> My first instinct was, I had underspecified the location of the car. The model seems to assume the car is already at the car wash from the wording. GPT 5.x series models behave a bit more on the spectrum so you need to tell them the specifics.
This makes little sense, even though it sounds superficially convincing. However, why would a language model assume that the car is at the destination when evaluating the difference between walking or driving? Why not mention that, it it was really assuming it?
What seems to me far, far more likely to be happening here is that the phrase "walk or drive for <short distance>" is too strongly associated in the training data with the "walk" response, and the "car wash" part of the question simply can't flip enough weights to matter in the default response. This is also to be expected given that there are likely extremely few similar questions in the training set, since people just don't ask about what mode of transport is better for arriving at a car wash.
This is a clear case of a language model having language model limitations. Once you add more text in the prompt, you reduce the overall weight of the "walk or drive" part of the question, and the other relevant parts of the phrase get to matter more for the response.
> However, why would a language model assume that the car is at the destination when evaluating the difference between walking or driving? Why not mention that, it it was really assuming it?
Because it assumes it's a genuine question not a trick.
While I don't think anyone has a plausible theory that goes to this level of detail on how humans actually think, there's still a major difference. I think it's fair to say that if we are doing a brute force search, we are still astonishingly more energy efficient at it than these LLMs. The amount of energy that goes into running an LLM for 12h straight is vastly higher than what it takes for humans to think about similar problems.
In the research group I am, we have usually try a few approach to each problem, let's say we get a:
Method A) 30% speed reduction and 80% precision decrease
Method B) 50% speed reduction and 5% precision increase
Method C) 740% speed reduction and 1% precision increase
and we only publish B. It's not brute force[1], but throw noodles at the wall, see what sticks, like the GP said. We don't throw spoons[1], but everything that looks like a noodle has a high chance of been thrown. It's a mix of experience[1] and not enough time to try everything.
One thing we know for sure is that humans learn from their interactions, while LLMs don't (beyond some small context window). This clear fact alone makes it worthless to debate with a current AI.
I can't for the life of me understand why people think it's OK to send and even expect plaintext email in 2026. There's so much content that requires formatting and non-Unicode support in order to make sense. Formatted text, lists, in-line graphs or images, tables, equations or other mathematical formulae, all of these benefit from a controlled layout that plaintext just doesn't offer or can barely approximate. Why would you limit your email communication like this?
The definition of "text" itself is quite vague. Is a code sample text? What about a series of code samples intercalated with common language descriptions? What about a numbered list? What about a poem?
Also, just because I send some text as HTML doesn't prevent in any way the receiver from formatting that however they want. I'm just adding some display hints, that their email client may or may not ignore.
What about it? Your e-mail composer will tell you when you put something in your message which isn't plaintext and offer to convert your message to HTML for you to keep writing it.
You were claiming earlier that one shouldn't add formatting to their email, such as emphasizing using italics as it is the recipient who should decide how to format the email text. This is a completely different discussion from whether email that does happen to only require plain text should be sent as plain text or HTML.
I should have expressed myself better. If plaintext is adequate for an e-mail, then it should be plaintext. That's what I suggested. Of course, if you need to use other features, such as the italics you mention, then plaintext doesn't cut it.
I think the poster above may have accidentally worded their response a little too personally, but their point is valid and not against neurodivergent people (or, at least, there is a version that is close to their argument that is so).
It's perfectly fine to ask people to change be careful in their correspondence to a specific person to avoid certain issues.
It's not fine, however, even for neuro divergent people, to expect social norms to change for everyone to match their particular preferences.
If we read the original article as representing a request from the author to specifically not answer emails to them by apologizing for replying late, that's a perfectly fine request that anyone corresponding with them should follow (once they become aware of it). If we read it as a general recommendation to everyone to change this clear social norm, then it's not fine, the justification given (one person finds this puts some kind of pressure on them, and others might too) is not strong enough to warrant everyone else changing their behavior pre-emptively.
This completely misses the point of why the complex numbers were even invented. i is a number: it is one of the 2 solutions to the equation x^2 = -1 (the other being -i, of course). The whole point of inventing the complex numbers was to have a set of numbers for which any polynomial has a root. And sure, you can call this number (0,1) if you want to, but it's important to remember that C is not the same as R².
Your whole point about Taylor series is also wrong, as Taylor series are not approximations, they are actually equal to the original function if you take their infinite limit for the relevant functions here (e^x, sin x, cos x). So there is no approximation to be talked about, and no problem in identifying these functions with their Taylor series expansions.
I'd also note that there is no need to use Taylor series to prove Euler's formula. Other series that converge to e^x,cos x, sin x can also get you there.
>The whole point of inventing the complex numbers was to have a set of numbers for which any polynomial has a root
Think about what this implies.
You have an operation, like exponentiation, that has limits. Something squared can never be negative if you are talking about any real number.
In terms of Sets, you essentially have an operation that produces results only in a finite subset of the overall set. And so the inverse of that operation, when applied to the complement of that finite subset, is undefined.
However you can introduce another (ordered) set in complement to your original set and combine them to form a new set, with operations that define how you move around the values of those sets. So in the case of imaginary numbers, you basically redefine all your reals as "real number + 0 i". And now you have a way to apply that inverse operation to the complement of the finite subset, which means you can get answers to the roots of the polynomial.
And in defining the operation of multiplication, you essentially define a way to move around the 2 dimensional set now. And moving around 2 dimensions is exactly the same thing as rotation+scaling.
And note that when you say sqrt(-1) = i, you basically assume that the complex plane is 2d. There is nothing that is stopping you from making a complex plane 3d or 4d or nd. So sqrt(-1) can also be j, or it can be k. To know what it is, you have to specify the axis of the plane when you specify the sqrt operation, which again, brings it back to the concept of rotations.
And thats my whole point, there is nothing special about i, its simply just a construct that bakes in rotations through any way you wanna define it.
>our whole point about Taylor series is also wrong, as Taylor series are not approximations, they are actually equal to the original function if you take their infinite limit
Looking back at what I wrote, I worded it very poorly.
I don't have a problem with any math involved, not trying to say that Eulers identity is not valid.
What Im trying to say is that all the definitions sort of assume that if you have some operation that you can do on real numbers, and have a result, if you just plug in a complex value, it all works out, so people think that i behaves like a number. I personally don't think that this is the case, specifically about i behaving like a number just because those results work out.
For example, even without Taylor series, you can prove Eulers identity using the limit formula for e(x). The idea is that you have (1+xi/n)^n as n goes to infinity, but because you baked in the rotation as a multiplication in the definition, all you are doing is starting at 1+0i and doing smaller and smaller rotations to get to some value, and the limit of that value is essentially the unit vector rotated by a certain angle. So naturally the cos and sin equivalence arises.
My issue is that the limit equation for e, in the case of the reals, take e x times in multiplication and then compute the limit equation, and you get equivalence. But in the case of the complex, you don't really have any idea what it takes something to ith power, but you can compute the limit equation, and so you end up with a definition of what it means to take something to the ith power.
My argument is that its not really applicable - not that its wrong, but the fact that its not defining exponentiation to the ith power in the sense that i has "number like" qualities like real numbers do. You would have to prove that an equivalence
What is really happening is that you never really escape the real numbers, and your complex numbers are just simplified operations that rotate/scale a number, like rotation matricies do through multiplication, and that in the nature of the definition of those rotations, you get stuff like Eulers identity, which is somewhat pointless because like I mentioned - the value of e (i.e 2.7) is never really used to compute anything in regards to complex numbers in polar form of re^ix, all you care about is r and the x which is the angle.
And for this reason, I don't consider i a number, so the analytic/smooth interpretations to me are meaningless.
> And in defining the operation of multiplication, you essentially define a way to move around the 2 dimensional set now. And moving around 2 dimensions is exactly the same thing as rotation+scaling.
Again, this is not how complex numbers were defined. The only original goal was to come up with a number that can solve the equation x^2 + 1 = 0, so that (R + {this number}, + , *), becomes an algebraically closed field. Once you've set this goal, there is really a single simple choice for how the operations will work, because everything else is already constrained. If x^2 + 1 = 0, we already know that:
So the formula for complex number multiplication comes out of the arithmetic of real numbers, extended with this extra entity defined simply by being a root of x^2 + 1. The fact that this operation happens to represent a rotation in the RxR plane is "an accident" (I'm sure there are deep ties that make this necessary, probably related to the structure of polynomials themselves).
And while you can define other algebraically closed fields that include the reals as a subfield, the complex numbers are the simplest such set. R^n for n>2 is clearly more complex, for example. So there is a clear reason to prefer sqrt(-1) = i, and thus ending up with a 2d vector space.
> What Im trying to say is that all the definitions sort of assume that if you have some operation that you can do on real numbers, and have a result, if you just plug in a complex value, it all works out, so people think that i behaves like a number. I personally don't think that this is the case, specifically about i behaving like a number just because those results work out.
Again, complex multiplication is not some intentional construction, it is baked into how we defined the complex numbers the first time we did. Again, our goal was to find (well, define) the solution(s) of the equation x^2 + 1 = 0. The fact that we can plug in this x to any formula that involves other numbers just falls out of this goal, it's not an additional assumption. In the case of e(x), this is simpler to see with the power series formula:
e(nx) = sum(1 + nx + (nx)^2/2! + (nx)^3/3! + (nx)^4/4! + ...)
but, by definition, x^2 + 1 = 0, so x^2 = -1, so the formula becomes:
e(nx) = sum(1 + nx - n^2/2! - xn^3/3! + n^4/4! + ...) =
= sum(1 - n^2/2! + n^4/4! ...) + sum(xn - xn^3/3! + xn^5/5! ...) =
= sum(1 - n^2/2! + n^4/4! ...) + x sum(n - n^3/3! + n^5/5! ...) =
= cos n + x sin n
Note that this falls out of the properties of e(x), cos(x), and sin(x) for real numbers, and the single property of i that it is a solution of x^2 + 1 = 0.
I also think that the definition that e(x) is "take e x times in multiplication and then compute the limit" is any more intuitive. I certainly don't think that `e^2 = e(2) = lim (1 + 2/n) ^ n, with n-> infinity` is any more intuitive than the definition of `e(2i) = lim (1 + 2i/n)^n, with n -> infinity`.
> which is somewhat pointless because like I mentioned - the value of e (i.e 2.7) is never really used to compute anything in regards to complex numbers in polar form of re^ix, all you care about is r and the x which is the angle
This is also not really true, because the value of e is deeply tied to the values of cos x and sin x. This also becomes visible if you want to compute 2^(ix). 2^(ix) = e ^ ix (log_e 2) = cos (x log_e 2) + i sin (x log_e 2), using log_e to denote the natural logarithm to make it clearer that it is related to e. So the value of e itself is still there in the formula, even if we discount the relationship between e and cos and sin.
The things is, there are no accidents in math. If you end up with formulas that look like something else, that something else was a defining part of the original definition, whether it was obvious or not.
You would agree that in the quest to solve x^2+1=0, we had to introduce another dimension, with a multiply operator that lets us move through that dimension. All im saying is that introducing another dimension and ability to move in that dimension is the same thing as rotation and scaling.
As for e, the point im trying to make is that if you look at what it means to take something to the power of something when it comes to reals, there is a clear definition. But taking something to imaginary power is meaningless it iself. For reals, the power operator has a strict definition with multiplication and division for rationals, and generically extended to reals through limits. And to compute a limit means that you have to have continuity and smoothness. So by extension, to compute exponentiation, you have to have continuity and smoothness, and vice versa.
My argument is that there is no guarantee of continuity/smoothness on the complex plane - one can define it as such (i.e the analytic view) but in my stricter philosophical view, to make something with similar properties like real numbers, you have to have all the analogous operations work within the numbers itself. I.e you have to be able to define what 2i^3i is without ever referencing anything from the real numbers.
This is not done for complex plane - to define something to the i power you need to borrow definitions from the real plane.
As such, you can't define exponentiation to the i, because you can't compute limits. Computing the taylor series expansion equivalence or limit formulas in the way me and you presented them are "holograms" - i.e meaningless results.
And it seems that way. Say that you ignore taylor series formulation, skip it completely, and define the polar form to be r(10)^ix = cos(x)+isin(x), where r is radius and x is angle. JUST BECAUSE YOU CAN.
Would you lose anything? Not really. You would still have a way to compute 2^i - it would just be 10^(ilog(2)) -> r = 1, angle=log(2). I.e the map still exists in quite a good shape.
Basically it doesn't matter if r(10)^ix = cos(x)+isin(x) or r(e)^ix = cos(x)+isin(x), as counterintuitive as it may seem, which furthers my point that exponentiation operation to imaginary numbers is not defined.
So without exponentiation, all you are left with is basically a more strict rigid construction of another set with an ordering property, a multiplication operation definition, (i^2 = -1), and the resultant orthogonality that represents a cartesian plane.
And if you think thats silly, consider the fact that the algebraic view of complex numbers doesn't even consider calculus on them to be valid.
> And it seems that way. Say that you ignore taylor series formulation, skip it completely, and define the polar form to be r(10)^ix = cos(x)+isin(x), where r is radius and x is angle. JUST BECAUSE YOU CAN.
The thing is, you can't do this. Your numbers will not work out correctly. For example, (e^(pi*i))^i = 1/e^pi is a direct consequence of how exponentiation works and the definition of i. If you define 10^(xi) = cos x + i sin x, you will get:
(e^(pi\*i))^i =
= (10^(log_10 e \* pi \* i))^i
= (cos (log_10 e \* pi) + i sin (log_10 e \* pi)) ^ i
= 10^(log_10 $exp)i
= cos $exp + i sin $exp
I very much doubt that sin(cos (log_10 e * pi) + i sin (log_10 e * pi)) is 0, so this number can't possibly be equal to the real number 1/e^pi. So, with your definition, the property (a^x)^y = a^(x*y) doesn't hold for all a, x, y. So, your definition doesn't represent the exponential function at all.
I think the point is to look at it the other way: we invented/chose i so that exponentiation (and all other functions) maintain the same properties for the complex numbers as they have for the reals. The whole goal of defining the complex numbers is to extend R in a way that maintains all of the properties of functions on R.
> You would agree that in the quest to solve x^2+1=0, we had to introduce another dimension, with a multiply operator that lets us move through that dimension.
I don't really agree, no. In the analytic definition, there is no second dimension. Sure, C is isomorphic to R×R, but that is not how you construct it. C is not R×R, it's R+{a + b * i | a,b in R, b≠0} in this view. Just like Z is N+{a * -1 | a in N, a≠0}. You introduce one new number that you need to solve the equation, and then all of the numbers needed to make the new construction a field again. You don't introduce a new notion of multiplication, C uses the exact same multiplication operation as R does, or at least as polynomials over R do, as I showed.
The fact that C is isomorphic to R×R, and that multiplication of numbers in C is isomorphic to scaling+rotation of vectors in R×R, is not part of the construction.
I do agree that there are no accidents in math, so I imagine this isomorphism is related to some more fundamental relationship between polynomials, 2d vectors, and rotation matrices - because our construction of C is strictly motivated by polynomials.
> For reals, the power operator has a strict definition with multiplication and division.
I don't agree with this statement. It's true for the integers, but already for the the rationals it loses any direct relationship to multiplication and division (how do you get 4^(1/2) = 2 by repeated multiplication?). And for the reals, it's completely gone. We can't even define many properties of real-valued exponentials - we don't even know if e^pi is an irrational number, for example.
> When you search for something like taylor series or limit form of e and you see a way to compute e^i what you are doing is basically using operators designed for real numbers and extending them to the complex numbers (when you substitute i for where normally a real number would be in the power term.
We've already agreed that there are no accidents in math. So just as the multiplication of polynomials is fundamentally linked with rotations of 2d real-valued vectors, we must accept that real exponentiation is fundamentally linked to complex exponentiation, otherwise the formulas wouldn't work the way they do.
Note also that calculus is not limited to operations on the number line - you can take the derivatives or integrate or calculate limits of n-dimensional curves, and even differentiate over surfaces and n-dimensional manifolds more generally. Smoothness and continuity are anyway part of the structure of C, regardless what definition of it you use.
> And if you think thats silly, consider the fact that the algebraic view of complex numbers doesn't even consider calculus on them to be valid.
I don't understand what you mean by this. Calculus is well defined for functions over any field of size continuum, and that is exactly what <C,+,*,0,1> is in the algebraic view.
>C uses the exact same multiplication operation as R does
Not quite. If it did, the -i*-i would be i^2, not -1. And yes, I totally agree that C is R+, not RxR. The point is that you still introducing something extra with some rules, where you introduce the concept of geometric orthogonality into i^2 = -1, whether that is your intention or not.
>For reals, the power operator has a strict definition with multiplication and division for rationals, and generically extended to reals through limits.
I accidentally swapped reals and rationals there. The whole point was to highlight that exponentiation for real exponents relies on limits which relies on continutity.
>So just as the multiplication of polynomials is fundamentally linked with rotations of 2d real-valued vectors, we must accept that real exponentiation is fundamentally linked to complex exponentiation, otherwise the formulas wouldn't work the way they do.
Don't agree.
Multiplication of polynomials involves operations that are clearly defined. When you do (a+bi)^2, you have defined what it means to multiply complex numbers in their construction, without needing to use any such formula from real numbers.
Exponentiation where you have i exponent however, is not defined solely in the complex field.
>I don't understand what you mean by this. Calculus is well defined for functions over any field of size continuum, and that is exactly what <C,+,*,0,1> is in the algebraic view.
Algebraic view is basically the idea of that numbers are only defined if there is some relation that expresses their definition.
For example, 1+2=3, locks all 3 number down. 1 is 3-2, 2 is 3-1 and 3 is 1+2.
pi or e on the other hand, are "something else", because there is no algebraic formula that defines them. To do so you have to invoke the computation of limits, which is an analytic view, not algebraic.*
> Not quite. If it did, the -i-i would be i^2, not -1.
It is i², though, but that is equal to -1. Just like 22 = 2² = 4. I also maintain that the historical view is that this something extra comes from the properties of polynomials and their roots, not from geometric orthogonality.
> Exponentiation where you have i exponent however, is not defined solely in the complex field.
We can take another tack for defining the complex exponential function, if you'd prefer. One of the definitions of the exponential function is that e^x is the only function that respects f'(x) = f(x) (well, up to constant multiplication).
So, we need to look for a function f(z) such that f'(z) = f(z). There are various ways to do this (for example, using the Taylor series expansion and noting that all of the f derived n times factors are equal to f(z), which yields the power series definition). You don't need to appeal to limits of e^x to get there this way.
> Algebraic view is basically the idea of that numbers are only defined if there is some relation that expresses their definition.
Understood, you are using a different sense of "algebraic" than what I was - I was thinking more of the abstract algebraic definition of C.
Still, the sense you are using seems to be the concept of algebraic numbers, which is more formalized - the algebraic numbers are all those that represent the root of a polynomial with integer or rational coefficients. Interestingly, while pi and e are not algebraic numbers, i or i+7 are still algebraic.
However, I'm not sure what the point of bringing this up is. Exponentiation is simply not defined over the algebraic numbers, especially not e^x where x is algebraic - so if we restrict ourselves to algebraic numbers, e^i is not defined, true, but neither is e^1. And while 2^2 is defined, of course, I'm not even sure you can define 2^sqrt(2), so I wouldn't be surprised if 2^i doesn't make sense either.
Either way, the algebraic numbers are not "a way of thinking about numbers", they are a restricted subset of what is generally meant by "number", and many famous and useful results from many branches of math do not work over this subset (for example, you can't even use the same set of algebraic numbers to refer to the length of a circle and the length of a square).
I still think you misunderstanding what im saying. i don't have a problem with the math working out, its that i have a problem with the math being able to be applied in the first place.
Let me try a different way.
Suppose you start with just the i number line with the rules that exist. You have zero, integer i's, rational is, and even irrational i's. All seems good. Then you start to define operations. i*i is undefined (because it goes to -1, and the reals are outside of your domain that you are working with currently). And this means you can't effectively do any sort of futher work in defining exponentiation, or limits, because you can have purely complex polynomials with just undefined terms.
So like you said, complex is R+, not something like RxR. The definition of complex numbers is intrinsic to real numbers - its an enhancement on real numbers. And by extension, all the math works out when you do taylor series with e^i and such.
But this pretty much means its a rigid definition, i.e you are defining something in a certain construction to supplement reals.
And as for geometric claim, my argument with that is that just like when you have x, and then you add <x,y> in some form and way, you are defining geometry. So in defining a+bi, you are defining geometry.*
Note that "reality" is not quantized in any existing theory. Even in QM/QFT, only certain properties are quantized, such as mass or charge. Others, like position or time, are very much not quantized - the distance between two objects can very well be 2.5pi planck lengths. And not only are they not quantized, the math of these theories does not work if you try to discretize space or time or other properties.
Having tried to use that, I can assure you that no one actually (a) remembers their passphrase, nor (b) is willing to type it in when it does come up. It's a fun idea, but it's actually much worse UX than even a secure password.
This makes little sense, even though it sounds superficially convincing. However, why would a language model assume that the car is at the destination when evaluating the difference between walking or driving? Why not mention that, it it was really assuming it?
What seems to me far, far more likely to be happening here is that the phrase "walk or drive for <short distance>" is too strongly associated in the training data with the "walk" response, and the "car wash" part of the question simply can't flip enough weights to matter in the default response. This is also to be expected given that there are likely extremely few similar questions in the training set, since people just don't ask about what mode of transport is better for arriving at a car wash.
This is a clear case of a language model having language model limitations. Once you add more text in the prompt, you reduce the overall weight of the "walk or drive" part of the question, and the other relevant parts of the phrase get to matter more for the response.
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