Examples of Examples
As a companion to the Specificity Sequence, heres’s a place to share memorable examples of examples from your life or from popular culture.
As a companion to the Specificity Sequence, heres’s a place to share memorable examples of examples from your life or from popular culture.
The very funny HBO Series Hello Ladies has a running joke where the characters ask each other to “name three” examples of a questionable claim they just made. Here are two compilations.
Reminds me of that scene from Family Guy: https://youtu.be/UjtiAkakogM?t=40
Or this one from Brooklyn Nine Nine, that I inexplicably can’t find an actual video clip of
https://i.imgur.com/ulXy2LD.png
LOL great one
In The 5-Second Level and SOTW: Be Specific, Eliezer mentions being influenced by this example:
Examples of “proof by theory”
That someone has a theory that supports something is evidence for something.
Examples
1. Once 3 people tell us something, we believe it. Some people think it, so it’s true. Even knowing they are in cahoots and trying to manipulate us. I cannot source the study, but try it. It is scarily effective.
2. Ancel Keys formulated his dietary fat / heart disease hypothesis in the 1950s. Over a period of 3-4 years he moved from “hypothesis” to “almost certain” even though no new evidence arose in support of the hypothesis. It appears that every time he wrote on the issue, he noted that he himself, a very intelligent and credible authority, believed the theory, which seemed to weigh in favour of the theory. He cited his own previous papers which then added to the weight of the case, in his mind. [Keys may also have been influenced by the fact that his chief rival John Yudkin believed that sugar was the chief culprit, which view was therefore clearly wrong (theory in this case as anti-evidence). We are still sorting through the wreckage of his catastrophe].
3. Teenage fashions in clothes and politics. Teenagers are very concerned about acceptance by the group, and at the same time they have little experience and knowledge. So they seek cues from those around them as to what fashion statements and political opinions are acceptable. They are seeking cues from those around them, who are just as clueless as they are. Result: strongly held but more or less random fashions and opinions. One late teen recently told me he considers himself fortunate indeed to have been born at that one magic time when his peer group adhered to basically every right and true political and social opinion.
4. Contagion in financial markets. Didier Sornette has had some success in modeling the structure of financial bubbles and crashes based on the premise that speculators are very anxious about the direction of prices and highly uncertain about them at the same time. They have very little good information about future prices. In Sornette’s model, traders take cues from traders they are in contact with, resulting in violently fluctuating “phase changes” in investor opinion leading to log-periodic hyper-exponential price moves. Again the opinions of other traders are taken as data when in fact they have little information content.
When I was taught the incompleteness theorem (proof that there are true mathematical claims that cannot ever be proven), I wished for an example of one of its unprovable claims. Math is a very strange territory. You will often find proofs of the existence of extraordinary things, but no instance of those extraordinary things. You can know with certainty that they’re out there, but you might never get to see one. Without examples, we must always wonder if the troublesome cases can be confined to a very small region of mathematics and maybe this big impressive theorem will never really actually impinge on our lives in any way.
The problem is, an example of incompleteness would have to be a true claim that nobody could prove. If nobody could prove it, how would we recognise it as a true claim?
Well, how do we know that the sun will rise again tomorrow? We know that it rose before, many times, it’s never failed, there’s no reason to suspect it wont rise again. We don’t have a metaphysical proof that the sun will rise again tomorrow, but we don’t really need one. There is no proof, but the evidence is overwhelming.
It occurred to me that we could say a similar thing about the theorem P ≠ NP. We have tried and failed to prove or disprove it for so long that any other field would have accepted that the evidence was overwhelming and moved on long ago. A physicist would simply declare it a law of reality.
I was quite happy to find my example. It wasn’t some weird edge case. It’s a theorem that gets used every day by computer scientists to triage their energies, see, if you can prove that a problem you’re trying to solve is equivalent or stronger than a known NP problem, you would be well advised to assume it’s unsolvable, even though we wont ever be able to prove it (although, admittedly, we haven’t been able to prove that we wont ever be able to prove it, that too seems fairly evident, if not guaranteed)
...why not just take Goodstein’s theorem, which can be expressed in Peano Arithmetic but not proven in it—yet it can be proven in ZF. I assume we both agree that the continuum hypothesis isn’t clearly true or false, but if you happen to have an opinion there you could take that. Or, for that matter, if you believe that Peano Arithmetic or ZFC or whatever your favorite formal theory is true with high probability, then a consistency statement in that theory is an unprovable statement that you believe! You can write down a statement that is a translation of “Will this turing machine that searches until it finds a contradiction halt?”
Secondly, you say
...but both math and physics have solved centuries old conjectures, even close to the modern day. The parallel postulate was generally thought to be necessary for consistent geometry for 2 millenia; Fermat’s last theorem took 3 centuries; The 36 officers problem took 2 centuries (and the general case took another 50); Catalan’s conjecture took 1.5; The Poincaré conjecture and the four color theorem, took 1 century. Quantum mechanics and relativity overturned major bodies of work that had been around for centuries. Gravitational waves took a century to discover from when they were proposed; the estimates of the age of the universe had issues that took like 75 years to work out; and the Higgs boson and pentaquarks took like 50 years to discover.
The mathematical examples are especially impressive because of how old and developed core fields like number fucking theory are. Whereas the entire concept of turing machines has only been around since the 30s (1 century ago) and P vs NP is from 1970 (60 years ago).
When the entire field is only a century old, and the subfield (complexity theory) and problem in question is only around 50 years old, you really shouldn’t update that much on the problem being unprovable.
The only examples of unprovable statements that were found ‘in the wild’ I’ve heard of are consistency statements and the continuum hypothesis (mathematical logic probably has plenty but I consider that cheating, in the same way that undecidable problems in computer science are cheating). So the base rate of unprovable statements in non-logic fields is quite low, even if we go ahead and count every 50 year old famous important open problem as unprovable.
Get back to me in another century (if we’re still alive) and I’ll consider it plausible. But to think there’s more than a small chance of unprovability is to forget history.
Good objection to accepting that a conjecture is unprovable without proof.
Though not a good objection to accepting it’s probably true and moving on to work on more fruitful stuff. Do you have a theory of productivity in mathematics? (In undergrad I never encountered one, the stance seemed to be “don’t think too hard about usefulness or else you might fall out of pure mathematics and into physics or engineering. Anyway, it’s basically random. There’s just no way to estimate the usefulness of any given theorem! It’s all fine! Follow your heart! (into my phd programme)”) Do you have an argument against empirical mathematics? (Something like “stable probabilities for proof outcomes are difficult, even for reasonable people, to agree on”. Afaik this is just not a question people have been asking.)
I’m a curious person who finds pure math interesting, but I still care a lot about application. You can definitely estimate the usefulness of math, in a way like how people estimate the importance of pure math.
However, my understanding is that basically none of the famous conjectures are important because the statement itself is useful. Rather, we expect that if we can answer the question we should also have a better developed theory that will then have various applications (or on the pure end, let us understand various things).
There’s a whole bunch of unknown containments in complexity theory, and (reportedly) the theory currently just doesn’t seem to have good approaches to tackle the problems. This suggests that if we can resolve whether P=NP, we should also have some much better tools that can resolve a whole lot more and allow us to build up more theory.
I put like 80% on P ≠ NP since the experts think it’s likely true. Extra weight is given to people like Scott Aaronson, who also believe similarly I think. This is how I deal with fields I don’t know enough to have my own opinions on.
Empirical mathematics is fine, but once you get to famous outstanding conjectures the track record isn’t so good. For most random statements I cook up, empirics will vastly narrow down the possibilities and agree with proofs. But there are famous examples of this failing for famous conjectures in number theory, which a priori seems more amenable to empirics than computational complexity or, say, algebraic geometry. This suggests that once you’ve weeded out the easy stuff, empirics are less useful for what’s left.
But what’s the empirical (as in, similar to checking a number theory conjecture on all the integers up to however much you can compute) evidence for P ≠ NP, besides humans not finding the necessary algorithms? It’s not like someone has an algorithm that seems to have asymptotics of O(f(n)) and where proving it does would resolve the conjecture (Levin’s universal search doesn’t count on account of being too ridiculously slow to get empirical evidence about).