Of course you can't bring me a bucket of negative apples. That's the point. It very much is a philosophy and doesn't always relate to reality. Reality is, you can never really have a bucket of negative apples.
I just disagree with your example and I dislike it because I've literally had an argument with an engineer (an engineer!!!) about whether negative numbers were real who insisted on sticking to this sort of example. Negative indicates direction, this isn't really abstract. If it's increasing your balance of apples, it's positive. If it's decreasing your balance of apples, it's negative. Many people understand this quite well in the sense of monetary debt. I "possess" negative $xxxx, because that's the amount I owe someone else, in turn the holder of the debt "possesses" positive $xxxx because they expect to receive it at some point in the future.
If this is philosophy it's the barest levels, and it's philosophy that any culture with a concept of ownership and debt would have no difficulty with.
Bucket of negative apples, hmm. I feel slightly sheepish to say this but I immediately visualised it to be "a bucket that you intend to put n apples in, but didn't." Thank god I'm in a semi-technical field now :D
Edit: language.
The complex number notation describes an object in space or an action to be performed on objects in space. Other than it being more compact, it's not substantively different than a statement describing an action like "travel 50m northeast, then turn left, then travel 40m".
In electrical engineering, imaginary/complex numbers are critical in circuit analysis once you move into the AC domain. You can try doing it with sines and cosines, but that's infeasible as soon as you move beyond anything basic. [1] But when you transform them into complex numbers using Euler's formula (one of the most beautiful things in maths, IMHO, especially Euler's identity) [2], it becomes "easy".
Then there's also the Fourier Transform [3] which transforms time-domain signals to frequency-domain.
So my visualisation of imaginary numbers is something involving circles and spirals being twisted and shifted all over the place. Hard to explain, but I've reached the point where I feel I have a good grasp on them. :)
[1] http://hyperphysics.phy-astr.gsu.edu/hbase/electric/impcom.h... [2] https://en.wikipedia.org/wiki/Euler%27s_formula [3] https://betterexplained.com/articles/an-interactive-guide-to...
Oh yes no doubt j is very useful - but that's what I meant by it being part of the Toolbox. To visualise it as an entity is something else though, and I know that you're not really supposed to do that, especially when it was 'invented' to be useful in the first place.
Perhaps with more experience and perseverance, I might have reached an understanding like yours. I do find myself regretting for giving up on engineering so soon, but hey, I can still take my time to understand what I couldn't before. Thanks for the links :)
Developing an intuition about what rotation a particular complex number will give you is more complex, admittedly. But this is enough to visualize the effects, if not precisely mentally predict the result.
In hindsight, I think that I approached it wrong, and it was far too simplistic anyway. I've been reading popular maths books lately and it's made me realise that there are more ways than just visualisation to understand something. Ironically (a happy one!) this is making me excited about maths for the first time in my life.
A key distinction here is that one has a defined implementation. Making that distinction is philosophy. Talking within the framework of implementation is mathematics.
And it's not really my fault that nobody can bring me negative apples. It's not like I made the rules! Sheesh! ;-)
I just don't see the value in conflating both terms (philos and math) when there's a distinction pointed out above.
You can't bring me a bucket of negative apples. You can bring representations of future apples, or something like that. However, negative apples do not actually exist. In all these years, nobody has brought me a basket of negative apples.
In the moment where you have an empty bucket and a debt of apples, I'd call that a bucket of negative apples.
You can call it negative apples, if you want. However, it's just an empty bucket. ;-)
In other words, it's all in your head. A real, physical, "negative apple" does not exist in this world. Only the human philosophical concept of "owing someone apples" or similar such things. That's the guys' point. It doesn't matter whether you can bring representations of negative apples because those all depend on a philosophical bedrock.