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Ishaan comments on Open thread, Jan. 26 - Feb. 1, 2015 - Less Wrong
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Comments (431)
When your goal is to do good mathematics (or good epistemology, but that's a separate discussion) you really want to do that "restrict your attention" thing.
Human intuition is to treat assumptions as part of a greater sistem. "It's raining" is one assumption, but you can also implicitly assume a bunch of other things, like rain is wet., to arrive at statements like "it's raining => wet".
This gets problematic in math. If I tell you axioms "A=B" and "B=C", you might reasonably think "A=C"...but you just implicitly assumed that = followed the transitive property. This is all well and good for superficial maths, but in deeper maths you need to very carefully define "=" and its properties. You have to strip your mind bare of everything but the axioms you laid down.
It's mostly about getting in the habit of imagining the universe as completely nothing until the axioms are introduced. No implicit beliefs about how things aught to work. All must be explicitly stated. That's why it's helpful to have the psychology of "putting building blocks in an empty space" rather than "carving assumptions out of an existing space".
I mean, that's not the only way of thinking about it, of course. Some think of it as an infinite number of "universes" and then a given axiom "pins down" a subset of those, and I guess that's closer to "assumption" psychology. It's just a way of thinking, you can choose what you like.
The real important thing is to realize that it's not just about making operations that conserve truth values..,that all the mathematical statements are arbitrarily constructed. That's the thing I didn't fully grasp before...I thought it was just about "suppose this is true, then that would be true". I thought 1+1=2 was a "fact about the actual universe" rather than a "tautology" - and I didn't quite grasp the distinction between those two terms. Until I broke free of this limitation, I wasn't able to think thoughts like "how would geometry be if the parallel postulate isn't true?", because, well, "obviously (said my incorrect intuition) the parallel postulate is factual and how can you even start considering how things would look without it?"
..as I write this, I'm realizing that this is a really hard misconception to explain to one who has never suffered from it, because the misconception seems rather bizarre in hindsight once you are set right. Maybe you just intuitively get it and so aren't seeing why some people would be led astray by thinking of it as an assumption.
Reading your reply to me, you do seem to have your thoughts correct, and you seem to gravitate toward the "pin down" way of thinking, so I think for you it is perfectly okay to mentally refer to them as assumptions. But it confused me.
I think I see what you mean. I would probably describe it not as a difference in the properties of axioms/assumptions themselves, but rather a difference in the way they are used and manipulated, a difference in the context.
I do not recall a realization similar to yours, however, perhaps because thinking in counterfactuals and following the chain of consequences comes easy to me. "Sure, let's assume A, it will lead to B, B will cause C, C is likely to trigger D which, in turn, will force F. Now you have F and is that what you expected when you wanted A?" -- this kind of structure is typical for my arguments.
But yes, I understand what you mean by blocks in empty space.
I don't think this is really the same skill as following counterfactuals and logical chains and judging internal consistency. Maybe the "parallel postulate" counterfactual was a bad example.
It's more the difference between
"Logic allows you to determine what the implications of assumptions are, and that's useful when you want to figure out which arguments and suppositions are valid" (This is where your example about counterfactuals and logical chains comes in) [1]
and
"Axioms construct / pin down universes. Our own universe is (hopefully) describable as a set of axioms". (This is where my example about building blocks comes in) [2]
And that's a good way of bridging [1] and [2].
I am not too happy with the word "universe" here because it conflates the map and the territory. I don't think the territory -- "our own universe", aka the reality -- is describable as a set of axioms.
I'll accept that you can start with a set of axioms and build a coherent, internally consistent map, but the question of whether that map corresponds to anything in reality is open.
I very strongly do. I think the universe is describable by math. I think there exist one or more sets of statements that can describe the observable universe in its entirety. I can't imagine the alternative, actually. What would that even be like?
That's actually the only fundamental and unprovable point that I take on faith, from which my entire philosophy and epistemology blossoms. ("Unprovable" and "faith" because it relies on you to buy into the idea of "proof" and "logic" in the first place, and that's circular)
I don't necessarily think we can find such a set of axioms. mind you. I can't guarantee that there are a finite number of statements required, or that the human mind is necessarily is capable of producing/comprehending said statements, or even that any mind stuck within the constraints of the universe itself is capable. (I suppose you can take issue with the use of the word "describable" at this point). But I do think the statements exist, in some platonic sense, and that if we buy into logic we can at least know that they exist even if we can't know them directly. (In the same sense that we can often know whether or not a solution exists even if it's impossible to find)
No "universally compelling arguments in math and science" applies here: I can't really prove it to you, but I think anyone who believes in a lawful, logical universe will come around to agree after thinking about it long enough.
What if it requires an infinite set of statements to specify? Consider the hypothetical of a universe where there are no elementary particles but each stage is made up of something still simpler. Or consider something like the Standard Model but where the constants are non-computable. Would either of these fit what you are talking about?
Yes, that would fit in what I am talking about. I have a bad habit of constantly editing posts as I write, so you might have seen my post before I wrote this part.
Such a universe wouldn't even necessarily be "complicated". A single infinite random binary string requires an infinitely long statement to fully describe (but we can at least partially pin it down by finitely describing a multiverse of random binary strings)
Yes, thank you, I don't think that was there when I read it. I'm not sure then that the statement that universe runs on math at that point has any degree of meaning.
It seems self evident once you get it, but it's not obvious.
In the general population you get these people who say "well, if it's all just atoms, whats the point"? They don't realize that everything has to run on logic regardless of whether the underlying phenomenon is atoms or souls or whatever. (Or at least, they don't agree. I shouldn't say "realize" because the whole thing rests on circular arguments.)
It also provides sort of ontological grounding onto which further rigor can be built. It's nice to know what we mean when we say we are looking for "truth".
Interesting. We seem to have a surprisingly low-level (in the sense of "basic") disagreement.
A couple of questions. Does your view imply that the universe is deterministic? And if "I can't guarantee ... even that any mind stuck within the constraints of the universe itself is capable" then I am not sure what does your position actually mean. Existing "in some platonic sense" is a very weak claim, much weaker than "the universe runs on math" (and, by implication, nothing else).
No, randomness is a thing.
Practically, it means we'll never run into logical contradictions in the territory.
Theoretically, it means we will never encounter a phenomenon that in theory (in a platonic sense) cannot be fully described. In practice, we might not be able to come up with a complete description.
In a platonic sense, the territory must have at least one (or more) maps that totally describes it, but these maps may or may not be within the space of maps that minds stuck within the constraints of said territory can create.
As the only claim that I've been taking on faith and the foundation for all that follows, it is meant to be a weak claim.
I'm trying to whittle down the principles I must take on faith before forming a useful philosophy to as small a base as possible, and this is where I am at right now.
Descartes's base was "I think before I am", and from there he develops everything else he believes. My base is "things are logical" (which further expands into "all things have descriptions which don't contain contradictions")
Maps require a mind, a consciousness of some sort. Handwaving towards "platonic sense" doesn't really solve the issue -- are you really willing to accept Plato's views of the world, his universals?
The problem is that, as stated, this claim (a) could never be decided; and (b) has no practical consequences whatsoever.
Think of it this way: Godel's incompleteness theorem demonstrates there will always be statements about the natural numbers that are true, but that are unprovable within the system. It's perfectly okay for us to talk about those hypothetical statements as existing in the "platonic" sense, even though we might never really have them in the grasps of our minds and notebooks.
Similarly, it's okay for us to talk about a space of maps even while knowing we can't necessarily generate every map in that space due to constraints on us that might exist. I haven't actually read any Plato, so I might be misusing the term. I'm just using the word "platonic" to describe the entire space of maps, including the ungraspable ones. "Platonic" is merely to distinguish those things from things that actually exist in the territory.
part a) I endorse Dxu's defense of what I said, and see my reply to him for my objections to what he said.
part b) I disagree in principle with the idea that the validity of things depends on practical consequences, However, the whole point here is to create a starting point from which the rest of everything can be derived, and the rest of everything does have practical consequences
(it may be fair to say that there is no practical reason to derive them from a small starting point, but that is questioning the practicality of philosophy in general)
So, you're talking about things you can, basically, imagine.
In which sense do "ungraspable maps" exist, but herds of rainbow unicorns gallivanting on clouds do not?
I concur with your disagreement :-) but here we have TWO things: (1) unprovable and unfalsifiable; and (2) of no practical consequences.
Consider the claim that there is God, He created the universe, but then left forever. The same two things could be said of this claim as well.
Really? If I wrote a physics engine in, let's say, Java, is that not a(n approximate) map of physical reality? I would say so. Yet the physics engine isn't conscious. It doesn't have a mind. In fact, the simulation isn't even dependent on its substrate--I could save the bytecode and run it on any machine that has the JVM installed. Moreover, the program is entirely reducible to a series of abstract (Platonic) mathematical statements, no substrates required at all, and hence no "minds" or "consciousness" required either
In what sense is the physics engine described above not a map?
Hence, I assume, Ishaan's use of the word "faith".
In practice, it means we will never find something in the territory that is logically contradictory. (Not that we're likely to find such a thing in the first place, of course, but if we did, it would falsify Ishaan's claim, so it's not unfalsifiable, though it is untestable. Seeing that Ishaan has stated that he/she is taking this claim "on faith", though, I can't see that untestability is a big issue here.)
Personally, I disagree with Ishaan's approach of taking anything on faith, even logic itself. That being said, if you really need to take something on faith, I have trouble thinking of a better claim to do so with than the claim that "everything has a logical description".
True, but it was written by someone with a conscious mind (you), just as a map drawn on paper by a cartographer was drawn by someone with a mind.
An interesting question. No, I am not sure I want to define maps this way. Would you, for example, consider the distribution of ecosystems on Earth to be a map of the climate?
I tend to think of maps as representations and these require an agent.
I don't understand what this means -- logic is in the mind. Can you give me an example that's guaranteed to be not a misunderstanding? By a "misunderstanding" I mean something like the initial reaction to the double-slit experiment: there is a logical contradiction, the electron goes through one slit, but it goes through both slits.
Let me unpack "faith" a little bit, then, because it's not like regular faith. I only use the word "faith" because it's the closest word I know to what I mean.
I agree with the postmodern / nihilist / Lesswrong's idea of "no universally compelling arguments" in morality, math, and science. Everything that comes out of my mind is a property of how my mind is constructed.
When I say that I take logic "on faith", what I'm really saying is that I have no way to justify it, other than that human minds run that way (insert disclaimers about, yes, I know human minds don't actually run that way)
I don't have a word to describe this, the sense that I'm ultimately going to follow my morality and my cognitive algorithm even while accepting that is not and cannot be justification for them outside my own mind. (I kinda want to call this "epistemic particularism" to draw an analogy from political particularism, but google says that term is already in use and I haven't read about it so I am not sure whether or not it means what I want to use it for. I think it does, though.)
I think there would exist a way to logically describe the universe Eliezer would find himself in.
I disagree with Eliezer here. If the people in this universe want to use "2", "3", and "+" to describe what is happening to them, then their "3" does not have the same meaning as our "3" We are referring to something with integer properties, and they are referring to something with other properties. I think Wittgenstein would have a few choice words for Eliezer here (although I've only read summaries of his thoughts, I think he's basically saying what I'm saying).
I don't think Eliezer should be interpreted as admitting that the territory might be illogical. I think he just made a mistake concerning what definitions are. (I'm not saying your interpretation is unreasonable. I'm saying that the fact that your interpretation is reasonable is a clue that Eliezer made a logical error somewhere, and this is the error I think he made. (I'd be curios to know if he'd agree with me that he made an error in saying 2+2=3 is not a re-definition. Judging from his other writing I suspect he would.)
(And again, it's circular because it has to be. The fact that your perfectly logical interpretation of Eliezer basically just invoked the Principle of Explosion indicates the statements themselves contain a logical error, but none of this works if you don't buy into logic to begin with. You're throwing out logic, and I'm convincing you that this is illogical - which is a silly thing to do, but still.)
Eliezer's weird universe is still in the space of logic-land. We've just constructed a different logical system, where 2+2=3 because they are different;y defined now. It's not like Eliezer is simultaneously experiencing and not experiencing three earplugs or something. An illogical world isn't merely different from our world - it's incomprehensible and indescribable nonsense insofar as our brain is concerned. If you're still looking at evidence and drawing conclusions, you're still using logic. (Inb4 paraconsistent and fuzzy logic, the meta rules handling the statements still use the same tautology-contradiction structure common to all math)