathode-ray tube,
the first generation of electronic technology
[e-motor/transistor]
-- will be applied in the first electronic calculators (1939).
- Axiomatization,
1899[<>]:
David Hilbert[>] (Germany) formally axiomatizes geometry
- Logic [<>]:
The reliance on rigorous axiomatization and deduction
(by logicists as well as mathematists)
raised interest in formal languages and deductive systems.
The systems themselves became objects of mathematical study
(based on the mathematical school? - cf. [x]).
Such efforts became known as metamathematics
[x].
-
«Hilbert and his followers held that the only meaningful, or 'contentful',
parts of mathematics consist of finitary assertions about finitary objects, like
natural numbers. This includes particular statements like '234 + 123 = 357'
and generalizations like 'a + b = b + a', made with free variables. It does not
include statements, like 'for every n there is a p greater than n, such that p
and p + 2 are both prime', that contain bound variables ranging over an
infinite domain. The infinitary, or 'ideal', parts of mathematics, including
analysis and set theory, have value only in the role of facilitating the
production of finitary, contentful statements. In each case, we need to be
assured that the use of ideal mathematics does not yield anything incorrect
about the finitary part. (Instrumentalism.) The Hilbert programme called for
each branch of mathematics to be formalized and for the formalisms to be
studied metamathematically. Noting that the subject-matter of
metamathematics - sequences of characters - is finitary, Hilbert declared that
metamathematics be conducted with only finitary means. Then, once the
consistency of a formal deductive system is established, the system can
confidently be used to produce finitary results. (Consistency proofs.)
»
[x].
-
«In some cases, such as Hilbert and his followers, [metamathematics]
was part of a formalist philosophical agenda, sometimes called the Hilbert
programme. (Formalism.) Others, like Heyting, produced axiomatic versions
of the logic of intuitionism and intuitionistic mathematics, in order to contrast
and highlight their revisionist programmes (see Brouwer).
-
A variation on the mathematical theme took place in Poland under
Lukasiewicz and others. Logic itself became the branch of mathematics to be
brought within axiomatic methodology. Systems of propositional logic, modal
logic, tense logic, Boolean algebra, and mereology were designed and analysed.
-
A crucial development occurred when attention was focused on the
languages and the axiomatizations themselves as objects for direct
mathematical study. Drawing on the advent of non-Euclidean geometry,
mathematicians in this school considered alternative interpretations of their
languages and, at the same time, began to consider metalogical questions
about their systems, including issues of independence, consistency,
categoricity, and completeness. Both the Polish school and those pursuing
the Hilbert programme developed an extensive programme for such
'metamathematical' investigation. (Metalanguage; metalogic.) Eventually,
notions about syntax and proof, such as consistency and derivability, were
carefully distinguished from semantic, or model-theoretic counterparts, such
as satisfiability and logical consequence.
»
[x].
-
«The ensuing metamathematical research culminated with Gödel's
incompleteness theorems [>],
which dealt a serious blow to the Hilbert programme»
[x].
|
 Encod. + Telecom + Keyb., 1899-1901:
Donald Murray's (Australia) telegraph with typewriter-like keyboard
[qwerty/Z3].
It used a Baudot-like [<]
5-bit code but with NUL, LF, CR, DEL.
It was adopted by Western Union, and,
1930, by CCITT as ITA2 [>].
 Encod., 1901:
libraries of piano rolls (machine readable data!):
W S B Mathews writes "The Possibility of Self-Playing Instruments"
as editorial in "Music: a monthly magazine, devoted to the art, science,
technic and literature of music." Chicago, Illinois, July 1901:
«The perfecting of what are called "self-playing" instruments, of
the Aeolian and Pianola type, is going on at a great rate, and the
vast popularity of this kind of ready-made music is bound to have
a material influence upon popular musical culture in this country.
In the long run the piano self-player is likely to occupy the
leading position, numerically at least, because one of these
instruments can be bought for a couple of hundred dollars or less,
and then every household will be able to play any music for which
they care sufficiently to buy or hire from the circulating library
the necessary rolls.»
[quote from message
by Joyce Brite cf]
- Telecom, 1901: Guglielmo Marconi's wireless transatlantic transmission.
- Algrebra, 1900:
Betrand Russell [>] (Wales)
works on binary relations as foundation for functions.
He identified the concept of "domain" and the dual notion
(nameless, christianed "converse domain" in Principia Mathematica[>],
also there proposed "field" for domain = converse domain)
Notat.: uses Peano's a b for OR
(but not for relation sum?),
a b for relation composition
and -a for relation complement [CBR]
- Axiomatization, 1901/02 [<>]:
Russell[<>]
and, independently, Zermelo discover ``Russell's paradox'' of set theory.
Type Theory 1903[>]:
This leads Russell to introduce his theory of types [x]
distiguishing between sets and classes.
«An elegant version of the theory, called simple type theory,
was introduced by Ramsey.» [x]
- Technology, 1906: Lee De Forest's triode tube can amplify radio signals,
follows the diode, based on the Edison effect.
- Philosophy, 1907: L E J Brouwer's intuitionism,
a development from constructive mathematics.
«Intuitionism stresses that mathematics has priority over logic,
the objects of mathematics
are constructed and operated upon in the mind by the mathematician, and
it is impossible to define the properties of mathematical
objects simply by establishing a number of axioms»
[ref].
-
Also Poincaré (France 1854-1912):
«Poincaré was absolutely correct, however, in his criticism
of those like Russell who wished to axiomatise mathematics
were doomed to failure.
The principle of mathematical induction, claimed Poincaré, cannot be logically deduced.
He also claimed that arithmetic could never be proved consistent if one defined arithmetic by a system of axioms as Hilbert had done.
These claims of Poincaré were eventually shown to be correct».
- Axiomatization, 1908 [<>]:
Zermelo's axiomatization of set theory.
The theory now known as Zermelo-Fraenkel set theory
is the result of some modifications and clarifications,
due to Skolem, Fraenkel, and von Neumann, among others
[x].
- Telecom, 1910: Krum's teleprinter (based on Pearne's idea):
typewriter-style keyboard to enter outgoing messages,
paper roll for printing incoming messages.
- Theory, 1910: Axel Thue's (Norway) paper on the word problem for finitely presented semigroups.
- Encod., 1911: The Chinese telegraphic code book (CTC)
assigns 4 digit code numbers for 9800 Chinese characters.
[see here for CTC
and here for other code books]
  1920 [<>]:
Emil Post[>] (logician, Poland/USA), in his PhD thesis,
proved completeness and the consistency of the propositional calculus in Principia Mathematica
by introducing the truth table method
(for "true" and "false", but also for an arbitrary finite number of truth values).
Notation:
A new idea was to give a framework for inference/deduction systems
based on a finite process of symbol manipulation specified by productions.
«It would be fair to say that Post's thesis marks the beginning of proof theory»[>].
|
- Society, 1920s: About the radio: "The wireless music box has no imaginable commercial value.
Who would pay for a message sent to nobody in particular?"
- Meaning, 1923[<>]:
C K Ogden and I A Richards, in "The Meaning of Meaning",
analyzed sixteen definitions of "meaning". They reduced the function
of language to the referential and the emotive.
- Theory, 1924: «Schonfinkel introduces the idea of "Currying" functions, an idea anticipated by Frege [1893]».
- Cyber, 1924[>]:
Jakob von Uexküll's[>]
feedback loop (``function loop'')
organism[sensors/effectors]
<-> environment[sensables/effectables]
[SuB 323].
- Classes, 1924: Otto Jespersen (Danish linguist, 1860-1943)
ranks of subordination
("He(primary) writes(secondary) dreadfully(tertiary)"
or "very(tertiary) cold(secondary) water(primary)")
- Endod. + Telecom, 1925: Bell Lab's analog wire photo facsimiles
- Processor, 1925/28: Vannevar Bush's (MIT) [>]
Differential Analyzer funded by the Rockefeller Foundation:
mechanical analog [next: Mallock]
processor [tabulator/TM].
for numeric integration and differentiation
Three-fold punch card inputs:
for ``programming the relays'' [what does this mean?],
for coefficients,
and for initial values of the integration.
10,000-fold torsion amplification by electric motors.
- Theory,
1928:
Wilhelm Ackermann's [>] (Germany)
example of a non-primitive recursive but effectively computable function,
Hilbert-type-2 function (function with numbers and a number theoretic function as parameters,
later simplified to Ackermann's function) [<].
«This discovery led to classification of recursion schemas [Peter 1951, 1957]
and led Herbrand to search for a notion of effectively calculable functions».
- Logic, 1930 [<>]:
Haskell Curry's thesis (supervised by Hilbert[<]) Grundlagen der kombinatorischen Logik.
Curry's main work was in mathematical logic with particular interest in the theory of
formal systems and processes. He formulated a logical calculus using inferential rules.
His non-Hilbterian formalist philosophy of mathematics
«depends on a historical thesis that as a branch of mathematics develops,
it becomes more and more rigorous in its methodology,
the end-result being the codification of the branch in formal deductive systems.
Curry claimed that assertions of a mature mathematical theory should be construed
not so much as the results of moves in a particular formal deductive system ...
but rather as assertions about a formal system. ...
For Curry, then, mathematics is an objective science, and it has a subject matter
-- formal systems. In effect, mathematics is metamathematics.»
[x].
- Comp. Theory + Encod., 1931 [<>]:
Kurt Gödel shows the incompleteness
of any consistent axiomatization of arithmetics
(or classical analysis, set theory, or other sufficiently rich formal systems
[x])
-- the own consistency cannot be derived from/within itself
(let alone in a finitary fragment, as Hilbert[< had stipulated).
Gödel used an encoding of formulae as numbers (Gödel numbering),
and of function on them as logical operations.
This result raised questions about the encoding of formulae and the computability of deductions
in the middel of the 1930s.
«There were a number of characterizations of computability,
developed more or less independently, by logicians like
Gödel (recursiveness), Post, Church (lambda-definability),
Kleene, Turing (the Turing machine), and Markov (the Markov algorithm).»
[x].
- Encod., early 1930: CCITT's 5-bit ITA2 (International Telegraph Alphabet #2),
modifying Western Union's Murray code, with SPACE, BEL
[</>].
Used to "read" programs of the Manchester/Ferranti Mark 1.
Based on ITA2: 6-bit CCITT#4 code, 6-bit TTS (teletypesetting) code, 6-bit+parity codes, and 7-bit code with always 4 bits on.
7-bit CCITT#3 code with always 3 bits on,
a variation of Moore code.
- PL, 1932: K Menger's rules of spelling for the paranthesis-free notation [CC].
- Commerce, early 30's: IBM's 90% monopoly on puch card systems.
1932-36: Antitrust case about punch card machines against IBM and Remington-Rand.
- Theory, 30's: Abstract lattice theory (Birkhoff et.al.) [Math]
- Calculator, 1933: Mallock Machine,
an electrical analog [Differential Analyzer/Manchaster]
calculating device [Fahrdiagraph/with relays]
to solve linear simultaneous equations.
- Cyber, 1933[<]:
W B Cannon's concept of "homoeostasis" (of an organism's inner milieu) [SuB 136].
Cf. homoeostatic meshs 1961.
- Meaning, when? [<>]:
Louis Hjelmslev's (Danish linguist 1899-1965)
distinction of the meaning of signifiers
into denotative (Saussure's signified)
and connotative [Sem].
- Syntax, 1933[<>]:
Bloomfeld's analysis of sentences' constituent structure
- Syntax, 1934[<>]:
Stephen C Kleene's (USA) [>]
rules of spelling for formulae with paranthesis in most general form [CC].
|
Principle, 1934 [<]:
Binary Logic -
Arithmetics not by counting but by logic operations
[<>]
on binary numbers.
Note that older telegraph terminology talked of
marks and spaces instead of bits.
Attanasoff's idea (in a tavern)
of using base-two numbers for electronic computing
by direct logical action and not by enumeration
Zuse's considerations started in 1934 lead to the hypothesis "data processing starts with the bit"
(which he called "yes/no status" at that time,
and wrote "+"/"-" in his 1945 Plankalkül).
|
- 1934/35 [<>]:
Gerhard Gentzen (German logician, 1909-1945),
«in his fundamental paper of 1935,
expounded a radically new way of formalizing logic - natural deduction,
which he carried out for both classical and intuitionistic first-order logic.»
Natural deduction, the first non-axiomatic formalization of logic,
was «introduced independently by S Jaskowski in 1934
and Gerhard Gentzen in 1935» [x].
«All previous mathematical logicians -
including Frege, Russell and Whitehead, Hilbert, and Heyting (intuitionism,
mathematical) - had formalized logic axiomatically, their method being
modelled on the misleading analogy of formal theories. In these
formalizations, certain logically valid formulae were assumed as axioms, from
which a minimum of rules of derivation preserving logical validity yielded the
rest. This older method required ad hoc definitions of derivability from a set
of premisses (since not all rules of derivation preserved truth under a given
interpretation of the schematic letters); it often demanded much ingenuity to
obtain the formal theorems. Worse, it concentrated philosophical and logical
attention on the notion of logical truth in place of that of logical consequence.
»
«In the same paper,
Gentzen developed another method of formalization, a sequent calculus»
[x].
«He showed that any proof in either of these systems can be converted
to a proof in the other. His cut elimination theorem
- still undoubtedly the best theorem in proof theory -
showed that any sequent calculus proof can be
converted into a tableau (or truth-tree) in which formulae are steadily broken
down, not built up.» [x].
With this, proof theory[<] is usually reckoned to begin
as a discipline in its own right.
|
- Syntax, 1935[<>]:
Ajdukiewicz's (Polish logician) categorial grammar
- Meaning, 1935[<>]:
Alfred Tarski's (Poland/US, 1902-1983, pupil of Lukasiewicz[<])
The Concept of Truth in Formalized Languages
- one of the foundation stones of model theory.
He is «one of the pioneers in the study of formalized logical systems
as purely algebraic structures. He emphasized
the difference between the metalanguage, used to talk about these structures,
and the formal language [object language] whose syntax formed the system being studied»
[x].
-
Compositional semantics by truth-conditions:
«A sentence's truth is determined systematically by the
satisfaction of its parts; thus Tarski could show how to formally derive, from
the axioms and rules of the theory, sentences (so-called 'T-sentences') which
state what might intutitively be regarded as the conditions under which any
sentence of the language is true»
[x].
-
«Model-theoretic and truth-theoretic semantics provide the two leading
versions of truth-conditional semantics for natural language»
[x].
Notat.: He introduced "^" for conjunction (``and'') [TEP].
The meaning of the syntactic phrase "A ^ B"
is the logical conjunction of the meanings of "A" and of "B":
M[A ^ B] = M[A] and M[B]
- Commerce, 1935: The Social Security Act brings IBM a government contract to maintain employment records for 26 millon people.
Market of over 4 billion punch cards p.a.
- Calculator, 1935:
The Manchester Differential Analyser,
an analog [prev.: Mallock]
e-motor driven device with a disc-and-wheel device integrator component
based on the Differential Analyzer [realy?].
- Calculator, 1936[<>]:
Benjamin Burack (psychologist, Chicago) constructed first electrical logic machine:
light bulbs display the logical relationships between a collection of switches
(relays?) - i.e., based on circuit logic[<]
- Calculator, [when?]:
In the pre-war time several mechanical devices for calculating encryptions and decryptions were in use.
E.g. the German Enigma, typewriter-like,
3 (later 4 or 5) wheels, was capable of some 159 trillion possible combinations.
- 1936: Dudley's (Bell) speech coding and synthesis
- Marvin Camras's magnetic recording techniques
- Classific., when? [<>]:
Wittgenstein's question: what is the criterion for the class of games or work of art
[CvP]?
There is no sharp criterion, only a similarity in various, varying sets of features.
- Concepts are characterized by family resemblance.
«Except for technical terms in mathematics, Wittgenstein maintained that
for most concepts, meaning is determined not by definition, but by family resemblance.
Such terms can be defined only in terms of similarities and representative "prototypes"».
This sparked research into prototypes (JL Austin, L Zadeh, F Lounsbury),
resulting in "prototype theory" [>].
|
1882 [>])
meant that there are mathematical object
which cannot be "reached" by applying operations
= which cannot be constructed algebraically
[-> which cannot be represented uniquely by (finite compositions of) symbols]
" for logical OR?
" for "is an element of",
and invented "
" ("C"onsequence?) for implication
and existence quantor "
" (reversed "E") [
[