There seems to actually be real momentum behind this attempt as reviving Less Wrong. One of the oldest issues on LW has been the lack of content. For this reason, I thought that it might be worthwhile opening a thread where people can suggest how we can expand the scope of what people write about in order for us to have sufficient content.
Does anyone have any ideas about which areas of rationality are underexplored? Please only list one area per comment.
(This is in the same general area as casebash's two suggestions, but I think it's different enough to be worth calling out separately.)
Most of the material on LW is about individual rationality: How can I think more clearly, approximate the truth better, achieve my goals? But an awful lot of what happens in the world is done not by individuals but by groups. Sometimes a single person is solely responsible for the group's aims and decision-making, in which case their individual rationality is what matters, but often not. How can we get better at group rationality?
(Some aspects of this will likely be better explored for commercial gain than for individual rationality, since many businesses have ample resources and strong motivation to spend them if the ROI is good; I bet there are any number of groups out there offering training in brainstorming and project planning, for instance. But I bet there's plenty of underexplored group-rationality memespace.)
Optimizing group norms for effectiveness. Could also be phrased as "team-level rationality."
There are certain group norms (or "cultural practices or attitudes") that are generally good to have in place, irrespective of what the goal of the group is. Many of these are so obvious and natural that almost all human cultures develop them organically. Some of them are more controversial, because they border on politicized topics. Some of them are yet undiscovered.
I would further editorialize that Less Wrong has historically been paralyzed by insinuations of phygishness whenever the topic of optimizing for group norms comes up. I find this annoying. You can't have the results of the (fictional) Bene Gesserit or Mentats, or the Beisutsukai Order, or for that matter the (actual) Navy SEALS, NASA Apollo program, gold medalist Olympic team, or McKinsey-level consulting firm without committing to the idea that you're going to be establishing a novel set of group norms geared toward optimizing some specific purpose.
As a group we're going to find it difficult to obtain extraordinary results if we rely on ordinary cultural technologies.
I upvoted you because I noticed that the term "team level rationality" piquing my interest. Is "team level" or "group rationality" emphasized or taught in follow-on CFAR workshops?
This seems like a potential area of low-hanging fruit where existing "executive team coaching program" content could be adapted. Somebody hypothesized that the growing popularity of local meetups and professional growth sapping LW readership. Group effectiveness content, especially in the context of the world-class teams/names/organizations that you listed, could potentially be immediately implemented in local meetups and in professional capacities.
I don't doubt however that as difficult as it has been for a community to generate individual rationality content, group effectiveness content is even harder to generate due to a perceived smaller set of individuals capable of proven & effective group enhancement, longer timeframes to realize group results and outline group experiments, plus a will and capability to explain said technique progressions.
EDIT: Kahneman's "Thinking, Fast and Slow" has an anecdote about the "leaderless group challenge" as the inspiration for his illusion of validity cognitive bias. The group challenge is an example of the type of activity, often described as "team building exercises", that could be adapted specifically to raise small collective acuity and coordination effectiveness. As far as I'm aware, no widely available content exists specific outside of specific business, military or other domain-specific niches.
Another indirect tangent is the "Checklist Manifesto" by Atul Gawande, drawing on his experience with medical errors (especially in high-performing OR units). Although this a huge step in the right direction, it still doesn't quite get to the root of formulating and internalizing a set of practices specific to enhancing collective effectiveness (even in a small groups).
Non-bayesian reasoning. Seriously, pretty much everything here is about experimentation, conditional probabilities, and logical fallacies, and all of the above are derived from bayesian reasoning. Yes, these things are important, but there's more to science and modeling than learning to deal with uncertainty.
Take a look at the Wikipedia page on the Standard Model of particle physics, and count the number of times uncertainty and bayesian reasoning are mentioned. If your number is greater than zero, then they must have changed the page recently. Bayesian reasoning tells you what to expect given an existing set of beliefs. It doesn't tell you how to develop those underlying beliefs in the first place. For much of physics, that's pretty much squarely in the domain of group theory / symmetry. It's ironic that a group so heavily based on the sciences doesn't mention this at all.
Rationality is about more than empirical studies. It's about developing sensible models of the world. It's about conveying sensible models to people in ways that they'll understand them. It's about convincing people that your model is better than theirs, sometimes without having to do an experiment.
It's not like these things aren't well-studied. It's called math, and it's been studied for thousands of years. Everything on this site focuses on one tiny branch, and there's so much more out there.
Apologies for the rant. This has been bugging me for a while now. I tried to create a thread on this a little while ago and met with the karma limitation. I didn't want to deal with it at the time, and now it's all coming back to me, rage and all.
Also, this discussion topic is suboptimal if your aim is to explore new areas of rationality, as it presumes that all unexplored areas will arise from direct discussion. It should have been paired with the question "How do we discover underexplored areas of rationality?" My answer is to that is to encourage non-rational discussion where people believe, intuitively or otherwise, that it should be possible to make the discussion rational. You're not going to discover the boundaries of rationality by always staying within them. You need to look both outside and inside to see where the boundary might lie, and you need to understand non-rationality if you ever want hope of expanding the boundaries of rationality.
End rant.
Rationality is about more than empirical studies. It's about developing sensible models of the world. It's about conveying sensible models to people in ways that they'll understand them. It's about convincing people that your model is better than theirs, sometimes without having to do an experiment.
Hmm, I'm not sure I understand what you mean. Maybe I'm missing something? Isn't this exactly what Bayesianism is about? Bayesianism is just using laws of probability theory to build an understanding of the world, given all the evidence that we encounter. Of course that's at the core just plain math. E.g., when Albert Einstein thought of relativity, that was an insight without having done any experiment, but it is perfectly in accordance with Bayesianism.
Bayesian probability theory seems to be all we need to find out truths about the universe. In this framework, we can explain stuff like "Occam's Razor" in a formal way, and we can even include Popperian reasoning as a special case (a hypothesis has to condense probability mass on some of the outcomes in order to be useful. If you then receive evidence that would have been very unlikely given the hypothesis, we shift down the hypothesis' probability a lot (=falsification). If we receive confirming evidence that could have been explained just as well by other theories, this only slightly upshifts our probability; see EY's introduction.) But maybe this is not the point that you were trying to make?
I also think that EY is not Bayesian sometimes. He often assigns something 100 per cent probability without any empirical evidence, but because simplicity and beauty of the theory. For example that MWI is correct interpretation of QM. But if you put 0 probability on something (other interpretations), it can't be updated by any evidence.
Hmm, I'm quite confident (not 100%) that he's just assigning a very high probability to it, since it seems to be the way more parsimonious and computationally "shorter" explanation, but of course not 100% :) (see Occam's razor link above for why Bayesians give shorter explanations more a priori credence.)
Regarding Kuhnianism: Maybe it's a good theory of how the social progress of science works, but how does it help me with having more accurate beliefs about the world? I don't know much about it, so would be curious about relevant information! :)
Is there a single book or resource you would recommend for learning how group theory/symmetry can be used to develop theories and models?
I work in fluid dynamics, and I've mainly seen group theory/symmetry mentioned when forming simplifying coordinate transformations. Fluid dynamicists call these "dimensionless parameters" or "similarity variables". I am certain other fields use different terminology.
See my response below to WhySpace on getting started with group theory through category theory. For any space-oriented field, I also recommend looking at the topological definition of a space. Also, for any calculus-heavy field, I recommend meditating on the Method of Lagrange Multipliers if you don't already have a visual grasp of it.
I don't know of any resource that tackles the problem of developing models via group theory. Developing models is a problem of stating and applying analogies, which is a problem in category theory. If you want to understand that better, you can look through the various classifications of functors since the notion of a functor translates pretty accurately to "analogy".
I have no background in fluid dynamics, so please filter everything I say here through your own understanding, and please correct me if I'm wrong somewhere.
I don't think there's any inherent relationship between dimensionless parameters and group theory. The reason being that dimensionless quantities can refer to too many things (i.e., they're not really dimensionless, and different dimensionlessnesses have different properties... or rather they may be dimensionless, but they're not typeless). Consider that the !∘sqrt∘ln of a dimensionless quantity is also technically a dimensionless quantity while also being almost-certainly useless and uninterpretable. I suppose if you can rewrite an equation in terms of dimensionless quantities whose relationships are restricted to have certain properties, then you can treat them like other well-known objects, and you can throw way more math at them.
For example, suppose your "dimensionless" quantity is a scaling parameter such that scale * scale → scale (the product of two scaling operations is equivalent to a single scaling operation). By converting your values to scales, you've gained a new operation to work with due to not having to re-translate your quantities on each successive multiplication: element-wise exponentiation. I'd personally see that as a gateway to applying generating series (because who doesn't love generating series?), but I guess a more mechanics-y application of that would be solving differential equations, which often require exponentiating things.
Any time you have a set of X quantities that can be applied to one another to get another of the X quantities, you have a group of some sort (with some exceptions). That's what's going on with the scaling example (x * x → x), and that's what's not going on with the !∘sqrt∘ln example. The scaling example just happens to be a particularly simple example of a group. You get less trivial examples when you have multiple "dimensionless" quantities that can interact with one another in standard ways. For example, if vector addition, scaling, and dot products are sensible, your vectors can form a Hilbert space, and you can use wonderful things like angles and vector calculus to meaningful effect.
I can probably give a better answer if I know more precisely what you're referring to. Do you have examples of fluid dynamicists simplifying equations and citing group theory as the justification?
Thanks for the detailed reply, sen. I don't follow everything you said, but I'll take a look at your recommendations and see after that.
I can probably give a better answer if I know more precisely what you're referring to. Do you have examples of fluid dynamicists simplifying equations and citing group theory as the justification?
Unfortunately, the subject is rather disjoint. Most fluid dynamicists would have no idea that group theory is relevant. My impression is that some mathematicians have interpreted what fluid dynamicists have done for a long time in terms of group theory, and extended their methods. Fluid dynamicists call the approach "dimensional analysis" if you reduce the number of input parameters or "similarity analysis" if you reduce the number independent variables of a differential equation (more on the latter later)
The goal generally is dimension reduction. For example, if you are to perform a simple factorial experiment with 3 variables and you want to sample 8 different values of each variable, you have 8^3 = 512 samples to make, and that's not even considering doing multiple trials. But, if you can determine a coordinate transformation which reduces those 3 variables to 1, then you only have 8 samples to make.
The Buckingham Pi theorem allows you to determine how many dimensionless variables are needed to fully specify the problem if you start with dimensional quantities. (If everything is dimensionless to begin with, there's no benefit from this technique, but other techniques might have benefit.)
For a long list of examples of the dimensionless quantities, see Wikipedia. The Reynolds number is the most well known of these. (Also, contrary to common understanding, the Reynolds number doesn't really say anything about "how turbulent" a flow is, rather, it would be better thought of as a way to characterize instability of a flow. There are multiple ways to measure "how turbulent" a flow is.)
For a "similarity variable", I'm not sure what the best place to point them out would be. Here's one example, though: If you take the 1D unbounded heat equation and change coordinates to \eta = x / \sqrt{\alpha t} (\alpha is the thermal diffusivity), you'll find the PDE is reduced to an ODE, and solution should be much easier now. The derivation of the reduction to an ODE is not on Wikipedia, but it is very straightforward.
Dimensional analysis is really only taught to engineers working on fluid mechanics and heat transfer. I am continually surprised by how few people are aware of it. It should be part of the undergraduate curriculum for any degree in physics. Statisticians, particularly those who work in experimental design, also should know it. Here's an interesting video of a talk with an application of dimensional analysis to experimental design. As I recall, one of the questions asked after the talk related the approach to Lie groups.
For an engineering viewpoint, I'd recommend Langhaar's book. This book does not discuss similarity variables, however. For something bridging the more mathematical and engineering viewpoints I have one recommendation. I haven't looked at this book, but it's one of the few I could find which discusses both the Buckingham Pi theorem and Lie groups. For something purely on the group theory side, see Olver's book.
Anyhow, I asked about this because I get the impression from some physicists that there's more to applications of group theory to building models than what I've seen.
Consider that the !∘sqrt∘ln of a dimensionless quantity is also technically a dimensionless quantity while also being almost-certainly useless and uninterpretable.
This is an important realization. The Buckingham Pi theorem doesn't tell you which dimensionless variables are "valid" or "useful", just the number of them needed to fully specify the problem. Whether or not a dimensionless number is "valid" or "useful" depends on what you are interested in.
Edit: Fixed some typos.
Regarding the Buckingham Pi Theorem (BPT), I think I can double my recommendation that you try to understand the Method of Lagrange Multipliers (MLM) visually. I'll try to explain in the following paragraph knowing that it won't make much sense on first reading.
For the Method of Lagrange Multipliers, suppose you have some number of equations in n variables. Consider the n-dimensional space containing the set of all solutions to those equations. The set of solutions describes a k-dimensional manifold (meaning the surface of the manifold forms a k-dimensional space), where k depends on the number of independent equations you have. The set of all points perpendicular to this manifold (the null space, or the space of points that, projected onto the manifold, give the zero vector) can be described by an (n-k)-dimensional space. Any (n-k)-dimensional space can be generated (by vector scaling and vector addition) of (n-k) independent vectors. For the Buckingham Pi Theorem, replace each vector with a matrix/group, vector scaling with exponentiation, and vector addition with multiplication. Your Buckingham Pi exponents are Lagrange multipliers, and your Pi groups are Lagrange perpendicular vectors (the gradient/normal vectors of your constraints/dimensions).
I guess in that sense, I can see why people would make the jump to Lie groups. The Pi Groups / basis vectors form the generator of any other vector in that dimensionless space, and they're obviously invertible. Honestly, I haven't spent much time with Lie Groups and Lie Algebra, so I can't tell you why they're useful. If my earlier explanation of dimensionless quantities holds (which, after seeing the Buckingham Pi Theorem, I'm even more convinced that it does), then it has something to do with symmetry with respect to scale, The reason I say "scale" as opposed to any other x * x → x quantity is that the scale kind of dimensionlessness seems to pop up in a lot of dimensionless quantities specific to fluid dynamics, including Reynold's Number.
Sorry, I know that didn't make much sense. I'm pretty sure it will though once you go through the recommendations in my earlier reply.
Regarding Reynold's Number, I suspect you're not going to see the difference between the dimensional and the dimensionless quantities until you try solving that differential equation at the bottom of the page. Try it both with and without converting to dimensionless quantities, and make sure to keep track of the semantics of each term as you go through the process. Here's one that's worked out for the dimensionless case. If you try solving it for the non-dimensionless case, you should see the problem.
It's getting really late. I'll go through your comments on similarity variables in a later reply.
Thanks for the references and your comments. I've learned a lot from this discussion.
Social skills - this skills are incredibly important for actually getting anything done in the real world. The biggest issue I see with discussing this topic is that it will inevitably lead to discussion of PUA. This will force us to either censor the conversation or to have people put off from Less Wrong by this. In particular, it could cause less women to contribute to this site.
I believe that there are many areas where we can use the framework of prediction and calibration to assess expert experience. I think that's true in many domains and can allow us to have verified expertise in those domains in a way that's very different from verifying knowledge with college degrees.
This is more of a practical suggestion than a theoretical one, but what if we had an instant message feature? Some kind of chat box like google hangouts, where we could talk in a more immediate sense to people rather than through comment and reply.
As an addendum, and as a way of helping newer members, maybe we could have some kind of Big/Little program? Nothing fancy, just a list of people who have volunteered to be 'Bigs,' who are willing to jump in and discuss things with newer members.
A 'little' could ask their big questions as they make their way through the literature, and both Bigs and Littles would gain a chance to practice rationality skills pertaining to discussion (controlling one's emotions, being willing to change one's mind, etc.) in real time. I think this would help reinforce these habits.
The LessWrong study hall on Complice is nice, but it's a place to get work done, not to chat or debate or teach.
Like others pointed out, there's a Slack channel administered by Elo, a lesswrong IRC, and a SSC IRC. (I'm sometimes present in the first, but not the other two; I don't know how active they are now.)
As an addendum, and as a way of helping newer members, maybe we could have some kind of Big/Little program? Nothing fancy, just a list of people who have volunteered to be 'Bigs,' who are willing to jump in and discuss things with newer members.
Is the idea here pairing (Alice volunteers as a Big and is matched up with Bob, they swap emails / Hangouts / etc. and have one-on-one conversations about rationality / things that Bob doesn't understand yet) or in-need matching (Alice is the Big on duty at 7pm Eastern time, and Bob shows up in the chat channel to ask questions that Alice answers), or something else?
This also made me think of the possibility of something like "Dear Prudence"; maybe emails about some question that are then responded to in depth, or maybe chat discussions that get recorded and then shared, or so on.
(Somewhat tangential, but there are other things you can overlay on top of online communities in order to mimic some features of normal geographic communities, which seem like they make them more human-friendly but require lots of engagement on the part of individuals that may or may not be forthcoming.)
Thanks for the info - I'll check out some of the chat channels. I had no idea they existed.
As for the idea, I hadn't thought it through quite that far, but I was picturing something along the lines of your second suggestion. Any publicized and easily accessible way of asking questions that doesn't force newer members to post their own topics would be helpful.
I remember back when I was just starting out on LessWrong, and being terrified to ask really stupid questions, especially when everyone else here was talking about graduate level computer science and medicine. Having someone to ask privately would've sped things up considerably.
The debating community doesn't have the goal of arguments being in touch with reality. The only thing that matters is whether a judge will accept the argument.
When it comes to thinking about whether a scientific paper makes an argument that's likely robust, that's quite different.
The skill of not being convinced by persuasive arguments that aren't in touch with reality is valuable.
I think that it would be worthwhile seeing what we can learn from the debating community.
Consider the article Flowsheet Logic and Notecard Logic. I suspect most of the things we would learn would be antipatterns, but it's still useful to have negative examples (especially when those examples are widespread).
A big gap I see is memory. Having read a few books on learning and memory, I think what's been posted on LessWrong has been fragmented and incomplete, and we're in need of a good summary/review of the entire literature. There's a lot of confusion on the subject here too, e.g., this article seems to think spaced repetition and mnemonics are mutually exclusive techniques, but they're not at all. When I used Anki I frequently used mnemonics as well. The article seems to be an argument against bad flash cards, not spaced repetition in general. Probably over a year ago I did start writing a sequence on memory enhancement, but it is a low priority task for me it and do not anticipate completing it any time soon.
The case of ugh fields that subjectively seem useful.
For example, I don't care much about base rates for violence occurring to any defined population subgroup. I think it most useful to me, personally, to 1) consider my own (and sometimes other people's) safety my own responsibility at any given moment, 2) to bite the bullet when it appears that I have misjudged the situation. However, I am not sure if these limits were set 'rationally', or simply because I do not like to think about the subject.
Or to put it another way, "suppose thinking about something is SCARY beyond its actual worst impact on some instrumental goal, is there any practical point where you jury-rig some heuristics so that the goal is okay enough?"
It could be a difficult endeavour but I'd love to see what we can do with what we already have on LW. I don't see any easily-discoverable links to (for example) the Repository repository. Would anyone be kind as to share links to some pages they believe are useful, but are not easily reachable?
Here is a post with a few recommendations in the comments, which seem interesting but I don't really know if the recommendations are still good, or have been superseded by fresher material.
Here is an interesting analysis by Jonah Sinick.
Here there is a collection of lectures about economics.
Here should be more, but I trust that the veterans could fill this in higher numbers and higher quality than I possibly could.
"Repository repository" -- a post listing various "repository" posts, like the "Solved Problems Repository", the "Useful Concepts Repository", the "Mistakes Repository", and the "Good things to have learned" post.
I think life extension should be discussed more here.
Many rationalists disappointment me with respect to life extension. Too many of them seem to recognize that physical conditioning is important, yet very few seem to do the right things. Most rationalists who understand that physical conditioning is important think they should do something, but that something tends to be almost exclusively lifting weights with little to no cardiovascular exercise. (I consider walking to barely qualify as cardiovascular exercise, by the way.) I think both are important, but if you only could do one, I'd pick cardio because it's much easier to improve your cardiovascular capacity that way. (Cardiovascular capacity/VO2max correlates well with longevity, as discussed here.) I'm not alone in the belief that cardio is much more important; similar things have been said for a long time. I'd recommend Ken Cooper's first book for more on this perspective.
The inability for rationalists to regularly do cardiovascular exercise probably stems from similar problems that cause cryocrastination. I'd like to see more on actually implementing cardiovascular exercise routines. I have some notes on this which could help. Off the top of my head I can remember that there's evidence morning runners tend to maintain the habit better and that there's evidence that exercising in a group helps with compliance. I personally find Beeminder to help a little bit, but not much.
It's unclear to me how rationality and life extension are related. Are you thinking about the following, or something different?
Lots of philosophical / cultural effort has been put into accepting the inevitability of death, but this is mistakenly used to accept the nearness of death despite changing technology meaning that's in play. Rationality helps carve out the parts of that which are no longer appropriate.
Life extension is one of the generic instrumental goods, in that whatever specific goals you have, you can probably get more of them with a longer life than a shorter one. This makes it a candidate as a common interest of many causes.
Rationality habits are especially useful in life extension research, because of the deep importance of reasoning from uncertain data; 30-year olds can't quite wait for a 60-year study of intermittent fasting to complete in order to determine whether or not they should do intermittent fasting starting when they are 30.
I have been thinking about all three things. I have strong connections with life extension community and we often discuss such topics.
I am planning to write about how much time you could buy by spending money on life extension, on personal level and on social level. I want to show that fighting aging is underestimated from effective altruistic point of view. I would name it second most effective way to prevent sufferings after x-risks prevention.
I have a feeling that as most EA-people are young they are less interested in fighting aging, as it is remote to them, and they also will survive until Strong AI anyway, which will either kill them or make immortal (or even something better, which we can't guess).
I have a feeling that as most EA-people are young they are less interested in fighting aging, as it is remote to them, and they also will survive until Strong AI anyway, which will either kill them or make immortal (or even something better, which we can't guess).
There's a general point that lots of futurists are the sort of people who would normally be very low time preference (that is, they have a low internal interest rate) but who behave in high time preference ways because of their beliefs about the world, and this causes lots of predictable problems and is not obviously the right way to cash out their beliefs about the world. (For example, consider the joke of 'the Singularity is my retirement plan,' which is not entirely a joke if you expect AI to hit in, say, 2040 but for you to be able to start collecting from an IRA in 2050.)
Maybe the right approach is that it's worth explicitly handling the short, medium, and long time horizons and investing effort along each of those lines. Things like life extension that make more sense in long time horizon worlds are probably still worth investing in, even if there's only a 10-30% chance we actually have that long.
We have posts like http://lesswrong.com/lw/jrt/lifestyle_interventions_to_increase_longevity/
Listening to other people's lived experience especially through your own privilege. You don't get to define words without listening to the people who experience them.
The consequences of our beliefs about status and signalling. For example given how pervasive signaling is in our lives should we optimize our lives for the most status etc.
We know that we care about status. We know that we can't talk to people about that in real life. Should we then make status our motivating terminal value?
See my response below to WhySpace on getting started with group theory through category theory. For any space-oriented field, I also recommend looking at the topological definition of a space. Also, for any calculus-heavy field, I recommend meditating on the Method of Lagrange Multipliers if you don't already have a visual grasp of it.
I don't know of any resource that tackles the problem of developing models via group theory. Developing models is a problem of stating and applying analogies, which is a problem in category theory. If you want to understand that better, you can look through the various classifications of functors since the notion of a functor translates pretty accurately to "analogy".
I have no background in fluid dynamics, so please filter everything I say here through your own understanding, and please correct me if I'm wrong somewhere.
I don't think there's any inherent relationship between dimensionless parameters and group theory. The reason being that dimensionless quantities can refer to too many things (i.e., they're not really dimensionless, and different dimensionlessnesses have different properties... or rather they may be dimensionless, but they're not typeless). Consider that the !∘sqrt∘ln of a dimensionless quantity is also technically a dimensionless quantity while also being almost-certainly useless and uninterpretable. I suppose if you can rewrite an equation in terms of dimensionless quantities whose relationships are restricted to have certain properties, then you can treat them like other well-known objects, and you can throw way more math at them.
For example, suppose your "dimensionless" quantity is a scaling parameter such that scale * scale → scale (the product of two scaling operations is equivalent to a single scaling operation). By converting your values to scales, you've gained a new operation to work with due to not having to re-translate your quantities on each successive multiplication: element-wise exponentiation. I'd personally see that as a gateway to applying generating series (because who doesn't love generating series?), but I guess a more mechanics-y application of that would be solving differential equations, which often require exponentiating things.
Any time you have a set of X quantities that can be applied to one another to get another of the X quantities, you have a group of some sort (with some exceptions). That's what's going on with the scaling example (x * x → x), and that's what's not going on with the !∘sqrt∘ln example. The scaling example just happens to be a particularly simple example of a group. You get less trivial examples when you have multiple "dimensionless" quantities that can interact with one another in standard ways. For example, if vector addition, scaling, and dot products are sensible, your vectors can form a Hilbert space, and you can use wonderful things like angles and vector calculus to meaningful effect.
I can probably give a better answer if I know more precisely what you're referring to. Do you have examples of fluid dynamicists simplifying equations and citing group theory as the justification?
Thanks for the detailed reply, sen. I don't follow everything you said, but I'll take a look at your recommendations and see after that.
Unfortunately, the subject is rather disjoint. Most fluid dynamicists would have no idea that group theory is relevant. My impression is that some mathematicians have interpreted what fluid dynamicists have done for a long time in terms of group theory, and extended their methods. Fluid dynamicists call the approach "dimensional analysis" if you reduce the number of input parameters or "similarity analysis" if you reduce the number independent variables of a differential equation (more on the latter later)
The goal generally is dimension reduction. For example, if you are to perform a simple factorial experiment with 3 variables and you want to sample 8 different values of each variable, you have 8^3 = 512 samples to make, and that's not even considering doing multiple trials. But, if you can determine a coordinate transformation which reduces those 3 variables to 1, then you only have 8 samples to make.
The Buckingham Pi theorem allows you to determine how many dimensionless variables are needed to fully specify the problem if you start with dimensional quantities. (If everything is dimensionless to begin with, there's no benefit from this technique, but other techniques might have benefit.)
For a long list of examples of the dimensionless quantities, see Wikipedia. The Reynolds number is the most well known of these. (Also, contrary to common understanding, the Reynolds number doesn't really say anything about "how turbulent" a flow is, rather, it would be better thought of as a way to characterize instability of a flow. There are multiple ways to measure "how turbulent" a flow is.)
For a "similarity variable", I'm not sure what the best place to point them out would be. Here's one example, though: If you take the 1D unbounded heat equation and change coordinates to \eta = x / \sqrt{\alpha t} (\alpha is the thermal diffusivity), you'll find the PDE is reduced to an ODE, and solution should be much easier now. The derivation of the reduction to an ODE is not on Wikipedia, but it is very straightforward.
Dimensional analysis is really only taught to engineers working on fluid mechanics and heat transfer. I am continually surprised by how few people are aware of it. It should be part of the undergraduate curriculum for any degree in physics. Statisticians, particularly those who work in experimental design, also should know it. Here's an interesting video of a talk with an application of dimensional analysis to experimental design. As I recall, one of the questions asked after the talk related the approach to Lie groups.
For an engineering viewpoint, I'd recommend Langhaar's book. This book does not discuss similarity variables, however. For something bridging the more mathematical and engineering viewpoints I have one recommendation. I haven't looked at this book, but it's one of the few I could find which discusses both the Buckingham Pi theorem and Lie groups. For something purely on the group theory side, see Olver's book.
Anyhow, I asked about this because I get the impression from some physicists that there's more to applications of group theory to building models than what I've seen.
This is an important realization. The Buckingham Pi theorem doesn't tell you which dimensionless variables are "valid" or "useful", just the number of them needed to fully specify the problem. Whether or not a dimensionless number is "valid" or "useful" depends on what you are interested in.
Edit: Fixed some typos.