irchans

As a mathematician, I am looking forward to the day when nearly all of undergraduate mathematics and many of the lower level grad school math books are encoded in Lean (a proof checking language). Often I find that theorems are not stated precisely enough and I have trouble finding the exact statement of a theorem without digging through math books in my library. It would be nice if we put the physics and chemistry books into Lean also.

Simplifying all of math is another endeavor, but I imagine that you could have a bunch of LLMs trying to shorten existing Lean proofs.

Here is another list of ideas in math made by a lesser math person. (top 100 most useful)

counting zero integer decimal positional notation 100, 1000, … the four arithmetic operations + – * / fractions decimal notation 0.1, 0.01, … basic propositional logic (Modus ponens, contrapositive, If-then, and, or, nand, …) negative numbers equivalence classes equality & substitution basic algebra – idea of variables, equations, … the idea of probability commutative and associative properties distributive property powers (squared, cubed,…), – compound interest (miracle of) scientific notation 1.3e6 = 1,300,000 polynomials first order predicate logic infinity irrational numbers De Morgan’s laws statistical independence the notion of a function square root (cube root, …) inequalities (list of inequalities) power laws (i.e. abac=ab+c ) Cartesian coordinate plane basic set theory random variable probability distribution histogram the mean, expected value & strong law of large numbers the graph of a function standard deviation Pythagorean theorem vectors and vector spaces limits real numbers as limits of fractions, the least upper bound continuity Rn, Euclidean Space, and Hilbert spaces (inner or dot product) derivative correlation central limit theorem, Gaussian Distribution, Properties of Guassains. integrals chain rule modular arithmetic sine cosine tangent π, circumference, area, and volume formulas for circles, rectangles, parallelograms, triangles, spheres, cones,… linear regression Taylor’s theorem the number e and the exponential function Rolle’s theorem, Karush–Kuhn–Tucker conditions, derivative is zero at the maximum the notion of linearity Big O notation injective (one-to-one) / surjective (onto) functions imaginary numbers symmetry Euler’s Formula eiπ+1=0 Fourier transform, convolution in time domain is the product in the frequency domain (& vice versa), the FFT fundamental theorem of calculus logarithms matrices conic sections Boolean algebra Cauchy–Schwarz inequality binomial theorem – Pascal’s triangle the determinant ordinary differential equation (ODE) mode (maximum likelihood estimator) cosine law prime numbers linear independence Jacobian fundamental theorem of arithmetic duality – (polyhedron faces & points, geometry lines and points, Dual Linear Program, dual space, …) intermediate value theorem eigenvalues median entropy KL distance binomial distribution Bayes’ theorem 210≈1000 compactness, Heine – Borel theorem metric space, Triangle Inequality Projections, Best Approximation 1/(1−X)=1+X+X2+… partial differential equations quadratic formula Reisz representation theorem Fubini’s theorem the ideas of groups, semigroups, monoids, rings, … Singular Value Decomposition numeric integration – trapezoidal rule, Simpson’s rule, … mutual information Plancherel’s theorem matrix condition number integration by parts Euler’s method for numerical integration of ODEs (and improved Euler & Runge–Kutta) pigeon hole principle

mathematical used less often: Baire category theorem, Banach Spaces, Brouwer Fixed Point Theorem, Carathéodory’s Theorem, Category Theory, Cauchy integral formula, calculus of variations, closed graph theorem, Chinese remainder theorem, Clifford algebra (quaternions), Context Free Grammars, countable vs uncountable infinity, Cramer’s Rule, cohomology, Euclidean algorithm, fundamental group, Gauss’ Law, Grassmannian algebra , Graph Theory, Hahn-Banach Theorem, homology, Hairy Ball Theorem, Hölder’s inequality, inclusion-exclusion, Jordan Decomposition, Kalman Filters, Markov Chains (Hidden Markov Models), modules, non-associative algebras, Picard’s Great Theorem, Platonic/Euclidean solids, Principle of Induction, Probabilistic Graphical Models (Bayesian Networks, Markov Random Fields), Pontryagin duality, Quaternions, Spectral Theorem, Sylow p subgroup, repeating decimals equal a fraction, ring ideals, sine law, tensors, tessellation, transcendental numbers, Uniform Boundedness Theorem, Weierstrass approximation theorem. From http://artent.net/2012/11/27/100-most-useful-theorems-and-id...

For years I've had an idea in my head to add a Forth to the old game Master of Orion so that I could automate some of my common actions in the game. There is an open source GIThub called 1oom with source code for the game and I thought I would just write my own Forth. Well, I've recently been using AI's to code, so I asked GPT and it suggested that I use ficl rather than writing my own Forth. It helped me add ficl to 1oom. I've added Forth words to access most of the game variables and I've just begun adding forth words that can manipulate the game state. It's been fun. I had not written much, if any, Forth code in the last 30 years, so it's been a fun trip down memory lane.

I remember seeing the Amiga bouncing ball demo and being amazed.

https://www.youtube.com/watch?v=-ga41edXw3A

The Amiga was the second computer that I ever owned. The first was a C64 that I bought freshman year. (My father bought a TRS-80 before that.)

Certainly most universities now have a very strong liberal bias. I think most science departments were left leaning in the 1950s, but it is stronger now. (I think colleges and universities have always been more progressive than the general populous.) The administrations of universities are now very strongly Democrat leaning. I think that Trump just sees a lot of Democrat run institutions and thinks, "Why should the government support these institutions run almost entirely by Democrats."

I'm retired. I hope to get a 3% per year income from my savings every year after inflation and taxes. If my state implemented a 1% wealth tax on savings each year, I would go bankrupt in 20 years. I am hoping that I will live 20 years.

I also "had an Iron Man moment last week" as a mathematician. I've been doing joint math research with two friends (professors) on a project for several years. Last week, I decided to explore part of our research using chat GPT. 1) I would have a thought. 2) Present it to GPT. 3) Ask GPT to write theorems that were easy to prove and put those proofs into LaTeX. (I always have to check the proofs carefully.) 4) Then I would ask it to generate code (I mostly use Mathematica the language.) 5) Running the code would help verify the proofs. It would also inspire more thoughts and I would go back to step 2). This worked very well, but at one point I could not bound a particular expression and I was not understanding it well, so I got out the pencil and paper and redid the derivation myself which helped a lot. This whole process worked about 10 times faster than doing it without GPT. At the end of a couple of hours I had around 20 pages of correct proofs and all the code needed to do numerical simulations related to the proofs.

ikrima, Do you have any links to your research or paper titles?

How about:

(* Mathematica Notation, Assume x>0 ) If[ x < 10^(-10), 1+ x/2, ( order x^2 error ) If[ x> 10^10, Sqrt[x], ( order 1/Sqrt[x] error ) (else) Sqrt[x +1] ] ] ( I guess the If statements take too much time. *)

Us humans do have difficulty with safety. Sometimes we are able to overcome that problem to an extent. Here are some the few examples where humans have done well with safety: FAA commercial airlines, Soyuz, ISS, Shinkansen trains, US Nuclear power post 3 mile island, Vaccines, and the Falcon 9.

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

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.

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.

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.

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

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

Very cool.

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.

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.

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 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.

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.

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.)

I also wrote a brute force, kinda dumb, depth-first Suduoku solver. I think it took me a few hours in C or C++. At every node in the search tree, I just checked to make sure no constraints were violated. If I remember correctly, it would run in less than 30 seconds and it would always find a solution if a solution existed. (It might have been a lot less than 30 seconds. I only remember thinking, "That was quick".)

The article is a very enjoyable history of how Cuneiform was deciphered by a small group of often competing scholars.

Oh yes, I also will listen to NPR and other radio news when I'm driving.

I forgot. I also watch NBC Nightly News and "What about it?" on YouTube.

Mostly I read Slashdot, HN, and the Wall Street Journal. I read the Economist when I feel like paying for it (sometimes I get a deal like $50 for the digital version for a year). I did Ground News for a year.

What kind of chocolate ? :) (humor)