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I think that finding proofs for open mathematical questions should count as critical thought. (See https://medium.com/%40cognidownunder/three-erdős-problems-fe...)

That's impressive, but it isn't thought, anymore than neurons in a dish that learn to play Tetris have thoughts, or if you spent eons painstakingly calculating what the TPU did with the model to come up with the same output tokens, but via pen and paper instead.

When the TPU does it, is the TPU thinking? Where does the critical thinking take place in the endless pages of matrix math that eventually evaluates into the same token output as the TPU?


AI makes it possible for someone who has never written code to generate a program that does what they want. One of my friends wanted to simulate a 7,9 against a dealer 10 upcard in the card game blackjack. GPT was able to write the simulation for him in javascript/html. So it took a 0.001x coder and turned him into a 0.2x coder.

Was it actually correct? How would they tell?

This morning a person posted a question to the Reddit group r/Mathematica (https://www.reddit.com/r/Mathematica/comments/1s1fin2/can_ho...).

I asked GPT to write code to address their question and the code was quite acceptable drawing the circle and finding the correct intersection point. It would have take me about 40 minutes to write the code, so I would not have done it myself.

Currently, GPT is great for writing short programs. The results often have a bug or two that is easy to fix, but it's much faster to have GPT write the code. This works fine for projects that are less than 100 lines of code where you just want something that works.


This take was accurate about 2 years ago, up until perhaps one year ago. Current capabilities far exceed what you are outlining, for example using Claude Opus models in a harness such as Claude Code or OpenCode.

I think this 6 set Venn Diagram is nice because I made it. :)

http://162.243.213.31/wp-content/uploads/2012/08/Ven3.png


How did you make this? It is nice. Why does the red line have a heart-shaped bounce/curve in the open white space instead of just being round?


Type "Polar plot 6 curves of the form r = (1 + Sin[2^(k - 1) t]/2^(k - 1)) where k =1, 2, .., 6 and t=0 to 2 Pi each curve should be a different color" into ChatGPT5.

https://chatgpt.com/share/690f675d-c340-8013-b598-41fe487b4e...

It has the nice properties that you can do any number of sets (in theory) and all the boundary intersections are either osculating or perpendicular.


Looks like sinusoids in polar coordinates


> I became a mathematician. From this childhood exposure, entropy was the first mathematical "concept" beyond arithmetic that I understood.

Very cool.


Even after reading the article, I think there are reasonable ways to set up a low cost system that uses zero-knowledge proofs to "prove" your age without disclosing your identity. I do think that you will need trusted entities and the system will only stop most, maybe 80 or 90 percent of children under 18 from seeing porn. But, if you do this, then maybe 99% of kids under the age of 14 will have a lot of difficulty viewing porn which is a good thing. There may be valid a slippery slope argument for not setting up the age validation system even if everything I said above is true.


On the other hand: Are you willing to pay hundreds of millions for developing the biggest data leak in human history, killing websites like Wikipedia in the process, while stopping only 10% of underage children from seeing porn?

The current systems being put in place in the UK are privacy-invading and ineffective. In my opinion they are worse than not having anything at all. I might be willing to change my viewpoint if something better comes along, but if a proper solution was so easy, why haven't we seen a peer-reviewed reference design yet? What's stopping the nerds from nerding harder?


> On the other hand: Are you willing to pay hundreds of millions for developing the biggest data leak in human history

The comment you are replying to was talking about ZKP based systems. In those systems you don't show any identity information to the websites that you are trying to prove your age to.

Those systems can be made leak proof by making the party that you have to show identification to be some party that already has your identity information. For example it can be the government agency that issues your driver's license.


But these systems then are trivial to bypass by a person that publishes their private key for others to use as impersonation. If the site can't determine if the same id is used for multiple requests, they can't prevent it. And if the gov isn't able to see which site is requesting the data, neither can it.


Systems like the EU's digital identity wallet use hardware-based security. The private keys are generated by the secure element in your smartphone or something equivalent on a smart card, and any operations that need the keys during a verification are done in that secure element.


IIRC the new EU spec doesn't actually require using "secure elements" that could limit the user, only says they should be used if present. It shouldn't be hard to find some device where the hardware isn't present or is insecure to extract the keys from.

Or people could just proxy requests to the device, even with a reasonable rate limit in place, one donor could provide access for over a dozen people each day.


Nobody is using the EU wallet and perhaps trusting it is a mistake.


So, I think that I can avoid a lot of the data leak problems.

There might be a simple way to do this with a crypto-currency. If possession of a credit card is considered proof of age, then possession of cryptocurrency should also be considered proof of age. Maybe the user could play $0.01 to the porn site using crypto currency to prove that he is over 18. If done properly, no one, not even the government, would know who the user was.

Here is another idea.

You have independent stores where the clerks can sell proof-of-age certificates to people. These certificates are essentially just 20 random Base64 characters. By law, the independent stores are not allowed to identify the customer (who pays with cash). The store clerk is only permitted to issue certificates to people who appear to be over the age of 18, no id required. The store keeps a list of every certificate that they have sold along with the month in which the certificate was sold so that the certificates can expire after several months.

Now I claim that it is possible to create open source zero knowledge proof software that runs on a server for each store, a few government certifying authorities, the porn websites, and on the users computer so that as long as the stores don't identify the users, no one will be able to identify the user. The government will not be able to tell which certificate was used to access the porn. The government will not have access to the certificates. It will not be able to tell which store issued the certificate. The porn site will not learn the certificate of the user nor will it know his identity.

Also, the number of lines of code needed for each program of the five needed open source programs will be less than 1000 lines, maybe less than 100 lines.

I think that all of this could be done at a cost of about $50,000 to develop the software plus the cost of running the servers. I feel like I could write all the code for less than that.

The system is not perfect. You have to trust the stores to not identify the customers and to do a decent job of identifying who is over the age of 18. Some kids will get certificates by copying their parent's code or copying the code of an older friend. Some 16 year olds will look like they are 18 and they will be able to buy proof of age certificates. But, over 80% of kids under 17 will not be able to view porn.

Despite the low cost and effectiveness of the idea above, I am not sure that it is a good idea. I don't like the government censoring content.


Yeah, I think even if we only manage to delay the "age of first porn viewing" to something like 14-15, thats probably a win.


Maybe, but as a parent, I believe its an embarrassment to expect to radically retrofit a society in such ways as to make up for my own negligent lack of responsibility for my own children, which I do take quite seriously. Not to mention the myriad of resultant unintended consequences which invariably arise when such systems(of which i'm quite familiar) are brought to bear. Though I do speak from such a position of professional neutrality, as I would gain no benefit at all from implementing such a ubiquitously mandated system. Perhaps if things were different, I'd think otherwise.


In my opinion "we need mandatory age verification" is an admission that we can't really address the overarching issue of parents that can't/won't parent at a good enough level. Narcissistic parenting without any added access to questionable content on a smartphone is still... narcissistic parenting. The definition of "parent better" differs between people and is often non-negotiable, even way before anything involving CPS occurs. Not to mention, the content being withheld will become available at adulthood anyway, and can still be harmful if the person has not been given the tools to navigate it well.

Admittedly the bar is far higher with ubiquitous social media and smartphones. I'm not sure a parenting license system would ever work out in practice. Yet a lot of issues stemming from upbringing can cause irreversible harm and I don't feel like those root causes are brought up that much in the broader discussion about mental health symptoms.

It pains me to think that some amount of debilitating childhood trauma is unavoidable, but content restriction at least sounds like an actionable problem that doesn't require uprooting the fabric of society to correct.


My son accidentally brought a knife to school at age 12 -- maybe a 4 inch blade. When he realized that he had a knife in his backpack, he told his teacher. He was suspended from school for about 3 days and we had a fairly pleasant conversation with the principal after the suspension.


When I was at university, one of my classmates was a cop. He was petrified because that day, due to schedule issues, he had all his cop stuff locked in the trunk of his personal car. At the time, having that sort of weaponry on campus was a big deal. He would have been better off comming to class in full uniform (the exception for cops would not apply if he wasnt on duty or at least in uniform.) He knew what might happen if someone discovered his handgun/taser/mace was on campus.


I remember back in elementary school the YW in my class brought a huge kitchen knife with him in his backpack. He showed it to me. Later that day, he slightly cut himself with it in the toilet over a broken heart or something like that. Next day he was back to school. We called him sleeping bag because he was wearing his pants so low


I myself have been suspended for having a "weapon". The weapon in question was a bent paper clip. No I'm not kidding.


You probably remember when the cops would get called if you were caught with a cell phone or a pager at school.


When I was in eighth grade my advanced algebra teacher said "I wonder if you always did your homework in pen, would you make fewer mistakes." That was 45 years ago. Now I am a mathematician and I have done 95% of my math in pen since then. I'm not sure how much it helped, but as you said maybe I think a bit more before I write because I don't like scratching out mistakes.


This might be the best article that I have read this year. It might change the way that I program. I really want to write up examples for each of the ideas presented: Monotonicity, Pre- and post-conditions, Invariants, Isolation, Induction, and Proof-affinity as a quality metric. I think it's easy to find these ideas in my code, but I want to try to practice writing proofs "in my head" as I code to make my code better. It is not apparent to me now how this will work, but if I write up some examples, I imagine it will become clearer.


I believer there is an error in

"Readers comprehend “the boy hit the ball” quicker than “the ball was hit by the boy.” Both sentences mean the same, but it’s easier to imagine the object (the boy) before the action (the hitting). All brains work that way. (Notice I didn’t say, “That is the way all brains work”?)".

It should be, "the SUBJECT (the boy) before the action (the hitting)." (I added caps for emphasis.)


> "the boy hit the ball"

In this sentence, boy=subject, hit=verb, ball=object.

> All brains work that way.

If language sentence structure reflects how brains think, then that's not entirely true. While most languages are SVO (subject-verb-object), not all are. Japanese is SOV (subject-object-verb), while biblical Hebrew is/was VSO (verb-subject-object). I'm sure there are other variations.

EDIT: it just occurred to me that Japanese SVO is syntactically similar to Forth/RPN.


Forth :) It's been like 30 years since I the last time I wrote a Forth program that was more than 10 lines long---fond memories.


Not to mention passive contructions are way overused (oops, I just used one). At least if the sentence mentions the acting subject, an active construction is shorter and less stuffy.


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