> sometimes you need the excellent dry theory and sometimes you need the more concrete but messy application, and in truth you will always vacillate between the two - this is the former
Fully agree. Axler says it quite clearly that the book is intended for a second course in linear algebra. While not an applied text, I find it close enough that it allows you to subsequently go back to your applied material and see it with new eyes.
I'm actually right now working through this book (3rd edition). It's pretty much the most abstract and most clear and simple maths books I've worked through so far (I'm not a mathematician).
Personally I find it suits my needs perfectly, even though initially it can be intimidating. But once you start getting the hang of it I think it can allow you to build a much deeper intuition for things than a more applied text.
No. But it's reasonable to presume he's a compromised source. That doesn't mean he wants to be, nor that everything he's saying has been dictated to him. But his public communications would be, at the very least, monitored and vetted.
at this point he is at the mercy of his FSB handler, either he posts what he is told/expected to or he loses usefulness and joins Russell 'Texas' Bentley as a cautionary tale.
His continued existance in Russia antagonizes the US Intelligence Community. That itself has value to Russia. Not tremendous value, but value nevertheless.
As other commenters have pointed out in one way or another, the problem seems to actually be that this simplistic model of voter choice can't capture all the structure of the real world that humans can quickly infer from the setup. Things like: state elections have millions of voters, 55/45 is actually a decisive, not a narrow win etc.
In a generic setup, imagine you have a binary classifier that outputs probabilities in the .45-.55 range - likely it won't be a really strong classifier. You would ideally like polarized predictions, not values around .5.
Come to think of it, could this be an issue of non-ergodicity too ( hope I'm using the term right)? i.e. state level prior is not that informative wrt individual vote?
This is not a matter of class balance that much. If you want to predict which of two parties somebody will vote with, the most natural framing is that of binary classification.
For that you need to threshold your predictions. Ideally you'd like your model to generate a bimodal distribution so that you can threshold without many false positives etc.
I definitely have an inner monologue im some situations, reading is a good example. I can speed read in which case I don't really have a perception of the sounds, but if I'm closely reasing something then I do have a sense of the sounds if words as I'm reading.
The idea of thinking independtly like that though seems unbearably slow to me (although lots of very clever people report doing it, so obviously it isn't for them!)
> The idea of thinking independtly like that though seems unbearably slow to me
So, as a person who definitely has an inner monologue, I absolutely agree. It's not like I'm literally sounding out words in my mind all the time. The vast majority of things I do I do without explicit use of language.
I think my understanding (mostly coming from Chomsky and in disagreement when OP) is that language is a mechanism for thinking that is mostly not accessible through conscious introspection.
> mostly coming from Chomsky and in disagreement when OP
Is Chomsky's idea that language is not necessarily the same as spoken language, it is an alternate brain mechanism that provides structure to thought as opposed to being the wild west of fluid/analog non-discrete/non-symbolic type of information processing?
The idea would be that language first evolved as a mechanism for thought and only later became used for communication. Spoken language would still be connected to the underlying mechanism though.
My conscious thoughts are verbalized in my head, and it is somewhat slow, but I also have a sense of intuition, which is very fast, though works best in silence. I can pull things from intuition into conscious thought, but explaining why I feel something is the same slower process.
So thinking about something isn't a one speed operation, but being able to communicate those thoughts is.
I think almost exclusively through inner monologue, and I find I can't speed read at all. If I'm not vocalizing I'm not thinking, so when I try to not vocalize in order to speed read I don't retain anything. It's like my brain is incapable of processing the words if they aren't being vocalized.
> The idea of thinking independtly like that though seems unbearably slow to me
While I seem to be a fast reader relative to people I know, I very much feel my reading speed is limited by sounding the words in my mind, so I agree - it's near-unbearably slow.
It seems obvious to me that thinking critically about what you read or hear takes effort and time. I wouldn't call it unbearable, though, because the alternative is polluting your mind with unvetted notions.
Linguistic thinker: I reach conclusions by feel, pretty much jumping to them, and then sniffing out the "warp trail" from that jump and putting that into words. It doesn't feel like inventing post facto rationalizations, but rather retracing the reasoning that happened in the background. And I do need to do that step - the conclusion doesn't seem "stable" unless I trace it back like this.
Words are also based on the same intuition, you just verbalize it and restrict yourself to what you have words for instead of all concepts and thoughts.
For example I do math entirely without any words in my process, and I easily got a master in math that way, barely had to study. No problem, you just translate it at the end. Limiting myself to words just makes it harder to think freely.
That just sounds so different to how I solve math problems. I was naturally good at maths, but it was always from monologue kind of bruteforcing different solutions until one of them seemed to work. I guess it might be a reason also why I find LLMs really exciting since I feel like if I can do it, LLM should be able to do it. I don't feel like I am doing anything special.
I always had problem with trusting my intuition or gut so I was worse in a lot of other real life things however. But math seemed abstract and solvable by words and brute force.
I wish things just magically came to me, but I think I always have to go through things with my inner monologue.
Like if I was to do multiplication in my head e.g. with same numbers, for example 62 x 62. I would have to go through it as a monologue.
I first remind myself of the strategy to do it, which is first I will do 60 x 60. Then it is 3600, then I add 2 x 60, 3720, and then there is 2 x 62 left, but I have to keep reminding myself occasionally what the last numbers were, that initial multiplication was 62 x 62, then I got 3720, and now I have to add 124... okay lets go 3820, 24 left, now 3844. Of course it is easier to remember as I am typing this, but in my head I have to keep reminding myself. And now I am not sure if I did a mistake so I go verify that on the calculator.
> I first remind myself of the strategy to do it, which is first I will do 60 x 60. Then it is 3600, then I add 2 x 60, 3720, and then there is 2 x 62 left, but I have to keep reminding myself occasionally what the last numbers were, that initial multiplication was 62 x 62, then I got 3720, and now I have to add 124... okay lets go 3820, 24 left, now 3844. Of course it is easier to remember as I am typing this, but in my head I have to keep reminding myself. And now I am not sure if I did a mistake so I go verify that on the calculator.
I do multiplication in my head similarly, just strip out the words, the numbers just flickers through and operations happens by themselves and I'm done in a second or two when I'm not rusty. Now that I'm rusty I do it more like (60x60=3600, 2x2=4, 60x2=120, 3600 + 4 + 120 + 120 = 3844), without doing any words, I just did that in my head right now and I am 100% sure not a single word, just the steps, I do sometimes verbalize the numbers so what I wrote in those parentheses is the most verbal my process for mathing that out gets.
Edit: Looking at that, I think it might be easier for someone to correct your thinking if you think in words, but thinking without words is way faster and way more creative since it removes the restriction of only thinking about concepts you have words for.
Edit2: I think your verbalization there is a ritual for the concepts to get to you. For me all I need to do is see 62x62 and imagine I want to solve it and all those thoughts flow to me automatically, basically a shortcut instead of having a large verbal ritual to piece together the concepts. I never did math verbally the way it is taught, so I am not sure how people think that way, to me this was always automatically this way.
If you can or ever were able to do this in 2 seconds that is unimaginable to me. Not in a doubting way, I just have no idea how I could do it in 2 seconds. It would take me 20 to 30s and at least. But most people wouldn't even try to multiply this top of their head.
Also this multiplying of two digit numbers I was never taught, I just came to a strategy that I explained with my thoughts. And I used to do these for fun as a kid. But I would only get faster than that if it was something I had memorized, but not numbers from scratch. And I had to constantly try to repeat numbers in my head that I had stored for addition down the road.
What is usually tougher top of my head and especially if I am tired at all, is something like 68 x 68. Then here first I have to decide whether I go from 60 x 60 or 70 x 70. Since it's that close to 70, I think 70 is more likely be easier. So I think okay 4900 - that's from memory right. Then I start to think what I have to take off from 4900 and what is 68 x 68 lacking compared to 70 x 70. So I will think that if I add 2 x 68, it will make it 68 x 70 and then I have 2 x 70 missing. So I need to deduct 140 and 136. So now at this point, this is much easier done in writing but here I frequently have to repeat numbers or redo some steps because I am not certain or I forget. But otherwise 4900 - 140, this comes easily instantly 4760 - maybe that's how it is for you with 62 x 62. And then now I take 100 off, it's 4660, and further 36 it would be 4624. Right now testing this in my head it took more than a minute because I wasn't sure whether I could just use the logic that I was thinking out of the box and I'm quite tired.
What if you were to have to do something more difficult e.g. 3 digits multiplication or 4?
I got to 4624 in 8 seconds on that now, and I haven't done significant amounts of mental math in over a decade. The 60x8 etc took a couple of seconds each but other than that it just happens automatically. A decade ago each of those would take a fraction of a second and it takes about the same time to calculate as to write it, I tend to calculate the numbers left to right instead of right to left as you are taught since you write it left to right, also more useful when calculating approximations since I just do what I normally do but stop earlier.
> Not in a doubting way, I just have no idea how I could do it in 2 seconds
Words are just slow. I never did math with words, so arithmetics is fast, and so is all the other things you learn in a math degree. I think word based processing in math is a big reason people have such a tough time with it, it really makes it much harder to think, like you are bogged down in a fog instead of up and free and agile with clear sight of everything.
Maybe it was easier for people to get past this stage when calculators were less prevalent. Calculators lets people get away with keeping their arduous word based math instead of just internalizing the concepts.
> What if you were to have to do something more difficult e.g. 3 digits multiplication or 4?
When on tests I write down some intermediate values. I am not a human calculator, I am just fast at math reasoning and being relatively fast at arithmetic's comes for free from that.
I did some physics tests without a calculator, you can approximate all those special functions using regular math logic you learn in high school. I did still ace the test, it isn't that hard once you have internalized it all.
I read the words and then imagine them until I internalize them. I tend to space out a lot when I do that, so I don't really listen during lessons but I tend to learn most things during the lesson that way.
Makes me really bad at following instructions though, since unlike math or physics you shouldn't internalize them instead you should just execute the words and that is really annoying, my brain really doesn't want to think in words. so try to translate everything...
Anyway, I think internalizing things might be harder or impossible if you haven't already done so with all the previous steps. If all your thinking and knowledge is word based then it is hard to break that, and vice versa, I can't think about math in terms of words really. I can slowly translate some things to words but I can't really drive thought with the words.
I used to do math olympiads in high school and the concepts I saw there were always something I hadn't seen before. I always used inner monologue to figure out solutions though. I had never practiced much for math olympiads, I performed relatively well for my area, not good enough to make it to international levels though.
I can't follow verbal instructions as well though, I couldn't listen to the teacher etc. But I think it's because my inner monologue takes all the focus so I just follow my inner monologue which is probably completely another topic than what the lesson is about. I can't really focus away from my inner monologue. When I try, then I would just have philosophical meta discussion about my inner monologue. But also this meant that I couldn't actually learn the subjects as well during the class. I either had to learn from my free time or not at all.
Also have to watch films with subtitles, because audio language I have troubles focusing on.
I think all I have is my inner monologue, no good visual imagery, or other types of "thinking". I think makes me really poor at navigation as well. I never remember how to get somewhere. And also in general for anything 3d, like 3d games I will perform bad at, awareness wise.
Might sound weird but does the look of the operator matter? Say × vs x vs • vs *, I would assume it doesn't but I'm curious if there's a visual change, for lack of a better term.
No, I translate that to the same concept in my head and then I do it.
Edit: Thinking more about it, in my head the numbers are numbers, but the operators are invisible, not sure what to call it but I don't think of the multiplication operator as anything tangible, I just know it is a multiplication of the two numbers.
> That is thinking slow for you?? How do you reach conclusions or how do you know the reasons why you reached the conclusion?
Are you thinking that using words during thinking helps with reaching conclusions and knowing why you reached the conclusion?
For me it's just there, the thoughts, the facts, the logic, the sequence, the connections, etc. Expressing all of that in words happens afterward (if it happens).
I don't even know how to think without words. I thought thinking literally means using words in sequence to problem solve. At least I used to think so. If someone says they are going to think about something I have always thought that they were going to focus on their inner monologue of sequence of words.
Not OP but I also don't think with an internal monologue most of the time. For me it's often more like mentally manipulating abstract shapes or quantities and trying to make them fit together. When I'm writing software I'm literally thinking about pointers and bytes etc, not thinking about the words "pointer" and "byte". This is highlighted by the fact that after intense programming sessions I have dreamed about code, like I am computer memory and I'm being allocated by a memory manager or something.
Sometimes I do explicitly think with an internal monologue, though. Like if I'm debugging something I'll sometimes narrate what the program is doing in words. Also if I'm trying to figure out how some event happened I'll try to tell a story in my mind. It helps then as it forces me to serialise things.
When I read there is sometimes an internal monologue. When I write, there isn't. I don't talk like this. I think it's quite clear sometimes when people write with an internal monologue as their text reads like speech (which can be a good or bad thing, depending on its purpose).
Reading your comment, I don't subvocalize at all. In order to get through my college degree, I taught myself speed reading. Now, I just naturally do it. My eye flicks to the middle of every 5-10 word chunk and flies though the text. If you did speech synthesis at that speed, it'd be a incomprehensible chipmunk. So I don't subvocalize at all. It'd be way too slow. To suppress subvocalization I used to hum (in my mind) instead, but I don't need to anymore. If something is hard to understand I will slow way down, then the subvocalization might kick in. When typing, I do subvocalize everything, since I can't write that fast.
I think this post/paper was more about your personal thought process rather than reading though. I very rarely have any internal monologue. In fact, the rare times I do have one are usually very awkward social situations where I wanted to say something but don't. Otherwise never. My whole life has been that way. An internal monologue sounds like a nightmare to be honest. Constant talking that nobody else can hear? No thank you.
I read much faster when I can focus on reading instead of vocalizing; in this case I am no longer internally vocalizing the text. If I do vocalize what I read it goes a lot slower, but then it helps me to synthesize conplex concepts in a text.
Thanks for your comment. It just occurred to me that I have an inner voice narrating the text when I am reading in English.
This does not happen when I read in my first two languages.
This explains why I read slower when I read in English.
I don't have an internal monologue at all (I am able to speak to myself in my head if I want to, but I seldom have a reason to). When I read, the information just gets uploaded to my brain. I don't vocalize words in any way, silently or otherwise. I don't read one word at a time either. When I read something quickly I can "feel" that my understanding of the material is lagging "the cursor", sometimes by even a paragraph at a time.
> There are many flavors of this kind of accounting. Another version of free attribution would allow the company to take that entire 30-pound batch of “33% recycled” pouches and split them even further:
> A third of them, 10 pounds, could be labeled 100% recycled — shifting the value of the full batch onto them — so long as the remaining 20 pounds aren’t labeled as recycled at all.
I'd be fairly cautious in assuming that these more recent coauthored pieces by Chomsky represent his views very accurately. They're pretty crude by his standards and don't really read like his writing (though he may well have signed off on them in some sense).
There is some overlap with the content of recent interviews, but only some. In particular, the fussing over what is or isn’t “true intelligence” comes from Watumull and isn’t consistent with Chomsky’s past comments on AI over several decades.
> would generate way too much workload to follow up and thereby cost time and money for better leads
Having lived a decade in NL, my impression was that keeping costs down is the top priority. Unless you have a serious chronic condition or were in an accident, good luck getting somebody to take a look at you.
(irl, after a while you learn to push, exaggerate symptoms etc. or just go back home to get tests and treatment).
Yep, the loudest people get the most help. There's no good solution for that other than to become a bitchy 'client'. It's unfortunate that despite the promises, you still have to 'use it correctly' if you want those good outcomes as a patient.
Fully agree. Axler says it quite clearly that the book is intended for a second course in linear algebra. While not an applied text, I find it close enough that it allows you to subsequently go back to your applied material and see it with new eyes.