Invention or Discovery?

“A scientist searches for knowledge of the natural world through a specific methodology: the scientific method of hypotheses and experimental validation. Scientists have a clear goal: to describe natural phenomena with rational explanations based on repeatable experimentation and consensus. Speculation is allowed only insofar as it leads to verifiable predictions.”

— Marcelo Gleiser (“The Island of Knowledge”, 2014)

Excerpted from Marcelo Gleiser's book: "The Island of Knowledge", wherein he discusses whether mathematics is an invention or a discovery and why it matters:

“The display of wonder and regularity of Nature – day and night, seasons and tides, a Moon with phases, planets that return, the life and death cycle of plants and animals, gestation periods – requires a methodic counting and organizing as a means to gain some level of control over what is otherwise distant and unapproachable, the trends of a world evolving in ways clearly beyond human power. How else would pattern-seeking humans order their sense of reality if not through a language capable of describing these patterns, of analyzing them, of exploring their repetition as a learning tool? The mathematization of Nature, and the ordering of observed trends in terms of laws, is one of the distinctive achievements of our species.

“The power of mathematics comes from its being detached from physical reality, from the abstract treatment of its quantities and concepts. It starts in the outside world, the world as it is perceived by our senses, when we identify approximately circular and triangular forms in Nature, or learn how to count and measure distances and time. But then mathematics takes a simplifying step and lifts these asymmetric shapes from Nature and idealizes them as symmetric, so that we can more easily construct mental relations with them. These relations and their progeny may or [may] not be applicable back to the study of Nature. If they are, they may be used in a scientific model of some kind. If not, they may remain forever locked in the abstract realm of ideas they inhabit. This transplanting of forms and numbers from Nature, which allows for the abstract manipulation of number and form, is also why mathematics is always an approximation to reality and never reality as it is.

In the mind’s eye: A beautiful image is a poor reflection of truth.

“Nature’s creative power often hides behind asymmetries and not symmetries. Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line. The richness is found not in isolating order above everything else, but in contrasting order and disorder, symmetry and asymmetry, as complementary players in the ways we describe Nature. Symmetry … are excellent approximations to what we are attempting to describe. The danger, and the origin of the Platonic fallacy, is to believe that the symmetries are an imprint of Nature instead of an explanatory device we conceived to describe what we see and measure.

“There is a very productive alliance between the human brain and its mathematical attempts to make sense of reality. Mathematical results are not snapshots of some transcendent truth but a very human invention. The nexus of our quest for knowledge is not to be found outside of us but within us. Theorems in abstract mathematics, even if apparently completely disconnected from immediate reality, are the products of logical rules and concepts constructed with our minds. Our minds function in specific ways that reflect the embodiment of cognitive tools, which facilitate the development of abstract conceptual tools. We create the mind games of pure mathematics in the convolutions of our neocortex. And our neocortex is the result of eons of evolution driven by the pressures of natural selection and genetic variability, where the link between creature and environment is essential.

“The discussion of mathematics being an invention or a discovery, … points more to the importance of the human brain as a rare and wondrous oddity in the Universe than to the elusive truths written in some imponderable abstract realm. The cause of celebration is not “out there” or “up above” or in the “mind of God” but in this small mass we humans carry within our cranial cavity.

References:

1. Gleiser, Marcelo (2014). The Island of Knowledge: The Limits of Science and the Search for Meaning. Basic Books.
2. Lakoff, George and Nuñez, Rafael (2001). Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being. Basic Books.
3. Baum, Eric B. (2006). What is Thought? A Bradford Book.

Quit, Stick or Pivot?

How does one know when to quit, stick, or pivot? Let's hear from the experts:

“Many of life’s failures are people who did not realize how close they were to success when they gave up.”

— Thomas Edison (1847-1931)

“Extraordinary benefits accrue to the tiny minority of people who are able to push just a tiny bit longer than most.”

— Seth Godin, “The Dip” (2007)

“Quitters never win and winners never quit.”

— Vince Lombardi (1913-1970)

“Never, never, never, never —
in nothing, great or small, large or petty —
never give in, except to convictions of honour and good sense.”

— Winston Churchill (1874-1965)

“If at first you don’t succeed, try, try again. Then quit.
There’s no point in being a damn fool about it.”

— W.C. Fields (1880-1946)

“Winners quit all the time.
They just quit the right stuff at the right time.”

— Seth Godin, “The Dip” (2007)

“You’ve gotta know when to hold ’em, know when to fold ’em.
Know when to walk away, know when to run.”

— Don Schlitz, “The Gambler” (1976)

“The people who are the best in the world specialize at getting really good at the questions they don’t know.”

— Seth Godin, “The Dip” (2007)

“Don’t be trapped by dogma - which is living with the results of other people’s thinking. Don’t let the noise of other’s opinions drown out your own inner voice. And most important, have the courage to follow your heart and intuition. They somehow already know what you truly want to become. Everything else is secondary.”

— Steve Jobs (1955-2011)

“Our life is an apprenticeship to the truth…
that there is no end in nature,
but every end is a beginning.”

— Ralph Waldo Emerson (1803-1882)

“Rise and rise again and again ...
until lambs have become lions
and the rule of darkness is no more.”

— Maitreya The Friend of all Souls (1943-2012)

“I have fought the good fight,
I have finished the race,
I have kept the faith.”

— St. Paul ("The Bible", 2 Timothy 4:7)

“Being able to persist is not the most important thing –
the ability to start over is.”

— F. Scott Fitzgerald (1896-1940)

“The greatest accomplishment is not in never failing,
but in rising again after you fall.”

— Vince Lombardi (1913-1970)

Life has endless ways of presenting its journeymen with contradictions that beg resolution, and/but gives little clues in the ways of solving them. ‘Nuff said.

References:

1. Godin, Seth (2007). The Dip: A Little Book That Teaches You When to Quit (and When to Stick). Penguin Group.
2. Ries, Eric (2011). The Lean Startup: How Today's Entrepreneurs Use Continuous Innovation to Create Radically Successful Businesses. Crown Business.
3. Horowitz, Ben (2014). The Hard Thing About Hard Things: Building a Business When There Are No Easy Answers. Harper Business.
4. Bass, Thomas (2000). The Predictors: How a Band of Maverick Physicists Used Chaos Theory to Trade Their Way to a Fortune on Wall Street. Holt.

Causal Entropic Forces

“You have to die. Every day.
Until the Omega is destroyed.”

— “Edge of Tomorrow: Live, Die, Repeat.” (2014)

Omega has the ability to control time. “Whenever an Alpha is killed, the Omega starts the day over again, but you see, this time it can remember what's going to happen,” explained Rita. And an enemy that knows the future can't lose.

Rita Vrataski (Emily Blunt) and Dr. Carter (Noah Taylor) are telling Bill Cage (Tom Cruise) his combat mission.

Alex Wissner-Gross and Cameron Freer recently proposed “a causal generalization of entropic forces” that they showed can induce certain patterns of behavior with some very striking characteristics. One would not guess those outcomes by looking purely at the constraint that produces them. Underlying this set of intriguing behaviors is simply the computational capability to integrate over all possible futures so as to maximize the rate of entropy production over an entire trajectory. The observed behavior is considered to emerge from such a computation, without explicit goal-driven programming. In our view, these behaviors bear striking resemblance with examples we have seen in Swarm Intelligence (e.g., ant colonies, bird flocking, animal herding, bacterial growth, and fish schooling) or Artificial Life (aka A-Life or “life-as-it-might-be”). A key technical difference, it seems, lies in their underlying optimization algorithms. For instance, the entropic force F is determined by: F = T ∇Sτ, where T is the reservoir temperature, S is the entropy, and τ is the time horizon. Further development of interesting applications based on causal entropic forces is undertaken by Sergio Hernandez, who has also produced an interesting collection of videos along with code samples to demonstrate how it could be made to work.

“Physical agents driven by causal entropic forces might be viewed from a Darwinian perspective as competing to consume future histories,” according to the interpretation of Wissner-Gross and Freer. “In practice, such agents might estimate causal entropic forces through internal Monte Carlo sampling of future histories generated from learned models of their world. Such behavior would then ensure their uniform aptitude for adaptiveness to future change due to interactions with the environment, conferring a potential survival advantage.” This is powerful stuff, indeed. Certainly something we should look into more closely before releasing MVP into the trading jungle.

This may be a good time to examine, by way of an analogy with modern physics, how our approach to arbitrage trading based on speculated models might work. In 2009, while stranded in the south of France towards the end of his summer vacation, Erik Verlinde found time to put forth a heuristic argument starting from first principles that shows Newton's law of gravitation naturally arises in a theory in which space emerges through a holographic scenario. The key idea is that gravity is essentially a statistical effect, and a manifestation of entropy in the universe. More specifically, Verlinde uses the holographic principle to consider what is happening to a small mass at a certain distance from a bigger mass, say a star or a planet. Moving the small mass a little means changing the information content (or entropy) of a hypothetical holographic surface between both masses. This change of information is linked to a change in the energy of the system. Then, using statistics to consider all possible movements of the small mass and the energy changes involved, Verlinde finds movements toward the bigger mass are thermodynamically more likely than others. This effect can be seen as a net force pulling both masses together. This is called an entropic force, as it originates from the most likely changes in information content. This means that, in order to understand motion, one must keep track of the amount of information.

The central notion needed to derive gravity is information, measured in terms of entropy. Information causes motion. Gravity is thus an emergent phenomenon, and not one of the fundamental forces as previously thought, which is perhaps why gravity is so hard to be unified with the other three fundamental forces. Verlinde’s conceptual model is robust so far and has demonstrated superior explanatory power in a holographic context in which space is emerging:

1. it can derive the familiar Newton’s law of gravitation (which nobody has been able to do for 300 years);
2. it can explain inertia (which nobody had thought needed explaining);
3. it recovers Newton’s second law of motion (i.e., the familiar F = ma); and
4. it can generalize to a relativistic situation and derive Einstein’s equations.

Newton’s Law of Gravitation (accepted without explanation for 300 years).

Following Verlinde’s line of reasoning, other physicists are able to carry out immediate follow-on work in cosmology. For example, the mysterious dark energy is obviated when considering an alternative interpretation of the accelerated expansion of the universe as resulting from an entropic force naturally arising from information storage on the holographic horizon surface screen. We can think of Verlinde’s heuristic argument as representing a speculated model for the “origin of gravity.” After all, it is based on the postulated holographic principle and has not been rigorously tested in experiments. However, by simply adhering to the direct implications of the speculated model, one can deduce certain immediate “stylized facts” about the universe: some of which we have earlier observed empirically (e.g., Newton's law of gravitation), but others we have not even imagined a connection exists (e.g., dark energy). The analogy carries over effortlessly to the financial universe. Arbitrage trading based on the speculated model is possible if multiple sets of empirical observations, both old and new, can be nicely tied together this way and help calibrate internal parameters of the speculated model for added precision.

In 1946, as Stanislaw Ulam, a mathematician then working at Los Alamos, was convalescing from an illness and playing solitaires, he pondered an interesting question: What are the chances that a Canfield solitaire laid out with 52 cards will come out successfully? After spending a lot of time trying to estimate them by pure combinatorial calculations, Ulam wondered whether a more practical method than "abstract thinking" might not be to lay it out say one hundred times and simply observe and count the number of successful plays. Thus began the first attempts to practice the “Monte Carlo Method,” as it was later named by John von Neumann after he learned the method from Ulam. The use of Monte Carlo methods requires a large amount of random numbers, and spurs the development of pseudorandom number generators that can produce pseudo-random sequences which are “random enough” in a certain sense to be useful.

A Kindred Spirit: The Busy Count of Monte Carlo.

Back at Los Alamos, they began to plan actual calculations using the first computer, ENIAC. In von Neumann’s formulation of the neutron diffusion problem, each neutron history is analogous to a single game of Solitaire, and the use of random numbers to make the choice along the way is analogous to the random turn of the cards. But after a series of “games” have been played, how does one extract meaningful information? For each of the thousands of neutrons, the variables describing the chain of events are stored, and this collection constitutes a numerical model of the process being studied. The collection of variables is analyzed using statistical methods identical to those used to analyze experimental observations of physical processes. Once can thus extract information about any variable that was accounted for in the process.

Richard Feynman, who had worked at Los Alamos administering the computation group during the war years, was developing his “sum over histories” version of quantum mechanics at Cornell after the war. In Feynman’s formulation, a probability amplitude is associated with an entire motion of a particle as a function of time, rather than simply with a position of the particle at a particular time. Feynman’s path integral method is mathematically equivalent to Schroedinger’s wavefunction and Heisenberg’s matrix formulation. As Feynman wrote in 1948, “there is a pleasure in recognizing old things from a new point of view,” and that “there is always the hope that the new point of view will inspire an idea for the modification of present theories.” Although it is much harder to compute, the path integral is often more intuitive for thinking about particle interactions without actually having to compute the integral.

Feynman: The Man with a Life of Many Paths.

The process of sampling alternative paths reveals essential features of quantum mechanics, one of which is the inclination of electrons to “explore all paths”. Freeman Dyson recalled Feynman describing his new method this way: “the electron does anything it likes; it just goes in any direction at any speed, forward and backward in time, however it likes, and then you add up the amplitudes and it gives you the wavefunction.” The electron is a free spirit indeed. In fact, the electron is so free-spirited that it refuses to choose which path to follow – so it tries them all.

Path Integral: Summing over all possibilities. (Image Credit: Einstein Online).

In a simulated universe, one has the luxury to try every single path, i.e., mind-numbingly and painstakingly exploring every unique permutation of events, in order to realize optimal coordination and control. All such efforts can be viewed as part of a giant search process, bounded in space by maximal causal entropy and in time by minimum coordination latency. To the outside world, what really matters, from a macroscopic viewpoint, is the net sum of all such internal microscopic efforts, mostly unseen, unknown, and unappreciated. But they all add up to generate a final observable difference. That bit of difference is all that really matters in the end. What would you do in this world if you have but one life to live?

“Two roads diverged in a wood, and I –
I took the one less traveled by, and that has made all the difference.”

— Robert Frost (“The Road Not Taken”, 1920)

References:

1. Wissner-Gross, Alex and Freer, Cameron (2013). Causal Entropic Forces. Physical Review Letters, 110 (168702), 1-5. Retrieved from: http://www.alexwg.org/publications/PhysRevLett_110-168702.pdf and http://www.alexwg.org/publications/PhysRevLett_110-168702_SupplementalMaterial.pdf
2. Metropolis, Nicholas, and Ulam, Stanislaw (1949). The Monte Carlo Method. Journal of the American Statistical Association, Vol. 44, No. 247, pp. 335-341. Retrieved from: http://web.maths.unsw.edu.au/~peterdel-moral/MetropolisUlam49.pdf
3. Metropolis, Nicholas (1987). The Beginning of the Monte Carlo Method. Los Alamos Science (Special Issue), pp. 125-130. Retrieved from: http://jackman.stanford.edu/mcmc/metropolis1.pdf
4. Keim, Brandon (2009, March 10). Humans No Match for Go Bot Overlords. Wired. Retrieved from: http://www.wired.com/2009/03/gobrain/
5. Feynman, Richard P. (2006). QED: The Strange Theory of Light and Matter. Princeton University Press. Available as: Douglas Robb Memorial Lectures from the University of Auckland.
6. Feynman, Richard P. (1948). Space-Time Approach to Non-Relativistic Quantum Mechanics. Rev. Mod. Phys. 20 (2): 367-387. Retrieved from: http://web.ihep.su/dbserv/compas/src/feynman48c/eng.pdf
7. Taylor, Edwin F. and Vokos, Stamatis and O’Meara, John M. and Thornber, Nora S. (1998). Teaching Feynman’s Sum-Over-Paths Quantum Theory. Computers in Physics, Vol. 12, No. 2, pp. 190-199. Retrieved from: http://www.aip.org/cip/pdf/vol_12/iss_2/190_1.pdf
8. Verlinde, Erik P. (2011). On the Origin of Gravity and the Laws of Newton. JHEP 04, 29. Retrieved from: http://link.springer.com/article/10.1007%2FJHEP04%282011%29029

Island Economy

Quarks and leptons are considered the fundamental particles of which matters in the physical universe are composed. So one might ask: what are the elemental components of securities or other financial instruments that make up our financial universe?

Kenneth Arrow and Gerard Debreu developed an approach, called the Time-State Paradigm, which characterizes promised future payments in terms of both the times at which payments are to be made and the states of the world that must occur for payments to be made.

An island economy powered by the almighty coconuts. (Image Credit: Haemimont Games).

Let's start with the simplest example involving both time and uncertainty. For simplicity, we consider an island economy whose only commodity is the coconut. There is no concept of money on the island, and so the coconut has become its native unit of currency. Naturally, the only productive investment on the island is growing coconut trees. The yield of the coconut trees in the next season is primarily influenced by island weather over the next year. If the weather is good (i.e., sunny), the coconut trees produce a bounty of 100 coconuts. If the weather is bad (i.e., rainy), the coconut trees instead produce only 70 coconuts.

For purpose of forecasting the island economy, we consider two time periods: (i) today, and (ii) a year from today. We also consider two possible future states of the island: (a) good weather; and (b) bad weather. Keep in mind that the states of the island are mutually exclusive (only one can occur), and exhaustive (one of them must occur).

According to Arrow-Debreu, there are a total of three elemental time-state claims in the island economy:

• One coconut today
• One coconut a year from today if the weather is good
• One coconut a year from today if the weather is bad

These are called atomic time-state claims. Any investment vehicle can be considered to be composed of such atomic claims. For ease of exposition, these claims can be alternately described as follows:

• One "Current Coconut" (CC)
• One "Sunny Coconut" (SC)
• One "Rainy Coconut" (RC)

A Friday market exists on the island that allows such time-state claims to be traded efficiently at zero cost. Dealers in the Friday market stand ready to trade such atomic claims and do so without cost as a service to the island.

For example, a "fair-weather dealer" specializing in "Sunny Coconuts" would be willing to trade (or swap): 0.4 coconuts today for 1.0 sunny coconut a year from now, or 1.0 sunny coconut a year from now for 0.4 coconuts today.

In standard financial terms, we can understand this obligation from the certificate issued by a coconut tree grower that reads as follows:

A "Sunny Coconut" security issued by a coconut grower and traded on the Friday market by a "fair-weather dealer".

This piece of paper would be called a security. Should the Island Credit-Rating Agency care to examine the property of the coconut growers (i.e., coconut trees) and is able to determine that no more than 100 pieces of such securities have been issued, and that there are no other claims on the growers' assets in the event of good weather, this security would be rated AAA (or "Triple-A") and can be considered "default-free".

The security in question, under these conditions, represents a property right in an atomic time-state claim. It is thus an "atomic security", or an Arrow-Debreu security. The price for this "Sunny Coconut" security is 0.4 current coconuts, because the dealer stands ready to trade this number of current coconuts for the security. The ability to make a trade is thus central to the definition of a price.

Outside of an island economy, dealers in the real world would charge more to sell a security than they pay to buy it. The difference between the ask and bid price is called the spread, and provides compensation for market-making. In practice, the mid-point between the bid and ask price is often used as a surrogate for "the price".

To complete our picture of the island economy, there is another dealer on the island who specializes in "Rainy Coconuts". This "foul-weather dealer" is willing to trade (or swap): 0.5 coconuts today for 1.0 rainy coconut a year from now, or 1.0 rainy coconut a year from now for 0.5 coconuts today.

The trading environment on the island now comprises of dealer-operated markets for trading: (i) CC and SC, and (ii) CC and RC. Note that each such trade has the characteristics of an investment – today's coconuts are traded for the prospect of coconuts in the future. For example, one who purchases a SC security can be said to have invested 0.4 coconuts today to obtain 1.0 coconut in the future if the weather is good. Similarly, one who purchases a RC security can be said to have invested 0.5 coconuts today to obtain 1.0 coconut in the future if the weather is bad.

But what of other possible types of trades on the island? For example, swapping sunny coconuts for rainy coconuts (i.e., SC for RC)? How might this be accomplished? And what are the terms of the trade?

Consider someone who wishes to trade 1.0 sunny coconut for some number of rainy coconuts. One way to do this is to trade 1.0 sunny coconut for 0.4 current coconuts, while at the same time trading 0.4 current coconuts for 0.8 rainy coconuts. The net result is to swap 1.0 sunny coconut for 0.8 rainy coconuts. If a dealer were to offer any other terms of trade (i.e., other than 1.0 sunny coconut for 0.8 rainy coconuts), an astute trader would spot an opportunity to exploit this dealer by engaging in a combination of trades that could provide:

1. net receipt of coconuts in at least one time and state; and
2. no net payout of coconuts at any other time and state.

Such an opportunity, termed an arbitrage, is rare and fleeting in a well-functioning capital market. An arbitrage is thus a "coconut machine", and every trader's dream. When an opportunity of this type arises, traders will rush to exploit it, in the process causing dealers to adjust their terms of trade until arbitrage disappears. A set of swap terms that does not permit arbitrage is considered "arbitrage-free".

In a broad sense, every security transaction can be considered a swap. When one side of the swap involves current coconuts (e.g., as in the purchase of an atomic security), it is called an investment. Thus one invests current coconuts in the hope of obtaining more coconuts in the future. But the swap of sunny coconuts for rainy coconuts involves no net current coconuts and is not considered an investment; thus these types of swaps are referred to as "zero-investment strategies". Arbitrage would be a zero-investment strategy; it does not tie up capital for longer than is necessary to execute the combination of trades.

What is interesting here is that although markets are being made in only future atomic time-state claims, it is possible to synthesize trades involving any current and future claims. If one can trade each possible future atomic time-state claim for current units of a numeraire (e.g., coconuts), then any desired trade can be accomplished. Think of it as traversing a connected graph “from point to point” using only its “hub-and-spoke". Thus a set of atomic security prices is sufficient for accomplishing any desired trade.

In the real world, most traded securities represent patterns of payments over many states. After all, people rarely make explicit agreements for payments contingent on just a single state of the world. Therefore, one can think of these securities as “packages of atomic time-state claims”. For example, a correctly-priced forward contract for coconuts (rain or shine) would promise to pay 0.4444 coconuts next year no matter what, in exchange for a promise from the counterparty to deliver 1.0 coconut if the weather has been good and nothing otherwise. Furthermore, by combining existing securities, one can synthesize a security that does not exist. The result is often termed a derivative security, as it is derived from existing securities.

Nobody knows coconuts like I do!

It is difficult to imagine everyone agreeing on the probabilities associated with various future states of the world. Not even the wisest coconut tree growers on the island can accurately forecast the weather over the next year. Therefore, it is believed that security prices reflect a set of “consensus probabilities,” representing weighted averages of investors’ participation in the market. The differences among the investors arise from individual preferences, circumstances, and predictions. Nevertheless, they can all agree on one thing: an additional coconut is likely to be worth more in a state of scarcity than in a state of plenty. In other words, sunny coconuts are cheap and rainy coconuts are expensive. This is how William Sharpe describes the concept of risk premium: “it is not risk per se that that is likely to be rewarded in a well-functioning capital market – only the risk of doing badly in bad times.”

References:

1. Sharpe, William F. (1993). Nuclear Financial Economics. Research Paper 1275, Stanford University. Retrieved from: http://web.stanford.edu/~wfsharpe/art/RP1275.pdf