Thursday, 19 March 2015
Friday, 28 November 2014
Why We Need a Proof Assistant in Law and Finance.
Generalising the "Proof Assistant" for Understanding the Models of Law and Finance.
With oh so many subjects available to study & so many amalgamations going on at both the subject and institutional level, maybe we ought to think about what's "worth doing" and "why". There is a trend at the higher-end of mathematics to build "proof assistants" (see Voevosky's videos at his website at the Institute of Advance Studies), It's clear to him that building a communication tool to computers so that proofs can be checked WILL eventually be the way mathematics will be taught in the general culture someday. This is absolutely determined, because how else can the mathematical space be explored with such complex proofs, NOBODY can be certain if they are true? A machine-based (including a quantum machine-based) tool that does most of the mechanical "checking" quickly would help ensure that complex proofs are true and accurate. Most complex proofs after all have no way of being checked to 100% accuracy by mere humans.
Apply this same idea to much more difficult subjects such as law and finance [Remember Von Neumann's comment? "Anyone who thinks mathematics is difficult, has not yet experienced real life."], and the concept of a "law and finance 'proof assistant'" becomes both theoretically and practically interesting.
Now, imagine the irony if building that sort of machine were really, really difficult? Voevosky states that the only computer language that could take on the formalities of his Homotopy Type Theory (HOTT) is Coq. So everybody and his mother is now writing in Coq to get to a universal proof assistant.
But Gross, Chlipala & Spivak (http://adam.chlipala.net/papers/CategoryITP14/CategoryITP14.pdf) say that doing rather easy category theory in Coq is pretty hard! So they've written some short-cuts to make the use of Coq less burdensome.
This brings me to my point that category theory which was invented [see Eilenberg & Mac Lane 1945) so that we could compare complex theories in mathematics could be made to be extremely useful for COMPARATIVE LAW and a genuine understanding of how COMPLEX FINANCIAL INSTRUMENTS actually work. In effect, the hinge is that legal systems are models and in the extreme are isomorphic categories which can be compared using functors. Same can be said about financial instruments. From a legal and financial perspective, the MOST COMPLICATED instrument in the financial universe is the MORTGAGE because it has centuries of legal strata embedded within it (legal-historical interpretations are about 2000 years) and its current re-interpretation via ASSET-BACKED SECURITIES REGULATIONS as an underlying asset of pass-through or senior-subordinated note structures has been further complexified with CREDIT RISK RETENTION REGULATIONS. All of these legal-financial rules are complex models that need to be re-arranged into MODEL-TYPES than can be formally compared. Otherwise, really, just as Voevosky says about complex proofs, we have no chance at all in understanding how these instruments actually work in the real world.
For a way to get started in this approach, I recommend reading David Spivak's (2013/2014) A Category Theory for Scientists. There's an old version freely available on the web and the MIT edition is also quite convenient.
With oh so many subjects available to study & so many amalgamations going on at both the subject and institutional level, maybe we ought to think about what's "worth doing" and "why". There is a trend at the higher-end of mathematics to build "proof assistants" (see Voevosky's videos at his website at the Institute of Advance Studies), It's clear to him that building a communication tool to computers so that proofs can be checked WILL eventually be the way mathematics will be taught in the general culture someday. This is absolutely determined, because how else can the mathematical space be explored with such complex proofs, NOBODY can be certain if they are true? A machine-based (including a quantum machine-based) tool that does most of the mechanical "checking" quickly would help ensure that complex proofs are true and accurate. Most complex proofs after all have no way of being checked to 100% accuracy by mere humans.
Apply this same idea to much more difficult subjects such as law and finance [Remember Von Neumann's comment? "Anyone who thinks mathematics is difficult, has not yet experienced real life."], and the concept of a "law and finance 'proof assistant'" becomes both theoretically and practically interesting.
Now, imagine the irony if building that sort of machine were really, really difficult? Voevosky states that the only computer language that could take on the formalities of his Homotopy Type Theory (HOTT) is Coq. So everybody and his mother is now writing in Coq to get to a universal proof assistant.
But Gross, Chlipala & Spivak (http://adam.chlipala.net/papers/CategoryITP14/CategoryITP14.pdf) say that doing rather easy category theory in Coq is pretty hard! So they've written some short-cuts to make the use of Coq less burdensome.
This brings me to my point that category theory which was invented [see Eilenberg & Mac Lane 1945) so that we could compare complex theories in mathematics could be made to be extremely useful for COMPARATIVE LAW and a genuine understanding of how COMPLEX FINANCIAL INSTRUMENTS actually work. In effect, the hinge is that legal systems are models and in the extreme are isomorphic categories which can be compared using functors. Same can be said about financial instruments. From a legal and financial perspective, the MOST COMPLICATED instrument in the financial universe is the MORTGAGE because it has centuries of legal strata embedded within it (legal-historical interpretations are about 2000 years) and its current re-interpretation via ASSET-BACKED SECURITIES REGULATIONS as an underlying asset of pass-through or senior-subordinated note structures has been further complexified with CREDIT RISK RETENTION REGULATIONS. All of these legal-financial rules are complex models that need to be re-arranged into MODEL-TYPES than can be formally compared. Otherwise, really, just as Voevosky says about complex proofs, we have no chance at all in understanding how these instruments actually work in the real world.
For a way to get started in this approach, I recommend reading David Spivak's (2013/2014) A Category Theory for Scientists. There's an old version freely available on the web and the MIT edition is also quite convenient.
Wednesday, 12 November 2014
S&P 500 in 392 Weeks: Scale Invariance Test Coming
In one of Mandelbrot's original works on "scale invariance," he studied the cotton markets around the mid- to late 19th century and found that the shape of the price versus time graphs were similar no matter whether you measured the average price per week or month. A discovery of any kind of invariance is an important fact about how the way the world works. No matter how quirky the invariance is, any theory worth being called a theory needs to explain the invariance's existence.
The above graph is not scale invariant. But it might be multi-scale invariant. Much depends on what will happen in the next week or so.
Towards a Homotopy Type Theory for Law and Finance
1. Imagine a homotopy diagram for law and
finance involving contracts, torts and criminal law, as well as the media,
culture, justice, fairness. The universe
of discourse is represented by an oval that looks like the cosmic background
radiation map (LOL) and it is divided in half so that we have a starting frame
(ideal initial conditions) between one part on the left which is an unjust and
unfair society and another part on the right which is a just and fair
society. Criminal litigation is a
partition that moves from right to left with the ideal as the central line
axis. Thus, societies can maximize or
minimize the unjust-unfair part in relation to the just-fair part. Each successful prosecution deforms the two
parts such that a just-fair prosecution in the unjust-unfair part tends to
decrease the unjust-unfair part and increase the just-fair part. The old way of talking about the connection
between the two parts is to call it a "fibration" between
"manifolds"--but those are the physicists and maths whizzos who don't
have a handle on the niceties of social theories. Now, the fibration are just functional
connections between the two parts, and it turns out, all that you need to know
that could ever really happen between the two parts are embedded in the
fibration. In Homotopy Type Theory, the
fibrations are the essence of the "covering space" between the two
parts. We can start to work out certain kinds of equivalences.
2. Now, assume criminal prosecutions are
"transport functions" between the two parts of the oval.
3. Bizarrely, (and this is a big guess) very
dense litigation and all forms of risk of loss (default in the widest possible
sense) are functors and act as covering spaces between the two parts.
4. Implication: you don't need to know the substance of each
criminal prosecution, just the fact that it is being done, that deforms the two
parts towards or away from the ideal state of society.
5. Please note that the term ‘ideal state’ here
does not mean Plato’s ideal good state; it means a perfectly continuous
geometric construct of the intuition that does not require anything at all except a few arrows and some ovals.
Sunday, 9 November 2014
Fault Tree Analysis is the Teleology that Ontology Needs; Dr. Kent Stephens' Classic Paper
Gosh! Here's one of the great papers of the 20th century that very few people have even heard of. I'm serious. I think this paper ranks higher than Akerloff's information asymmetry paper on a market for "lemons", and just a tad below Claude Shannon's masters thesis on information theory.
http://files.eric.ed.gov/fulltext/ED095588.pdf
This is Kent G. Stephens paper on "Fault Tree Analysis."
Once I gave a 2 day seminar in London to a delegation of Russian academics from Moscow State University who were in the department of engineering and organisations. The first day was a total disaster because they said they wanted something "on practical project management". So, that evening I produced some slides about "and-logic" and "or-logic" and combined it with a flow diagram on "critical project analysis and implementation." I said, "This work comes largely from Dr. Kent Stephens." And before I could finish my sentence, the Head of the Department, a very sharp tongued professor, said, "Yes, we know all his work in our department, and we can see that you put much effort OVERNIGHT to bring to us today your original thoughts. Thank you." And I was dismissed! The point of this story is that this was the only time in over 20 years of using Dr Kent G. Stephens ideas that anyone had ever said they knew him and his ideas.
The reason I think this paper is one of the most important papers in the 20th century is because it is the first and only paper I know of that successfully combines cultural value analysis to figuring out how organisations FAIL! Back in the day, Dr. Stephens had assistants with questionnaires ask individuals in an organisation particular sorts of "valued questions" to determine what we now call the "critical path" within an organisation. He'd figure out the critical path of communications and were most of the errors occurred that jammed up the organisation. if Aristotle were alive, he'd be very proud of Dr. Stephens' work because it's been used to fix a lot of otherwise "failing institutions". And unlike the BS consulting you see 99.99% of the time, the good doctor and his team would come up with fantastically elegant solutions. E.g., he took a failing elementary-to-high school that was in the bottom 5 of California to the top 10 in one year!
His paper is important to keep in mind if one embarking on building an "ontology of an organisation". Too many times, I see ontologies being built without a fundamental understanding of the TELEOLOGY of the human actors. A complete ontology needs to understand teleology deeply, and I think Dr. Stephens helps us a long way in this regard.
http://files.eric.ed.gov/fulltext/ED095588.pdf
This is Kent G. Stephens paper on "Fault Tree Analysis."
Once I gave a 2 day seminar in London to a delegation of Russian academics from Moscow State University who were in the department of engineering and organisations. The first day was a total disaster because they said they wanted something "on practical project management". So, that evening I produced some slides about "and-logic" and "or-logic" and combined it with a flow diagram on "critical project analysis and implementation." I said, "This work comes largely from Dr. Kent Stephens." And before I could finish my sentence, the Head of the Department, a very sharp tongued professor, said, "Yes, we know all his work in our department, and we can see that you put much effort OVERNIGHT to bring to us today your original thoughts. Thank you." And I was dismissed! The point of this story is that this was the only time in over 20 years of using Dr Kent G. Stephens ideas that anyone had ever said they knew him and his ideas.
The reason I think this paper is one of the most important papers in the 20th century is because it is the first and only paper I know of that successfully combines cultural value analysis to figuring out how organisations FAIL! Back in the day, Dr. Stephens had assistants with questionnaires ask individuals in an organisation particular sorts of "valued questions" to determine what we now call the "critical path" within an organisation. He'd figure out the critical path of communications and were most of the errors occurred that jammed up the organisation. if Aristotle were alive, he'd be very proud of Dr. Stephens' work because it's been used to fix a lot of otherwise "failing institutions". And unlike the BS consulting you see 99.99% of the time, the good doctor and his team would come up with fantastically elegant solutions. E.g., he took a failing elementary-to-high school that was in the bottom 5 of California to the top 10 in one year!
His paper is important to keep in mind if one embarking on building an "ontology of an organisation". Too many times, I see ontologies being built without a fundamental understanding of the TELEOLOGY of the human actors. A complete ontology needs to understand teleology deeply, and I think Dr. Stephens helps us a long way in this regard.
GDP-Derivatives: A Global Risk Management Tool; What do we mean by "invariance up to isomorphism"
Economic statistics are compiled and written by bureaucrats who get fired only if they show they haven't been doing any work, so is it any surprise that their figures should be revised?
http://www.nytimes.com/2014/11/07/business/economy/doubting-the-economic-data-consider-the-source.html?partner=rss&emc=rss&_r=2
One of the problems in financial engineering is getting a set of figures that the world can agree on. This is what I call the problem of finding the invariance. One way to think about Category Theory is that it's all about finding invariance at the level of isomorphism, or more forcefully, of finding what is uniquely true and accurate because it is indicated by gestural arrows that point directly at it. At a visceral level, notice how when we point at something saying that "it's right there", we are in a state of "understanding is not merely a pointing but an extension of the pointing as part of the activities of the world." Category Theory at the level of functoriality tells us "it's all about the pointing" so the object itself is not at necessary, or again, to put it more forcefully, the object is completely defined by the infinite number of pointings that we have of that object, so the substantiality of the object disappears! We don't need the object at all, because now we know it completely in its infinite possibilities of being. This sounds very abstract (and it is) but in everyday life, we do this "gestural understanding" all the time, whenever we eat, sleep, converse, enjoy a drink...all of these "things" are invariances at the level of isomorphism. But the problem of government statistics...
is that they get revised and so we have tremendously long time-lags in response to "certainties of announcements" that affect our buying and selling decisions.
In 1999, I had worked with an ex-Merrill Lynch derivatives trader to create a "GDP-derivative" which basically would allow you to take take bets on the GDP of any nation in the world. Of all the derivatives that could help humanity manage its "spaceship resources", I thought a GDP derivative would be the best. It would mean essentially that a globally active company (or any other legal entity including a state) could manage its risk. So, if say you wanted to hedge Brazilian GDP risk, you could. The conceptual design for this product was pretty EZ. All you have to do is think "swap", i.e., the cash flows of a buyer and seller in relation to the data regarding the GDP figure. For 'proving out' the instrument, we just made a table of natural buyers versus sellers, and listed the sectors underneath each heading, and thought through which companies would be "natural buyers and sellers" given different scenarios of "expected GDP". Anyone doing a masters level course on quantitative finance should be able to knock up this model in a leisurely afternoon.
Anyway, the problem we had was the "revisions" on GDP data. Since these numbers came out 6 to 18 months after the first announcement, it became difficult to "match up" reality. In the language I use today, I'd say, "We couldn't get a simple isomorphism and therefore, no invariance." Without an invariance (an agreement on the GDP-figures), our model would not work. Of course, that was back then, before we had Google data. Now, I'm pretty sure we could crunch up our own GDP-index in order to create the GDP-derivative. Then it's a matter of selling and marketing...
http://www.nytimes.com/2014/11/07/business/economy/doubting-the-economic-data-consider-the-source.html?partner=rss&emc=rss&_r=2
One of the problems in financial engineering is getting a set of figures that the world can agree on. This is what I call the problem of finding the invariance. One way to think about Category Theory is that it's all about finding invariance at the level of isomorphism, or more forcefully, of finding what is uniquely true and accurate because it is indicated by gestural arrows that point directly at it. At a visceral level, notice how when we point at something saying that "it's right there", we are in a state of "understanding is not merely a pointing but an extension of the pointing as part of the activities of the world." Category Theory at the level of functoriality tells us "it's all about the pointing" so the object itself is not at necessary, or again, to put it more forcefully, the object is completely defined by the infinite number of pointings that we have of that object, so the substantiality of the object disappears! We don't need the object at all, because now we know it completely in its infinite possibilities of being. This sounds very abstract (and it is) but in everyday life, we do this "gestural understanding" all the time, whenever we eat, sleep, converse, enjoy a drink...all of these "things" are invariances at the level of isomorphism. But the problem of government statistics...
is that they get revised and so we have tremendously long time-lags in response to "certainties of announcements" that affect our buying and selling decisions.
In 1999, I had worked with an ex-Merrill Lynch derivatives trader to create a "GDP-derivative" which basically would allow you to take take bets on the GDP of any nation in the world. Of all the derivatives that could help humanity manage its "spaceship resources", I thought a GDP derivative would be the best. It would mean essentially that a globally active company (or any other legal entity including a state) could manage its risk. So, if say you wanted to hedge Brazilian GDP risk, you could. The conceptual design for this product was pretty EZ. All you have to do is think "swap", i.e., the cash flows of a buyer and seller in relation to the data regarding the GDP figure. For 'proving out' the instrument, we just made a table of natural buyers versus sellers, and listed the sectors underneath each heading, and thought through which companies would be "natural buyers and sellers" given different scenarios of "expected GDP". Anyone doing a masters level course on quantitative finance should be able to knock up this model in a leisurely afternoon.
Anyway, the problem we had was the "revisions" on GDP data. Since these numbers came out 6 to 18 months after the first announcement, it became difficult to "match up" reality. In the language I use today, I'd say, "We couldn't get a simple isomorphism and therefore, no invariance." Without an invariance (an agreement on the GDP-figures), our model would not work. Of course, that was back then, before we had Google data. Now, I'm pretty sure we could crunch up our own GDP-index in order to create the GDP-derivative. Then it's a matter of selling and marketing...
Friday, 7 November 2014
The End of Education Monopolies, Long Live Personal Learning Assistants
1. Suppose all information about every subject is at your fingertips. Why would you bother to go to University, indeed, high school, if only to learn the social rituals, find mating partners, travel together, get a good job? Human-to-human interfaces are good for some things, but for certainty of info the human-to-machine interface is about 20x better. See, Dr. Kent Stephens studies on the ICBM in the 1950s.
2. In the year 2000, I was in Pasadena at a certain prestigious University whose name will remain unnamed and I was speaking to the Dean who said that their Uni had just received $35 million from a billionaire-developer-alumna to build a beautiful new hi-tech law school building. The building was about 25 to 30 stories and they had their own television channel, and all lecture halls had at least two digital video cameras and the data from the lectures were sent to a Media Centre, where media workers developed the content into broadcast. The Dean said to me, "What will we (law school bricks and mortar) do when we put all these courses onto two disks?" I said, "Isn't one more than enough?"
3. The disintermediation of universities hasn't occurred YET because no one has yet figured out how to make a University within your own simple point and click powers. There's too much content on the Web and it's not at all clear what if anything but the certainty of exchange approximates reality.
4. I propose the Personal Learning Assistant. The PLA grows with you and mirrors your hobbyist interests (non-profit and at a cost of consumption) and professional specialised knowledge (for profit and chargeable at globally competitive rates). The PLA is not you, but it's pretty close to being your Web-clone. It can apply for jobs, do jobs, and even multiply in terms of identifiable profiles on the Web. BTW, the PLA does not simply exist in digital code, but will have significant impact on your physical being and others. For example, who'll find the best heart specialist just in time, get the appropriate therapeutic apps to manage your incipient diabetes, run every sort of psycho-chemical tests and ensure that you are aware of the survival odds at the next traffic junction other than your PLA? Who's your Guardian Angel and Protector? If you want to go on "auto-PLA" then you can turn down the control to subliminal 0,005 second input-outputs. What's the point of a university education if you can have this much fun at 10 magnitudes above and below the normal medium of perception? I guess this is what web-based education has to offer. Open the doors of perception and get a universal education. BTW building your own PLA immediately answers questions about long-term social welfare.
Thursday, 6 November 2014
Ebola's Decay in Contagion
My toy model of Ebola mortality where the doubling occurs every 20 days with an arbitrary start date of January 11th 2014 predicted 4,096 deaths for the 9th of October 2014, and the official WHO statistic was 4,033 for the 10th of October 2014. Assuming the same rate of doubling, the toy model predicted 8,192 deaths for the 29th of October 2014. The official figure hovered around 5,000 deaths for the 1st of November 2014. Thus, the toy model is dead. If the official figures are accurate then the doubling factor in days has increased to around 40 days, that is, about 100 deaths per day. While sad in an absolute sense, this figure is very good news globally because it shows that there is a decay in contagion.
The first chart above shows some correlation to the news effect of Ebola on VIX (a volatility or "fear" indicator).
The second chart shows history of potential pandemics from the 1950s.
So far humanity appears to be missing the bullet and technological and policy responses, no matter how awkward at first, appear to be protective enough for the species.
Sunday, 26 October 2014
On the Laws of Immortality (of the Virtual Sort)
On the Laws of Immortality (of the Virtual Sort).
I'm wondering whether we need to have an educational course that anticipates a "religious - technological future that presumes virtual immortality." See, reference to Tipler's Physics of Immortality below. I have no idea what to call this course. The course would be built on anticipating fundamental technological breakthroughs aimed at the ultimate teleology or Omega Point and would try to figure out their consequences in terms of the law. This isn't about patent law or innovations--the best course for that is at the Law School at Stanford University. This is a course that would bring back Aristotle's causa of teleology so that Ethics and Politics could be discussed in relation to the causae of Form (mathematical and visual technologies), Efficient (the material implementation of such breakthroughs from the micro to the macro) and the Substantive (the uniqueness of breakthrough). Permit me an example.
Yesterday, I was speaking with a young friend who sells computers by day and plans film-making at night. His next project is a 15 minute short on "How to become a Roman Emperor." He tells me that there is a breakthrough that has occurred that will require ALL ELECTRONIC equipment to be redone. The fundamental breakthrough will enable a mobile phone to carry 1,000 terrabytes of memory. And it will mean that a lot of programming to make things "compact" will become unnecessary. Maybe we won't need any programming language at all, and maybe a lot of processes we take for granted to check, validate and verify our electronic memories will become obsolete.
Now, what kind of universe of legal discourse would that innovation imply?
This technology will be out within 4 or 5 years.
Reference: Tipler, F.T. (2000) The Physics of Immortality. All the fundamental breakthroughs according to Tipler (who was a top rated physicist until he published this CRAZY book that makes him look looney--but I think it's just a great example of taking a rather simple idea very seriously), will be about achieving an Omega Point (Theilhard de Chardin's idea back in the 1930's) that everything in life will be resurrected. Now, resurrection in a physics sense is a White Hole where an infinitude of memories can be played back. Tipler describes the evolution of the universe from black hole singularity to white hole resurrection. To get to a white hole, all the energy of the universe will be used to store all events. It's a CRAZY idea because it is an ultimate teleology and modern science does not like teleology at all.
I'm wondering whether we need to have an educational course that anticipates a "religious - technological future that presumes virtual immortality." See, reference to Tipler's Physics of Immortality below. I have no idea what to call this course. The course would be built on anticipating fundamental technological breakthroughs aimed at the ultimate teleology or Omega Point and would try to figure out their consequences in terms of the law. This isn't about patent law or innovations--the best course for that is at the Law School at Stanford University. This is a course that would bring back Aristotle's causa of teleology so that Ethics and Politics could be discussed in relation to the causae of Form (mathematical and visual technologies), Efficient (the material implementation of such breakthroughs from the micro to the macro) and the Substantive (the uniqueness of breakthrough). Permit me an example.
Yesterday, I was speaking with a young friend who sells computers by day and plans film-making at night. His next project is a 15 minute short on "How to become a Roman Emperor." He tells me that there is a breakthrough that has occurred that will require ALL ELECTRONIC equipment to be redone. The fundamental breakthrough will enable a mobile phone to carry 1,000 terrabytes of memory. And it will mean that a lot of programming to make things "compact" will become unnecessary. Maybe we won't need any programming language at all, and maybe a lot of processes we take for granted to check, validate and verify our electronic memories will become obsolete.
Now, what kind of universe of legal discourse would that innovation imply?
This technology will be out within 4 or 5 years.
Reference: Tipler, F.T. (2000) The Physics of Immortality. All the fundamental breakthroughs according to Tipler (who was a top rated physicist until he published this CRAZY book that makes him look looney--but I think it's just a great example of taking a rather simple idea very seriously), will be about achieving an Omega Point (Theilhard de Chardin's idea back in the 1930's) that everything in life will be resurrected. Now, resurrection in a physics sense is a White Hole where an infinitude of memories can be played back. Tipler describes the evolution of the universe from black hole singularity to white hole resurrection. To get to a white hole, all the energy of the universe will be used to store all events. It's a CRAZY idea because it is an ultimate teleology and modern science does not like teleology at all.
Friday, 24 October 2014
Mapping Ebola Event Risk by Postulating a Financial Markets Ontology
Mapping Ebola Event Risk by Postulating a Financial Markets Ontology.
1. As we said last time, correlation is not causation. This is an important distinction in science. If we wish to do science in law and finance, then using a Category Theory approach, we should translate our statements made of legal and financial terms into SCIENTIFIC PROPOSITIONS that adhere to at least a FIRST ORDER LOGIC. Translated into this first order logic, we can check our premises and inspect our deductions and inferences for their weaknesses and soundness. If you would like to know what scientific language would look like, you can turn to Spivak's A Category Theory for Scientists (Old Version) 2013-14 which is freely available on the Web. In brief, theories and models within theories are written in propositional form using "arrow-language", thus, f:A-->B, where f designates the name of the arrow between A and B. If you trace Spivak's work who is at MIT, you'll note that he wrote a paper together with Robert E. Kent on "Ologs", which are basically "ontological statements" of real processes. Robert E. Kent (who's affiliation to any academic institution appears non-existent) has written on the INSTITUTIONAL APPROACH which is a version of Category Theory that is translated for use among people who work and play in the area of INSTITUTIONAL THEORY. As far as I can tell with my almost total ignorance of the field, the first successful transplantation of a fundamental device from Category Theory into Institutional Theory was by Dimaggio & Powell in their extremely well-cited paper (over 27,000 citations so far) on "The Iron Cage Revisited: Institutional Isomorphism and Collective Rationality in Organizational Fields" (1983) see Jistor. They used the concept of isomorphism which in Category Theory means f:A->B and g:B->A, so that the morphisms f and g which represent "processes" link the two objects A and B in a manner such that the objects are isomorphic, that is, that A and B are unique up to isomorphism. Even if you don't quite understand what this "definition" means, you should understand that isomorphism is the way Category Theorists think about "equivalence." It is all process driven.
2. Now Robert E. Kent's work on the Institutional Approach is part of a blossoming field called ONTOLOGIES. An ontology is simply a model of real processes involving people and machines, especially computational devices that can be linked. The link between Category Theory and Ontology goes back to a prescient genius named Goguen who was at the University of California San Diego. His paper on "A Categorical Manifesto" 1991 is a must read. And the paper co-authored with Burstal on: INSTITUTIONS: ABSTRACT MODEL THEORY FOR SPECIFICATION AND PROGRAMMING (1992). This paper lays out a fantastic theory of institutions that is rigged according to category theory formalities. Now how do these theories relate to the Ebola versus Financial Charts above?
3. Well, the point is that it is genuinely difficult to understand what the above charts MEAN unless we have some kind of FINANCIAL MARKET ONTOLOGY buzzing away in the back of our minds. That ontology would need to be made explicit in order for us to have an explicit understanding of the EVENT RISK that Ebola (as an event) poses onto the minds of financial traders.
4. To do that which is required in 3 may seem very complicated indeed! But I don't think so. We will examine that space--the space of financial market ontology--in further blogs.
Wednesday, 22 October 2014
Let's Get Rid of Causation in Finance.
This chart comes from Zerohedge today. It makes me laugh. Is the world one? That is, it is so interconnected in terms of web-based and email connections, that messages from the market manipulators instantaneously infects our own wet-ware so that we unconsciously push buttons that amplify particular messages in the noosphere? "Correlation is not causation" so says Zerohedge repeating the mantra that distinguishes the knee-jerk Palovian-Skinnerian stimulus-response reductionism of our poor little brain from the rationality and decision-making processes of the game-theoretic monsters that can crunch big data until a particular solution fits the circumstances just inside the horizon of noncomputability (NP-Hard problems). The best papers still in this area where SPIRITUALITY meets PHYSICS are by Emilie Nother -- Einstein's tutor in infinite-dimensional Hilbert space and lovely well-loved teacher who only wanted to teach but the idiotic German Universities could not even give her a proper teaching post. In the long run of a future history, Germany in the late 19th and 20th centuries will be castigated not only for Nazism but also because its chauvinistic attitude against women--and Noether will be remembered far into the future, farther than when people forget about Hitler. When Noether died, Einstein wrote something to the effect that she had discovered the first spiritual law of physics. There are a couple Noether Theorems, but the one FINANCE THEORETICIANS SHOULD PAY ATTENTION TO but do not is the one that allows one to move from an energy conservation law to dimensional space. How she moves from correlation-covariance is very similar to what the finance theorists from Bachelier to Miller tried with almost brute force. Basically, Noether moves from correlation through covariance using symmetries. Symmetries as we know in the early to first half of the 20th century is the algebra of group theory, and allegedly, our standard model is based in the ideal on our methods developed in group theory. Group theory to me looks like it depends on identity, associativity and invertibility (reversal). I was studying this area for a while back in 2006-07, and then imagine my delight and surprise to find a theory that could encompass and ground group theory using only identity and associativity! That is, why study Group Theory when Category Theory could do all of group theory and go even much farther?
So now we can come back to the two graphs above. How do we compare them? You might say, "Do a correlation analysis and determine the variance." OK. Then you can do the "co-variance analysis" and come up with a number you can compare against other co-variances. But this assumes a standard deviation metric underneath, in other words, a normal distribution. As everyone knows in finance except the crazy financial regulators, there is no such animal in the de facto. "Volatility" is not a good measure. What would be a much measure would be a fractal dimension a la Mandelbrot. If we got use to a Mandelbrotian measure, we'd have a much better feel for the "jitteriness" and "emotionality" of prices, and very importantly, we'd have much better metrics and therefore, a better language to gauge and communicate our intimations and observations of what's actually going on.
To realize these intuitions, I believe we have to build from the ground up, and link the above charts to particular geometries, and to talk about them properly, we would need a new vocabulary that would allow us to make comparisons between the charts and the contexts that surround them in an absolutely precise way.
While we would never say Chart 1 causes Chart 2 or vice versa, we could say without any problem that Chart 1 is isomorphic to Chart 2, and then we would be forced to make explicit exactly WHAT THE PROCESS that makes the isomorphism up. If we could do that, we would not ever again bother with the concept of "causation" in finance.
A Category Theory of Financial Instruments and
Financial Institutions - A Fundamental
Ontology of Law and Finance
-Draft Abstract-
In an early paper examining the regulation
of private placement memorandum (PPM) regulation, we found that there are in
general two approaches to financial regulations: (1) the regulation of the
behaviours of the financial institution; and (2) regulation of the financial
instruments. PPMs are interesting
financial instruments since they sit in-between two extremes of financial
regulatory types: (1) purely private form of financial intermediation (that is,
at the extreme, we have bilateral financial contracts) and (2) highly
ritualized guidance on what can be communicated in the raising of capital (that
is, prospectus-type regulations) stemming from the Securities Act of 1933 and
Securities Exchange At of 1934. The
unnatural divide between the regulation of financial institutions and the
regulation of financial instruments has played itself out in terms regulations
which aim at adjusting the incentives relating to particular types of
businesses within a financial institutional framework. In general, the play out has been a border
conflict between banking regulations and capital market regulations. This is not to say that any particular
jurisdiction uses one type of financial regulation exclusively to the exclusion
of the other, but rather there is a combination or admixture of financial
regulations, that in total, ascribe to one general tendency or the other. Recall the legal theorists who attempted to
justify financial regulations on the basis that “Law Matters”, stating in
particular that law matters in that it set outs the initial permit or license
to practice a certain form of business. As
we shall see, if we apply certain fundamental risk and return models, which the
financial industry itself uses to measure its own financial instruments, we can
distinguish different sorts of businesses according to these risk and return
financial instrument components. Thus,
from an extreme financial institutional perspective, which applies financial
theory to its own institutional design, behaviours and assessment of
behaviours, financial instrument theory applies laws to accomplish the
institution’s particular objectives and the discipline and guidance therefore
are simple financial theory models of risk and return, which captures the
market definition of money. From the
institutional perspective apply financial theory, an institution is simply a
financial instrument with certain risk and return characteristics. Thus, financial institutions regarding
themselves as financial instruments are constrained by the “rules of the game”
of finance, which are basically arbitrage (“the law of one price”) which forever
use laws such as contracts, and every other sort of law and regulation, as
merely instrumental. There are
researchers who assert that there should be a legal theory of finance and use
legal ideas to promote the logical priority of law to finance. There is no
argument with this thesis if we add the distinction that law is the context in
which finance exists. However, it is not
entirely plausible to say that finance exists because of law, nor is it
plausible to assert that law is absolutely necessary for finance to exist in
the world. In any case, the purpose of
this paper is not to adjust or determine where the horse and carriage can
become one or the other—we believe that is actually an obvious distinction--but
rather to understand if possible the system, model or ecology in which law and
finance co-exist. As a theoretical
approach, we shall combine the financial instrument view and the financial
institutional view into one totality, and call that totality, a law and finance
ontology. Our view, which is slightly
complicated because of the distinctness of disciplines and variety of
sub-disciplines involved in law and finance, is to start with the simplest
ideal financial instrument and then to ask ourselves what would happen to that
financial instrument in the real world of law and finance.
This model of moving from a well-defined
financial instrument (which is the particular
in the Aristotelian sense of a genus-specie category or more modernly, the event) to how this financial instrument
operates in the real world (the generalized reality of abstract continuity) is
in effect a study of the reality of finance given a legal context in the form
of a mapping or under a mapping technology.
Thursday, 21 August 2014
“If-then’s” generate “choices” or “how to free yourself of other’s causal claims”.
I read a bit of news—mainstream, twitter & FB--and I have this desperate urge to review a bit of logic.
For example, when someone declares "Do it my way or the highway," you don't have to worry. You can reconstruct that statement into:
If it is x, then it must be y.
It is either not x or it is y.
For example:
If he wears brown shorts, then he must be a terrorist.
Either he does not wear brown shorts or he is a terrorist.
Translating the above threat, “Do it my way or the highway,” means “If you don’t do it my way then you must take the highway.” The phrase “do it my way” is negated in the antecedent.
Let’s go further. How about the statement:
“If you believe in me, then you will go to heaven”?
Translates to:
“Either you don’t believe in me or you will go to heaven.”
That’s proper.
But suppose that statement gets warped into:
“If you don’t believe in me then you will go to hell.”
Which translated into “or” form becomes:
“Either you believe in me or you will go to hell.”
Bottom-line: when you get into arguments about choices, watch out for causation type statements as blame that turn out to be dichotomies. Once you see this rule operating, you don't have to take sides.
You are free to mosey along.
I read a bit of news—mainstream, twitter & FB--and I have this desperate urge to review a bit of logic.
For example, when someone declares "Do it my way or the highway," you don't have to worry. You can reconstruct that statement into:
If it is x, then it must be y.
It is either not x or it is y.
For example:
If he wears brown shorts, then he must be a terrorist.
Either he does not wear brown shorts or he is a terrorist.
Translating the above threat, “Do it my way or the highway,” means “If you don’t do it my way then you must take the highway.” The phrase “do it my way” is negated in the antecedent.
Let’s go further. How about the statement:
“If you believe in me, then you will go to heaven”?
Translates to:
“Either you don’t believe in me or you will go to heaven.”
That’s proper.
But suppose that statement gets warped into:
“If you don’t believe in me then you will go to hell.”
Which translated into “or” form becomes:
“Either you believe in me or you will go to hell.”
Bottom-line: when you get into arguments about choices, watch out for causation type statements as blame that turn out to be dichotomies. Once you see this rule operating, you don't have to take sides.
You are free to mosey along.
Friday, 19 July 2013
Default Invariance: The Three Approximations of Legal and Financial Reality
I took a break from blogging to work on a theory paper entitled, "Default Invariance, A Naive Category Theory of Law and Finance." You can see its abstract on www.ssrn.org. I think the idea of taking a simple logical structure implied by the simplest form of a legal-financial phenomenon, namely, a financial contract with a one-period payment, and looking to the topological space implied by its terminal object will forever change the way we do law and finance. In the paper there are three approximations of law and finance that correspond to the structure implied by the terminal objects, Pay, Not-Pay and Pay & Not-Pay. In the simplest rough and ready terms, these terminal objects imply a point, a line ("risk homological chain complex") and a cyclic matrix ("a ring structure"). Each approximation defines the context-environment of the legal financial structures. And I'm happy to say that we can may make explicit conceptual calculations which improve on ("correct") the works of three Nobel laureates in economics, Arrow, Debreu and Sharpe. Arrow-Debreu-Sharpe basically set out the contingent claim model (Debreu, by the way applied abstract algebraic topological methods in his seminal work, A Theory of Value, which got rid of probability for one-period claims) which underlies everything we know and do in risk management, corporate governance, portfolio theory and practically, everything else we call "modern finance theory." So, one way to read my Default Invariance paper is that it puts Arrow-Debreu-Sharpe into the perspective of a naive category theory, and shows our syntactical structures pre-dispose our conceptual calculations. Anyway, you can read the paper to find out for yourself. It's got 35 original diagrams that are meant to help "re-wire" one's own internal mapping of how the law and finance world works.
Fourth Approximation: Taking Parts and Partitions Seriously
If we wanted to study law and finance as a physical process, we might find that there is a Darwinian-light version to the selection of laws and financial products that appears to apply. Recall Darwin used the principle random selection for the adaptation of certain macro-features appearing to differentiate species according to external environments.
Consider Lancelot Law Whyte (1965) Internal Factors in Evolution, cited by John Bonner (20 July 2013) "Evolution, by chance?" New Scientist, 26-27, 26. As Bonner states, "His [Whyte's] thesis was straightforward. Not only is there selection of organisms in the environment--Darwinian natural selection, which is eternal--but there is also continuous internal selection during development. Maybe the idea was too simple and straightforward to have taken root." Bonner then goes on to state his own thesis, "This fits in neatly with my contention that the shape of microorganisms is more affected by randomness than for large, complex organisms. Being small means very few development steps, with little or internal selection. The effect of a mutation is likely to be immediately evident in the eternal morphology, so adult variants are produced with large numbers of different shapes and there is an increased chance that some of these will be untouched by natural selection. Compare this with what happens in a big, complex organism--a mammal, say. Only those mutations that occur at a late stage of development are likely to be viable--eye or hair colour in humans are obvious examples. Any unfavourable mutation will likely be eliminated by internal selection." He points out the evidence that the shapes of microorganisms are "less likely to be culled by natural selection" by citing Radiolaria (50,000 species) and diatoms (100,000 species) and Foraminifera (270,000 species). Then he states, "If you are a strict adaptionist, you have to find a separate explanation for each shape. If you favour my suggestion that their shapes arose through random mutation and there is little or no selection, the problem vanishes." [p. 27]
What structure is implied by Bonner's internal versus external environment selection thesis? I find his terminology a bit confusing. For what is the external environment of a micro-organism? Isn't everything outside it in a sense a micro-structure and therefore, could be in it, as well? Perhaps we can clarify the thesis by translating the situation into a morphism f: A-->2. Imagine the object A population with lots and lots of elements but having one partition such that you can maximise or minimise either part. The f-morphism are injections to either of the two elements in 2. So long as the partition exists, the 2 separate values will exist in 2. So, we don't need the internal versus external division. In category theory, there is a theorem which just gets rid of all "internal diagrams" so that anything and everything that can be possibly expressed can be done with external diagrams only. I think the same can be said about Whyte's and Bonner's thesis above. In other words, Darwin's random selection to adaption is preserved in the structure of parts and partition via a morphism f: A-->2.
The Fourth Approximation is taking the Third Approximation of Pay and Not-Pay as parts with a partition. The structured implied from this terminal object is "Continuous Contingencies" (CC) to "Infinitely Discrete Randomness" (IDR). This sounds extraordinarily vague, but what it means is that which is undifferentiable can be made into discrete unit choice. In syntactic form: g: CC-->IDR. This is similar to the conceptual step of moving from "God as ubiquitous being" to "eating a properly cooked vegetarian meal is a morally correct choice of being."
I took a break from blogging to work on a theory paper entitled, "Default Invariance, A Naive Category Theory of Law and Finance." You can see its abstract on www.ssrn.org. I think the idea of taking a simple logical structure implied by the simplest form of a legal-financial phenomenon, namely, a financial contract with a one-period payment, and looking to the topological space implied by its terminal object will forever change the way we do law and finance. In the paper there are three approximations of law and finance that correspond to the structure implied by the terminal objects, Pay, Not-Pay and Pay & Not-Pay. In the simplest rough and ready terms, these terminal objects imply a point, a line ("risk homological chain complex") and a cyclic matrix ("a ring structure"). Each approximation defines the context-environment of the legal financial structures. And I'm happy to say that we can may make explicit conceptual calculations which improve on ("correct") the works of three Nobel laureates in economics, Arrow, Debreu and Sharpe. Arrow-Debreu-Sharpe basically set out the contingent claim model (Debreu, by the way applied abstract algebraic topological methods in his seminal work, A Theory of Value, which got rid of probability for one-period claims) which underlies everything we know and do in risk management, corporate governance, portfolio theory and practically, everything else we call "modern finance theory." So, one way to read my Default Invariance paper is that it puts Arrow-Debreu-Sharpe into the perspective of a naive category theory, and shows our syntactical structures pre-dispose our conceptual calculations. Anyway, you can read the paper to find out for yourself. It's got 35 original diagrams that are meant to help "re-wire" one's own internal mapping of how the law and finance world works.
Fourth Approximation: Taking Parts and Partitions Seriously
If we wanted to study law and finance as a physical process, we might find that there is a Darwinian-light version to the selection of laws and financial products that appears to apply. Recall Darwin used the principle random selection for the adaptation of certain macro-features appearing to differentiate species according to external environments.
Consider Lancelot Law Whyte (1965) Internal Factors in Evolution, cited by John Bonner (20 July 2013) "Evolution, by chance?" New Scientist, 26-27, 26. As Bonner states, "His [Whyte's] thesis was straightforward. Not only is there selection of organisms in the environment--Darwinian natural selection, which is eternal--but there is also continuous internal selection during development. Maybe the idea was too simple and straightforward to have taken root." Bonner then goes on to state his own thesis, "This fits in neatly with my contention that the shape of microorganisms is more affected by randomness than for large, complex organisms. Being small means very few development steps, with little or internal selection. The effect of a mutation is likely to be immediately evident in the eternal morphology, so adult variants are produced with large numbers of different shapes and there is an increased chance that some of these will be untouched by natural selection. Compare this with what happens in a big, complex organism--a mammal, say. Only those mutations that occur at a late stage of development are likely to be viable--eye or hair colour in humans are obvious examples. Any unfavourable mutation will likely be eliminated by internal selection." He points out the evidence that the shapes of microorganisms are "less likely to be culled by natural selection" by citing Radiolaria (50,000 species) and diatoms (100,000 species) and Foraminifera (270,000 species). Then he states, "If you are a strict adaptionist, you have to find a separate explanation for each shape. If you favour my suggestion that their shapes arose through random mutation and there is little or no selection, the problem vanishes." [p. 27]
What structure is implied by Bonner's internal versus external environment selection thesis? I find his terminology a bit confusing. For what is the external environment of a micro-organism? Isn't everything outside it in a sense a micro-structure and therefore, could be in it, as well? Perhaps we can clarify the thesis by translating the situation into a morphism f: A-->2. Imagine the object A population with lots and lots of elements but having one partition such that you can maximise or minimise either part. The f-morphism are injections to either of the two elements in 2. So long as the partition exists, the 2 separate values will exist in 2. So, we don't need the internal versus external division. In category theory, there is a theorem which just gets rid of all "internal diagrams" so that anything and everything that can be possibly expressed can be done with external diagrams only. I think the same can be said about Whyte's and Bonner's thesis above. In other words, Darwin's random selection to adaption is preserved in the structure of parts and partition via a morphism f: A-->2.
The Fourth Approximation is taking the Third Approximation of Pay and Not-Pay as parts with a partition. The structured implied from this terminal object is "Continuous Contingencies" (CC) to "Infinitely Discrete Randomness" (IDR). This sounds extraordinarily vague, but what it means is that which is undifferentiable can be made into discrete unit choice. In syntactic form: g: CC-->IDR. This is similar to the conceptual step of moving from "God as ubiquitous being" to "eating a properly cooked vegetarian meal is a morally correct choice of being."
Tuesday, 25 September 2012
Lecture 1 post hoc notes - Legal Aspects of Corporate Finance
Lecture 1: Legal Aspects of Corporate Finance
Guest instructors: Professor Edmond Curtin and PhD Candidate Rezarte Vukatana
I walked in a few minutes late with a bundle of papers and just started talking about THEORY as if it were the most natural thing in the world. I told them about a Russian table tennis star whose training regime included 6 hours of chalk and blackboard theory everyday. But the main point came from Hohfeld's definition of theory: "A theory is not even a theory unless it can be used by practitioners in their practice." I don't think I introduced myself but I did introduce Edmond and later Rezi. I mentioned a few themes:
(1) WEAK EQUIVALENCE as the subtle equivalence of thoughts;
(2) the UNITY OF SCIENCE CRITERION as the main ground for adjudicating theories-- a theory should be judged on how it helps us understand the unity of all knowledge of being;
(3) sign, symbol (Edmond mentioned "signifier" pointing to the picture of the green man in the exit sign--everyone turned to look);
(4) HOHFELD the undergrad chemistry student turned professor of Yale Law School who in early 20th century wrote only 6 articles and invented a periodic table for the law - 4 JURAL OPPOSITES and 4 JURAL CORRELATIVES with enormous theoretical effects;
(5) CORBIN and WILLISTON who wrote encyclopediac tomes on contracts law, and how Corbin (a Hohfeldian student) took just one jural correlative, rights versus duties, and turned that tiny almost trivial legal distinction into 7 (or was it 9?) volumes of contract law;
(6) And Where are Contracts anyway? shock horror to the civil law students ["on paper", "after the signature" they say] but no, says the common law jurisprudentem--CONTRACTS EXIST IN THE MIND [Edmond]; horror of horrors, is this the pure subjectivism, relativism and thus, total discretionary totalitarianism of the law?;
(7) Why some questions within professional discourse make no sense ("What's north of the north pole, eh?"] and is there a way of understanding that transcends the bounds of discourse? Later, the astute Russian student answering a question about "material information" asked a rhetorical question about the distinguishment of various risks. Then I told a long story about Yuanjia, the Great Wun Chin master, who when cajoled by a Japanese martial artist that there are levels in the artistry of tea, replied, "The tea makes no such distinctions and is thoroughly enjoyed."
Thankfully, Edmond gave us a brief rendition on some of the essential legal principles of DERIVATIVES--how they actually create MORE RISK and MORE ANXIETY, and never less.
Rezi described part of her PhD dissertation research--theory of self-fulfilling prophecy a la Merton (?) and how this can be used to help explain the strange behaviours of very complex nodes of financial system called intermediated securities accounts.
I passed around 3 LLM dissertations for the students' inspection, and gave them a homework assignment.
I filed some prospectuses at LLMCFL2012@gmail.com [if you want the password, you need to contact me] with my notes, and asked the students to write 2,000 words on (1) the risks of the prospectus transaction (either Salvatore Ferragamo or Prada); and (2) determine whether and what parts of the selected prospectus would need to be changed under the Directive 2010/73 Nov 2010. They'll need to review about 600 to 800 pages and email me their work by 12noon Monday. Nice shock therapy.
Tuesday, 11 September 2012
Extreme Philosophy: On the Limits of Self-Referential Truth: Why Paradox Has Been Binned By Naive Category Theory
1. Here are two papers of EXTREME PHILOSOPHICAL SIGNIFICANCE:
[1] Lawvere, F. William, "Diagonal arguments and cartesian closed categories with Author Commentary," Lecture Notes in Mathematics, 92 (1969), 134-145, available at:
http://www.tac.mta.ca/tac/reprints/articles/15/tr15.pdf
[2] Yanosky, Noson (2003) "A Universal Approach to Self-Referential Paradoxes, Incompleteness and Fixed Points," available at: http://arxiv.org/pdf/math/0305282v1
2. Unless you've studied a bit of category theory, i.e., read Lawvere and Schanuel (2008, 2nd edition) and Lawvere and Roseburgh (2003), Lawvere [1] will be very obscure even with Lawvere's commentary. But take a look and get a feel. Then, look at Yanosky [2] which explains in a more breezy (but precise) way what the genius Lawvere was up to, and even more cleverly in order to reach a "wider audience", dropped category theory altogether and explains Lawvere's discoveries in easy enough "set and function" language.
3. I realize that category theory is not for everybody (yet) and recently, in the literature, there is a push-back accusing category theory of making "foundational claims" that are unjustified. For example, that the entirety of mathematics can be put on a category theory footing and replace set theory as the fundamental theory which all other theories must bow down to. But I don't think category theory as it is practiced sets out to make any really big claims like these--that would be the job of propogandists. Rather it "solves" some rather apparent fundamental problems by "resolving" the problems into a diagrammatic logic. If you buy the diagrams as BEING DENOTIVE then you might also see how category theory IS linked to Aristotle's great work On Categories. Mac Lane in a footnote joked about how the title "category theory" came from "purloining words from the philosophoers, Aristotle and Kant" [pp. 29-30 of Categories for the Working Mathematician]. He doesn't say anything more about this jokey link. But if you read and understand Aristotle's motive in his Categories, you can see immediately that Aristotle set up foundational problems so they can be resolved. He analysed knowledged into what might be called "said-of" and "thing-in" and asked what are those abstractions that are primary, that is, what are those properties that are extended and therefore, must be. He listed 10 categories [what they are appears arbitrary] and he showed how you can use these primary categories to categorize everything else, that is, that which is not so extended and universal. Now, this mental-conceptual move to abstraction in order to solve a particular problem is a natural function. Lawvere & Schanuel in Conceptual Mathematics explain this movement in terms of isomorphisms: e.g. think of how you can understand what's happening in a film even after walking into the cinema late. In media res, you know Humphrey Bogart is playing a particular character and Audrey Hepburn is playing another character, and when you sort out who's who in the film, suddenly, you can follow the plot in the film with the actors as playing their roles. Similarly, being born in the middle of things, we open our eyes, stretch our arms and legs, and explore the universe, fully confident that we will be able to sort EVERYTHING out. This confidence comes from something pretty powerful within ourselves that enables us to gain knowledge. And the point here is that knowledge isn't at its rock bottom paradoxical. It is in all likelihood isomorphic.
4. Lawvere [1] takes a swipe at the propogandists who have been using some of the great theoretical work of theorists (such as Russell, Cantor, Godel, Tarski) and turned them into very general claims about the nature of paradox at the heart of knowledge. To put this into a general philosophical context, Aristotle's optimism was founded on his discovery of a general scientific method which if simply re-iterated, would eventually uncover all the mysteries of the universe. It was based on observing that which is and translating those into propositions which could be understood. If at the heart of heart of "proposition making" we have paradox, then this whole enterprise is doomed to failure. So, burdened with the prospect of failure, why start the programme of knowledge?
5. The answer by Lawvere [1] and Yanofsky [2] shows why the propogandists of paradox are simply wrong. In Yanofsky's terms, Lawvere's great little paper [1] has been largely ignored by category theorists and philosophers alike because it is written in a forbidding unpopular formalism. Yanofsky translates the results of Lawvere's paper by saying the classical paradoxes of self-referential truth (e.g. Liar's paradox, Russell paradox, Godel's incompleteness and so on) are just instances of overstepping the limitations of a discourse ("discourse" is my term). There must be a way of limiting what a discourse can say about itself. This "problem" comes up in law and finance whenever they try to talk about themselves. I call it the problem of structure. That is, there is no such question in law and finance that says, "What is the structure of law? What is the structure of statements about finance?" There is no call for self-consciousness within laws or financial practice. Rather, the call for such professional consciousness comes from without. But there is a way of understanding such questions about professional discourses from a category theory perspective. And not only do the questions about the structure of law and finance make sense, they actually direct in some fashion a resolution to answers about the structure of law and finance.
For example, one of the things I have been harping on in this blog is that there is a fundamental structure to law and finance in the forms of an individual unit which I have dubbed the "financial contract" and the "great cycle of default invariance." From these structures, we can explain a lot of current practice at the individual-to-individual level of financial transactions on up to historical and contemporary nausea of continuously impending financial catastrophes. It's all a matter of "mapping" and translating apparent limitations within the discourse of law and finance into a notation which allows for mental journeys and conceptual calculations. By the way, one of the virtues of seeing how paradoxes are slain in [1] and [2] is that we can recover a sense of optimism that Aristotle once had in the unity of science. Again, I say, judge the value of a theory by its contribution to the unity of science.
Monday, 14 May 2012
On the Unity of Science: Law, Finance and the Philosophy of Category Theory
1. We should measure our progress as a species of knowledge by how well individual disciplines meet the criterion of the unity of science. This criterion was first stated as an almost urgent request by Edward O. Wilson with his concept of consilience.
2. There are so many analogies between one subject and another, the vocabularies (as objects may be different), but the way in which these vocabularies are used (the morphisms) are so similar that they might as well be said to be the same. What do we mean by "same" or better, if we capture the water colourist wash, and call it by its technical name, "weakened equivalence"?
3. Mazur (June 12, 2007) in "When is one thing equal to some other thing," htt://www.math.harvard.edu/%7Emazur/preprints/when_is_one.pdf writes in tribute to one of the founders of category theory, Mac Lane, sets out an approachable essay on the question of the meaning of equivalence. This is the deep point where all our equations and assertions in science sink to. He sets out three approaches of how we have answered this question.
4. The first is the "bureau of standards" where by convention we can point to something in a designated office that is an equivalent exemplar. [Id @ 4-5.] The second is a type of universal quantification as in Frege's definition of cardinality. [Id @ 5.] And the third is a compromise where "we indicate what we do rather than what we say we do when quizzed about our foundations." [Ibid.] I call this third method a promiscuous stitching, using the same needle and thread or glue may be all we need to make appropriate connections between subjects, disciplines and fields of knowledge.
5. In mathematics, you can "package" entire mathematical theories either as (1) formal systems a la the David Hilbert programme or as (2) categories. [Id @6.] On the one hand, the formal systems go all the way back to Euclid and are much admired under the rubric of axiomatization. On the other hand, categories are a relatively recent invention (1945 with a paper by Samuel Eilenberg and Saunders Mac Lane was more an announcement of new technique than a new view of mathematics) and its method of a sparse vocabulary and sketches of arrows betrays its deep goal which is to reveal structure. Mathematicians, like others engaged in doing or performing in their particular discourse, don't really "axiomize" but rather "play games with conviction."
6. Somewhere in "Tool and Object, A History and Philosophy of Category Theory," (2007), Kromer quotes Bill Lawvere (my preferred radical category theorist) for saying something to the effect, "the point is not to achieve maximal abstraction, but an optimal abstraction, a just-right abstraction that works appropriately at the level where it is most needed and used." Of course, I am attributing a certain line of argument to Lawvere which I do not think he would disagree with. Lawvere was motivated to find a theory of physics, to explain how things worked, but his work on the philosophy of category theory takes him on exoduses into Hegel. I believe it is this urge to find "synthesis" with simple tools that motivates him. He has been accused of being both revolutionary and idiosyncratic. Revolutionary for advocating that all of mathematics can be thought of as a category of category theory. And idiosyncratic because for such a great mathematician, he and Schanuel wrote a best selling book entitled, Conceptual Mathematics, A First Introduction to Category Theory (2009 2nd Edition) wherein you don't need any university level mathematics to understand. In fact, I recommend this book to all my law and finance students who are interested in pursuing the application of category theory to the field of law and finance.
7. The main points about the test for the unity of science (consilience) is that the most appropriate method for pursuing a rigorous apprehension of science (i.e., the three approaches stated by Mazur: bureaucratic standard, universal axiomization or balanced compromse (I call "promiscuous stitching") may be something so natural and simple that even our high school students can be engaged in this entreprise.
8. For theories of law and finance, we see that there has been an influence of the latest trend or fashion from other fields that have filtered into the vocabulary of the legal theorist or financial theorist. For example in the last 5 to 10 years, in both fields there is an emphasis at least in the titles to papers on the concept of "complexity" and "behaviours." This is not to say these concepts are red-herrings. From my view, they are just another batch of ideas that come from a few equations. Another example, what would Hart's programme of primary and secondary laws be without the notions of first order and second order logics emanating from the Cambridge logicians in the early part of the 20th century? Not that Hart genuinely meant to implement the same programme, but the inspiration for an orderly resolution of the definition of the meaning of law was certainly intended to take the script from the philosophy department--and these were the ideas pre-Wittengstein. In finance, the initial idea of covariance goes back to Bachelier's PhD dissertation (1905) and then developed as various methods for "curve fitting" against time horizons. Very little work has been done on how the various theories of law and finance might be approached in a unified way. But here the stumbling block may have been the limited view on the number of approaches to reach unification. I do not mean by "unification" a form of axiomatiion or foundational premises evolved and expanded in a universalistic sense that may have endeared Spinoza. Rather we have a very powerful alternative which is a kind of Kantian insight that the intellectual revolution begins with a recognition that
"There are only two possible ways in which synthetic representations and their objects...can meet one another. Either the object (Genenstand) alone must make the representation possible, or the representation alone must make the object possible." [quoted from Mazur supra @ 20.]
9. In law and finance theory, I would (and will) argue that one of the significant leaps in our imagination of how law and finance work together is to recognize the structure of something called "default invariance." This is captured by or very conveniently set out with a category theory approach. Default invariance permeates all financial contracts, and all states of the financial-regulatory-political system.
2. There are so many analogies between one subject and another, the vocabularies (as objects may be different), but the way in which these vocabularies are used (the morphisms) are so similar that they might as well be said to be the same. What do we mean by "same" or better, if we capture the water colourist wash, and call it by its technical name, "weakened equivalence"?
3. Mazur (June 12, 2007) in "When is one thing equal to some other thing," htt://www.math.harvard.edu/%7Emazur/preprints/when_is_one.pdf writes in tribute to one of the founders of category theory, Mac Lane, sets out an approachable essay on the question of the meaning of equivalence. This is the deep point where all our equations and assertions in science sink to. He sets out three approaches of how we have answered this question.
4. The first is the "bureau of standards" where by convention we can point to something in a designated office that is an equivalent exemplar. [Id @ 4-5.] The second is a type of universal quantification as in Frege's definition of cardinality. [Id @ 5.] And the third is a compromise where "we indicate what we do rather than what we say we do when quizzed about our foundations." [Ibid.] I call this third method a promiscuous stitching, using the same needle and thread or glue may be all we need to make appropriate connections between subjects, disciplines and fields of knowledge.
5. In mathematics, you can "package" entire mathematical theories either as (1) formal systems a la the David Hilbert programme or as (2) categories. [Id @6.] On the one hand, the formal systems go all the way back to Euclid and are much admired under the rubric of axiomatization. On the other hand, categories are a relatively recent invention (1945 with a paper by Samuel Eilenberg and Saunders Mac Lane was more an announcement of new technique than a new view of mathematics) and its method of a sparse vocabulary and sketches of arrows betrays its deep goal which is to reveal structure. Mathematicians, like others engaged in doing or performing in their particular discourse, don't really "axiomize" but rather "play games with conviction."
6. Somewhere in "Tool and Object, A History and Philosophy of Category Theory," (2007), Kromer quotes Bill Lawvere (my preferred radical category theorist) for saying something to the effect, "the point is not to achieve maximal abstraction, but an optimal abstraction, a just-right abstraction that works appropriately at the level where it is most needed and used." Of course, I am attributing a certain line of argument to Lawvere which I do not think he would disagree with. Lawvere was motivated to find a theory of physics, to explain how things worked, but his work on the philosophy of category theory takes him on exoduses into Hegel. I believe it is this urge to find "synthesis" with simple tools that motivates him. He has been accused of being both revolutionary and idiosyncratic. Revolutionary for advocating that all of mathematics can be thought of as a category of category theory. And idiosyncratic because for such a great mathematician, he and Schanuel wrote a best selling book entitled, Conceptual Mathematics, A First Introduction to Category Theory (2009 2nd Edition) wherein you don't need any university level mathematics to understand. In fact, I recommend this book to all my law and finance students who are interested in pursuing the application of category theory to the field of law and finance.
7. The main points about the test for the unity of science (consilience) is that the most appropriate method for pursuing a rigorous apprehension of science (i.e., the three approaches stated by Mazur: bureaucratic standard, universal axiomization or balanced compromse (I call "promiscuous stitching") may be something so natural and simple that even our high school students can be engaged in this entreprise.
8. For theories of law and finance, we see that there has been an influence of the latest trend or fashion from other fields that have filtered into the vocabulary of the legal theorist or financial theorist. For example in the last 5 to 10 years, in both fields there is an emphasis at least in the titles to papers on the concept of "complexity" and "behaviours." This is not to say these concepts are red-herrings. From my view, they are just another batch of ideas that come from a few equations. Another example, what would Hart's programme of primary and secondary laws be without the notions of first order and second order logics emanating from the Cambridge logicians in the early part of the 20th century? Not that Hart genuinely meant to implement the same programme, but the inspiration for an orderly resolution of the definition of the meaning of law was certainly intended to take the script from the philosophy department--and these were the ideas pre-Wittengstein. In finance, the initial idea of covariance goes back to Bachelier's PhD dissertation (1905) and then developed as various methods for "curve fitting" against time horizons. Very little work has been done on how the various theories of law and finance might be approached in a unified way. But here the stumbling block may have been the limited view on the number of approaches to reach unification. I do not mean by "unification" a form of axiomatiion or foundational premises evolved and expanded in a universalistic sense that may have endeared Spinoza. Rather we have a very powerful alternative which is a kind of Kantian insight that the intellectual revolution begins with a recognition that
"There are only two possible ways in which synthetic representations and their objects...can meet one another. Either the object (Genenstand) alone must make the representation possible, or the representation alone must make the object possible." [quoted from Mazur supra @ 20.]
9. In law and finance theory, I would (and will) argue that one of the significant leaps in our imagination of how law and finance work together is to recognize the structure of something called "default invariance." This is captured by or very conveniently set out with a category theory approach. Default invariance permeates all financial contracts, and all states of the financial-regulatory-political system.
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