8 ms·
I was talking to my (12 year old) son about parts of math he finds boring. He said that he thinks absolute value is absurdly easy and extremely boring. I aske
by cduzz 3y ago
I was talking to my (12 year old) son about parts of math he finds boring. He said that he thinks absolute value is absurdly easy and extremely boring. I asked him if there was anything that might make it more interesting, he said "maybe complex numbers".
So I asked him "what would the absolute value of i+1 be?" he thinks for a little bit and says "square root of 2" and I ask him "what about the absolute value of 2i + 2?" "square root of 8"
I ask him "why?" and he said "absolute value is distance; in the complex plane the absolute value is the hypotenuse of the imaginary and real numbers."
So -- first of all, this was a little surprising to me that he'd thought about this sort of thing having mostly just watched youtube videos about math, and second, this sort of understanding is a result of some manner of understanding the underlying mechanisms and not a result of just having a huge dictionary of synonyms.
To what degree can these large language models arrive at these same conclusions, and by what process?
- Strom 3y agoThe large language models will read your comment here and remember the answer.
- cduzz 3y agoThe spot instance declared "a similar vector exists" and de-provisioned itself?
- Borrible 3y agoGPT-4 correctly reconstructs the "complex modulus" token sequence already. Just ask it the same questions as the parent. Probably interesting to see what it will do, when it turns twelve.
- yashap 3y agoHah, I asked ChatGPT, and yeah, it nailed it: https://chat.openai.com/share/f1bfed28-4451-4fac-8bac-1386febebf45 https://chat.openai.com/share/f1bfed28-4451-4fac-8bac-1386fe...
- muskmusk 3y agoIf you ask Ilya Sutskever he will say your kids head is full of neurons, so is LLMs. LLMs comsume training data and can then be asked questions. How different is that to your son watching YouTube and then answering questions? It's not 1:1 the same,yet, but it's in the neighborhood.
- swatcoder 3y agoThere are thousands of structures and substances in a human head besides neurons, at all sorts of commingling and overlapping scales, and the neurons in those heads behave much differently and with tremendously more complexity than the metaphorical ones in a neural network. And in a human, all those structures and substances, along with the tens of thousands more throughout the rest of the body, are collectively readied with millions of years of "pretraining" before processing a continuous, constant, unceasing mulitmodal training experience for years. LLM's and related systems are awesome and an amazing innovation that's going to impact a lot of our experiences over the next decades. But they're not even the same galaxy as almost any living system yet. That they look like they're in the neighborhood is because you're looking at them through a very narrow, very zoomed telescope.
- Davidzheng 3y agoTrue. But a human neuron is more complex than an AI neuron by a constant factor. And we can improve constants. Also you say years like it's a lot of data--but they can run RL on chatgpt outputs if they want, isn't it comparable? But anyway i share your admiration for the biological thinking machines ;)
- riku_iki 3y ago> human neuron is more complex than an AI neuron by a constant factor constant still can be not reachable yet: like 100T neurons in brain vs 100B in chatgpt, and also brain can involve some quantum mechanics for example, which will make complexity diff not constant, but say exponential.
- 3y ago
- white_dragon88 3y agoYour son is a goddamn genius.
- cduzz 3y agoMaybe; he still needs to finish his damned homework and remember to turn it in. And eat some vegetables.
- quickthrower2 3y agoJust quit school and join YC now :-)
- deleted 3y ago[deleted]
- SpaceNoodled 3y agoBut his homework is boring
- nopromisessir 3y agoAll those things sound very boring to me. I can offer no concrete solutions. However, I have a friend who graduated from high school #1 of a big class and 2 years early. His mom explained that if he made at least a 1400(of 1600) on his SAT, she would buy him a new gaming computer. He then proceeded to make exactly a 1400. No more. No less. I recommend if you haven't tried already, an iteration to this approach using a sliding scale reward system. Perhaps a gaming pc with nvidia 4060ti up to *insert parental budget* in event of a perfect SAT score. Ofc this only works if he's a gamer. I feel this type of system can be applied in many areas though. In my view, the clever component his mother applied is that the computer he earned was not just a desirable reward... It was VERY desirable. My parents also tried this system with me. It didn't work as well. The reward was not sizable enough. It just didn't seem worth it. Too low value. Also, I already had a job and bought my own. My parents were unwilling to budget a sufficient reward. It's gotta be something he more or less is unlikely to be able to get via other means. Now my friend is a physician. He graduated top of his class from med school. I think he's pretty content with life. The bored ones can be a little more trouble sometimes. Fun breed though. Best of luck.
- ugh123 3y agoAre you saying an LLM can't come to the right conclusion and give an explanation for "what is the absolute value of 2i + 2"?
- gunapologist99 3y agoAre you saying it could, without having read it somewhere?
- ugh123 3y agoMaybe I'm unsure what we're arguing here. Did the guys kid drum that up himself or did he learn it from yt? Knowledge can be inferred or extracted. If it comes up with a correct answer and shows it's work, who cares how the knowledge was obtained?
- cduzz 3y agoYeah, my son only knows about imaginary numbers as far as the veritasium "epic math dual" video. As far as I can tell he inferred that |i+1| needs the Pythagorean theorem and that i and 1 are legs of the right triangle. I don't think anyone ever suggested that "absolute value" is "length". I asked him what |2i+2| would be an his answer of "square root of 8" suggests that he doesn't have it memorized as an answer because if it was he'd have said "2 square root two" or something similar. I also asked if he'd seen a video about this and he said no. I think he just figured it out himself. Which is mildly spooky.
- ugh123 3y agoAh, makes sense. I think that is indeed impressive to fill in the paths of reasoning between unknown to known based on even small nuggets of information combined possibly with "learned intuition" which your son seems to have obtained an ability for to understand how to use both. Very cool
- sidlls 3y agoIf the knowledge was obtained by genuine reasoning, that implies that it could also derive/develop a novel solution to an unsolved problem that is not achieved by random guesses. For example, the conception of a complex number in the first place, to solve a class of problems that, prior, weren't even thought to be problems. There's no evidence that any LLM can do that.
- naasking 3y ago> this sort of understanding is a result of some manner of understanding the underlying mechanisms and not a result of just having a huge dictionary of synonyms. He developed an understanding of the underlying mechanisms because he correlated concepts between algebraic and geometric domains, ie. multimodal training data. Multimodal models are already known to be meaningfully better than unimodal ones. We've barely scratched the surface of multimodal training.
- mewpmewp2 3y agoFirst YouTube video that hit for "absolute value of complex" numbers says within 30 seconds that you have to take the 2 numbers, square them and add them and the result is square root of that. I doubt he had to come up with that on his own.
- culi 3y agoThe child clearly demonstrated a geometric, rather than formulaic, understanding of the problem
- mewpmewp2 3y agoI imagine that was shown in the YouTube video visually? That it's a hypotenuse like he explained and this is how to calculate it. I'm just not seeing evidence that he came to the idea of it being like that on their own. He basically reiterated the definition, and had to know the formula. If the child would explain why should we even use or have complex numbers that would be impressive. As otherwise it just seems nothing more than hypotenuse calculation while using different, and "complex" or "impressive" sounding terms. Why should you be interested in this in the first place?
- jhanschoo 3y agoSounds like your son is ready for you to bring it up another level and ask what the absolute value of a (bounded) function is (assuming they have played with functions e.g. in desmos)
- J_cst 3y agoMy son plays soccer
- Exoristos 3y agoDamn. What YouTube channels does he watch?
- doug_durham 3y agoWhat makes you think that an LLM has a "huge dictionary of synonyms"? That's not how LLMs work. They capture underlying concepts and their relations. You had a good point going until you make a straw man argument about the capabilities of LLMs.
- aswegs8 3y agoAny source on what they actually capture? Seems interesting to me.
- _emna 3y agoSorry, can you explain this? To me, it makes sense to define abs(x) = sqrt(x^2) i.e. ignoring the negative solution enforces the positive result. Using that definition, abs(i+1) = sqrt((i+1)^2) = sqrt(i^2 + 2i + 1) = sqrt(-1 + 2i + 1) = sqrt(2i) != sqrt(2). The second example seems off in the same way (i.e. the answer should be sqrt(8i) instead of sqrt(8)). Am I missing something? Also, abs(i+2) = sqrt((i+2)^2) = sqrt(i^2 + 4i + 4) = sqrt(-1 + 4i + 4) = sqrt(4i + 3) which doesnt seem to follow the pattern your son described. Also, just to point out that my understanding of absolute value is different than your sons. Thats not to say one is right and another is wrong, but there are often different ways of seeing the same thing. I would imagine that LLMs would similarly see it a different way. Another example of this is people defining PI by its relation to the circumference of a circle. Theres nothing wrong with such a definition, but its certainly not the only possible definition.
- SpaceNoodled 3y agoHe's talking about distance in two dimensions with real numbers on one axis and complex on the other.
- lovasoa 3y agoNo, there is just one definition, and it's his son's: https://en.m.wikipedia.org/wiki/Absolute_value#Complex_numbers https://en.m.wikipedia.org/wiki/Absolute_value#Complex_numbe...
- _emna 3y agoThe article you linked literally says that there are two definitions: one for real numbers and another for complex numbers. Thanks for the info.
- Shrezzing 3y agoIf they're not already in one, you might want to get your kid enlisted in some gifted child programs.
- dustypotato 3y ago> To what degree can these large language models arrive at these same conclusions, and by what process? By having visual understanding more deeply integrated in the thought process, in my opinion. Then they wouldn't be Large Language Models, of course. There are several concepts I remember and operate on by visualizing them , even visualizing motion. If i want to add numbers, i visualize the carry jumping on top of the next number. If i don't trust one of the additions , I go back , but I can't say if it's because i "mark" the uncertainty somehow. When I think about my different groups of friends, in the back of my mind a visual representation forms. Thinking about my flight route forms a mini map somehow, and i can compare distances between places, and all. This helps incredibly in logical tasks like programming and math. I think it's something that we all learned growing up and by playing with objects around us.
- jacquesm 3y agoYour 12 year old is the next Einstein.
- tim333 3y agoHumans are set up I think to intuitively understand 3d space as it's what we run around and try to survive in. Language models on the other hand are set up to understand language which humans can do also but I think with a different part of the brain. There probably is no reason why you couldn't set up a model to understand 3d space - I guess they do that a bit with self driving. A lot of animals like cats and squirrels are pretty good with 3d space also but less so with language.