One of the first formal proof systems. This is actually understandable!

This is Ciro Santilli-2020 definition of the foundation of mathematics (and the only one he had any patience to study at all).

TODO what are its limitations? Why were other systems created?

Engineering analysis is a systematic process used to evaluate and solve problems in engineering contexts. It involves applying mathematical and scientific principles to understand the behavior of systems, materials, and processes, helping engineers to design, optimize, and improve products and systems. Key components of engineering analysis include: 1. **Problem Definition**: Clearly identifying the problem to be solved or the question to be answered. 2. **Modeling**: Creating mathematical or computational models that represent the physical system or process.

Hydrogen pinch refers to a situation in the hydrogen economy or hydrogen supply chain where there is a significant imbalance between hydrogen supply and demand. This can occur when demand for hydrogen exceeds the available supply, leading to increased prices and potential shortages. The concept of a "pinch" is commonly used in resource and energy economics to describe constraints on supply that can impact pricing and availability.

Psychopolitical validity is a concept that merges psychological insights with political theory to evaluate how personal experiences and identities intersect with broader political contexts. It emphasizes understanding how individual psychological factors—such as emotions, motivations, and beliefs—inform and are informed by political beliefs and actions. The term reflects a recognition of the interplay between the psyche and political systems, suggesting that personal experiences of oppression, identity formation, and social interactions can shape political attitudes and behaviors.

In philosophy, the term "state of affairs" refers to the way the world is at a particular time, often in relation to the existence or non-existence of certain facts, conditions, or situations. The concept is primarily associated with the fields of metaphysics and philosophy of language. A state of affairs comprises a specific arrangement of objects and their properties, as well as the relationships that hold between them.

Given stuff like arxiv.org/pdf/2107.12475.pdf on Erdős' conjecture on powers of 2, it feels like this one will be somewhere close to computer science/Halting problem issues than number theory. Who knows. This is suggested e.g. at The Busy Beaver Competition: a historical survey by Pascal Michel.

Naming taxonomic ranks like genus, domain, etc. is a fucking waste of time, only useful before we developed molecular biology.

All that matters is the tree of clades with examples of species in each clade, and common characteristics shared by the clade.

And with molecular biology, we can build those trees incredibly well for extant species. When extinct species are involved however, things get more complicated.

Spin structure is a concept from topology and theoretical physics that arises in the context of manifold theory, particularly in relation to spin manifolds. In mathematics, a spin structure is typically defined on a manifold that enables the definition of spinors, which are mathematical objects that generalize the notion of complex numbers and vectors.

We ust use the if mod notation definition as mentioned at: math.stackexchange.com/questions/4305972/what-exactly-is-a-collatz-like-problem/4773230#4773230

Described at: arxiv.org/pdf/2107.12475.pdf where a relation to the Busy beaver scale is proven, and the intuitive relation to the Collatz conjecture described. Perhaps more directly: demonstrations.wolfram.com/CollatzSequenceComputedByATuringMachine/

Amazingly confirms the wave particle duality of quantum mechanics.

The effect is even more remarkable when done with individual particles such individual photons or electrons.

Richard Feynman liked to stress how this experiment can illustrate the core ideas of quantum mechanics. Notably, he night have created the infinitely many slits thought experiment which illustrates the path integral formulation.

Vs: image: the codomain is the set that the function might reach.

The image is the exact set that it actually reaches.

E.g. the function:could have:

Note that the definition of the codomain is somewhat arbitrary, e.g. $x^2$ could as well technically have codomain:even though it will obviously never reach any value in ${\mathbb{R}}^2$.

The exact image is in general therefore harder to characterize.

The inaugural that predicted the Josephson effect.

Published on Physics Letters, then a new journal, before they split into Physics Letters A and Physics Letters B. True Genius: The Life and Science of John Bardeen mentions that this choice was made rather than the more prestigious Physical Review Letters because they were not yet so confident about the results.

Paywalled by Elsevier as of 2023 at: www.sciencedirect.com/science/article/abs/pii/0031916362913690

A Cartesian product that carries over some extra structure of the input groups.

E.g. the direct product of groups carries over group structure on both sides.