MATHEMATICIANS TO PROBABILITY THEORY - cvut.czeuler.fd.cvut.cz/~hyksova/esu2007_hyksova.pdf · •...

CONTRIBUTION OF CZECH MATHEMATICIANS TO PROBABILITY THEORY ESU 5, July 24, 2007

Magdalena Hykšová

CTU Prague

Bernard Bolzano (1781 – 1848)

Tomáš Garrigue Masaryk (1850 – 1937)

Václav Šimerka (1818 – 1887)

Emanuel Czuber (1851 – 1925)

Bohuslav Hostinský (1884 – 1951)

Karel Rychlík (1886 – 1968)

Otomar Pankraz (1903 – 1976)

• contributions to the development of probability theory as a mathematical discipline (Emanuel Czuber, Bohuslav Hostinský)

• contributions dealing with the foundations of probability theory (Karel Rychlík, Otomar Pankraz)

• contributions dealing with interpretations of probability theory (Bolzano, Masaryk, Šimerka, Czuber, Pankraz)

• interesting rather from the philosophical or epistemological point of view

• in common: conception of probability theory as a substantial tool for philosophical and scientific cognition

• recognition of the significance of probability theory for epistemology

• understanding its significance for our everyday life

Still interesting, enlightening and inspiring

School mathematics:

Probability theory

≡ throwing dices or coins, drawing balls ...

≡ artificial examples without any connection to reality

⇒ unloved discipline

Probability theory ≡ one of the most interesting and important disciplines

• close relation to our everyday considerations:

hypothesis about our surroundings and about ourselves

• global warming and other ecological issues • human evolution • psychological processes • reasons and causes of our illnesses • credibility of historical events, witnesses, partners, ...

→ trustworthiness of these hypotheses

(one of the important motivations for the development of probability theory)

To which extent the available evidence supports a hypothesis in question?

To which extent the available evidence supports a hypothesis in question?

?

Probability theory ≡ one of the most interesting and important disciplines

• close relation to our everyday considerations: hypothesis about our surroundings and about ourselves → trustworthiness of these hypotheses

• we are surrounded by randomness o organic world – tissue cells, vegetations, people themselves ...o inorganic world – molecules of gas and liquid, crystals ... o random meetings, accidents o illness – chance for healing, surviving o defects of materials o failures of railway safety systems, nuclear power stations ...

substantial for physics, biology, medicine, humanities, engineering ...

≡ unique opportunity to persuade scientists from other branches about the necessity of mathematics and its methodologies

FOUNDATIONS OF PROBABILITY THEORY Andrei Nikolajevič Kolmogorov (1903 – 1987) Grundbegriffe der Wahrscheinlichkeitsrechnung, 1933 axiomatization of probability theory

⇒ acceptance of probability theory as a “true” mathematical discipline

First reactions: Reviews: 1933 – 35 K. Dörge, W. Feller, H. L. Rietz, G. Doetsch

Papers:

1934 Z. Łomnicki and S. Ulam, E. Hopf, J. L. Doob

Textbooks:

1937 Harald Cramér: Random Variables and Probability Distrib.

REACTIONS OF CZECH MATHEMATICIANS

Karel Rychlík (1886 – 1968) professor of mathematics at the Czech Technical University

private associate professor at Charles University in Prague

University lecture:

1933/34 , 1936/37 Introduction to probabilty calculus (from the axiomatic point of view)

Facultative lecture at Charles University

Textbook:

1938 Introduction to probability calculus

[1937 Harald Cramér: Random Variables and Probability Distrib.]

OTOMAR PANKRAZ (1903 – 1976)

1930 – 31 actuary

1931 – 39 assistant of Karel Rychlík at the Czech Tech. University in Prague

Interest in the development of probability theory

1931 review of Mises’ book Wahrscheinlichkeitsrechnung (detailed analysis, quotes also critics, supports Mises)

1933 Zur Grundgleichung für den zeitlichen Zerfall der statistischen Kollektivs (journal Aktuary Sciences)

1935 associate prof. at Charles Univ. in Prague (1933 paper)

1938 associate prof. at the Czech Technical Univ. in Prague

1939 On Probability Axioms (Rozpravy Jednoty pro vědy pojistné)

1940 On the Concept of Probability (ČPMF)

Critique of Kolmogorov‘s Axioms

Inspired by Hans Reichenbach (Wahrscheinlichkeitslehre, 1935)

• Probability introduced as a one-argument function P(A) → conditional probability (two-argument function) introduced by

an additional definition aside from the axioms:

( )( ) ( )AP A BP B P A

∩=

→ it is shown that it also satisfies the axioms (for a fixed A)

• distinguishing conditional and unconditional probabilities does not have a logical reason

• in the proof that satisfies the axioms, A is supposed to be constant – it is not satisfactory

( )AP B

PROBABILITY = TWO-ARGUMENT FUNCTION

Critique of Kolmogorovov‘s Axioms

Inspired by Hans Reichenbach (Wahrscheinlichkeitslehre, 1935)

• Probability introduced as a one-argument function P(A) → conditional probability (two-argument function) introduced by

an additional definition aside from the axioms:

( )( ) ( )AP A BP B P A

∩=

→ it is shown that it also satisfies the axioms (for a fixed A)

• distinguishing conditional and unconditional probabilities does not have a logical reason

• in the proof that satisfies the axioms, A is supposed to be constant – it is not satisfactory

( )AP B

PROBABILITY = TWO-ARGUMENT FUNCTION

FUNDAMENTAL CONCEPT = CONDITIONAL PROBABILITY

Fundamental concept = conditional probability In the life: „I’m sure I could be a movie star, if I could get out of this place”

(Pianist – Billy Joel)

Fundamental concept = conditional probability In the life: „I’m sure I could be a movie star, if I could get out of this place”

(Pianist – Billy Joel)

His probability for becoming a movie star is high – but conditionally on his getting out of this place

Every prediction, above all every probability evaluation is

conditional; not only by a mentality or psychology of the

individual in question, but also – and above all – by the

degree of knowledge ... (Bruno de Finetti, 1974)

All probabilities are conditional

In the life: • Every event occurs under certain conditions • No dice is perfect • No board is absolutely flat • • •

Propositions in scientific theories

Causal proposition: „An event B occurs ⇔ an event A occurs before it“ consequence cause

Propositions in scientific theories

Causal proposition: „An event B occurs ⇔ an event A occurs before it“ consequence cause Randomness proposition ..... VX : „An event B occurs ⇔ one of elementary events of a set C

occurs before it

{ , , , ...C }ξ η ϕ= ... arbitrary set (possible causes of the event B)

[|C|=1 → causal proposition]

Event B:

Set C of possible causes: • The structural engineer made an erroneous calculation • The geologist elaborated an erroneous opinion • The site manager did not keep the project • The supplier provided bad material • The neighbor damaged the subsoil when extending a cellar

• • •

Who should be arrested?

Event B

Randomness proposition ..... VX : „An event B occurs ⇔ one of elementary events of a set C

occurs before it“

{ , , , ...C }ξ η ϕ= ... arbitrary set

Statistical theory: by means of randomness propositions , , .X YV Vdeals with random events X, Y, ...

Every statistical theory can be regarded as a set of certain randomness propositions: { }, , ...X YV VΩ =

Characterization of relations between propositional forms with the help of sets

A theory which deals with random events X, Y, ... can be regarded as a system of randomness propositions VX ,VY ,...

Propositional forms: X YV V⇒ .... X YV ⇒ ¬V ....

Domains of truth:

X YM M⊆

X YM M∩ = ∅

X YV ⇒ ¬V X YV V⇒

X YM M∩ = ∅ X Y XM M M∩ =

X YV ⇒ ¬V X YV V⇒ X p YV V⇒


( )( )X Y

X

M MM

μμ

∩



( )( )X Y

X

M MM

μμ

∩

Probability calculus:

prop. form VX with the random event X with domain of truth MX the set of realizations MX



( )( )X Y

X

M MM

μμ

∩ ... ( )( ) |( )

P X Y P Y XP X∩ =

Probability calculus:

prop. form VX with the random event X with domain of thruth MX the set of realizations MX probabilistic implication ⇔ conditional probability

X p YV ⇒ V ( , ) ( )YP Y X P X=

• Erroneous calc. of the structural engineer • Erroneous opinion of the geologist • The site manager did not keep the project • The supplier provided bad material • The neighbor damaged the subsoil

when extending a cellar • • •

Hypothesis

Available evidence

Evidence B: C:

Evidence: all ravens I have ever observed were black

Evidence: all ravens I have ever observed were black Hypothesis: all ravens are black

INDUCTIVE LOGIC Deductive logic: premises logically entail the conclusion, their truth provides a guarantee of the truth of the conclusion Inductive logic (Rudolf Carnap, 1958) the aim: to describe and justify inductive conclusions (are not fully guaranteed by premises)

Specification of a measure of the degree to which an evidence E supports a hypothesis H

= inductive (logical) probability of H supported by E

P(E|H)P(H)P(E H)P(H|E) ( ) ( )P E P E∧= = for P(E) 0≠

⇒ all probabilities are conditional (it is impossible to speak about the probability of a hypothesis, only about its probability based on a given evidence)

Similar approach: Karl Popper, 1959: The Logic of Scientific Discovery, App. IV Alan Hájek, 2003: What Conditional Probability Could not Be

PROBABILITY INTERPRETATIONS

Within the probability theory, two directions are being developed: a philosophical one and a mathematical one. They are usually not progressing together. Mathematical side of the subject keeps a great advance before the philosophical one... A closer connection of both directions seems to me to be in a mutual interest.

Emanuel Czuber, 1923: Die philosophische Grundlagen der

Wahrscheinlichkeitsrechnung

The theory of probability has a mathematical aspect and a foundational aspect. There is a remarkable contrast between the two. While an almost complete consensus and agreement exists about the mathematics, there is a wide divergence of opinions about the philosophy.

With a few exceptions ... all probabilists accept the same set of axioms for the mathematical theory, so that they all agree about what are the theorems. Yet in the twentieth century at least, four strikingly different interpretations of this mathematical calculus have been developed, and each of them has adherents today. This book will give a detailed account of these interpretations ...

Gillies, D., 2000: Philosophical Theories of Probability


EPISTEMOLOGICAL INTERPRETATIONS

Probability = degree of knowledge or belief (dependent on the individual)


EPISTEMOLOGICAL INTERPRETATIONS

Probability = degree of knowledge or belief (dependent on the individual)

OBJECTIVE INTERPRETATIONS

Probability = feature of the objective material world (independent of the individual, nothing to do with human knowledge or belief)

LOGIAL INTERPRETATION

BERNARD BOLZANO (1781 – 1848), 1837 Tomáš Garrigue Masaryk (1851 – 1925), 1883 Johannes von Kries (1853 – 1928), 1886 William Ernst Johnson (1858 – 1931), 1921 John Maynard Keynes (1883 – 1946), 1921 Ludwig Wittgenstein (1889 – 1951), 1921 EMANUEL CZUBER (1851 – 1925), 1923 Friedrich Waismann (1896 – 1959), 1930 Harrold Jeffreys (1891 – 1989), 1939 OTOMAR PANKRAZ (1903 – 1976), 1939 Rudolf Carnap (1891 – 1970), 1950

Probability = degree of rational belief

LOGIAL INTERPRETATION

BERNARD BOLZANO (1781 – 1848), 1837 Tomáš Garrigue Masaryk (1851 – 1925), 1883 Johannes von Kries (1853 – 1928), 1886 William Ernst Johnson (1858 – 1931), 1921 John Maynard Keynes (1883 – 1946), 1921 Ludwig Wittgenstein (1889 – 1951), 1921 EMANUEL CZUBER (1851 – 1925), 1923 Friedrich Waismann (1896 – 1959), 1930 Harrold Jeffreys (1891 – 1989), 1939 OTOMAR PANKRAZ (1903 – 1976), 1939 Rudolf Carnap (1891 – 1970), 1950

Probability = degree of rational belief

Probability theory = extension of deductive logic

BERNARD BOLZANO (1781 – 1848)

* October 5, 1781 in Prague 1796 – 1804 studies: philosophy, mathematics, theology (Charles University in Prague) 1804 religion teacher at Charles U. 1820 suspended (alleged dispense

of improper ideas) since 1820 lived mainly outside of Prague † December 18, 1848 in Prague


Lehrbuch der Religionswissenschaft, ein Abdruck der Vorle-sungshefte eines ehemaligen Religionslehrers an einer katho-lischen Universität, von einigen seiner Schüler gesammelt und herausgegeben. Sulzbach, 1834. Published anonymously, texts of Bolzano’s religious lectures at the university before his deposal in 1820.

Wissenschaftslehre. Versuch einer ausführlichen und größten-theils neuen Darstellung der Logik mit steter Rücksicht auf deren bisherige Bearbeiter. Sulzbach 1837 [finished around 1830]

Probability theory = extension of deductive Logic


Lehrbuch der Religionswissenschaft, 1834

Bolzano builds the probability theory in order to defend the Holy Scripture against attempts to shake the belief In neuerer Zeit hat man aber auf verschiedene Art gesucht, den historischen Glauben, besonders in Hinsicht auf Wunder, wan-kend zu machen, und behauptet, dass Erzählungen von Wun-dern, vornehmlich solchen, die sich vor vielen Jahrhunderten ereignet haben, nie strenge erweislich wären. Dergleichen Be-hauptungen haben z. B. Joh. Crayg, Dav. Hume, Bolinbroke, J. J. Rousseau, G. F. Gahrdt, Im. Kant u. m. A. Vorgetragen...

Wer einige Kenntnisse in der Buchstabenrechnung hat, wird auch noch folgende mathematische Sätze leicht zu verstehen vermögen, die ich nur darum hier beifügen will, weil sie zur gründlichen Widerlegung jener Einwürfe dienen, die selbst von Mathematikern, z. B. von Joh. Crayg, gegen die Möglichkeit der historischen Beglaubigung eines Wunders mit einem Anscheine von Gelehrsamkeit vorgebracht werden sind.

JOHN CRAIG (†1731)

1696 Theologiae christianae principia mathematica

– Matematické základy křesťanské teologie (publ. 1699) Investigated the reliability of a historical testimony with the special emphasis on the story of Christ

For example:

Probability of a history transmitted over a distance D in a time T by M successive witnesses:

= + − + ⋅ + ⋅2 22 21( ) qkP x M s T D

t d

x – probability attached to the first witness

s – the suspicion (negative) attached to each of the remaining witnesses

k – suspicion arising over a time t

q – suspicion arising over a distance d

s, k, q < 0

Second comming of Christ

Craig: this event would coincide with the disappearance of faith (Gospel of Luke, 18.8)

Probability of the testimony of the four Gospels would not vanish entirely until the year 3150 (modern recalculation: 3156)

INTERPRETATION OF CRAIG’S PROBABILITY 1986 S. M. Stigler: John Craig and the Probability of History: from the Death of Christ to the Birth of Laplace A simple interpretation that shows that Craig proceeded as a highly sophisticated statistician of the 20th century

Craig‘s probability ≠ probability in our sense

=¬

Pr ( | )log Pr ( | )

E HP E H

E – witness (evidence) at present H – hypothesis, event P – Craig‘s „probability“ Pr – probability in „our“ sense Pr (H ) – apriori probability independent of a witness

Second comming of Christ

Craig: this event would coincide with the disappearance of faith (Gospel of Luke, 18.8)

Probability of the testimony of the four Gospels would not vanish entirely until the year 3150 (modern recalculation: 3156)

Attempt to prevent the society from an adverse effect of the opinion often dispensed by various sects that the Second Coming was imminent

→ criticized by theologians as well as mathematicians

Nevertheless, in his 1986 paper Stigler argued that Craig’s work was misunderstood and substantially underestimated. He emphasized that there did not exist a unique definition of probability at that time and Craig’s concept differed from the later one measured on the interval [0,1]. Using today tools, Stigler proposed an interpretation of Craig’s probability as the logarithm of the likelihood ratio, which shows him to have been operating intuitively as a highly sophisticated twentieth-century statistician.


Wissenschaftslehre, 1837

Probability theory = extension of deductive Logic

Relative validity of a proposition M with respect to propositions A, B, C, D,...

|set of all cases where besides A,B,C,D,... a proposition M is true| =

|set of all cases where propositions A,B,C,D,... are true|

§161*

... so fragen wir bei der vergleichungsweisen Gültigkeit des Satzes M hinsichtlich auf gewisse andere Sätze A, B, C, D,... nach dem Verhältnisse, in welchem die Menge der Fälle, bei welchen die A, B, C, D,... wahr werden, zur Menge der Fälle stehet, bei welchen nebst A, B, C, D,... auch noch M wahr wird. Wahrscheinlichkeit aber nenne ich dieses Verhältniß, weil es mir däucht, daß wir ... unter der Wahrscheinlichkeit wirklich nichts Anderes, als ein solches Verhältniß zwischen gegebenen Sätzen verstehen, ohne vorauszusetzen, daß diese Sätze eben von einem denkenden Wesen vorgestellt und geglaubt werden müßten.




(m X ) ... Measure for the set of the cases where X is true

Bolzano‘s probability = the degree of justification of hypothesis M on the basis of an evidence E A B C= ∧ ∧ ∧K:

( ( )) ( )( | ) ( ) (L

Lm M A B C m M EP M E m A B C m E

∧ ∧ ∧ ∧ ∧= =∧ ∧ ∧

)




(m X ) ... Measure for the set of the cases where X is true

Bolzano‘s probability = the degree of justification of hypothesis M on the basis of an evidence E A B C= ∧ ∧ ∧K:

( ( )) ( )( | ) ( ) (L

Lm M A B C m M EP M E m A B C m E

∧ ∧ ∧ ∧ ∧= =∧ ∧ ∧

)

E A B C= ∧ ∧ ∧ • • •

Premises:

E1 ... no traffic jam occurs

E2 ... the chief will not want any additional work

( )1 2 pE E∧ ⇒ H

Premises:

E1 ... no traffic jam occurs

E2 ... the chief will not want any additional work

E3 ... I will not get stuck in a lift

( )1 2 3 pE E E ′∧ ∧ ⇒ H

1 2 3E E E E= ∧ ∧ ∧ • • •

Jan Berg, 1987 Introduction to the new edition of Wissenschaftslehre:

Bolzano was therefore the first philosopher who framed the concept of the inductive probability Comparison of theories of Bolzano, Wittgenstein and Carnap

Probability = a relation between a hypothesis and its evidence

⇒ Bolzano‘s probability concept has formal properties of the concept of a conditional probability

Emanuel Czuber, 1923: Die Philosophischen Grundlagen der Wahrscheinlichkeitsrechnung Cites Bolzano, von Kries, ... Erste Tagung für Erkenntnislehre der exakten Wissenschaften Prag, 15. – 17. September 1929

Conference on Epistemology of exact sciences iniciated by Wiener Kreis and Gesellschaft für empirische Philos. in Berlin on the occasion of Tagung der Deutschen Physikalischen Gesellschaft und der Deutschen-Mathematiker Vereinigung

1930 Annalen der Philosophie taken over by Carnap and Reichenbach and published as Erkenntnis

→ own publication series of the Vienna Circle

Erste Tagung für Erkenntnislehre der exakten Wissenschaften, Prague, September 15. – 17., 1929

• Hans Hahn read the program declaration Wissenschaftliche Weltauffassung – der Wiener Kreis (Scientific Conception of the World – Vienna Circle) critique of pseudo-sciences aim: unified science purified from a “slag of historical languages” tool: logical analysis of a language Carnap: expresions have a sense only inside a (artificially

created) language deductivesystem

• lectures devoted to the foundations of mathematics, logic and science in general

• then most important representatives of logical probabilities met there

• ideas of Bolzano, Keynes, Wittgenstein, Waismann, ... met there

• Erkenntniss I (editoři: Rudolf Carnap, Hans Reichenbach)

Explicit citation of Bernard Bolzano:

Philipp Frank: Eröffnungssprache Fried. Waismann: Logische Analyse des Wahrscheinlichkeitsbegriffs Walter Dubislaw: in der Discussion

TOMÁŠ GARRIGUE MASARYK (1850 – 1937) * March 17, 1850 in Hodonín (Göding; in Moravia)

1865–69 German grammar school in Brünn 1869–72 Grammar school in Vienna 1872–76 University in Vienna (PhDr.)

Philosophy, Philology; Zimmermann, Brentano

1876–77 University Leipzig Charlotte Garrigue – ♥1878, USA

1878 Privat associate prof. in Wien 1882 Professor at Charles Univ. in Prag

1891–93, 1900–14 Austrian Council 1914–18 Exil (France, Russia – Czechoslovak legion)

1918–36 President of Czechoslovakia

† 14. September 1937 in Lány

TOMÁŠ GARRIGUE MASARYK (1850 – 1937) University in Vienna, Associate professor, 1878: Suicide as the Social Phenomenon of Present Time (5 philosophers with the greatest influence: David Hume) Charles University in Prag, entrance lecture, Oktober 16, 1882: David Hume’s Scepsis and the Probability Calculus. A contribution to the History of Logic and Philosophy (published: in Czech 1883, 45 pp., in German 1884, 15 pp.) Translation, 1883: D. Hume: Eine Untersuchung über die Prinzipien der Moral (An Enquiry Concerning the Principles of Morals) ... Deutsch von TGM

David Hume’s Scepsis and the Probability Calculus. A contribution to the History of Logic and Philosophy, 1884

The fundamental idea of Hume’s Scepsis:

Experience and reason contradict each other

All sciences based on experience are unsure and logically groundless

To the contrary, mathematics is an absolutely safe Science

Mathematics alone deserves our confidence, sciences based on experience are unsafe since the understanding of causal connections evades us...

[probability that I will wake up tomorrow again based only on my previous experience: n/(n+1) ... Laplace]

Attempts to disprove Hume’s scepsis – a historical outline:

• Scottish school – Thomas Reid (1763), James Beattie (1770), James Oswald (1766)

• Immanuel Kant (1724 –1804) Kritik der reinen Vernunft, 1781

• Friedrich Eduard Beneke (1798 – 1854) System der Metaphysik und Religionsphilosophie, 1840 – psychological arguments

• Johann Georg Sulzer (1720 – 1779), 1755 D. Hume: Philosophische Versuche Ueber Die Menschliche Erkenntniss, 1755 – anonyme Übersetzung + Bemerkungen the first German attempt to disprove Hume’s scepsis, similar arguments as Beneke

Inductive logik, probability theory – in the connection to Hume • Johann Georg Sulzer (1755) • Moses Mendelssohn (1756) • Joseph Marie Degérando (1804) • Sylvestre-François Lacroix (1816) • Siméon Denis Poisson (1837)

Inductive logik, probability theory – in general • Gottfried Wilhelm Leibniz (1703) • Jacob Bernoulli (1713) • Pierre-Simon Laplace (1749 – 1827) • Adolphe Quetelet (1846) • Rudolf Herschel (1850) All these newer treatises are missing an explicit reference to Hume; they are therefore missing, I would like to say, a true point.

Hume himself spoke much about probability, but it seems that he did not know the mathematical rules of probability calculus, since he was not able to distinguish subjective and objective probability, and it is therefore understandable how he came to his sceptical theory of induction...

What is missing:

Bernard Bolzano:

Lehrbuch der Religionswissenschaft, 1834

– Hume is explicitely cited

Wissenschaftslehre, 1837

– the foundations of inductive logic

1886 – 87 fight for the truth: supposititious old Czech manuscripts

wurde er auf einen Schlag einer breiteren Öffentlichkeit bekannt, als er sich in den Streit um zwei angeblich aus dem Mittelalter stammende, in Wirklichkeit aber zu Anfang des 19. Jahrhunderts gefälschte Handschriften einschaltete. In der Zeitschrift Athenaeum ließ er die Gegner der Echtheit der Königinhofer und Grünberger Handschriften zu Wort kommen und vertrat vehement die Meinung, dass eine moderne Nation sich nicht auf eine erfundene Vergangenheit berufen solle.

Königinhofer Handschrift

1817 Two supposedly old Czech manuscripts were found: • Dvůr Králové nad Labem (Königinhof an der Elbe)

(Václav Hanka, end of the 13th century)

• Zelená Hora (Grünberg, 9 th – 10 th century)

1858 Anonymous author called them falsificates (newspaper) → great scandal → the newspaper publisher was given a prison sentence → amnesty from the emperor Franz Joseph → other authors responded with a vehement defense of manuscripts

An important role in the national revival

Paintings inspired by the manuscripts

Mikoláš Aleš

Josef Mánes

1886 New discussion: Athaeneum (published by Masaryk)

Jan Gebauer (philologist, literary historian) Philological reasons for the falsification:

• grammatical “ oddities“ – deviations from of the Czech grammar of that time

• concurrent occurrence of “suspicious“ forms in manuscripts and in the works from the 19th century (before the discovery of the manuscripts)

Other reasons (Masaryk, Gebauer, Goll etc.): • historical • sociological • aesthetical • paleographial

1967 falsification definitively proved

Jan Gebauer (philologist, literary historian) Philological reasons for the falsification:

• grammatical “ oddities“ – deviations from of the Czech grammar of that time

• concurrent occurence of “suspicious“ forms in manuscripts and in the works from the 19th century (before the discovery of the manuscripts)

Josef Kalousek (historian) and other defenders of authenticity: these oddities and coincidence are only accidental

August Seydler [physicist, Masaryk’s friend], 1886: What is the probability that they are really accidental?

1. “Accidental“ oddities

According to Gebauer: KH contains ca. 5400 words, among them 700 oddities probability that they are all accidental:

<⋅ 91

3 10 P

We have 3 480 Million equally weighted arguments against one to believe that all oddities can be attributed to the mere chance. 2. Concurrent occurrence of “suspicious“ forms in

manuscripts and in the works from the 19th century

< 141

10 P

The oddities and coincidence require an explanation

< ⋅⋅ 9 11 1

3 10 10 P 4

Example: A man who does not know the laws of nature learned from historical sources, how long has the sun raised every day (about 2 000 000 times). Suddenly he starts to doubt, whether the sun raises tomorrow again. Was will we think about him? This man has 1 500-times more reasons for his doubt than for the hypothesis of the accidental occurance of the oddities in the manuscripts! And he has 50 million times more reasons for his doubts than we have for the hypothesis of the accidental concurren occurance of suspicious forms in manuscripts and in the works from the 19th century!

EMANUEL CZUBER (1851 – 1925) January 19, 1851 born in Prague

1875 supply teacher at the IInd German realschule in Prague 1876 assistant professor at the Technical University in Prague:

compensatory calculus, later probability theory

1877 full professor at IInd German Realschule in Prague 1886 professor of mathematics at the Technical University in Brünn 1890–91 rector of Technical University in Brünn 1891 professor of mathematics at Technical University in Wien

lectures on probability theory, insurance-technical course 1894–95 rector of the Technical University in Wien 1903–13 dean of the Faculty of Applied Mathematics and Physics 1918 honorary doctor in München August 22, 1925 Gnigl bei Salzburg

Die Entwicklung der Wahrscheinlichkeitstheorie und ihrer Anwendungen, 1899, 271 pp.

1st chapter: foundations of probability theory from the historical and philosophical point of view (besides well-known names cites e.g. J. von Kries and C. Stumpf)

Further parts: various applications of probability theory

Each topic contains the outline of the historical development

The greatest stress: idea formation, philosophical aspect

Die philosophischen Grundlagen der Wahrscheinlichkeitsrechnung, Teubner, 1923, 343 pp. Solely devoted to the philosophical foundations of probability theory

Logical interpretation of probability

Stress on its significance for epistemology and natural philosophy

Cites also Bernard Bolzano

SUBJECTIVE INTERPRETATION VÁCLAV ŠIMERKA (1818 – 1887), 1882

Frank Plumpton Ramsey (1903 – 1933), 1925

Bruno de Finetti (1906 – 1985), 1937

Probability = degree of belief of a particular individual

Realistic approach

– deals with real concepts, with a subjective acceptance or rejection of hypothesis

... corresponds to our everyday considerations



Bruno de Finetti (1906 – 1985), 1937

Leonard Jimmie Savage (1917 – 1971), 1954


Important Role: conditional probabilities

Aposterior probability P(H|E) = degree of belief in a hypothesis H based on the evidence E (situation, circumstances, witnesses)

Bayes theorem: P(H)P(E|H)P(H|E)= ( )P E



Bruno de Finetti (1906 – 1985), 1937



Important Role: conditional probabilities

Aposterior probability P(H|E) = degree of belief in a hypothesis H based on the evidence E (situation, circumstances, witnesses)

Bayes theorem: P(H)P(E|H)P(H|E)= ( )P E

Problem: numerical formulation

One of possible solutions: analogy of a betting system

VÁCLAV ŠIMERKA (1818 – 1887)

Studies: Philosophy in Prague Theology in Hradec Králové

1845 Priest

1852 Study of physics → Grammar school in České Budějovice (supply professor)

1862 Parish priest Slatina u Žamberka

1886 Parish priest Jenšovice u Vysokého Mýta

VÁCLAV ŠIMERKA (1818 – 1887) Síla přesvědčení. Pokus v duchovní mechanice. ČPMF 1882 Die Kraft der Ueberzeugung. Ein mathematisch-philos. Versuch. Sitzungsberichte der Philosophisch-Historischen Classe der Kaiserlichen Akademie der Wissenschaften 104 (1883), Wien

Concept of Conviction Similarly as white, grey, black, red, blue etc. are summarized by the concept of color, also the concepts of feeling, conjecture, possibility, probability, hypothesis, belief, knowledge, certainty etc. summarized by one concept, namely conviction.

A new, first term in the sequence: empty mind (a proposition in question is completely unknown or arguments for and against are in equilibrium)

How can the conviction be expressed by numbers For this purpose the probability calculus is exceptionally convenient, since our conviction about the possibility of an event increases in the same rate as does the mathematical probability, that is, everything is more believable, the more it seems to be probable. The terms in the sequence in Nr. 1, namely empty mind, feeling, ... up to knowledge and certainty can therefore be expressed by numbers between 0 and 1, where 0 corresponds to none, 1 the highest conviction. Causes or sources of the conviction = „grounds“ their power = „probability“ Grounds corresponding to v=0 are called empty

An imperfection of a conviction = difference between the complete knowledge and the given

conviction v:

1 , 1v thus vε ε= − = −

1ε = ... complete imperfection (empty mind) Convictions ν, ν', …,

1 , ' 1 ', '' 1 '', ...v v vε ε ε= − = − = − corresponding imperfections

Resuling power of conviction V: 1 (1 )(1 ')(1 '')V v v v− = − − − L

[The imperfection of a human conviction = a product of imperfections of its grounds]

' ''v v v= = = =L 0 ... empty mind, V=0

empty grounds provide no belief

Conviction product: 1 (1 )(1 ')(1 '')V v v v− = − − − L

' ''v v= = =L 0 ...... V v=

In an empty mind every ground enroots with its full power This is attested not only by the experience from schools and common men, many of which believe even very shaky novels and stories, but also the experiences of missionaries who give evidence that Christianity enroots the best in the nations with disordered minds, when their original superstitions were rebut, without being substituted by anything else; otherwise is it much more difficult. ... The empty mind can therefore be deceived by false grounds, what would be otherwise not so simple. It is clear that this is the basis of the old immoral principle: slander, something will stick in the memory.

The objection that the calculation with conviction is not reliable because it is based on probability which provides only a possibility, is disproved by the experience with insurance companies.

Also here the old rule holds:

Every calculus is better than no calculus.

Moreover, it can not harm our century overfull of materialism to calculate with something spiritual. Because of these and similar reasons I believe that I will not remain a lonely labourer on this new field.

Moreover, it can not harm our century overfull of materialism to calculate with something spiritual. Because of these and similar reasons I believe that I will not remain a lonely labourer on this new field. Cited by:

Tomáš Garrigue Masaryk (1850 – 1937) Report Logika, Athenaeum, 1884 „an ingenious piece of work“

Moreover, it can not harm our century overfull of materialism to calculate with something spiritual. Because of these and similar reasons I believe that I will not remain a lonely labourer on this new field. Cited by:

Tomáš Garrigue Masaryk (1850 – 1937) Report Logika, Athenaeum, 1884 „an ingenious piece of work“ Further development independent of Šimerka Frank Plumpton Ramsey (1903 – 1933), 1925

Bruno de Finetti (1906 – 1985), 1937


KAREL VOROVKA (1879 – 1929)

1897 – 1901 Studies: Charles University in Prague, → Professor at the Realschule

1921 Associate prof.: Philosophy of natural sciences

1927 Charles University, Faculty of Natural Sciences Professor: philosophy of natural sciences

Filosofický dosah počtu pravděpodobnosti, 1912 Philosophical comprehension of probability calculus

Critique of the inductive logic and subjective interpretations (also Šimerka)

CONTRIBUTIONS TO THE DEVELOPMENT OF PROBABILITY THEORY

• Geometrical probability

• Emanuel Czuber (1851 – 1925)

• Bohuslav Hostinský (1884 – 1951) • Markov chains

• Bohuslav Hostinský (1884 – 1951)

GEOMETRICAL PROBABILITY

Classical definition of probability: – based on combinatoric considerations – for example: probability that a duck

comes twice in two throws:

number of cases favourable to the event P = number of all cases (equally possible)

Geometrical probability concept: Extension of classical definition of probability to situations where the set of elementary events is uncountable

Example – lost keys:

Example – lost keys: What is the probability that the keys are lying on AB?

| || |ABP AC=

Example – lost wallet:

What is the probability that I hit the black circle inside the target provided I hit the target at all?


Area of the black circleP = Area of the biggest circle


Area of the black circleP = Area of the biggest circle

Probability that a point X hitting the set B hits also the set A:

↑ ↑Measure of the set AP(X A | X B) = Measure of the set B

Points on lines, curves ... measure: length

( | ) CDP X CD X ABAB

↑ ↑ =

Points in regions in plane/space ... measure: area/volume

↑ ↑m(A)P(X A | X B) = m(B)

Lines in plane measure:

: cos sin 0r x y qϕ ϕ+ − =

B

dqdϕ∫∫

Probability that a line ( , )q ϕ , randomly chosen in M,

is contained also in M1 (provided M1 is a subset of M):

11(( , ) | ( , ) ) A

A

dqd

P q A q Adqd

ϕ

ϕ ϕϕ

↑ ↑ =∫∫

∫∫

ROOTS OF THE THEORY • Louis Leclerc, Comte de Buffon (1707 – 1788), 1777

Needle and several other problems

• Pierre Simon de Laplace (1749 – 1827), 1812

• Isaac Todhunter (1820 – 1884), 1857

• Since 1865: British journal Mathematical Questions with Their Solutions from the ‘Educational Times‘: various problems and exercises concerning geom. probability

James Joseph Sylvester (1814 – 1897) Morgan William Crofton (1821 – 1895) Thomas Archer Hirst (1830 – 1892) Arthur Cayley (1821 – 1895) and others

• French mathematicians:

Gabriel Lamé (1795 – 1870) Joseph Bertrand (1822 – 1900) Joseph-Émile Barbier (1839 – 1889)

EMANUEL CZUBER (1851 – 1925)

Zur Theorie der Geometrischen Wahrscheinlichkeiten, 1884 Sitzungsberichte der mathematisch-naturwissenschaftlichen Classe der kaiserlichen Akademie der Wissenschaften Wien 90, 719 – 742

Geometrische Wahrscheinlichkeiten und Mittelwerte, 1884

The first monograph solely devoted to geometrical prob.

State of the art + original ideas

Historical development from Buffon up to “present”

Explicit reference to French and English predecessors

→ bridge between France and England

Expanding results of Crofton (greatest credit)

Theoretical expositions + examples

Meeting problem (problem XXII, pp. 45–47) Formulation, incorrect solution: H. Laurent: Traité du calcul des probabilités, 1873 A and B are set to meet at a given place. Unfortunately, they forgot the exact time of their meeting, they only remember that is is between 2 and 3 PM. They both choose their arrival time uniformly in this interval and each waits 10 minutes (or until 3, whichever comes first) for the other to come. What is the probability that they meet?

A and B are set to meet at a given place. Unfortunately, they forgot the exact time of their meeting, they only remember that is is between 2 and 3 PM. They both choose their arrival time uniformly in this interval and each waits 10 minutes (or until 3, whichever comes first) for the other to come. What is the probability that they meet? The sample space: { }Ω , : 2 , 3A B A Bt t t t= ≤ ≤ Favourable cases: { } { }= − ≤ , meet | | 1/ 6A BA B t t

A and B are set to meet at a given place. Unfortunately, they forgot the exact time of their meeting, they only remember that is is between 2 and 3 PM. They both choose their arrival time uniformly in this interval and each waits 10 minutes (or until 3, whichever comes first) for the other to come. What is the probability that they meet? The sample space: { }Ω , : 2 , 3A B A Bt t t t= ≤ ≤ Favourable cases: { } { }= − ≤ , meet | | 1/ 6A BA B t t

The area of the shaded area (and thus the probability):

( )= − =

25 1( , meet) 1 6 3P A B 16

Die Entwicklung der Wahrscheinlichkeitstheorie und ihrer Anwendungen, 1899 (Jahresbericht der DMV, 1– 271)

Development of probability theory and its applications

First part: foundations of probability theory

– a great attention paid also to geom. prav. Wahrscheinlichkeitsrechnung und ihre Anwendung auf Lebensversicherung, 1903 (Teubner, Leipzig)

[2nd ed. 1908 (vol. 1) and 1910 (vol. 2), 3rd ed. 1914]

Separate chapter: geometrical probability

3rd edition: references to A. A. Markov, Louise Bachelier,

discussion of Bertrand paradox

BOHUSLAV HOSTINSKÝ (1884 – 1951)

1907 PhDr. (philosophy) → supply grammar school teacher

1908/09 study in Paris

1912 Charles University in Prague: private associate professor for higher mathematics

1920 Masaryk University in Brno Faculty of Natural Sciences full professor

1920–51 founder and director of the Institute of Theoret. Physics

1921–22, 1927–28, 1945 dean of the faculty

Scientific interest

• Analytical and differential geometry (mainly 1906 – 1915)

• Mathematical analysis

• Probability theory (since 1917) o Geometrical probability (since 1917 – Buffon problem) o Markov chain, time development of systems

• Physics (since 1917) o Light reflection on a planar curve o Mechanic a electromagnetic vibrations o Motion of a point in a power field o Kinetic gas theory (motivation for geom. probability)

Geometrical probability

• Buffon’s problem (1917, 1920) Nové řešení Buffonovy úlohy o jehle. Rozpravy ČAVU 26(1917) Sur une nouvelle solluation du probléme d’aigulle. Bull. Sci. Math. 44 (1920)

• Sur les probabilités géométriques (1925). Spisy Brno, 26 pp. Extension of the works: M. W. Crofton: On the theory of local probability, Phil. Trans. A158 (1868), 181–199 E. Czuber: Geometrische Wahrscheinlichkeiten und Mittelwerte, Leipzig 1884; Zur Theorie der geom. Wahrscheinlichkeiten, Sitzungsber. Wien 1884, 719-742.

• Geometrické pravděpodobnosti (1926)

The first and for a long time the only Czech book on geometrical probability

References to Czuber

Buffon’s needle problem

α↑ ⇔ ≤ ll D sin2x

Favourable cases:

l l .0

sin2favS dπ

α α= =∫

( )

2

2P d dππ↑ = =

⋅l l lD

Assumptions underlying the “classical solution”:

Parallel lines are drawn on a boundaryless board

Probability that the mid point of the needle hits a region of the area ε is proportional to this area and independent of the position of the region

The axis of the needle can fall into any direction with the same probability

Hostinský, 1917, 1920:

No real experiment can satisfy all these assumptions

Parallel lines are drawn on a square table board, the experiment requires the needle to fall on the board

→ probability that the mid point of the needle hits a square of a given area nearby the edge of the table is lower than the probability that it hits a square of the same area nearby the middle

Hostinský, 1917, 1920:

No real experiment can satisfy all these assumptions

Parallel lines are drawn on a square table board, the experiment requires the needle to fall on the board

→ probability that the mid point of the needle hits a square of a given area nearby the edge of the table is lower than the probability that it hits a square of the same area nearby the middle

Solution based on Poincaré’s method of arbitrary functions → classical solution as a limit case

1920: Hostinský sent the paper to Bulletin des Sciences Mathém.

→ discussion in the correspondence with Maurice Fréchet

→ could have awoke Fréchet’s interest in probability theory

Geometrical probability

• Buffon problem (1917, 1920) Nové řešení Buffonovy úlohy o jehle. Rozpravy ČAVU 26(1917) Sur une nouvelle solluation du probléme d’aigulle. Bull. Sci. Math. 44 (1920)

• Sur les probabilités géométriques (1925). Spisy Brno, 26 pp. Extension of the works: M. W. Crofton: On the theory of local probability, Phil. Trans. A158 (1868), 181–199 E. Czuber: Geometrische Wahrscheinlichkeiten und Mittelwerte, Leipzig 1884; Zur Theorie der geom. Wahrscheinlichkeiten, Sitzungsber. Wien 1884, 719-742.

• Geometrické pravděpodobnosti (1926)

The first and for a long time an only Czech book on geometrical probability

References to Czuber

Structure analysis of defects and impurities in material

Tissue probes Geological exploration Oil search, oil well

CONTRIBUTIONS TO THE DEVELOPMENT OF PROBABILITY THEORY

• Geometrical probability

• Emanuel Czuber (1851 – 1925)

• Bohuslav Hostinský (1884 – 1951) • Markov chains

• Bohuslav Hostinský (1884 – 1951)

MARKOV CHAIN

Stochastic discrete-state and discrete-time process Discrete set of states { }1 2 3, , , , ,K Kns s s s Discrete time Probability of a transition from a state ei to a state ej is

independent of the way how the system has come to ei

Shuffling cards problem Poincaré, 1908, 1912 A player shuffles q cards Probabilities of particular permutations S1, S2, ..., Sq! :

p1 + p2 + ⋅⋅⋅+ pq! = 1, pi ≠ 0 for all i Probability of a particular ordering at the end of repeated shuffling

→∞=( ) 1lim !

nn

p q

J. Hadamard: Sur le battage des cartes. CR 185 (1927), 5–9

The proof of Poincaré’s assertion by two new ways – repeated a method that had already been used by

Paul Lévy: Calcul des Probabilités, 1925 (textbook)

After the publication of Hadamard’s paper, Lévy writes to Frechet, that he came to the idea to use this method by reading a paper by Hostinský

1927: Frechet visits Prague, meets Hostinský, draws his attention to Hadamard’s paper

1928: Frechet writes to Hostinský that he has found a similar idea in the book of the insurance mathematician from Brno: F. M. Urban: Grundlagen der Wahrscheinlichkeitsrechnung und der Theorie des Beobachtungsfehler, Leipzig, 1923

1928: International congress of mathematicians in Bologna Hostinský, Hadamard: cards problem (publ. Compte Rendus, 1928) still at the congress draw György Pólya their attention to a 20 years older work of Andrej Andrejevič Markov, containing similar investigation

– thus the concept of Markov chain emerged and then spread immediately – Hostinský used this concept in a paper published in 1929

But a similar method was used already by Louis Bachelier: Thesis, 1900

– financial considerations – in fact the first processing of Brown motion in this way

1936: got acquainted with the papers of Hostinský and Hadamard

→ irritated letter to Hostinský, claiming the priority 1931: Bachelier cited by Kolmogorov,

consequently by Hostinský (1931) According to A. P. Juškevič (encyclopedia entry about Markov in the Dictionary of scientific biography): The first such method is already contained in the treatise of Francise Galtona from 1889

Beginnings of Markov chains

Francis Galton 1889 Louis Bachelier 1900 Andrej Andrejevič Markov 1906/7 Bohuslav Hostinský 1917/20 | György Pólya | F. M. Urban 1923

↓ P.P. Lévy 1925 (M. Fréchet) J. Hadamard 1927 J. Hadamard 1928, B. Hostinský 1928 – Bologne

Andrej Andrejevič Markov, 1906/07: Analysis of the text of Evžen Oněgin

a piece containing 20 000 phones, divided (keepin the order) into 200 groups consisting of 100 phones → determined the number of vowels in each group Hostinský:

Physical applications of Markov chains

– Brownian motion – Ergodic principle

Applications today

• Physics • Queuing theory • Railway safety systems • Testing • Internet applications • Statistics • Mathematical biology • Gambling • Music

• • •

Probability is Probability is everywhere everywhere around us ... around us ...

MATHEMATICIANS TO PROBABILITY THEORY - cvut.czeuler.fd.cvut.cz/~hyksova/esu2007_hyksova.pdf · •...

Documents

Transcript of MATHEMATICIANS TO PROBABILITY THEORY - cvut.czeuler.fd.cvut.cz/~hyksova/esu2007_hyksova.pdf · •...