On Scientific Models

Interview with David Stainforth [climate scientist], 7 The Reasoner No. 2 (February [sic] 2, 2013):

KS [Katie Steele]: What is the significance of your emphasis on policy here? Are you saying that you now devote a lot of time to science communication, or, rather, that you approach your work as a climate scientist in a different way, i.e., with an eye to policy relevance?
DS: The latter. The attention to policy has lead to a shift in emphasis in my scientific work—from modelling and running simulations to the proper interpretation of the data output of these model simulations.
In the past I set up and did a lot of runs (simulations) of these large complex climate models called General Circulation Models [GCMs]. This involves a lot of time and a lot of hard work in getting these computer models up and running. . . these simulations are difficult to produce. But I have done my time in this respect. The climateprediction.net project that I was involved in is still running, and that’s great, but it is up to others now to facilitate the simulations.
The important issue for me now is this: these climate model simulations produce vast output, and there are so many questions about how to analyse these big data sets. . . In short, what does it all mean? Why run these simulations? We need to really think about what we can get out of these climate models and how the results should be presented. It is tempting to just keep making the models more and more complicated and apparently derive more and more detailed predications [sic] of the type that policy-makers want. Moreover, the power of computers has its own allure. . . such shiny sophisticated machines that seem to offer endless opportunities for fast and powerful problem-solving. . . for the mathematically-minded, there is a temptation to create more and more complicated models. We need to be very careful, however, about faithfully representing what we actually know about the future climate on the basis of model simulations.
KS: I see. So what do you think the models can be used for? Do they yield predictions? Is the ‘pulling back’ just a matter of being more modest about the precision of the predictions that we can derive from climate models? Or should we use climate models in quite a diff erent way altogether?
DS: I think it is best to think of climate models as research tools—they are useful for understanding interactions between diff erent parts of the climate system. Of course, we don’t want to give up on predicting the climate, but we need to be realistic about climate prediction, and in particular, multi-decadal climate prediction, which is of interest to policy-makers. Climate models should be seen as just one of the inputs that allow us to formulate scenarios of how the climate could change in response to diff erent  forcings. [Forcings are external forces that change the dynamics of the system; a prominent forcing is increased carbon dioxide emissions.] We should aim to formulate scenarios that collectively tell us how the climate could change over time, and give us a general indication of the sensitivity of the climate system.
The main point here is that the output of climate models should not be taken at face value—as predictions of future climate—and presented in more or less unadulterated form to the public and to policy-makers.
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Is there a moral here for the economic analysis of evidence, inference, and proof? I think so.
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Law and Economics

It's almost enough to make me believe in law and economics:

Ethan Bronner, Law Schools’ Applications Fall as Costs Rise and Jobs Are Cut NYTimes (Jan. 31, 2013):

Law school applications are headed for a 30-year low, reflecting increased concern over soaring tuition, crushing student debt and diminishing prospects of lucrative employment upon graduation.

The New York Times

As of this month, there were 30,000 applicants to law schools for the fall, a 20 percent decrease from the same time last year and a 38 percent decline from 2010, according to the Law School Admission Council. Of some 200 law schools nationwide, only 4 have seen increases in applications this year. In 2004 there were 100,000 applicants to law schools; this year there are likely to be 54,000.

Death of Paul Rice

I deeply regret (and I am embarrassed) that I have been so out of touch that I did not know that Professor Paul Rice passed away last July. His Evidence casebook was marvelous and comprehensive. In the iteration of the casebook I knew, it covered common law evidence as well as evidence codifications. I marveled at his apparent ability to cover such a vast amount of material in his Evidence course.

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The "Human Brain Project"

The European Union has decided to invest in a big way in research on the human brain. See James Kanter, [Two] Science Projects [Each] to Receive Award of a Billion Euros, NYTimes (Jan. 28, 2013):

BRUSSELS — Projects to imitate the brain and to develop new materials for information technology have won awards of about 1 billion euros each that will be announced Monday by the European Commission.
The awards, the largest of their kind ever made by the European authorities and equivalent to about $1.35 billion each, are aimed at helping innovative industries in the European Union and nonmember countries like Switzerland.   [snip, snip]         

The Human Brain Project aims to create the most accurate simulation to date of the brain and its functions. The project could help aid diagnoses of diseases, help with the testing of new drugs, and develop supercomputing techniques modeled on the brain.

The project involves scientists from 87 institutions and will be led by Henry Markram, a professor at École Polytechnique Fédérale de Lausanne in Switzerland. Partners in that project include the Institut Pasteur in France, I.B.M. in the United States and SAP in Germany.

        

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Workshop on Predictive Coding

DESI V Workshop at ICAIL 2013 

Standards for Using Predictive Coding

and Other Machine Learning Algorithms in E-discovery

June 14, 2013

Consiglio Nazionale delle Ricerche, Rome, Italy

DESI V site: http://www.umiacs.umd.edu/~oard/desi5/

DESI V call for submissions: http://www.umiacs.umd.edu/~oard/desi5/Desi5Cfs.pdf

ICAIL 2013 site: http://icail2013.ittig.cnr.it/

 

Purpose

The DESI (Discovery of Electronically Stored Information) workshop series addresses the problem of searching effectively and efficiently for relevant documents, in the context of litigation or investigation, across increasingly complex, enterprise-wide collections within corporate and institutional settings.   The workshops discuss the use of AI and other advanced forms of search techniques in legal settings, as cost-efficient alternatives to traditional Boolean and manual searching.  At DESI V, we intend to focus on best practices and standards for using predictive coding and other forms of machine learning in e-discovery.

Submissions

We invite participation from e-discovery stakeholders and practitioners from the law, government, and industry, along with researchers on process quality, information retrieval, human language technology, human-computer interaction, artificial intelligence, and other fields connected with e-discovery. The dialogue at the workshop will be expected to center around how lawyers are currently using such techniques, how better protocols can be developed that will satisfy the interests of the legal community, and what open questions exist that would benefit from further research into optimizing the use of these techniques in a variety of legal and investigatory settings.

We encourage the submission of research papers and position papers on both supporting technologies for e-discovery (search, text classification, etc.) and on efforts to develop best practices and standards for use of these technologies. Accepted position papers and accepted research papers will be made available on the Workshop's Web page and distributed to participants on the day of the event, and some speakers may be selected from among those submitting position papers. See the Call for Submissions, http://www.umiacs.umd.edu/~oard/desi5/Desi5Cfs.pdf, for further details.  Any questions may be addressed to Doug Oard (oard@umd.edu).

Important Dates

• Research papers due: May 1, 2013
• Position papers due: May 8, 2013
• Accept/Reject notification for research papers: May 15, 2013
• Preliminary Agenda posted: May 22, 2013
• Camera-ready research papers due: May 22, 2013
• ICAIL Conference: June 10-13, 2013
• DESI V Workshop: June 14, 2013

Organizing Committee

Jason R. Baron, National Archives and Records Administration, USA
Jack G. Conrad, Thomson Reuters, Switzerland
Dave Lewis, David D. Lewis Consulting, USA
Debra Logan, Gartner Research, UK
Douglas W. Oard, University of Maryland, USA
Fabrizio Sebastiani, Istituto di Scienza e Tecnologia dell'Informazione, Italy

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An Important Message about Fuzzy Logic

The recent award of a major scientific prize to Lotfi Zadeh provoked many admirers from around the world to send along their congratulations on a discussion list devoted to fuzzy logic and soft computing. In so doing, these well-wishers also made a variety of comments about fuzzy logic. Professor Zadeh eventually responded with a substantive comment. (See below.) His comment is important and enlightening in several different ways. Although I will let his comment speak for itself, I do wish to emphasize the astonishing reach of fuzzy logic that Zadeh highlights. It is also worth mentioning that the use of fuzzy logic is apparently accelerating rather than plateauing. There is thus reason to think and to hope that an increasing number of legal scholars (in addition to luminaries such as Kevin Clermont and Lothar Phillips) will decide to use fuzzy logic to explore reasoning about and in law. It is high time that they do so!   

The message from Professor Zadeh: 

Dear members of the BISC Group:

    The BBVA Award has rekindled discussions and debates regarding what fuzzy logic is and what it has to offer. The discussions and debates brought to the surface many misconceptions and misunderstandings. A major source of misunderstanding is rooted in the fact that fuzzy logic has two different meanings -- fuzzy logic in a narrow sense, and fuzzy logic in a wide sense. Informally, narrow-sense fuzzy logic is a logical system which is a generalization of multivalued logic. An important example of narrow-sense fuzzy logic is fuzzy modal logic. In multivalued logic, truth is a matter of degree. A very important distinguishing feature of fuzzy logic is that in fuzzy logic everything is, or is allowed to be, a matter of degree. Furthermore, the degrees are allowed to be fuzzy. Wide-sense fuzzy logic, call it FL, is much more than a logical system. Informally, FL is a precise system of reasoning and computation in which the objects of reasoning and computation are classes with unsharp (fuzzy) boundaries. The centerpiece of fuzzy logic is the concept of a fuzzy set. More generally, FL may be a system of such systems. Today, the term fuzzy logic, FL, is used preponderantly in its wide sense. This is the sense in which the term fuzzy logic is used in the sequel. It is important to note that when we talk about the impact of fuzzy logic, we are talking about the impact of FL. Intellectually, narrow-sense fuzzy logic is an important part of FL, but volume-wise it is a very small part. In fact, most applications of fuzzy logic involve no logic in its traditional sense. 

    What is not widely recognized within the scientific community and the general public, is that fuzzy logic has become a vast enterprise.There are over 280,000 papers in the literature with fuzzy in title. There are 25 journals with fuzzy in title. There are close to 25,000 fuzzy-logic-related patents issued or applied for in the United States and Japan. There is a long list of applications ranging from digital cameras to fraud detection systems. Particularly worthy of note, on one end, is the fuzzy logic subway system in Sendai, a city of over 1 million in Japan. On the other end, numerically, is Omron's 120 million fuzzy logic blood pressure meters.

    Most, but not all of the constituents of fuzzy logic are what are called FL-generalizations of traditional, bivalent-logic-based systems of reasoning and computation. Examples. Fuzzy arithmetic, fuzzy cluster analysis, fuzzy differential equations, fuzzy control, fuzzy linear programming, etc. FL-generalization of a theory or a formalism, T, involves introduction into T of the concept of a fuzzy set, followed by addition of related concepts and techniques. FL-generalization may be applied to any field, any theory, any system, any formalism and any algorithm. The  fundamental importance of FL-generalization derives from the fact that in the real world almost all classes have unsharp (fuzzy) boundaries. As a consequence, FL-generalization opens the door to construction of better models of reality.

    It is of interest to observe that the impact of FL-generalization is growing in visibility and importance in mathematics -- a field in which precision plays a quintessential role. We see a growing number of papers with fuzzy in title in many branches of mathematics, among them topology, algebra, differential equations, group theory, set theory, and functional analysis. What may come as a surprise to many is that Math.Sci.Net database lists over 22,383 papers with fuzzy in title. I did not anticipate that this will happen when I wrote my first paper on fuzzy sets. My expectation was that the concept of a fuzzy set would find its main applications in the realm of soft, human-centered sciences. 

    When it comes to practical application of fuzzy logic, there is a major source of misunderstanding. Fundamentally, fuzzy logic is aimed at precisiation of what is imprecise. But in many of its applications fuzzy logic is used, paradoxically to imprecisiate what is precise. In such applications, there is a tolerance for imprecision, which is exploited through the use of fuzzy logic. Precisiation carries a cost. Imprecisiation reduces cost and enhances tractability. This is what I call the Fuzzy Logic Gambit. What is important to note is  that precision has two different meanings: precision in value and precision in meaning. In the Fuzzy Logic Gambit what is sacrificed is precision in value, but not precision in meaning. More concretely, in the Fuzzy Logic Gambit imprecisiation in value is followed by precisiation in meaning. An example is Yamakawa's inverted pendulum. In this case, differential equations are replaced by fuzzy if-then rules in which words are used in place of numbers. What is precisiated is the meaning of words.

        Some critics have been saying that fuzzy logic is a passing fad. This assessment of fuzzy logic fails to recognize that the world we live in is, in large measure, a world of fuzzy classes, and that science has much to gain from shifting its foundation from classicalAristotelian logic to fuzzy logic. Comments are welcome.
                         Regards to all,

                         Lotfi

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Tacit Inferential Reckoning - The Dung Beetle and the Human Animal

If the dung beetle has tacit inferential knowledge, or capacity, why not human beings?
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Fuzzy Logic and Fuzzy Thinking in Law

I have been thinking about Lotfi Zadeh and his fuzzy logic. Indeed, I have been thinking, off and on, about Zadeh's fuzzy logic for decades. The law is full of fuzzy thinking. If there is a precise way to think about fuzzy ideas, legal scholars, judges, etc., should use it. On reflection and re-reflection and re-re-reflection etc., I do believe that Zadeh has given us important tools for radically better ways of thinking about fuzzy thinking.

It has often been said that the standard probability calculus can do all that fuzzy logic does. But such claims sometimes have the air of gratuitous dicta: one wants to see the proof in such probabilistic pudding. The fuzziness and vagueness of legal concepts (and many other types of concepts) usually can only awkwardly be characterized in terms of uncertainty. Legal principles, concepts, etc., often employ vague notions. There is a difference between a vague concept and an uncertain proposition. Probability theory sometimes deals well with uncertain propositions - but not so well with vague and fuzzy concepts.   In many of its iterations, Zadeh's fuzzy logic does not seem to reach beneath the surface of our fuzzy thinking (e.g., our conventional legal thinking), but seems just to accept the fuzziness of ordinary thinking. It might be said that fuzzy logic, in some its iterations, just describes the natural behavior of fuzzy terms, operators ("and" "or" etc.), and the like, and does not attempt to identify the causes, bases, or foundations of the fuzzy ideas we have and use.  This creates a puzzle: How can a logic that does not purport to identify the foundations, sources, or causes of fuzzy thinking give us useful new thoughts? (This question assumes that fuzzy thinking is a sort of disease.)  

The answer, I think, lies in the notion of tacit knowledge. There is genuine knowledge buried in some or much of our "ordinary" fuzzy thinking. (If that were not the case, few of us would survive even for one day.) Fuzzy logic's proven successes suggest that fuzzy logic may offer a way to uncover, or display, much "innate," or tacit, human knowledge.

These are admittedly deep and possibly murky waters, and I confess I do not have the ability to swim through them easily. I console myself with the thought that the acquisition of knowledge is a collective human enterprise and that there will be others who may be able to build on some of the paltry number of insights I may have acquired over the years. But if it turns out I have not learned much of enduring value about fuzzy thinking, the efforts I have made may have been worth the candle - because people can learn by studying other people's errors. &&&     The dynamic evidence page
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A Religious Liberty Clinic at Stanford Law School

Ethan Bronner, At Stanford, Clinical Training for Defense of Religious Liberty NYTimes (Jan. 21, 2013):

Backed by two conservative groups, Stanford Law School has opened the nation’s only clinic devoted to religious liberty, an indication both of where the church-state debate has moved and of the growth in hands-on legal education.

Begun with $1.6 million from the John Templeton Foundation, funneled through the Becket Fund for Religious Liberty, the school’s new Religious Liberty Clinic partly reflects a feeling that clinical education, historically dominated by the left’s concerns about poverty and housing, needs to expand.

“The 47 percent of the people who voted for Mitt Romney deserve a curriculum as well,” said Lawrence C. Marshall, the associate dean for clinical legal education at Stanford Law School. “My mission has been to make clinical education as central to legal education as it is to medical education. Just as we are concerned about diversity in gender, race and ethnicity, we ought to be committed to ideological diversity.” Mr. Marshall became a hero to liberals for his work to exonerate death penalty inmates when he was a professor at Northwestern Law School a decade ago.

[snip, snip]

“In framing our docket, we decided we would represent the believers,” said James A. Sonne, the clinic’s founding director, explaining that the believers, rather than governments, were the ones in need of student lawyers to defend them. “Our job is religious liberty rather than freedom from religion.”

Mr. Sonne, who grew up the son of a psychoanalyst in a nominally Episcopalian home near Cherry Hill, N.J., converted to Roman Catholicism while a student at Duke University. He went on to Harvard Law School and later a professorship at Ave Maria School of Law, a Catholic institution. He acknowledges the political coloration of much of the religious-freedom debate but says he does not want his clinic to be seen as a program for conservatives.

His first four students — a Mormon, a Methodist, a Catholic and someone brought up as a Seventh-day Adventist — agree, saying they were drawn to the clinic by the profound questions it raises and the real lawyering it offers, from meeting a potential client to appellate review.

[snip, snip]

Barry Lynn, the executive director ofAmericans United for Separation of Church and State, said he was “shocked that a major law school would accept a gift from Becket,” which he described as “a group that wants to give religious institutions or individuals a kind of preferential treatment, even if that hurts a third party.”

But Hannah C. Smith of Becket, who took part in a panel discussion here on Monday to observe the clinic’s opening, said what liberals like Mr. Lynn call the strict wall of separation is found nowhere in the Constitution. Her group, she said, is working to show that “there are certain God-given rights that existed before the state. God gave people the yearning to discover him. Religious freedom means we have to protect the right to search for religious truth free from government intrusion.”

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Campbell's Law

During a C-Span broadcast of a discussion of Charles J. Wheelan's Naked Statistics: Stripping the Dread from the Data (2013), a member of the audience - a statistician - asked Wheelan if he knew about Campbell's Law. I turned to Wikipedia, of course, and found the following entry (see http://en.wikipedia.org/wiki/Campbell's_law):

Campbell's law is an adage developed by Donald T. Campbell:^[1]

"The more any quantitative social indicator is used for social decision-making, the more subject it will be to corruption pressures and the more apt it will be to distort and corrupt the social processes it is intended to monitor."

The social science principle of Campbell's law is sometimes used to point out the negative consequences of high-stakes testing in U.S. classrooms.

What Campbell also states in this principle is that "achievement tests may well be valuable indicators of general school achievement under conditions of normal teaching aimed at general competence. But when test scores become the goal of the teaching process, they both lose their value as indicators of educational status and distort the educational process in undesirable ways. (Similar biases of course surround the use of objective tests in courses or as entrance examinations.)"^[1]

Campbell's law was published in 1976 by Donald T. Campbell, an experimental social science researcher and the author of many works on research methodology. Closely related ideas are known under different names, e.g. Goodhart's law, and the Lucas critique. Another concept related to Campbell's law emerged in 2006 when UK researchers Rebecca Boden and Debbie Epstein published an analysis of evidence-based policy, a practice espoused by Prime Minister Tony Blair. In the paper, Boden and Epstein described how a government that tries to base its policy on evidence can actually end up producing corrupted data because it, "seeks to capture and control the knowledge producing processes to the point where this type of ‘research’ might best be described as ‘policy-based evidence’."^[2] (Boden and Epstein 2006: 226)

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Would Lon L. Fulller's students have given him high marks on teaching evaluation forms? I doubt it. I vividly remember hearing my fellow law students complain that Fuller just created confusion in the classroom.

What would or should Fuller have done to raise his teaching evaluation scores (if he cared about them)? I suppose he would have simplified the parable of the Speluncean Explorers and he would given a multiple-choice exam in the jurisprudence course he taught.

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Lotfi Zadeh Wins Major Award

LOTFI A. ZADEH

The BBVA Foundation Frontiers of Knowledge Award in the Information and Communication Technologies (ICT) category has been granted in this fifth edition to the electrical engineer Lotfi A. Zadeh, “for the invention and development of fuzzy logic.” This “revolutionary” breakthrough, affirms the jury in its citation, has enabled machines to work with imprecise concepts, in the same way humans do, and thus secure more efficient results more aligned with reality. In the last fifty years, this methodology has generated over 50,000 patents in Japan and the U.S. alone. 

On hearing of the award, Zadeh remarked that it meant a lot to him for several reasons: “First, because fuzzy logic has been somewhat controversial. Some people have greeted it with enthusiasm but others have been skeptical. It also has a special significance for me because I am a great admirer of Spain and the Spanish people. I’d therefore like to take this opportunity to express my deep appreciation to all those who were involved in my receiving this award, particularly Luis Magdalena and Enric Trillas of the European Centre for Soft Computing in Mieres, who were among those putting forward my nomination."

Classical logic, based on class membership, imposes that an element should strictly belong ir not belong to a clearly demarcated set, like for instance the set of even numbers. But reality is a lot more complex. Hence we have groups, classes and sets whose boundaries are blurred, like that of “good basketball players.” To belong to this set, a basketball player must “be tall” and “shoot well”, but these concepts are imprecise. A binary system would specify, for example, that “be tall” equates to “measure more than 185 cm” and discard all players below this height, regardless of their shooting prowess. But fuzzy logic, like a human coach, would find room in the set of good players for one who measures 184 cm but is an excellent shooter. In this sense, what fuzzy logic does is bridge the gap between classical logic and the real world.

This indeed is what Zadeh was seeking when the began the research that led him to fuzzy logic: “As an engineer, I was always convinced that mathematics held the answers to almost any problem, but I also realized that classical mathematics was constrained by its inability to tolerate imprecision.” To get over this shortcoming, Zadeh turned to the human model: “We humans have a remarkable capability to reason and make decisions in an environment of uncertainty and incompleteness of information (…). The principal objective of fuzzy logic is the formalization of this capability.”

Human beings intuitively apply fuzzy logic to their decisions, juggling imprecise data and weighing up each relevant element. Zadeh’s contribution was to apply such logic to the decision-making processes of systems and computers, so they cease to operate as mere calculating machines and become capable of evaluating degrees and shades of reality and deciding accordingly in an autonomous or semi-autonomous fashion (with little or no human intervention). 

According to the jury, the contributions of Lotfi A. Zadeh (Baku, Azerbaijan, 1921) have been “enthusiastically adopted by industry, where thousands of engineers have designed a whole plethora of complex and intelligent systems (…)."

But Zadeh’s work has also changed the face of numerous industrial processes, where it has simplified design, providing more efficient products that are easier to use and more tractable to change, while bringing down production costs. 

A seminal paper

En 1965, Lotfi Zadeh articulated fuzzy sets for the first time in a paper that would come to be among the most cited of the 20th century, with over 35,000 mentions. And the next step from there was the development of fuzzy logic, a brilliant contribution to extending the frontiers of knowledge. Indeed Zadeh is defined in the jury’s citation as the founder of “a new field of research which has proved powerful in many application domains.”

The controversy around fuzzy logic began with the name: “The word fuzzy has a pejorative connotation in English, and this turned out to be a handicap when it came to gaining the acceptance of the scientific community. But it was the word that came closest to what I had in mind. In Asia, however, they don’t have problems with the word fuzzy, so they were more receptive to my work. They also have a culture that accepts shades of grey, as opposed to the western – Cartesian – tradition where everything is either black or white.”

This was perhaps the reason, he speculates, that one of the earliest applications of his concept was the automated subway system in the Japanese city of Sendai. 

Fuzzy logic opened the door to machine understanding of such imprecise instructions as “brake smoothly” or “refrigerate until the air is cool,” which would be instantly understood by any human being acquainted with the system, but are utterly impenetrable for a conventional computer program. The conceptual shift was so abrupt that Zadeh initially had to face the skepticism of many scientist colleagues, until the success of the practical applications of his theory dissipated all such doubts. 

Zadeh’s work has enabled us to communicate with machines through an increasingly natural, human language. 

The laureate, still working at the age of 91, sees this as the most promising research avenue in the fuzzy logic field, and hopes to author some further advance that will connect computers and systems more closely with natural language.   

International jury

The jury in this category was chaired by George Gottlob, Professor of Computer Science at the University of Oxford (United Kingdom), with Ramón López de Mántaras, Director of the Artificial Intelligence Research Institute of the Spanish National Research Council (CSIC) acting as secretary. Remaining members were Oussama Khatib, Professor in the Artificial Intelligence Laboratory in the Computer Sciences Department of Stanford University (United States), Rudolf Kruse, Head of the Department of Knowledge Processing and Language Engineering at Otto-von-Guerike-Universität Magdeburg (Germany), Mateo Varelo, Director of the Barcelona Supercomputing Center (Spain) and Joos Vandewalle, Head of the SDC Division in the Department of Electrical Engineering at the Katholieke Universiteit Leuven (Belgium).

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The Second Amendment - NRA-Style - at Work

Martha Neil, Two Men with Assault Rifles Legally Take a Walk Down the Street in Portland, Ore. WSJ (Jan. 10, 2013):

"Two 22-year-old men committed no crime this week by openly carrying assault rifles, slung over their shoulders, in a residential neighborhood of Portland, Ore., known for its row of antique stores.

"But the Wednesday afternoon stroll by Steven M. Boyce and Warren R. Drouin created a stir, resulting in a lockdown by at least one school, the Oregonian reports.

"Police warned the two that there would undoubtedly be 911 calls over the weapons, but didn't take other action because what they were doing was legal. Although Portland bans individuals from having loaded firearms in public places, the two men are exempted from that ban under state law because they have concealed-carry licenses, the newspaper explains."

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Good guys with guns, I guess.

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The Lessons of "The Parable of the Ox"

There are all sorts of lessons of John Kay's delightful The Parable of the Ox (July 25, 2012). The original (underlying) hypothesis of crowd intelligence perhaps should not be entirely ignored.

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What odd, imaginative nanocreatures, we are...

"...[W]e're odd little creatures, like some cosmic nanobacteria — diminishingly small to the point that it makes us seem utterly irrelevant. Yet collectively we have the tools and conceptual ability to gain an astonishingly clear picture of what this vast universe looks like on a scale more immense than the most astronomers of just a half century ago would have imagined." The Republican, Skywatch: 2012's discoveries in astronomy overwhelming (Jan 2, 2013).

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2013 Summer School on Law and Logic

The dates for the Harvard Law School - European University Institute 2013 Summer School on Law and Logic are now settled: July 15 - 26, 2013. Venue: "The school will take place at the Badia Fiesolana, in San Domenico di Fiesole, which is located in the hills above Florence, at a short distance from both Florence and Fiesole." My law school will be one of the sponsors of this summer program. See http://lawandlogic.org/

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Formalizing Visual Thinking? Formalizing Insight?

In an interesting interview in the most recent issue of the ever-interesting electronic journal The Reasoner, Mateja Jamnik of the University of Cambridge argues that it might be possible to automate the invention of the kinds of visual images, or representations, that mathematicians sometimes or often use to get insights into the development of mathematical proofs. Her argument is significant in part because, if successful, her argument undercuts one of Roger Penrose's arguments against purely computational theories of the mind (or brain). Beyond that Jamnik's argument raises the ever-interesting question of why visual models (sometimes considered "informal") are as important as they seem to be in "exact sciences" such as mathematics and physics.

Here are some snippets from this fascinating interview:

Mateja Jamnik: I think that a proof is a social construct in mathematics. I think the history of mathematics shows us that where mathematicians—famous mathematicians—came up with a solution or a proof of a problem or a theorem,and presented it to other famous mathematicians,as long as they convinced them, they trusted that that was the correct proof. Andit was only when the logicians came along that they formalised the logic of that domain and were able to either prove or disprove the proof, to actually formalise it. And that’s what we mean when we talk about the formal proof, that you can verify that it is correct, that it follows from a set of axioms and so on. Whereas mathematicians, I don’t think, are very interested in that.They are really more interested in the insight in the proof, and trying to understand why the theorem holds, and that’s why I say it’s a social construct because as long as they convince enough people that it’s correct, nobody’s going to go and check it out to see whether it is or not. I mean, as long as fellow mathematicians believe that they understand and that they trust the process of the proof, then they’re fine. And history of mathematics is full of examples like that, where there are proofs that were thought to be proofs for fifty years and even by very famous people, and they were disproved and it was shown that they were not proofs at all. Whereas, from a formalist point of view, formal proof has a very technical meaning, which is that it follows from a set of axioms. 

[snip, snip]

ML [interviewer Mary Leng]: So what interests me in your work is that you’re saying that these methods, though visual,can be formalized.

MJ: Yes, that’s exactly what I’m trying to do.

ML: Whereas some of the thought in thinking about diagrammatic reasoning in philosophy is to say that there’s this element of our cognition of mathematics that isn’t formalizable. I suppose that’s something that Penrose is pushing at as well, in his claim that there are diagrammatic proofs that cannot be automated, because he wants to push the idea that there’s something special about us, that we’re not purely computational.

MJ: That’s right. It is. But what I’m trying to show is that all these so-called ‘informal’ methods, they can be formalized. And so I’m looking into the use of diagrams. In fact I just came from a conference on diagrams where everybody was coming from areas of mathematics, philosophy, cognitive science, cognitive psychology, computer science, artificial intelligence,and basically we have a common interest, which is to study diagrams, and the applications and theoretical foundations of the use of diagrams. And there are lots of people who come up with representations—diagrammatic representations—that they formalize and they use reasoning on. So it’s totally possible. And also my work for my PhD, on DIAMOND [Jamnik’s interactive theorem prover, which made use of diagrams to construct proofs of theorems], was in the domain of natural number arithmetic, where the proofs were not at all like the normal logical proofs. In fact, I would say that one of the hypotheses that came out of that work was that people use something like what we call ‘schematic proof’ to find a solution to a problem. So basically, you look at a few examples—concrete examples—of your problem and you solve them, and then you spot the pattern and you generalize that pattern, and you try to make an argument about how that pattern is a justification for the general statement for all cases. So what I did with natural number arithmetic was that I would represent these theorems and statements in mathematics using diagrams, and then use just manipulations of concrete cases of diagrams—for some natural number like 5, 6, or whatever—and then spot the pattern and generalize this into a program which, basically upon input, will produce a solution for that particular case. We call this—this program, this general pattern, this procedure—we call this ‘schematic proof’. My hypothesis is that this is one possible model of how people do proofs, and there’s plenty of evidence of that from history.

[snip, snip]

ML [interviewer Mary Leng]: This brings me to the issue about Penrose, because Penrose wants to say that we’re fundamentally different from machines. You mention in your book (Mathematical Reasoning with Diagrams, 2001) Penrose’s scepticism about the possibility of modelling diagrammatic reasoning in computers, and I suppose behind all that is the thought that we want to find things that we can do that computers can’t. So if it turns out that we can model this reasoning in automated settings, then that speaks against this idea that we’re so different.

MJ: Yes absolutely. I think that we don’t understand reasoning enough to be able to make claims like this. So what spurred me on to say something about Penrose in my book is because he presented this example about a cube—about something which is innately human reasoning that machines wouldn’t be able to do. He presented this example of a proof that the sum of the first n hexagonal numbers is n-cubed. He presented this visual proof which basically says, “I’ll give you an example for a cube, that is of size three, and if you split it up in this way you can see that those are the first three hexagonal numbers, and if you continue doing that, then you understand that the general theorem holds.” And that’s precisely what, for example, the theorem about the odd natural numbers, in 2-D, is. It’s an analogue of that in 2-D. And I thought, there’s something very procedural about this. You know, and we see, I suppose he’s appealing to the fact, the visual effect, that you sort of ‘splatter’ the 3-D onto 2-D to see that those three shells of the cube form a hexagonal number. But you can think of it in the same way about the odd natural numbers. You have that the sum of the first n odd natural numbers is n-squared. You can take a square of size three and then you split it into ‘ells’, and you see that each ell is the subsequent odd natural number. And that’s exactly an analogue of what Penrose presented. And obviously I showed we can capture that, you know? DIAMOND could easily do that. OK it doesn’t have a 3-D interface, so technically you can’t, but in principle, there’s nothing that would stop it really. So I suspect that Penrose is asking, really, “Could the computer come up with an idea like that?” and we don’t know that yet.

ML: So just to clarify, DIAMOND is a proof checker rather than a theorem prover?

MJ: It’s an interactive theorem prover, so it means that the user constructs the sample cases, so it means the user has the insight, really, into what the proof should look like. So the computer’s not coming up with an insight.

ML: In your book you mention the hope that you could actually develop a computer program that could do the insight steps. How have things moved in that regard?

MJ: I haven’t moved much in that direction yet, because I’ve been looking at a different direction with reasoning with diagrams, but that would definitely be the next step, to put some sort of search procedure on top of all these visual methods and geometric manipulations of diagrams and check whether anything interesting comes up. Now of course Penrose would probably say, the computer doesn’t have the insight. But where does that come from in a mathematician? It comes from experience, it comes from...well we don’t know. That’s why I’m interested in modelling this type of reasoning.

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