StephenR

2014 Less Wrong Census/Survey

All done.

Astray with the Truth: Logic and Math

Believing that a theory is true that says "true" is not a thing theories can be is obviously silly.

Oh okay. This is a two-part misunderstanding.

I'm not saying that theories can't be true, I'm just not talking about this truth thing in my meta-model. I'm perfectly a-okay with models of truth popping up wherever they might be handy, but I want to taboo the intuitive notion and refuse to explicate it. Instead I'll rely on other concepts to do much of the work we give to truth, and see what happens. And if there's work that they can't do, I want to evaluate whether it's important to include in the meta-model or not.

I'm also not saying that my theory is true. At least, not when I'm talking from within the theory. Perhaps I'll find certain facets of the correspondence theory useful for explaining things or convincing others, in which case I might claim it's true. My epistemology is just as much a model as anything else, of course; I'm developing it with certain goals in mind.

I were talking about physical systems, not physical laws. Computers, living cells, atoms, the fluid dynamics of the air... "Applied successfully in many cases", where "many" is "billions of times every second"

The math we use to model computation is a model and a tool just as much as computers are tools; there's nothing weird (at least from my point of view) about models being used to construct other tools. Living cells can be modeled successfully with math, you're right; but that again is just a model. And atoms are definitely theoretical constructs used to model experiences, the persuasive images of balls or clouds they conjure notwithstanding. Something similar can be said about fluid dynamics.

I don't mean any of this to belittle models, of course, or make them seem whimsical. Models are worth taking *seriously*, even if I don't think they should be taken *literally*.

Then ZFC is not one of those cores ones, just one of the peripheral ones. I'm talking ones like set theory as a whole, or arithmetic, or Turing machines.

The best example in the three is definitely arithmetic; the other two aren't convincing. Math was done without set theory for ages, and besides we have other foundations available for modern math that can be formulated entirely without talking about sets. Turing machines can be replaced with logical systems like the lambda calculus, or with other machine models like register machines.

Arithmetic is more compelling, because it's very sticky. It's hard not to take it literally, and it's hard to imagine things without it. This is because some of the ideas it constitutes are at the core cluster of our categories, i.e. they're very sticky. But could you imagine that *some agent* might a) have goals that never require arithmetical concepts, and b) that there could be models that are non-arithmetical that could be used toward some of the same goals for which we use arithmetic? I can imagine ... visualise, actually, both, although I would have a very hard time translating my visual into text without going very meta first, or else writing a ridiculously long post.

Astray with the Truth: Logic and Math

Many of them render considering them not true pointless, in the sense all my reasoning and senses are invalid if they don't hold so I might as well give up and save computing time by conditioning on them being true.

I call these sorts of models *sticky*, in the sense that they are pervasive in our perception and categorisation. Sitcky categories are the sort of thing that we have a hard time not taking literally. I haven't gone into any of this yet, of course, but I like it when comments anticipate ideas and continue trains of thought.

Maybe a short run-long run model would be good to illustrate this stickiness. In the short run, perception is fixed; this also fixes certain categories, and the "degree of stickiness" that different categories have. For example, chair is remarkably hard to get rid of, whereas "corpuscle" isn't quite as sticky. In the long run, when perception is free, no category needs to be sticky. At least, not unless we come up with a more restrictive model of possible perceptions. I don't think that such a restrictive model would be appropriate in a background epistemology. That's something that agents will develop for themselves based on their needs and perceptual experience.

Many of them are directly implemented in physical systems

Different mathematical models of human perceptual experience might be perfectly suitable for the same purpose.. Physics should be the clearest example, since we have undergone many different changes of mathematical models, and are currently experiencing a plurality of theories with different mathematics in cosmology. The differences between classical mechanics and quantum mechanics should in particular show this nicely: different formalisms, but very good models of a large class of experiences.

you cant have one of them without ALL the others.

I think you slightly underestimate the versatility of mathematicians in making their systems work despite malfunctions. For instance, even if ZFC were proved inconsistent (as Edward Nelson hopes to do), we would not *have* to abandon it as a foundation. Set theorists would just do some hocus pocus involving ordinals, and voila! all would be well. And there are several alternative formulations of arithmetic, analysis, topology, etc. which are all adequate for most purposes.

You might try to imagine an universe without math.

In the case of some math, this is easy to do. In other cases it is not. This is because we don't experience the freefloating perceptual long term, not because certain models are necessary for all possible agents and perceptual content.

Astray with the Truth: Logic and Math

That was a wonderful comment. I hope you don't mind if I focus on the last part in particular. If you'd rather I addressed more I can accommodate that, although most of that will be signalling agreement.

To assert P is equivalent to asserting "P is true" (the deflationary theory in reverse). That is still true if P is of the form "so and so works". Pragmatism is not orthogonal to, or transcendent of, truth. Pragmatists need to be concerned about what truly works.

I'll note a few things in reply to this:

- I'm fine with some conceptual overlap between my proposed epistemology and other epistemologies and vague memes.
- You might want to analyse statements "P" as meaning/being equivalent to "P is true," but I am not going to include any explication of "true" in my epistemology for that analysis to anchor itself to.
- Continuing the above, part of what I am doing is tabooing "truth," to see if we can formulate an epistemology-like framework without it.
- What "truly works" is more of a
*feeling*or a*proclivity*than a proposition, until of course an agent develops a*model*of what works and why.

What is right is contextual truth.

I agree with you here absolutely, modulo vocabulary. I would rather say that no single framework is universally appropriate (problem of induction) and that developing different tools for different contexts is shrewd. But what I just said is more of a model inspired by my epistemology than part of the epistemology itself.

Astray with the Truth: Logic and Math

Applying it to what problem? (If you mean the physics posts you linked to, I need more time to digest it fully)

No, not that comment, I mean the initial post. The problem is handling mathematical systems in an epistemology. A lot of epistemologies have a hard time with that because of ontological issues.

Nobody actually conceptualises science as being about deriving from thinking "pink is my favority color and it isn't" -> "causality doesn't work".

No, but many people hold the view that you can talk about valid statements as constraining ontological possibilities. This is including Eliezer of 2012. If you read the High Advance Epistemology posts on math, he does reason about the particular logical laws constraining how the physics of time and space work in our universe. And the view is very old, going back to before Aristotle, through Leibniz to the present.

Astray with the Truth: Logic and Math

If you call for a core change in epistemology it sounds like you want more than that. To me it's not clear what that more happens to be.

I'm going to have to do some strategic review on what exactly I'm not being clear about and what I need to say to make it clear.

In case you don't know the local LW definition of rationality is : "Behaving in a way that's likely to make you win."

Yes, I share that definition, but that's only the LW definition of instrumental rationality; epistemic rationality on the other hand is making your map more accurately reflect the territory. Part of what I want is to scrap that and judge epistemic matters instrumentally, like I said in the conclusion and addendum.

Still, it's clear I haven't said quite enough. You mentioned examples, and that's kind of what this post was intended to be: an example of applying the sort of reasoning I want to a problem, and contrasting it with epistemic rationality reasoning.

Part of the problem with generating a whole bunch of specific examples is that it wouldn't help illustrate the change much. I'm not saying that science as it's practised in general needs to be radically changed. Mostly things would continue as normal, with a few exceptions (like theoretical physics, but I'm going to have to let that particular example stew for a while before I voice it outside of private discussions).

The main target of my change is the way we conceptualise science. Lots of epistemological work focuses on idealised caricatures that are too prescriptive and poorly reflect how we managed to achieve what we did in science. And I think that having a better philosophy of science will make thinking about some problems in existential risk, particularly FAI, easier.

Astray with the Truth: Logic and Math

I think there's a bit of a misunderstanding going on here, though, because I am perfectly okay with people using classical logic if they like. Classical logic is a great way to model circuits, for example, and it provides some nice reasoning heuristics.There's nothing in my position that commits us to abandoning it entirely in favour of intuitionistic logic.

Intuitionistic logic is applicable to at least three real-world problems: formulating foundations for math, verifying programmes, and computerised theorem-proving. The last two in particular will have applications in everything from climate modeling to population genetics to quantum field theory.

As it happens, mathematician Andrej Bauer wrote a much better defence of doing physics with intuitionistic logic than I could have: http://math.andrej.com/2008/08/13/intuitionistic-mathematics-for-physics/

Astray with the Truth: Logic and Math

I've added an addendum. If reading that doesn't help, let me know and I'll summarise it for you in another way.

Astray with the Truth: Logic and Math

Intuitionistic logic can be interpreted as the logic of finite verification.

Truth in intuitionistic logic is just provability. If you assert A, it means you have a proof of A. If you assert ¬A then you have a proof that A implies a contradiction. If you assert A ⇒B then you can produce a proof of B from A. If you assert A ∨ B then you have a proof of at least one of A or B. Note that the law of excluded middle fails here because we aren't allowing sentences A ∨ ¬A where you have no proof of A or that A implies a contradiction.

In all cases, the assertion of a formula must correspond to a proof, proofs being (of course) finite. Using this idea of finite verification is a nice way to develop topology for computer science and formal epistemology (see Topology via Logic by Steven Vickers). Computer science is concerned with verification as proofs and programmes (and the Curry-Howard isomorphism comes in handy there), and formal epistemology is concerned with verification as observations and scientific modeling.

That isn't exactly a specific example, but a class of examples. Research on this is currently very active.

I entered "Other: electic mixture" on the survey. On my Facebook profile, I elaborate this as "classical liberalism, Rawlsian liberalism, reactionary, left-libertarianism, conservatism, and techno-futurism." Ideologies are for picking apart, not buying wholesale. I gather a variety of them together and cut away the rotten parts like moldy cheese. What's left is something much more workable than the originals.