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I am reading Anti-Tech Revolution by Theodore Kaczynski, in the second chapter the author talks about how technology will cause problems in society and before he begins the discussion, he introduces proposition. Here is the first:

Proposition 1. In any environment that is sufficiently rich, self propagating systems will arise, and natural selection will lead to the evolution of self-propagating systems having increasingly complex, subtle, and sophisticated means of surviving and propagating themselves.

This I felt was an additional assumption one would make after accepting the Darwinian proposition of natural selection... so I wondered.. does there exist any fleshed out sub field of biology which have axiom systems from which experimental predictions can be worked out from?

Definition of an axiomatic system:

An axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems.

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  • $\begingroup$ The proposition is wrong. In the context of explaining the origin of life, it has been shown that in some environments self-replicating molecules can arise... but we are not certain that such conditions ever existed on Earth. $\endgroup$
    – Roger V.
    Commented Jun 29, 2021 at 7:46
  • $\begingroup$ Hmm how does that make it wrong though..? It doesn't state the conditions, just leaves it to be defined somewhere elese 'sufficiently rich' $\endgroup$
    – user56920
    Commented Jun 29, 2021 at 9:10
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    $\begingroup$ Well, if we want to be mathematically rigorous, it is not "wrong", but "meaningless" without definition of what is sufficiently rich. However, I think what is implied is that such conditions are common and rather general to system composition and its other properties, while in practice there may be only one combination of chemicals and external parameters that produces self-replicators. $\endgroup$
    – Roger V.
    Commented Jun 29, 2021 at 9:25
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    $\begingroup$ I feel like it's worth noting that Ted Kaczynski is better known as "The Unibomber". He carried out a bombing campaign across the US resulting in the murder of three people. He's currently imprisoned and serving 8 life sentences. This doesn't necessarily disqualify his neo-luddite ideas, but they ought to be approached with a heavy dose of skepticism. The point of his book is not to forward ideas about mathematical biology, but to predict tech-caused apocalypse (with his solution being the incitement of societal collapse and rebirth). $\endgroup$
    – Brickman
    Commented Jul 1, 2021 at 13:54
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    $\begingroup$ I agree with the general sentiment but it's the unabomber, not the unibomber. @Brickman $\endgroup$
    – user56920
    Commented Jul 3, 2021 at 20:58

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It's hard for me to know the degree to which the Kaczynski quote is wrapped up in what you are interested in, so I will answer your question very narrowly without considering that quote:

does there exist any fleshed out sub field of biology which have axiom systems from which experimental predictions can be worked out from?

Yes, fields exist in which such axiomatic systems exist (as I understand them). Probably the best example is mathematical population genetics, which was mostly worked out before we had molecular methods, but which has been enormously useful in the interpretation and formulation of hypotheses regarding such data as it has accumulated. For a somewhat less mathematically involved introduction, see also here.

It uses Mendelian inheritance (or even simpler systems) as a set of basic assumptions, and tries to figure out what should happen in evolution across generations based on that.

I make no claims about other fields of biology, though probably there can be good arguments that such fields exist.

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In a way, all biological research is based on certain axioms, just as all scientific study is bound certain foundational principles. The proposition you shared sounds like something straight out of General Systems Theory (GST), an inherently interdisciplinary field based on the premise that complex systems share common organizing principles, and that these principles can be discovered and modeled mathematically. GST also attempts to follow certain axioms. Such as:

Every system is bounded by space and time, influenced by its environment, defined by its structure, and expressed through its functioning. A system may be more than the sum of its parts if it expresses synergy or emergent behavior... (more GST here)

And many of these these principles (or axioms, if you prefer) are universal across biology. For example, the concept of the relationship between structure and function is fundamental in some way to every biological discipline I can think of.

Systems biology is another concept related to GST that can be applied a few different ways. The most common is probably by those who simply favor a top-down research approach of complex biological systems, motivated by the knowledge that there are certain biological phenomena that cannot be fully understood or predicted through reductionist study of a system's component parts alone. Of course, top-down and bottom-up approaches are complementary, and can seldom replace one another.

There are also those who take a more GST-based approach to biology. For example, I once worked with a mathematician who took some well characterized mathematical models describing road traffic patterns and applied them to molecular genetics, ultimately generating an improved model of how RNA polymerases travel along a DNA strand during transcription. But one could also draw parallels or generalities (as your proposition does) from looking at different types of evolving systems, like biological populations, communities, nations, religions, corporations, economies, and so on.

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I am not completely sure, but I think mathematical biology, bioinformatics and/or computational biology are what you are looking for. However, the assumptions in these fields are not always fundamental "truths" like with mathematical axioms or Einsteins postulates. The issue with axiomatizing biology is that the systems that are studied, unlike algebras or fundamental forces, are complex, composite and non-homogeneous. That makes reductionistic approaches generally quite unproductive.

However, that does not mean all hope is lost. Usually, in the cases of, for instance, cancer modelling and swarming behaviour, the assumptions are simplifications that allows us to create mathematical models that can make predictions regarding those systems.

The field I think most closely resembles what you may be looking for is evolutionary game theory. This is an application of game theory where players have been replaced by populations and strategies have been replaced by genetic traits. This approach has contributed to the understanding of sexual selection, co-evolution and more.

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On the question "does there exist any fleshed out sub field of biology which have axiom systems from which experimental predictions can be worked out from?", we do have axioms for FOL elementary theory groups (Shoenfield, pg. 22) in which predictions (FOL proofs?) in a certain sub-areas of biology (e.g. genetic code) can be had, as alluded to in "In this section we review some work describing the genetic code in groupoid and group theory terms".

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