Thomas Barrios

Unit distances result is very exciting, but re: “math is solved” — humans regularly solve long-open problems, and yet infinitely more interesting open problems remain.

Wild that an LLM autonomously disproved the unit-distance conjecture 🤯 But it’s also striking to me that, almost immediately after seeing the construction, a human mathematician was able to improve it further. Speaks to the potential of human–AI collaboration in math, QED 🔲

If you are a mathematician, then you may want to make sure you are sitting down before reading further.

We usually repeat that correlation doesn’t imply causation, which is true. But the reverse, causation doesn’t imply correlation, is ALSO true. Let X~N(0,1) and Y=X^2. In this example, X causes Y. But the correlation between X and Y is Corr(X,Y)=Corr(X,X^2)=Cov(X,X^2)/(σ_X*σ_(X^2))=(E[X*X^2]-E[X]*E[X^2])/(σ_X*σ_(X^2))=0. Why?👇 The function h(x)=x^3 is odd: (-x)^3=-x^3 and the distribution is symmetric around 0, f is even: f(x)=f(-x) Thus (x^3)*f(x) is odd (odd*even=odd function) and the integral of an odd function over a symmetric interval is 0. Thus E[X^3]=∫X^3*f(x)dx=0 (the integral is from -♾️ to ♾️) The same occurs with the function g(x)=x because g(-x)=-x=-g(x) (it’s odd). So E[X]=∫X*f(x)dx=0 (the integral is again from -♾️ to ♾️) Thus Corr(X,Y)=0 because Cov(X,X^2)=0. Thus we have proven that causation DOES NOT IMPLY correlation. Q. E. D. ▪️