 If you balance a 16-wheeler upright on its end, I guess it would be some sort of semi-stable equilibrium. Sometimes, when I discover a new concept, I begin seeing it in all sorts of interesting places. The checkout line at the supermarket, the user interface for a certain app on my phone. It's amazing how one lens can allow you to understand so much about the world, or at least to think about it in new and helpful ways. For example, in many functions, there's an ultimate maximum value, but there may also be smaller peaks, places where, if you couldn't see enough of the function, you might think we're the max. That's relatively boring, as far as mathematics is concerned. But the concept of the local maximum describes many interesting phenomena, and has been bouncing around in my head recently, coloring how I think about certain things. What kind of things? Part one, algorithms. This is my much-abused, hard-working robot vacuum cleaner. As you can see, he's particularly obsessed with cleaning this very small space under my cheap IKEA chair. Almost every time I run him through my apartment, he'll do a decent job tidying the entryway in the kitchen, then he'll find his way to this spot and just clean it incessantly until I come over and free him. The algorithms he uses to navigate are very simple and usually very efficient at covering the full extent of the space he has to clean, but if you feed this specific geometry into them, they just bounce around inside it aimlessly for hours. The space under the IKEA chair is a local maximum for the robot's navigation algorithms. The ultimate goal of the algorithms is to vacuum as much floor space as possible, and they're normally fantastic at doing that, but when they reach this location, every other direction the robot might go seems like a losing proposition compared to this tiny section of carpet. Local maximum are an issue for any software designed to maximize some variable, collectively called hill climbing algorithms because they climb the hill of a function to arrive at its peak. If you imagine a program trying to crawl along this function to get to the highest possible value, depending on how far it's willing to explore a downward trend, it may get stuck on this little hill here, failing to recognize the much larger mountain over there. Local maximum are a real problem for these sorts of search algorithms, which are used in many different fields, software, finance, anywhere that finding the optimum or largest value of a function is useful. Speaking of which, Part 2, Local Maxima in Science. Local Maxima aren't restricted to the simulations that scientists sometimes use to model the universe. The same phenomenon can be observed in the universe itself. We've discussed the most well-known example of this in Episode 85, Evolution. There are many simple mechanisms which would be massively advantageous for any organism which managed to evolve them. Mechanisms which appear nowhere in the incredibly diverse landscape of life on this planet. The reason we don't see animals zipping around on wheels instead of legs, or plants that harvest different wavelengths of light with more energy is because evolution is a sort of hill climbing algorithm, one which has identified local maxima of survival and replication and doesn't allow wide departures from those maxima. There's no incomplete version of a wheel which is better for locomotion than a leg, even a leg that isn't very good. Any organism that developed some mutation in the direction of a wheel would suck compared to its legged brethren. It'd get hunted down faster by predators. It'd be less adept at chasing down prey or searching for sustenance. It'd likely be selected for extinction in a hot second, forced out of the game by other non-wheeled organisms, never having the time to really develop that wheel thing enough to be competitive. Another way to think about local maxima is in the context of energy and equilibria. If you flip this graph upside down, we can imagine it as a series of valleys of different depths. If you drop a ball into some terrain that looks like this, it'll roll around and eventually come to rest. It might drop to the lowest point, but it might also come to rest in this little divot. This is a semi-stable equilibrium point. If you nudge the ball a little in either direction, it'll return to this low spot. If you were to push it hard enough, however, it would find its way out of the pocket and continue on down to the lowest energy state. Hill climbing, valley rolling, local max, local min, tomato, tomata. When you start talking about energy states and equilibria, you really throw the door open to a very broad application of this idea. Physics, chemistry, just about every field has an example of a semi-stable equilibrium that has the potential to become a runaway reaction, given a hard enough push. If you think about it, everything that's flammable sort of exists in a local minimum energy state. And if you're ever brave enough to check out the Wikipedia page for false vacuum, maybe the whole universe is. It's not always a bad thing. Part 3, local maxima in culture. Cumulative cultural evolution theory posits a ratchet-like hill climbing process of culture where good ideas are communicated through generations and honed over time, becoming more and more useful, a stochastic process with selection for adaptive traits. It's not the craziest theory I've heard. The results of cultural experimentation are passed on through tradition, transmitting centuries of trial and error in a body of custom that often goes unquestioned, sometimes persisting for long periods of time without any clear understanding of how our particular custom works, or even what it's for. Why do you hold your breath to cure hiccups? Because it raises the CO2 concentration of your blood, triggering an asphyxiation reaction which circumvents the normal diaphragm control system. We didn't even know what carbon dioxide was before the 1700s, and you probably didn't know the specific reason that holding your breath helps with hiccups, but you still do it, as your ancestors have for millennia. If you're a cook, you probably use a dozen such tidbits of cultural know-how every time you step into the kitchen without even thinking, from how to hold a knife to salting pasta water. Do it wrong, and you're apt to get an earful from whoever's nearby about the proper way to do things. As with our other examples, cultures are prone to local maxima. Traditions and memes that have long outlived their usefulness, but continue to reward compliance and punish deviation. Just about everyone agrees that it would be better for the US to transition to the metric system, yet doing so would cost some very rich companies a lot of money and annoy a lot of crotchety people. So, here we are. We can imagine how amazing it would be for the whole world to use the same standard units and fasteners and everything else. We can envision how simple and wonderful it would be, but to get there, it would have to get worse before it got better, so it's not going to happen. Part four, local maxima in human psychology. Some say that those who don't study history are doomed to repeat it, but the reality of the situation is actually much less rosy. Even those who study history manage to screw up in very much the same ways as their predecessors. As well and good to be a free thinker, we need people to question the prevailing cultural wisdom, regardless of the stigma associated with being a deviant. But trying to work everything out from first principles isn't just a question of reinventing the wheel. It incurs the risk of falling into well-trodden local maxima of human psychology. Self-reinforcing ideologies that are shaped in a way attractive to the human psyche that we've only managed to escape with a great deal of collective hard work and careful consideration. That is, through the hill climbing mechanism of culture. Reject the prevailing cultural wisdom and, well, flat earth conspiracy theories have gained quite a following in recent years and have been roundly mocked for holding backwards beliefs. But reading the flatter society's wiki, it seems that those who subscribe to the idea see themselves as free-thinking skeptics, valuing experimental evidence and careful considerations of facts that are available, refusing to accept the unthinking cultural dogma of around earth when they clearly see evidence to the contrary. Racist and sexist retrace the same longest-proven arguments about supposedly fundamental differences between types of humans, convinced that they're just asking hard questions that are too spicy for today's easily offended PC culture, even though those ideas lost scientific credibility decades ago. Anti-vaxxers refuse to accept the prevailing medical consensus and raise the alarm that vaccines may cause autism again and again, demanding that someone in the scientific community rigorously investigate the effects of so many vaccinations on children. You know, like the hundreds of studies which have found no such link, but you know, this time for real. The trick is, these are self-reinforcing paradigms. They provide seemingly intuitive explanations for phenomena and discount more commonplace explanations as myths for unthinking sheeple. When you've found a whole cluster of beliefs that seem to reinforce and validate each other, alternate explanations will automatically fail in comparison because they can't address every facet of the paradigm simultaneously. If you're trying to convince someone that the earth is round, merely showing them a video of a rocket launch isn't sufficient. They have defensible reasons to doubt that evidence, lenses distort images, and it doesn't explain any of the other reasons that they believe. The deviation in flight path of Flight UA 270, the result of the Bedford level experiment, or their subjective experience of a seemingly flat planet. To overcome the inertia of a local maximum, you have to explain all these phenomena better than the flat earth paradigm all at once. And to be honest, people don't usually afford the attention span necessary to get pushed towards a higher global maximum. If you refuse to trust the wisdom of the crowd pulling you that direction, it's technically possible you'll find something even higher. Maybe your hill climbing algorithm really is better than cultures. Maybe trying to reason from first principles will shake you loose at the errors of your ancestors. Or maybe you'll get stranded on a local maximum that culture has tried before and surpassed, vacuuming over the same square of carpet over and over. At least he seems happy. Both culture and individuals can get stuck in local maxima and there's no real way of knowing if we've found our way to the truth or just up. What examples of local maxima can you think of? Please leave a comment below and let me know what you think. And speaking of maxima, 20,000 subscribers. Yup. I'm planning on doing some sort of Google hangout session where we can do Q&A or just talk for a while. So if you're interested in that, keep an eye on this channel and on Twitter, Facebook, Instagram. Thank you very much for watching. Don't forget to subscribe while I share. And don't stop thunking.