People who have a math degree, in what context were you first taught how to work with first order logic? That is, to interpret mathematical formulas that use quantifiers, to negate them, and to prove them?

I can only give four options, so please discuss especially if your situation was not covered. Maybe you never did learn this!

#math #mathematics #logic

Let's Verify Step by Step compares process and outcome supervision on MATH. The process-reward model reaches 78.2% best-of-1860 vs 72.4% for outcome. But that gap narrows fast at small N, where most deployments actually live.

https://benjaminhan.net/posts/20260512-lets-verify-step-by-step/?utm_source=mastodon&utm_medium=social

#Paper #LLMs #Reasoning #Mathematics #ICLR #OpenAI #AI

Can I share a happy math(s)-related tidbit?

Over a decade ago, I watched my son suffer through AP calculus, and as I looked over the curriculum and the book, it seemed like it really *sucked*. They threw a bunch of rules and operations at the students, fast, with little effort to explain or justify.

An example that stood out was showing the notation "dy/dx" and saying it was simply indicating a derivative, and should *not* be viewed as a fraction. Then, later, in differential equations, they move the dx to other side of the equation, just like a fraction.

What is a student to do? Could we perhaps back to the intuitive notion of "infinitesimals" that motivated Newton and Leibniz? The notion was pilloried at the time because it wasn't rigorously justified, and could lead to issues. It took decades to develop ideas of limits to justify Newton's and Leibniz's ideas.

Somehow I came across "non-standard analysis" which makes infinitesimals mathematically rigorous, and it's really cool. Also came across a lovingly-written textbook, available for free here:

https://people.math.wisc.edu/~hkeisler/calc.html

However, I still couldn't quite fathom how the problem was solved. Non-standard analysis uses "numbers that are smaller than the smallest real number" even though, I knew from my training, the real numbers are dense. Between zero and any number, however small, there is a real number. Indeed, an infinite number of them.

Finally I started reading the epilogue to the textbook above. The idea (or one idea) is to create a class of mathematical objects, each of which is an infinite sequence that diverges, but at a specific rate. For instance, one is {1, 2, 3, ..., n, ...} while another is {1^2, 2^2, 3^2, ..., n^2, ...}. Both tend to infinity, but the latter is "bigger". We can define operations (addition, multiplication, etc.) element-wise and develop appropriate comparators (>, <, =). And then they can serve as numbers: we can do all the things with them that we do with numbers.

The inverse of a diverging "number" is an infinitesimal. And indeed, for any real number, however small, even a "large" infinitesimal will eventually be smaller. So it's true: every infinitesimal is smaller than the smallest real number (or at least a fixed small real number). Yet they function as numbers.

This is just a cool idea and it brings me joy to have the issue resolved, even though it has zero direct impact on my life.

#mathematics #nonstandardanalysis

🧠 “Is #Intelligence a mathematical structure?”🔢 – #Zoomposium with #GittaKutyniok

The key to the next generation of intelligent systems – On computability, limitations, and the future of AI

📎https://philosophies.de/index.php/2026/04/24/intelligenz-mathe-struktur/

🎥https://youtu.be/SbLf8oBAv90

#ArtificialIntelligence #AI #Mathematics #NeuralNetworks #DeepLearning #MachineLearning #Intelligence #CognitiveScience #MathematicsAndAI #AIResearch #ExplainableAI #Computability #NeuromorphicComputing #Science

Theorem of the Day (May 12, 2026) : The Hockey Stick Identity
Source : Theorem of the Day / Robin Whitty
pdf : https://www.theoremoftheday.org/Binomial/Hockeystick/TotDHockeystick.pdf
notes : https://www.theoremoftheday.org/Resources/TheoremNotes.htm#272

#mathematics #maths #math #theorem @Theoremoftheday

How Unknowable Math Can Help Hide Secrets
Source : Quanta Magazine / Ben Brubaker

https://www.quantamagazine.org/how-unknowable-math-can-help-hide-secrets-20260511/
#mathematics #maths #math

What's the most interesting meal you can serve, consisting entirely of foods and drinks mentioned in well-known theorems or constructions of #mathematics?

You can make a roughly balanced meal using
• the Ham Sandwich Theorem, for protein and starch
• Conway's game of Sprouts, for vitamins
• the Blancmange Function, for a dairy-based dessert.

https://en.wikipedia.org/wiki/Ham_sandwich_theorem
https://en.wikipedia.org/wiki/Sprouts_(game)
https://en.wikipedia.org/wiki/Blancmange_curve

But it's not too inspiring, flavour-wise. Can it be made more interesting?

Can you solve it? #permutation #combination #maths #puzzle #iit #jee #gate #olympiad

https://makertube.net/w/ivBTSDAecKmY9RLdT16oSw