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."
