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hen comments on Self-Congratulatory Rationalism - Less Wrong
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True, and a discovery like that might require us to make some pretty fundamental changes. But I don't think Morpheus could be right about the universe's relation to math. No universe, I take it, 'runs' on math in anything but the loosest figurative sense. The universe we live in is subject to mathematical analysis, and what reason could we have for thinking any universe could fail to be so? I can't say for certain, of course, that every possible universe must run on math, but I feel safe in claiming that we've never imagined a universe, in fiction or through something like religion, which would fail to run on math.
More broadly speaking, anything that is going to be knowable at all is going to be rational and subject to rational understanding. Even if someone has some very false beliefs, their beliefs are false not just jibber-jabber (and if they are just jibber-jabber then you're not talking to a person). Even false beliefs are going to have a rational structure.
That is a fact about you, not a fact about the universe. Nobody could imagine light being both a particle and a wave, for example, until their study of nature forced them to.
People could imagine such a thing before studying nature showed they needed to; they just didn't. I think there's a difference between a concept that people only don't imagine, and a concept that people can't imagine. The latter may mean that the concept is incoherent or has an intrinsic flaw, which the former doesn't.
In the interest of not having this discussion degenerate into an argument about what "could" means, I would like to point out that your and hen's only evidence that you couldn't imagine a world that doesn't run on math is that you haven't.
For one thing, "math" trivially happens to run on world, and corresponds to what happens when you have a chain of interactions. Specifically to how one chain of physical interactions (apples being eaten for example) combined with another that looks dissimilar (a binary adder) ends up with conclusion that apples were counted correctly, or how the difference in count between the two processes of counting (none) corresponds to another dissimilar process (the reasoning behind binary arithmetic).
As long as there's any correspondences at all between different physical processes, you'll be able to kind of imagine that world runs on world arranged differently, and so it would appear that world "runs on math".
If we were to discover some new laws of physics that were producing incalculable outcomes, we would just utilize those laws in some sort of computer and co-opt them as part of "math", substituting processes for equivalent processes. That's how we came up with math in the first place.
edit: to summarize, I think "the world runs on math" is a really confused way to look at how world relates to the practice of mathematics inside of it. I can perfectly well say that the world doesn't run on math any more than the radio waves are transmitted by mechanical aether made of gears, springs, and weights, and have exact same expectations about everything.
"There is non trivial subset of maths whish describes physical law" might be better way of stating it
It seems to me that as long as there's anything that is describable in the loosest sense, that would be taken to be true.
I mean, look at this, some people believe literally that our universe is a "mathematical object", what ever that means (tegmarkery), and we haven't even got a candidate TOE that works.
edit: I think the issue is that Morpheus confuses "made of gears" with "predictable by gears". Time is not made of gears, and neither are astronomical objects, but a clock is very useful nonetheless.
I don't see why "describable" would necessarily imply "describable mathematically". I can imagine a qualia only universe,and I can imagine the ability describe qualia. As things stand, there are a number of things that can't be described mathematically
Example?
Qualia, the passage of time, symbol grounding..
Absolutely, it's a fact about me, that's my point. I just also think it's a necessary fact.
What's your evidence for this? Keep in mind that the history of science is full of people asserting that X has to be the case because they couldn't imagine the world being otherwise, only for subsequent discoveries to show that X is not in fact the case.
Name three (as people often say around here).
Well, the most famous (or infamous) is Kant's argument the space must be flat (in the Euclidean sense) because the human mind is incapable of imagining it to be otherwise.
Another example was Lucretius's argument against the theory that the earth is round: if the earth were round and things fell towards its center than in which direction would an object at the center fall?
Not to mention the standard argument against the universe having a beginning "what happened before it?"
I don't intend to bicker, I think your point is a good one independently of these examples. In any case, I don't think at least the first two of these examples of the phenomenon you're talking about.
I think this comes up in the sequences as an example of the mind-projection fallacy, but that's not right. Kant did not take himself to be saying anything about the world outside the mind when he said that space was flat. He only took himself to be talking about the world as it appears to us. Space, so far as Kant was concerned, was part of the structure of perception, not the universe. So in the Critique of Pure Reason, he says:
So Kant is pretty explicit that he's not making a claim about the world, but about the way we percieve it. Kant would very likely poke you in chest and say "No you're committing the mind-projection fallacy for thinking that space is even in the world, rather than just a form of perception. And don't tell me about the mind-projection fallacy anyway, I invented that whole move."
This also isn't an example, because the idea of a spherical world had in fact been imagined in detail by Plato (with whom Lucretius seems to be arguing), Aristotle, and many of Lucretius' contemporaries and predecessors. Lucretius' point couldn't have been that a round earth is unimaginable, but that it was inconsistent with an analysis of the motions of simple bodies in terms of up and down: you can't say that fire is of a nature to go up if up is entirely relative. Or I suppose, you can say that but you'd have to come up with a more complicated account of natures.
And in particular he claimed that this showed it had to be Euclidean because humans couldn't imagine it otherwise. Well, we now know it's not Euclidean and people can imagine it that way (I suppose you could dispute this, but that gets into exactly what we mean by "imagine" and attempting to argue about other people's qualia).
No, he never says that. Feel free to cite something from Kant's writing, or the SEP or something. I may be wrong, but I just read though the Aesthetic again, and I couldn't find anything that would support your claim.
EDIT: I did find one passage that mentions imagination:
I've edited my post accordingly, but my point remains the same. Notice that Kant does not mention the flatness of space, nor is it at all obvious that he's inferring anything from our inability to imagine the non-existence of space. END EDIT.
You gave Kant's views about space as an example of someone saying 'because we can't imagine it otherwise, the world must be such and such'. Kant never says this. What he says is that the principles of geometry are not derived simply from the analysis of terms, nor are they empirical. Kant is very, very, explicit...almost annoyingly repetitive, that he is not talking about the world, but about our perceptive faculties. And if indeed we cannot imagine x, that does seem to me to be a good basis from which to draw some conclusions about our perceptive faculties.
I have no idea what Kant would say about whether or not we can imagine non-Euclidian space (I have no idea myself if we can) but the matter is complicated because 'imagination' is a technical term in his philosophy. He thought space was an infinite Euclidian magnitude, but Euclidian geometry was the only game in town at the time.
Anyway he's not a good example. As I said before, I don't mean to dispute the point the example was meant to illustrate. I just wanted to point out that this is an incorrect view of Kant's claims about space. It's not really very important what he thought about space though.
There's a difference between "can't imagine" in a colloquial sense, and actual inability to imagine. There's also a difference between not being able to think of how something fits into our knowledge about the universe (for instance, not being able to come up with a mechanism or not being able to see how the evidence supports it) and not being able to imagine the thing itself.
There also aren't as many examples of this in the history of science as you probably think. Most of the examples that come to people's mind involve scientists versus noscientists.
See my reply to army above.
Hold on now, you're pattern matching me. I said:
To which you replied that this is a fact about me, not the universe. But I explicitly say that its not a fact about the universe! My evidence for this is the only evidence that could be relevant: my experience with literature, science fiction, talking to people, etc.
Nor is it relevant that science is full of people that say that something has to be true because they can't imagine the world otherwise. Again, I'm not making a claim about the world, I'm making a claim about the way we have imagined, or now imagine the world to be. I would be very happy to be pointed toward a hypothetical universe that isn't subject to mathematical analysis and which contains thinking animals.
So before we go on, please tell me what you think I'm claiming? I don't wish to defend any opinions but my own.
Hen, I told you how I imagine such a universe, and you told me I couldn't be imagining it! Maybe you could undertake not to gainsay further hypotheses.
I found your suggestion to be implausible for two reasons: first, I don't think the idea of epistemically significant qualia is defensible, and second, even on the condition that it is, I don't think the idea of a universe of nothing but a single quale (one having epistemic significance) is defensible. Both of these points would take some time to work out, and it struck me in our last exchange that you had neither the patience nor the good will to do so, at least not with me. But I'd be happy to discuss the matter if you're interested in hearing what I have to say.
You said:
I'm not sure what you mean by "necessary", but the most obvious interpretation is that you think it's necessarily impossible for the world to not be run by math or at least for humans to understand a world that doesn't.
This is my claim, and here's the thought: thinking things are natural, physical objects and they necessarily have some internal complexity. Further, thoughts have some basic complexity: I can't engage in an inference with a single term.
Any universe which would not in principle be subject to mathematical analysis is a universe in which there is no quantity of anything. So it can't, for example, involve any space or time, no energy or mass, no plurality of bodies, no forces, nothing like that. It admits of no analysis in terms of propositional logic, so Bayes is right out, as is any understanding of causality. This, it seems to me, would preclude the possibility of thought altogether. It may be that the world we live in is actually like that, and all its multiplicity is merely the contribution of our minds, so I won't venture a claim about the world as such. So far as I know, the fact that worlds admit of mathematical analysis is a fact about thinking things, not worlds.
What do you mean by "complexity"? I realize you have an intuitive idea, but it could very well be that your idea doesn't make sense when applied to whatever the real universe is.
Um, that seems like a stretch. Just because some aspects of the universe are subject to mathematical analysis doesn't necessarily mean the whole universe is.
For my purposes, complexity is: involving (in the broadest sense of that word) more than one (in the broadest sense of that word) thing (in the broadest sense of that word). And remember, I'm not talking about the real universe, but about the universe as it appears to creatures capable of thinking.
I think it does, if you're granting me that such a world could be distinguished into parts. It doesn't mean we could have the rich mathematical understanding of laws we do now, but that's a higher bar than I'm talking about.
You can always "use" analysis the issue is whether it gives you correct answers. It only gives you the correct answer if the universe obeys certain axioms.
I don't know what you mean by "run on math". Do qualia run in math?
It's not my phrase, and I don't particularly like it myself. If you're asking whether or not qualia are quanta, then I guess the answer is no, but in the sense that the measured is not the measure. It's a triviality that I can ask you how much pain you feel on a scale of 1-10, and get back a useful answer. I can't get at what the experience of pain itself is with a number or whatever, but then, I can't get at what the reality of a block of wood is with a ruler either.
Then by imagining an all qualia universe, I can easily imagine a universe that doesn't run on math, for some values of an"runs on math"
I don't think you can imagine, or conceive of, an all qualia universe though.
You don't get to tell me what I can imagine, though. All I have to do is imagine away the quantitative and structural aspects of my experience.
I rather think I do. If you told me you could imagine a euclidian triangle with more or less than 180 internal degrees, I would rightly say 'No you can't'. It's simply not true that we can imagine or conceive of anything we can put into (or appear to put into) words. And I don't think it's possible to imagine away things like space and time and keep hold of the idea that you're imagining a universe, or an experience, or anything like that. Time especially, and so long as I have time, I have quantity.
That looks likes typical mind fallacy
I don't know where you are getting yourmfacts from, but it is well known that people's abilities at visualization vary considerably, so where's the "we"?
Having studied non euclidean geometry, I can easily imagine a triangle whose angles .sum to more than180 (hint: it's inscribed on the surface of a sphere)
Saying that non spatial or in temporal universes aren't really universes is a True Scotsman fallacy.
Non spatial and non temporal models have been serious proposed by physicists; perhaps you should talk to them.
It depends on what you mean by "imagine". I can't imagine a Euclidian triangle with less than 180 degrees in the sense of having a visual representation in my mind that I could then reproduce on a piece of paper. On the other hand, I can certainly imagine someone holding up a measuring device to a vague figure on a piece of paper and saying "hey, I don't get 180 degrees when I measure this".
Of course, you could say that the second one doesn't count since you're not "really" imagining a triangle unless you imagine a visual representation, but if you're going to say that you need to remember that all nontrivial attempts to imagine things don't include as much detail as the real thing. How are you going to define it so that eliminating some details is okay and eliminating other details isn't?
(And if you try that, then explain why you can't imagine a triangle whose angles add up to 180.05 degrees or some other amount that is not 180 but is close enough that you wouldn't be able to tell the difference in a mental image. And then ask yourself "can I imagine someone writing a proof that a Euclidian triangle's angles don't add up to 180 degrees?" without denying that you can imagine people writing proofs at all.)
These are good questions, and I think my general answer is this: in the context of this similar arguments, being able to imagine something is sometimes taken as evidence that it's at least a logical possibility. I'm fine with that, but it needs to be imagined in enough detail to capture the logical structure of the relevant possibility. If someone is going to argue, for example, that one can imagine a euclidian triangle with more or less than 180 internal degrees, the imagined state of affairs must have as least as much logical detail as does a euclidian triangle with 180 internal degrees. Will that exclude your 'vague shape' example, and probably your 'proof' example?
The idea of rational understanding rests on the fact that you are separated from the object that you are trying to understand and the object itself doesn't change if you change your understanding of it.
Then there the halting problem. There are a variety of problems that are NP. Those problems can't be understood by doing a few experiments and then extrapolating general rules from your experiments. I'm not quite firm with the mathematical terminology but I think NP problems are not subject to thinks like calculus that are covered in what Wikipedia describes as Mathematical analysis.
Heinz von Förster makes the point that children have to be taught that "green" is no valid answer for the question: "What's 2+2?". I personally like his German book titled: "Truth is the invention of a liar". Heinz von Förster headed the started the Biological Computer Laboratory in 1958 and came up with concepts like second-order cybernetics.
As far as fictional worlds go, Terry Pratchetts discworld runs on narrativium instead of math.
That's true as long there no revelations of truth by Gods or other magical processes. In an universe where you can get the truth through magical tarot reading that assumption is false.
That's not obvious to me. Why do you think this?
I also don't understand this inference. Why do you think revelations of truth by Gods or other magical processes, or tarot readings, mean that such a universe would a) be knowable, and b) not be subject to rational analysis?
It might depend a bit of what you mean with rationality. You lose objectivity.
Let's say I'm hypnotize someone. I'm in a deep state of rapport. That means my emotional state matters a great deal. If I label something that the person I'm talking to as unsuccessful, anxiety raises in myself. That anxiety will screw with the result I want to achieve. I'm better of if I blank my mind instead of engaging in rational analysis of what I'm doing.
Logically A -> B is not the same thing as B -> A.
I said that it's possible for there to be knowledge that you can only get through a process besides rational analysis if you allow "magic".
I'm a little lost. So do you think these observations challenge the idea that in order to understand anyone, we need to assume they've got mostly true beliefs, and make mostly rational inferences?
That depends on your definition of "math".
For example, consider a simulated world where you control the code. Can you make it so that 2+2 in that simulation is sometimes 4, sometimes 15, and sometimes green? I don't see why not.
I think you're conflating the physical operation that we correlate with addition and the mathematical structure. 'Green' I'm not seeing, but I could write a computer program modeling a universe in which placing a pair of stones in a container that previously held a pair of stones does not always lead to that container holding a quadruplet of stones. In such a universe, the mathematical structure we call 'addition' would not be useful, but that doesn't say that the formalized reasoning structure we call 'math' would not exist, or could not be employed.
(In fact, if it's a computer program, it is obvious that its nature is susceptible to mathematical analysis.)
Is "containing mathematical truth" the same as "running on math"?
I guess I could make it appear that way, sure, though I don't know if I could then recognize anything in my simulation as thinking or doing math. But in any case, that's not a universe in which 2+2=green, it's a universe in which it appears to. Maybe I'm just not being imaginative enough, and so you may need to help me flesh out the hypothetical.
If I write the simulation in python I can simple define my function for addition:
Unfortunately I don't know how to format the indention perfectly for this forum.
We don't need to go to the trouble of defining anything in Python. We can get the same result just by saying
If I use python to simulate a world than it matters how things are defined in python.
It doesn't only appear that 2+2=green but it's that way at the level of the source code that depends how the world runs.
But it sounds to me like you're talking about the manipulation of signs, not about numbers themselves. We could make the set of signs '2+2=' end any way we like, but that doesn't mean we're talking about numbers. I donno, I think you're being too cryptic or technical or something for me, I don't really understand the point you're trying to make.
What do you mean with "the numbers themselves"? Peano axioms? I could imagine that n -> n+1 just doesn't apply.
Math is what happens when you take your original working predictive toolkit (like counting sheep) and let it run on human wetware disconnected from its original goal of having to predict observables. Thus some form of math would arise in any somewhat-predictable universe evolving a calculational substrate.
That's an interesting problem. Do we have math because we make abstractions about the multitude of things around us, or must we already have some idea of math in the abstract just to recognize the multitude as a multitude? But I think I agree with the gist of what you're saying.
Just like I think of language as meta-grunting, I think of math as meta-counting. Some animals can count, and possibly add and subtract a bit, but abstracting it away from the application for the fun of it is what humans do.