I agree with you in part, many things in math have little direct relation to reality, or at least daily life for most people.
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.
I was trained as an engineer and the abstract part of maths used to bug me a lot. We were often told to just accept them as they are, that just like everything else they add to your Toolbox. My biggest bugbear is imaginary numbers, can't seem to get my head around what they are but I accept that this is due to my limited imagination. Well I read that they can't be imagined because it's a human construct - honestly I don't know what to say or think.
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
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. :)
Computer, mostly hardware (so heavy on electronics rather than programming)
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 :)
You can and should visualize imaginary and complex numbers. Whoever told you otherwise was wrong. See my other post. Plot a complex number (say 2+4i) and then try two operations on it: multiply by 2+2i (this will rotate the point to the left and scale it); and try adding a complex number (say 1-3i) which will have the effect of translating the original point.
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.
I was thinking about your post yesterday, thanks for it :) Sorry, I didn't clarify with what I meant by "visualising." When you think of an abstract entity, like a foreign word for example, you can sort of grasp what it means when you have it translated. But you can't do that with the imaginary number. You can only get a 'feel' of what it is when it's set in context, like the expressions you gave, and even then the application is still in a math world (geometric (?) space.)
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.
Imaginary numbers just have an unfortunate name. The complex numbers (having a real and imaginary component) are objects on a 2-dimensional plane. Using complex numbers lets you describe, in algebraic terms, geometric entities and operations. Adding a+bi is translation. Multiplying by i is rotation by 90˚ counterclockwise. Multiplying by a, a real valued scalar, is scaling. Multiplying by bi is rotation and scaling. Multiplying by a+bi is rotation and scaling again, but the angle of rotation will be something other than 90˚.
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".
Imagine a flying unicorn shooting rainbows out of it's face. I can imagine that, even though it's a human construction, so I reject the argument that you can't imagine human constructions.
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.