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I agree. We're very proud of our community ♥

Exactly!

A further thought: the center is an illusion that only emerges because the many possible positions are collapsed into a one-dimensional spectrum. It's not in the ballot results. That means in the actual high-dimensional space all positions are equally extreme (if each question is treated as a binary), because the dimensions aren't related to each other. That means the "center" isn't any better/less dogmatic than the extremes, it's a side effect of how you present the data of ballot results in your experiment. If the extremes are supposedly biased, then it's a bias that is created in the presentation of the result. Some kind of cluster analysis would be necessary instead, to figure out what distribution of positions actually exists. ...In this particular setup. I do think that irl political issues are a lot more interdependent, so this model isn't a perfect analog. Although a simple left-right spectrum is still a simplistic way of viewing politics.

@@gorgenfol That's a very good point, and well articulated. Indeed, once you see each individual position as the result of radical left-right choices, *everyone* is an "extremist" in that sense of the word. So you've pointed out a missing nuance in my analysis, and I appreciate that. The point I'm trying to make is incredibly difficult to convey because we're stuck with all these labels and connotations. The word "extremism" has multiple meanings that collide with each other here. Thanks for pointing that out.

I want to start by saying how much I respect and appreciate your response. Even though you disagree with many points, you still applaud my decision to make & publish the video. That's a rare and refreshing mix of candor and open-mindedness. Really cool. 1. You make a very valid point. The model in the video implies that every position is equally extreme, in a purely technical sense. Purely based on that model, I shouldn't single out the "all blue" and "all red" positions as more extreme than any other position. Thank you for pointing that out. In my defense, I do explicitly acknowledge in the video that it's "possible that this really represents your personal opinion". It's just statistically unlikely, but that's true for any other specific position as well. What I'm learning from this, is that I mangled at least two different meanings of the word "extremist" and I should have been more precise about that. 2. Several other people have made the same remark in the comments: the binary choices are not independent. This is obviously true to some degree, but so far I'm not convinced that the correlation between different choices is huge. I have a hunch that the correlation is much less pronounced than you suggest. For example, I just don't see how my own opinion on gun control is necessarily dictated by my opinion on climate change, or vice versa. There are dozens of other examples I could use. Maybe you're talking about people who base their opinions on identity rather than on the content of the matter. But that makes my main point even more important: Identity is a *horribly* bad motivator for large-scale social change. I'm really curious what you think about this. 3. Your first point 3 is where I instinctively disagree with you the most. "Not every person can make an informed opinion" sounds like you're saying that a large group of people are incapable of thinking freely about social issues that impact their own lives. That's a dangerous idea. The entire point of a democracy is that most people know best what's good for them. Nobody else can make that assessment in their place. Simplistic ideological talking points are not an acceptable substitute for that. Maybe I'm misreading what you're saying, so again I'm interested in your thoughts. 5. You say that you are politically active. I'm willing to bet money that there are plenty of issues on which you disagree with "your" politicians, with other campaigners for the same party, and with the official party line. If I'm right about that, I wouldn't count you as an extremist at all (in the non-technical sense of the word, see point 1). The insult, if there is one, is targeted at people who blindly follow a single party no matter what. Interestingly, I feel exactly the same way about people who blindly follow the "middle" no matter what. In a way, the opposite of "extremism" is not "centrism". Extremism and centrism are very similar. I would argue that the opposite of extremism is pluralism. Or to put it in religious terms, I'm arguing against both monotheistic theocracy *and* against forced atheism. I prefer polytheism instead.

Thank you for your thoughtful comment. I'm not arguing for centrism at all. Instead, I'm arguing that centrism doesn't exist. There is no single label that can accurately cover the huge pile of positions around the middle. This goes for "centrist", "moderate left" and "moderate right". The only labels that do have some merit, in my estimation, are the labels for extreme positions where you can basically predict what someone will say before they say it. Luckily, those extremes are not heavily populated. They are very loud, which is why we need more visibility for the diverse, non-labeled, and definitely not centrist majority "in the middle".

Thank you so much! The goal was to shed a different light on this topic, and it's good to hear that it works for you.

@@AllAnglesMath my friend, I do not say this lightly: it has been years since something positively affected my critical thinking about life issues as much as this video. I've changed my mind on issues plenty over the years, of course, but my general gamut of skepticism hasn't changed much, except being punctuated with big ideas that make me fully reexamine how I think. I'd say the last time I really had to sit and stop and reconsider HOW I think was YEARS ago when I was confronted with Bayes Theorem in medicine and how often the "obvious" result is very wrong by Bayesian reasoning... and very dangerous when human lives are in that margin. I wouldn't call myself politically undereducated either, but this seriously tempered some of my biases. I just showed my (boomer, original hippie) mother both videos, and she had the same reaction of "[vid1] I see what he's saying! Makes sense. Nicely said. [Vid2] ....wow. you weren't kidding. That is stunning. That's decades of wisdom such a succinct, visual way." ........ I would like to add the conversation continued and trailed off with her griping "if people actually would sit down and WATCH this damn thing..." so congrats you're probably gonna end up posted in a Facebook group soon

@@lexinwonderland5741 Wow, your comment has just made my day/week/month. I was aware that this is a controversial topic, and I'm getting quite a bit of pushback in some of the other comments. Thank you so much for sharing this, it means a lot. Can I ask you, when you say you have spent "years teaching" important ideas with a mathematics underpinning, what was your profession? Was it a challenge to reach your students or colleagues with that approach? It's interesting to me from an educational perspective. I'm obviously happy that you want to share this with other people. All the best to your "boomer originally hippie" mom 🌻😉

I like the comparison to PCA, that's a really nice angle that I hadn't considered yet. But I would still maintain that many of these positions are independent of each other. Can you reasonably argue that someone's positions on gun control and climate change are necessarily correlated? Even within a single issue like climate change, you can hold many different combinations of opinions (e.g. being convinced of the problem but not of the proposed solutions; or being convinced that climate change is a hoax but still being in favor of wind energy for geopolitical reasons). I don't think people's opinions are that tightly correlated. I do agree with your comment on the way democracy is currently implemented. No system will ever be perfect of course.

NotQuiteAllAnglesMath

I would never claim to have solutions here. Just shedding some light from a different angle. "Politics is not just a bunch of answers on a questionnaire." True, but I wasn't talking about the way democracy is (currently) implemented. I was making a point about people's opinions, and the very coarse-grained low-dimensional way they are portrayed.

@@AllAnglesMath I'm sorry, I came off as a jerk. I mostly agree with the video. However, I feel like the video implies some nefarious reasons for the existence of the political spectrum when it just seems to be a naturally emerging ordering (Note: every set can be well-ordered so it's not weird that such an ordering exists) of how you would like to be governed. If the video was similar but about political parties instead I would wholly agree. But every party could be ordered on the political spectrum and that ordering will not be useless. I'm also not against more axes to measure nuances.

@@nonoliveleftgreen I appreciate your candor. I agree that the spectrum probably has a more natural origin, like a combination of human nature, clusters of issues, etc. So it's not necessarily true that a bunch of power-hungry politicians got together and decided to divide everyone into brackets. But the thing is: once the spectrum exists, the temptation to start bracketing everyone becomes almost irresistible (both for politicians, media & marketeers, but also for the voters themselves). So it's good to remind ourselves from time to time that this is just a coarse-grained summary, and that there's a *lot* more going on beyond the line.

@@AllAnglesMath Yeah. I agree.

That's a great explanation! Not only because it's correct, but also because it's easy to visualize on the Pascal triangle. Nice!

@@AllAnglesMath I wonder, how is this usually explained?

@@shiftsky7130 You can also explain it in terms of binary choices. Each number in the triangle tells you how many ways you can choose k out of n objects. Think of each object as a bit that you can set to 1 or 0 depending on whether you choose that object or not. When you sum this over an entire row, you get n bits that can all be either 1 or 0, so 2^n in total. But your explanation is way more intuitive and that's why I prefer it.

@@AllAnglesMath That makes so much sense! I must admit, that explanation is closer to the material at hand, and thus provides more insight once someone understands it

Your disappointment stems from high expectations, for which I'm very grateful. No need to apologize for not liking the video. I knew this wouldn't be for everyone, and I decided to publish it anyway. I would love to respond to your criticisms, but I don't know if a back-and-forth discussion in a comment section is the right way to do that. One thing I have to point out though: I'm absolutely not encouraging "mindless centrism". I'm actually doing the opposite. I argue that the "centrist" label is meaningless. Most people are not centrists. The low-dimensional spectrum merely makes it seem that way. I would never call myself a "centrist"; like I said in the video, I actually don't even like that label. The opposite of extremism is *not* centrism or moderation or compromise. Rather, it's plurality/diversity.

@@AllAnglesMath I appreciate and understand the bandwidth dilemma, where you lack time to give quality responses everywhere, but I think you're doing a great job honestly. 👍 I appreciate the time you've taken already. I think maybe there is nuance to the words "centrist" and "moderate" that might be particular to the American political system. I'm not certain. In my view, "moderates" are voters that have moderate view points. These are the people with varied opinions that would follow a sort of Gaussian distribution like you demonstrated. "Centrists" are those that take the middlest position from what they see as two extremes. They group tightly around the midpoint in the same way the other two camps group around the extremes. If you rephrased the video using the term "moderate" rather than saying they land near the "center" or "middle" of the line then I think that fixes what left me misinterpreting your point. I think you're talking about moderates, because moderates don't fall anywhere in the left-right dichotomy, as you illustrated. It's a higher dimensional space of possible viewpoints. I think simply evoking the term "center" or "middle" is what lead me to conclude you were encouraging centrism. (Where people pick the middlest viewpoints.) Moderates have varied opinions and don't answer according to a party line each time. I think this is incredibly important and healthy for a political system. Centrists do not have varied view points because they take whatever position is the apparent average of the two party lines. I think this is incredibly damaging to a political system. If group think is responsible for people voting mostly "left" or mostly "right", centrists would vote mostly right down the middle. Your model requires binary choices, so centrists don't exist in this case, but if each respondant had a continuum of choices, you'd see people cluster far left, far right, and exactly down the middle. My argument is that most people in all three camps suffer from the same group think. I'm glad you're not advocating for people taking the middlest option to spite the extreme positions. 👍 If you make future videos that deal with political issues in another country, it might be worth the time to ask a mathematician from that country for their feedback on how you've presented things. There may be nuance that you've missed.

@@AllAnglesMath I hope the push back from your viewers hasn't gotten you down at all. I think your videos really are awesome. Not every video is going to land how you hoped with your audience so don't let my opinion about how you presented a complicated and nuanced topic get you down. I hope to see many more videos from you going forward. 🙂

@@jneal4154 Wow, I totally appreciate your effort to clarify your thoughts. Your second comment captures what I was trying to convey. Other people have the same response as you did, so perhaps I didn't express everything sufficiently clearly. It's a topic full of labels that come with a lot of baggage. So, to be absolutely clear: I don't like the term "centrism" at all. On one hand, I respect people's right to be genuinely disinterested in politics. I can appreciate that people want to get on with their lives and that they don't feel like taking a stand on anything. But I am not one of those people. I do have very outspoken opinions. It's just that my opinions "cancel" each other after projecting onto the 1D line. That's frustrating in a way, because it means you don't belong anywhere on that line. My conclusion is that the line itself is the problem. I also still want to respond to your first comment, but you've basically already formulated what my response would be, to a large degree. Maybe it was a misunderstanding around terms like "centrist" and "moderate". So instead, I have a question for you. Imagine replacing representative democracy with direct democracy. This eliminates the parties & career politicians. It forces voters to think about each issue individually. It also makes "strategic voting" unnecessary (like your own strategic vote against Trump). In such a system, would your positions on various issues be heavily correlated? In other words: is my assumption of a high-dimensional space correct, or would you see that this space clusters into perhaps 3 or 4 dimensions?

Fair enough. This video is very different from our usual fodder, I admit. But I stand by my decision to publish it. The topic is important enough to try and tackle it from an unusual perspective. And don't worry, we won't become one of the "plethora".

There are definitely examples of issues that correlate very strongly. But I think most of them dont. If you look at polls about people's preferences and their voting behaviors, you will be surprised to see how much diversity there really is. Maybe you can give some specific examples of issues that would be naturally correlated? I'm curious to see if we can pry them apart without having to stretch them too much.

Great point. In countries where the voting system is completely different, polarization is often also less pronounced.

First past the post has been moving steadily *away* from a two party system in the UK since the 1950s. 3rd or "Other" parties, notably Plaid Cymru, Sinn Feinn, the SNP, and even the Liberal Democrats have seen increased vote share until today. In 1951 97 per cent of British voters voted either Conservative or Labour. In 2005 the vote share of the two major parties fell below 75%, a figure only really changed since Brexit. If the local elections and polling are anything to go off, that vote share is going to be closer to 50% than 75%. Great video @AllAnglesMath

Good points, although I do want to point out that my characterization of the gun rights issue is binary only for the sake of this video. If we consider all possible greyscale positions in between, the main point of the video only becomes even more outspoken.

Maybe, more to the point, it's not clear all of these positions originate from a high dimensional space of axioms. Or even worse, maybe they do, but the cost of having to reason through every axiom is high enough that people develop heuristics, the same way we don't check the exact height of every step in a staircase before walking up. It's likely that the space of heuristics is small. So, really what's being asked is do you agree on a smaller set of heuristics which lead to a similar set of positions. It's true that 1 heuristic is unlikely to be enough, but it's not clear that we have too many more than 10 for the day to day issues in our lives. The difficulty is that the number of actual life paths is so large that government needs polices which capture all of them, but our heuristics are only well tuned for a single path where they may fit well (if one is lucky). So, in a representative democracy, this is solved by selecting a representative set of heuristics via a individual representatives of certain groups. Historically, this made sense to do geographically because heuristics tend to be similar, given similar day to day training data. In some sense, now, it's not so clear because so many interactions are long range, and parties have capitalized on this to break apart community blocks into voting blocks like you point out. This in principle can be mitigated by rank choice voting with open primaries, so that the space of heuristics is well represented, and the winner is just the mean position. Similarly the larger the number of representatives, the higher the dimension of the final projection. In any case, what you've identified is what makes government and more broadly society such a difficult organizational problem to solve.

​@@dhuinter Ranked voting seems like a pretty good idea, but I think a better method of optimising heuristic representation would be allowing each voter to choose ("approve") as many candidates as they want. In ranked voting, I think voters would be likely to treat their '1' rank as their only meaningful vote, since it will count the most and, since most people (at least in the American system which seems to be the one you're talking about) would be most familiar with the two """main""" candidates, would end up exploited by the same bifurcations we see today. That's not to say people won't rank candidates, but I don't think they'll rank them based on their own beliefs (most of which they likely cannot be arsed to think to much about) but rather on the endorsed ranking's of "their" political party. If people are not encouraged to 'choose' a candidate and are instead encouraged to consider which are acceptable and which are not, I think the people would be inclined to choose a more diverse set of candidates since they have no other way of expressing their political views (which normally would have been forced to conform to a single campaign's views/policies in order to pick it as a favourite either for a #1 vote in a ranked system or for their only vote in the current system) except by choosing every candidate representing the heuristics they care about. In this system, third (and fourth and fifth and so on) parties would be much easier to choose since doing so does not prevent one from choosing any other better-advertised or more famous candidate. I think being able to choose as many candidates as one pleases would also mitigate the non-voting party since it relieves the pressure of making a vote --- you don't have to pretend like you are especially fond of or at all represented by any particular candidate. But the most important feature of allowing voters to choose as many candidates as they please is that the result will ALWAYS be the candidate approved of by the most people (assuming people use the system properly, which they are more likely to do that other systems, as I have tried to explain).

Nah fam

Let's look at a fictional system that has a good dozen of parties for choice (not a dysfunctional binary one). And lets also assume these parties actually have different opinions on different matters - there are many different topics of discussion in the world, this part is fact, not assumption. Now imagine your figurative hall again where people walk in and look to see under what banner they want to walk. If we follow your logic, even in this simple example, we are still taking a dozen samples from a very high dimensional space, and then project them onto a two-dimensional line. It is still silly.

I suppose you could describe the way each individual plays the political game as another spectrum: On the one end, players are conscientiousness toward their own true political beliefs with the goal of bringing the real world closer in line with their ideal, and on the other, players adopt and mimic the party line with the goal of seeking tribal acceptance. Measuring that distribution might tell us something about the upper limits of democracy, but the first ceiling we're likely to hit is due to the mechanics of our electoral systems. In the US, for example, the possibility of conscientious voters achieving political representation is foreclosed by our first-past-the-post election rules because tribal voting is more likely to achieve plurality than a distribution of each individual's personal preferences. And that fact feeds-back into a voter's analysis whether or not to "settle" for casting their vote for the most closely aligned political party. So is the issue you're pointing out more a problem with individuals or with the system itself?

@@antozon5070 The game here is encoding politics with a vector. The amount of components that vector has describes its dimensionality. At one extreme we can have a 1d vector; {0} is left wing and {1} is right wing. At the other you can have a vector, like in this video, with every single policy preference; a massive column vector {{1},{0}, {0}, ...and on and on you go}. The author of the video argued that you need this full mega-hyperdimensional vector to truly encode politics. But not only is it unwieldy, you have the further issue that volume grows when dimensionality grows. A nightmare. You're pointing out that just taking the dot product and projecting onto a 1d line cannot encode all that data. I accept that. In mathematical terms the true dimensionality of something is its "rank" which is governed by something call "linear independence" in its basis matrix. Can't really explain all that in a YT comment... Put it this way, what about a vector which encodes the identity of the voter {class, gender, ethnicity, income, age, education}? What's the least dimensionality in the input we would need to accurately predict the full policy output? In maths terms we would take this low dimensional vector describing the identity of the voter, multiply it by a matrix (a long series of vectors) which would output a huge vector containing all the voter's policy preferences. Yes, information may be lost (not necessarily though, depends on linear dependence / null space) but regardless, with only a handful of dimensions you can get a predictor of the final policy vector. Meaning you can encode all the information in a succinct form. And this is possible precisely because political preferences aren't arbitrary. If people picked their references randomly like is suggested here then you would need the full mega vector. Sorry, wrote too much...

@davidmurphy563 : That's an interesting hypothesis, and well worth considering. I mean, people clearly are tribal; we all fall for that. But I would propose some counter-evidence. In many multi-party democracies, people tend to move around a lot between (very) different parties. Their votes slosh around from far right to far left and everything in between. This indicates that people are less tribal than we might give them credit for. Also, in many poll results I have seen, voters are always much closer to the center than the parties they voted for. Your argument is that the goal of democracy is to handle struggles between conflicting groups. I would argue that that's the means, not the goal. The goal is to organize and execute tasks that are too big for individuals or small groups to handle. The goal is to protect people's rights. The goal is to run society. Conflict resolution is merely a means to get there. Thanks for the thoughtful comments. Keep them coming!

@@tophat593 I like your explanation a lot. Someone else in the comment section phrased this in terms of PCA (Principal Component Analysis). I think that's what you're also aiming at. Maybe if we performed such a dimensionality reduction on people's political opinions, we would find that identity is the biggest contributor. That would be cool, but it still wouldn't mean that you can reduce people to that single aspect. The other components of a PCA are also really important.

Yes, exactly. And when you take any of those tests, you are positioning yourself in a very high-dimensional space, not a 1D line.

Sigue siendo una ilusión

I respectfully disagree. The fact that democracy is currently broken, doesn't mean that it's fundamentally unworkable. We need to fix it, not throw it away. There's nothing better you could replace it with.

@@AllAnglesMath Autocracy or anarchism, both are way better alternatives to democracy

@@AllAnglesMath I assume, you are from the USA - I conclude that from your assumption of a 2-party system and the colors for left and right. The winner-takes-all voting system that you have over there for the presidential elections strongly favors a 2-party system in the sense that it almost necessarily evolves into a 2-party system. I think, changing the voting system might be one of the most important things that one could do to "fix" the system - maybe something based on ranked choice voting. There's actually some quite interesting math behind different voting systems - but I'm sure, you know this already.

@@andreaspatsalides1914 Nope - autocracy means oppression and anarchy means chaos.

@@MusicEngineeer My bad, I confused autocracy with aristocracy. Anarchy also means no ruler, not necessarily chaos, a very different definition. Both definitions are acceptable, I was not refering to chaos

In that case you could still use the same symbol, and it just means the same thing as the symbol we currently use.

@@AllAnglesMath Yes, I still agree. I'm only wondering what kind of "a thing it just means" to connect a predicate by a logical "or" uncountably many times like in the expression exists x (x > 0) where x "runs through all" real numbers if we cannot even rewrite this expression into the form x1 > 0 or x2 > 0 or x3 > 0 or ....

@@bookofproofs Yeah, it's an interesting question. I don't immediately see a way to break into it, which makes it even more interesting.

Thank you so much!

My parents are responsible for the fact that I exist, so I will pass on the message 😉

I keep getting ads for cat food, even though we don't have pets 🤔

Thank you so much! If you're interested in monoids, we also have a more elaborate video with examples from computer science.

That's an excellent point. In fact, in the upcoming video about the SVD, we do indeed show a diagonal rectangular matrix, and we explicitly call it diagonal. My bad. I should have been more clear; the point was that in this specific video, we're only dealing with square matrices. Thanks for the correction!

No, hemos utilizado nuestra propia biblioteca para la renderización de animaciones. Está escrito en Python. Usamos OpenCV para producir el video final.

Thank you!

There was no way to do it in less time. Thanks for having such confidence in our videos 🤟

@@AllAnglesMath yeah of course when its about this channel the longer the better :)

There's a great story about a teacher who was explaining this to his students. He said: there are languages in the world in which a double negation becomes an even stronger negation (as in "I didn't see nobody"), but then other languages in which a double negation cancels into a positive, just like (-1)*(-1)=1. But, he claimed, there are no languages anywhere on earth in which a double *positive* cancels to become a negation. At that point, one of his students shouted: Yeah, right!