It seemed to me that the author was describing his instinctive mental representation of numbers, and not that mental math is only achieved by using shape-analogs.
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.
Why not make all the integers square blocks? Then everything fits together. It seems strange what's going on with the odd numbers. Especially adding something to 9 is even stranger. Seems arbitrary rather then instinctive.
not necessarily. only if you think in terms of cou ting does your sense make priority. if I were to think in terms of multiplication - circles are more useful for a lot things.
Circles cannot compose under addition or multiplication. You can combine two blocks to form a new block. You can combine two integers to form a new integer.
You cannot compose two circles to form a new circle.
Composition can either be multiplication or addition. In short integers are monoidal under both multiplication and addition, circles are not as you can't physically combine circles to form a new circle without mutating the shape of the circle itself.
This means circles are a bad shape compared with blocks to use for addition and multiplication.
You seem to be carrying the impression that this is deliberately constructed -- it's not, this is just part of the author's intuitive representational system. He's not "making it" one way or the other. It's made; he's perceiving it.
I literally asked if it was a condition or if it was deliberately constructed. If I asked that question, how am I implying it was deliberate? Did you read my post?
I used learned behavior as a synonym to deliberate construction in the sense that we construct things from what we learn.
But let's say you're right. If that's the case, then you're entire response is off base. You claim I am implying it's "Deliberately constructed." How could I imply something was "Deliberately constructed" if I asked it was "learned" or "related to synesthesia." Neither of those options involve "deliberate construction."
Er. Why don't you read my post again. I don't understand how you derived all that from what I wrote. I literally said this method was inefficient and asked how about whether it's a condition or he learned it.
As another commenter pointed out, you literally did not.
In fact, you went so far as to imply that it's constructed and not synesthesia.
As the same commenter pointed out, if multiple people have interpreted your comment in a particular way, regardless of your intentions, then it's maybe time to step back and re-evaluate.
>In fact, you went so far as to imply that it's constructed and not synesthesia.
No that implication is your and the other commenters imagination. READ my comment again.
> As the same commenter pointed out, if multiple people have interpreted your comment in a particular way, regardless of your intentions, then it's maybe time to step back and re-evaluate.
So serious. If multiple commenters interpreted my intention wrong then that's my bad wording is off. I mean it's not a huge deal. If I correct my mistake, and I inform you of my true intention, What then is the big deal?
Your telling me to take a step back and re-evaluate as if problems with my wording violated criminal law? I clarified my intent all you need to do is address it.
IN addition to this, there is the factor of actual grammatical English language interpretation vs. popular interpretation. There is something to say that the English definition and intent of my sentence holds equal ground to a popular misinterpretation.
I don't have a visual representation for numbers, but numbers with a 9 at then end like say 29 in my mind is always transformed in to 30-1 , so instinctively (nobody teaches me this) computations like 29 + 15 = (30 -1) + 15 = 30 +15 -1 =45-1. This makes it more easy for multiplications 2915= 3015 - 15 . I could apply this for numbers ending with 7 or 8 but it does not fill natural for me as 9.
Oh that's a symbolic reflex. totally normal, I have shortcuts in my brain like that too. But the Article describes a visual entity... a block with a dent (the number 9) then the dent orange peeling another number until it's smaller.
It would be even better if he visualized the universe as a quantum superposition of states that collapses to a single answer when supplied a problem. That way he could solve things which NFAs can!
No why doesn't he just visualize everything as blocks like he does with even numbers. Why throw a dent in the number 9 and make it act like a can opener?
I'm not being pretentious here. Literally his model looks like some over engineered contraption.
I suspect he’s describing how it is, not claiming that it provides any utility. It’s just someone sharing some aspect of their experience with the universe, not someone prescribing a technique.
I can see how you concluded the latter, since this site frequently have posts with this style of title where the upshot is an implied “and why you should too”. This time, though? This time it’s just someone’s blog where they’re telling you how it is for them.
As far as I can tell, they didn’t share it on here and they’re not proselytizing the experience. It just is.
Seems like enough people have the same confusion that it might be a worthwhile exercise to evaluate whether you are effectively communicating what you believe yourself to be.
Hence my response. It clarifies my intent. Your response to me indicates that you are now aware of what I'm communicating. Thus it might be a worthwhile exercise to address that rather then go on some needless tangent on some misinterpreted wording.
Additionally grammar matters. Technically the english meaning of my sentences do not imply what most people believed I said.
I think it's more of the latter. The shapes are not there to help him do arithmetic in a more efficient way. The shapes are there just because that's how numbers are represented in the author's brain.
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).
Minus the contextual thing... colors are monoidal under RGB. Its a 3 dimensional vector with all the properties of numbers.
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.
I show this to my kids constantly. I love how it plants the seeds of concepts like square numbers and divisibility in a show that is ostensibly about just addition and subtraction
Damn, I let my kids watch Alphablocks as toddlers but I never noticed Numberblocks! At 7 and 8 they're very good readers but the older one gets bored with arithmetic homework very easily. I've failed as a parent and mathematician.
> the older one gets bored with arithmetic homework very easily
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:
We just watched their episode on zero and it was fantastic. It's a hard concept to explain and I think they did it beautifully. My toddler is addicted.
I had a box of these in elementary school, too. I am not great at math but these things ... I didn't get the point. They bored the hell out of me whenever we had to use them. I had some great experiences by stacking them and making jenga-like tower builds, though. I think they even came with a small booklet that showed you some basic builds. But using them to do math stuff always felt unnecessary and tedious because it was much simpler just to do it on paper or in my head.
I was very anti-screen time, but Number Blocks changed my perspective. My kids have really accelerated their ability to process numbers just by watching this show.
This is how I visualize numbers. 5 and 10 are the primary blocks. 7 is a 5 with a 2 on top that could either slide off or take 3 from another number. Great to see it being taught.
My visualizations are definitely heavily base-10-centric. I did some experiments with visualizing numbers in other bases while doing this writeup, and I've come to the conclusion that I just convert each digit to its base 10 equivalent and add. My mental representation for 0x6a, for example, takes on the same shape as (6 * 16) + 10 in my head.
There is a really distinct feeling I have about the fact that 2 times 8=16 and 3 times 6=18. Really hard to describe but something like 8 and 6 being siblings fighting about who is stronger/bigger.
This is similar to how I see it. I have a hard time explaining how 7 is a tipsy number that's about the fall into 3 and 4, and how 9 has a voracious appetite to take away a number from another and you can't stop him. It all started when I was younger and my mom told me to bring every number down to 2s and 3s, and to always be adding or subtracting idle numbers (just numbers without any operators).
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.
That description is eerily close to how I feel. Seven for me is definitely a very "loosely bound" number, and it wants to separate along the three and the four.
Things like numbers having personalities is actually a type of synesthesia! Apparently the specific name for it is "ordinal linguistic personification" [0] What the author of this article is describing is also a type of synesthesia. You should look into it!
I do the same thing! Though my shapes are different, it's the same. My wife is tremendously bad at math and I kept telling her you have to picture things and she said I don't think of it that way, I just see the writing of the number itself and I say "well, that's why you're bad at math!".
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.
I do something very similar, but with power-2 numbers (and have done since my childhood, long before I knew what power-2 numbers were).
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.
Negatives are the same shape as their positive counterpart, but the inverse. So where there would normally be a shape for the number, there's instead a void or impression that wants to be filled.
That does sound like synesthesia. I have sound -> touch synesthesia. There's some discussions about whether a pretty big percentage of people have some form or another actually
Welp... now i understand why I've only encountered like one other person that seemed to immediately understand what I was talking about when I used the word "thought-shapes".
So, a few question to all the number-as-shapes-in-head-representators out there: What happens in front of your inner eye when you do more complicated operations like exponentials, modulo, ...? Do you have distinct visualisation for certain ways to represent a number (roots, fractions and so on) too? And do these representations help you when you solve a problem where you don't have to "count" anything, like when you have to write a proof or something?
I don't see things the exact same way as the author, though similar (simplest way I could describe it is things like addition are filling up tanks of liquid of [usually] 10^n size, though a little more amorphous and yet "jelly" than what you would normally think of as liquid? I'm finding it hard to describe).
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.
There's a book which describes studies done on this topic [1]. I haven't read it in full, but I found it useful to know how other people have looked at this issue.
Cool! If the author is reading, have you looked at synaesthesia, and do you think it applies here? The idea of addition having both a visual appearance and a kinetic feel is alien to me (perhaps partly because I avoid mental arithmetic like the current plague). But it's apparently reasonably common to have colour and spatial associations with numbers.
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 now pretty convinced that this could be classified as some form of synesthesia, though I'd never thought of it that way before. When I'm actually doing math I tend to _feel_ the shapes of the numbers and their interactions, rather than it being a highly visual sensation like some others experience. The visualizations of the shapes only really happen when I slow down and focus.
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.
I have synaesthesia and your description of numbers just feels right to me. I have the fairly common association of letters and numbers with colours, which (at least with people I've shared this with) often is stronger up to 10, with further numbers typically blending the properties of their digits.
Wow, this is utterly alien to me. I have always had a head for numbers but they are never shapes.
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.
Do you have perfect pitch? What frequencies do the numbers have? How high is the frequency of a really large number like 1000? What about fractions and irrational numbers?
I don't, though I did play piano for 10 years starting at age 6. I think the frequencies are around middle C or so, though I haven't really checked.
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.
I am good at remembering numbers & phone numbers too!. And its similar to what you mentioned. I can recite my whole phonebook if needed, even for numbers punched 15+ years ago.
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)
Sadly I no longer know my friends phone numbers, since the advent of smart phones. But I still know the phone numbers of anyone I regularly dialed as a child, 30 years ago.
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?
Since Stephen Hawking’s movement was limited for much of his life, he claimed that he had learned to do more math quickly in his head via visualizing geometry. Seems similar.
As a person "self-diagnosed" with aphantasia, I feel cheated knowing that other people have a built in cheat-sheet. No wonder why I was struggling with memorizing things like the multiplication table in school.
FWIW, I don't have aphantasia and consider myself a very visually-oriented person when it comes to math, but doing precise arithmetic by visualizing the quantities is mind-boggling to me. I memorized the multiplication tables using a song we learned in school.
This is something I feel like we're going to have to realize more and more in society over the next decades. That a lot of people simply have genetic cheats that others are missing. At the moment we kind of pretend it's nurture to a large degree. Should we 'unbias' the world to make it more equal for everyone regardless of genetic cheat? (If so how?) What's the correct adjustment?
There’s no such thing and fortunately your vision is completely wrong.
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.
I thought I was aphantasic, but after a coaching session with AphantasiaMeow, I'm sure I'm hypophantasic.
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!
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.
I don’t think there’s anything wrong with your approach - you don’t have to ‘think’ about the solution because it’s already there. I don’t know if that translates to an actual reduction in mental fatigue, but if it works for you then changing it will no doubt cause at least short term strain.
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.
Author here - like one of the other commenters said, I don't think there's anything wrong with your approach, or any way of thinking for that matter. I don't think there's anything particularly "right" or advantageous about the way my brain works either. I don't have any reason to believe I'm better at math than the average engineer - definitely not a math prodigy or super genius or something like that.
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 :)
I think the really great thing you did here, was just lay it out. So little is said/shown on this topic that it's really valuable to just get people conscious of their own process, so that they can compare and contrast.
I mean ... just as an example, what happens if what you are adding are not numbers?
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.
In India we learn tables (multiplication tables, but we just call them tables) from 1 to 10, and later till 20. Each one has this format,
1x1=1
1x2=2
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.
Strings with string concatenation form a monoid [1] (natural numbers with addition form a commutative monoid, and integers with addition form an abelian group). Incidentally, that was literally the topic of the first class in my first CS semester. :)
Off-topic language quirk: It seems odd to me that monoids can be 'commutative' while groups can be 'abelian' but both adjectives mean the same thing. Alas, if the language were consistent this joke wouldn't exist:
I'm surprised the author depends on a single visual model for numbers. To provide another data point, my ordinal numbers are brightly colored (maybe an association with award ribbons I'd get in elementary school); cardinals with no extra structure are specific clusters of bluish cubes (8 is 2x2x2); when ring structure shows up, they're malleable grids (maybe because the distributive law's self similarity made learning the times table easier); for just multiplication, colored lights in a poset (so L-function identities "look like" intricate lattices of Christmas lights). For general groups, not much: I just feel like I'm playing a board game.
It's hard to imagine having only one instinctive visualization for integers.
I don't see his way of viewing numbers as particularly efficient. It's very inn-efficient. It's an anomaly for sure but I would hesitate to call it a talent or super human ability.
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.
I relate to the OP on a fundamental level although the literal expression would be different for me. I do not think it has any relation to speed. It is not a deliberate step. It would be slower to mimic this behavior, but if you have it by default it's just kind of there.
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 fairly agree with this. I still wonder how the author visualizes irrational numbers, exponential functions, etc. and more importantly, proves some (even simple) theorems with this kind of visualization.
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).
You might be right about this particular version of synesthesia not being too useful but I have music->visual (shape, color, texture, distance, location) synesthesia that I can turn on and off (only when smoking even small amounts of pot) and it’s a huge advantage when trying to do anything music-related.
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.
One thing that helps at lot with programming is my tendency to visualize branches and dependencies as graphs/trees as I read/write code. This makes aberrations and code smells extremely obvious. A dirty hack makes you go from something that looks like a beautiful fine-toothed comb to a comb with a cancerous tumor on it.
Neat. Some of us can't see things in our heads at all (aphantasia), so we definitely can't do things this way.
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.
While writing this article, I learned that Ed Catmull has aphantasia. It's amazing to me that someone with a Turing award for work on computer graphics can't mentally "see" those graphics when he closes his eyes. It'd be really eye-opening to somehow get his (or anyone else's) mental state into my own brain, just to try it out for a little.
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.
I caution against looking at numbers in any single way. The more different ways you can visualize math concepts, the better. Practice seeing them in different ways.
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.
While I don't think this is bad advice I don't really think it is along the same lines as what the author is describing.
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.
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.
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.
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?
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.
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.
Author here: was definitely not trying to frame this as a tutorial or anything like that. I don't think that my "methods" have any particular advantages. It's just how my brain works.
> I caution against looking at numbers in any single way.
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.
I can taste words. Meaning some words immediately remind me of something I have eaten before. I can logically understand why some words taste like the food because they sound like the name of a food but some words don't even come close still they remind me of a certain food. I guess I am alone because I haven't found anyone who feels this way.
I’ve often wondered what it would be like to not have aphantasia, or experience synesthesia. Not keen on mind-altering substances like LSD though…
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.
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.
I have this as well, including the loops for time units (day, week, year) and varying “camera” perspectives. Sometimes it’s labelled as “space-sequence synesthesia” (or “space-time”, which sounds cooler but doesn’t cover the non-time sequences like temperature). Always fun to find someone else with it in the wild!
I have a similar freakish ability, but mine has to do with writing. I can basically ~see~ approximately three pages of prose in my mind's eye while writing. It only works under certain conditions, but it feels just like I'm transcribing something rather then doing any kind of deliberate thinking. People are shocked what comes out of me, and even more so when they see how quickly it happens. You would need to see me in person to experience the full effect, but my body does not match my words. Imagine the biggest lumberjack you've ever seen describing the petals of a flower with such high precision that it takes your breath away. That's me. I've started to slowly nurture this talent, because it finally occurred to me that it might be special.
I do something similar when writing longer things like papers. I'll think about the general topics I want to cover and the order, highlights, etc., and then I can write 10 or 20 pages more or less as a continuous flow.
Interesting, I have a similar ability but for code. I write some of the best code when I'm not at a computer because it doesn't take any time to refactor and I can play freely. A lot of the time when I write new code it's just transcribing the systems I created from memory. It's similar for reading code, being able to keep a lot of system complexity and behavior in my head at once. I wonder how common this is for mastery in other circumstances, like sports or art.
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.
> You would need to see me in person to experience the full effect, but my body does not match my words.
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.
Oh my god! I had the same experience in the late 2010s as a tiny little emo girl. The absolute shock in CS group projects with a bunch of guys who were upset that they “got put with the girl” when I had the whole thing done in an hour was chefs kiss.
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.
I think I was rolled by a min-maxxer: 20 INT and probably around 18 CHA (based on the fact that I could get 100 10 year olds to cooperate as one of them), but damn my CON and STR are so low I have constant pain debuffs to all my rolls. It's BS.
Wow. I can’t "see" anything in my head. I think I have aphantasia (though I’m not really sure, I can see flashs of things but not keep them).
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>
That’s interesting. I never “see” anything like that in my mind, but a few times in high school, I was explaining some geometric proof to a classmate; at the end, he noted that my verbal explanation and my finger pointing at the chart were compatible but out of sync with hand motion being about twice as fast, and then went to repeat both verbal and finger pointing, albeit at a slower rate, and they matched perfectly.
This happened more than once though not many times, and I considered it weird but kind of forgot about it.
This description makes me think I might have a mental canvas like this as well, except it might be a headless browser or something :)
in my 20's I went through a numerology phase and began taking the digital roots of everything, it became a habit and now I can't not do it. I developed a really similar sort of visual mechanical sense for the digits 0-9 where the digits click together as if they were magnets and the closer they are to 5 the more they repel their own parts (eg two 5's might easily disintegrate to snap into a nearby pair of 3's). it's really interesting to hear about other versions of this sort of thing.
I find it interesting that in the UK a primary school child (say aged about 7) would trivially know that "80 + 4" is 84, but for the problem "4 x 20 + 10 + 7 = ?", might require quite a lot of effort to work out that the answer is 97.
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".
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[ 4.6 ms ] story [ 251 ms ] threadWas 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.
Composition can either be multiplication or addition. In short integers are monoidal under both multiplication and addition, circles are not as you can't physically combine circles to form a new circle without mutating the shape of the circle itself.
This means circles are a bad shape compared with blocks to use for addition and multiplication.
But let's say you're right. If that's the case, then you're entire response is off base. You claim I am implying it's "Deliberately constructed." How could I imply something was "Deliberately constructed" if I asked it was "learned" or "related to synesthesia." Neither of those options involve "deliberate construction."
Yes, it would be grand if our minds worked in a rational, logical fashion. But that's not even remotely representative of reality.
In fact, you went so far as to imply that it's constructed and not synesthesia.
As the same commenter pointed out, if multiple people have interpreted your comment in a particular way, regardless of your intentions, then it's maybe time to step back and re-evaluate.
No that implication is your and the other commenters imagination. READ my comment again.
> As the same commenter pointed out, if multiple people have interpreted your comment in a particular way, regardless of your intentions, then it's maybe time to step back and re-evaluate.
So serious. If multiple commenters interpreted my intention wrong then that's my bad wording is off. I mean it's not a huge deal. If I correct my mistake, and I inform you of my true intention, What then is the big deal?
Your telling me to take a step back and re-evaluate as if problems with my wording violated criminal law? I clarified my intent all you need to do is address it.
IN addition to this, there is the factor of actual grammatical English language interpretation vs. popular interpretation. There is something to say that the English definition and intent of my sentence holds equal ground to a popular misinterpretation.
I'm not being pretentious here. Literally his model looks like some over engineered contraption.
I can see how you concluded the latter, since this site frequently have posts with this style of title where the upshot is an implied “and why you should too”. This time, though? This time it’s just someone’s blog where they’re telling you how it is for them.
As far as I can tell, they didn’t share it on here and they’re not proselytizing the experience. It just is.
Additionally grammar matters. Technically the english meaning of my sentences do not imply what most people believed I said.
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.
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:
https://beastacademy.com/
https://www.singaporemath.com/
https://ssddproblems.com/
While I don't remember the colors or anything, I still visualize addition like this I think.
7-3 I found interesting because those are modulus complements in base 10
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...
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.
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...
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.
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.
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.
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?
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!
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.
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.
https://sites.google.com/site/steveyegge2/math-every-day
https://www.cantorsparadise.com/the-unparalleled-genius-of-j...
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 :)
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.
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.
edit: But actually after that I used something called "Math-U-See", which used physical blocks to develop intuition. That was pretty cool.
[1] https://en.m.wikipedia.org/wiki/Monoid
Q: What's purple and commutes?
A: An abelian grape.
It's hard to imagine having only one instinctive visualization for integers.
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).
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.
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...
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.
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.
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.
They are describing how their brain naturally sees numbers: which is commonly referred to as synesthesia.
'Cause I am very curious how the author experiences imaginary and complex numbers ... or even negative integers, irrationals, and transcendental numbers.
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.
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!
With that being said I wish my mind was blank sometimes, I wish my inner monologue would shut up every now and then. :)
I’m curious why these differences happen and to what degree it’s difference in thought versus difference in conscious perception of thought.
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.
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.
https://m.imgur.com/gallery/zcW36gH
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?
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.
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.
https://www.youtube.com/watch?v=AbASOcqc1Ss
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>
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".