Prospects for a Green Mathematics

contribution to the Mathematics of Planet Earth 2013 blog by John Baez and David Tanzer

It is increasingly clear that we are initiating a sequence of dramatic events across our planet. They include habitat loss, an increased rate of extinction, global warming, the melting of ice caps and permafrost, an increase in extreme weather events, gradually rising sea levels, ocean acidification, the spread of oceanic “dead zones”, a depletion of natural resources, and ensuing social strife.

These events are all connected. They come from a way of life that views the Earth as essentially infinite, human civilization as a negligible perturbation, and exponential economic growth as a permanent condition. Deep changes will occur as these idealizations bring us crashing into the brick wall of reality. If we do not muster the will to act before things get significantly worse, we will need to do so later. While we may plead that it is “too difficult” or “too late”, this doesn’t matter: a transformation is inevitable. All we can do is start where we find ourselves, and begin adapting to life on a finite-sized planet.

Where does math fit into all this? While the problems we face have deep roots, major transformations in society have always caused and been helped along by revolutions in mathematics. Starting near the end of the last ice age, the Agricultural Revolution eventually led to the birth of written numerals and geometry. Centuries later, the Enlightenment and Industrial Revolution brought us calculus and eventually a flowering of mathematics unlike any before. Now, as the 21st century unfolds, mathematics will become increasingly driven by our need to understand the biosphere and our role within it.

We refer to mathematics suitable for understanding the biosphere as green mathematics. Although it is just being born, we can already see some of its outlines.

Since the biosphere is a massive network of interconnected elements, we expect network theory will play an important role in green mathematics. Network theory is a sprawling field, just beginning to become organized, which combines ideas from graph theory, probability theory, biology, ecology, sociology and more. Computation plays an important role here, both because it has a network structure—think of networks of logic gates—and because it provides the means for simulating networks.

One application of network theory is to tipping points, where a system abruptly passes from one regime to another. Scientists need to identify nearby tipping points in the biosphere to help policy makers to head off catastrophic changes. Mathematicians, in turn, are challenged to develop techniques for detecting incipient tipping points. Another application of network theory is the study of shocks and resilience. When can a network recover from a major blow to one of its subsystems?

We claim that network theory is not just another name for biology, ecology, or any other existing science, because in it we can see new mathematical terrains. Here are two examples.

First, consider a leaf. In The Formation of a Tree Leaf by Qinglan Xia, we see a possible key to Nature’s algorithm for the growth of leaf veins. The vein system, which is a transport network for nutrients and other substances, is modeled by Xia as a directed graph with nodes for cells and edges for the “pipes” that connect the cells. Each cell gives a revenue of energy, and incurs a cost for transporting substances to and from it.

The total transport cost depends on the network structure. There are costs for each of the pipes, and costs for turning the fluid around the bends. For each pipe, the cost is proportional to the product of its length, its cross-sectional area raised to a power α, and the number of leaf cells that it feeds. The exponent α captures the savings from using a thicker pipe to transport materials together. Another parameter β expresses the turning cost.

Development proceeds through cycles of growth and network optimization. During growth, a layer of cells gets added, containing each potential cell with a revenue that would exceed its cost. During optimization, the graph is adjusted to find a local cost minimum. Remarkably, by varying α and β, simulations yield leaves resembling those of specific plants, such as maple or mulberry.


A growing network

Unlike approaches that merely create pretty images resembling leaves, Xia presents an algorithmic model, simplified yet illuminating, of how leaves actually develop. It is a network-theoretic approach to a biological subject, and it is mathematics—replete with lemmas, theorems and algorithms—from start to finish.

A second example comes from stochastic Petri nets, which are a model for networks of reactions. In a stochastic Petri net, entities are designated by “tokens” and entity types by “places” which hold the tokens. “Reactions” remove tokens from their input places and deposit tokens at their output places. The reactions fire probabilistically, in a Markov chain where each reaction rate depends on the number of its input tokens.


A stochastic Petri net

Perhaps surprisingly, many techniques from quantum field theory are transferable to stochastic Petri nets. The key is to represent stochastic states by power series. Monomials represent pure states, which have a definite number of tokens at each place. Each variable in the monomial stands for a place, and its exponent indicates the token count. In a linear combination of monomials, each coefficient represents the probability of being in the associated state.

In quantum field theory, states are representable by power series with complex coefficients. The annihilation and creation of particles are cast as operators on power series. These same operators, when applied to the stochastic states of a Petri net, describe the annihilation and creation of tokens. Remarkably, the commutation relations between annihilation and creation operators, which are often viewed as a hallmark of quantum theory, make perfect sense in this classical probabilistic context.

Each stochastic Petri net has a “Hamiltonian” which gives its probabilistic law of motion. It is built from the annihilation and creation operators. Using this, one can prove many theorems about reaction networks, already known to chemists, in a compact and elegant way. See the Azimuth network theory series for details.

Conclusion: The life of a network, and the networks of life, are brimming with mathematical content.

We are pursuing these subjects in the Azimuth Project, an open collaboration between mathematicians, scientists, engineers and programmers trying to help save the planet. On the Azimuth Wiki and Azimuth Blog we are trying to explain the main environmental and energy problems the world faces today. We are also studying plans of action, network theory, climate cycles, the programming of climate models, and more.

If you would like to help, we need you and your special expertise. You can write articles, contribute information, pose questions, fill in details, write software, help with research, help with writing, and more. Just drop us a line.


This post appeared on the blog for Mathematics of Planet Earth 2013, an international project involving over 100 scientific societies, universities, research institutes, and organizations. They’re trying to have a new blog article every day, and you can submit articles as described here.

Here are a few of their other articles:

The mathematics of extreme climatic events—with links to videos.

From the Joint Mathematics Meetings: Conceptual climate models short course—with links to online course materials.

There will always be a Gulf Stream—and exercise in singular perturbation technique.

13 Responses to Prospects for a Green Mathematics

  1. John Merryman says:

    I think some of the more significant problems are not that complicated.

    The major economic fallacy is that we treat money as a commodity, when in fact it is a contract. To the banks, it is a commodity, one which they manufacture by expanding demand, ie. debt, but to the larger society, it is drawing rights on productivity. Essentially a promise and if we collectively make more promises to ourselves than we are willing and able to keep, we will eventually be disappointed. As it is, we are raping the planet to make good on those promises. If people understood money as a form of public utility, a collective contract, they would be far more careful what value they took from social relations and environmental resources to convert into currency. So clarifying this would use peoples own self interest to store value back in society and the environment.
    Another issue is this top down theology. Logically a spiritual absolute would be the essence of being from which we rise, not a moral ideal from which we fell. Monotheism serves to validate top down monolithic political models. Remember it was the polytheists who first formalized democracy. When the gods argue, it is reflected in the political models. Sometimes it is useful for society to act as one, but it should be kept in context, not treated as inerrant.

    As it is, the only public debates about spirituality are the pros and cons of a theistic greybeard. Consciousness, whether as a function of the mind or society, is a network. Oneness, not one.

  2. Toussaint says:

    As a biotechnology student interested in mathematics and its applications, I have read this article with great interest! Yet I think it is way ahead of the basic problem.

    Too many times I see people choosing to be ignorant and have chosen to throw facts aside and listen to fallacies. While intellectuals will possibly appreciate this article and the value of mathematics to tackle our problems many “common” people won’t care.

    What am I talking about? Everywhere [for completeness - I'm from an african country and been in 4 countries there and now i'm in india] I keep “preaching” (people seem to react better to preachers than teachers) that they should be more careful about the numbers of kids they give birth to, more efficient in the utilization of any resources and they should try to be productive with the little available! Here are the answers I get:

    1. I say: we don’t have much space on earth nor resources to support all these kids, limit your births. They say: God created the earth to be filled, don’t you think he has a plan? Now how do you tell someone the importance of mathematics to solve this pressing problem when they don’t even think there is a problem? To those not from a judeo-christian faith, i get something more rational but flawed: people die, we have to replace them. True but at what rate? I’m not saying that everyone from a judeo-christian faith has those point of views(being a christian myself) but that the not-so-much educated branch of the population easily fall for those fallacies.

    2. I have a brother doing business studies and he tells me their goal is to create *unlimited* wealth. And when I ask him and his peers where do they get unlimited resources to create such wealth, they say it’s not their problem, they only “make” money! Some even go as far as saying engineers/scientists aren’t doing well enough to bring technological changes to make more wealth. Like the previous commenter has said, this attitude of looking at money like something we can make as we want than truly need is participatating in killing this planet.

    I think a significant and more powerful change will happen when the majority of people on earth will realize: hey my kids won’t have some place to live in the near future if i don’t make some changes in my thinking and ways of life. While science has brought and continues to bring powerful changes in the lives of many people it will never find a cure for purposeful ignorance and stupidity. Let’s begin by fighting those and maybe politicians who tend to follow the public opinion or the guy with money’s opinion will bring in more support for network theory!

    • John Baez says:

      Toussaint:

      Too many times I see people choosing to be ignorant and have chosen to throw facts aside and listen to fallacies. While intellectuals will possibly appreciate this article and the value of mathematics to tackle our problems many “common” people won’t care.

      I agree. This article was written for the Mathematics of Planet Earth 2013 blog, so we are writing for an audience of professional mathematicians who are interested in environmental issues, but perhaps not already working on these issues. We are trying to convince these mathematicians that mathematics is starting to change in important and exciting ways, and that they can be part of this change.

      This is a completely different job than getting large numbers of people to realize that “hey, my kids won’t have some place to live in the near future if i don’t make some changes in my thinking and ways of life.” What we are doing in this article is probably much less important! But, it’s also something that needs to be done.

      And, speaking just for myself, it’s something I feel better at doing. I think about this a lot: should I completely quit being a mathematician and focus all my energy on reaching a large audience? So far the answer is “no”, for lots of reasons. One reason is that I’ve already spent several decades trying to get good at communicating to mathematicians. So, I’m fairly well known among mathematicians but not very well known to a broad audience. Thus, I have a comparative advantage when I’m trying to reach mathematicians, which I’d probably lose if I switched to a larger audience.

      I’m not under the delusion that mathematics is especially important when it comes to solving the world’s biggest problems—at least, not in the short term. If I were an expert on sewage systems I would talk to people who work on those, and I’d make a bigger impact… in the short term.

      Mathematics is very powerful, but it often takes decades or more to work. Since I believe we’ll still be around then, I think mathematicians should turn to work on math related to understanding the biosphere.

    • John Baez says:

      Toussaint wrote:

      I have a brother doing business studies and he tells me their goal is to create *unlimited* wealth. And when I ask him and his peers where do they get unlimited resources to create such wealth, they say it’s not their problem, they only “make” money!

      Since he’s your brother, could you slap some sense into him for me?

      • John Merryman says:

        Professor Baez,
        While I certainly agree mathematicians are better at thinking deeply, than broadly, given the help certain members of the community assisted in the creation of the derivatives bubble at the core of modern finance, there is some degree of complicity. Would it be possible to begin to devise an effective financial system, not dedicated to the creation and storage of vast amounts of notational value? Possibly by emphasizing that it is a network of contractual obligations and not created value? People might again understand rights and responsibilities are inextricably bound.

      • Toussaint says:

        While this article was written for mathematicians, there are really a lot of other professions that will benefit from reading this article!

        Professor Baez wrote:

        Mathematics is very powerful, but it often takes decades or more to work. Since I believe we’ll still be around then, I think mathematicians should turn to work on math related to understanding the biosphere.

        I couldn’t agree more! And this is one of the things I like about mathematicians: they don’t wait for “a right time” to get work done. If it can be done now, then let’s do it even if the impact will be in 50 years or so! As a matter of fact, in biotechnology, we’ve been exploring possibilities of shifting from the current way of learning things and shift to a more mathematical way. For example when learning biochemistry, we memorize biological pathways (no petri nets, not even mentioning them,…), same thing in genetics (no mention of information theoretic aspects or randomness of the genetic code), and ecology(the volterra model), and so on. Therefore, this article is really timely and thanks for sharing professor!

        Professor Baez wrote:

        Since he’s your brother, could you slap some sense into him for me?

        Starting today!

    • davidtweed says:

      I’ll just note that there’s another view (not one shared by John, but we’re a broad church): maybe people aren’t persuadable, and that understanding stuff so we can deal with the results as best we can. As an analogy, suppose there’s a community building up on what appears to be an “completely inert” volcano that you think is merely dormant. You could try to persuade people to live elsewhere, but it might be that for every person you persuade that “a dormant volcano will wake up some time” and leave, two others arrive to take up their place. In that case, maybe the best thing you can do is become an expert on how lava flows, how to divert it as much as you can, how crowds behave in order to expeditiously evacuate people once an eruption happens, etc. It’d clearly be better to learn the way to persuade people to leave before the eruption, but that just may not be acheivable with actual, ornery “I know what I want to believe, now I’ll manufacture a rationalization” human beings.

      • Toussaint says:

        davidtweed wrote:

        In that case, maybe the best thing you can do is become an expert on how lava flows, how to divert it as much as you can, how crowds behave in order to expeditiously evacuate people once an eruption happens, etc.

        Interesting proposition! It has the triple advantage that: 1/ it stimulates more intense studies (about volcanoes in this case) 2/ the money that could be used to spend time relocating those people or talking to them can be transferred to building those structures 3/ When the volcano finally wakes up, they will maybe have a little more appreciation of science.

        The other side is talking some sense into people can not be abandoned completely because we can’t know if by the time the volcano wakes up, we’ll have enough engineering in place to protect them.

        When I remember the case in Italy about the scientists who couldn’t make an accurate prediction about a coming seism and how the community reacted after it happened, alternatives like these are worth it because they can save scientists some bad time. But I have no idea how to shelter people against a seismic wave. Maybe folks working on mechanical waves can think about this.

        • davetweed says:

          I agree it’s certainly something that, if you believe persuasion is possible, that’s the best course of action. (I’ll also note I may be biased in that I’ve never managed to persuade anyone of anything in my life… (I’ve often, though not always, thought I was right but never been actually able to persuade someone to actually change either their views or behaviour.))

      • John Baez says:

        David wrote:

        I’ll just note that there’s another view (not one shared by John, but we’re a broad church): maybe people aren’t persuadable, and that understanding stuff so we can deal with the results as best we can.

        I think some people can be persuaded into taking action on environmental problems before those problems hit them in the face. I’m not really sure why some people are more easily persuaded than others… I can make guesses, but if I got serious about trying to persuade large numbers of people I’d avoid relying on intuition and start reading research on this topic and maybe doing my own.

        Digressing somewhat, I was impressed by the MIT Technology Review article about how in the 2012 US election, the Obama campaign used the experimental method to assess which tactics worked. This is not a political endorsement, Marvin! This is just my interest in any attempt to do things very systematically:

        When Jim Messina arrived in Chicago as Obama’s newly minted campaign manager in January of 2011, he imposed a mandate on his recruits: they were to make decisions based on measurable data. But that didn’t mean quite what it had four years before. The 2008 campaign had been “data-driven,” as people liked to say. This reflected a principled imperative to challenge the political establishment with an empirical approach to electioneering, and it was greatly influenced by David Plouffe, the 2008 campaign manager, who loved metrics, spreadsheets, and performance reports. Plouffe wanted to know: How many of a field office’s volunteer shifts had been filled last weekend? How much money did that ad campaign bring in?

        But for all its reliance on data, the 2008 Obama campaign had remained insulated from the most important methodological innovation in 21st-century politics. In 1998, Yale professors Don Green and Alan Gerber conducted the first randomized controlled trial in modern political science, assigning New Haven voters to receive nonpartisan election reminders by mail, phone, or in-person visit from a canvasser and measuring which group saw the greatest increase in turnout. The subsequent wave of field experiments by Green, Gerber, and their followers focused on mobilization, testing competing modes of contact and get-out-the-vote language to see which were most successful.

        The first Obama campaign used the findings of such tests to tweak call scripts and canvassing protocols, but it never fully embraced the experimental revolution itself. After Dan Wagner moved to the DNC, the party decided it would start conducting its own experiments. He hoped the committee could become “a driver of research for the Democratic Party.”

        To that end, he hired the Analyst Institute, a Washington-based consortium founded under the AFL-CIO’s leadership in 2006 to coördinate field research projects across the electioneering left and distribute the findings among allies. Much of the experimental world’s research had focused on voter registration, because that was easy to measure. The breakthrough was that registration no longer had to be approached passively; organizers did not have to simply wait for the unenrolled to emerge from anonymity, sign a form, and, they hoped, vote. New techniques made it possible to intelligently profile nonvoters: commercial data warehouses sold lists of all voting-age adults, and comparing those lists with registration rolls revealed eligible candidates, each attached to a home address to which an application could be mailed. Applying microtargeting models identified which nonregistrants were most likely to be Democrats and which ones Republicans.

        The Obama campaign embedded social scientists from the Analyst Institute among its staff. Party officials knew that adding new Democratic voters to the registration rolls was a crucial element in their strategy for 2012. But already the campaign had ambitions beyond merely modifying nonparticipating citizens’ behavior through registration and mobilization. It wanted to take on the most vexing problem in politics: changing voters’ minds.

        The expansion of individual-level data had made possible the kind of testing that could help do that. Experimenters had typically calculated the average effect of their interventions across the entire population. But as campaigns developed deep portraits of the voters in their databases, it became possible to measure the attributes of the people who were actually moved by an experiment’s impact. A series of tests in 2006 by the women’s group Emily’s List had illustrated the potential of conducting controlled trials with microtargeting databases. When the group sent direct mail in favor of Democratic gubernatorial candidates, it barely budged those whose scores placed them in the middle of the partisan spectrum; it had a far greater impact upon those who had been profiled as soft (or nonideological) Republicans.

        That test, and others that followed, demonstrated the limitations of traditional targeting. Such techniques rested on a series of long-standing assumptions—for instance, that middle-of-the-roaders were the most persuadable and that infrequent voters were the likeliest to be captured in a get-out-the-vote drive. But the experiments introduced new uncertainty. People who were identified as having a 50 percent likelihood of voting for a Democrat might in fact be torn between the two parties, or they might look like centrists only because no data attached to their records pushed a partisan prediction in one direction or another. “The scores in the middle are the people we know less about,” says Chris Wyant, a 2008 field organizer who became the campaign’s general election director in Ohio four years later. “The extent to which we were guessing about persuasion was not lost on any of us.”

        One way the campaign sought to identify the ripest targets was through a series of what the Analyst Institute called “experiment-informed programs,” or EIPs, designed to measure how effective different types of messages were at moving public opinion.

        The traditional way of doing this had been to audition themes and language in focus groups and then test the winning material in polls to see which categories of voters responded positively to each approach. Any insights were distorted by the artificial settings and by the tiny samples of demographic subgroups in traditional polls. “You’re making significant resource decisions based on 160 people?” asks Mitch Stewart, director of the Democratic campaign group Organizing for America. “Isn’t that nuts? And people have been doing that for decades!”

        An experimental program would use those steps to develop a range of prospective messages that could be subjected to empirical testing in the real world. Experimenters would randomly assign voters to receive varied sequences of direct mail—four pieces on the same policy theme, each making a slightly different case for Obama—and then use ongoing survey calls to isolate the attributes of those whose opinions changed as a result.

        Politicians working on either side of the global warming debate could use tactics like these to figure out what persuades people. But being trained as a scientist, I feel much happier trying to discover the truth than trying to become an expert at persuading people of my point of view whether it be true or false.

        • John Merryman says:

          Professor,
          “I think some people can be persuaded into taking action on environmental problems before those problems hit them in the face.”
          Some of us are like the spring, always reaching out to possibilities. Others are like the fall, contracting back into what is solid and secure, even if it is just our own tiny shell. It is an eternal cycle of expansion and consolidation. We reach for the future, but we only grasp the past.
          The current wave is, in terms of energy resources expanded, the largest human civilization will ever know. Given what it has done to the atmosphere and the planet, that might not be such a bad thing.
          The question then, will be what shape the coming contraction takes. Structures tend to grow wholisitically, but come apart discretely. To the extent society mimics biology, religion is a society’s vision of itself, government is its central nervous system and finance is its circulatory system. Given we have been looking outward for the past few centuries, now that we have to start looking inward, in the coming winter of our times, how can this produce the seeds of the next spring, amid the decay of the fall?
          Government used to be private enterprise. It was called monarchy and the conservatives of the era thought mob rule could never work. Now we are being ruled by titans of finance. Is it wise to have the circulatory system of the economic ecosystem in private hands, or could it be more effectively managed as a public utility? The way democracy works is by pushing power down to the levels it is most responsive, yet still able to maintain some degree of regulation. What if finance could be structured on such a bottom up model? Local community banks serving local needs and forming regional cooperatives. All the while operating with the understanding of currency as a form of public contract and not private property. We own our houses, businesses and cars, but not the roads connecting them and no one cries socialism over that. If we treated money as a medium similar to roads, we would find we needed far less of it, as people would begin to understand value as tangible and not notational.
          Then society can grow as much as its foundation can support.

  3. David Lyon says:

    If “Limits to Growth” is correct, we have already overshot the sustainable population and resource extraction levels of Earth. According to their mathematical models, mean living standards will peak by around 2020. Food production and clean carbon fuel per capita will begin to decline in the 2020′s, leading to an accelerating deterioration of the global environment, followed by a collapse in population. This mathematical model was made in 1972, when there perhaps was still time to become a sigmoidal civilization that gently approaches sustainability rather than a bell-shaped one that overshoots and collapses. However, “Limits to Growth” had a limited impact upon behavior and the political power of professors and scientists seems to have decreased over the past 40 years. Perhaps a follow-up study “Limits to “Limits to Growth”” should be published!

    http://gailtheactuary.files.wordpress.com/2011/10/limits-to-growth-forecast.png

    If we’re within a decade of the beginning of the end and new mathematics takes decades to have an impact, as John Baez says, then I’m worried that this green mathematics push, while noble, is too late to salvage the situation. Perhaps the knowledge of how to create a stable and sustainable civilization can be our legacy to the next flowering, as we have benefited from the Greeks and Romans before us. Ancient Mediterranean civilization seems to have followed a similar pattern of exponential growth followed by exponential collapse.

    http://i451.photobucket.com/albums/qq235/guaporense/LeadPollutionandShipwrecksdefinitiveversion-1.jpg

    The beautiful form of these curves leads one to believe that there is indeed a mathematical framework which can predict the fate of human civilizations using only a few parameters. But would knowledge of network theory, especially if it were discovered after 1 AD, have saved the Roman Empire?

    • John Baez says:

      Thanks for your thoughtful comments, David Lyons. You write:

      Perhaps a follow-up study “Limits to “Limits to Growth”” should be published!

      There have been a number of followups. The most recent book seems to be:

      • Donella Meadows, Dennis Meadows and Jorgen Randers, Limits to Growth: The 30-Year Update, Chelsea Green Publishing Company, 2004.

      There was also a 40-year followup conference in 2012:

      • Videos of Perspectives on Limits to Growth, a conference in 2012.

      More importantly:

      If we’re within a decade of the beginning of the end and new mathematics takes decades to have an impact, as John Baez says, then I’m worried that this green mathematics push, while noble, is too late to salvage the situation.

      I don’t think it makes sense to talk about ‘the beginning of the end’ without specifying what will end. Technological civilization—the electric power grid, computers, and so on—is very unlikely to disappear in less than 50 years, even if some major collapse makes its scope shrink to a few lucky areas. I think there’s just too much momentum for our civilization to suddenly disappear, short of something like an enormous nuclear war or big asteroid impact. So, I’m not very interested in sudden collapse scenarios. If you want a grim but more plausible scenario, try these books:

      • John Michael Greer, The Long Descent: A User’s Guide to the End of the Industrial Age, New Society Publishers, 2008.

      • John Michael Greer, The Ecotechnic Future: Envisioning a Post-Peak World, New Society Publishers, 2009.

      I think these books should be fascinating even for people who utterly disagree with Greer’s predictions. But given your own pessimism about the future of our current civilization, you might wind up agreeing with him.

      I think it’s quite possible that we’re too late for certain things. That’s why the mission statement of Azimuth goes like this:

      While many groups are trying to prevent global warming before it gets worse, there is a powerful array of political and economic forces lined up against them. This makes it easy to fall into apathy. But this apathy is a serious mistake. It comes from a shallow analysis of the situation.

      Suppose, for the sake of argument, that the groups trying to prevent global warming, in fact fail. What next? Will the world come to an end? Will everyone die? Will all species go extinct? The answer is obviously no.

      If the world were certainly coming to an end, we’d be off the hook. We could say: “Nothing I do will really have any effect, so I might as well just relax and enjoy myself”.

      But we are not off the hook. Even if a disaster of some sort is certain, there are different degrees of disaster, and it’s our responsibility to minimize the disaster.

      While this is phrased in a way that focuses on global warming, the same philosophy applies to other problems. Even if it’s ‘too late’ to do some things, we still have a duty to do other things.

      Now, you could agree with this but still argue that developing new mathematics is a bad choice of action! Maybe I should be doing something else, something that pays off sooner.

      Perhaps the knowledge of how to create a stable and sustainable civilization can be our legacy to the next flowering, as we have benefited from the Greeks and Romans before us.

      That would count as a success for me. If indeed our civilization is unsustainable and doomed to crash—let me not try to estimate the chances of this right now!—perhaps it would be good to have some people laying the ground for the next one. But it would also be good to have people working to mitigate the suffering and irreversible damage (loss of cultural and biological information) caused by the collapse of this one. Maybe the second is more important than the first.

      John Michael Greer has suggestions for what to do if you think our current civilization is headed for a slow but permanent decline as resources run out: a “long decline”.

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