The first line of your answer is something I hadn’t thought of before and strikes me as really good advice.
I don’t really agree with your point on fundamentals. I’m a big “fundamentals” guy—I did a Math Ph.D. and despite coming in with a big head start, it took me a disproportionately long time to even start research because I had to build up everything from scratch in my own words (which is just not feasible nowadays). I say this to indicate that I already err on the side of being overly-focused on fundamentals. (Unrelated but out of curiosity—which specific mathematical controversies are you referring to?)
But using fundamentals for anything more than addressing fringe theories (like homeopathy and flat earth) would strike me over-extrapolating from partial knowledge of a system. For example, I don’t believe questions of “how will X affect Y aspect of the economy/ a person’s health/ the global climate” can be fully addressed by fundamentals, but people (not saying you) will often make them seem like they can be by using some simple model or assuming that we we know is all there is to know.
Yes, it takes a crazy amount of time, so I choose my battles. In case of homeopathy and evolution I actually got the background knowledge regardless of controversy, as I was already interested in natural science. (Also, my wife is a biochemist, so I often defer to her expertise.)
For political/economical things, my rule of thumb is to listen to people who mostly do the “boring” work. Such as, collecting information on corruption, which 99.99% of time everyone ignores, and 0.01% explodes into a huge political scandal. When the scandal happens, I trust the opinion of the guy who worked on all the non-scandalous stuff over anyone who only pays attention to the “cool” stuff.
Basically, you need to know what is “the business as usual”, and compare to that. For example, most people enter politics in order to make some kind of personal profit. But most of them stay within certain lines, and some of them go way beyond them. So instead of “did X embezzle money at Y”, a better question is “did X embezzle the usual amount of money at Y, or did something extraordinary happen here”? If you ask the right question, you will see the greatest criminals. If you ask the wrong question, your attention will go where the media want it to go. Yeah, in a perfect world all crime would be prosecuted, but we don’t live in that world. (For example the problem with Trump is not that he is bad, but that he is bad beyond usual. Some people believe that introducing random change to the system is good. I believe most of the time it only makes the system worse.)
With math… well, there are topics that other people consider controversial, such as Cantor’s diagonal argument or the Axiom of Choice. My approach is that instead of “joining a side”, I want to get a better insight into how these things work, on the mechanical level. (LOL, you probably know about these things much more than I do, but telling me the answer wouldn’t help me much unless I can derive it myself. And that’s kinda my point: I wouldn’t spend my effort on trying to agree or disagree with you, but rather on trying to get to your level, at least in some very narrow area.)
The method of “put the numbers (that you can enumerate) in a table and invert the diagonal” provides a number that is outside the original set, which can be used in multiple ways. Like, if you put the rational numbers in the table, the inverted diagonal will be an irrational number. If you put the computable numbers in the table, the inverted diagonal will be an uncomputable number. So even if someone claims to be coming from a finitist or whatever other perspective, my intuition would be like “you guys basically redefined ‘number’ to mean X, and I am not going to argue definitions here, but anyway, when you put all Xs in the table, the inverted diagonal is going to be a non-X”.
The real fun, from my perspective, is Skolem’s paradox. I think I approximately understand the point. The first-order logic cannot express certain concepts (even quite simple ones, such as standard natural numbers). So you should not think of the axioms as describing “the Platonic world”, but rather as applying to different models. (Feel free to think about the Platonic world as containing models of sets, rather than the sets directly.) Then instead of “wow, this is a mysterious paradox”, you should approach it as a hacker and say “wow, there seem to be malicious models that technically follow the letter of the axioms, while completely ignoring the original intentions of their authors… sounds like a lot of fun”. And then, whenever you hear a statement about something, you should be like “I wonder how this statement works in one of those malicious models”. -- And, specifically, I wonder how exactly Cantor’s diagonal argument works in one of those malicious models, but I am not there yet. I am reading a textbook that I have already read previously, except this time, I am like a hacker looking for a possible exploit. Like, when it says “natural numbers”, I go like “I wonder what happens if those are nonstandard natural numbers”.
(And I plan to approach the axiom of choice the same way. Like, the axiom is neither “true” nor “false”, there are simply models that are compatible with it, and models that are incompatible, and I want to get an intuitive understanding of both.)
This will take me a lot of time, and perhaps I will never finish it in my lifetime, but that’s okay because I enjoy math. :)
I’ll PM you about the math stuff when I have a chance. I would guess that the “devil” in Cantor diagonalization that you’re unearthing is closely related to the impossibility of semantic closure in a mathematical language. I’m not a set theorist but I also spent some time thinking about these things and getting partial answers that I was okay walking away with.
Do you have any sources you’d be willing to share for politics and economics?
I suspect this is all connected somehow: set theory, first-order logic, Gödel’s incompleteness theorem, etc. I plan to read a textbook on logic next. Then model theory.
Sorry, my sources for politics are local, so unless you are interested in the political situation of Slovakia (short version: it sucks; slightly longer version: some people believe that voting for populists is relatively harmless because they are all talk no action, well this is mostly true until suddenly it’s not and then all competent people just scream helplessly) I don’t have anything to offer. And even here, the source were the people (the kind that does the boring stuff) I spend a lot of time talking to, not any written source.
One detail maybe: some people in politics may be easier to approach than they seem. (Then again, that’s my local experience, maybe not true in USA.) For example, once I successfully invited our member of parliament to our local Less Wrong meetup, to tell us about some then current cause. It was surprisingly simple: I just messaged her on Facebook like “hey, we have a meetup of some smart people who like to discuss various things, and we would be happy to hear your opinion on X”, and she was like “actually, I have a free evening that day, so unless something unexpected happens, I would be happy to meet you”.
If you put the computable numbers in the table, the inverted diagonal will be an uncomputable number.
In fact, diag-a1. if you put the computable numbers in the table, and diag-a2. pretend that the assignment is computable also, and diag-a3. (optionally) pretend that procedure for looking up the row of specific number is computable also,
then diag-l1. the inverted diagonal is computable by explicit construction, diag-l2. the inverted diagonal does not belong to the table,
thus diag-t1. we have the contradiction (based on diag-l1 and diag-l2), diag-t2. (if diag-a3 holds) the contradicting place is computable as well.
I agree with your view but that specific example was wrong!
The inverted diagonal differs from every element in the table for any ordering of the computables (by construction). Try to actually pseudocode a program to get the nth digit of the nth computable in your arrangement without invoking a halting oracle.
Retrieve I-th computable from table: computable by assumption.
Get K-th digit from a given computable: computable by definition.
Get I-th digit of I-th computable: computable as composition of (1) and (2).
Invert given digit: trivially computable.
Get an inverted I-th digit of I-th computable: computable as composition of (3) and (4).
I’ve actually written formal refutation of any bijection between N and 2N in Idris 2.
Demonstration
total stands for functions which are defined everywhere on their stated domain (as opposed to partial), and computations for which are proven to always halt (if there’s no such guarantee, function is covering). %default total means that all functions below must be total.
Void type means that a contradiction was obtained from given arguments.
%default total
data Bijection : Type -> Type -> Type where
||| A bijection between two types.
||| @ a the first type
||| @ b the second type
||| @ fwd mapping from first type to the second
||| @ bck mapping from second type to the first
||| @ fb_producer proof that fwd . bck = id
||| @ bf_producer proof that bck . fwd = id
DoMap : {0 a,b : Type} -> (fwd: a -> b) -> (bck: b -> a)
-> (fb_producer: (av : a) -> (bck $ fwd av) = av)
-> (bf_producer: (bv : b) -> (fwd $ bck bv) = bv)
-> Bijection a b
map_fneq : {0 a,b : Type} -> {f : a->b} -> {g : a->b} -> f = g -> (v:a) -> f v = g v
map_fneq (Refl {x = f}) v = Refl
bool_inv : {v : Bool} -> (v = not v) -> Void
bool_inv {v = True} prf = uninhabited prf
bool_inv {v = False} prf = uninhabited prf
diagonal : (Bijection Nat (Nat -> Bool)) -> Void
diagonal (DoMap f g _ bf_prf) = bool_inv conflict where
H : Nat -> Bool
H i = not $ f i i
h_hi_by_idx : f (g H) (g H) = H (g H)
h_hi_by_idx = map_fneq (bf_prf H) (g H)
h_hi_by_def : H (g H) = (not $ f (g H) (g H))
h_hi_by_def = Refl
conflict : f (g H) (g H) = (not $ f (g H) (g H))
conflict = trans h_hi_by_idx h_hi_by_def
I interpreted your algorithm for listing computables to be something like “enumerate the Turing machines that output ‘.’ then 0s and 1s and list what they print”, without worrying about the fact that some computables repeat. This technically lists all computables even though some machines will get stuck in some non-halting behavior that prints nothing after printing finitely many 0s and 1s. But you can’t figure out the nth digit of an arbitray thing in the list because you don’t always know if a machine will continue to print 0s or 1s or get indefinitely “stuck”.
It seems that you actually meant that you could have an algorithm that begins by enumerating the Turing machines that print computables and never get stuck in some non-halting configuration. This can’t be done.
If you have any surjection: N→S⊂(N→{0,1}) and diagonalize against it, you know the result is not in S. This fact doesn’t depend on the actual nature of the surjection N→S, just that S is the image. Here S is the computables.
I interpreted your algorithm for listing computables to be something like “enumerate the Turing machines that output ‘.’ then 0s and 1s and list what they print”, without worrying about the fact that some computables repeat.
I’m pretty sure my argument did not mention how computables are listed at all, rather proving that for any specific listing the inverse-diagonal is computable as well.
If you have any surjection: N→S⊂(N→{0,1}) and diagonalize against it, you know the result is not in S. This fact doesn’t depend on the actual nature of the surjection N→S, just that S is the image. Here S is the computables.
Yes. However, it’s the specific choice of set “computables” which creates the contradiction: I agree with “inverse-diagonal for rationals is an irrational number” and like.
Once again: for any “user-provided” computable table of computable digit sequences, I can, in finite time, get value for any specific position in table; therefore, each digit of inverse sequence is computable; therefore, I conclude that the inverse-diagonal sequence is itself computable (if I’m not mistaken in definitions).
My claim is that such a table does not exist because it leads to a contradiction. If you add it as an assumption, you can obtain a false conclusion because the assumption itself can never hold.
The first line of your answer is something I hadn’t thought of before and strikes me as really good advice.
I don’t really agree with your point on fundamentals. I’m a big “fundamentals” guy—I did a Math Ph.D. and despite coming in with a big head start, it took me a disproportionately long time to even start research because I had to build up everything from scratch in my own words (which is just not feasible nowadays). I say this to indicate that I already err on the side of being overly-focused on fundamentals. (Unrelated but out of curiosity—which specific mathematical controversies are you referring to?)
But using fundamentals for anything more than addressing fringe theories (like homeopathy and flat earth) would strike me over-extrapolating from partial knowledge of a system. For example, I don’t believe questions of “how will X affect Y aspect of the economy/ a person’s health/ the global climate” can be fully addressed by fundamentals, but people (not saying you) will often make them seem like they can be by using some simple model or assuming that we we know is all there is to know.
Yes, it takes a crazy amount of time, so I choose my battles. In case of homeopathy and evolution I actually got the background knowledge regardless of controversy, as I was already interested in natural science. (Also, my wife is a biochemist, so I often defer to her expertise.)
For political/economical things, my rule of thumb is to listen to people who mostly do the “boring” work. Such as, collecting information on corruption, which 99.99% of time everyone ignores, and 0.01% explodes into a huge political scandal. When the scandal happens, I trust the opinion of the guy who worked on all the non-scandalous stuff over anyone who only pays attention to the “cool” stuff.
Basically, you need to know what is “the business as usual”, and compare to that. For example, most people enter politics in order to make some kind of personal profit. But most of them stay within certain lines, and some of them go way beyond them. So instead of “did X embezzle money at Y”, a better question is “did X embezzle the usual amount of money at Y, or did something extraordinary happen here”? If you ask the right question, you will see the greatest criminals. If you ask the wrong question, your attention will go where the media want it to go. Yeah, in a perfect world all crime would be prosecuted, but we don’t live in that world. (For example the problem with Trump is not that he is bad, but that he is bad beyond usual. Some people believe that introducing random change to the system is good. I believe most of the time it only makes the system worse.)
With math… well, there are topics that other people consider controversial, such as Cantor’s diagonal argument or the Axiom of Choice. My approach is that instead of “joining a side”, I want to get a better insight into how these things work, on the mechanical level. (LOL, you probably know about these things much more than I do, but telling me the answer wouldn’t help me much unless I can derive it myself. And that’s kinda my point: I wouldn’t spend my effort on trying to agree or disagree with you, but rather on trying to get to your level, at least in some very narrow area.)
The method of “put the numbers (that you can enumerate) in a table and invert the diagonal” provides a number that is outside the original set, which can be used in multiple ways. Like, if you put the rational numbers in the table, the inverted diagonal will be an irrational number. If you put the computable numbers in the table, the inverted diagonal will be an uncomputable number. So even if someone claims to be coming from a finitist or whatever other perspective, my intuition would be like “you guys basically redefined ‘number’ to mean X, and I am not going to argue definitions here, but anyway, when you put all Xs in the table, the inverted diagonal is going to be a non-X”.
The real fun, from my perspective, is Skolem’s paradox. I think I approximately understand the point. The first-order logic cannot express certain concepts (even quite simple ones, such as standard natural numbers). So you should not think of the axioms as describing “the Platonic world”, but rather as applying to different models. (Feel free to think about the Platonic world as containing models of sets, rather than the sets directly.) Then instead of “wow, this is a mysterious paradox”, you should approach it as a hacker and say “wow, there seem to be malicious models that technically follow the letter of the axioms, while completely ignoring the original intentions of their authors… sounds like a lot of fun”. And then, whenever you hear a statement about something, you should be like “I wonder how this statement works in one of those malicious models”. -- And, specifically, I wonder how exactly Cantor’s diagonal argument works in one of those malicious models, but I am not there yet. I am reading a textbook that I have already read previously, except this time, I am like a hacker looking for a possible exploit. Like, when it says “natural numbers”, I go like “I wonder what happens if those are nonstandard natural numbers”.
(And I plan to approach the axiom of choice the same way. Like, the axiom is neither “true” nor “false”, there are simply models that are compatible with it, and models that are incompatible, and I want to get an intuitive understanding of both.)
This will take me a lot of time, and perhaps I will never finish it in my lifetime, but that’s okay because I enjoy math. :)
I’ll PM you about the math stuff when I have a chance. I would guess that the “devil” in Cantor diagonalization that you’re unearthing is closely related to the impossibility of semantic closure in a mathematical language. I’m not a set theorist but I also spent some time thinking about these things and getting partial answers that I was okay walking away with.
Do you have any sources you’d be willing to share for politics and economics?
I suspect this is all connected somehow: set theory, first-order logic, Gödel’s incompleteness theorem, etc. I plan to read a textbook on logic next. Then model theory.
Sorry, my sources for politics are local, so unless you are interested in the political situation of Slovakia (short version: it sucks; slightly longer version: some people believe that voting for populists is relatively harmless because they are all talk no action, well this is mostly true until suddenly it’s not and then all competent people just scream helplessly) I don’t have anything to offer. And even here, the source were the people (the kind that does the boring stuff) I spend a lot of time talking to, not any written source.
One detail maybe: some people in politics may be easier to approach than they seem. (Then again, that’s my local experience, maybe not true in USA.) For example, once I successfully invited our member of parliament to our local Less Wrong meetup, to tell us about some then current cause. It was surprisingly simple: I just messaged her on Facebook like “hey, we have a meetup of some smart people who like to discuss various things, and we would be happy to hear your opinion on X”, and she was like “actually, I have a free evening that day, so unless something unexpected happens, I would be happy to meet you”.
In fact,
diag-a1. if you put the computable numbers in the table, and
diag-a2. pretend that the assignment is computable also, and
diag-a3. (optionally) pretend that procedure for looking up the row of specific number is computable also,
then
diag-l1. the inverted diagonal is computable by explicit construction,
diag-l2. the inverted diagonal does not belong to the table,
thus
diag-t1. we have the contradiction (based on diag-l1 and diag-l2),
diag-t2. (if diag-a3 holds) the contradicting place is computable as well.
I agree with your view but that specific example was wrong!
The inverted diagonal differs from every element in the table for any ordering of the computables (by construction). Try to actually pseudocode a program to get the nth digit of the nth computable in your arrangement without invoking a halting oracle.
Retrieve I-th computable from table: computable by assumption.
Get K-th digit from a given computable: computable by definition.
Get I-th digit of I-th computable: computable as composition of (1) and (2).
Invert given digit: trivially computable.
Get an inverted I-th digit of I-th computable: computable as composition of (3) and (4).
I’ve actually written formal refutation of any bijection between N and 2N in Idris 2.
Demonstration
totalstands for functions which are defined everywhere on their stated domain (as opposed topartial), and computations for which are proven to always halt (if there’s no such guarantee, function iscovering).%default totalmeans that all functions below must be total.Voidtype means that a contradiction was obtained from given arguments.I interpreted your algorithm for listing computables to be something like “enumerate the Turing machines that output ‘.’ then 0s and 1s and list what they print”, without worrying about the fact that some computables repeat. This technically lists all computables even though some machines will get stuck in some non-halting behavior that prints nothing after printing finitely many 0s and 1s. But you can’t figure out the nth digit of an arbitray thing in the list because you don’t always know if a machine will continue to print 0s or 1s or get indefinitely “stuck”.
It seems that you actually meant that you could have an algorithm that begins by enumerating the Turing machines that print computables and never get stuck in some non-halting configuration. This can’t be done.
If you have any surjection: N→S⊂(N→{0,1}) and diagonalize against it, you know the result is not in S. This fact doesn’t depend on the actual nature of the surjection N→S, just that S is the image. Here S is the computables.
I’m pretty sure my argument did not mention how computables are listed at all, rather proving that for any specific listing the inverse-diagonal is computable as well.
Yes. However, it’s the specific choice of set “computables” which creates the contradiction: I agree with “inverse-diagonal for rationals is an irrational number” and like.
Once again: for any “user-provided” computable table of computable digit sequences, I can, in finite time, get value for any specific position in table; therefore, each digit of inverse sequence is computable; therefore, I conclude that the inverse-diagonal sequence is itself computable (if I’m not mistaken in definitions).
My claim is that such a table does not exist because it leads to a contradiction. If you add it as an assumption, you can obtain a false conclusion because the assumption itself can never hold.