Puzzle 1: I write down two different numbers that are completely unknown to you, and hold one in my left hand and one in my right. You have absolutely no idea how I generated these two numbers. Which is larger?
You can point to one of my hands, and I will show you the number in it. Then you can decide to either select the number you have seen or switch to the number you have not seen, held in the other hand. Is there a strategy that will give you a greater than 50% chance of choosing the larger number, no matter which two numbers I write down?
At first it seems the answer is no. Whatever number you see, the other number could be larger or smaller. There’s no way to tell. So obviously you can’t get a better than 50% chance of picking the hand with the largest number—even if you’ve seen one of those numbers!
But “obviously” is not a proof. Sometimes “obvious” things are wrong!
It turns out that, amazingly, the answer to the puzzle is yes! You can find a strategy to do better than 50%. But the strategy uses randomness. So, this puzzle is a great illustration of the power of randomness.
If you want to solve it yourself, stop now or read Quanta magazine for some clues—they offered a small prize for the best answer:
• Pradeep Mutalik, Can information rise from randomness?, Quanta, 7 July 2015.
Greg Egan gave a nice solution in the comments to this magazine article, and I’ll reprint it below along with two followup puzzles. So don’t look down there unless you want a spoiler.
I should add: the most common mistake among educated readers seems to be assuming that the first player, the one who chooses the two numbers, chooses them according to some probability distribution. Don’t assume that. They are simply arbitrary numbers.
I’d seen this puzzle before—do you know who invented it? On G+, Hans Havermann wrote:
I believe the origin of this puzzle goes back to (at least) John Fox and Gerald Marnie’s 1958 betting game ‘Googol’. Martin Gardner mentioned it in his February 1960 column in Scientific American. Wikipedia mentions it under the heading ‘Secretary problem’. Gardner suggested that a variant of the game was proposed by Arthur Cayley in 1875.
Actually the game of Googol is a generalization of the puzzle that we’ve been discussing. Martin Gardner explained it thus:
Ask someone to take as many slips of paper as he pleases, and on each slip write a different positive number. The numbers may range from small fractions of 1 to a number the size of a googol (1 followed by a hundred 0s) or even larger. These slips are turned face down and shuffled over the top of a table. One at a time you turn the slips face up. The aim is to stop turning when you come to the number that you guess to be the largest of the series. You cannot go back and pick a previously turned slip. If you turn over all the slips, then of course you must pick the last one turned.
So, the puzzle I just showed you is the special case when there are just 2 slips of paper. I seem to recall that Gardner incorrectly dismissed this case as trivial!
There’s been a lot of work on Googol. Julien Berestycki writes:
I heard about this puzzle a few years ago from Sasha Gnedin. He has a very nice paper about this
• Alexander V. Gnedin, A solution to the game of Googol, Annals of Probability (1994), 1588–1595.
One of the many beautiful ideas in this paper is that it asks what is the best strategy for the guy who writes the numbers! It also cites a paper by Gnedin and Berezowskyi (of oligarchic fame).
Okay, here is Greg Egan’s solution, paraphrased a bit:
Pick some function such that:
•
•
• is monotonically increasing: if then
There are lots of functions like this, for example
Next, pick one of the first player’s hands at random. If the number you are shown is compute Then generate a uniformly distributed random number between 0 and 1. If is less than or equal to guess that is the larger number, but if is greater than guess that the larger number is in the other hand.
The probability of guessing correctly can be calculated as the probability of seeing the larger number initially and then, correctly, sticking with it, plus the probability of seeing the smaller number initially and then, correctly, choosing the other hand.
Say the larger number is and the smaller one is Then the probability of guessing correctly is
This is strictly greater than since so .
So, you have a more than 50% chance of winning! But as you play the game, there’s no way to tell how much more than 50%. If the numbers on the other players hands are very large, or very small, your chance will be just slightly more than 50%.
Here are two more puzzles:
Puzzle 2: Prove that no deterministic strategy can guarantee you have a more than 50% chance of choosing the larger number.
Puzzle 3: There are perfectly specific but ‘algorithmically random’ sequences of bits, which can’t predicted well by any program. If we use these to generate a uniform algorithmically random number between 0 and 1, and use the strategy Egan describes, will our chance of choosing the larger number be more than 50%, or not?
But watch out—here come Egan’s solutions to those!
Egan writes:
Here are my answers to your two puzzles on G+.
Puzzle 2: Prove that no deterministic strategy can guarantee you have a more than 50% chance of choosing the larger number.
Answer: If we adopt a deterministic strategy, that means there is a function that tells us whether on not we stick with the number x when we see it. If we stick with it, if we swap it for the other number.
If the two numbers are and with then the probability of success will be:
This is exactly the same as the formula we obtained when we stuck with with probability but we have specialised to functions valued in
We can only guarantee a more than 50% chance of choosing the larger number if is monotonically increasing everywhere, i.e. whenever But this is impossible for a function valued in To prove this, define to be any number in such that such an must exist, otherwise would be constant on and hence not monotonically increasing. Similarly define to be any number in such that We then have but
Puzzle 3: There are perfectly specific but ‘algorithmically random’ sequences of bits, which can’t predicted well by any program. If we use these to generate a uniform algorithmically random number between 0 and 1, and use the strategy Egan describes, will our chance of choosing the larger number be more than 50%, or not?
Answer: As Philip Gibbs noted, a deterministic pseudo-random number generator is still deterministic. Using a specific sequence of algorithmically random bits
to construct a number between and means takes on the specific value:
So rather than sticking with with probability for our monotonically increasing function we end up always sticking with if and always swapping if This is just using a function as in Puzzle 2, with:
if
if
So all the same consequences as in Puzzle 2 apply, and we cannot guarantee a more than 50% chance of choosing the larger number.
Puzzle 3 emphasizes the huge gulf between ‘true randomness’, where we only have a probability distribution of numbers and the situation where we have a specific number generated by any means whatsoever.
We could generate using a pseudorandom number generator, radioactive decay of atoms, an oracle whose randomness is certified by all the Greek gods, or whatever. No matter how randomly is generated, once we have it, we know there exist choices for the first player that will guarantee our defeat!
This may seem weird at first, but if you think about simple games of luck you’ll see it’s completely ordinary. We can have a more than 50% chance of winning such a game even if for any particular play we make the other player has a move that ensures our defeat. That’s just how randomness works.
What is the most agreed-on figure for our future carbon budget?
My answer:
Asking “what is our future carbon budget?” is a bit like asking how many calories a day you can eat. There’s really no limit on how much you can eat if you don’t care how overweight and unhealthy you become. So, to set a carbon budget, you need to say how much global warming you will accept.
That said, here’s a picture of how we’re burning through our carbon budget:
It says that our civilization has burnt 60% of the carbon we’re allowed to while still having a 50-50 chance of keeping global warming below 2 °C.
This chart appears in the International Energy Agency report World Energy Outlook Special Report 2015, which is free and definitely worth reading.
The orange bars show CO_{2} emissions per year, in gigatonnes. The blue curve shows the fraction of the total carbon budget we have left to burn, based on data from the Intergovernmental Panel for Climate Change. The projection of future carbon emissions is based on the Intended Nationally Determined Contributions (INDC) that governments are currently submitting to the United Nations. So, based on what governments had offered to do by June 2015, we may burn through this carbon budget in 2040.
Our civilization’s total carbon budget for staying below 2 °C was about 1 trillion tonnes. We have now burnt almost 60% of that. You can watch the amount rise as we speak:
Quoting the International Energy Agency report:
The transition away from fossil fuels is gradual in the INDC Scenario, with the share of fossil fuels in the world’s primary energy mix declining from more than 80% today to around three-quarters in 2030 […] The projected path for energy-related emissions in the INDC Scenario means that, based on IPCC estimates, the world’s remaining carbon budget consistent with a 50% chance of keeping a temperature increase of below 2 °C would be exhausted around 2040, adding a grace period of only around eight months, compared to the date at which the budget would be exhausted in the absence of INDCs (Figure 2.3). This date is already within the lifetime of many existing energy sector assets: fossil-fuelled power plants often operate for 30-40 years or more, while existing fossil-fuel resources could, if all developed, sustain production levels far beyond 2040. If energy sector investors believed that not only new investments but also existing fossil-fuel operations would be halted at that critical point, this would have a profound effect on investment even today.
Since we seem likely to go above 2 °C warming over pre-industrial levels, it would be nice to make a similar chart for a carbon budget based on 3 ° C warming. The Trillionth Tonne website projects that with current trends we’ll burn 1.5 trillion tonnes, for a warming of 3 °C in a cautious scenario, by 2056.
But: we would never burn the 1.5 trillionth tonne if emissions dropped by 1.2% per year from now on. And we’d not even burn the trillionth tonne if they dropped by 2.6% per year.
• Workshop on Mathematical Trends in Reaction Network Theory, 1-3 July 2015, Department of Mathematical Sciences, University of Copenhagen. Organized by Elisenda Feliu and Carsten Wiuf.
Looking through the abstracts, here are a couple that strike me.
First of all, Gheorghe Craciun claims to have proved the biggest open conjecture in this field: the Global Attractor Conjecture!
• Gheorge Craciun, Toric differential inclusions and a proof of the global attractor conjecture.
This famous old conjecture says that for a certain class of chemical reactions, the ones coming from ‘complex balanced reaction networks’, the chemicals will approach equilibrium no matter what their initial concentrations are. Here’s what Craciun says:
Abstract. In a groundbreaking 1972 paper Fritz Horn and Roy Jackson showed that a complex balanced mass-action system must have a unique locally stable equilibrium within any compatibility class. In 1974 Horn conjectured that this equilibrium is a global attractor, i.e., all solutions in the same compatibility class must converge to this equilibrium. Later, this claim was called the Global Attractor Conjecture, and it was shown that it has remarkable implications for the dynamics of large classes of polynomial and power-law dynamical systems, even if they are not derived from mass-action kinetics. Several special cases of this conjecture have been proved during the last decade. We describe a proof of the conjecture in full generality. In particular, it will follow that all detailed balanced mass action systems and all deficiency zero mass-action systems have the global attractor property. We will also discuss some implications for biochemical mechanisms that implement noise filtering and cellular homeostasis.
Manoj Gopalkrishnan wrote a great post explaining the concept of complex balanced reaction network here on Azimuth, so if you want to understand the conjecture you could start there.
Even better, Manoj is talking here about a way to do statistical inference with chemistry! His talk is called ‘Statistical inference with a chemical soup':
Abstract. The goal is to design an “intelligent chemical soup” that can do statistical inference. This may have niche technological applications in medicine and biological research, as well as provide fundamental insight into the workings of biochemical reaction pathways. As a first step towards our goal, we describe a scheme that exploits the remarkable mathematical similarity between log-linear models in statistics and chemical reaction networks. We present a simple scheme that encodes the information in a log-linear model as a chemical reaction network. Observed data is encoded as initial concentrations, and the equilibria of the corresponding mass-action system yield the maximum likelihood estimators. The simplicity of our scheme suggests that molecular environments, especially within cells, may be particularly well suited to performing statistical computations.
It’s based on this paper:
• Manoj Gopalkrishnan, A scheme for molecular computation of maximum likelihood estimators for log-linear models.
I’m not sure, but this idea may exploit existing analogies between the approach to equilibrium in chemistry, the approach to equilibrium in evolutionary game theory, and statistical inference. You may have read Marc Harper’s post about that stuff!
David Doty is giving a broader review of ‘Computation by (not about) chemistry':
Abstract. The model of chemical reaction networks (CRNs) is extensively used throughout the natural sciences as a descriptive language for existing chemicals. If we instead think of CRNs as a programming language for describing artificially engineered chemicals, what sorts of computations are possible for these chemicals to achieve? The answer depends crucially on several formal choices:
1) Do we treat matter as infinitely divisible (real-valued concentrations) or atomic (integer-valued counts)?
2) How do we represent the input and output of the computation (e.g., Boolean presence or absence of species, positive numbers directly represented by counts/concentrations, positive and negative numbers represented indirectly by the difference between counts/concentrations of a pair of species)?
3) Do we assume mass-action rate laws (reaction rates proportional to reactant counts/concentrations) or do we insist the system works correctly under a broader class of rate laws?
The talk will survey several recent results and techniques. A primary goal of the talk is to convey the “programming perspective”: rather than asking “What does chemistry do?”, we want to understand “What could chemistry do?” as well as “What can chemistry provably not do?”
I’m really interested in chemical reaction networks that appear in biological systems, and there will be lots of talks about that. For example, Ovidiu Radulescu will talk about ‘Taming the complexity of biochemical networks through model reduction and tropical geometry’. Model reduction is the process of simplifying complicated models while preserving at least some of their good features. Tropical geometry is a cool version of algebraic geometry that uses the real numbers with minimization as addition and addition as multiplication. This number system underlies the principle of least action, or the principle of maximum energy. Here is Radulescu’s abstract:
Abstract. Biochemical networks are used as models of cellular physiology with diverse applications in biology and medicine. In the absence of objective criteria to detect essential features and prune secondary details, networks generated from data are too big and therefore out of the applicability of many mathematical tools for studying their dynamics and behavior under perturbations. However, under circumstances that we can generically denote by multi-scaleness, large biochemical networks can be approximated by smaller and simpler networks. Model reduction is a way to find these simpler models that can be more easily analyzed. We discuss several model reduction methods for biochemical networks with polynomial or rational rate functions and propose as their common denominator the notion of tropical equilibration, meaning finite intersection of tropical varieties in algebraic geometry. Using tropical methods, one can strongly reduce the number of variables and parameters of biochemical network. For multi-scale networks, these reductions are computed symbolically on orders of magnitude of parameters and variables, and are valid in wide domains of parameter and phase spaces.
I’m talking about the analogy between probabilities and quantum amplitudes, and how this makes chemistry analogous to particle physics. You can see two versions of my talk here, but I’ll be giving the ‘more advanced’ version, which is new:
• Probabilities versus amplitudes.
Abstract. Some ideas from quantum theory are just beginning to percolate back to classical probability theory. For example, the master equation for a chemical reaction network describes the interactions of molecules in a stochastic rather than quantum way. If we look at it from the perspective of quantum theory, this formalism turns out to involve creation and annihilation operators, coherent states and other well-known ideas—but with a few big differences.
Anyway, there are a lot more talks, but if I don’t have breakfast and walk over to the math department, I’ll miss those talks!
You can learn more about individual talks in the comments here (see below) and also in Matteo Polettini’s blog:
• Matteo Polettini, Mathematical trends in reaction network theory: part 1 and part 2, Out of Equilibrium, 1 July 2015.
Today I’m going to this workshop:
• Higher-Dimensional Rewriting and Applications, 28-29 June 2015, Warsaw, Poland.
Many of the talks will be interesting to people who are trying to use category theory as a tool for modelling networks!
For example, though they can’t actually attend, Lucius Meredith and my student Mike Stay hope to use Google Hangouts to present their work on Higher category models of the π-calculus. The π-calculus is a way of modelling networks where messages get sent here and there, e.g. the internet. Check out Mike’s blog post about this:
• Mike Stay, A 2-categorical approach to the pi calculus, The n-Category Café, 26 May 2015.
Krzysztof Bar, Aleks Kissinger and Jamie Vicary will be speaking about Globular, a proof assistant for computations in n-categories:
This talk is a progress report on Globular, an online proof assistant for semistrict higher-dimensional rewriting. We aim to produce a tool which can visualize higher-dimensional categorical diagrams, assist in their construction with a point-and-click interface, perform type checking to prevent incorrect composites, and automatically handle the interchanger data at each dimension. Hosted on the web, it will have a low barrier to use, and allow hyperlinking of formalized proofs directly from research papers. We outline the theoretical basis for the tool, and describe the challenges we have overcome in its design.
Eric Finster will be talking about another computer system for dealing with n-categories, based on the ‘opetopic’ formalism that James Dolan and I invented. And Jason Morton is working on a computer system for computation in compact closed categories! I’ve seen it, and it’s cool, but he can’t attend the workshop, so David Spivak will be speaking on his work with Jason on the theoretical foundations of this software:
We consider the linked problems of (1) finding a normal form for morphism expressions in a closed compact category and (2) the word problem, that is deciding if two morphism expressions are equal up to the axioms of a closed compact category. These are important ingredients for a practical monoidal category computer algebra system. Previous approaches to these problems include rewriting and graph-based methods. Our approach is to re-interpret a morphism expression in terms of an operad, and thereby obtain a single composition which is strictly associative and applied according to the abstract syntax tree. This yields the same final operad morphism regardless of the tree representation of the expression or order of execution, and solves the normal form problem up to automorphism.
Recently Eugenia Cheng has been popularizing category theory, touring to promote her book Cakes, Custard and Category Theory. But she’ll be giving two talks in Warsaw, I believe on distributive laws for Lawvere theories.
As for me, I’ll be promoting my dream of using category theory to understand networks in electrical engineering. I’ll be giving a talk on control theory and a talk on electrical circuits: two sides of the same coin, actually.
• John Baez, Jason Erbele and Nick Woods, Categories in control.
If you’ve seen a previous talk of mine with the same title, don’t despair—this one has new stuff! In particular, it talks about a new paper by Nick Woods and Simon Wadsley.
Abstract. Control theory is the branch of engineering that studies dynamical systems with inputs and outputs, and seeks to stabilize these using feedback. Control theory uses “signal-flow diagrams” to describe processes where real-valued functions of time are added, multiplied by scalars, differentiated and integrated, duplicated and deleted. In fact, these are string diagrams for the symmetric monoidal category of finite-dimensional vector spaces, but where the monoidal structure is direct sum rather than the usual tensor product. Jason Erbele has given a presentation for this symmetric monoidal category, which amounts to saying that it is the PROP for bicommutative bimonoids with some extra structure.
A broader class of signal-flow diagrams also includes “caps” and “cups” to model feedback. This amounts to working with a larger symmetric monoidal category where objects are still finite-dimensional vector spaces but the morphisms are linear relations. Erbele also found a presentation for this larger symmetric monoidal category. It is the PROP for a remarkable thing: roughly speaking, an object with two special commutative dagger-Frobenius structures, such that the multiplication and unit of either one and the comultiplication and counit of the other fit together to form a bimonoid.
• John Baez and Brendan Fong, Circuits, categories and rewrite rules.
Abstract. We describe a category where a morphism is an electrical circuit made of resistors, inductors and capacitors, with marked input and output terminals. In this category we compose morphisms by attaching the outputs of one circuit to the inputs of another. There is a functor called the ‘black box functor’ that takes a circuit, forgets its internal structure, and remembers only its external behavior. Two circuits have the same external behavior if and only if they impose same relation between currents and potentials at their terminals. This is a linear relation, so the black box functor goes from the category of circuits to the category of finite-dimensional vector spaces and linear relations. Constructing this functor makes use of Brendan Fong’s theory of ‘decorated cospans’—and the question of whether two ‘planar’ circuits map to the same relation has an interesting answer in terms of rewrite rules.
The answer to the last question, in the form of a single picture, is this:
(Click to enlarge.) How can you change an electrical circuit made out of resistors without changing what it does? 5 ways are shown here:
You can remove a wire with a resistor on it if one end is unattached. Again, it doesn’t do anything.
You can take two resistors in series—one after the other—and replace them with a single resistor. But this new resistor must have a resistance that’s the sum of the old two.
You can take two resistors in parallel and replace them with a single resistor. But this resistor must have a conductivity that’s the sum of the old two. (Conductivity is the reciprocal of resistance.)
Finally, the really cool part: the Y-Δ transform. You can replace a Y made of 3 resistors by a triangle of resistors But their resistances must be related by the equations shown here.
For circuits drawn on the plane, these are all the rules you need! This was proved here:
• Yves Colin de Verdière, Isidoro Gitler and Dirk Vertigan, Réseaux électriques planaires II.
It’s just the beginning of a cool story, which I haven’t completely blended with the categorical approach to circuits. Doing so clearly calls for 2-categories: those double arrows are 2-morphisms! For more, see:
• Joshua Alman, Carl Lian and Brandon Tran, Circular planar electrical networks I: The electrical poset EP_{n}.
Even the biggest European oil and gas companies are calling for a carbon tax! Their motives, of course, should be suspect. But they have realized it’s hopeless to argue about the basics. They wrote a letter to the United Nations beginning:
Dear Excellencies:
Climate change is a critical challenge for our world. As major companies from the oil & gas sector, we recognize both the importance of the climate challenge and the importance of energy to human life and well-being. We acknowledge that the current trend of greenhouse gas emissions is in excess of what the Intergovernmental Panel on Climate Change (IPCC) says is needed to limit the temperature rise to no more than 2 degrees above pre-industrial levels. The challenge is how to meet greater energy demand with less CO2. We stand ready to play our part.
It seems there are just a few places, mostly former British colonies, where questioning the reality and importance of man-made global warming is a popular stance among politicians. Unfortunately one of these, the United States, is a big carbon emitter. Otherwise we could just ignore these holdouts.
Given all this, it’s not so surprising that Pope Francis has joined the crowd and released a document on environmental issues:
• Pope Francis, Enyclical letter Laudato Si’: on care for our common home.
Still, it is interesting to read this document, because unlike most reports we read on climate change, it addresses the cultural and spiritual dimensions of this problem.
I believe arguments should be judged by their merits, not the fact that they’re made by someone with an impressive title like
His Holiness Francis, Bishop of Rome, Vicar of Jesus Christ, Successor of the Prince of the Apostles, Supreme Pontiff of the Universal Church, Primate of Italy, Archbishop and Metropolitan of the Roman Province, Sovereign of the Vatican City State, Servant of the servants of God.
(Note the hat-tip to Darwin there. )
But in fact Francis has some interesting things to say. And among all the reportage on this issue, it’s hard to find more than quick snippets of the actual 182-page document, which is actually quite interesting. So, let me quote a bit.
I will try to dodge the explicitly Christian bits, because I really don’t want people arguing about religion on this blog—in fact I won’t allow it. Of course discussing what the Pope says without getting into Christianity is very difficult and perhaps even absurd. But let’s try.
I will also skip the extensive section where he summarizes the science. It’s very readable, and for an audience who doesn’t want numbers and graphs it’s excellent. But I figure the audience of this blog already knows that material.
So, here are some of the passages I found most interesting.
He discusses how St. Francis of Assisi has been an example to him, and says:
Francis helps us to see that an integral ecology calls for openness to categories which transcend the language of mathematics and biology, and take us to the heart of what it is to be human. Just as happens when we fall in love with someone, whenever he would gaze at the sun, the moon or the smallest of animals, he burst into song, drawing all other creatures into his praise.
[…]
If we approach nature and the environment without this openness to awe and wonder, if we no longer speak the language of fraternity and beauty in our relationship with the world, our attitude will be that of masters, consumers, ruthless exploiters, unable to set limits on their immediate needs. By contrast, if we feel intimately united with all that exists, then sobriety and care will well up spontaneously. The poverty and austerity of Saint Francis were no mere veneer of asceticism, but something much more radical: a refusal to turn reality into an object simply to be used and controlled.
On the responses to ecological problems thus far:
The problem is that we still lack the culture needed to confront this crisis. We lack leadership capable of striking out on new paths and meeting the needs of the present with concern for all and without prejudice towards coming generations. The establishment of a legal framework which can set clear boundaries and ensure the protection of ecosystems has become indispensable, otherwise the new power structures based on the techno-economic paradigm may overwhelm not only our politics but also freedom and justice.
It is remarkable how weak international political responses have been. The failure of global summits on the environment make it plain that our politics are subject to technology and finance. There are too many special interests, and economic interests easily end up trumping the common good and manipulating information so that their own plans will not be affected. The Aparecida Document urges that “the interests of economic groups which irrationally demolish sources of life should not prevail in dealing with natural resources”. The alliance between the economy and technology ends up sidelining anything unrelated to its immediate interests. Consequently the most one can expect is superficial rhetoric, sporadic acts of philanthropy and perfunctory expressions of concern for the environment, whereas any genuine attempt by groups within society to introduce change is viewed as a nuisance based on romantic illusions or an obstacle to be circumvented.
In some countries, there are positive examples of environmental improvement: rivers, polluted for decades, have been cleaned up; native woodlands have been restored; landscapes have been beautified thanks to environmental renewal projects; beautiful buildings have been erected; advances have been made in the production of non-polluting energy and in the improvement of public transportation. These achievements do not solve global problems, but they do show that men and women are still capable of intervening positively. For all our limitations, gestures of generosity, solidarity and care cannot but well up within us, since we were made for love.
At the same time we can note the rise of a false or superficial ecology which bolsters complacency and a cheerful recklessness. As often occurs in periods of deep crisis which require bold decisions, we are tempted to think that what is happening is not entirely clear. Superficially, apart from a few obvious signs of pollution and deterioration, things do not look that serious, and the planet could continue as it is for some time. Such evasiveness serves as a licence to carrying on with our present lifestyles and models of production and consumption. This is the way human beings contrive to feed their self-destructive vices: trying not to see them, trying not to acknowledge them, delaying the important decisions and pretending that nothing will happen.
On the risks:
It is foreseeable that, once certain resources have been depleted, the scene will be set for new wars, albeit under the guise of noble claims.
He writes:
Everything is connected. Concern for the environment thus needs to be joined to a sincere love for our fellow human beings and an unwavering commitment to resolving the problems of society.
Moreover, when our hearts are authentically open to universal communion, this sense of fraternity excludes nothing and no one. It follows that our indifference or cruelty towards fellow creatures of this world sooner or later affects the treatment we mete out to other human beings. We have only one heart, and the same wretchedness which leads us to mistreat an animal will not be long in showing itself in our relationships
with other people. Every act of cruelty towards any creature is “contrary to human dignity”. We can hardly consider ourselves to be fully loving if we disregard any aspect of reality: “Peace, justice and the preservation of creation are three absolutely interconnected themes, which cannot be separated and treated individually without once again falling into reductionism”.
Technoscience, when well directed, can produce important means of improving the quality of human life, from useful domestic appliances to great transportation systems, bridges, buildings and public spaces. It can also produce art and enable men and women immersed in the material world to “leap” into the world of beauty. Who can deny the beauty of an aircraft or a skyscraper? Valuable works of art and music now make use of new technologies. So, in the beauty intended by the one who uses new technical instruments and in the contemplation of such beauty, a quantum leap occurs, resulting in a fulfilment which is uniquely human.
Yet it must also be recognized that nuclear energy, biotechnology, information technology, knowledge of our DNA, and many other abilities which we have acquired, have given us tremendous power. More precisely, they have given those with the knowledge, and especially the economic resources to use them, an impressive dominance over the whole of humanity and the entire world. Never has humanity had such power over itself, yet nothing ensures that it will be used wisely, particularly when we consider how it is currently being used. We need but think of the nuclear bombs dropped in the middle of the twentieth century, or the array of technology which Nazism, Communism and other totalitarian regimes have employed to kill millions of people, to say nothing of the increasingly deadly arsenal of weapons available for modern warfare. In whose hands does all this power lie, or will it eventually end up? It is extremely risky for a small part of humanity to have it.
The basic problem goes even deeper: it is the way that humanity has taken up technology and its development according to an undifferentiated and one-dimensional paradigm. This paradigm exalts the concept of a subject who, using logical and rational procedures, progressively approaches and gains control over an external object. This subject makes every effort to establish the scientific and experimental method, which in itself is already a technique of possession, mastery and transformation. It is as if the subject were to find itself in the presence of something formless, completely open to manipulation. Men and women have constantly intervened in nature, but for a long time this meant being in tune with and respecting the possibilities offered by the things themselves. It was a matter of receiving what nature itself allowed, as if from its own hand. Now, by contrast, we are the ones to lay our hands on things, attempting to extract everything possible from them while frequently ignoring or forgetting the reality in front of us. Human beings and material objects no longer extend a friendly hand to one another; the relationship has become confrontational. This has made it easy to accept the idea of infinite or unlimited growth, which proves so attractive to economists, financiers and experts in technology. It is based on the lie that there is an infinite supply of the earth’s goods, and this leads to the planet being squeezed dry beyond every limit. It is the false notion that “an infinite quantity of energy and resources are available, that it is possible to renew them quickly, and that the negative effects of the exploitation of the natural order can be easily absorbed”.
The idea of promoting a different cultural paradigm and employing technology as a mere instrument is nowadays inconceivable. The technological paradigm has become so dominant that it would be difficult to do without its resources and even more difficult to utilize them without being dominated by their internal logic. It has become countercultural to choose a lifestyle whose goals are even partly independent of technology, of its costs and its power to globalize and make us all the same. Technology tends to absorb everything into its ironclad logic, and those who are surrounded with technology “know full well that it moves forward in the final analysis neither for profit nor for the well-being of the human race”, that “in the most radical sense of the term power is its motive – a lordship over all”. As a result, “man seizes hold of the naked elements of both nature and human nature”. Our capacity to make decisions, a more genuine freedom and the space for each one’s alternative creativity are diminished.
The technocratic paradigm also tends to dominate economic and political life. The economy accepts every advance in technology with a view to profit, without concern for its potentially negative impact on human beings. Finance overwhelms the real economy. The lessons of the global financial crisis have not been assimilated, and we are learning all too slowly the lessons of environmental deterioration. Some circles maintain that current economics and technology will solve all environmental problems, and argue, in popular and non-technical terms, that the problems of global hunger and poverty will be resolved simply by market growth. They are less concerned with certain economic theories which today scarcely anybody dares defend, than with their actual operation in the functioning of the economy. They may not affirm such theories with words, but nonetheless support them with their deeds by showing no interest in more balanced levels of production, a better distribution of wealth, concern for the environment and the rights of future generations. Their behaviour shows that for them maximizing profits is enough.
Ecological culture cannot be reduced to a series of urgent and partial responses to the immediate problems of pollution, environmental decay and the depletion of natural resources. There needs to be a distinctive way of looking at things, a way of thinking, policies, an educational programme, a lifestyle and a spirituality which together generate resistance to the assault of the technocratic paradigm. Otherwise, even the best ecological initiatives can find themselves caught up in the same globalized logic. To seek only a technical remedy to each environmental problem which comes up is to separate what is in reality interconnected and to mask the true and deepest problems of the global system.
Yet we can once more broaden our vision. We have the freedom needed to limit and direct technology; we can put it at the service of another type of progress, one which is healthier, more human, more social, more integral. Liberation from the dominant technocratic paradigm does in fact happen sometimes, for example, when cooperatives of small producers adopt less polluting means of production, and opt for a non-consumerist model of life, recreation and community. Or when technology is directed primarily to resolving people’s concrete problems, truly helping them live with more dignity and less suffering. Or indeed when the desire to create and contemplate beauty manages to overcome reductionism through a kind of salvation which occurs in beauty and in those who behold it. An authentic humanity, calling for a new synthesis, seems to dwell in the midst of our technological culture, almost unnoticed, like a mist seeping gently beneath a closed door. Will the promise last, in spite of everything, with all that is authentic rising up in stubborn resistance?
Near the end he calls the for the development of an ‘integral ecology’. I find it fascinating that this has something in common with ‘network theory':
Since everything is closely interrelated, and today’s problems call for a vision capable of taking into account every aspect of the global crisis, I suggest that we now consider some elements of an integral ecology, one which clearly respects its human and social dimensions.
Ecology studies the relationship between living organisms and the environment in which they develop. This necessarily entails reflection and debate about the conditions required for the life and survival of society, and the honesty needed to question certain models of development, production and consumption. It cannot be emphasized enough how everything is interconnected. Time and space are not independent of one another, and not even atoms or subatomic particles can be considered in isolation. Just as the different aspects of the planet—physical, chemical and biological—are interrelated, so too living species are part of a network which we will never fully explore and understand. A good part of our genetic code is shared by many living beings. It follows that the fragmentation of knowledge and the isolation of bits of information can actually become a form of ignorance, unless they are integrated into a broader vision of reality.
When we speak of the “environment”, what we really mean is a relationship existing between nature and the society which lives in it. Nature cannot be regarded as something separate from ourselves or as a mere setting in which we live. We are part of nature, included in it and thus in constant interaction with it. Recognizing the reasons why a given area is polluted requires a study of the workings of society, its economy, its behaviour patterns, and the ways it grasps reality. Given the scale of change, it is no longer possible to find a specific, discrete answer for each part of the problem. It is essential to seek comprehensive solutions which consider the interactions within natural systems themselves and with social systems. We are faced not with two separate crises, one environmental and the other social, but rather with one complex crisis which is both social and environmental. Strategies for a solution demand an integrated approach to combating poverty, restoring dignity to the excluded, and at the same time protecting nature.
Due to the number and variety of factors to be taken into account when determining the environmental impact of a concrete undertaking, it is essential to give researchers their due role, to facilitate their interaction, and to ensure broad academic freedom. Ongoing research should also give us a better understanding of how different creatures relate to one another in making up the larger units which today we term “ecosystems”. We take these systems into account not only to determine how best to use them, but also because they have an intrinsic value independent of their usefulness.
He concludes by discussing the need for ‘ecological education’.
Environmental education has broadened its goals. Whereas in the beginning it was mainly centred on scientific information, consciousness-raising and the prevention of environmental risks, it tends now to include a critique of the “myths” of a modernity grounded in a utilitarian mindset (individualism, unlimited progress, competition, consumerism, the unregulated market). It seeks also to restore the various levels of ecological equilibrium, establishing harmony within ourselves, with others, with nature and other living creatures, and with God. Environmental education should facilitate making the leap towards the transcendent which gives ecological ethics its deepest meaning. It needs educators capable of developing an ethics of ecology, and helping people, through effective pedagogy, to grow in solidarity, responsibility and compassionate care.
Even small good practices can encourage new attitudes:
Education in environmental responsibility can encourage ways of acting which directly and significantly affect the world around us, such as avoiding the use of plastic and paper, reducing water consumption, separating refuse, cooking only what can reasonably be consumed, showing care for other living beings, using public transport or car-pooling, planting trees, turning off unnecessary lights, or any number of other practices. All of these reflect a generous and worthy creativity which brings out the best in human beings. Reusing something instead of immediately discarding it, when done for the right reasons, can be an act of love which expresses our own dignity.
We must not think that these efforts are not going to change the world. They benefit society, often unbeknown to us, for they call forth a goodness which, albeit unseen, inevitably tends to spread. Furthermore, such actions can restore our sense of self-esteem; they can enable us to live more fully and to feel that life on earth is worthwhile.
Part of the goal is to be more closely attentive to what we have, not fooled into thinking we’d always be happier with more:
It is a return to that simplicity which allows us to stop and appreciate the small things, to be grateful for the opportunities which life affords us, to be spiritually detached from what we possess, and not to succumb to sadness for what we lack. This implies avoiding the dynamic of dominion and the mere accumulation of pleasures.
Such sobriety, when lived freely and consciously, is liberating. It is not a lesser life or one lived with less intensity. On the contrary, it is a way of living life to the full. In reality, those who enjoy more and live better each moment are those who have given up dipping here and there, always on the look-out for what they do not have. They experience what it means to appreciate each person and each thing, learning familiarity with the simplest things and how to enjoy them. So they are able to shed unsatisfied needs, reducing their obsessiveness and weariness. Even living on little, they can live a lot, above all when they cultivate other pleasures and find satisfaction in fraternal encounters, in service, in developing their gifts, in music and art, in contact with nature, in prayer. Happiness means knowing how to limit some needs which only diminish us, and being open to the many different possibilities which life can offer.
yet keeping warming below 2°C seems ever more difficult:
The big international climate negotiations to be concluded in Paris in December 2015 bring these issues to the forefront in a dramatic way. Countries are already saying what they plan to do: you can read their Intended Nationally Determined Contributions online!
But it’s hard to get an overall picture of the situation. Here’s a new report that helps:
• International Energy Agency, World Energy Outlook Special Report 2015: Energy and Climate Change.
Since the International Energy Agency seems intelligent to me, I’ll just quote their executive summary. If you’re too busy for even the executive summary, let me summarize the summary:
Given the actions that countries are now planning, we could have an increase of around 2.6 °C over preindustrial temperature by 2100, and more after that.
A major milestone in efforts to combat climate change is fast approaching. The importance of the 21st Conference of the Parties (COP21) – to be held in Paris in December 2015 – rests not only in its specific achievements by way of new contributions, but also in the direction it sets. There are already some encouraging signs with a historic joint announcement by the United States and China on climate change, and climate pledges for COP21 being submitted by a diverse range of countries and in development in many others. The overall test of success for COP21 will be the conviction it conveys that governments are determined to act to the full extent necessary to achieve the goal they have already set to keep the rise in global average temperatures below 2 degrees Celsius (°C), relative to pre-industrial levels.
Energy will be at the core of the discussion. Energy production and use account for two-thirds of the world’s greenhouse-gas (GHG) emissions, meaning that the pledges made at COP21 must bring deep cuts in these emissions, while yet sustaining the growth of the world economy, boosting energy security around the world and bringing modern energy to the billions who lack it today. The agreement reached at COP21 must be comprehensive geographically, which means it must be equitable, reflecting both national responsibilities and prevailing circumstances. The importance of the energy component is why this World Energy Outlook Special Report presents detailed energy and climate analysis for the sector and recommends four key pillars on which COP21 can build success.
The use of low-carbon energy sources is expanding rapidly, and there are signs that growth in the global economy and energy-related emissions may be starting to decouple. The global economy grew by around 3% in 2014 but energy-related carbon dioxide (CO2) emissions stayed flat, the first time in at least 40 years that such an outcome has occurred outside economic crisis.
Renewables accounted for nearly half of all new power generation capacity in 2014, led by growth in China, the United States, Japan and Germany, with investment remaining strong (at $270 billion) and costs continuing to fall. The energy intensity of the global economy dropped by 2.3% in 2014, more than double the average rate of fall over the last decade, a result stemming from improved energy efficiency and structural changes in some economies, such as China.
Around 11% of global energy-related CO2 emissions arise in areas that operate a carbon market (where the average price is $7 per tonne of CO2), while 13% of energy-related CO2 emissions arise in markets with fossil-fuel consumption subsidies (an incentive equivalent to $115 per tonne of CO2, on average). There are some encouraging signs on both fronts, with reform in sight for the European Union’s Emissions Trading Scheme and countries including India, Indonesia, Malaysia and Thailand taking the opportunity of lower oil prices to diminish fossil-fuel subsidies, cutting the incentive for wasteful consumption.
Nationally determined pledges are the foundation of COP21. Intended Nationally
Determined Contributions (INDCs) submitted by countries in advance of COP21 may vary in scope but will contain, implicitly or explicitly, commitments relating to the energy sector. As of 14 May 2015, countries accounting for 34% of energy-related emissions had submitted their new pledges.
A first assessment of the impact of these INDCs and related policy statements (such as by China) on future energy trends is presented in this report in an “INDC Scenario”. This shows, for example, that the United States’ pledge to cut net greenhouse-gas emissions by 26% to 28% by 2025 (relative to 2005 levels) would deliver a major reduction in emissions while the economy grows by more than one-third over current levels. The European Union’s pledge to cut GHG emissions by at least 40% by 2030 (relative to 1990 levels) would see energy-related CO2 emissions decline at nearly twice the rate achieved since 2000, making it one of the world’s least carbon-intensive energy economies. Russia’s energy-related emissions decline slightly from 2013 to 2030 and it meets its 2030 target comfortably, while implementation of Mexico’s pledge would see its energy-related emissions increase slightly while its economy grows much more rapidly. China has yet to submit its INDC, but has stated an intention to achieve a peak in its CO2 emissions around 2030 (if not earlier), an important change in direction, given the pace at which they have grown on average since 2000.
Growth in global energy-related GHG emissions slows but there is no peak by 2030 in the INDC Scenario. The link between global economic output and energy-related GHG emissions weakens significantly, but is not broken: the economy grows by 88% from 2013 to 2030 and energy-related CO2 emissions by 8% (reaching 34.8 gigatonnes). Renewables become the leading source of electricity by 2030, as average annual investment in nonhydro renewables is 80% higher than levels seen since 2000, but inefficient coal-fired power generation capacity declines only slightly.
With INDCs submitted so far, and the planned energy policies in countries that have yet to submit, the world’s estimated remaining carbon budget consistent with a 50% chance of keeping the rise in temperature below 2 °C is consumed by around 2040—eight months later than is projected in the absence of INDCs. This underlines the need for all countries to submit ambitious INDCs for COP21 and for these INDCs to be recognised as a basis upon which to build stronger future action, including from opportunities for collaborative/co-ordinated action or those enabled by a transfer of resources (such as technology and finance). If stronger action is not forthcoming after 2030, the path in the INDC Scenario would be consistent with an an average temperature increase of around 2.6 °C by 2100 and 3.5 °C after 2200.
National pledges submitted for COP21 need to form the basis for a “virtuous circle” of rising ambition. From COP21, the energy sector needs to see a projection from political leaders at the highest level of clarity of purpose and certainty of action, creating a clear expectation of global and national low-carbon development. Four pillars can support that achievement:
1. Peak in emissions – set the conditions which will achieve an early peak in global
energy-related emissions.
2. Five-year revision – review contributions regularly, to test the scope to lift the level of ambition.
3. Lock in the vision – translate the established climate goal into a collective long-term emissions goal, with shorter-term commitments that are consistent with the long-term vision.
4. Track the transition – establish an effective process for tracking achievements in
the energy sector.
The IEA proposes a bridging strategy that could deliver a peak in global energy-related
emissions by 2020. A commitment to target such a near-term peak would send a clear message of political determination to stay below the 2 °C climate limit. The peak can be
achieved relying solely on proven technologies and policies, without changing the economic and development prospects of any region, and is presented in a “Bridge Scenario”. The technologies and policies reflected in the Bridge Scenario are essential to secure the long-term decarbonisation of the energy sector and their near-term adoption can help keep the door to the 2 °C goal open. For countries that have submitted their INDCs, the proposed strategy identifies possible areas for over-achievement. For those that have yet to make a submission, it sets out a pragmatic baseline for ambition.
The Bridge Scenario depends upon five measures:
• Increasing energy efficiency in the industry, buildings and transport sectors.
• Progressively reducing the use of the least-efficient coal-fired power plants and
banning their construction.
• Increasing investment in renewable energy technologies in the power sector from
$270 billion in 2014 to $400 billion in 2030.
• Gradual phasing out of fossil-fuel subsidies to end-users by 2030.
• Reducing methane emissions in oil and gas production.
These measures have profound implications for the global energy mix, putting a brake on growth in oil and coal use within the next five years and further boosting renewables. In the Bridge Scenario, coal use peaks before 2020 and then declines while oil demand rises to 2020 and then plateaus. Total energy-related GHG emissions peak around 2020. Both the energy intensity of the global economy and the carbon intensity of power generation improve by 40% by 2030. China decouples its economic expansion from emissions growth by around 2020, much earlier than otherwise expected, mainly through improving the energy efficiency of industrial motors and the buildings sector, including through standards for appliances and lighting. In countries where emissions are already in decline today, the decoupling of economic growth and emissions is significantly accelerated; compared with recent years, the pace of this decoupling is almost 30% faster in the European Union (due to improved energy efficiency) and in the United States (where renewables contribute one-third of the achieved emissions savings in 2030). In other regions, the link between economic growth and emissions growth is weakened significantly, but the relative importance of different measures varies. India utilises energy more efficiently, helping it
to reach its energy sector targets and moderate emissions growth, while the reduction of
methane releases from oil and gas production and reforming fossil-fuel subsidies (while
providing targeted support for the poorest) are key measures in the Middle East and Africa, and a portfolio of options helps reduce emissions in Southeast Asia. While universal access to modern energy is not achieved in the Bridge Scenario, the efforts to reduce energy related emissions do go hand-in-hand with delivering access to electricity to 1.7 billion people and access to clean cookstoves to 1.6 billion people by 2030.
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This is called the Landauer limit, and it came after many decades of arguments concerning Maxwell’s demon and the relation between information and entropy.
In fact, these arguments are still not finished. For example, here’s an argument that the Landauer limit is not as solid as widely believed:
• John D. Norton, Waiting for Landauer, Studies in History and Philosophy of Modern Physics 42 (2011), 184–198.
But something like the Landauer limit almost surely holds under some conditions! And if it holds, it puts some limits on what organisms can do. That’s what David Wolpert spoke about at our workshop! You can see his slides here:
• David Wolpert — The Landauer limit and thermodynamics of biological organisms.
You can also watch a video:
Hi! Want to talk? Okay, go into that other room and watch TV, then come back when you’re done and we’ll talk about it.
But no matter: our workshop on Information and Entropy in Biological Systems was really exciting! I want to make it available to the world as much as possible. I’m running around too much to create lovingly hand-crafted summaries of each talk—and I know you’re punishing me for that, with your silence. But I’ll keep on going, just to get the material out there.
Marc Harper spoke about information in evolutionary game theory, and we have a nice video of that. I’ve been excited about his work for quite a while, because it shows that the analogy between ‘evolution’ and ‘learning’ can be made mathematically precise. I summarized some of his ideas in my information geometry series, and I’ve also gotten him to write two articles for this blog:
• Marc Harper, Relative entropy in evolutionary dynamics, Azimuth, 22 January 2014.
• Marc Harper, Stationary stability in finite populations, Azimuth, 24 March 2015.
Here are the slides and video of his talk:
• Marc Harper, Information transport and evolutionary dynamics.
We managed to get a video of Ostling’s talk, but not Harte’s. Luckily, you can see the slides of both. You can also see a summary of Harte’s book Maximum Entropy and Ecology:
• John Baez, Maximum entropy and ecology, Azimuth, 21 February 2013.
Here are his talk slides and abstract:
• John Harte, Maximum entropy as a foundation for theory building in ecology.
Abstract. Constrained maximization of information entropy (MaxEnt) yields least-biased probability distributions. In statistical physics, this powerful inference method yields classical statistical mechanics/thermodynamics under the constraints imposed by conservation laws. I apply MaxEnt to macroecology, the study of the distribution, abundance, and energetics of species in ecosystems. With constraints derived from ratios of ecological state variables, I show that MaxEnt yields realistic abundance distributions, species-area relationships, spatial aggregation patterns, and body-size distributions over a wide range of taxonomic groups, habitats and spatial scales. I conclude with a brief summary of some of the major opportunities at the frontier of MaxEnt-based macroecological theory.
Here is a video of Ostling’s talk, as well as her slides and some papers she recommended:
• Annette Ostling, The neutral theory of biodiversity and other competitors to maximum entropy.
Abstract: I am a bit of the odd man out in that I will not talk that much about information and entropy, but instead about neutral theory and niche theory in ecology. My interest in coming to this workshop is in part out of an interest in what greater insights we can get into neutral models and stochastic population dynamics in general using entropy and information theory.
I will present the niche and neutral theories of the maintenance of diversity of competing species in ecology, and explain the dynamics included in neutral models in ecology. I will also briefly explain how one can derive a species abundance distribution from neutral models. I will present the view that neutral models have the potential to serve as more process-based null models than previously used in ecology for detecting the signature of niches and habitat filtering. However, tests of neutral theory in ecology have not as of yet been as useful as tests of neutral theory in evolutionary biology, because they leave open the possibility that pattern is influenced by “demographic complexity” rather than niches. I will mention briefly some of the work I’ve been doing to try to construct better tests of neutral theory.
Finally I’ll mention some connections that have been made so far between predictions of entropy theory and predictions of neutral theory in ecology and evolution.
These papers present interesting relations between ecology and statistical mechanics. Check out the nice ‘analogy chart’ in the second one!
• M. G. Bowler, Species abundance distributions, statistical mechanics and the priors of MaxEnt, Theoretical Population Biology 92 (2014), 69–77.
Abstract. The methods of Maximum Entropy have been deployed for some years to address the problem of species abundance distributions. In this approach, it is important to identify the correct weighting factors, or priors, to be applied before maximising the entropy function subject to constraints. The forms of such priors depend not only on the exact problem but can also depend on the way it is set up; priors are determined by the underlying dynamics of the complex system under consideration. The problem is one of statistical mechanics and it is the properties of the system that yield the correct MaxEnt priors, appropriate to the way the problem is framed. Here I calculate, in several different ways, the species abundance distribution resulting when individuals in a community are born and die independently. In
the usual formulation the prior distribution for the number of species over the number of individuals is 1/n; the problem can be reformulated in terms of the distribution of individuals over species classes, with a uniform prior. Results are obtained using master equations for the dynamics and separately through the combinatoric methods of elementary statistical mechanics; the MaxEnt priors then emerge a posteriori. The first object is to establish the log series species abundance distribution as the outcome of per capita guild dynamics. The second is to clarify the true nature and origin of priors in the language of MaxEnt. Finally, I consider how it may come about that the distribution is similar to log series in the event that filled niches dominate species abundance. For the general ecologist, there are two messages. First, that species abundance distributions are determined largely by population sorting through fractional processes (resulting in the 1/n factor) and secondly that useful information is likely to be found only in departures from the log series. For the MaxEnt practitioner, the message is that the prior with respect to which the entropy is to be maximised is determined by the nature of the problem and the way in which it is formulated.
• Guy Sella and Aaron E. Hirsh, The application of statistical physics to evolutionary biology, Proc. Nat. Acad. Sci. 102 (2005), 9541–9546.
A number of fundamental mathematical models of the evolutionary process exhibit dynamics that can be difficult to understand analytically. Here we show that a precise mathematical analogy can be drawn between certain evolutionary and thermodynamic systems, allowing application of the powerful machinery of statistical physics to analysis of a family of evolutionary models. Analytical results that follow directly from this approach include the steady-state distribution of fixed genotypes and the load in finite populations. The analogy with statistical physics also reveals that, contrary to a basic tenet of the nearly neutral theory of molecular evolution, the frequencies of adaptive and deleterious substitutions at steady state are equal. Finally, just as the free energy function quantitatively characterizes the balance between energy and entropy, a free fitness function provides an analytical expression for the balance between natural selection and stochastic drift.
Here are the slides of the talk I’m giving on Monday to kick off the Categorical Foundations of Network Theory workshop in Turin:
This is a long talk, starting with the reasons I care about this subject, and working into the details of one particular project: the categorical foundations of networks as applied to electrical engineering and control theory. There are lots of links in blue; click on them for more details!