I am a successful software developer and I’m terrible at math. To me, 6+3 is not an interaction between two different anything, rather, it’s a key in a hash table where I’ve stored “9” as the value. All arithmetic is rote memory recall for me. I work with complex numbers by just breaking them down into multiple steps.
Now I’m wondering if I should challenge my brain to do this differently.
With that being said, trying to think a different way for the challenge of it is definitely interesting. Reading through some of the other comments here and trying to taste words or replicate other people's minds is a weird, fun exercise :)
https://sites.google.com/site/steveyegge2/math-every-day
https://www.cantorsparadise.com/the-unparalleled-genius-of-j...
I would argue his way of thinking of numbers makes him slower at doing calculations.
When you create a 2D visual representation of a number system you want to choose a shape that has the same properties as numbers. Namely the shape must be monoidal under composition. This allows you to keep one type of shape
For example (int + int = int). When you compose two triangles together you get a parallelogram, so triangles are actually kind of bad as you would need to classify several different types as numbers. (triangle + triangle = parallelogram) The only shape that I can think of that is monoidal under arithmetic composition is rectangular quadrilaterals with at least two parallel sides.
Examples: Rectangles, parallelograms, and trapezoids each can be composed to form another shape in its own class. With rectangles likely being the most efficient representation as they are fully symmetrical (to compose two trapezoids to form a new trapezoid one trapezoid has to be inverted, this does not happen with rectangles).
So his even number visual representation is quite good (it uses blocks) but his odd number representation is all over the place and seems arbitrary. Just look at 9. It involves "orange peeling" another number just to shove it into the little dent. His system involves mutating, rotating and changing the shape of each "number" in order to perform composition. This costs more "brainpower" to do and is the main reason why I don't classify his ability as a "gift".
It's highly inefficient. I think many HNers are mistaking it for a super human ability. I don't agree. This is more of an interesting ability then it is a talent.
But that's just a guess. Would actually like to see a quantitative measure of how fast he is at adding numbers under his system. This would definitively answer the question.
Certain calculations are actually faster because i begin to have faith in my feeling of the math over doing an actual calculation - with the same type of confidence i have when recalling a times table for example. Still, it usually doesnt get me all the way to an answer
There are certain mathematical rules that you can probably identify that are related to my internal expressions and how they "fit" together. For example, I do not know without calculating what "25 x 15" is, but I have an idea of what the answer feels like. anything below 100 or over 1000 feels outright OCD level out-of-place. Numbers like 114, 201, etc, feel dirty and incomplete. we can identify in this scenario that the shape / feeling of the answer for me is related to an intuition for the mathematical principle that the product of two numbers that are divisible by 5 is also divisible by 5 - but at no point did I deliberately evoke that rule when conceiving of a possible answer. Also this is a simple example, this intuition runs beyond my knowledge and ability to formally explain the principles. In reality, many such principles (learned or inferred) come together at once to feed my internal expression of the answer. A calculator says 375 is the answer, though 325 and 475 feel about the same
I do not think it makes me better at getting correct answers, but it does help me accept an answer as being correct when looking at it also feels right. It's most useful when identifying errors. There is a big help when you see "15 x 25 = 356" and without thinking you can feel internally like something is out of place, dirty, needs attention (this applies to advanced topics as well). As I said above though, more than the correct answer can have the same or similar feeling - so it is prone to false negatives
With something like math, intuition based guess work that has room for false negatives is hardly that useful overall. So maybe the only real edge it can provide is in working with novel concepts where you have to guess a direction to explore and hope you uncover something useful. That is an unfounded hypothesis though.
I have a similar impression when reading posts elsewhere about categorical structures in programming: they are repetitive and mostly trivial (actually, the category theory without context is trivial).
For example, a string concat can be understood as an addition operation:
1 + 0 = 1 (identity)
1 + 1 = 2
1 + 2 = 3
2 + 1 = 3 (communitive)
"a" + "" = "a" (identity)
"a" + "a" = "aa"
"a" + "b" = "ab"
"b" + "a" = "ba" (non-communitive)
There's this whole intuition about addition itself that can be applied to something other than integers, and being able to reason about that is applicable to how you design software, particularly function interfaces.
Just as a note, my mother made me memorize the multiplication table when I was a kid, and I had ended up memorizing additions just through sufficient practice. I was able to intuit what additions and multiplications meant, but for the purpose of taking tests in school or doing homework, additions just pop out as answers because of the memorization. It wasn't until much later in life that I started encountering ideas such as, what if you were adding something other than numbers.
It's hard to imagine having only one instinctive visualization for integers.
First number is 1, so its table of 1. Then x as multiplier sign. Then a count from 1 to 10. Then = sign. Then the result. We kids are supposed to write each line in left to right direction, then move to next line.
We use paper with square tables or graph on it. Most of the time, kids simply write 1, move to next line, again write 1, all the way till 10th line. Then we move to next column, write x, then move up, x all the way till 1st line. Then 1,2,3, in next column, = in next column coming up. Then the answers going down.
OP does what they do in number blocks.
I guess with enough practice they are both fine for solving known problems. I think our way is better for programming, and his way is "better" for physical building.
I also think there’s no need for people to feel like they need to be some math or grammar prodigy to get by in life. It’s perfectly fine to outsource your mental functions, including memory to a calculator, notebook or PKM system like Obsidian.
I know what you mean, I also "see" code in a similar way as the OP author explains numbers.
It's though mostly "blocks" that interact with other "blocks" and a large application is comprised of probably hundreds of blocks organised in specific shapes with interaction lines between them.
This helps me spot "poor" application design when blocks that should be separate are actually intertwined (tightly-coupled or concerns not separated).
It's sometimes hard to describe these in architecture documents or PR's as it seems not everyone is seeing the program on this level.
If I am bored, or trying to fall asleep, I picture reversing a linked list or a bubble sort.
It's fun, isn't it? One of the saddest things about aging is I never again will be a 17 year old blonde girl who looked like Rapunzel with a decade of programming experience in the mid-00s. The dissonance drove people insane.
I “booksmarted” so many guys in college who didn’t get that in real life you can put points into other stats like chr and it doesn’t take away from your int.
So I totally have to write the code to reason about it. I didn’t knew people could imagine portions of code so it explains some things. I’m not sure it bothers me or even if that’s abnormal because it’s always been like that so I can manage that. But it’s tiring.
But it also forced me to learn to be concise and to express the fullness of my thoughts through langage (or code). Which is really useful in this job.
Also, I have ADHD which I know from my psychiatrist affects short term memory. I wonder if "picturing" things happens in the same brain région than short term memory. It would explain a lot of things. Maybe I’m just some individual with broken RAM and I had to compensate with "overclocking" my CPU of thoughts. </personal-theory>
https://www.youtube.com/watch?v=OPTOCwQoYR4&t=29m23s
This is how my toddlers are learning. It's really good.
Is he/she getting bored because it's too easy, or frustrated because it's too hard? I was crazy bored with arithmetic homework and I later took 2 years of math from a local university while I was still in high school because my high school ran out of high-enough-level math classes for me. Look up the story of Gauss in elementary school for a much more extreme example.
Don't assume your kid is behind at math because they don't like arithmetic homework! They could be too far ahead! Useful links if you suspect that might be the case:
While I don't remember the colors or anything, I still visualize addition like this I think.
Sometimes numbers are for quantifying a pile of things, and 255 and 256 are basically the same.
Sometimes numbers are for cryptographically signing things, and 255 is extremely secure while 256 is completely vulnerable.
Sometimes numbers are for arranging tournaments, and 256 is a tremendously useful number while 255 is super annoying and you should look for another.
Sometimes numbers are stored in a single byte, and 256 (=0) is the friendliest number you will ever know, while 255's words are BACKED BY NUCLEAR WEAPONS.
Sometimes infinity is a useful number, sometimes it's not. Sometimes 1/2 is a useful number (pies), sometimes it's not (babies). Sometimes sqrt(-1) is a useful number, sometimes it's not. Sometimes the sum of all positive integers equals -1/12; sometimes that's stupid.
All of these situations may call for visualizing numbers differently.
This sort of thing reminds me of an article I read a while back about how some people don't have an inner monologue when they're thinking which I assumed everyone did and found wildly strange trying to think about how other people think. This article is also equally confusing to me.
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!
'Cause I am very curious how the author experiences imaginary and complex numbers ... or even negative integers, irrationals, and transcendental numbers.
You misunderstand. This person is talking bout how they see the numbers in their minds eye meaning this is how their brain works. As a visual thinker I can relate to how there's an uncanny ability to see things as shapes or things.
The unique thing I think I have is that I visualize long strings of digits as notes on a musical scale. 735 is high-low-middle. I have found I can retain strings of up to 15 or so digits in short-term memory by chunking them into triplets and memorizing them as arpeggio chords, or by their relative positions.
I see numbers as notes. So each unique number - say my college ID, my aunt's cell number or my driving license is a MIDI tune in my head.
Also, visually each number is like a 'identity' - not a numeral. When I had a image processing class, running an edge detector on binary images was fun. I could guess the potrait/image just looking at the numbers (just like how you'd guess animal shapes in a connect-the-dots game)
I have all my important numbers memorized - SIN, multiple credit cards, driver's license, library card, health cards, all my financial account numbers, and the same for my partner. Do you do that too?
Only digits have notes associated with them. A number like 1000 is just four notes in a row. Irrational numbers again are just composed of their digits.
Was this learned by him or is this some sort of synesthesia condition?
I've never thought about it before, but while I definitely don't have as distinct models as the author, I do understand and agree with an instinct around numbers "fitting" together to make tens, and it definitely informs how I break down e.g. triple digit mental addition.
I experienced similar things growing up. For my case, it was usually colors. Each number was associated with a specific shade of color, but in my case it was less about the numbers themselves; it was more contextual. Eg. The number four represented different colors depending on whether it was describing the time of day, the number of floors on a building, or amount in currency.
I had brought this up in my youth only to be met with derision and threatened with being labeled "abnormal" by the authority figures, so I worked to suppress and hide this aspect. (South Korean society had a lot of backwards ideas in the 90s).
All of this is because our eyes have 3 color detectors, RGB. The reality is, RGB is just a mapping. Actual color is a singular number scalar based off the wavelength of light. So colors are actually a GOOD choice for number representation.
The real question is how do colors compose in your brain? If you encountered two numbers 8 and 9. Could you add those numbers just thinking in terms of colors? What about for large numbers, there must be large enough numbers where no color mapping exists as colors have limited range in our spectrum of vision. How does your brain picture 9999999999999999999999?
Even if it's contextual if your brain follows consistent logic during composition of numeric entities it's still a valid analogy. The can opener number 9 just seems completely over engineered.
There is no such a thing as a base or perfect model, nor goal to reach, by cheating or not.
People who are usually referred as bad at math just need another perspective. They might not understand the dominant perspective.
Compare eg. Groethendick or Lebesgue work with their contemporary fellas. And then ask yourself: why are some people more comfortable and fruitful with one perspective but not some other. Is there some constructions of some fields that will suit better one or another group of the population. Do our brains internal structure mature at the same age… etc.
And having none is very different from having a tiny amount - I think if I was motivated I could train to have phantasic abilities.
So not always a genetic cheat sheet - just something we don't talk about or train people in, when we could. I wouldn't be able to swim either if I'd never been trained!
Exponentials get represented as a third dimension; where basic arthimetic is 1 or 2d depending on the context, exponentials go into a third dimension if that makes sense.
Modulo is the leftovers / splash-out when I pour one number into several smaller containers.
Fractions are simply fractional amounts of a tank of liquid (i.e. 2/3 is simply a measuring cup filled to the 2/3 line type of thing), but I can't ever picture them very accurately for weird fractions. "Improper" fractions are basically the same as modulo.. almost as if they're unstable in my head and automatically "pour" themselves into more tanks that fill as needed until some remainder is left.
I don't have a visualization for roots, which is probably why I'm generally so bad at them.
The representations helped in engineering school for getting a "feeling" about a formula; it was often very easy to notice if an equation I was massaging had gone off the rails. For a pure proof however (not that I did much of that), it was useless.
Your modulo visualization helps me, I think.
[1] - https://www.amazon.in/Visual-Thinking-Mathematics-Marcus-Gia...
If someone repeats a "tasty" word a couple times, could that make you want to eat a meal with that taste?
Also: are there "disgusting" words? Does the word "vomit" taste like it, or like something totally different?
Came across the “tongue knows” meme recently and it is wild for me — perhaps as close as I might ever get! Curious if anyone else with aphantasia has the same reaction?
> Your tongue knows exactly how everything you look at will feel. > Try it! Look at the table leg. You know what it will feel like if you lick it. Imagine licking a football. Or the couch. Whether you have or haven’t actually licked these things, when you imagine it, your tongue knows. It knows.
On the contrary there is enough evidence that top mathematical competitions (IMO, Schweitzer, Putnam competition etc.) have been dominated by Russians & Slavics for a very long time. Same holds for Fields & Abel prizes. The link between phonetics & notations have little to do with proficiency. Basic arithmetic upto counting 5 maybe - when numbers are like 一 ニ 三. But beyond that the association seems tenuous. Math isn't just arithmetic.
Asian kids have a lot of study drills. The amount of after-class homework that I see kids doing makes me sad. Arithmetic gets better with practice. No magic formula
https://www.npr.org/sections/krulwich/2011/07/01/137527742/c...
This is the same thing as map reading or what we do in programming. The thing that's disappearing in this comic: https://heeris.id.au/2013/this-is-why-you-shouldnt-interrupt...
I also realized early on that I could count way faster if I fought the urge to say the numbers in my head because the idea of the number would still be there. I started by saying (eh eh eh eh eh) in my head instead of (one two three four five). Eventually you can do things like run your finger across a comb and instantly know how many bristles you passed - that gives you a tactile response for each number rather than the words themselves. If you count by 2s 3s or 5s you can go even faster (which is what the circle is doing in the article). Shortening the "time" axis of the counting.
There's something very rhythmic about counting beats that line up with powers of 2 and I'm able to count things extremely quickly and precisely without even thinking about the numbers I'm counting. When I want to remember how much I've counted, I simply think back at where I am in the rhythm and come up with the results in a strange vibes-ey way I'm not really able to describe (for example, I'll just intuitively 'know' the difference between having counted 32 beats and 64 beats, and then I can use that knowledge to hone in on the precise number I'm at using a sort of mental binary search).
I'm sure someone with more knowledge of musical theory or neurology could provide a better explanation, but it feels like I'm somehow taking advantage of whatever part of the brain keeps track of beats and rhythms in music, then using it to count.
Edit: I just tried this technique while listening to music and, as expected, I completely lost the ability to count in this way. Almost immediately I lost track, before I even hit 16.
I explained it (poorly) to my wife once and she made fun of me about it. Well until our son told us years later out of the blue that it's how he sees numbers.
[0] https://en.m.wikipedia.org/wiki/Ordinal_linguistic_personifi...
Sometimes the domain is already graphical - and I take every opportunity to make the code match the visual layout, ex:
https://github.com/danthedaniel/gameoflife-rs/blob/master/sr...
/// Count living cells adjacent to a cell in the matrix.
#[inline]
#[rustfmt::skip]
fn alive_neighbors(&self, x: i32, y: i32) -> u8 {
[
self[(x - 1, y - 1)], self[(x + 0, y - 1)], self[(x + 1, y - 1)],
self[(x - 1, y + 0)], /* selected cell */ self[(x + 1, y + 0)],
self[(x - 1, y + 1)], self[(x + 0, y + 1)], self[(x + 1, y + 1)],
]
.iter()
.fold(0, |total, &neighbor| total + (neighbor != 0) as u8)
}I'm also curious if you are an unusually quick calculator compared to others you know. Synaesthetes can sometimes turn their condition into a talent, like the famous Shereshevsky who had a photographic memory; every experience was utter sensory overwhelm, making mundane information very memorable.
I'm not an exceptionally quick calculator as far as I'm aware, though I've never tried to measure. I don't have strong associations with numbers greater than ten (though certain classes of numbers like multiples of five tend to have forms in some contexts), so I do arithmetic on larger numbers digit-by-digit, which is inherently kind of slow.
But then I read Moonwalking with Einstein, about a journalist's attempt to win the US memory championship. Most of the book is just about that, but he does spend a chapter critizing Daniel Tammet, basically accusing him of lying about his condition and using simple mnemonic tricks to do his "feats of memory". His case is very compelling (e.g., Tammet being inconsistent in answering how numbers "look like" in different interviews, forum posts by Tammet on mnemonic techniques before he was famous, etc).
If I remember well some natural talented musicians (that didn't do a lot of formal training), also see notes/music in colors.
This made me think if I had some special talent relates to this kind of visualization but the only thing I associate with colors are places, which might explain my good orientation skills but nothing more than that.
I see time differently, days of the week, yearly calendar, distance units, temperature. All these, maybe more I can't recall now, visualize different from just numbers. E.g. the year is a loop. If I want to recall a month name, I always see a part of that loop, and the camera is not fixed. I'm fairly certain my mind didn't come up with this on its own, but there were some visual that got paired with it. Same with numbers I suspect, but this one is more obvious.
"5" and "6" are definitely guys.
The evens tend to be a bit kinder than the odds. Hasn't helped me with arithmetic, though.
My weird thing is that they all have a color. 0 is gray, 1 is blue, 2 is yellow, 3 is red, 4 is green, 5 is blue again, 6 is purple, 7 is red, 8 is orange, 9 is yellow. After 9, it's just the last digit that typically "colors" the number in my head.
And I get what you mean about numbers having a personality: https://news.ycombinator.com/item?id=30365289
Your comment made me laugh out loud. 7 isn't that rebellious in my universe.
[1] https://4.bp.blogspot.com/-Et6_8IvPOW0/VEPMsOiyVAI/AAAAAAAAP...
Such a fascinating read, thank you! I'd love to read/see more about those other numbers with unique forms, and also features of the way numbers combine. (like the way you described 7+3 or 9+x), I want a part 2! Thanks again.
Beyond 10, it's mostly classes of numbers that have unique forms, rather than the numbers themselves. So 15, 25, and 35 to some extent are somewhat of a stair pattern made of three squares, and they want to interlock nicely with each other. Things like powers of two also take on these sorts of interlocking forms, so maybe it's just numbers where I've memorized their doubled versions over time.
Low multiples of three from around 18 through 27 take on a sort of blobby trefoil shape and want to be divided into three parts. Higher, obvious multiples of three like 333 or 666 are similarly tri-lobed, but each lobe is a bit spikier in a way. I don't really have any strong associations with operations for these apart from splitting them into three. Again, I wonder if this is a sort of learned association from multiples of three that I encounter a lot.
7-3 I found interesting because those are modulus complements in base 10
Although now that I think about it there is still some element of what's described in this article. There's no visual shape involved in the way I model numbers, but it resonates to think of 7 as "10 with a 3 missing", but also as "5 with a 2 on it". The concepts are built in reference to their closest multiple of 5, and slide between different equivalent forms as necessary in calculations.
By the way, the way I do mental math without images feels like it is using sounds and words for the short-term storage and recall. The language brain seems good at putting something aside for a minute and then bringing it back afterwards with a low chance of error, like repeating something someone just said back to them verbatim even though you weren't really listening.
The one method I am sure _doesn't_ work well for mental math is picturing the grade-school algorithms on an imaginary sheet of paper. For whatever reason it is very error-prone. I once did an informal (definitely unscientific) survey on this (30 or so people IRL plus like 100 reddit users) and iirc there was a strong correlation between "imagining the pen-and-paper algorithm", "being bad at mental math", and "not liking math". Wish I still had the data from that -- all I remember is roughly confirming my hunch that those were related. I also wrote a blog post about this a few years ago (https://alexkritchevsky.com/2019/09/15/mental-math.html) but I wish I had included the survey information in there, it would have been much more interesting.
Interesting that we share some conceptual similarities in how we think about numbers, but they're expressed through different pathways (language vs. visual.) I wonder if the people who imagine pen-and-paper stuff when doing mental math just don't have these pathways set up, and instead recall memories of math-adjacent experiences in lieu of another internal representation of numbers.
There is no other feeling in my mind which give me that amount of understading,but it is an understanding which cannot be formulate properly to other person. I feel it as an phenomena which origin started in my mind and grow there for my whole life (which is highly correlated with my introvert pov). Thanks for that thoughts, all best.
https://en.wikipedia.org/wiki/Number_form
https://www.discovermagazine.com/health/the-rare-humans-who-...
In France, "97" is said "Quatre vinght dix sept", i.e. 4x20+10+7. This is apparently acceptable to the brain as a final answer, there's no way to collapse it to "90+7".
Instead of forms I have color associations with digits.
And I suspect that many people have those...
What is your color associations (if any) with 0,1,2,3,4,6,7,8 and 9 digits?
Your sort of just build everything you need out of analogs. It makes me think that if we were not indocrinated into numbers from an early age, we'd end up inventing them as an abstraction to the sort of thing you have to do when you're trying to avoid numbers.
Another one I suggest trying is expressing and exploring linear regression without reference to probability theory.
I remember as a child trying to draw the shape of the number “line” (it curls and twists) and being surprised that I was unable to do so.
This has never seemed to have given me any advantage or disadvantage in learning more complex maths than one gets in primary school. But since so much arithmetic is done in numbers less than 100 (or scaled down to that range) it does make a lot of things easier.
This is almost how adding, subtracting numbers is taught in schools here. I didn't notice this pattern when was a kid (if this was the pattern back then) but after having sat down with both my kids I can see the pedagogy clearer.
This actually inspired me to keep up with the literature the kids have to go through when they are learning math. I am a little surprised that I find it so enjoyable. There are methods I completely forgot and methods I can grasp easily and add to my existing knowledgebase..
There is always something to learn, even if it is elementary!
At the same time, 8 is sometimes a 10 with a notch with room for 2..
Question: Did you find math easy from the beginning of school? I had immense trouble, it just didn't make sense no matter how long I looked at the shapes.. What logic was there behind the 6 shape and the 1 shape together makes the 7 shape..
Do you see negative numberse as different from positive ones, or do you just see them as positive numbers that have to be subtracted? Depending on the context, they're just a subtraction, other times I see them as white and black dots, and when there is equilibrium it is a grey plane of 0, and when add some to the plane, that number of that sign will be visible, for example, you have 0 + 3 = 3 white ones, then you add two black ones ( 0 + 3 + (-2) ) = 1 and the two black ones merge with the white ones and become grey and then you see only the one white one that 's left.. It goes that if you then add two more black ones, ( 0 + 3 + (-2) + (-2) ) = -1 one of the black ones merge with the remaining white one, and there is one black one left on the plane.
I eventually got the basics.. these days, I still prefer using letters instead of the actual numbers, and deal with the abstract instead of the concrete and just let the computer do the calculation if a concrete value is needed.
Wrote about my experience a while ago: https://paradite.com/2017/09/17/bilingual-numbers/
On a related note, I wouldn't be surprised if synesthesia is actually more common than we currently think because a lot of people who have it think it's normal or never thought anything of it.
The consequence is, I don't think there is a 'right' visualization for numbers. You either have an exact model of mathematics in your brain, or you have some approximation thereof, which by definition has to be wrong in some way (but is easier to get / reason about). There is only one true model (if we all agree to use the same axioms, etc), but an infinite amount of approximations, which make them each unique. Fun to think about.
The world we live in is a mental model created by our brains, and the data that underlies the model is supplied by our senses. The model we make will differ depending on which of our senses is dominant.
For example, my primary sense is vision. When I read fiction, I often see pictures in fiction my head. I can be thrown out of a story because the pieces don't fit together, and I find myself saying "You can't get there from here!"
But vision isn't everyone's primary sense. My SO is a good example. She's extremely nearsighted. Without her glasses, anything farther than about 2' from her face is a blur. Her primary sense is hearing. When she asks me a technical question, my first impulse is to grab pencil and paper and draw a diagram. That conveys nothing to her, so I need to find a different metaphor based on hearing to describe the underlying concept.
I corresponded with a chap years back whose primary sense was touch. He felt holes in arguments. And in the oddest case I recall hearing of, there was a chap who could not find his way to the office in the morning. He was not stupid, and was a trained engineer. But testing revealed he was not visual at all. Landmarks conveyed nothing to him. He did have a strong kinesthetic sense. So he was driven from his home to his office in a low slung sports car that transmitted every dip and curve in the toad to the passengers. Thereafter, he could find his way to the office with no problem,. because his body remembered what the drive felt like.
I've spent a fair it of time over the years exploring where people are coming from in discussions. "Yes, I understand what you believe. You've made that quite clear and explicit. My question is why you believe it? How did you adopt this belief? What makes it emotionally satisfying to you?' Belief systems like religion and politics live on an emotional level, and aren't usually amenable to rational argument but cause they aren't rational in origin.
What our primary sense is and how that affects your view of the world my be more critical than you assume. No, you aren't representative, and others may not share your experience.
Would they start with 2D numbers (like our “complex” numbers)? Would fractions be the default way of counting instead of whole numbers?
Or something completely different, like thinking of color shades instead of “words” when thinking of numbers as we do.
While I'm an analog guy and I'm thinking in geometry. For example, if I sleep 7 hours at 10 pm, I'm getting up at 4 + 1 = 5, (the opposite of 10 is 4 which equals six hours).
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