Bernhard Riemann

From Bernhard Riemann’s Philosophical Fragments:

  1. The purposefulness of biological structures does not find its fulfillment in a single organism. The reasons for the structure of the human organism are clearly to be sought in the constitution of the whole surface of the Earth, organic nature included.
  2. Organic motions repeat themselves innumerably, both in different individuals simultaneously, and in the life of an individual or of a generation, one after another. In each case, therefore, there is not any particular, but one common cause to be assumed, for the purposefulness which already lies in them of its own accord.
  3. Biological structures, both in the lives of particular individuals (with humans and animals), and in the lives of particular species (with plants and embryos) undergo no development. The cause for their purposefulness is to be sought for in a simultaneously progressing thought process.
After removal of this (organic) purposefulness, there still remains, as is generally recognized with humans and animals, and with plants in Fechner's opinion, a closed system of interlocked, variable purpose and effect relations; and this purposefulness in them is explained by a unified thought process.

Back to Biology

Let’s return to our earlier discussion of evolutionary upshifts, taking the move to land as our current example. The move to land required a whole constellation of new technologies, for both plants and animals.

To move to less moist areas, plants had to develop waxy cuticles to prevent water from escaping through their exteriors and stomata to guard against water loss through leaves. As they developed to larger sizes, they had to hold themselves upright, and develop xylem and phloem to carry water and nutrients to the higher parts of the plant. The changes in reproduction were enormous as well. For the earliest land plants, the bryophytes (e.g., mosses), fertilization occured outside the adult plant and required a moist environment: male and female gametes are released, which must combine in the surrounding environment. The development of seed-bearing plants (sporophytes) was an incredible advancement: fertilization took place within the adult plants, producing diploid seeds which could withstand dry conditions. Combined, all these changes demonstrate an increasing degree of independence from the environment with the move to land. True seeds, being diploid (having two sets of chromosomes) are not comparable, quantitatively, with the haploid gametes dispersed by mosses: this reproductive change is a qualitative one.

Animals similarly needed to control their internal water environment, and freed themselves from the need for water in reproduction. Amphibians, which have characteristics of both water and land animals, require water for reproduction; the eggs both lack a waterproof shell, and are fertilized outside the female’s body, in the water. In contrast, reptiles have waterproof eggs and fertilization occurs internally – inside the body of the female. They also have scaly skin that prevents water loss. With the eventual development of warm-blooded mammals, land animals became independent of the temperature of their surroundings as well. Because of their increased independence from the environment, land animals are themselves new environments for specialized parasites living within them, in a more specifically connected way. Again, the completely different approach to reproduction as animal life has advanced is an example of a qualitative change.

Slide 1
Early plants release spores (gametes), like eggs and sperm. (LPACTV)
Slide 2
Fertilization occured in the ambient environment, requiring water. (LPACTV)
Slide 3
In seed plants, fertilization occurs within the plant, allowing waterproof seeds. (LPACTV)
Slide 4
Frog eggs. Since fertilization occurs outside the female, the eggs cannot have hard shells. (Geoff Gallice)
Slide 5
Mating frogs – fertilization is actually occuring outside: eggs and sperm are released simultaneously. (Erik Paterson)
Slide 5
Crocodile eggs with leathery shells, suited for land. Crocodile fertilization is internal. (Kevin Walsh)
Example Frame

Qualitative change

Previously, we had looked at the kinds of qualitative change seen in the introduction of new degrees of freedom in milling machines, and the different sizes of infinites. We'll now continue to look at how to represent new technologies (both human and biological) that create states of existence that could not have existed without their introduction, where a whole class of new plants, animals or economic products become possible because of the power of newly developed technologies. The biological examples just given are excellent illustrations of non-quantitative differences.

The work of Bernhard Riemann will give us a way to investigate increased connectedness and new active powers — new transcendentals. Understanding his approach to complex functions and the work of Niels Abel will be our focus. To start, we'll look at what complex numbers and complex functions are.


Next: Complex Magnitudes