it's also why you shouldn't hold equity in your own company any longer than necessary (e.g., an apple employee shouldn't tie up the majority of their networth in apple stock).
stock options/rsus are a great example of some sort of cognitive bias (maybe it's the endowment effect?).
give somebody $1000 worth of shares in their company, and a lot of folks will hang on to them. but if you gave them $1000 cash to invest, they almost certainly would not choose to dump all of that money into their own company.
Tax-law also makes is sticky: StockX -> Money -> StockY means a portion of is lost as capital-gains tax on the money step. StockY might be better... but is it something that will perform better-enough to be worth the switching costs? (At this point logarithms and spreadsheets start getting involved.)
In contrast, starting with Money and then choosing between StockX or StockY is an easier choice.
I like creatine when I’m lifting because it reduces soreness. For some people that effect helps them push further and train harder, for me it’s just proactive pain relief
There’s a human connection element to old radio that streaming will never replicate. And I worry that it’s gone from modern automated radio too.
It used to be if you were driving down the road and heard a song you liked
Hearing a song you liked on the radio hit harder then playing it yourself because you knew that there was another human nearby at that moment in time that had picked it out and played it. You were physically tethered to them through invisible waves through the sky. Lonely road trips felt less lonely. Late night work shifts felt less alienating because there was somebody ELSE working late.
It absolutely does - allegedly there’s 6 million listeners to the biggest breakfast radio show in the UK and 4 million to the second biggest. That’s 10 million in a country of about 55 million adults.
In the 90s I knew a guy in high school who walked around the school in a black trenchcoat with portable speakers tucked inside that were continuously playing Western movie themes. I can only imagine how that bit of creativity would go over these days.
I suppose it was rare enough and somewhat fun in the 90s. Maybe that guy was doing this for others to enjoy as well, or to appear singular (whether it worked or not).
Nowadays it wouldn't appear particularly exotic (portable speakers are ubiquitous) and people playing loud music (from speakers or phones) seem to do this for themselves without caring about the surrounding people, that's just highly egotistical and self-centered. It's usually not music I personally like too.
That's not fun, it's just annoying. There's already too much noise in our cities.
But if you do it right, playing (singular) music for others, reading the room and all, I'm sure it can still work.
Sometimes, but not too often, in Berlin of all places you encounter someone biking around with a massive speaker attached to their bike blaring (obviously) techno
Not always a bike. Sometimes just a trolley or a handcart or a custom jerry-rigged contraption.
In the US "boom boxes" went out with the 1970s. They need to get with the times.
Okay, in all seriousness: boom boxes were a thing in the 1970s and early 1980s. There are still people who use the latest generation of them, but thankfully not nearly as many. These days people cranking music in a public place are generally doing it as part of a group that all want to hear the same thing. (and in back then the group wanting to hear the same music was the common case)
When I opened https://radio.garden/ and spun the globe the first time, I tuned to a radio station somewhere in Africa, and it gave me tingles to think that in Lagos at that same moment, some bus driver was surely listening to the same music.
Never in my life has radio in my country played music I wanted to listen to. The first thing I did when I bought my first car was disconnect the antenna entirely.
Yes, and I agree with a sibling comment that livestreaming is the modern equivalent, if not in ease or quality but in spirit. It's also why, despite having many movies in my library, I enjoy turning on TCM to watch old classics. I know that thousands of people at any given time are probably watching the same piece of history I am, right then.
You can still listen to the radio while driving, many people still do that, I still do that. I agree though that in essence streaming is a very alienating experience.
Many people do, but I don't get it. The typical station has a tiny playlist that they have had on repeat for a decade now. I haven't turned the radio on in years, and never was a regular listener, but I can still tell you what song will be next if you turn it on (for the stations that are played most often in public)
Yeah, but most current corporate radio sucks. It’s not a human broadcasting and allowing their idiosyncratic tastes to sneak in. Nationwide each station is just shuffling one of three or four playlists of inoffensive songs picked because nobody hates it (but nobody loves it)
I don't know. I mean if that is the criteria then livestreams surely fix this because there is another human at that moment in time also and you can even interact with them. Or do you like the passive interaction that they should just play music and shut up
This still exists today. When a new song gets released on YouTube for example a group of people close to it in latent space gets it recommended. By seeing a recommendation of a song in your feed you are part of a cohort of people with similar tastes.
That is the core issue. Only a preselected cohort are exposed to new media. That limits common experiences to digital islands rather than shared among the masses.
proofs by counterexample are effective but ultimately unsatisfying. they get you to an answer but they don't help help you understand and bend you r mind into seeing how the math works and lead you on to the new set of questions.
and for now as long humans are going to judge of what counts as an elegant or illuminating proof, there's going to be work for human mathematicians
I would only agree partially. There are counterexamples that are not illustrative, but it is fairly common that in thinking about how to construct a counterexample you gain a more thorough understanding of the original problem and at least one fundamental issue which prevents the conjecture from being true.
Right; see Lakatos. In its roughest form, you study the structure of whatever counterexamples you find, add those as (negated) preconditions to your proposition, rinse and repeat until you have a true statement. If the proposition remains useful, you now have a new definition.
Counterexamples are literally the only way to show a "for all" statement is false. (Non-constructive proofs by contradiction work by showing a counterexample must exist.)
Hell, one could even go further into the past and refer to the thankless work of pre-computer era mathematicians who sweated over manual calculations in order to disprove various prime related conjectures: https://en.wikipedia.org/wiki/Mersenne_conjectures
You seem to have an objection to non-intutionist mathematics in general, a position that was once held by many an illustrious mathematician but is relatively fringe in the contemporary academic community. Mathematical facts don't have to be intellectually satisfying or make sense to you, the human, rather it is up to you to wrap your mind around discovered mathematical facts.
Are you thinking of proof by contradiction, which is rejected by constructionism?
[Dis]proof by counterexample is the most straightforward way to show a statement to be false. What better way is there to disprove a general statement like 'all x are y' than finding an 'x' that isn't 'y'?
It’s very straightforward, but it often doesn’t (and here didn’t) fully satisfy the curiosity that was embedded in the original problem. Why did the Jacobian conjecture seem to be true? Is there some underlying symmetry that’s very slightly broken? Or perhaps there’s all kinds of counterexamples, and the intuitive pattern is only real for certain kinds of functions which happen to predominate in our intuition. Then how should we adjust our intuitions to better capture the space of possible polynomial functions?
Those are all good questions, but I don't really understand what alternative you or the OP are looking for actually resolving an untrue conjecture, besides a counter example.
I would frame it differently. The existence of compact counterexamples to a true-seeming conjecture suggests that there’s some deeper understanding waiting to be discovered. Fuzz testing for theorems, if that makes sense. I hope mathematicians in 2036 will be able to explain in detail why the Jacobian conjecture was false and identify which similar, true conjectures the community’s intuition was pointing towards.
We can take a simpler example. Let's say someone conjectures that all linear maps are isomorphic if they have the same domain and codomain*. A counterexample is easy to find, but true insight would be to notice that all linear maps with the same domain and codomain that are not isomorphic map some non-zero elements to zero. That is much more interesting than just finding a counterexample. Although, that isn't to say that finding a counterexample is not very interesting.
*statements only apply to maps whose domain is finite-dimensional
I suspect an AI, possibly a successor to current LLMs, will achieve that by the early 2030s. It might help illuminate many mysteries in math and beyond for us all.
There are many cases where it's possible to prove that counterexamples must exist, without identifying a specific example. This kind of proof provides more insight into the problem than simply finding a counterexample.
As a constructivist: we don't disagree :) We just distinguish between "don't disagree" and "agree." Constructive mathematics says it's fine if you want to claim that there's not no counterexample -- you just can't use that in a situation that demands an actual counterexample (like an algorithm that produces a result). This tends to guide people towards looking for results that don't require this kind of indirection, since they apply more broadly and in more kinds of logics -- orthodox constructive results are kind of a lowest common denominator of consistent truth and remain broadly compatible with most axioms, while nonconstructive results often fail in particular models. Which seems like a pretty sane stance to me, but maybe I'm too thoroughly indoctrinated to see how unreasonable it is :P
(Note that this is about excluded middle. There ARE constructive logics with interpretations of excluded middle, e.g. some forms of classical linear logic, but they do not play as nicely with other logics. Constructivists often reject even weak forms of choice for largely the same reasons--there are some forms of choice that are constructively valid in some logics, but these results often fail to hold true in more conventional logics. And the same is true for a whole host of related notions that proof assistants like Rocq reject by default, propositional extensionality (which says that two proofs of the same proposition are equal) and function extensionality (which says functions are equal whenever their results are equal on all the arguments in their domain -- which might seem obviously acceptable until you realize that it's false in most programming languages!) being prominent but much less discussed examples. It's all about remaining broadly compatible with lots of different types of reasoning, not because people think the reasoning is invalid per se).
Maybe I was unclear. The part that I meant a constructivist would disagree was the subjective part: "This kind of proof (existence of a counterexample) provides more insight into the problem than simply finding a counterexample."
I find it hard to believe that a constructivist would agree that across the board, proof of existence of X is more interesting than a construction of X. Isn't that the point!?
A direct counterexample is more "informative" in a very literal sense (its truth value doesn't collapse). But the extra proof relevant content we can use here is not that large -- all it means in this case is that we can directly compute the object and its Jacobian, a well as two points evaluating to the same result. That's nice, but it's not that interesting by itself unless I can use the exact constructed form to prove other interesting stuff (and we can! Most of the followup results that immediately followed from the disrpoof come from being able to directly transform this object into counterexamples to other conjectures; if we didn't have constructive proofs of thoe counterexamples, we wouldn't have such procedures). But being more interesting than a completely uninformative counterexample still doesn't mean it's inherently interesting or enlightening. If anything I'd indeed argue constructive arguments are generally less mysterious and magical than nonconstructive proofs -- in some sense, the constructive proof pulls back the curtain and shows you where the trick is.
In a constructive system, it’s often possible to refute a universal proposition without exhibiting a counterexample, by proving that the proposition implies falsehood.
The constructivist will still object that you can’t, from that, conclude that “…therefore a counterexample must exist,” without actually providing a counterexample. But the general principle I was describing still applies - a proof often gives you insight that an example by itself doesn’t.
To roughly repeat myself from a sibling comment: I may have been unclear. I didn't claim that the only way to disprove something in a constructive system is to produce a counterexample.
The part that I was referring to was the last statement from the OP: that "a proof of existence of a counterexample necessarily provides more insight than a counterexample". I can't imagine a constructivist would agree with that in general.
I'm the OP. In the cases where a proposition can be refuted by a proof that doesn't involve counterexamples, by its nature that proof will tell you something about the reason that the proposition is false.
Whereas a counterexample, on its own, proves the proposition false but doesn't necessarily tell you anything else.
The real difference in the constructive case is that there are fewer classes of proposition for which a proof without witnesses is possible.
(Edit: side note, I didn't explicitly say "necessarily" in my original comment. I suppose there could be exceptions, although I'm struggling to think of an example. Constructively speaking, the ball is in your court!)
I think maybe a better way of explaining it would be that an uninformative proof by definition needs to be based on proving that the set under consideration must be inhabited without ever defining an object in that set. This generally means you must show the set is inhabited by exploring some abstract properties of the set itself. A single counterexample, by contrast, by itself is a direct proof that the set is inhabited, so you don't necessarily learn any other interesting properties about the set. So it's not really about constructive vs. non-constructive, I think it's closer to e.g. the idea that point-free stuff tends to be more beautiful and meaningful than pointed stuff (which I think most mathematicians would agree with and which really has nothing to do with intuitionism per se).
In this case, I think part of the problem is that there was kind of no good reason to think the Jacobian conjecture was true in > 2 dimensions other than it being kind of hard to find counterexamples. So a really interesting disproof would be one that, e.g., was able to exhaustively classify the counterexamples, or showed why it seemed in practice to be hard to come up with functions violating the conjecture. AFAIK, this doesn't really accomplish either of those things, not even after you learn the procedure that constructed the function -- it kind of tells you why we should have expected to find a counterexample but not how rare such counterexamples are.
I think the constructive position is basically that people's entire issue with lack of excluded middle being absent is just that people like being able to say "P" instead of "~~P" because it sounds better, considering you can prove ~~P for all the classical propositions that use excluded middle.
> people like being able to say "P" instead of "~~P" because it sounds better
It depends on how old an intuitionist/constructionist you are. Back in the day, they were interested in logic as a description of correct reasoning. Brouwer saw LEM as a mistake in the foundations.
These days, the influence of formalization, including proof theory and model theory, has removed a lot of the teeth from that debate and made it possible to summarize as you have.
I studied this in the early 1980s, and my professor was definitely in the "this is a black and white issue" camp, although he came down on the classical side.
(Side note, I was once a back seat passenger in a car with my prof and Quine in the front seat, who was visiting at the time. Quine was famously committed to the idea that first order logic is the only kind worthy of the name.)
Ah, I didn't realize this was a generational thing. I am definitely a "new" intuitionist, so that probably greatly influences my perspective. I suppose that before results like this, the setoid model, etc. were known constructivism was indeed a much more hardline position to have to take!
Isn't the fact that you now know that the conjecture is false a huge help? At least it will help convince people to look at the conjecture more closely, no?
You, right now, have the ability to spend a few weeks studying the Jacobian conjecture and its counterexample and write up a blog post about what you think is special about this counterexample.
You could spend the rest of your life coming up with conjectures that look elegant but are ultimately false. Disproof by counterexample only works if it's false, and we shouldn't be satisfied with a false conjecture to begin with.
> and for now as long humans are going to judge of what counts as an elegant or illuminating proof, there's going to be work for human mathematicians
Considering ChatGPT was released only three and half years ago, and LLMs could do high school math only less than two years ago, I think this "for now" will not last very long.
Seems like a breakdown on the incentives / imperatives in the field? I hope that's not an over bold guess from a non-mathematician.
Couldn't people in principle continue to study a problem that's only been shown to break at one point? Prove something adjacent, or slightly weaker, or elaborate the counter example into a powerful explanatory framework?
Maybe I’m just not pure enough but I find the whole concept of proof by counterexample to be elegant, and I don’t see why proving that something must be true is superior to proving that it can’t be false.
It's elegant if all you're concerned with is whether a conjecture is true or false. Answered, move along!
But mathematics is not a collection of facts. Mathematics is the study of abstraction. And what do you learn from a single data point? What can you abstract from that?
That's why just being a counterexample isn't really interesting. There has to be more than "counterexample" for there to be something to abstract. Was it generated from an analysis of the problem? Can the counterexample be generalized to explore the problem further? Is the counterexample a surprise in a way that suggests something is missing from current understanding?
Being a counterexample doesn't mean that something isn't interesting to a mathematician. But it's also not the interesting part.
You have a weird definition of "smuggling". I say it outright. Because I follow it up with a description of the actual practice of math: it's the study of abstraction. That's not a frame. That's just what math is.
This is a description of one portion of math... one that's very easy to get tunnel vision towards when undertaking a very formal undergraduate mathematical education. And that's especially true if it was alongside a computer science education, which is precisely the branch of math concerned with formal systems being used in calculation.
I've done both of those things. I know what you get taught. But I've kept my math education going for the 25 years since then. I've talked to practicing mathematicians about what they do. I've learned a lot about the scope of math.
As an aside: most people really dislike it when I say that they should be much more precise about different numerical systems. The integers are not a subset of the rationals. They are entirely different constructions, but there is an isomorphism between integers and a subset of the rationals that preserves the integers' ring structure within that subset of the rationals and a few other aesthetic concerns. You can see why no one wants to communicate like this, even if they acknowledge it's technically correct. So I know all about pushing symbols around.
But I also know that pushing symbols around isn't the whole story. Pushing symbols around is only useful as a final check. Do you want to validate that 1+2=3? Pushing symbols around can help. But how do you decide that the ideas behind 1, 2, 3, +, and = are worth having precise and compact representations?
Math doesn't just use formal systems to generate proofs. It's not enough for symbols to be arranged neatly according to some rules. Math is also the process of creating the sets of symbols and their rules and communicating to other people why this set of rules and symbols is interesting. What ideas get preserved when you are working with this system? What is it an abstraction over?
> The integers are not a subset of the rationals. They are entirely different constructions, but there is an isomorphism between integers and a subset of the rationals that preserves the integers' ring structure within that subset of the rationals and a few other aesthetic concerns.
This actually strikes me as a very formal perspective.
Considering it from an informal perspective, it's a bit more fuzzy isn't it. As you say there are many isomorphic things, and when we say The Integers it's not actually clear which one of them we mean. Maybe we mean one of them today and another tomorrow. Often times it doesn't matter, and so we don't clarify the question.
Like you could imagine defining the BootstrapNaturals then the BootstrapIntegers then the BootstrapRationals then use them to define the Reals. And then say that the Naturals, Integers and Rationals are defined as subsets of the Reals. This would be one way to put the common view of the Naturals as being a subset of the Reals on a solid formal foundation. It's rarely done ig because it's seen as obviously unproblematic to be a bit handwavy.
Another criticism of the common construction of numbers we could pose, inspired by object oriented programming, is that they fail at "information hiding". In programming an object should ideally not expose its internals. But in mathematics we may define 0 as say the empty set, making set operations on numbers syntactically valid which is kinda strange.
But yeah I think everyone has a sort of implicit understanding that 0 isn't actually the empty set. That it's merely a sort of hmm... thought experiment? That considering it 0 is a limited time offer, for the duration of the definition phase?
Maybe we come back to the isomorphism after all. "The natural numbers are something isomorphic to this set stuff I will now do"
And manias are so foundational to the relatively young field of AI that they made the term “AI winter”. We should brace for the third (I think) AI winter soon
When the current mania ends that will be more like a tech/software winter than just AI. I expect the public to have a complete distrust for tech companies and politics to push for strict regulations
That has ever been the case. As soon as it works reliably, it's not "AI" any more. Take spellcheckers or collaborative filtering as examples, but there are lots more. Hofstadter in G.E.B. said it well:
> There is a related “Theorem” about progress in AI: once some mental function is programmed, people soon cease to consider it as an essential ingredient of “real thinking”. The ineluctable core of intelligence is always in that next thing which hasn’t yet been programmed. This “Theorem” was first proposed to me by Larry Tesler, so I call it Tesler’s Theorem: “AI is whatever hasn’t been done yet.”
In this moment, the opposite is happening. Everything is getting called "AI", whether it uses LLMs to prompt LLMs about how to prompt LLMs, uses "conventional" machine learning, or just looks mysterious enough that they can expect the market to not ask questions.
I am reminded of "game AI", which for the most part has historically been just giant decision trees, encoded one way or another, because if you hook up any sort of real AI to a game entity or collection of game entities that does any sort of learning or training, even simple 1980s-era reinforcement learning, it turns out the game entities will roflstomp the human players, and the human players aren't interested in paying for that experience. We've been calling those collections of if statements and for loops "AI" for a long time, though, because who wants to hear about how deliberately stupid their opponents are?
Sure, or maybe the actual applications of “AI” are small and unobtrusive, like the dictation of doctor’s notes example, and it’s not actually the massive revolution it’s claimed to be.
I struggle to reconcile this attitude with, among other things, the massive number of exploits being discovered and the pretty straightforward utility for coding. Have you used codex recently?
Perhaps, but I think what people mean by intelligence is something that learns and adapts. If LLMs couldn’t do in context learning I don’t think people would think of them as AI, more as a kind of queryable database via natural language. There are other algorithms that learn and adapt, but in much more narrow circumstances that most people won’t obviously interact with.
Right, and my point is that now that is the bar. If you'd told me in 2008 that markov chains would be smart enough to one-shot even a trivial video game from scratch I would certainly have called that AI even if you had to do some major rigamarole to get it to work.
Because AI meant "make computers smart like people".
We taught the computer to spell check, but the computer still didn't feel smart like a person, just smart like a machine so that obviously wasn't AI. We taught it to do algebra, same thing. With LLMs though, now it really does feel like an artificial human, so this time it really is AI.
Right, but I remember when spellcheck first started to be a thing, and people were like "Wow proofreaders are out of a job, ai is so cool", but then it became normal.
a feature that i thought would be good for llm-centric languages would be making something like python's doctest prominent where you put simple little unit tests in the docstring of the function rather than in some testing module someplace else.
it would make it easy for humans to easily stub out tests with a docstring description and the tests that would guarantee certain behaviors. for the machine, it'd make it easy to add in new tests because the function+tests are in the same context window
shouldn't finding a way to kill yourself be pretty easy, even without legal DWD? you can overdose on OTC meds, or get a knife, or (in America) pick up a gun.
None of these things guarantee death and if you fail they come with some awful complications. Overdosing on Morphine is probably the easiest, most humane way to die.
