# Karl Weierstrass, Richard Dedekind and Georg Cantor

### From Biomatics.org

Weierstrass, Dedekind, and Cantor used three distinct approaches to describe the irrational numbers in terms of the rational numbers, thereby constructing a mathematical entity from a more primitive notion. The question thus arises as to what portion of mathematics and logic can be algorithmically constructed from a set of primitive notions and furthermore precisely what is this set of primitive notions.

If molecules are "smart" and evolve mathematical systems, do they also perform similar constructions? Do they even have use of irrational numbers? If so, then what strategy do they use to construct them?

## Contents |

# Primitive Notions

## Numbers

St. Augustine of Hippo (A.D. 354–430) wrote " Numbers are the Universal language offered by the deity to humans as confirmation of the truth." Similar to Pythagoras, he too believed that everything had numerical relationships and it was up to the mind to seek and investigate the secrets of these relationships or have them revealed by divine grace.

## Elementary catastrophes

Catastrophe theory analyses *degenerate critical points* of the potential function — points where not just the first derivative, but one or more higher derivatives of the potential function are also zero. These are called the germs of the catastrophe geometries. The degeneracy of these critical points can be *unfolded* by expanding the potential function as a Taylor series in small perturbations of the parameters.

When the degenerate points are not merely accidental, but are structurally stable, the degenerate points exist as organising centres for particular geometric structures of lower degeneracy, with critical features in the parameter space around them. If the potential function depends on two or fewer active variables, and four or fewer active parameters, then there are only seven generic structures for these bifurcation geometries, with corresponding standard forms into which the Taylor series around the catastrophe germs can be transformed by diffeomorphism (a smooth transformation whose inverse is also smooth). These seven fundamental types are now presented, with the names that Thom gave them.

## Network topology

Network topology is the study of the *arrangement* or *mapping* of the elements (links, nodes, etc.) of a network, especially the physical (real) and logical (virtual) interconnections between nodes.^{[1]} ^{[2]} ^{[3]}

A local area network (LAN) is one example of a network that exhibits both a physical topology and a logical topology. Any given node in the LAN will have one or more links to one or more other nodes in the network and the mapping of these links and nodes onto a graph results in a geometrical shape that determines the physical topology of the network. Likewise, the mapping of the *flow of data* between the nodes in the network determines the logical topology of the network. It is important to note that the physical and logical topologies *might* be identical in any particular network but they also may be *different*.

Any particular network topology is determined only by the graphical mapping of the configuration of physical and/or logical connections between nodes. LAN Network Topology is, therefore, technically a part of graph theory. Distances between nodes, physical interconnections, transmission rates, and/or signal types may differ in two networks and yet their topologies may be identical^{[2]}.

## The Elements

Periodic table showing atomic, van der Waals and ionic radii for all elements with their unique colors. |

## Amino Acids

All amino acids have the same general formula:

**The twenty amino acids found in biological systems:
**

All proteins are linear chains composed of these 20 amino acids.

## Twenty Elemental Algebras

The structures of the 20 amino acids serve as small fundamental algebraic systems

## Organic Chemistry Reactions

http://www.organic-chemistry.org/namedreactions/ .

## Emotion classification

Ekman devised a list of basic emotions from cross-cultural research on the Fore tribesmen of Papua New Guinea. He observed that members of an isolated, stone age culture could reliably identify the expressions of emotion in photographs of people from cultures with which the Fore were not yet familiar. They could also ascribe facial expressions to descriptions of situations. On this evidence, he concluded that some emotions were basic or biologically universal to all humans.^{[7]} The following is Ekman's (1972) list of basic emotions:

However in the 90s Ekman expanded his list of basic emotions to 15, notably including a greater range of positive emotions (e.g. Ekman 1999).

## Elemental Uncertainties

Harmonic analysis is the branch of mathematics which studies the representation of functions or signals as the superposition of basic waves. It investigates and generalizes the notions of Fourier series and Fourier transforms. The basic waves are called " harmonics", hence the name "harmonic analysis." In the past two centuries, it has become a vast subject with applications in areas as diverse as signal processing, quantum mechanics, and neuroscience. The classical Fourier transform on Rn is still an area of ongoing research, particularly concerning Fourier transformation on more general objects such as tempered distributions. For instance, if we impose some requirements on a distribution f, we can attempt to translate these requirements in terms of the Fourier transform of f. The Paley-Wiener theorem is an example of this. The Paley-Wiener theorem immediately implies that if f is a nonzero distribution of compact support (these include functions of compact support), then its Fourier transform is never compactly supported. This is a very **elementary form of an uncertainty principle** in a harmonic analysis setting. See also classic harmonic analysis. Fourier series can be conveniently studied in the context of Hilbert spaces, which provides a connection between harmonic analysis and functional analysis.