This reminds me of how I became a better guitar player by learning piano. I had been playing guitar (classical & metal mostly) for about 15 years before learning piano. Upon learning some basic piano, I found the guitar fretboard came alive in entirely new ways. I believe this to be a result of the piano keyboard having a visually clear and uniform organization of notes, enabling me to "see" how music theory operated in a more visceral way.
Learning philosophy and then logic and then computer programming taught me math. Or at least taught me how to understand math in a way I’d never been able to in all my many years of education. It makes me wonder how many kids are missing out because we’re focused on teaching math one specific way even though it’s quite clear the way we’re teaching it just doesn’t “click” for a lot of students.
Interesting because this idea touches on a few things:
1) Effective data visualization is extremely beneficial for the learning process;
2) Combining sensory experiences together enhance learning process;
For instance, on piano you can see AND hear the intervals on a scale.
Strangely though, outside of elementary school there’s less emphasis on tactile learning—and no football is not the same thing. Art teaches the mind how to think outside the box and in creative ways. It also taps into the subconscious and allows us to learn and communicate better.
I first thought the exact same thing as you. But this argument stretched to it's conclusion means no one know maths. Which very well might be your point. But then it loses any useful meaning. Even professional mathematicians are specialized in a small sub field of mathematics and there are huge parts they don't really know.
Coming from the opposite direction, I'm quite curious as to where the linear algebra and combinatorics fit in. As a former mathematician (switched to computing after one post-doc) who has been trying to learn about music and to play the piano, I'm trying to make sense of how music and mathematics relate.
There's some stuff with group actions going on - I think you can think of a dihedral group acting on sets of triads or something along those lines and there are some articles out there about that, but I feel like I don't have a good mapping in my head as to the relationship between music and mathematics.
I can see combinatorics relating in the sense that exploring the space of potential compositions can be approached as a ‘combination of’ type problem for different elements.. chords as combinations of notes, chord progressions as combinations/orderings of chords, likewise for the order of notes in an arpeggio.. rhythms/grooves can be thought of as like combinations of emphasised beats (emphasis on 2 & the and of 3 vs. emphasis on 1 & the and of 3 for example) at different levels of beat-subdivision.. then the sounds each emphasised beat can take come from a discrete set (kick, snare etc.) so can be thought of as another combination-of type problem. many possibilities.
Perhaps this is not what the GP meant though, I’m curious about however it applies. No idea about where linear algebra could fit in, I have yet to see those parallels myself yet but interesting to think how it could relate
W.r.t. combinatorics: I was thinking of tuning systems and scales. I have always thought about the theory, but never written it down. Some other people actually did a much better job than me: http://andrewduncan.net/cmt/
W.r.t. linear algebra: mostly in relation to sound synthesis, but also in relation to music composition, theory of overtones, piano tunings... You ask, I'll explain.
Sound synthesis: mostly when I was playing with PD (Pure Data). Linear algebra comes up all the time in sampling theory.
Music composition: this is a bit more far fetched, but I always thought of composition as a combination of vectors which can vary semi-independently. We want to pick some plane through that vector space and move within the plane. For example, we can vary in harmonic complexity, melodic complexity, rhythmic complexity, dynamics, tempo. But it wouldn't make sense to do all at the same time. So we pick one plane, say harmonic complexity and dynamics, and move within that plane. This allows the listener to get used to the 'space' that is opened by the composition. We can then slowly add more vectors, to keep the interest.
A very visual representation of this is the trackpads that you can see on certain modular synthesizers. Often there is a complex setup, but the variables of the setup are reduced to two dimensions, which allows the composers to vary within this projected plane.
Piano tunings: well, basically you have a vector of 230 piano strings of which the overtones are all slightly out of tune. To tune a piano 'correctly', you need to minimise set of equations of strings and overtones (matrix).
> When all my first-grade classmates were memorizing math facts like “2 + 3 = 5”
> I was called out during class when my teacher saw me fiddling with my fingers during a classroom exercise
From their LinkedIn profile, it appears that the author went to school in the US. I'm not from the US, but my impression of US schools is the complete opposite of one where students are taught to "memorize math facts" or one where students get "called out" for doing something in a different way.
In fact, from the many workbooks that I see on the net, my impression is one where a lot of pain is taken to ensure that students understand the material without resorting to rote memorization.
Also, my impression is that students, if they show independence of thought or potential in an area, are given a lot of support to pursue it from within the schooling system in the form of being put on accelerated paths, freedom to attend classes of higher grades, etc.
In contrast, many other countries have rigid grade structures, where students are forced to stick to the level of discourse in their grade in school, and can only pursue their interest outside the schooling system.
Perhaps some of the European systems have it better, but this flexibility to rise beyond your grade within the school system is something that isn't available in a LOT of countries.
I'm not from the US, but my impression of US schools is
You can't generalize US schools. There are a lot of them, and they generally answer to different authorities, local school districts, state school districts, religious (Catholic) schools do their own thing, etc.
I think there's significant variance, but I don't think this is necessarily the norm. Everyone must memorize some basic tables as well as standard algorithms, but I wouldn't say it ever struck me as a problem.
The only exception to this for me was pre-calc, in which they taught us trigonometry rules without explaining it's basic relationships to the unit circle. Our calculus teacher was dumbfounded the next year when we had no intuitive understanding for these things. It took an hour to correct. No big loss. Shrug.
The algorithmic flexibility is part of ‘common core’ which is in most US schools now. However the method teaching algorithmic flexibility is often forcing the kids to use each of the several algorithms and marking their answers wrong if they didn’t use the technique requested by a specific question.
I remember having some issues the year when I switched schools and states in 3rd or 4th grade (not sure exactly). The school district we moved to had kids memorizing all their multiplication tables and would do 2 min tests to ensure they were memorized and not thought about. I did terribly on these tests because I had not memorized them the year or a half year before like the other kids since I was at a different school. My mother got upset that I was being failed in math because she knew I was good at math. So she took me with her to parent-teacher day made me do double-digit multiplication in my head in front of my teacher. This was something no other kid in my class could do and made it apparent to my teacher that I should not be failing math.
As an adult, I have mixed feelings about this experience. My mother used to brag about this she had me do it in front of her friends, other parents with kids my age. I distinctly remember during this in a Perkins while trying to enjoy my breakfast. For me, this was simply a trick involving a larger “working memory” as opposed to “rote memory”. Being that it was still a matter of being a “trick” that required practice albeit a different kind of practice it becomes something I thought of as a trained monkey scenario.
This has come up a lot I’m my life. I worked in construction for a while out of high school and my boss would treat me this way as well. He would have me be his human calculator to help him figure out measurements. One time this was going on in a complicated octagonal room where we were trying to lay out brackets for a projection screen. He asked me to run some numbers for him and I refused. I did this because I understood what he thought he was doing but knew that his method was flawed. So I refused to give him the wrong answer, even though it was the right answer to the numbers he gave me. It caused I really big fight where he began talking down to me and tried to put me in my place.
Another Forman was there who witnessed the whole thing and I ended up doing a job with that other guy like 6 months later, but he had not forgotten the incident. He brought it up and gave me a long speech about how I should have just given my boss what he asked for instead of the correct measurements. Something I, to this day, steadfastly disagree with. Fast forward a year or two and my boss was fired due to his incompetence in measuring and ordering the materials that cost the company many tens of thousands of dollars.
Intelligence and thinking skills, in general, are practiced and exercised. However, they are not trained-monkey tricks. It is a method of living well and garnering fullness from life. And to no other end is it acceptable to waste such a thing.
I've been looking at music over the last year or so, bits of basic music theory. As a child I had no idea how linked to maths it was, and how much easier music is once you realise that it's just applied mathematics and touch typing
I've been doing the same. Are there any resources you've found particularly helpful? As a true beginner, the Music Matters YT channel does a great job of giving enough of the Why's without going so deep as to completely lose me.
My music theory teacher used to say that that music is the only art that's also a science. I think this headline should not be surprising to anyone who reads music.
I see philology/linguistics as quite science-y too. You can argue that's just the building blocks of the art (literature), but so is the science-y side of music (which we call "musical language" in Spanish conservatories, see the parallel?) and mostly deals with "syntax" and historical accidents inherited from the western Common Practice Period.
The artsy side of music is very much un-sciency and mostly rationalized post-hoc.
I like learning of people's mnemonic sequences. Really nice how TFA was able to adapt their strategy to new, previously unencountered ideas like negative, hexadecimal.
I'd truly like to know how Von Neumann remembered everything.
> I'd truly like to know how Von Neumann remembered everything.
I've been hearing claims of von Neumann's perfect memory for a long time on HN, but I've been unable to find primary sources to back them except for a single quotation by Herman Goldstine. Could the claim of "perfect memory" have been an exaggeration?
Surely von Neumann had an exceptional memory, but truly remembering everything is extremely rare (some say impossible). If he did, I imagine that the psychologists at Princeton or the IAS would have caught wind of the mathematician with a perfect memory and wouldn't have missed a chance to examine him; all the more so since "true photographic memory has never been demonstrated to exist" [1].
Music provided a framework for assigning numbers to fingers that I didn't grow up with. So I find myself counting using my fingers naturally but sometimes start with my index finger, and other times start with my thumb. I think his system is more intuitive and could be more useful but I'm not sure if I can relearn it as an adult and make it stick without a lot of practice! As an aside, my mother is French and learned math in France, and in her school she learned rather different methods for working out addition and multiplication. I remember doing a double-take the first time I saw how she worked out a problem when I was in grade school.
Hooray for any tool (or workflow) that makes it easier to work with otherwise-abstract formulas! This way sounds to me like a handy method for everyday in-mind adding and subtracting too.. I definitely aspire to upgrade my understanding of notes and frequencies to the point where I can connect them to the math!
