The mammals are the class of vertebrate animals characterized by the presence of mammary glands, which in females produce milk for the nourishment of young; the presence of hair or fur; and endothermic or "warm-blooded" bodies. The brain regulates endothermic and circulatory systems, including a four-chambered heart. Mammals encompass some 5,500 species, distributed in about 1,200 genera, 152 families and up to 46 orders, though this varies with the classification scheme.

The Classification Problem in an area of science or mathematics is the problem of separating a large class of objects into smaller classes, and giving a criterion for determining whether a particular object is or is not in a particular class.
One of the most famous attempts at classification in biology is Carolus Linnaeus's famous classification of living things by class, order, genus, and species. An example in physics is the classification of the physical world into matter and energy, and the classification of matter according to its atomic number and energy according to its wavelength. In mathematics, one successful solution to a classification problem is the classification of vector spaces by their dimension. Two vector spaces are isomorphic if and only if they have the same dimension. Another example is the classification of two dimensional manifolds: every two dimensional manifold is a connected sum of spheres, tori, and projective planes. For the intelligent human species, classification is a big problem as much as a big solution.

The most commonly cited definition of "species" was first coined by Ernst Mayr. By this definition, called the biological species concept or isolation species concept, species are "groups of actually or potentially interbreeding natural populations which are reproductively isolated from other such groups".

The definition of a species given above as taken from Mayr, is somewhat idealistic. Since it assumes sexual reproduction, it leaves the term undefined for a large class of organisms that reproduce asexually. Biologists frequently do not know whether two morphologically similar groups of organisms are "potentially" capable of interbreeding. Further, there is considerable variation in the degree to which hybridization may succeed under natural and experimental conditions, or even in the degree to which some organisms use sexual reproduction between individuals to breed. Consequently, several lines of thought in the definition of species exist:

Typological species 

A group of organisms in which individuals are members of the species if they sufficiently conform to certain fixed properties. The clusters of variations or phenotypes within specimens (ie: longer and shorter tails) would differentiate the species. This method was used as a "classical" method of determining species, such as with Linnaeus early in evolutionary theory. However, we now know that different phenotypes do not always constitute different species (e.g.: a 4-winged Drosophila born to a 2-winged mother is not a different species). Species named in this manner are called morphospecies.

Morphological species 

A population or group of populations that differs morphologically from other populations. For example, we can distinguish between a chicken and a duck because they have different shaped bills and the duck has webbed feet. Species have been defined in this way since well before the beginning of recorded history. This species concept is much criticised because more recent genetic data reveal that genetically distinct populations may look very similar and, contrarily, large morphological differences sometimes exist between very closely-related populations. Nonetheless, most species known have been described solely from morphology.

Biological / Isolation species 

A set of actually or potentially interbreeding populations. This is generally the most useful formulation for scientists working with living examples of the higher taxa like mammals, fish, and birds, but meaningless for organisms that do not reproduce sexually. It does not distinguish between the theoretical possibility of interbreeding and the actual likelihood of gene flow between populations and is thus impractical in instances of allopatric (geographically isolated) populations. The results of breeding experiments done in artificial conditions may or may not reflect what would happen if the same organisms encountered each other in the wild, making it difficult to gauge whether or not the results of such experiments are meaningful in reference to natural populations.

Mate-recognition species 

A group of organisms that are known to recognise one another as potential mates. Like the isolation species concept above, it applies only to organisms that reproduce sexually. Unlike the isolation species concept, it focuses specifically on pre-mating reproductive isolation.

Phylogenetic / Evolutionary / Darwinian species 

A group of organisms that shares an ancestor; a lineage that maintains its integrity with respect to other lineages through both time and space. At some point in the progress of such a group, members may diverge from one another: when such a divergence becomes sufficiently clear, the two populations are regarded as separate species.

Microspecies 

Species that reproduce without meiosis or mitosis so that each generation is genetically identical to the previous generation.

Darwin's Theory of Evolution - A Theory In Crisis
Darwin's Theory of Evolution is a theory in crisis in light of the tremendous advances we've made in molecular biology, biochemistry and genetics over the past fifty years. We now know that there are in fact tens of thousands of irreducibly complex systems on the cellular level. Specified complexity pervades the microscopic biological world. Molecular biologist Michael Denton wrote, "Although the tiniest bacterial cells are incredibly small, weighing less than 10-12 grams, each is in effect a veritable micro-miniaturized factory containing thousands of exquisitely designed pieces of intricate molecular machinery, made up altogether of one hundred thousand million atoms, far more complicated than any machinery built by man and absolutely without parallel in the non-living world."