Historically, most mathematicians have believed that mathematical truths such as "2+5=7" exist independently of human minds, being universally and eternally true. Mathematicians believe they are discovering properties of, say, prime numbers, rather than merely inventing them. This view of mathematics dates back to Pythagoras (500 BC) and Plato (400 BC). It is often called “Platonism”. In order ...

Quantum mechanics is different from most other branches of science in that it uses complex numbers in a fundamental way. Complex numbers were originally created as a mathematical curiosity: i = √−1 was the asserted “imaginary” solution to the polynomial equation x2 = −1. As time went on, an entire mathematical edifice was constructed with these “imaginary” numbers. Complex numbers have kept lon...

Mathematical models could be helpful in assisting the Indian Government’s new initiative of issuing biometric cards to its citizens. In this note, we look into the role of mathematical models in estimating the missing, non-enumerated population numbers, estimating annual numbers of cards required by age, gender and regions in India. The linkage between National Population Register and biometric...

Suppose that A is an infinite set. Constructing a denumerable subset {a1, a2, a3, . . .} of A, which is the same as showing that there is an injection f : Z −→ A, requires more than just mathematical induction. In these notes we explore some consequences of basic definitions con cerning infinite sets with the aid of mathematical induction only. We show that the set of rational numbers, and more...

We study mathematical expectations of Betti numbers of configuration spaces of planar linkages, viewing the lengths of the bars of the linkage as random variables. Our main result gives an explicit asymptotic formulae for these mathematical expectations for two distinct probability measures describing the statistics of the length vectors when the number of links tends to infinity. In the proof ...

Non-Crossing Tree partitions are newer mathematical objects that have recent applications in genetics and mathematical biology. We explore several interesting connections between these partitions and the more commonly studied non-crossing set partitions. While non-crossing set partitions are counted by the Catalan numbers, we prove that non-crossing tree partitions in Binary trees are counted b...

The Fibonacci sequence is one of the most beautiful and febulas mathematical concepts. Number turn in place that you have probably seen nature. a series numbers follows unique integers sequence. These generate patterns can be found all real life. are also called golden ratio. section used for architect, art, Space exploration, etc. Keywords: Numbers, Golden ratio

Church-Turing Thesis: Claim that every computable function can be computed by a Turing machine. Computability theory: Study of computable functions on the natural numbers. Continuum hypothesis: Conjecture that there are only two sizes of infinite sets of real numbers. Database: Finite, typically relational structure. First order logic: Mathematical model of the part of language built up from th...

Leibniz described imaginary roots, negatives, and infinitesimals as useful fictions. But did he view such 'impossible' numbers mathematical entities? Alice Bob take on the labyrinth of current scholarship.