The Limitations of Set Theory in Understanding Biological Evolution

Table of Links

Abstract and Introduction

Part I. A Definition of Life

Part II. The first Miracle: The emergence of life is an expected phase transition – TAP and RAF.

Part III. The Second Miracle: The evolution of the biosphere is a propagating, non-deducible construction, not an entailed deduction. There is no Law. Evolution is ever-creative

Part IV. New Observations and Experiments: Is There Life in the Cosmos?

Conclusion and Acknowledgments

Figures and References

Conclusion

We have sought the source of life in all our creation myths among all the peoples of the earth, perhaps back to Neanderthal 500,000 years ago. The issue of the Origin of Life as a scientific problem arose with Pasture’s claim: Life only comes from life, (55).

Experimental efforts have been underway since Haldane and Oparin, (56,57), then the famous Miller Urey experiments in 1953, (58). Intense efforts based on the conviction that life must be based on template replication of polynucleotides have been carried out, (4,5,6). So far, no case of molecular reproduction has been found on this sensible pathway.

The concept of the emergence of collectively autocatalytic sets was first formulated in 1971, (14). Such reproducing chemical sets of DNA, of RNA, and of peptides have been engineered using evolved polymer sequences. Xavier has now found small molecule collectively autocatalytic sets containing no DNA, RNA or peptide polymers in all 6700 prokaryotes, (11,12). This strongly suggests, but does not prove, that such systems may be the earliest form of molecular reproduction in the universe.

Well established mathematical theory demonstrates that such systems can arise as a first-order phase transition in a universe where the diversity and atomic complexity of molecules increases. If this is correct, as we claim, the emergence of life is an expected phase transition in the evolving universe.

Since Newton, basic science has rested on the powerful Newtonian Paradigm. This paradigm requires a prestated and knowable phase space of all the values of the relevant variables. But living organisms are Kantian Wholes that achieve Catalytic Closure, Constraint Closure, and Spatial Closure. Stunningly, we can use no mathematics based on Set Theory – all of mathematics it seems – to deduce the ever-creative evolution of the biosphere.

Life is an expected miracle in the universe whose ways of becoming are literally numberless. The 20th Century saw the emergence of the atomic Age, the mushroom cloud, and mutually assured destruction. With Gödel, the 20th century also saw the End of Certainty, (59). In this, the first quarter of the 21st Century, we begin barely to glimpse the astonishing blossoming creativity of the biosphere of which we are members. We are, truly, Of Nature, not Above Nature.

Acknowledgments

We are truly grateful to Ingemar Ernberg, Carlos Gershenson, Wim Hordijk, Niles Lehman, and Ricard Solé for advice and comments.