Mathematics, Science, and Phylogenetics Compared: An Essay

R. H. Zander

Res Botanica,

A Missouri Botanical Garden Web site

February 11, 2011



Mathematics, Science, and Phylogenetics Compared:
An Essay


R. H. Zander, Missouri Botanical Garden, February 11, 2011


Science, mathematics and phylogenetics may be compared with reference to patterns in nature.


Mathematics is consistent in all dimensions, bounded only by Gödel’s Incompleteness Theorem. It is axiomatic, using lemmas (proven statements used as logical stepping stones) and theorems (statements proven on basis of previous true statements). It is entirely deductive (mathematical induction is not the same as inductive reasoning, it is actually a rigorous deduction). It ignores causes and real entities. And it is doxastic in that we have faith that it is true because a deduction must be true if based on a true first principle. It is also somewhat magical in that mathematical concepts, no matter how convoluted or pyramidal, work well every time in real situations.


Science  (including evolutionary systematics as distinct from phylogenetics) is paraconsistent in that there may be contradictory theories but we need them to deal with nature, e.g. wave and particle theories of light, polythetic species descriptions, different species concepts. It does not, however, ignore any fundamental dimension, i.e., one not reducible to component terms, like length and force. It is non-axiomatic and corrigible in that scientific theory is expected to be continuously tested by new facts, but facts are not tested by theories, which instead must subsume relevant facts or be changed. It is based on hypotheses and theories involving causal explanations with a footprint in natural phenomena. It uses both deductive and inductive reasoning, e.g. theory of evolution using deduction from e.g. fossils and polyploidy, and induction to extend the theory to all evidence and all species. It is founded on empiricism in rejecting hidden causes and unnamed or unobservable entities. And it is nondoxastic in that we require that all theories and assumptions be falsifiable or at least have some demonstrable causal connection with natural phenomena. The only quasi-fundamental patterns in science are certain Laws and Principles of physics, and even these are presented in such as way as to be subject to falsification. The magic is in the greatly advanced understanding of nature through scientific theorization based on all available facts, and competition between alternative theories.


Phylogenetics projects an aura of the exactitude and certainty of mathematics. It is, however, not consistent because its apparently fundamental patterns are generated only by sister-group analysis. A whole dimension, accessible through ancestor-descendant analysis, is ignored, yet is critical to evolutionary theory as being directly involved in inferences of “descent with modification.” Phylogenetics is axiomatic in the sense that sister-group trees are taken as lemmas and traits or distributions mapped thereon as theorems (though these are incorrectly called “hypotheses”). Facts, as cladograms of similarity, are tested against theories (past classifications). It is largely deductive, citing the “hypothetico-deductivism” of Popper but substituting observed pattern for hypothesis. It rejects empiricism in rejecting or at least relegating non-phylogenetically informative data and in relying on unnamable “shared ancestors” as hidden causes. And, it is doxastic in that adherents believe that sister-group analysis of evolution is the only acceptable basis for classification because it reflects facts of present-day evolutionary relationships, not theory. Phylogenetics is neither mathematics nor science because of the categorical prescriptions and proscriptions in axiomatic structuralism. A phylogenetic tree is linguistically “pseudo-referencing,” imbuing a representation with import it lacks, and there is no real magic in that.


This Essay is an extract from a paper submitted to a philosophy of science journal, hence the specialist jargon.