I think I have aphantasia and no inner monologue. Mind you, I can summon an inner voice to compose a sentence before saying it, but when I'm thinking about something being discussed and someone asks me what my thoughts are so far... I never have any idea what to say. My mind is blank! It's always blank! There are never any discernable words or images in there to give you. If I need to communicate my thoughts, I have to spend significant amounts of time translating to words and choosing words before I can actually summarise what I was thinking, which is much more nebulous to me than words or images.

My 'thoughts' are closer to a mouse cursor changed into an hourglass while waiting for a computation to finish than 'First we need to do <XYZ>, but to do <XYZ> we need <X>, <Y>, and <Z>. To get <X>, <Y>, and <Z>, we need to ...'

I find it really hard to operate in live/in-person discussions because of this. I physically end up just as silent and blank as my mind!

I find this kind of stuff, including the authors article, weirdly fascinating. I try to do what other people describe, such as yourself, and it really is impossible. It just makes no sense to me. I'm sure I have ways of thinking as well that probably baffle other people. It's all very strange.

With that being said I wish my mind was blank sometimes, I wish my inner monologue would shut up every now and then. :)

I was surprised when I learned that everybody doesn’t have a spatial calendar. Mine is a rectangle with the first six months on top and the last six months returning in the opposite direction on the bottom, forming a cycle. (I also “feel” mathematics and code spatially.)

I’m curious why these differences happen and to what degree it’s difference in thought versus difference in conscious perception of thought.

Some people can think in numbers in a way that does not require visual representation or any kind of representation, and as such, it is also possible for such a person to express the pure idea in different ways, including numbers as shapes as the author is doing.

'Cause I am very curious how the author experiences imaginary and complex numbers ... or even negative integers, irrationals, and transcendental numbers.

Negative numbers are just like the positive ones, but kind of... the opposite. Like the indentation formed if you pressed the positive number into clay or sand or something. It's like they want to be filled or take away from something else rather than adding onto other forms.

RE how I think about imaginary or complex numbers, in short, I don't :)

I've never studied much higher math, and don't have any reason to think that I'd be particularly good at it.

The author describes thinking of 9 as floating around looking for a 1 to chomp off another number. This is very clearly designed to support good intuitions about adding in base 10, but it produces bad intuitions about binary numbers, multiplication, polynomials, etc. If faced with myriad other problems that involve 9, like, say: "which is bigger: 2^9 or 9^2?" or "how should we store words from an alphabet with 9 characters in memory?" or "how can we distribute 9 things equally?" or "for which n is 9^n + 2 prime?" or "how should we expect an atom to act if it has 9 electrons?", a completely different way of looking at the number 9 is warranted. In that last case, the exact opposite is true: 9 is desperately trying to rid itself of a 1, not find another 1 to grab.

The author isn't describing a random system they came up with to deal with numbers.

They are describing how their brain naturally sees numbers: which is commonly referred to as synesthesia.

What's interesting about it is that it may show some evidence of synesthesia being education or culture dependent. Using base10 for numbers is taught at an early age, when the brain is forming a huge number of associations for learning, but I don't think it's been shown that synesthesia arises purely from "nature" (genetic) origins. So with the right thoughts, anyone may be able to push themselves into this state.

Author coming up with a system, and the system arising naturally are not mutually exclusive as far as we know. After all, what is natural vs... vs what, really?

I still think you're missing the point a bit. I don't think the author is doing this as some sort of trick or by design. I think they're literally describing how they visualize numbers in their head. Reading through other peoples' comments seems to support my conclusion on this.

Maybe they visualize other number relations differently in their head. To me, I could not do math in my head like this and it makes very little sense to me. I don't even really get what they're describing to be honest with you. I visualize numbers in my head as the number symbol you'd write down.