This is a pet peeve of mine, Axioms as assumptions (or self-evident truths) seem to be a very prevalent mode of thinking in educated people not exposed to much formal maths.
Well, it’s hard to articulate. There’s of course nothing wrong with assumptions per se, since axioms indeed are assumptions, my peeve is with the baggage that comes with it. People say things like “what if the assumptions are wrong?”, or “I don’t think that axiom is clearly true”, or “In the end you can’t prove that your axioms are true”.
These questions would be legitimate if the goal were physical truth, or a self-justifying absolute system of knowledge or whatever, but in the context of mathematics, we’re not so interested in the content of the assumptions as we are in the structure we can get out of them.
In my experience, this kind of thing happens most often when philosophically inclined people talk about things like the Peano axioms, where it’s possible to think we’re discussing some ideal entity that exists independently of thought, and disappears when people are exposed to, say, the vector space axioms, or some set of axioms of set theory, where it becomes clear that axioms aren’t descriptions but definitions.
Actually, you can ignore everything I’ve said above, I’ve figured out precisely what I have a problem with. It’s the popular conception of axioms as descriptive rather than prescriptive. Which, I suppose OP was also talking about when they mentioned building blocks as opposed to assumptions.
People say things like “what if the assumptions are wrong?”
That’s a valid question in a slightly different formulation: “what if we pick a different set of assumptions?”
“In the end you can’t prove that your axioms are true”
But that, on the other hand, is pretty stupid.
It’s the popular conception of axioms as descriptive rather than prescriptive.
Well, normally you want your axioms to be descriptive. If you’re interested in reality, you would really prefer your assumptions/axioms to match reality in some useful way.
I’ll grant that math is not particularly interested in reality and so tends to go off on exploratory expeditions where reality is seen as irrelevant. Usually it turns out to be true, but sometimes the mathematicians find a new (and useful) way of looking at reality and so the expedition does loop back to the real.
But that’s a peculiarity of math. Outside of that (as well as some other things like philosophy and literary criticism :-D) I will argue that you do want axioms to be descriptive.
I don’t think it’s “wrong” in the sense of “incorrect”… it’s just that if you don’t also realize that axioms are arbitrarily constructed “universes” and that all math takes place in the context of said fictional “universes”, you kind of miss the deeper point. Thinking of them as assumptions is a simple way to teach them to beginners, but that’s a set of training wheels that aught to be removed sooner rather than later, especially if you are using axioms for math.
And , handy side effect, your intuition for epistemology gets better when you realize that. (In my opinion).
if you don’t also realize that axioms are arbitrarily constructed “universes”
Well, they are a set of assumptions on the basis of which you proceed forward. Starting with a different set will land you in a different world built on different assumptions. But I see it as a characteristic of assumptions in general, I still don’t see what’s so special about axioms.
When you assume the parallel postulate, for example, you are restricting your attention to the class of models of geometry in which the parallel postulate holds. I don’t think that’s a useful way of thinking about other kinds of assumptions such as “the sun will rise tomorrow” or “the intended audience for this comment will be able to understand written English”.
(At least for me, I think that the critical axiom-related insight was the difference between a set of axioms and a model of those axioms.)
I don’t think that’s a useful way of thinking about other kinds of assumptions such as “the sun will rise tomorrow” or “the intended audience for this comment will be able to understand written English”.
What is useful depends on your goals. The difference is still not clear to me—e.g. by assuming that “the intended audience for this comment will be able to understand written English” you are restricting your attention to the class of situations in which people to whom you address your comment can understand English.
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]
Starting with a different set will land you in a different world built on different assumptions. But I see it as a characteristic of assumptions in general, I still don’t see what’s so special about axioms.
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 don’t think the territory—“our own universe”, aka the reality—is describable as a set of axioms.
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.
I very strongly do. I think the universe runs on math. I think there exist one or more sets of statements that can the universe in its entirety. I can’t actually imagine the alternative, actually.
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.
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)
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).
Does your view imply that the universe is deterministic?
No, randomness is a thing.
what does your position actually mean.
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.
a very weak claim,
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”)
the territory must have at least one (or more) maps
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?
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.
The problem is that, as stated, this claim (a) could never be decided; and (b) has no practical consequences whatsoever.
Maps require a mind, a consciousness of some sort.
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.
The problem is that, as stated, this claim (a) could never be decided; and (b) has no practical consequences whatsoever.
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.
Yes, all the logically consistent systems we can imagine, and more. (See the Godel analogy above for “and more”.)
In which sense do “ungraspable maps” exist, but herds of rainbow unicorns gallivanting on clouds do not?
You...can’t imagine logically coherent systems with rainbow unicorns on clouds?
Keep in mind, we’re making distinctions between “real tangible reality” and “the space of logically coherent systems”. Your ad-absurdum works by using the word “exist” to confound those two, in a “tree falls in the forest” sort of manner. I specifically used the word “platonic” hoping to separate those ideas. It’s merely an inconvenience of language that we don’t have the words to distinguish the tautological “reality” and “existence” of 1+1=2 from the reality of “look, there’s a thing over there”. People say “in Integers, there exists an odd number between every even number” but it’s not that sort of “existence”.
Maps require a mind, a consciousness of some sort.
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?
The problem is that, as stated, this claim (a) could never be decided;
Hence, I assume, Ishaan’s use of the word “faith”.
and (b) has no practical consequences whatsoever.
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”.
I disagree with Ishaan’s approach of taking anything on faith, even logic itself.
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.)
Suppose I got up one morning, and took out two earplugs, and set them down next to two other earplugs on my nighttable, and noticed that there were now three earplugs, without any earplugs having appeared or disappeared
I think there would exist a way to logically describe the universe Eliezer would find himself in.
(There are redefinitions, but those are not “situations”, and then you’re no longer talking about 2, 4, =, or +.) But that doesn’t make my belief unconditional.
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)
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.
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.
If I wrote a physics engine in, let’s say, Java, is that not a(n approximate) map of physical reality?
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.
we will never find something in the territory that is logically contradictory
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.
Can you give me an example that’s guaranteed to be not a misunderstanding?
You can never perceive red and not perceive red simultaneously, but if you could, that would embody a logical contradiction in the territory.
(Tree falls in the forest” type word play doesn’t count)
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.
That is not a contradiction in the evidence, that is simply a falsification of a prior hypothesis (as well as a violation of human physical intuition). However, If you were to insist upon retaining your old model of the universe after seeing the results of the experiment, then you would have a contradiction within your view of reality (which must accommodate both your previous beliefs and the new evidence)
This is the sort of thing I meant when I said earlier in the thread that the insight I’m referring to here is what led me to realize that there is nothing particularly odd about intuition-violating physics. There’s no reason the axioms of the universe need to be intuitive—they need only be logically consistent.
But, it’s good that you brought up this example: I think Eliezer’s example that Dxu linked, with 2+2=3, is similar to the double slit experiment—it’s violating prior intuitions and hypothesis about the world, not violating logic.
You can never perceive red and not perceive red simultaneously, but if you could, that would embody a logical contradiction in the territory.
I don’t understand. Perception happens in the mind, I don’t see anything unusual about the ability to screw up a mind (via drugs, etc) to the extent that it thinks it perceives red and does not perceive red simultaneously. Why would that imply a “logical contradiction in the territory”?
I’m not talking about it thinks it perceives red even when it doesn’t perceive red—that’s “tree falls in the forest” thinking. I’m talking about simultaneously thinking you perceive red and not thinking your perceive red.
But yes—you could screw up a mind sufficiently such that it thinks it’s perceiving red and not perceiving red simultaneously. Such a mind isn’t following the normal rules (and the rules of logic and so on arise from the rules of the mind in the first place, so of course you could sufficiently destroy or disable a mind such that it no longer things that way—there’s no deeper justification, so you are forced to trust the normal mental process to some degree...that’s what the “no universally compelling arguments and therefore you just have to yourself” spiel I was giving higher in the thread stems from).
I guess I bite the bullet, there is no real falsifying here? I did say you have to take it on faith to an extent because there is no other way. It’s a foundational premise for building an epistemic structure, not a theory as such.
Anyhow, I’m not sure we’re talking about the same thing anymore. If you don’t accept that the universe follows a certain logic, the idea of “falsifying” has no foundation anyway.
Well, I was thinking that in those other cases, you consider the other possibility (e.g., that nobody who reads my comment will understand it) and dismiss it as unlikely or unimportant. It doesn’t even make sense to ask “but what if it turns out that the parallel postulate doesn’t actually hold after all?”
This is a pet peeve of mine, Axioms as assumptions (or self-evident truths) seem to be a very prevalent mode of thinking in educated people not exposed to much formal maths.
What’s wrong with treating axioms as assumptions?
Well, it’s hard to articulate. There’s of course nothing wrong with assumptions per se, since axioms indeed are assumptions, my peeve is with the baggage that comes with it. People say things like “what if the assumptions are wrong?”, or “I don’t think that axiom is clearly true”, or “In the end you can’t prove that your axioms are true”.
These questions would be legitimate if the goal were physical truth, or a self-justifying absolute system of knowledge or whatever, but in the context of mathematics, we’re not so interested in the content of the assumptions as we are in the structure we can get out of them.
In my experience, this kind of thing happens most often when philosophically inclined people talk about things like the Peano axioms, where it’s possible to think we’re discussing some ideal entity that exists independently of thought, and disappears when people are exposed to, say, the vector space axioms, or some set of axioms of set theory, where it becomes clear that axioms aren’t descriptions but definitions.
Actually, you can ignore everything I’ve said above, I’ve figured out precisely what I have a problem with. It’s the popular conception of axioms as descriptive rather than prescriptive. Which, I suppose OP was also talking about when they mentioned building blocks as opposed to assumptions.
That’s a valid question in a slightly different formulation: “what if we pick a different set of assumptions?”
But that, on the other hand, is pretty stupid.
Well, normally you want your axioms to be descriptive. If you’re interested in reality, you would really prefer your assumptions/axioms to match reality in some useful way.
I’ll grant that math is not particularly interested in reality and so tends to go off on exploratory expeditions where reality is seen as irrelevant. Usually it turns out to be true, but sometimes the mathematicians find a new (and useful) way of looking at reality and so the expedition does loop back to the real.
But that’s a peculiarity of math. Outside of that (as well as some other things like philosophy and literary criticism :-D) I will argue that you do want axioms to be descriptive.
I don’t think it’s “wrong” in the sense of “incorrect”… it’s just that if you don’t also realize that axioms are arbitrarily constructed “universes” and that all math takes place in the context of said fictional “universes”, you kind of miss the deeper point. Thinking of them as assumptions is a simple way to teach them to beginners, but that’s a set of training wheels that aught to be removed sooner rather than later, especially if you are using axioms for math.
And , handy side effect, your intuition for epistemology gets better when you realize that. (In my opinion).
Well, they are a set of assumptions on the basis of which you proceed forward. Starting with a different set will land you in a different world built on different assumptions. But I see it as a characteristic of assumptions in general, I still don’t see what’s so special about axioms.
When you assume the parallel postulate, for example, you are restricting your attention to the class of models of geometry in which the parallel postulate holds. I don’t think that’s a useful way of thinking about other kinds of assumptions such as “the sun will rise tomorrow” or “the intended audience for this comment will be able to understand written English”.
(At least for me, I think that the critical axiom-related insight was the difference between a set of axioms and a model of those axioms.)
What is useful depends on your goals. The difference is still not clear to me—e.g. by assuming that “the intended audience for this comment will be able to understand written English” you are restricting your attention to the class of situations in which people to whom you address your comment can understand English.
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.
Yes, all the logically consistent systems we can imagine, and more. (See the Godel analogy above for “and more”.)
You...can’t imagine logically coherent systems with rainbow unicorns on clouds?
Keep in mind, we’re making distinctions between “real tangible reality” and “the space of logically coherent systems”. Your ad-absurdum works by using the word “exist” to confound those two, in a “tree falls in the forest” sort of manner. I specifically used the word “platonic” hoping to separate those ideas. It’s merely an inconvenience of language that we don’t have the words to distinguish the tautological “reality” and “existence” of 1+1=2 from the reality of “look, there’s a thing over there”. People say “in Integers, there exists an odd number between every even number” but it’s not that sort of “existence”.
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”.
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)
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.
You can never perceive red and not perceive red simultaneously, but if you could, that would embody a logical contradiction in the territory.
(Tree falls in the forest” type word play doesn’t count)
That is not a contradiction in the evidence, that is simply a falsification of a prior hypothesis (as well as a violation of human physical intuition). However, If you were to insist upon retaining your old model of the universe after seeing the results of the experiment, then you would have a contradiction within your view of reality (which must accommodate both your previous beliefs and the new evidence)
This is the sort of thing I meant when I said earlier in the thread that the insight I’m referring to here is what led me to realize that there is nothing particularly odd about intuition-violating physics. There’s no reason the axioms of the universe need to be intuitive—they need only be logically consistent.
But, it’s good that you brought up this example: I think Eliezer’s example that Dxu linked, with 2+2=3, is similar to the double slit experiment—it’s violating prior intuitions and hypothesis about the world, not violating logic.
I don’t understand. Perception happens in the mind, I don’t see anything unusual about the ability to screw up a mind (via drugs, etc) to the extent that it thinks it perceives red and does not perceive red simultaneously. Why would that imply a “logical contradiction in the territory”?
I’m not talking about it thinks it perceives red even when it doesn’t perceive red—that’s “tree falls in the forest” thinking. I’m talking about simultaneously thinking you perceive red and not thinking your perceive red.
But yes—you could screw up a mind sufficiently such that it thinks it’s perceiving red and not perceiving red simultaneously. Such a mind isn’t following the normal rules (and the rules of logic and so on arise from the rules of the mind in the first place, so of course you could sufficiently destroy or disable a mind such that it no longer things that way—there’s no deeper justification, so you are forced to trust the normal mental process to some degree...that’s what the “no universally compelling arguments and therefore you just have to yourself” spiel I was giving higher in the thread stems from).
But you said “that would embody a logical contradiction in the territory” and that doesn’t seem to be so any more.
My original question, if you recall, was for an example of something—anything—that would be falsify your theory.
I guess I bite the bullet, there is no real falsifying here? I did say you have to take it on faith to an extent because there is no other way. It’s a foundational premise for building an epistemic structure, not a theory as such.
Anyhow, I’m not sure we’re talking about the same thing anymore. If you don’t accept that the universe follows a certain logic, the idea of “falsifying” has no foundation anyway.
Well, I was thinking that in those other cases, you consider the other possibility (e.g., that nobody who reads my comment will understand it) and dismiss it as unlikely or unimportant. It doesn’t even make sense to ask “but what if it turns out that the parallel postulate doesn’t actually hold after all?”
Am I explaining myself any better?
Is my reply to Ishaan helpful?
It’s not very amenable to teaching.
The grandparent said “prevalent mode of thinking in educated people”—what’s convenient for teaching is not very relevant here.