Archive for the ‘Matherati’ Category

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Scenarios and possible worlds

Published by peter on 2009-07-01 in Decision theory, History, Mathematics and Matherati. 0 Comments

Herman Kahn was the inventor of scenario analysis, and he first applied it to analysis of US military options in the Cold War with the USSR in the 1950s. I was struck when I read his books by the fact that scenarios were developed in the same decade as possible worlds semantics for logical systems or languages, and both at a time when most people felt there was a choice of only two or three over-arching political systems. (In contrast, from the fall of the Berlin Wall until the Global Financial Crisis, most people probably thought there was no such choice, western capitalism having prevailed over all alternative systems.  Now, of course, nobody knows anything.)
Herman Kahn
I don’t think these simultaneous facts of scenarios and possible worlds were coincidences.  Which leads me to a question that has long intrigued me: just who did develop possible worlds semantics? Although the idea dates at least to Leibniz, Saul Kripke is usually credited, and so these semantics are often called Kripke frames.

But there are other candidates:

  • Richard Montague, logician and linguist, who published in 1960, but had presented his work at a conference at UCLA in 1955.
  • Carew Meredith and Arthur Prior in 1956. According to Jack Copeland, these two logicians developed the first possible worlds semantics for propositional modal logic in a one-page note dated 1956. Meredith was a near contemporary of Frank Ramsey at Winchester and Trinity College, Cambridge.
  • Charles L. Hamblin, whose London University PhD thesis submitted in October 1956 presents a possible worlds semantics for question-response interactions.
  • Hugh Everett, who presented the first formal possible-worlds theory of quantum mechanics in physics, in his 1956 Princeton University PhD thesis.
  • Jaako Hintikka, who developed a possible worlds semantics for logics of belief.  Although published in 1962, I understand Hintikka’s work was completed as early as 1957.
  • Stig Kanger, who published in 1957.
  • A. Bayart, a Belgian logician, who published in 1958 and 1959.

As I said, it is exceedingly odd that all these works were published around the same time. In addition, both Everett and Kahn were at Princeton University in the 1950s, although I don’t know if they overlapped. Also, Kahn had studied physics, so may have been aware of recent developments in the subject.

References:

A. Bayart [1958]: Correction de la logique modale du permier et du second ordre S5. Logique et Analyse, 1 (1): 28-45.

A. Bayart [1959]: Quasi-adéquation de la logique modale du second ordre S5 et adéquation de la logique modale du premier ordre S5. Logique et Analyse, 2 (6-7): 99-121.

B. J. Copeland [1999]: Notes towards a history of possible worlds semantics. pp. 1-14 in: The GoldBlatt Variations: Eight Papers in honour of Rob. Uppsala Prints and Preprints in Philosophy, Number 1. Uppsala, Sweden: Department of Philosophy, Uppsala University.

H. Everett [1956]: The Theory of the Universal Wave Function. Princeton, NJ, USA: Princeton University Press. Reprinted as pp. 3-140 of: B. S. DeWitt and R. N. Graham [1973]: The Many Worlds Interpretation of Quantum Mechanics. Princeton, NJ, USA: Princeton University Press. The main results of Everett’s PhD were published in: H. Everett [1957]: “Relative State” formulation of Quantum Mechanics. Review of Modern Physics, 29 (3): 454-462.

Robert Goldblatt [2006]: Mathematical modal logic:  a view of its evolution.  D. M. Gabbay and J. Woods (Editors): Handbook of the History of Logic. Volume 7.

C. L. Hamblin [1957]: Language and the Theory of Information. London, UK: PhD Thesis, Logic and Scientific Method Programme, University of London. Submitted October 1956, awarded 1957.

J. Hintikka [1962]: Knowledge and Belief. Ithaca, NY, USA: Cornell University Press.

H. Kahn [1960]: On Thermonuclear War. Princeton, NJ, USA: Princeton University Press.

H. Kahn [1965]: On Escalation: Metaphors and Scenarios. Pall Mall Press.

S. Kanger [1957]: Provability in Logic. Stockholm Studies in Philosophy. Stockholm, Sweden: Almqvist and Wiksell.

S. Kripke [1959]: A completeness proof in modal logic. Journal of Symbolic Logic, 24: 1-14.

S. Kripke [1963]: Semantical analysis of modal logic I: normal propositional calculus. Zeitschrift fur mathematische Logic und Grundlagen der Mathematik, 9: 67-96.

 

Achilles and the Tortoise

Published by peter on 2009-06-07 in Argumentation, Mathematics and Matherati. 0 Comments

An amusing account (at least to a mathematician) by Harvey Friedman of an encounter with eccentric Russian mathematician and dissident Alexander Yessenin-Volpin. Friedman supervised the Stanford PhD of John E. Hutchinson, who taught me calculus.  (Hat tip: AB)

Let me give an example. I have seen some ultrafinitists go so far as to challenge the existence of 2^100 as a natural number, in the sense of there being a series of ‘points’ of that length. There is the obvious ‘draw the line’ objection, asking where in

2^1, 2^2, 2^3, . . . , 2^100

do we stop having ‘Platonistic reality’? Here this . . . is totally innocent, in that it can be easily be replaced by 100 items (names) separated by commas.

I raised just this objection with the (extreme) ultrafinitist [mathematician Alexander] Yessenin Volpin during a lecture of his.  He asked me to be more specific.  I then proceeded to start with 2^1 and asked him whether this is ‘real’ or something to that effect.  He virtually immediately said yes.  Then I asked about 2^2, and he again said yes, but with perceptible delay. Then 2^3, and yes, but with more delay.  This continued for a couple of more times, till it was obvious how he was handling this objection.  Sure, he was prepared to always answer yes, but he was going to take 2^100 times as long to answer yes to 2^100 then he would to answering 2^1.  There is no way that I could get very far with this. (pp. 4-5).

Note: Of course, Friedman is wrong about the . . . being replaced by 100 items. We would expect it to be replaced with just 96 items, since 4 items in the list of 100 are already listed explicitly.
Reference:
Harvey M. Friedman [2002]: Lecture Notes on Philosophical Problems in Logic. Princeton University.

The decade around 1664

Published by peter on 2009-03-29 in Culture, History, Matherati, Probability theory, Prophecy, Science and Uncertainty. 0 Comments

We noted before that one consequence of the rise of coffee-houses in 17th-century Europe was the development of probability theory as a mathematical treatment of reasoning with uncertainty.   Ian Hacking’s history of the emergence of probabilistic ideas in Europe has a nice articulation of the key events, all of which took place a decade either side of 1664:

  • 1654:  Pascal wrote to Fermat with his ideas about probability
  • 1657: Huygens wrote the first textbook on probability to be published, and Pascal was the first to apply probabilitiy ideas to problems other than games of chance
  • 1662: The Port Royal Logic was the first publication to mention numerical measurements of something called probability, and Leibniz applied probability to problems in legal reasoning
  • 1662:  London merchant John Gaunt published the first set of statistics drawn from records of mortality
  • Late 1660s:  Probability theory was used by John Hudde and by Johan de Witt in Amsterdam to provide a sound basis for reasoning about annuities (Hacking 1975, p.11).

Developments in the use of symbolic algebra in Italy in the 16th-century provided the technical basis upon which a formal theory of uncertainty could be erected.  And coffee-houses certainly aided the dissemination of probabilistic ideas, both in spoken and written form.   Coffee houses may even have aided the creation of these ideas – new mathematical concepts are only rarely created by a solitary person working alone in a garret, but usually arise instead through conversation and debate among people each having only partial or half-formed ideas.
However, one aspect of the rise of probability in the mid 17th century is still a mystery to me:  what event or phenomena led so many people across Europe to be interested in reasoning about uncertainty at this time?  Although 1664 saw the establishment of a famous brewery in Strasbourg, I suspect the main motivation was the prevalence of bubonic plague in Europe.   Although plague had been around for many centuries, the Catholic vs. Protestant religious wars of the previous 150 years had, I believe, led many intelligent people to abandon or lessen their faith in religious explanations of uncertain phenomena.   Rene Descartes, for example, was led to cogito, ergo sum when seeking beliefs which peoples of all faiths or none could agree on.  Without religion, alternative models to explain or predict human deaths, morbidity and natural disasters were required.   The insurance of ocean-going vessels provided a financial incentive for finding good predictive models of such events.
Hacking notes (pp. 4-5) that, historically, probability theory has mostly developed in response to problems about uncertain reasoning in other domains:  In the 17th century, these were problems in insurance and annuities, in the 18th, astronomy, the 19th, biometrics and statistical mechanics, and the early 20th, agricultural experiments.  For more on the connection between statistical theory and experiments in agriculture, see Hogben (1957).  For the relationship of 20th-century probability theory to statistical physics, see von Plato (1994).
POSTSCRIPT (ADDED 2011-04-25):
There appear to have been major outbreaks of bubonic plague in Seville, Spain (1647-1652), in Naples (1656), in Amsterdam, Holland (1663-1664), in Hamburg (1663), in London, England (1665-1666), and in France (1668).   The organist Heinrich Scheidemann, teacher of Johann Reincken, for example, died during the outbreak in Hamburg in 1663.   Wikipedia now has a listing of global epidemics (albeit incomplete).
 
POSTSCRIPT (ADDED 2018-01-19):
The number 1664 in Roman numerals is MDCLXIV, which uses every Roman numeric symbol precisely once.  The number 1666 has the same property, and for that number, the Roman symbols are in decreasing order.
 
References:
Ian Hacking [1975]:  The Emergence of Probability: a Philosophical study of early ideas about Probability, Induction and Statistical Inference. London, UK: Cambridge University Press.
Lancelot Hogben [1957]: Statistical Theory. W. W. Norton.
J. von Plato [1994]:  Creating Modern Probability:  Its Mathematics, Physics and Philosophy in Historical Perspective.  Cambridge Studies in Probability, Induction, and Decision Theory.  Cambridge, UK:  Cambridge University Press.

Putting the "Tea" in IT

Published by peter on 2008-07-16 in Computer technology, Decision theory, Lead users, Marketing strategy and Matherati. 0 Comments

One of the key ideas in the marketing of high-tech products is due to Eric von Hippel of the MIT Sloan School, the idea that lead users often anticipate applications of new technologies before the market as a whole, and even before inventors and suppliers. This is because lead users have pressing or important problems for which they seek solutions, and turn to whatever technologies they can find to respond to their problems.
A good example is shown by the history of Information Technology. The company which pioneered business applications of the new computer technology in the early 1950s was not a computer hardware manufacturer nor even an electronic engineering firm, but a lead user, Lyons Tea Shops, a nationwide British chain of tea-and-cake shops.  Lyons specified, designed, built, deployed and operated their own computers, under the name of Leo (Lyons Electronic Office). Lyons, through Leo, was also the first to conceive and deploy many of the business applications which we now take for granted, such as automated payroll systems and logistics management systems. One of the leaders in that effort, David Caminer, has recently died at the age of 92. LEO was later part of ICL, itself later purchased by Fujitsu.
This post is intended to honour David Caminer, as a pioneer of automated business decision-making.