Which fields of learning have clarified your thinking? How and why?

16knb

12antigonus

9VincentYu

0vi21maobk9vp

7atorm

0beriukay

7shminux

0Arkanj3l

5JimL

1amcknight

4Cthulhoo

3Manfred

2jsteinhardt

2r_claypool

0Arkanj3l

0r_claypool

1fortyeridania

1vi21maobk9vp

1DanPeverley

0dbaupp

0DanPeverley

1Bo102010

1TimS

3daenerys

1Jayson_Virissimo

-2Arkanj3l

0TimS

0jsteinhardt

1TimS

-1lessdazed

0dbaupp

0lessdazed

0mwengler

0Suryc11

0RomeoStevens

0JoshuaZ

0[anonymous]

0vi21maobk9vp

4[anonymous]

2vi21maobk9vp

-1komponisto

2[anonymous]

4komponisto

0[anonymous]

5komponisto

1[anonymous]

0komponisto

0[anonymous]

0JoshuaZ

-2vi21maobk9vp

3JoshuaZ

0vi21maobk9vp

2[anonymous]

0vi21maobk9vp

0JoshuaZ

0komponisto

0[anonymous]

-1JoshuaZ

1vi21maobk9vp

0JoshuaZ

3Jack

0JoshuaZ

1vi21maobk9vp

-3AIDK

New Comment

Economics. I took a lot of econ, and loved it. It really is a powerful framework for understanding the world. Bastiat especially has a lot to teach aspiring rationalists. I would encourage anyone to study programming, even if it's just at the introductory level. Understanding what it means to explain things to a computer is just a great exercise in precise thinking.

Philosophy courses did, seminar-style analytic philosophy classes in particular. (I wouldn't say that history of philosophy classes altered the way I thought, though I can totally see how Hume might be shocking to someone very new to the subject.) Aside from the actual content I learned, I got the following out of them:

- The mental habit of condensing complicated lines of reasoning into minimal, fairly linear syllogisms, so that all of the logical dependencies and likely points of failure among the premises/inferences become much more obvious.
- Relatedly, an eagerness to search for ambiguities in arguments and to enumerate all their possible disambiguations, with an eye for the most charitable/defensible contenders.
- An appreciation for fine distinctions underlying seemingly straightforward concepts. (E.g., there are several related but distinct concepts that map onto the notion of a word or sentence's meaning.) These often have unexpected implications and/or vitiate seemingly plausible inferences.
- Not being allowed, on pain of embarrassment or a bad grade, to get away with BSing or relying on unacknowledged, controversial assumptions. You have to be up-front about precisely what you mean and what's at stake.
- Realization of the extreme rarity of knock-down arguments for any view, and the subsequent adjustment to the fact that assessing pretty much every philosophical question involves a robust trade-off of good and bad consequences. Sometimes every view on the table seems to imply something crazy, and you have to learn to accept that. And to accept that sometimes reality is crazy. (Yes, I know that the map is not the territory, etc.)
- If arguments are soldiers, then at least learning to let some of your soldiers die - and sometimes even putting them out of their misery, yourself! It's very common in philosophical writing to go through all of the failed arguments for your view before moving on to the ones you find more promising. Even then, it's expected that you highlight their most vulnerable spots.
- Epistemic humility. I'm much slower to draw hasty conclusions with high certainty on a given topic before I find out the best of what all sides have to offer. I definitely still form fast and intuitive judgments before investigating disputed subjects deeply, but I don't pretend that they're likely to be the last word or even novel contributions lacking high-level criticism.
- A much richer sense of the space of philosophical views. But then again, probably something analogous holds for most other disciplines (biologists are presumably better-tuned to the space of biological hypotheses). Still, though, philosophical-view-space intersects an unusually large number of things.

I don't know if you're in need of any of these things, or if you're likely to acquire them through a small handful of philosophy classes. Even if you are, whether or not you'd succeed greatly depends on the quality of your teachers and classmates.

The most useful course that I've taken so far was on game theory. It explains a lot of human behavior, and is applicable to real life. It can also be a soft introduction to mathematical proofs, and helps with some Less Wrong material. (Caveat: progress was slow because it attracted many non-mathematically inclined students.)

Brief comments about math and physics courses:

- Math (real analysis, linear algebra, number theory, combinatorics, abstract algebra): Learning each field of math greatly improved my abilities in other math fields and in physics, but they didn't help much with non-academic problems. (Statistics and probability theory may be more helpful, but I've only studied these in high school, so I can't comment.) However, learning real analysis was an excellent way to reveal the limits and faults of my intuition. (I was initially taught real analysis with the Moore method over two summers, which made this even more effective. Also, there are books devoted to counterexamples in real analysis.)
- Physics (mechanics, quantum theory, electricity/magnetism, astronomy): Compared to math, I found the type of thinking in physics to be more practical. It may also serve as an entry point to topics in math and programming. However, be careful of these contents in physics courses - I've seen a lot of unjustified hand waving about math and bad programming practices.

It may be a good thing that many non-mathematically inclined students were attracted - such an audience can slow down progressing to mathematically deeper topics and make lecturer spend time on the "physics vs. mathematics division" side of things.

I consider game theory and probability theory two topics that offer a lot of possibilities for point of view change; and in game theory the most important part not to miss is thinking about real utility functions...

Current sanity waterline is low enough that writing down the incentives can explain things that (somehow) are not yet considered universally obvious.

So, this doesn't have much to do with clearer thinking, but I strongly suggest taking a basic biology course that focuses on human anatomy and physiology, and a basic physics course that focuses on kinematics. LessWrong puts a lot of weight on the ability to think critically, but having a base of knowledge is also important. I'm always embarrassed by alumni from my alma mater who don't know how their own bodies work, or understand how energy and momentum are conserved.

I agree with the bio and physics. People criticize lab assignments for being boring and unscientific (largely the case only with a bad TA and bad methodology wherein it is partially YOUR fault if you "disprove" gravity in your lab). However, if you flip the thinking a bit, it is boring *because* you understand the working pieces of it. You learn, in a visceral way, that science is a thing. **A thing that you can do!** I remember our lab (with guidance, obviously) group derived the procedures for counting the population of a bunch of unicellular organisms just by thinking about the problems and the possible solutions. My experiences with TAs may be unique, but I think if you come at them the right way, there is a lot of value to be had.

A couple of points related to programming.

Learning it in high school was a deeply humbling experience for me, given how my repeated sureness of what my program was doing mismatched that of a computer actually running it. It basically taught me the importance of doubting, double checking and testing even the stuff unrelated to programming.

When I was tutoring physics years later, I would often tell the students: "if you think you understand a concept, you should be able to explain it to a computer in a programming language of your choice." This works better for simple kinematics than for solving Maxwell equations (too many tiny but essential details to take care of in the latter case), but the idea is the same: if you are simply guessing the teacher's password, you will get stuck pretty quickly.

On a different note, a short workshop on one of the variants of TRIZ was quite helpful (the classic TRIZ wasn't).

Game theory and...

Keep a journal. All your life. It is easy to harbor irrational thoughts and emotions in life. It is much more difficult when you must write them down. The exercise has multiple benefits. One learns to summarize, it clarifies thoughts and emotions, and provides cathartic relief. A healthy mind is one that can write; irrationality is a hundred times more difficult in print. Pity that today's journalists are in general such bad examples.

Do you have any suggestions about how to do this? How much to write, what kinds of things to write about, or anything like that? My guess is it's probably best to just try it and figure out what you like and what works, but maybe a bit of direction can help too.

I have a Ph.D. in physics, and, needless to say, during the long and painful process of achieving it, I've definitely become more rational (in the broader sense) and a clearer thinker. My achievements may sound very basics to LW people, but you can consider them like the foundations of rationality, I guess.

- Calculus and Algebra taught me how to clearly apply the hypothesis->proof->thesis process. It might look a not so great achievement, but it is indeed, since it reflects in how I treat everyday issues (it's like the first step toward rationality in some sense)
- Programming courses taught me to break down complex processes in simpler atomic steps.
- Lab courses have taught me how to treat the data. Again this skill has been since then applied on a day by day basis
- General Relativity and Quantum Mechanics taught me that reality in not what it intuitively appears to be. This is a very important teaching: don't always trust your intuition, even if you really
*feel*that you're right. - Doing research activity, finally, taught me to calibrate my judgement (I used to be more often wrong than I expected), collect and elaborate data by myself and that apparently all the referees of Physical Review love to nitpick.

Physics for me, of course. The upper-level physics labs (where you get to choose your own projects, start playing with NMR and cosmic-ray detectors, and are expected to display actual statistical rigor) were particularly valuable. Though of course you need most of the physics before and a maybe a statistics class to really appreciate them.

Computer science (not necessarily programming) teaches you to think about the world rigorously and gives you an idea of what you can and can't do and how to do the things you can. I'd recommend topics like theory of computation (computability/complexity), algorithms, randomness/randomized algorithms, plus cryptography / quantum computing if you have time. The lectures at http://www.scottaaronson.com/democritus/ cover a good collection of topics, although it may be useful to have a more in-depth understanding than just the lectures (certainly you should at least solve the problems he give).

Mathematics makes you better at rigorous thinking, problem solving, and developing intuition. The same goes for theory of computation / algorithms above, but math classes can be a good addition. At least for a while, the classes also provide you with some really good tools (they get increasingly divorced from reality after some point, usually at the graduate level). I'd recommend taking at least one class in each of combinatorics, analysis, algebra, and topology.

You said things that improve your general thinking, and for those I would give the above. Note that I'm not concentrating on usefulness for other things (like more direct usefulness) at all, although a lot of those classes are also generally useful. But here are some other topics that are just really really useful:

- LINEAR ALGEBRA -- this is actually just the most used set of tools in almost all technical fields
- statistics -- not crap about T-tests, although that is moderately useful to know; I mean how to manipulate/analyze randomness, statistical modeling, and probability theory (things like MIT's 18.440, 6.041, and 6.856, David Blei's Bayesian Nonparametrics class, and Martin Wainwright's high-dimensional statistics class; sorry, the last one doesn't have lecture notes, although the last two are also a little specific / advanced)
- statistical machine learning

I think I began to think much more clearly after I started reading about microeconomics. Opportunity cost in particular was an eye-opener.

I think part of it has been the content (opportunity cost especially), but another part has been the presentation. The economics writing I've read (from a couple textbooks, several popular books, and online articles/blogs) has generally done a good job of separating positive analysis and normative recommendations.

Programming was mentioned here, but surprisingly enough some aspects of system administration provided some insights that were useful to me later when studying mathematical logic. In general, even without all the AI thing, computers seem to have allowed people to repeat all the philosophical concepts in a new context..

The 20-th century foundation-of-reasoning discussion of proofs as objects (and how proofs depend on the set of the natural numbers..) can be nicely illustrated with chroot management; the old Aristotle idea of "first substance" of every object (unique but devoid of properties useful to use) is easily translated as DB id field.

For me, one of the most enlightening experiences I had in high school was learning how to play black-jack in the weeks after the AP Calculus Test. With the test done, our awesome teacher taught us all how to count cards and set up tables for us to play, and had us all keep track of our "winnings". Everyday, some students would play control, using a chart of statistically optimal moves to decide whether to hit, stand, split, or double down. I learned in a very intuitive way that making the correct decisions doesn't always lead to good results, and more importantly that that didn't invalidate the correct decisions. Control group pretty much always won overall, even if some kids got ahead in the short term it was better to play the way the math said to. We all enjoyed the experience, and learned some gambling and life skills while we were at it. She was definitely my favorite math teacher :)

Physics and recreational mathematics + computer science improved my mental abilities.

I took Physics for Engineers as a freshman in college. It's clear in retrospect that this class was designed to accomplish several things:

Force students who breezed through high school with little effort to work hard to maintain a good grade.

Weed out the students who don't have the intellectual firepower or stamina for engineering.

Teach a particular problem solving methodology. To get decent marks on a problem set, you had to

*always*draw a diagram,*always*start with appropriate equations,*always*derive the answer correctly (no numbers, just algebra + calculus), line by line.Work quickly and accurately. Tests and problem sets were always difficult enough to require the full amount of time allotted.

Teach physics. Only this goal was ever mentioned explicitly, of course.

This was a difficult class to do well in, and probably the class most responsible for people leaving the engineering program. The default schedule also had it co-requisite with Calculus II, another demanding class, and the two classes used each others' concepts - you'd have to understand one to do the other.

Going through this taught me all the intended lessons: I went through high school barely studying, but had to give up sleep to study and understand every homework problem to do well in this class. I wasn't sure I had an A in the course until after I got the final back.

Internalizing the lessons of that course set me up for success in later courses, and now professionally. Physics courses aren't about physics.

Project Euler is the other thing I think has changed my thought processes for the better. Being able to think of how to express an algorithm succinctly and correctly seems to help out in various situations, like training a new employee how to do a complex technical task.

To give better advice, it would be helpful to know what you are currently aiming to focus on. I don't have good advice if you plan on going into a STEM field.

But if you aren't going that route, I highly recommend taking enough statistics courses to get a good grip on the technical points of correlation, causation, power, population distribution, and regression.

Slightly more controversially, I think that a Philosophy of Science course would be helpful for you to acquire your own concept of what Science is and what purpose it serves. (If it doesn't cover Kuhn, then it isn't what I'm talking about, and ideally it would discuss something from Feyerabend).

And I think that some basis in On the Genealogy of Morals is essential to any discussion comparing value systems, but if the professor is not sympathetic to Nietzsche, then you might not get much value from it. In fact, if you have no exposure to Greek moral theory (esp. Aristotle), then I'm not sure that you will find Nietzsche interesting. And he writes in playful hyperbole, which means teasing out his meaning is a lot harder than understanding straightforward assertions.

Upvoted, and ditto.

One of the best courses I took in terms of taking a step back and looking at how science and knowledge actually work was a history of science since the enlightenment.

We read Kuhn's Structure of Scientific Revolutions and studied Popper. Learned about phlogiston theory, and the development of gas theory. We studied Darwin, as examined by his contemporaries, by reading the essays other scientists wrote about him and his works. Back then the big debate wasn't that evolution was "atheist" but rather on Darwin's *methodology* (deductivism v inductivism, basically)

tl:dr- History and Philosophy of Science is extremely worthwhile.

Besides Formal Logic and Statistics, Philosophy of Science was *the* best class I took during my time at university.

The program I got accepted into is proprietary to the university, and although somewhat STEM, it's as un-STEM as you can be while still being tangentially related.

In other words, a designer. This is the program overview from the faculty. In addition to the courses they mention here, I *have* to supplement with a STEM subject background, so doing something like statistics or computer programming is not out of line. Also, given that this is first year university, I get to sprawl over subjects outside of the faculty.

My fantasy was to do something like a Comprehensive Anticipatory Design Scientist, but that isn't necessarily a *goal*, per se, or an ends; more of a lifestyle of learning and finding the best ways to apply that learning to help people.

I have non-mathematically inclined tendencies (*note: that needs to be tabooed*), which is a source of shame for me. It severely limits my options. I've been taking up numerous nootropics and finding other solutions to compensate. But... I don't know. Is there a "math-sense" that you either have, or have not? Does anyone have an inspiring math success story?

If it's math-sense that you seek, take statistics courses until you think you could explain statistics to literature majors at a cocktail party. That's vastly more math-sense than most people have.

Also, classes in game theory (aka decision theory) could develop a different kind of quantitative thinking. But I don't think that is what most people mean when they say "Math-sense"

take statistics courses until you think you could explain statistics to literature majors at a cocktail party.

Is that supposed to be a lot or a little statistics?

The meme that liberal arts majors are almost always terrible at mathematics is incredibly dangerous to raising the sanity line. The meme is part of the same cluster of ideas as "Liking/Being good at math is weird and not for normal people." Sorry if I'm overreacting to the joke, but I really believe the meme is that dangerous.

If you weren't joking, sorry for misinterpreting you. The answer to your question, you want enough statistics training that you can deduce the really basic concepts (population, sample, null hypothesis) by yourself. Depending on the person, the focus of the professor, and the quality of the class, that could mean one really good class or a year long sequence. As I said above, your goal is to be able to explain a concept like regression to an interested lay audience.

I thought about this before posting: Evolutionary Psychology. I don't know that there are courses with that title, but look in bio and psych departments for offerings. Primate and Mammal psychology would also be valuable. Understanding things in evolutionary psychology have significantly enhanced my introspective attempts to become more accurate in understanding the world.

(my first thought was physics, 2nd philosophy of ethics where as a 17 year old I learned that there IS no "right" answer, that the best human minds ever are just giving their opinion.)

As a current college student, this post (and comments) is particularly relevant to me, so thank you.

However, what is the general consensus regarding studying (as far as undergrad. majoring in a field) government or political science as far as improving rationality? Recommended? Or not? And why?

[This comment is no longer endorsed by its author]

Math and computer science are the obvious choices. I'll second intro to microecon. It will dissolve a lot of confused thinking wrt psychology and interaction (transaction).

Programming has already been mentioned, but I'd like to note that different sorts of programming languages teach different things. Scheme-like languages cause a very different sort of thinking than C-like languages for example. That said, I think that shminux touched on the biggest lesson from programming. But it does produce other lessons like how to break tasks down into smaller, more manageable tasks, and how to investigate things that aren't doing what they are supposed to do.

So now onto other areas:

Psychology and cognitive science. Learned a lot about heuristics, fallacies and biases. Learned also that they apply to me (although I'm not sure I've internalized that as much as I should). Also, learned the important lesson that a good way of understanding complex systems is by how/when they go wrong.

Set theory: Did a really good job of showing how even reasonable sounding premises can lead to contradictions quite quickly. (If there's any general reason to not take Anselm-like arguments seriously it is this, aside from the specific issues with most of those sorts of arguments). Set theory also helps one see mathematics as a whole and see how different areas connect to each other. Also, big sets are *big*. Although I enjoyed math well before I studied set-theory, I first had the feeling of the numinous in a mathematical context when thinking about the cardinality of sets that can be made in ZFC. Almost embarrassingly, the first sets that really triggered this were sets produced simply using the axiom of substitution ( (N u P(N) u P(P(N)) u P(P(P(N))))....) (where N is the natural numbers and P is the powerset operaton) started triggering this feeling). After seeing all sorts of large cardinals, this now looks almost like a little child feeling awed from thinking about one thousand. Also, I used to be religious, and I suspect that one thing that helped become less religious was the fact that I found far more of the numinous in math and science than I did in religion.

Probability and combinatorics did a really good job teaching me how bad human intuition is about basic probability.

Astronomy taught me how just mind-bogglingly large the universe is. (Cue obvious Hitchhiker's references.)

Set theory also helps one see mathematics as a whole and see how different areas connect to each other.

For what it's worth, I strongly disagree. For a new student too much emphasis on foundations can be a major mental block when getting used to a new idea and especially a new circle of ideas. Set theory is used very informally in most of mathematics, as a notation. To learn more than this notation is mostly unnecessary for pure math, completely unnecessary for applications of math to other areas.

I first had the feeling of the numinous in a mathematical context

This kind of sentiment always reminds me of this.

Set theory also helps one see mathematics as a whole and see how different areas connect to each other.

For what it's worth, I strongly disagree. For a new student too much emphasis on foundations can be a major mental block when getting used to a new idea and especially a new circle of ideas. Set theory is used very informally in most of mathematics, as a notation. To learn more than this notation is mostly unnecessary for pure math, completely unnecessary for applications of math to other areas.

Well, the difference of mathematics from natural sciences that you need not only to build a good model of something, but also to describe it using a limited set of axioms (using proofs from there on).

For some people set theory is the area of mathematics which quickly reaches proofs that are accessible to our reasoning, but transcend our intuition. Sometimes even the notions used are easy to define formally but put strain on your imagination. For people inclined to mathematics this can be a powerful experience.

But that's nothing special about set theory. I prefer to think that the role of mathematics (at least the best kinds of mathematics) is to correct and extend our intuition, not to "transcend" it. But the kind of powerful experiences you describe were available two thousand years before the invention of set theory, and they're available all over modern math in areas that have nothing to do with set theory.

Not every student would benefit from learning set theory early beyond the universally needed understanding of injective/bijective mappings, but some would. It does depend on personality. It has some relation to cultural things.

"Role of mathematics" implies relatively long run; experience is felt in a very short run.

If you want to extend your intuition into an area nobody understands well, you often need to combine quite weak analogies and formal methods - because you need to do something to get any useful intuition.

There are many branches of science where you can get an amplified feeling of understanding somthing in the area where you don't have working intuition. There are three culture-related questions, though. First, how much (true or fake) understanding of facts you get from the culture before you know the truth? Second, how much do you need to learn before you can understand a result surprising to you? Third, is it customary to show the easiest-to-understand surprising result early in the course?

Of course, for different people in different cultural environments different areas of maths or natural sciences will be best. But it does seem that for some people the easiest way to get an example of reasoning in intuitively incomprehensible (yet) area is to learn set theory from easily accessible sources.

The construction of (other parts of) mathematics from set theory is a very important lesson in reductionism.

So important, in my view, that it outweighs the disadvantages of set theory that you often hear people complaining about.

I don't agree. Math is not made out of sets in the same way that matter is made out of atoms. In terms of reductionism differential equations are *more fundamental* than sets.

In terms of reductionism differential equations are more fundamental than sets.

Would you care to give an argument for this? This strikes me as wildly implausible, and my default interpretation is as a rhetorical statement to the effect of "boo set theory!!"

I've *never* seen set theory reduced to differential equations. On the other hand, the reduction of analysis (including differential equations) to set theory is standard and classical.

There are a lot of phenomena -- in mathematics, in the cosmos, and in everyday experience -- that you cannot understand without knowing something about differential equations. There are hardly any phenomena that you can't understand without knowing the difference between a cardinal and an ordinal number. That's all I mean by "fundamental."

But here is a joke answer that I think illustrates something. Differential equations govern most of our everyday experiences, including the experience of writing out the axioms for set theory and deducing theorems from them. And we can model differential equations in a first order theory of real numbers, which requires no set theory. A somewhat more serious point along these lines is made in some famous papers by Pour-El and Richards.

Is this a good way to think about set theory? Of course not. But likewise, the standard reduction to set theory does not illuminate differential equations. Boo set theory!

Like I suspected, this is rife with confusion-of-levels.

There are a lot of phenomena -- in mathematics, in the cosmos, and in everyday experience -- that you cannot understand without knowing something about differential equations. There are hardly any phenomena that you can't understand without knowing the difference between a cardinal and an ordinal number. That's all I mean by "fundamental."

That's like saying that you can get through life without knowing about atoms more easily than you can without knowing about animals, and so biology must be more fundamental than physics. Completely the wrong sense of the word "fundamental".

Differential equations govern most of our everyday experiences, including the experience of writing out the axioms for set theory and deducing theorems from them.

This is a classic confusion of levels. It's the same mistake Eliezer makes when he allows himself to talk about "seeing" cardinal numbers, and when people say that special relativity disproves Euclidean geometry, or that quantum mechanics disproves classical logic.

And we can model differential equations in a first order theory of real numbers, which requires no set theory

Your conception of "differential equations" is probably too narrow for this to be true. Consider where set theory came from: Cantor was studying Fourier series, which are important in differential equations.

But likewise, the standard reduction to set theory does not illuminate differential equations

...and nor does the reduction of biology to physics "illuminate" human behavior. That just isn't the point!

And we can model differential equations in a first order theory of real numbers, which requires no set theory

Your conception of "differential equations" is probably too narrow for this to be true.

Nope. It is literally possible to reduce the theory of Turing machines to real analytic ODEs. These can be modeled without set theory.

It is literally possible to reduce the theory of Turing machines to real analytic ODEs.

Okay, that sounds interesting (reference?), but what about the rest of my comment?

Here is Pour-El and Richards. Here is a more recent reference that makes my claim more explicitly. Both are gated.

What about the rest of my comment?

I'm not sure what to say. You've accused me of "confusing levels," but I'm exactly disputing the idea that sets are at a lower level than real numbers. Maybe I know how to address this:

But likewise, the standard reduction to set theory does not illuminate differential equations

...and nor does the reduction of biology to physics "illuminate" human behavior. That just isn't the point!

I don't know about human behavior, which isn't much illuminated by any subject at all. But the reduction of biology to physics absolutely does illuminate biology. Here's Feynman in six easy pieces:

Everything is made of atoms. That is the key hypothesis. The most important hypothesis in all of biology, for example, is that everything that animals do, atoms do. In other words, there is nothing that living things do that cannot be understood from the point of view that they are made of atoms acting according to the laws of physics. This was not known from the beginning: it took some experimenting and theorizing to suggest this hypothesis, but now it is accepted, and it is the most useful theory for producing new ideas in the field of biology.

You simply can't say the same thing -- even hyperbolically -- about the set-theoretic idea that everything in math is a set, made up of other sets.

Hilbert's 10th problem is about polynomial equations in integer numbers. This is a vastly different thing.

Yes, Hilbert's 10th Problem was whether there was an algorithm for solving whether a given Diophantine equation has solutions over the integers. The answer turned out to be "no" and the proof (which took many years) in some sense amounted to showing that one could for any Turing machine and starting tape make a Diophantine equation that has a solution iff the Turing machine halts in an accepting state. Some of the results and techniques for doing that can be used to show that other classes of problems can model Turing machines, and that's the context that Matiyasevich discusses it.

And we can model differential equations in a first order theory of real numbers, which requires no set theory

What signature do we need for it? Because in the first-order theory of real numbers without sets you cannot express functions or sequences.

For example, full theory of everything expressible about real numbers using "+, *, =, 0, 1, >" can be reolved algorithmically.

I'm not sure, presumably to "+*=01>" one adds a bunch of special functions. The "o-minimal approach" to differential geometry requires no sequences. Functions are encodable as definable graphs, so are vector fields.

As you note, for completeness reasons an actual o-minimal theory is not as strong as set theory; one has to smuggle in e.g. natural numbers somehow, maybe with sin(x). I could have made a less tendentious point with Godel numbering.

Again, this is meant as a kind of joke, not as a natural way of looking at sets. The point is that I don't regard von Neumann's {{},{{}},{{},{{}}}} as a natural way of looking at the number three, either.

Once you say that functions are definable graphs, you are on a slippery slope. If you want to prove something about "all functions", you have to be able to quantify over all formulas. This means you have already smuggled natural numbers into the model without defining their properties well...

When you consider a usual theory, you are only interested in the formulas as long as you can write - not so here, if you want to say something about all expressible functions.

And studying (among other things) effects of smuggling natural numbers used to count symbols in formulas into the theory is one of the easy-to-reach interesting things in set theory.

About natural numbers - direct set representation is quite unnatural; underlying idea of well-ordered set is just an expression of the idea that natural numbers are the numbers we can use for counting.

The true all-mathematical value of set theory is, of course to be a universal measure of weirdness: if your theory can be modelled inside ZFC, you can stop explaining why it has no contradictions.

I'm not sure one is more or less fundamental than the other. It does seem fair to say that as far as differential equations are concerned a completely different foundational setting wouldn't make any difference. So it isn't analogous to reductionism in that the behavior isn't brought about by the local interaction of pieces under the hood.

I'm not sure one is more or less fundamental than the other.

Really? You don't think the demonstrable reducibility of other branches of mathematics to set theory means anything?

It does seem fair to say that as far as differential equations are concerned a completely different foundational setting wouldn't make any difference.

This is actually a vacuous statement, because if it *did* make a difference, you wouldn't call it "a completely different foundational setting" of the *same subject*. Similarly, it wouldn't "make any difference" if, hypothetically, a 747 (as we understand it in terms of high-level properties) turned out to be made of something other than atoms; because by assumption the high-level properties of the thing we're reducing are *fixed*.

The important point is whether something *can* be reduced, not whether it *must* be.

I really don't know enough about programming to make a properly impressive analogy, but set theory is like a lower-level language or operating system *on top of which other branches can be made to run*.

I think the right analogy is not to building 747s out of parts, but to telling stories in different languages. The plot of "3 little pigs" has nothing to do with the English language, and the plot of Wiles's proof of Fermat's last theorem has nothing to do with set theory.

Really? You don't think the demonstrable reducibility of other branches of mathematics to set theory means anything?

Not in any strong sense, no. I reduce things to other fundamental frameworks also. One could for example choose categories to be one's fundamental objects and do pretty well. To extend your 747 analogy, this is closer to if we had two different 747s, one made of atoms and the other made from the four classical elements, and somehow for any 747 you could once it was assembled to decide to disassemble it into either atoms or earth, air, fire and water.

Well, we can disassemble every planet orbit to epicycles. Does that mean that our astronomical knowledge based on Newton's mechanics is worthless?

Epicycles are only a rough approximation that doesn't work very well, and it doesn't in any way give you Kepler's third law (that there's a relationship between the orbits). I'm also confused in that even if that were the case it wouldn't make or Kepler or Newton's mechanics worthless. What point are you trying to make?

Obviously Kepler's astronomical model is superior and that line might have been rhetorical flourish but the Ptolemaic, Copernican and Tychonic models were by no means "rough approximations" that don't "work very well". Epicycles worked very well which is part of why it took so long to get rid of them- the deviations of theory from actual planetary paths were so small that they were only detectable over long periods of time or unprecedented observational accuracy (before Brahe).

(I don't understand the grandparent's point either and agree that mathematical reduction to set theory is a different sort of thing from physical reduction to quantum field theory-- just pointing this one thing out.)

Yes, by doesn't work very well, I mean more "doesn't work very well when you have really good data." I should have been more clear.

Epicycles can give you arbitrary precision if you use enough of them... It is quite similar to Fourier transform.

My point is that in most cases you can disassemble a 767 into various colections of parts.

Think it will depend on what your other courses are. If you lean toward the mathemathical arena, then stuff like psychology or other social studies courses will give you an alternative to the way you think now. I have an idea that being a precise thinker is about being balanced in your areas of expertise. Anyway, it kind of depends on how you define precise thinking.

Did computer programming make you a clearer, more precise thinker? How about mathematics? If so, what kind? Set theory? Probability theory?

Microeconomics? Poker? English? Civil Engineering? Underwater Basket Weaving? (For adding...

depth.)Anything I missed?

Context: I have a palette of courses to dab onto my university schedule, and I don't know which ones to chose. This much is for certain: I want to come out of university as a problem solving

beast. If there are fields of inquiry whose methods easily transfer to other fields, it is those fields that I want to learn in, at least initially.Rip apart, Less Wrong!