Related papers: One Plus One Equals Two Ones: On Identity, Aggrega…
If we want to represent integers in base $m$, we need a set $A$ of digits, which needs to be a complete set of residues modulo $m$. When adding two integers with last digits $a_1, a_2 \in A$, we find the unique $a \in A$ such that $a_1 +…
Comparing clusterings is central to evaluating unsupervised models, yet the many existing similarity measures can produce widely divergent, sometimes contradictory, evaluations. Clustering similarity measures are typically organized into…
In many everyday categories (sets, spaces, modules, ...) objects can be both added and multiplied. The arithmetic of such objects is a challenge because there is usually no subtraction. We prove a family of cases of the following principle:…
In this paper, a class of combinatorial identities is proved. A method is used which is based on the following rule: counting elements of a given set in two ways and making equal the obtained results. This rule is known as "counting in two…
A classification is a surjective mapping from a set of objects to a set of categories. A classification aggregation function aggregates every vector of classifications into a single one. We show that every citizen sovereign and independent…
We consider a first-order logic for the integers with addition. This logic extends classical first-order logic by modulo-counting, threshold-counting and exact-counting quantifiers, all applied to tuples of variables (here, residues are…
This is a note showing that, contrary to our lasting belief, the nonadditivity X(1+2)=X(1)+X(2)+\alpha X(1)X(2) is not a true physical property. \alpha in this expression cannot be unique for a given system. It unavoidably depends on how…
Why does the physical 4-dimensional space have a 3 + 1 signature rather than a 4 + 0 or a 2 + 2 for its metric? We give a simple explanation based largely on a group-theoretic argument a la Wigner. Applied to flat spaces of higher…
Science and mathematics help people better to understand world, eliminating different fallacies and misconceptions. One of such misconception is related to arithmetic, which is so important both for science and everyday life. People think…
The multiplicative theory of a set of numbers (which could be natural, integer, rational, real or complex numbers) is the first-order theory of the structure of that set with (solely) the multiplication operation (that set is taken to be…
Given two real numbers $q_0,q_1>1$ satisfying $q_0+q_1\geq q_0q_1$ and two real numbers $d_0\ne d_1$, by a {double-base expansion} of a real number $x$ we mean a sequence $(i_k)\in \{0,1\}^{\infty}$ such that \begin{equation*}…
We indicate a way of distinguishing between structures, for which, we call two structures distinguishable. Roughly, being distinguishable means that they differ in the number of realizations each gives for some formula. Being…
The concept of measurement is discussed. It is argued that counting process in mathematics is also measurement which requires a basic unit. The idea of scale is put forward. The basic unit itself, which are composed of the infinitesimal of…
The unprecedented performance achieved by deep convolutional neural networks for image classification is linked primarily to their ability of capturing rich structural features at various layers within networks. Here we design a series of…
Assembly theory (AT) quantifies selection using the assembly equation and identifies complex objects that occur in abundance based on two measurements, assembly index and copy number, where the assembly index is the minimum number of…
We prove that the relation of bisimilarity between countable labelled transition systems is $\Sigma_1^1$-complete (hence not Borel), by reducing the set of non-wellorders over the natural numbers continuously to it. This has an impact on…
In this note, we construct and study an algebraic system similar to the natural numbers, but with noncommutative addition. The addition we introduce is a binary operation that commutes with itself in the sense of N. Durov. Neverheless, the…
To determine whether a number is congruent or not is an old and difficult topic and progress is slow. The paper presents a new theorem when a prime number is a congruent number or not. The proof is not necessarily any simpler or shorter…
The proof identity problem asks: When are two proofs the same? The question naturally occurs when one reflects on mathematical practice. The problem understandably can be seen as a challenge for mathematical logic, and indeed various…
Let $k\ge 2$ be an integer and let $A$ be a set of nonnegative integers. For a $k$-tuple of positive integers $\underline{\lambda} = (\lambda_{1}, \dots{} ,\lambda_{k})$ with $1 \le \lambda_{1} < \lambda_{2} < \dots{} < \lambda_{k}$, we…