Lemma 2.5.1.14

Since ducks can both swim and fly, each duck is found twice inC, once labeled as aflyer and once labeled as a swimmer. The typesAandBare kept disjoint inC, whichjustifies the name “disjoint union.”

facts, which are simply “path equivalences” in an olog. It isthe notion of path equivalences that make category theory so powerful.Apathin an olog is a head-to-tail sequence of arrows

Consider the aspectpan objectqhas››››—pa weightq. At some point in history, thiswould have been considered a valid function. Now we know that the same objectwould have a different weight on the moon than it has on earth. Thus as world-views change, we often need to add more information to our olog. Even the validityofpan object on earthqhas››››—pa weightqis questionable. However to build a modelwe need to choose a level of granularity and try to stay within it, or the whole modelevaporates into the nothingness of truth!

An aspect of a thingxis a way of viewing it, a particular way in whichxcan be regardedor measured. For example, a woman can be regarded as a person; hence “being a person”is an aspect of a woman. A molecule has a molecular mass (say in daltons), so “havinga molecular mass” is an aspect of a molecule. In other words, byaspectwe simply meana function. The domainAof the functionf:A—Bis the thing we are measuring, andthe codomain is the set of possible “answers” or results of the measurement.

We represent eachtype as a box containing asingular indefinite noun phrase.

Data gathering is ubiquitous in science. Giant databases are currently being minedfor unknown patterns, but in fact there are many (many) known patterns that simplyhave not been catalogued. Consider the well-known case of medical records. A patient’smedical history is often known by various individual doctor-offices but quite inadequatelyshared between them. Sharing medical records often means faxing a hand-written noteor a filled-in house-created form between offices.

As mentioned above category theory has branched out into certain areas of scienceas well. Baez and Dolan have shown its value in making sense of quantum physics, itis well established in computer science, and it has found proponents in several otherfields as well. But to my mind, we are the very beginning of its venture into scientificmethodology. Category theory was invented as a bridge and it will continue to serve inthat role.

All this time, however, category theory was consistently seen by much of the mathe-matical community as ridiculously abstract. But in the 21st century it has finally cometo find healthy respect within the larger community of pure mathematics. It is the lan-guage of choice for graduate-level algebra and topology courses, and in my opinion willcontinue to establish itself as the basic framework in which mathematics is done

In 1980 Joachim Lambek showed that the types and programs used in computerscience form a specific kind of category. This provided a new semantics for talking aboutprograms, allowing people to investigate how programs combine and compose to createother programs, without caring about the specifics of implementation. Eugenio Moggibrought the category theoretic notion of monads into computer science to encapsulateideas that up to that point were considered outside the realm of such theory.

Bill Lawvere saw category theory as a new foundation for all mathematical thought.Mathematicians had been searching for foundations in the 19th century and were reason-ably satisfied with set theory asthe foundation. But Lawvere showed that the categoryof sets is simply a category with certain nice properties, not necessarily the center ofthe mathematical universe. He explained how whole algebraic theories can be viewedas examples of a single system. He and others went on to show that higher order logicwas beautifully captured in the setting of category theory (more specifically toposes).It is here also that Grothendieck and his school worked out major results in algebraicgeometry.

The paradigm shift brought on by Einstein’s theory of relativity brought on the real-ization that there is no single perspective from which to view the world. There is nobackground framework that we need to find; there are infinitely many different frame-works and perspectives, and the real power lies in being able to translate between them.It is in this historical context that category theory got its start.

These theorems have not made theirway out into the world of science, but they are directly applicable there. Hierarchies arepartial orders, symmetries are group elements, data models are categories, agent actionsare monoid actions, local-to-global principles are sheaves, self-similarity is modeled byoperads, context can be modeled by monads.

No one would dispute that vector spaces are ubiquitous.But so are hierarchies, symmetries, actions of agents on objects, data models, globalbehavior emerging as the aggregate of local behavior, self-similarity, and the effect ofmethodological context.

I will use a mathematical tool calledologs, or ontology logs, to givesome structure to the kinds of ideas that are often communicated in pictures like theone on the cover. Each olog inherently offers a framework in which to record data aboutthe subject. More precisely it encompasses adatabase schema, which means a system ofinterconnected tables that are initially empty but into which data can be entered.

Agreementis the good stuff in science; it’s the high fives.But it is easy to think we’re in agreement, when really we’re not. Modeling ourthoughts on heuristics and pictures may be convenient for quick travel down the road,but we’re liable to miss our turnoff at the first mile. The danger is in mistaking ourconvenient conceptualizations for what’s actually there. It is imperative that we havethe ability at any time to ground out in reality.

Yes, my intention was to show the most easily replaced in dark and move it to the least easily replaced.

As a result, new disruptive innovations emerge that are more modular—or unbundled

And the result is a book, which is being released this month by Polity Press.

The bubbles then stick around for about 20 to 30 minutes, and you won’t be able to crack your knuckles again until they are gone — a sure sign that the sound is not caused by bursting bubbles, as was the leading hypothesis for many years.

If the space doctor’s ideas were wrong, your phone wouldn’t be able to tell where it was.

he did not have strategies that might alter thesituation

she saw herself asresponsible for and responsive to the needs of others.

one can appropriate cultural tools (an aspect of learning) without takingthem on internally (an aspect of identity).

maintenance.

reconceptualizes learning froman in-the-head phenomenon to a matter of engagement, participation, and member-ship in a community of practice.

eripheral trajectories never lead to full participation-rather, individuals on thistrajectory stay marginal to the practice over time.

In my research, we filmed an interaction between a couple and had each partner turn a rating dial as they watched their tape afterward. On this graph (at left), you can see how one couple rated their interaction. The blue dots represent the wife’s ratings over 15 minutes of conversation; the red dots represent the husband’s ratings. When you add them together, these ratings are a constant, which means that in this interaction, her gain is his loss and his gain is her loss. This is what’s called in game theory a “zero-sum game.” You’ve probably all heard of the concept. It’s the idea that in an interaction, there’s a winner and a loser. And by looking at ratings like this, I came to define a “betrayal metric”: It’s the extent to which an interaction is a zero-sum game, where your partner’s gain is your loss. On the other hand, by trust we really mean, mathematically, that our partner’s behavior is acting to increase our rating dial. Even though we’re disagreeing, my wife is thinking about my welfare, my best interests. When we scientifically tested these so-called trust and betrayal metrics, we found that a high trust metric is correlated with very positive outcomes, such as greater stability in the relationship. In a 20-year longitudinal study of couples in the San Francisco Bay Area that I recently completed with UC Berkeley psychologist Bob Levenson, we found that about 11 percent of couples had a zero-sum game pattern, like in that graph. Every six years, we would re-contact all of the couples in the study, and they would come back to Bob’s lab at Berkeley. Yet we noticed that many of the zero-sum couples weren’t coming back. I thought maybe they dropped out because they found the whole thing so unpleasant. Well, it turns out that they didn’t drop out. They died.

Interestingly, the investors’ expectations about the back-transfer from the trustee did not differ between the oxytocin and placebo recipients. Oxytocin increased the participants’ willingness to trust others, but it did not make them more optimistic about another person’s trustworthiness.

Conventional economic theory maintains that people will always behave in a purely self-interested manner. According to this worldview, it makes no sense to trust, whether in a trust game or in real life, as any trust will be exploited. The trustee will always keep her entire windfall for herself, so the investor would be better off not transferring any money in the first place. And yet when researchers like Joyce Berg and others have had people play the trust game with real monetary stakes, they have repeatedly found that the average investor will transfer half of her initial endowment and receive similar amounts in return. Through the trust game, researchers have also discovered a number of factors that seem to drive levels of trust. Familiarity breeds trust—players tend to trust each other more with each new game. So does introducing punishments for untrustworthy behavior, or even just reminding players of their obligations to each other.

To become a forgiving person, we have to practice forgiving smaller grievances. Then, when a bigger insult comes, we are ready, willing, and able to deal with it. Alternatively, like Delores, once we learn to forgive a major grievance, we can understand the value of limiting the power that pain and anger hold over us the next time we are hurt.

In a first analysis, data showed that people who took less than 10 seconds to decide how much to give gave approximately 15 percent more to the common pool than people who took longer than 10 seconds.

These definitions allow us toconsider the relation between learning and identity while still considering them to beindependent processes.

Coach J engagedconversations that fostered social relationships to sustain their effort in a difficultmoment As relationships were built, runners repeated important aspects of thepractice that allow them to gain competence.

They argue that the ability todo the math alone is not enough to support strong mathematical identities for stu-dents; rather, mathematical identities are tied to understanding and eng

This finding highlights the critical role of the nature ofthe relationship with the teacher/coach for opportunities both to develop practice-linked identities and to become better at the practice.

Our analyses show that relational resources sometimes served as akind of gateway to material and ideational resources (and thus as a gateway tolearning

three levels of analysisthat are required to offer an account of identity, including individuals, social interac-tion, and broader society.

Our analyses show that relational resources sometimes served as akind of gateway to material and ideational resources (and thus as a gateway tolearning).We als

. Our data seem to indicate that the personal relationshipbetween the coach and the athlete was central in determining athletes' access to theother material and ideational resources, as much of the teaching and learningoccurred in one-on-one interactions

Octavia displayed this same drive and independence, but at theend of practice she sat down on the grass for about 20 minutes to socialize with herteammates. At this point in the season, Octavia seemed to be participating in thepractice not for others but for herself, so that she might improve her performance.

We argue that Octavia's practice-linked identity with respect to track increasedover the course of the season. That is, she became more connected to the practice oftrack. This shift was evidenced both in her retrospective account in an interviewtoward the end of the year and in changes in her participation in practices and meetsfrom early in the season to later in the year. Co-occurring with these shifts werechanges in Octavia's goals in track, changes in her performance as she learned tohurdle, and changes in her social relationships with other members of the team

In this interaction, several ideationalartifacts about Octavia and about hurdling were made salient Importantly, sheactively participated in this exchange, deconstructing her own performance.

Connoisseurscallforthecontemplationofcomplexityalmostforitsownsake,orremindeveryonethatthingsaremoresubtlethantheyseem,orthanyoujustsaid.eattractivethingaboutthismoveisthatitisliterallyalwaysavailabletothepersonwhowantstomakeit.eoryisfoundedonabstraction,abstractionmeansthrowingawaydetailforthesakeofabitofgenerality,andsothingsarealways“morecomplicatedthanthat”—foranyvalueof“that”.

cf. actor-network theory

If you think everybody answers the same way, you may be an advocate of critical theory.

So how can you be beyond critical theory, given that it generally aims “to explain and transform all the circumstances that enslave human beings?“

Asu Ozdaglar

What does it mean to be an “item” or “computational object” within this collection? What is such a collection?

The fourth of the theories is as yet the least influential but seems to be gaining strength. Its key ideas are that human nature causes people to flourish more under some conditions than under others, and that social and political institutions should be organized to facilitate that flourishing. What, more specifically, are the conditions or “functionings” that enable people to flourish?

The innate qualities of intellectual pr operty, however, in combination with INTELLECTUAL PROPERTY: POLICY FOR INNOVATION 15   strong economic motivations have led U.S. intellectual property policy to operate according to rights - based, non - utilitarian theory, possibly as a result of lobbying (capture theory).

The “romantic conception of authorship” mentioned earlier as a formative trend of the rights - based theory of intellectual property is evident in the first pe rspective: t he notion that ideas are individual achievements and of indeterminate origin (not reliant on a process of building) (Fisher, 1999, Sect. II. B).

This is understatement to be sure, but the debate has been principally between two theories: a utilitarian policy theory, and a rights - based , non - utilitarian property theory (Long, 1995, n.pag.) .

The U . S . Co nstitution firmly grounds the proper role of intellectual property policy as utilitarian .

Keywords : anticommons, copyright, intellectual property, Lockean Proviso, patent, property rights, state of nature, trademark, utilitarian theory

The architect of Berlin's new main train station won a lawsuit against Deutsche Bahn AG in a court decision that forces the German s tate - owned rail company to replace the building's ceiling with his original design. Architect Meinhard von Gerkan sued Deutsche Bahn for distorting his plan by exchanging his ceiling, designed to resemble a cathedral's nave, for one made of flat metal des igned by another architect. German copyright rules protect the integrity of work by artists and architects.

In his ruling, judge Edmundo Rodríguez Achútegui recognized that Calatrava’s rights as author of the bridge had been infringed, but he ruled that the public utility of the addition took precedence over this private right. “In addition to constituting a singular artistic creation suitable for protection, the work is public one, offering a service to the citizens, and thus satisfies a public interest,” he said. “If we weigh these interests, the public must prevail over the private.

ecause such accumulation is less typical, the realm of intellectual property has less of the laborer/capitalist hierarchy of Marxist theory.

What makes number theory interesting is that problems that are simple to state (at least to mathematicians) are often fiendishly difficult to solve. The most famous such problem, Fermat's last theorem, was postulated in the 17th century. It took until 1993 to prove that it was true.