Half-closing my eyes and looking at the recent topic of morality from a distance, I am struck by the following trend.

In mathematics, there are no substantial controversies. (I am speaking of the present era in mathematics, since around the early 20th century. There were some before then, before it had been clearly worked out what was a proof and what was not.) There are few in physics, chemistry, molecular biology, astronomy. There are some but they are not the bulk of any of these subjects. Look at biology more generally, history, psychology, sociology, and controversy is a larger and larger part of the practice, in proportion to the distance of the subject from the possibility of reasonably conclusive experiments. Finally, politics and morality consist of nothing but controversy and always have done.

Curiously, participants in discussions of all of these subjects seem equally confident, regardless of the field's distance from experimental acquisition of reliable knowledge. What correlates with distance from objective knowledge is not uncertainty, but controversy. Across these fields (not necessarily within them), opinions are firmly held, independently of how well they can be supported. They are firmly defended and attacked in inverse proportion to that support. The less information there is about actual facts, the more scope there is for continuing the fight instead of changing one's mind. (So much for the Aumann agreement of Bayesian rationalists.)

Perhaps mathematicians and hard scientists are not more rational than others, but work in fields where it is easier to be rational. When they turn into crackpots outside their discipline, they were actually that irrational already, but have wandered into an area without safety rails.

By the standards of other fields, mathematics doesn't have "substantial" controversies, but here are a few examples, including the two mentioned by Morendil and Jonathan L.

1. A century ago there was controversy about constructive mathematics and about axioms like choice. I think these controversies were resolved by driving out the people with high standards into logic and theoretical CS (Informatics). The higher standards seem more useful in those places, but to call the current standards "uncontroversial" is, I think, misleading. Mathematicians claim to accept that constructive proofs or proofs in weaker systems are in some sense better, but in practice they never think it worth the loss of elegance. In recent decades, the desire to computerize calculation has increased the popularity of constructive proof some. For example, the original controversial non-constructive proof by Hilbert of his Basis Theorem (1890) is usually taught following Noether's very short proof (1921) rather than using Buchberger's constructive Gröbner bases (1965).

1b. Bourbaki and especially Grothendieck actually work in stronger systems than ZFC. It is not clear to me how controversial this is, but there have been awkward conversations (eg, on the FOM mailing list) about the status of Grothendieck's results.

2. Mathematicians have a strong consensus about the ideal of proof to which they aspire: formal ZFC, but they don't actually produce formal proofs. The real standard is informal proofs and a belief that they can be debugged to a formal proof. This is not a precise standard. People are generally nervous about very long proofs. They are not very nervous about the thousands of pages of SGA that go into Deligne's proof of the Ramanujan conjecture because most of that is modular. It breaks into small pieces which are easy to understand and which are reused in many places, thus extensively tested. But the proof of the classification of finite simple groups (famously condemned by Serre around 1980) is made up huge modules that are used for nothing else. In fact, it contained an error at a high level, that the organizers failed to observe that one of the assigned modules had not been delivered. There will be more confidence in the second generation proof because (1) it is projected to be only 5000 pages and (2) the same people will prove the modules as will use them.

2b. When someone rewrites a proof in his own words, it is much more reassuring than if he simply claims to have read and understood it. Modifying the proof, either for simplification or to prove something else is also helpful. But it is generally rude to express such doubts publicly. Here is someone getting in trouble for voicing such concerns. There are lots of decades old results that are controversial in some circles.

2c. It is quite common for top mathematicians to provide fewer details of proofs. Often other people write longer papers filling in the details. The status of the result before this is sometimes controversial, but again often not publicly.

3. The computer-generated proofs of the four-color theorem are controversial for two different reasons. One reason is that since they don't exist in a human mind, they don't provide insight. The other was that since they were of a different form than usual proofs, they were not trustworthy. In fact, there have been many generations of these proofs and I am told that each new one claims that the previous one missed some cases. The latest one generated a formal proof, so it is computer-checked and as defensible as anything. But the first controversy remains.

4. Physicists often make claims about mathematics. For example, that the scaling limit of the self-avoiding walk on the square grid exists; and that the limit has a particular fractal dimension. Is this a controversy that the mathematicians do not accept the physicists' "proof'? Or do they just mean different things by that word? Anyhow, the physicists go on to say that the limit is a conformal field theory, and the mathematicians reject this as a meaningless statement, though they try to salvage it, eg, through SLE.

5. People make calculation errors all the time. Long calculations can be controversial, but they probably should be trusted less than they are. It is generally better to compute many things in parallel, so that an error in the calculation infects all the answers and is immediately apparent.

As someone in academic sociology, this conflicts with my experience.

Seemingconfident is a matter of social skills, temperament, and the conditions of winning arguments with peers in the same area. But as best I can tell, in terms of assigning probabilities, sociologists can be divided into those who think the field (or at least the parts they're engaged in) to be scientific in the same manner as the physical sciences are scientific, but much harder and with greatly reduced ability to assure confidence in its findings, and a minority who think that society is so complex (especially when "meta" epistemological factors are brought in) that confidence in any of its conclusions beyond dumb facts is unwarranted and that science is a bad term for it. Claims people state with great confidence tend to be ones that enjoy broad agreement.It would be very surprising if hard scientists were mostly mutants.

Are you sure "distance from objective knowledge" is the best x-axis to make this observation? "The closer you get to humans, the worse the science gets" is a fairly common quote, supposedly (?) from W. Elsasser. That fields more directly involved with humans are less rational is not surprising : practitioners of such fields see the direct relevance to people's lives as a good thing and gain status from it. Attempts to make these fields more rational (carefully-defined terminology, good statistics, computerised data-gathering and analysis,...) will

appearto move such fields away from "directly relevant to people's lives" and towards "irrelevant academic ivory-tower practices"."Far from objective knowledge", "subjective" and "close to humans" are different ways of saying pretty much the same thing - though I agree that "close to humans" may make some reasons for the problem a bit more obvious.

If this was more about politics than verifiability, I'd expect professional ethicists to disagree more over substantive ethics than metaethics, while the opposite seems to be the case. (There could of course be other factors gumming up the works - I can't think of any other good contrasts off the top of my head.)

Metaethics is the far younger field. It will need time to come to some widely shared and agreed-upon results.

The earliest I've managed to trace the phrase "The closer you get to humans, the worse the science gets" is a blog comment by David Marjanović July 13 2007. He claimed it was a proverb. Someone else in the thread then claimed

"that's pretty much like a folk heuristic version of Walter M. Elsasser's 'Reflections on a Theory of Organisms: Holism in Biology'". I've been asking around for a better source, but it certainly doesn't seem to be something Elsasser said in those words. Marjanović is fond of it.Edit:Found an earlier rendition, "Scienticity is everywhere lower where the subject is closer." - from Donald Black. "Dreams of pure sociology." Sociological Theory 18:3 November 2000.What do you think about debates about which axioms or rules of inference to endorse? I'm thinking here about disputes between classical mathematicians and varieties of constructivist mathematicians), which sometime show themselves in which proofs are counted as legitimate.

I am tempted to back up a level and say that there is little or no dispute about conditional claims: if you give me these axioms and these rules of inference, then these are the provable claims. The constructivist might say, "Yes, that's a perfectly good non-constructive proof, but only a constructive proof is worth having!" But then, in a lot of moral philosophy, you have the

samesort of agreement. Given a normative moral theory and the relevant empirical facts, moral philosophers are unlikely to disagree about what actions are recommended. The controversy is at the level of which moral theory to endorse. At least, that's the way it looks to me.First-, or possibly second-order predicate logic has swept the board. Constructivism is just a branch of mathematics. Everyone understands the difference between constructive and non-constructive proofs, and while building logical systems in which only constructive proofs can be expressed is a useful activity, I think there are not many mathematicians who really believe that a non-constructive proof is worthless. There is some ambivalence toward such things as the continuum hypothesis and the axiom of choice, but those issues never seem to have any practical import outside of their own domain.

One reason for this is that, generally speaking, non-constructive proofs

canin fact be embedded into constructive logical systems. I think there is more controversy about what formal foundations we should endorse for mathematics. ZF(C) is good enough as a proof of concept, but type-theoretical and category-theoretical foundations seem to be better in terms of actually doing formalized mathematics in the real world.While that may be a reasonable justification for what mathematicians do, I think it is false as a historical claim about what caused mathematicians to do what they did. Mathematicians settled on their foundations ("No one can expel us from Cantor's Paradise," 1926) before they understood power and limits of constructive methods.

I'm curious if you are making a practical claim or a formal one.

How do you define the "bulk of the subject"? Take physics from wikipedia:

Having five competing theories seems like controversy to me. It's just that as a layperson I don't understand what the core of the controversy is about.

Molecular biology is the only one of those where I took classes at university. I don't think that the field is without controversy. It's just that outsiders don't get much information about it.

The stated goal of the human project was: Project goals were to among others:

They didn't determine all the basepairs of human DNA. After getting to 92% the declared their project a success. They then pretended that the other 8% doesn't matter. Happy coincidence isn't it? The stuff that's hard to sequence is also practically unimportant...

The extend to which those 8% matter is up for debate.

Determination of protein structures is a bit similar. Most of the protein sequences we have were created through crystellizing proteins. We don't really know exactly how big proteins move when they aren't crystallized and are surrounded by water and other stuff that's in the cell. It's practical to assume that we basically know how a protein looks like when we know how it looks like when it's crystellized.

For outsiders it's easy to understand history, psychology or sociology controversies. For the hard sciences it's hard for outsiders to understand the controversies within the field.

I think controversy is more closely related to the number of people who feel qualified to participate in discussions. The hard sciences exclude idiots by the very language they use. In softer subjects like sociology, or barely academic ones like politics, anyone can pick up the standard textbooks of the field, bullshit his or her way into a passing grade and then continue to muddy the waters until retirement.

When I discuss politics or morality with my friends (none of them are rationalists), it seems the most frequent problem is them confusing instrumental with terminal values. Since terminal values are subjective and it feels you have direct knowledge of them, values can be defended with unfaltering certainty as long as you think they are terminal. I have a faint memory of Eliezer specificly discussing this problem, maybe even on several occasions, so if someone can refer me to the posts, I am grateful.

Edit: this briefly discusses it, are there any others?

Maybe they just have different values than you do? Could you give an example?

Thanks for asking this question, I falsely believed my conclusions were obvious. Looked a long time for a satisfying answer and found none. I don't believe that the instrumental-terminal distinction is useful for human beings most of the time anymore, especially because we're really prone to this.

The problem of the instrumental-terminal distinction is that it ignores important aspects of human decision making.

I recommend a paper titled Emerging sacred values: Iran’s nuclear program in the journal Judgment and Decision Making. It's about the willingness of the Iranian to give up the right to a peaceful nuclear program.

It explain that human hold sacred as well as secular values. Sacred values aren't only existing within religion. The Iranian belief in Iran's right to a peaceful nuclear program also is a sacred value.

In Western culture a lot of people treat equality and democracy as sacred values. Many people who self label as skeptics treat principles like the scientific method or the theory of evolution as sacred values.

This assertion belongs to the sociology of science, so it's controversial. :)

What's a commonly accepted way to measure the variables above? For instance, Greg Chaitin seems to disagree with the assertion about mathematics. He brings up FLT; wasn't that a topic of controversy?

Quite.

I don't think he does. He's mostly talking about the beginnings of the establishment of the modern, controversy-free era of mathematics -- the solution of all the old controversies. Notice that he says that Gödel

discoveredincompleteness, Turingdiscovereduncomputability, hediscovered(an information-theoretic concept of) randomness. All these things remain discovered. Proofs of theorems absolutely settled the matter. There was no controversy about Wiles' proof of FLT, just people examining it very closely, finding (as far as I know) one fault, which Wiles fixed. Ordinary business of the day. Deolalikar still claims on his web page to have a proof of P != NP, but no-one else believes it.In contrast, has anyone ever claimed to have

discoveredsomething in sociology that remained uncontroversiallydiscovered?Yes.

I'm adding to that below, in rot13, but let me first ask another question: have you

lookedfor such claims, or is this strictly a rhetorical question?From about 5 minutes of looking: Zregba; frys-shysvyyvat cebcurpvrf; a discovery from sociology that's widely appealed to here on LW, in such contexts as the study of decision theories.

A Google search of things like /discovery sociology/ or "sociological discovery" didn't lead anywhere useful, but I don't know how one would systematically look without already having a lot more familiarity with the field than I do. That doesn't make it a rhetorical question though.

Fair enough. WP was my starting point - it didn't seem too hard to turn up relatively robust-looking claims.

It's the Appel-Haken 4CT proof I was actually thinking about, my bad. There was controversy about that not being a "proper" proof, as I recall, and it's been (unfavorably) compared to Wiles' proof in that respect (which helped me mix up the two - I'm no mathematician!).

My underlying question is "what counts as a controversy", and more directly "how would I go about checking the facts of your claim about the correlations between a field's distance to objective truth and proneness to controversy".

"A state of prolonged public dispute or debate." How prolonged? How much disputed? Look at the various disciplines I listed and see how they compare. Agreed, for mathematics, Appel-Haken was a controversy. Compared with politics, it was animated conversation over afternoon tea at the vicarage. Also, judging from the Wikipedia account, the controversy progressed steadily to a resolution.

If you want numbers and experiments, obviously I haven't done any of that, but just recounted what it seems to me that I have seen. You, or someone, would have to work out an objective measure of the existence and intensity of a controversy, and survey publications in various disciplines. I don't know if you could devise a method of detecting controversies just from citation patterns, but the more you could automate this the easier it would be to collect data.

Richard, do you think Pearlian causality is mathematics or something else? Because I think Pearlian causality is extremely controversial by your definition (to be fair not as a piece of math, but how applicable it is to practical scientific problems).

It's applied mathematics.

That is, you can erect the entire thing as pure mathematics, as you can with, say, probability and statistics, or rational mechanics. The motivation is to apply it to the real world, and the language may sound like it's talking about the real world, but that's just a way of thinking about the pure mathematics. Then to apply it to the real world, you need to step beyond mathematics and say what real-world phenomena you are going to map the mathematical concepts to.

Pearl is insistent that the concept of causality is primitive and not reducible to statistics, but I haven't ever read him philosophising about "what causes

reallyare". He just takes them as primitive and understood: do(X=x) means "set the value of X to x", although that is clearly an unsafe instruction to give an AGI. You would have to at least amplify it with something like "without having any influence on any other variable except via the causal arrows you are willing to allow might exist".There appears to be some dispute on this issue. I'd be interested to know his answer to the conundrum posed by Scott Aaronson, or for that matter the similar one I posed here. (I am not satisfied by any of the answers in either place.)

When you ask (in your koan) how the process of attributing causation gets started, what exactly are you asking about? Are you asking how humans actually came by their tendency to attribute causation? Are you asking how an AI might do so? Are you asking about how causal attributions are ultimately

justified? Or what?I think these are all aspects of the same thing: how might an intelligent entity arrive at correct knowledge about causes, starting from a lack of even the concept of a cause?

That seems like a very different question than, say, how humans actually came by their tendency to attribute causation. For the question about human attributions, I would expect an evolutionary story: the world has causal structure, and organisms that correctly represent that structure are fitter than those that do not; we were lucky in that somewhere in our evolutionary history, we acquired capacities to observe and/or infer causal relations, just as we are lucky to be able to see colors, smell baking bread, and so on.

What you seem to be after is very different. It's more like Hume's story: imagine Adam, fully formed with excellent intellectual faculties but with neither experience nor a concept of causation. How could such a person come to have a correct concept of causation?

Since we are now imagining a creature that has different faculties than an ordinary human (or at least, that seems likely, given how automatic causal perception in launching cases is and how humans seem to think about their own agency), I want to know what resources we are giving this imaginary Adam. Adam has no concept of causation and no ability to perceive causal relations directly. Can he perceive spatial relations directly? Temporal relations? Does he represent his own goals? The goals of others? ...

This is not an explanation: it is simply saying "evolution did it". An explanation should exhibit the mechanism whereby the concept is acquired.

That is one way of presenting the thought experiment.

Another way of presenting the thought experiment is to ask how a baby arrives at the concept. Then we are not imagining a creature that has different faculties than an ordinary human.

Another way is to imagine a robot that we are building. How can the robot make causal inferences? Again, "we design it that way" is no more of an answer than "God made us that way" or "evolution made us that way". Consider the question in the spirit of Jaynes' use of a robot in presenting probability theory. His robot is concerned with making probabilistic inferences but knows nothing of causes; this robot is concerned with inferring causes. How would we design it that way? Pearl's works presuppose an existing knowledge of causation, but do not tell us how to first acquire it.

That is part of the question. What resources does it need, to proceed from ignorance of causation to knowledge of causation?

I definitely agree that evolutionary stories can become non-explanatory just-so stories. The point of my remark was not to give the mechanism in detail, though, but just to distinguish the following two ways of acquiring causal concepts:

(1) Blind luck plus selection based on fitness of some sort. (2) Reasoning from other concepts, goals, and experience.

I do not think that humans or proto-humans ever

reasonedtheir way to causal cognition. Rather, we have causal concepts as part of our evolutionary heritage. Some reasons to think this is right include: the fact that causal perception (pdf) and causal agency attributions emerge very early in children; the fact that other mammal species, like rats (pdf), have simple causal concepts related to interventions; and the fact that some forms of causal cognition emerge very, very early even among more distant species, like chickens.Since causal concepts arise so early in humans and are present in other species, there is current controversy (right in line with the thesis in your OP) as to whether causal concepts are innate. That is one reason why I prefer the Adam thought experiment to babies: it is unclear whether babies already have the causal concepts or have to learn them.

EDIT: Oops, left out a paper and screwed up some formatting. Some day, I really will master markdown language.

Yes, it's (2) that I'm interested in. Is there some small set of axioms, on the basis of which you can set up causal reasoning, as has been done for probability theory? And which can then be used as a gold standard against which to measure our untutored fumblings that result from (1)?

Ah; not a true controversy then?

You might be interested in Bruno Latour's work on mapping controversies. The idea is to look at instances of "controversy" with eyes fully open rather than half-closed. There's one on string theory, for instance.

Just not much of a controversy.

So it's being done already! Any results yet? mappingcontroversies.net appears to all be research in progress.

While I don't think it changes the order of the fields, I can think of three versions of "how controversial is X" that might yield different answers: a person from field X might trust all its results and say it is uncontroversial; a person from field Y might say it is uncontroversially false; clearly there is controversy if we put the two people in a room.

Even at the extreme, physicists make lots of claims about mathematics that the mathematicians reject. Sometimes the claims are precise enough that the mathematicians accept them as conjectures (and aren't bothered that the physicists use the word "proof"), but often the mathematicians cannot even tell what the conjecture means and reject the label "mathematics." I gave some examples here.