A Mathematician's Approach to Evolution Theory

this is probably your main problem.

please elaborate your own understanding of what a time 4th dimension is.

 

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Not at all. After all semigroup theory has less axioms than group theory and yet the classification of groups is far from trivial and the classification of semigroups is a much harder problem.

The basics of biological evolution are:

0) An individual organism, hereafter an "individual", has a certain potential determined by factors of its internal state, some of which are heritable. "Individuals contain heritable state."
1) With access to suitable resources for sufficient time, individuals may fork into nearly identical pairs, individuals may bud off new individuals with nearly the same potential, or pairs of individuals may recombine that which determine their potential into new individuals. "Given optimum internal and external conditions, individuals may reproduce approximately their heritable state."
2) Individuals can transition to a final state where they no longer have the potential to reproduce or consume resources for reasons internal and external. "Individuals may die"
3) Populations contain a finite number of not-dead individuals, and if there are two or more then not all of which have the same heritable internal state. "Populations exhibit variation."
4) The resources available at any locale at a particular time are finite. "Reproduction cannot continue at a constant per individual per time unit rate forever."

Real biology is obviously more complicated than this, in that it deals with the structural, organizational and biochemical basis for these assertions. But at the biochemical level, for most organisms, axioms 0, 1, 2, and 3 are explained in terms of DNA and the Central Dogma of Molecular Biology† while 5 is a consequence of the local nature of physics. A separate field of Evolutionary Development concerns the history‡ of multi-cellular life on Earth and the molecular mechanism for growth and development of the individual and how such mechanisms are used in different populations.

From 0, 1 and 3, the percentage of individuals in a population with a particular heritable factor of internal state is subject to change.

From axioms 1, 2, 3, and 4 it follows that individuals A and B are need not have the same internal state and this internal state may contribute to a relative probability ( in both the Bayesian and frequentist interpretations of probability) of successfully reproducing more than the other. From axioms 0 and 3, the factors that cause this internal state to be the better reproducer may be heritable and if so, are more likely to be represented in the new additions to the population. Therefore from 3, the predominance of heritable internal factors that favor differential reproductive success tends to increase over time in populations that don't go extinct.

From 1, populations that depend on recombination reproduction will be most vulnerable to rapid changes in the percentage of individuals in a population with a particular heritable factor of internal state and from 4, individuals whose excursions are geographically or temporally non-overlapping are not in the dynamical population. Finally, organisms which lack any recombination may still introduce variation into the population via imperfect reproduction. Thus spatial, temporal barriers and other forms of bottlenecking can cause populations to diversify far from characteristic heritable traits of the original population.

Giving names to populations is called taxonomy, but it's necessarily messy when there is enough variation so that every individual is unique. Therefore "species" is not an absolute, but relative classification relative to other populations here and in the past. Thus "Origin of Species" describes both the changing of gross revealed traits in the fossil record, and forking of populations to the point that distant contemporaneous cousins from different populations no longer share environment, habits or physical ability to create viable offspring. Like the difference between a mathematical abstraction of a branching tree and a branch or manifold of finite diameter branching into two, trying to isolate "the branching event" in a finite population is often a problematical concept, yet obviously if we have two discrete populations where once there was one, such a "speciation event" happened.

// Added in Edit:

You might want to add additional axioms, like
5) Only a finite amount of time has elapsed since the first terrestrial organism.
6) The mechanisms of reproduction are themselves controlled by heritable factors.
7) Individuals from different populations compete for resources
8) Individuals from one population may be the resources for another.
9) Dead individuals from one population may be the resources for another.
10) DNA is the only important heritable factor.

But those axioms allow you to explore certain specialized cases of evolution, not the most general mathematical model. Indeed, computer models of evolution used for problem solving "genetic algorithms" compose a finite population of heritable state vectors satisfying axioms 0,2,3, and 4 and use one or more of the reproduction strategies of 1 to evolve the population to better solutions. Sometimes the problem involves a deterministic problem and sometimes a probabilistic problem and sometimes the problem involves head-to-head competition at games.

† Thus some pundit has coined the slogan "Evolution = DNA + time"
‡ Yes, "history," aka "stamp-collecting." Because in terrestrial life the number of heritable factors to internal state are so large, imperfections in reproduction are highly non-reversible. Thus each present-day organism has an internal state dictated by a large number of contingencies over the history of life on the planet. Also, death, the transition in axiom 2 is widely considered non-reversible with the consequence for populations: "extinction is forever"

 

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Is that all the respect that you have for the man that tutored Einstein on how to derive General Relativity?

"Mathematical science is in my opinion an indivisible whole, an organism whose vitality is conditioned upon the connection of its parts." - David Hilbert.

 

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With all due apologies to Henri Poincaré, the first to geometrize spacetime, I prefer to derive special relativity with the utmost generality, step-by-step, as outlined in The Quintessence of Axiomatized Special Relativity Theory.

 

Stephen Wolfram spent 20 years of his life trying to accomplish something like this with his 'cellular automata' 'A New Kind of Science'. The result was less than expected, even by Wolfram himself. If Wolfram failed at this (who compares in mathematical prowess to Hilbert), what makes you think another such attempt would succeed?

 

New mathematics research proves there's plenty of time for evolution
http://phys.org/news/2010-12-mathematics-plenty-evolution.html



http://www.paleolibrarian.info/2012/06/how-mathematics-can-prove-evolution.html
How Mathematics Can Prove Evolution:

For those of you who do not know the name, Dr. Jason Rosenhouse, I urge you to read an article this distinguished professor of mathematics wrote in 2001. It fully refutes all those supposedly “brilliant” minds who peddle Intelligent Design and suggest it to be a) appropriate and factual, b) an accurate refutation of Darwin and his theories, c) at least an equally valid set of scientific insights missed by the great many biologists, chemists, physicists, anthropologists, astronomers and geologists who either ignore or actually refute ID.

Here is a link to some of Rosenhouse’s publicationshttp://educ.jmu.edu/~rosenhjd/Evolution.html. The one specific title is, “HowAnti-Evolutionists Abuse Mathematics.”

The attack on Darwin from the real scientific community is zero. Natural Selection really isn't a "theory" although we define that word in science differently then lay folk. Natural Selection, like gravity, is actually a law and should be considered as such by everyone. It is only those who wish to refute natural selection in favor of Intelligent Design that do not accept the processes of evolution and the organic basis of all life, which are accidental but follow natural and mathematical laws.


ID theorists, much like the creationists before them, know they will not convince scientifically knowledgeable people. Instead, they market their ideas to a public untrained in both the methods and findings of science. And all too often theirs is the only viewpoint readily available.

 

http://educ.jmu.edu/~rosenhjd/Evolution.html

Science and Pseudoscience


The methods and findings of science can seem mysterious to lay audiences. The whole enerprise is viewed with suspicion in some quarters, given the propensity of many in the media to present scientific findings as a sort of non-divine revelation. On the other hand, everyone understands that in our culture statements are true to the extent they can be described as scientific. This fact makes it easy for charlatans to coopt the language of science to promote favored political agendas. Particularly insidious is the use of pseudoscience by many on the political right to justify narrow religious dogmas. Thus, we are treated to an endless barrage of claims that new findings in cosmology or biology prove God's existence, or that materialistic science has become obsolete. These claims are silly, but when presented in suitably technical dress they can appear impressive. The support of intelligent-design theory as an alternative to Darwin's theory of evolution is the latest example. Among professionals, intelligent-design is rejected owing to the manifest falseness of its arguments, coupled with its inability to produce useful results in day-to-day scientific work. ID's political defenders do not care about such things, however, and prefer to present Darwinism as dogmatic atheism in a lab coat. Such rhetoric is useful for arousing the emotions of lay audiences, but they have no connection with reality.

Many professionals respond to pseudoscience by ignoring it. This is a mistake, if for no other reason than the fact that funding decisions are made by politicians, who must be sensitive to their constituents, many of whom take pseudoscientific ideas very seriously indeed. It is not a good thing that a phony psychic passing off simple parlor tricks as communication with the dead has one of the highest rated shows on television. Nor is it good that many learn what they know of modern science from right-wing showmen like Rush Limbaugh or Sean Hannity (who once shamelessly described Darwinism as "liberal science".) Ultimately, the only way to defeat such nonsense is to confront it vigorously.

 

no, it's simply a time frame that is not or will not occur at this moment, referring to a past or a future.

 

evolution never ceases. existence is evolution.

 

He doesn't the tread is a sham attempt to argue a subject that belongs elsewhere here in the Physics & Math forum.

 

I realize. The approach was a little different than usual for most ID folks. Mathematical or physical arguments against Darwinian evolution usually take the form of an open or closed thermodynamic systems, consideration of random mutations being unrelated to evolutionary changes, harmful to the organism, and causing more disorder than benefit in an ordered living system.

Evolution can take a VERY long time. A mathematical prediction of how long it might take to adapt organisms to do things like going from swimming to walking, walking to flying, etc would be useful and also impress, but I haven't seen anything like that.

Mathematics already exists in for analyzing genetic variations leading to Darwinian speciation, but most of it does not appear to have derived of Hilbert. Hilbert had no computers to help with such a complex analysis, so it is doubtful he would have had very much to offer, to say the least. Einstein's problem was no doubt better suited to his abilities.

 

Math is only as good as its assumptions. Math is blind in the sense that it can begin with any assumptions and still come up with equations for those assumptions. For example, in video games, the physics engines allows for assumptions like infinite lives, anti-gravity properties or extreme material durability. These are not realistic assumptions in the real world, but help make the game more fun. Even such unrealistic assumptions, does not limit a mathematician from setting up equations to model it. One has to be careful about assuming anti-gravity equations, built on a weak conceptual foundation, means that math solves reality.

Applying math to evolution means no matter what premises you use, right or wrong, math can be supplied that can model this. Math can reinforce bad science. With the example of a mathematician helping Einstein, although he may have solved the math problem, the Grand unified theory is still not considered done, because some of the assumptions were not correct, on which the math was built. Gravity is not fully connected in the model due to unknown premises about the fundamental nature of gravity. Before the math, a conceptual foundation needs to be built, to make sure we are not using a game engine.

 

Math applied to life has to take into consideration water, since observation has proven that nothing, at any level of life, down to enzymes, works properly without water, nor can anything other solvent be substituted for water and cause life to return to the organics.

This is a cornerstone premise, but is not used in the evolution game engines. They tend to fixate on organics in a vacuum, which is not even real. It would be a waste of time for math to supply good equations for unrealistic assumptions; tuxedo on a monkey.

 

Correct.

 

Possibly the finest example of fundamental mathematics applied to the study of evolution does not come from Hilbert. This is widely known as the "molecular clock" theory:

http://discovermagazine.com/2012/jul-aug/05-obscure-formulas-that-rule-world

"CLOCKING EVOLUTION

K = 2NuP
K = u

To find the last common ancestor of any two species, scientists look to the molecular clock, which describes how quickly genetic mutations accrue. According to the clock equation, the rate at which mutations become permanent (K) equals the rate at which those mutations happen (u) times the population size (N) times the probability (P) that a mutation will become permanent, times 2. When dealing with mutations that confer no selective advantage, says Michael Steiper of Hunter College in New York, “the amount of genetic difference scales to time.” In other words, each gene is equally likely to become fixed, with probability 1/(2N), producing the simplified bottom formula. Molecular clocks allow scientists to trace the ancient tree of life, even when fossil evidence is scarce, and help find the branches that led to modern humans. You are collecting mutations (data for future anthropologists) right now."

 

Rigorous proofs are possible in the field of mathematics. Concepts like evolution cannot be proven with the same logical rigor.

The best you can do for evolution is to present the facts of fossil records & point out that evolution is an excellent explanation for them. Until/unless a better explanation is provided, Darwinian evolution should/must be accepted.

An excellent example is provided by the fossils starting with eohippus & ending with the modern horse. It is difficult to imagine a better explanation than Darwinian evolution, although religious folks accept god caused/created it as a the ultimate explanation for almost every event.

BTW: Even rigorous mathematical proofs only prove that the conclusion is consistent with the axioms. If any axiom can be successfully attacked, the proof fails.

 

Well, I think Eugene Shubert did want to get in a tussle with the axioms. So I presented some.

 

You're the only one who prefers your Quintessence of Axiomatized Bullshit. Or maybe you could show us where's it's being used beyond these illiterate threads you've been initiating on the Internet.

 

True

But the mathematical science weakness is that any conclusions reached are based on logic; fine, but the logic this science conclusions are based on are limited based on the knowledge known.

 

Of course someone like you would consider a strength to be a weakness.