(Note: I’ve edited some things in to be clearer on some points.)
Do you know where I could find proofs of the following?
“Normally we define exp(x) to be the limit of 1, 1+x, 1+x+x^2/2, it’ll never get within 1/ω of e.”
“If you make the novice mistake in fixing it of instead trying to define exp(x) as {1,1+x,1+x+x^2/2,...|}, you will get not exp(1)=e but rather exp(1)=3.”
These are both pretty straightforward. For the first, say we’re working in a non-Archimedean ordered field which contains the reals, we take the partial sums of the series 1+1+1/2+1/6+...; these are rational numbers, in particular they’re real numbers. So if we have one of these partial sums, call it s, then e-s is a positive real number. So if you have some infinitesimal ε, it’s larger than ε; that’s what an infinitesimal is. The sequence will not get within ε of e.
For the second, note that 3={2|}, i.e., it’s the simplest number larger than 2. So if you have {1,2,5/2,8/3,...|}, well, the simplest number larger than all of those is still 3, because you did nothing to exclude 3. 3 is a very simple number! By definition, if you want to not get 3, either your interval has to not contain 3, or it has to contain something even simpler than 3 (i.e., 2, 1, or 0). (This is easy to see if you use the sign-sequence representation—remember that x is simpler than y iff the sign sequence of x is a proper prefix of the sign sequence of y.) The interval of surreals greater than those partial sums does contain 3, and does not contain 2, 1, or 0. So you get 3. That’s all there is to it.
As for the rest of the comment… let me address this out of order, if you don’t mind:
In some ways I view them as the ultimate reality
See, this is exactly the sort of thinking I’m trying to head off. How is that relevant to anything? You need to use something that actually fulfills the requirements of the problem.
On top of that, this seems… well, I don’t know if you actually are making this error, but it seems rather reminiscent of the high school student’s error of imagning that there’s a single notion of “number”—where every notion of “number” they know fits in C so “number” and “complex number” become identified. And this is false not just because you can go beyond C, but because there are systems of numbers that can’t be fit together with C at all. (How does Q_p fit into this? Answer: It doesn’t!)
(Actually, by that standard, shouldn’t the surcomplexes be the “ultimate reality”? :) )
(...I actually have some thoughts on that sort of thing, but since I’m trying to point out right now that that sort of thing is not what you should be thinking about when determining what sort of space to use, I won’t go into them. “Ultimate reality” is, in addition to not being correct, probably not on the list of requirements!)
Also, y’know, you don’t necessarily need something that could be considered “numbers” at all, as I keep emphasizing.
Anyway, as to the mathematical part of what you were saying...
I still need to read more about surreal numbers, but the thing I like about them is that you can always reduce the resolution if you can’t solve the equation in the surreals. In some ways I view them as the ultimate reality and if we don’t know the answer to something or only know the answer to a certain fineness, I think it’s better to be honest about, rather than just fall back to an equivalence class over the surreals where we do know the answer. Actually, maybe that wasn’t quite clear, I’m fine with falling back, but after its clear that we can’t solve it to the finest degree.
I have no idea what you’re talking about here. Like, what? First off, what sort of equations are you talking about? Algebraic ones? Over the surreals, I guess? The surreals are a real closed field, the surcomplexes are algebraically closed. That will suffice for algebraic equations. Maybe you mean some more general sort, I don’t know.
But most of this is just baffling. I have no idea what you’re talking about when you speak of passing to a quotient of the surreals to solve any equation. Where is that coming from? And like—what sort of quotient are we talking about here? “Quotient of the surreals” is already suspect because, well, it can’t be a ring-theoretic quotient, as fields don’t have nontrivial ideals, at all. So I guess you mean purely an additive quotient? But that’s not going to mix very well with solving any equations that involve more than addition now, is it? Meanwhile what the surreals are known for is that any ordered field embeds in them, not something about quotients!
Anyway, if you want to solve algebraic equations, you want an algebraically closed field. If you want to solve algebraic equations to the greatest extent possible while still keeping things ordered, you want a real closed field. The surreals are a real closed field, but you certainly don’t need them just for solving equations. If you want to be able to do limits and calculus and such, you want something with a nice topology (just how nice probably depends on just what you want), but note that you don’t necessarily want a field at all! None of these things favor the surreals, and the fact that we almost certainly need integration here is a huge strike against them.
Btw, you know what’s great for solving equations in, even if they aren’t just algebraic equations? The real numbers. Because they’re connected, so you have the intermediate value theorem. And they’re the only ordered field that’s connected. Again, you might be able to emulate that sort of thing to some extent in the surreals for sufficiently nice functions (mere continuity won’t be enough) (certainly you can for polynomials, like I said they’re real closed, but I’m guessing you can probably get more than that), I’m not super-familiar with just what’s possible there, but it’ll take more work. In the reals it’s just, make some comparisons, they come out opposite one another, R is connected, boom, there’s a solution somewhere inbetween.
But mostly I’m just wondering where like any of this is coming from. It neither seems to make much sense nor to resemble anything I know.
(Edit: And, once again, it’s not at all clear that being able to solve equations is at all relevant! That just doesn’t seem to be something that’s required. Whereas integration is.)
“So if we have one of these partial sums, call it s, then e-s is a positive real number. So if you have some infinitesimal ε, it’s larger than ε; that’s what an infinitesimal is”—Are you sure this chain of reasoning is correct? Consider 1/2x. For any finite number of terms it will be greater than ε, but as x approaches ω, it should approach 1/2ω. Why can’t the partial sum get within 1/ω of e?
“But it seems rather reminiscent of the high school student’s error of imagining that there’s a single notion of “number”″ - okay, the term “ultimate reality” is a stretch. I can’t imagine all possible applications, so I can’t imagine all possible numbering systems. My point is that we don’t just want to use a seperate numbering system for each individual problem. We want to be philosophically consistent and so there should be broad classes of problems for which we can identify a single numbering system as sufficient. And there’s a huge set of problems (which I’m not going to even attempt to spec out here) for which surreals can be justified on a philosophical level, even if it is convenient to drop down to another number system for the actual calculations. Maybe an example will help, Newtonian physics and Special Relativity embedded in General Relativity. General relativity provides consistency for physics, even though we use one of the first two for the majority of calculations.
“I have no idea what you’re talking about here. Like, what?”
You’re right that it won’t be a nice neat quotient group. But here’s an example. N_0 - N_0 can equal any integer where N_0 is a cardinal, or even +/- N_0, but in surreal numbers it works as follows. Suppose X and Y are countable infinities. Then X—Y has a unique value that we can sometimes identify. For example, if X represents the length of a sequence and Y is all the elements in the sequences except one, then X—Y = 1. We can perform the calculation in the surreals, or we can perform it in the cardinals and receive a broad range of possible answers. But for every possible answer in the cardinals, we can find pairs of surreal numbers that would provide that answer.
(Hey, a note, you should probably learn to use the blockquote feature. I dunno where it is in the rich text editor if you’re using that, but if you’re using the Markdown editor you just precede the paragraph you’re quoting with a “>”. It will make your posts substantially more readable.)
Are you sure this chain of reasoning is correct?
Yes.
Consider 1/2x. For any finite number of terms it will be greater than ε, but as x approaches ω, it should approach 1/2ω.
What “terms”? What are you talking about? This isn’t a sequence or a sum; there are no “terms” here. Yes, even in the surreals, as x goes to ω, 1/(2x) will approach 1/(2ω), as you say; as I mentioned above, limits of functions of a surreal variable will indeed still work. But that has no relevance to the case under discussion.
(And, while it’s not necessary to see what’s going on here, it may be helpful to remember that if we if we interpret this as occurring in the surreals, then in the case of 1/2x as x→ω, your domain has proper-class cofinality, while in the case of this infinite sum, the domain has cofinality ω. So the former can work, and the latter cannot. Again, one doesn’t need this to see that—the partial sum can’t get within 1/ω of e even when the cofinality is countable—but it may be worth remembering.)
Why can’t the partial sum get within 1/ω of e?
Because the partial sum is always a rational number. A rational number—more generally, a real number—cannot be infinitesimally close to e without being e. (By contrast, for surreal x, 1/(2x) certainly does not need to be a real number, and so can get infinitesimally close to 1/(2ω) without being equal to it.)
You’re right that it won’t be a nice neat quotient group. But here’s an example. N_0 - N_0 can equal any integer where N_0 is a cardinal, or even +/- N_0, but in surreal numbers it works as follows. Suppose X and Y are countable infinities. Then X—Y has a unique value that we can sometimes identify. For example, if X represents the length of a sequence and Y is all the elements in the sequences except one, then X—Y = 1. We can perform the calculation in the surreals, or we can perform it in the cardinals and receive a broad range of possible answers. But for every possible answer in the cardinals, we can find pairs of surreal numbers that would provide that answer.
What??
OK. Look. I could spend my time attempting to pick this apart. But, let me be blunt, the point I am trying to get across here is that you are talking nonsense. This is babble. You are way out of your depth, dude. You don’t know what you are talking about. You need to go back and relearn this from the beginning. I don’t even know what mistake you’re making, because it’s not a common one I recognize.
Just in the hopes it might be somewhat helpful, I will quickly go over the things I can maybe address quickly:
N_0 - N_0 can equal any integer where N_0 is a cardinal, or even +/- N_0, but in surreal numbers it works as follows.
I have no idea what this sentence is talking about.
Suppose X and Y are countable infinities.
What’s an “infinity”? An ordinal? A cardinal? (There’s only one countably infinite cardinal...) A surreal or something else entirely? You said “countable”, so it has to be something to which the notion of countability applies!
This mistake, at least, I think I can identify. Maybe you should, in fact, look over that “quick guide to the infinite” I wrote, because this is myth #0 I discussed there. There’s no such thing as a unified notion of “infinities”. There are different systems of numbers, some of them contain numbers/objects that are infinite (i.e.: larger in magnitude than any whole number), there is not some greater unified system they are all a part of.
Then X—Y has a unique value that we can sometimes identify.
What is X-Y? I don’t even know what system of numbers you’re using, so I don’t know what this means.
If X and Y are surreals, then, sure, there’s quite definitely a unique surreal X-Y. This is true more generally if you’re thinking of X and Y as living in some sort of ordered field or ring.
If X and Y are cardinals, then X-Y may not be well-defined. Trivially so if Y>X (no possible values), but let’s ignore that case. Even ignoring that, if X and Y are infinite, X-Y may fail to be well-defined due to having multiple possible values.
If X and Y are ordinals, we have to ask what sort of addition we’re using. If we’re using natural addition, then X-Y certainly has a unique value in the surreals, but it may or may not be an ordinal, so it’s not necessarily well-defined within the ordinals.
If we’re using ordinary addition, we have to distinguish between X-Y and -Y+X. (The latter just being a way of denoting “subtracting on the left”; it should not be interpreted as actually negating Y and adding to X.) -Y+X will have a unique value so long as Y≤X, but X-Y is a different story; even restricting to Y≤X, if X is infinite, then X-Y may have multiple possible values or none.
For example, if X represents the length of a sequence and Y is all the elements in the sequences except one, then X—Y = 1.
Yeah, not going to try to pick this apart, in short though this is nonsense.
I’m starting to think though that maybe you meant that X and Y were infinite sets, rather than some sort of numbers? With X-Y being the set difference? But that is not what you said. Simply put you seem very confused about all this.
We can perform the calculation in the surreals, or we can perform it in the cardinals and receive a broad range of possible answers.
Are X and Y surreals or are they cardinals? Surreals and cardinals don’t mix, dude! It can’t be both, not unless they’re just whole numbers! You are performing the calculation in whatever number system these things live in.
You just said above you get a well-defined answer, and, moreover, that it’s 1! Now you’re telling me that you can get a broad range of possible answers??
If X is representing the length of a sequence, it should probably be an ordinal. As for Y… yeah, OK, not going to try to make sense of the thing I already said I wouldn’t attempt to pick through.
And if X and Y are sets rather than numbers… oh, to hell with it, I’m just going to move on.
But for every possible answer in the cardinals, we can find pairs of surreal numbers that would provide that answer.
There is, I think, a correct idea here that is rescuable. It also seems pretty clear you don’t know enough to perform that rescue yourself and rephrase this as something that makes sense. (A hint, though: The fixed version probably should not involve surreals.)
(Do surreal numbers even have cardinalities, in a meaningful sense? Yes obviously if you pick a particular way of representing surreals as sets, e.g. by representing them as sign sequences, the resulting representations will have cardinalities; obviously, that’s not what I’m talking about. Although, who knows, maybe that’s a workable notion—define the cardinality of a surreal to be the cardinality of its birthday. No idea if that’s actually relevant to anything, though.)
Even charitably interpreted, none of this matches up with your comments above about equivalence classes. It relates, sure, but it doesn’t match. What you said above was that you could solve more equations by passing to equivalence classes. What you’re saying now seems to be… not that.
Long story short: I really, really, do not think you have much idea what you are talking about. You really need to relearn this from scratch, and not starting with surreals. I definitely do not think you are prepared to go instructing others on their uses; at this point I’m not convinced you could clearly articulate what ordinals and cardinals are for, you’ve gotten everything so mixed up in your comment above. I wouldn’t recommend trying to expand this into a post.
I think I should probably stop arguing here. If you reply to this with more babble I’m not going to waste my time replying to it further.
I really appreciate the time you’ve put into writing these long responses and I’ll admit that there are some gaps in my understanding, but I don’t think you’ve understood what I was saying at all. I suspect that this is a hazard with producing informal overviews of ideas + illusion of transparency. One example, when I said equivalence classes, I really meant something like equivalence classes. Anyway, in regards to all the points you’ve raised it’d take a lot of space to respond to them all, so I think I’ll just add a link to my post when I get time to write it.
(Note: I’ve edited some things in to be clearer on some points.)
These are both pretty straightforward. For the first, say we’re working in a non-Archimedean ordered field which contains the reals, we take the partial sums of the series 1+1+1/2+1/6+...; these are rational numbers, in particular they’re real numbers. So if we have one of these partial sums, call it s, then e-s is a positive real number. So if you have some infinitesimal ε, it’s larger than ε; that’s what an infinitesimal is. The sequence will not get within ε of e.
For the second, note that 3={2|}, i.e., it’s the simplest number larger than 2. So if you have {1,2,5/2,8/3,...|}, well, the simplest number larger than all of those is still 3, because you did nothing to exclude 3. 3 is a very simple number! By definition, if you want to not get 3, either your interval has to not contain 3, or it has to contain something even simpler than 3 (i.e., 2, 1, or 0). (This is easy to see if you use the sign-sequence representation—remember that x is simpler than y iff the sign sequence of x is a proper prefix of the sign sequence of y.) The interval of surreals greater than those partial sums does contain 3, and does not contain 2, 1, or 0. So you get 3. That’s all there is to it.
As for the rest of the comment… let me address this out of order, if you don’t mind:
See, this is exactly the sort of thinking I’m trying to head off. How is that relevant to anything? You need to use something that actually fulfills the requirements of the problem.
On top of that, this seems… well, I don’t know if you actually are making this error, but it seems rather reminiscent of the high school student’s error of imagning that there’s a single notion of “number”—where every notion of “number” they know fits in C so “number” and “complex number” become identified. And this is false not just because you can go beyond C, but because there are systems of numbers that can’t be fit together with C at all. (How does Q_p fit into this? Answer: It doesn’t!)
(Actually, by that standard, shouldn’t the surcomplexes be the “ultimate reality”? :) )
(...I actually have some thoughts on that sort of thing, but since I’m trying to point out right now that that sort of thing is not what you should be thinking about when determining what sort of space to use, I won’t go into them. “Ultimate reality” is, in addition to not being correct, probably not on the list of requirements!)
Also, y’know, you don’t necessarily need something that could be considered “numbers” at all, as I keep emphasizing.
Anyway, as to the mathematical part of what you were saying...
I have no idea what you’re talking about here. Like, what? First off, what sort of equations are you talking about? Algebraic ones? Over the surreals, I guess? The surreals are a real closed field, the surcomplexes are algebraically closed. That will suffice for algebraic equations. Maybe you mean some more general sort, I don’t know.
But most of this is just baffling. I have no idea what you’re talking about when you speak of passing to a quotient of the surreals to solve any equation. Where is that coming from? And like—what sort of quotient are we talking about here? “Quotient of the surreals” is already suspect because, well, it can’t be a ring-theoretic quotient, as fields don’t have nontrivial ideals, at all. So I guess you mean purely an additive quotient? But that’s not going to mix very well with solving any equations that involve more than addition now, is it? Meanwhile what the surreals are known for is that any ordered field embeds in them, not something about quotients!
Anyway, if you want to solve algebraic equations, you want an algebraically closed field. If you want to solve algebraic equations to the greatest extent possible while still keeping things ordered, you want a real closed field. The surreals are a real closed field, but you certainly don’t need them just for solving equations. If you want to be able to do limits and calculus and such, you want something with a nice topology (just how nice probably depends on just what you want), but note that you don’t necessarily want a field at all! None of these things favor the surreals, and the fact that we almost certainly need integration here is a huge strike against them.
Btw, you know what’s great for solving equations in, even if they aren’t just algebraic equations? The real numbers. Because they’re connected, so you have the intermediate value theorem. And they’re the only ordered field that’s connected. Again, you might be able to emulate that sort of thing to some extent in the surreals for sufficiently nice functions (mere continuity won’t be enough) (certainly you can for polynomials, like I said they’re real closed, but I’m guessing you can probably get more than that), I’m not super-familiar with just what’s possible there, but it’ll take more work. In the reals it’s just, make some comparisons, they come out opposite one another, R is connected, boom, there’s a solution somewhere inbetween.
But mostly I’m just wondering where like any of this is coming from. It neither seems to make much sense nor to resemble anything I know.
(Edit: And, once again, it’s not at all clear that being able to solve equations is at all relevant! That just doesn’t seem to be something that’s required. Whereas integration is.)
“So if we have one of these partial sums, call it s, then e-s is a positive real number. So if you have some infinitesimal ε, it’s larger than ε; that’s what an infinitesimal is”—Are you sure this chain of reasoning is correct? Consider 1/2x. For any finite number of terms it will be greater than ε, but as x approaches ω, it should approach 1/2ω. Why can’t the partial sum get within 1/ω of e?
“But it seems rather reminiscent of the high school student’s error of imagining that there’s a single notion of “number”″ - okay, the term “ultimate reality” is a stretch. I can’t imagine all possible applications, so I can’t imagine all possible numbering systems. My point is that we don’t just want to use a seperate numbering system for each individual problem. We want to be philosophically consistent and so there should be broad classes of problems for which we can identify a single numbering system as sufficient. And there’s a huge set of problems (which I’m not going to even attempt to spec out here) for which surreals can be justified on a philosophical level, even if it is convenient to drop down to another number system for the actual calculations. Maybe an example will help, Newtonian physics and Special Relativity embedded in General Relativity. General relativity provides consistency for physics, even though we use one of the first two for the majority of calculations.
“I have no idea what you’re talking about here. Like, what?”
You’re right that it won’t be a nice neat quotient group. But here’s an example. N_0 - N_0 can equal any integer where N_0 is a cardinal, or even +/- N_0, but in surreal numbers it works as follows. Suppose X and Y are countable infinities. Then X—Y has a unique value that we can sometimes identify. For example, if X represents the length of a sequence and Y is all the elements in the sequences except one, then X—Y = 1. We can perform the calculation in the surreals, or we can perform it in the cardinals and receive a broad range of possible answers. But for every possible answer in the cardinals, we can find pairs of surreal numbers that would provide that answer.
(Hey, a note, you should probably learn to use the blockquote feature. I dunno where it is in the rich text editor if you’re using that, but if you’re using the Markdown editor you just precede the paragraph you’re quoting with a “>”. It will make your posts substantially more readable.)
Yes.
What “terms”? What are you talking about? This isn’t a sequence or a sum; there are no “terms” here. Yes, even in the surreals, as x goes to ω, 1/(2x) will approach 1/(2ω), as you say; as I mentioned above, limits of functions of a surreal variable will indeed still work. But that has no relevance to the case under discussion.
(And, while it’s not necessary to see what’s going on here, it may be helpful to remember that if we if we interpret this as occurring in the surreals, then in the case of 1/2x as x→ω, your domain has proper-class cofinality, while in the case of this infinite sum, the domain has cofinality ω. So the former can work, and the latter cannot. Again, one doesn’t need this to see that—the partial sum can’t get within 1/ω of e even when the cofinality is countable—but it may be worth remembering.)
Because the partial sum is always a rational number. A rational number—more generally, a real number—cannot be infinitesimally close to e without being e. (By contrast, for surreal x, 1/(2x) certainly does not need to be a real number, and so can get infinitesimally close to 1/(2ω) without being equal to it.)
What??
OK. Look. I could spend my time attempting to pick this apart. But, let me be blunt, the point I am trying to get across here is that you are talking nonsense. This is babble. You are way out of your depth, dude. You don’t know what you are talking about. You need to go back and relearn this from the beginning. I don’t even know what mistake you’re making, because it’s not a common one I recognize.
Just in the hopes it might be somewhat helpful, I will quickly go over the things I can maybe address quickly:
I have no idea what this sentence is talking about.
What’s an “infinity”? An ordinal? A cardinal? (There’s only one countably infinite cardinal...) A surreal or something else entirely? You said “countable”, so it has to be something to which the notion of countability applies!
This mistake, at least, I think I can identify. Maybe you should, in fact, look over that “quick guide to the infinite” I wrote, because this is myth #0 I discussed there. There’s no such thing as a unified notion of “infinities”. There are different systems of numbers, some of them contain numbers/objects that are infinite (i.e.: larger in magnitude than any whole number), there is not some greater unified system they are all a part of.
What is X-Y? I don’t even know what system of numbers you’re using, so I don’t know what this means.
If X and Y are surreals, then, sure, there’s quite definitely a unique surreal X-Y. This is true more generally if you’re thinking of X and Y as living in some sort of ordered field or ring.
If X and Y are cardinals, then X-Y may not be well-defined. Trivially so if Y>X (no possible values), but let’s ignore that case. Even ignoring that, if X and Y are infinite, X-Y may fail to be well-defined due to having multiple possible values.
If X and Y are ordinals, we have to ask what sort of addition we’re using. If we’re using natural addition, then X-Y certainly has a unique value in the surreals, but it may or may not be an ordinal, so it’s not necessarily well-defined within the ordinals.
If we’re using ordinary addition, we have to distinguish between X-Y and -Y+X. (The latter just being a way of denoting “subtracting on the left”; it should not be interpreted as actually negating Y and adding to X.) -Y+X will have a unique value so long as Y≤X, but X-Y is a different story; even restricting to Y≤X, if X is infinite, then X-Y may have multiple possible values or none.
Yeah, not going to try to pick this apart, in short though this is nonsense.
I’m starting to think though that maybe you meant that X and Y were infinite sets, rather than some sort of numbers? With X-Y being the set difference? But that is not what you said. Simply put you seem very confused about all this.
Are X and Y surreals or are they cardinals? Surreals and cardinals don’t mix, dude! It can’t be both, not unless they’re just whole numbers! You are performing the calculation in whatever number system these things live in.
You just said above you get a well-defined answer, and, moreover, that it’s 1! Now you’re telling me that you can get a broad range of possible answers??
If X is representing the length of a sequence, it should probably be an ordinal. As for Y… yeah, OK, not going to try to make sense of the thing I already said I wouldn’t attempt to pick through.
And if X and Y are sets rather than numbers… oh, to hell with it, I’m just going to move on.
There is, I think, a correct idea here that is rescuable. It also seems pretty clear you don’t know enough to perform that rescue yourself and rephrase this as something that makes sense. (A hint, though: The fixed version probably should not involve surreals.)
(Do surreal numbers even have cardinalities, in a meaningful sense? Yes obviously if you pick a particular way of representing surreals as sets, e.g. by representing them as sign sequences, the resulting representations will have cardinalities; obviously, that’s not what I’m talking about. Although, who knows, maybe that’s a workable notion—define the cardinality of a surreal to be the cardinality of its birthday. No idea if that’s actually relevant to anything, though.)
Even charitably interpreted, none of this matches up with your comments above about equivalence classes. It relates, sure, but it doesn’t match. What you said above was that you could solve more equations by passing to equivalence classes. What you’re saying now seems to be… not that.
Long story short: I really, really, do not think you have much idea what you are talking about. You really need to relearn this from scratch, and not starting with surreals. I definitely do not think you are prepared to go instructing others on their uses; at this point I’m not convinced you could clearly articulate what ordinals and cardinals are for, you’ve gotten everything so mixed up in your comment above. I wouldn’t recommend trying to expand this into a post.
I think I should probably stop arguing here. If you reply to this with more babble I’m not going to waste my time replying to it further.
I really appreciate the time you’ve put into writing these long responses and I’ll admit that there are some gaps in my understanding, but I don’t think you’ve understood what I was saying at all. I suspect that this is a hazard with producing informal overviews of ideas + illusion of transparency. One example, when I said equivalence classes, I really meant something like equivalence classes. Anyway, in regards to all the points you’ve raised it’d take a lot of space to respond to them all, so I think I’ll just add a link to my post when I get time to write it.