Re: The uncorrected simplifications/oversimplifications of Hamilton's Rule (Re: Removing Lewontin's Fallacy From Hamilton's Rule)
- From: name_and_address_supplied@xxxxxxxxxxx
- Date: Thu, 17 Nov 2005 01:10:27 -0500 (EST)
John Edser wrote:
> name_and_address_supplied@xxxxxxxxxxx wrote:_
> > > > I understand the reasons for this mistrust. I share your opinion here.
> > > > The problem is that this derivation is straight from Price's theorem,
> > > > which says everything and nothing.
>
> > > JE:-
> > > Price's theorem is not empirically based, i.e. while it is a valid
> > > proposition of mathematics it is not a valid proposition of science.
>
> > There are two issues here. Firstly, HR is a mathematically true
> > statement. It doesn't need to be empirically verified because it is
> > true by definition.
>
> JE:-
> There are no axiomatic truths within the sciences only within mathematics.
> The sciences are required to be empirically based but mathematics is not. HR
> as a proposition of science must be able to be verified or refuted within
> nature. It is not sufficient for HR to just remain "true by definition" i.e.
> only tautologically true like an axiom of mathematics because HR was
> created, employed and used to this day, to _account_ for the evolution of
> altruism (organism fitness altruism) within nature as supposedly, a valid
> theory of the biological sciences. A tautology is just a circular argument
> that is only true by definition so obviously it cannot be used in its own
> right as a theory of nature. Yet, over the past few months we have witnessed
> Felsenstein's derivation of HR from just tautological premises and his
> subsequent proof of its axiomatic truth when employing the Hardy Weinberg
> binomial expansion (in which all gene fitness epistasis remains deleted) of
> Hamilton's tautology for the diploid case. The conclusion is that no matter
> what you do HR always works where Felsenstein et al really think this must
> be a good thing. It isn't, it is a bad thing. How can an honored Professor
> of evolutionary theory be so incredibly naive? Again I would ask you or any
> other reader of integrity to please revisit Felsenstein's premises and
> ascertain for yourselves if they are just tautological and then make your
> findings known to sbe readers. So far this rather obvious request has gone
> unheeded.
I said there were two issues. On the one hand, Hamilton's rule is a
mathematical truism. On the other hand, there is what we can call
"Hamilton's hypothesis", which suggests that organisms should behave as
if they were rational agents pursuing the maximization of their
inclusive fitness. This is empirically testable -- i.e. falsifiable.
> > Secondly, the motivation for developing HR was to
> > have a predictive tool for social evolution. So there is an empirical
> > test we can apply, and that is to see if organisms behave as if they
> > were maximizing their inclusive fitness.
> > They might not be, so this
> > hypothesis is falsifiable.
>
> JE:-
> Please provide this test.
There are many possible tests. A good deal of effort is made by
evolutionary biologists to go out and gather data and examine the
goodness of fit for Hamilton's hypothesis. Rather than whining about
tautologies and Godel.
For example: Hamilton (1967, Science) used a kin selection approach to
determine what sex ratio should be employed by rational agents aiming
to maximise their inclusive fitness under local mate competition. His
prediction is that the unbeatable strategy for the sex ratio is
(N-1)/2N where N is the number of unrelated 'foundress' females
contributing offspring to the mating group. There has been a great deal
of interest in testing this theory, and lots of data has been gathered.
For example, fig wasps provide exactly the right sort of lifehistory,
and are readily amenable for testing the hypothesis. The fit between
empirical observation and theoretical prediction is very impressive. It
has been cited as among the best evidence for adaptive evolution in
nature, for example Frank (2003, Evolution).
> I put it to all sbe readers that no test to
> refutation can exist for HR because it was and remains a circular argument,
> i.e. a tautology improperly offered as a valid theory of nature in its own
> right. As I have previously pointed out the four conditional c to b
> propositions that Hamilton et al employ to explain WHY Hamilton's allele
> appears to spread on just a 100% relative basis are not biologically valid
> because all four of them fuse c to b to make just the one selectee removing
> any possible competition and selection. Only two unconditional WHY
> propositions actually exist which alone preserve the integrity of Hamilton's
> only two contesting selectees: the group centric b count and the organism
> centric c count, allowing a minimum of competition and thus selection. These
> are: unconditional altruism (any positive c) and unconditional selfishness
> (any negative c). When these are employed as the only valid WHY propositions
> in HR it remains impossible to separate them because rb>c remains
> mathematically equal to -rb<-c. All of these events were and remain entirely
> consistent with the proposition that HR was and remains, an empty tautology
> misused as a valid theory of nature for nearly 50 years.
HR can be used to make predictions. These predictions can be tested.
What's wrong with that?
> > Actually, I am more interested in a
> > quantitative test -- how good is the hypothesis that organisms should
> > behave as if maximizing their inclusive fitness at explaining our
> > empirical observations? Specifically, how much of the variance does
> > this hypothesis explain? What other hypotheses might we employ to
> > explain a greater proportion of the variance?
>
> JE:-
> Inclusive fitness cannot be a maximand because it is just a relative measure
> which means it is only a comparison of minimum of two events without any
> critical point of reference.
Perhaps -- but try telling that to the fig wasps!
> Because this is the case you cannot tell a
> maximand from a minimand when only using a relative fitness where these
> represent contrary self exclusive propositions of science. OTOH Total
> Darwinian Fitness (TDF) is not just a relative measure it is a refutable
> absolute assumption of nature, i.e. not just an absolute dictate. Like
> Einstein's c it is always maximized within Darwinism. Like the maximand c
> within M=Mc^2 where the velocity of light always travels as fast as possible
> with a maximum velocity of c TDF is always as large as possible so it cannot
> be selected to be reduced. The TDF ceiling is limited by the efficiency of
> each INDEPENDENT selectee so unlike c, the biological ceiling to TDF remains
> UNIQUE to each unit of selection.
>
> > > Converting sense in mathematics into sense within the sciences is quite
> > > simple in principle if not in practice: define within the mathematical
> > > proposition at least one constant algebraic term which can be verified
> > or
> > > refuted within nature.
>
> > Is this Edser's Theorem? Please provide some support for this
> > assertion.
>
> JE:-
> I have previously provided detailed examples on many different occasions.
> The general structure of my argument was and remains:
>
> If A varies with B (which is only tautological) THEN:
>
> 1) A = k+B as an additive.
> 2) A = kB as a non additive.
>
> The constant k can convert the tautology "A varies with B" into an
> empirically based testable proposition of science, if and only if, k
> represents an empirically based constant term. Quire clearly, if k is just
> another variable then "anything goes" which is OK for mathematics but not
> science. Arithmetic such as 2 + 2 = 4 only represents a valid tautological
> proposition of mathematics. It can be converted into a valid proposition of
> the sciences when it is multiplied by the constant "a" providing 2a + 2a =
> 4a which could represent the verified or refuted empirical proposition: "two
> apples plus two apples equals four apples". The problem has always been that
> the critical conversion of non empirically based axioms to empirical based
> processes is so obvious that it is mostly done automatically without any
> thought given over to the process. This has led to major errors within
> evolutionary theory as mathematicians confuse mathematics with biological
> science.
Oh John, that's pathetic.
HR is not a theorem stating how cost (c) and benefit (b) of a social
act relate to eachother. HR allows the costs and benefits to be varied
independently -- and the relatedness (r) too, for that matter. What HR
does do is tell us whether, given a particular combination of r, b and
c, the social act is favoured by selection. It tells us that the act is
favoured when r b > c.
> > > Hamilton's Rule was and remains an invalid
> > > proposition of science even if it is mathematically valid unless at
> > least
> > > one constant algebraic term is included on just one of the rule.
> > > > If we want to show that Hamilton's
> > > > rule is a general principle, this is the way we have to go. But if we
> > > > want to say anything about causality we need an explicit model, and
> > > > with that disappears our generality.
>
> > > JE:-
> > > Yes, Popper would agree with you because the most general of statements
> > are
> > > just epistemological perpetual motion machines, i.e. tautologies where
> > > causality remains 100% reversible. So far, Hamilton's Rule has not been
> > > supplied with any empirically based fitness limits which are absolutely
> > > required to break Hamilton's fitness tautology. Like Price's
> > mathematics, HR
> > > does not represent a valid proposition of science.
>
> > See above.
>
> JE:-
> See above.
>
>
> > Also, natural selection does not care about causation, only
> > correlation. A trait that is correlated with high fitness is favored,
> > whether it caused that high fitness or not.
>
> JE:-
> Incorrect. Natural selection was and remains, an entirely refutable non
> random process. Therefore much more than just a correlation has always
> existed between Darwinian theory and it's proposed effects e.g. the effect
> of non random patterns in gene freq. changes within one population. When
> Hamilton reduced Darwinian theory to just a tautology cause and effect
> become reversed within a heuristic gene centric argument.
Natural selection does not care about causation, only correlation. A
trait that is correlated with high fitness is favored, whether it
caused that high fitness or not.
> > > > So, learn to love the 1970
> > > > derivation, whilst keeping in mind exactly what it is for, and what it
> > > > is not for.
>
> > > JE:-
> > > That is why you have to love what Hamilton appears to be attempting but
> > > utterly hate what it has subsequently become. The errors within the rule
> > are
> > > and remain, gross. Hamilton et al show almost no understanding of just
> > > evolutionary theory basics: conserving levels of selection and correctly
> > > converting one level into another so they can be compared. Hamilton's
> > > mathematics treats its delicate biological levels of selection as if
> > they
> > > are just mince meat. A famous biochemist once joked: "don't worry about
> > the
> > > organism concept, they all look the same under the Wareing Blender....
>
> > Do you have any evidence to back up all this nonsense? Also, don't you
> > understand that kin selection approaches are mathematically equivalent
> > to levels of selection approaches?
>
> JE:-
> What is "nonsense" here is the amazing number of times I am required to
> repeat my arguments...
>
> Within HR the levels are not preserved or even correctly converted. Two
> uncorrected simplifications and one uncorrected oversimplifications remain
> to be corrected.
Show me that they are not correctly converted. Show me that 'putting
the population in Hamilton's blender', as it were, gives incorrect
predictions. Consider that perhaps they are correctly handled and
converted, and that you just can't follow the maths.
> Hamilton's Uncorrected Oversimplification of Darwinism:
>
> a) Neither rb>c or -rb<-c refers explicitly or even just implicitly
> to a minimum of one constant algebraic term so both remain oversimplified.
> This means the rule is reduced to be just an empty tautology where cause and
> effect are entirely reversible ensuring that unconditional altruism (any
> positive c) cannot be differentiated from unconditional selfishness (any
> negative c). This reduces Hamilton's proposition to just the useless
> statement that the reason Hamilton's allele appears to spread on just a 100%
> relative basis is not at all understood because in all cases, without
> exception, the relative increase may be caused by EITHER unconditional
> selfishness or unconditional altruism where the rule cannot tell which is
> operating in any one particular case.
HR does not attempt to predict, for example, b given r and c. What it
does do is tell us, given r, b and c, whether the social act described
by those values is favoured by selection. So all you have said above is
irrelevant.
> Hamilton's Uncorrected Simplifications of Darwinism:
>
> b) Because the one proactive actor is dealing in IBD proxy gene
> replications over organism generations conversion of Hamilton's group
> centric b fitness into a valid heuristic gene centric fitness within which
> gene replicate is only counted over organism generations and not _organism
> group_ generations, b must be divided by n which is the number of recipients
> that deal in the IBD genes for the actor. To convert b/n to this gene
> centric heuristic fitness it must be multiplied by r. The same must be done
> to c where n'= 1 (the number of actors which is defined as 1) and r' = 1
> (the relatedness of the actor to just itself is always 1). When this
> conversion process is done CORRECTLY the rule fails because it HR is rb/n >
> c. Please refer to my answer of your criticism of this argument below.
Sorry, I don't follow that at all. Is anyone else following this?
Perhaps someone could explain to me what he is on about?
> c) All gene fitness epistasis within HR was and remains, deleted,
> i.e. relatedness is has always been r^e and not just r within the
> _biological sciences_. Hamilton's deletion of e by just fixing it to 1 and
> then just forgetting all about it conveniently removed the following
> obstacles:
>
> i) When e>1 it prohibited anything but random
> mating, i.e. genome matching for the same altruistic alleles (Dawkins' Green
> Beard hypothesis) is prohibited simply because it costs more than it is
> worth since the cost of e increases geometrically as the gains from genome
> matching only increase arithmetically.
> ii) As e increases the probability that inclusive
> fitness can even start is reduced geometrically.
HR already allows for epistasis. True, HR was first derived
heuristically from a simple one-locus model. Here, epistasis wasn't
deleted, it was simply irrelevant. However, Hamilton (1970) provided
the general definition -- this allows for epistasis, and epistasis was
not deleted.
> Please refer my answer to your criticisms of these arguments below.
?
> > > Three major errors exist:
> > >
> > > 1)The rule remains a 100% relative and therefore just a tautological
> > > proposition of mathematics. Therefore none of the 4 _conditional_ WHY
> > > propositions, i.e. why Hamilton's allele appears to spread on just a
> > 100%
> > > relative basis when rb>c or -rb<-c which are:
> > >
> > > i) Altruism (positive c conditional to positive b)
> > > ii) Selfishness (negative conditional to a negative b)
> > > iii) Mutualism (negative c conditional to a positive b)
> > > iv) Spite (positive c conditional to a negative b)
> > >
> > > remain empirically valid. This is because, in every single case, the
> > > separation of the only two empirically based independent selectees that
> > > actually exist within HR which are:
> > >
> > > a) The one group centric b
> > >
> > > b) The one organism centric c
> > >
> > > become destroyed by the wareing blender of population genetics leaving
> > just
> > > the one group centric selectee soup produced when c is merged with b as
> > c
> > > takes all fitness responsibility away from b. Unless at least two
> > > independent selectee's remain defined (which means c cannot take any
> > fitness
> > > responsibility for b), the rule is not even a valid minimal proposition
> > of
> > > evolutionary theory. To satisfy this requirement at least two
> > independent
> > > selectees must be _defined_ and _remain conserved_ at all times to allow
> > a
> > > minimum of valid Darwinian competition.
> >
> > Demonstrate that this 'blending' leads to incorrect, i.e. empirically
> > indefensible, predictions.
> >
I notice you didn't respond to this. For one who whines continually
about established thought lacking empirical support (even when there
is an abundance of empirical support!), you are strangely loathe to
apply this thinking to your own random assertions.
> > > The only two unconditional WHY propositions which alone CAN conserve
> > > Hamilton's defined two selectees and maintain them throughout are:
> > >
> > > v) Unconditional Selfishness(any negative c).
> > > vii) Unconditional Altruism (any positive c).
> > >
> > > However, because the rule remains just a tautology, it becomes
> > impossible to
> > > tell them apart simply because rb>c remains mathematically equivalent to
> > > -rb<-c. This means the best the rule can offer is that while it can
> > measure
> > > when rb>c and can argue that one allele appears to increase relative to
> > > another it cannot say why this is the case because it cannot tell if
> > this is
> > > happening via unconditional selfishness or unconditional altruism.
> > > Note: unconditional mutualism is not even represented.
>
> > I'm not going to argue any more about this reflected rule nonsense. All
> > you have managed to demonstrate is your unfamiliarity with the concept
> > of negative numbers.
>
> JE:-
> You are NOT correct. Your stubborn refusal to argue this to a conclusion
> provides evidence of a lack of integrity on your part. I am willing to
> debate this point for as long as it takes to prove one of us wrong. What you
> cannot/refuse to understand is that reversing c to a negative c and rb to a
> negative rb by multiplying rb>c by -1 is *NOT* equivalent to not doing so
> and leaving HR as rb>c simply because rb and c remain *INDEPENDENT* and
> *NOT* *DEPENDENT* events.
John. You are on your own here, and you know it. I know, I know: even
in a minority of one the truth is still the truth and all that. But
we're fed up of this rant; you are not convincing anyone -- so please
stop.
> Example:
> If you have $10 independent of my $10, if we both lose our $10 then neither
> of us have $10 which means what he have remains the same on just a relative
> basis $0 = $0 but not on an absolute basis because $0 does not equal $10.
> OTOH if our money was dependently linked (as rb and c are within the 4 WHY
> propositions of HR) then when I lose my $10 you must gain it and vice versa.
> Multiplying this association by -1 validates your argument because it is
> entirely based on just the one independent event. This is why I provided an
> example using 3 independent events: you, me and a bank. The bank was
> necessary to prove that when we both lost our $10 it actually went somewhere
> else entirely which by definition was to the bank. The point about
> accountancy is that all movements of money are relative to MORE than just
> the one account holder where this is also true within fitness accounting
> within HR. Because you confused dependency with independency you ended up
> employing the Enron accounting technique which argued, just like you argued,
> that a negative credit can somehow remain a credit in the relative sense
> allowing you to write up a debt as equity which is an absurdity.
The link between your example and the logic of HR is obscure. If a bank
is needed, the bank can be included as a separate recipient in HR.
There is your point of reference. You seem to think that Hamilton
forgot all about the rest of the population in his models -- perhaps
you should go back and read his papers.
> Two INDEPENDENT account holders exist within HR and not just the one
> DEPENDENT account holder. These are:
>
> 1) Just the ONE b GROUP centric account holder.
>
> 2) Just the ONE c ORGANISM centric account holder.
>
> Because Hamilton's actor is dealing in genes IBD reproduced by proxy by n
> Darwinian recipients the group centric b must be must be CORRECTLY converted
> into an organism centric c just to be able to compare b "apples" with c
> "oranges". Only if this is done it can either b or c be converted into a
> heuristic gene centric fitness in which gene replicates are counted over
> organism generations.
You are using some sort of confused hybrid between a
levels-of-selection approach and a kin selection approach. Both are
equivalent, but you can't switch between them midway though a
calculation. Or at least, you will risk getting the wrong answer if you
do.
Go away and read about how a kin selection analysis is really done.
Then come back and astound us with your fantastic insight.
> > snip invalid argument from authority<
Learn some netiquette, John.
> > > 2) The deletion of n (the number of recipients)from the rule.
> > >
> > > Because Hamilton et al VERY STRICTLY define gene centricity to be the
> > number
> > > of genes replicated over organism generations (organism centric gene
> > > replications)
>
> > I would define "gene centricity" in terms of "the approach whereby we
> > focus on the gene as the unit of selection", rather than it being some
> > count of number of genes . . . and I believe Hamilton, who was a native
> > English speaker, would have followed me in this definition.
>
> JE:-
> But to be able use "the approach whereby we focus on the gene as the unit of
> selection" as a _valid_ inclusive FITNESS, genes as independent units of
> selection must be COUNTED hence the requirement of being able to identify,
> even as just an IBD probability, which gene was replicated from which
> supposed independent gene parent. HOW are genes defined to be counted within
> HR? Very strictly over organism generations, i.e. you do not count genes
> replicated over gene generations e.g. each cell mitosis or organism group
> generations (the time required to reproduce one group from a parent group)
> you only count genes over _organism_ generations. Because not a single
> independent gene fitness exists in nature counting genes over organism
> generations is just a heuristic exercise, i.e. an oversimplification of
> empirically based Darwinian theory.
Actually, HR refers to the impact of social behaviours on
*individuals'* fitnesses.
> > > and NOT organism group generations (organism group centric
> > > gene replications) then the group centric b must be converted to
> > organism
> > > centric by dividing b by n which is then converted to gene centric by
> > > multiplying b/n by r producing rb/n.
>
> > Not sure why you are doing this. So, taking an average, you have
> > suggested that each of the n recipients gets a benefit b/n, and we
> > multiply by their relatedness (assuming all are equally related, r) to
> > get rb/n as the inclusive fitness benefit accrued from each recipient.
> > But note, there are n of these recipients, so the total inclusive
> > fitness benefit is n * rb/n = rb.
>
> JE:-
> Yes but the total inclusive fitness benefit becomes a remade group centric
> benefit which counts the extra genes replicated by proxy over _organism
> group_ generations and not _organism_ generations. Woodgold made the same
> error. You cannot validly compare a gene centric count of Hamilton's gene
> replicated over organism group generations with a count of the same gene
> over organism generations because they are entirely different fitness types.
> The point is b cannot be validly compared to c unless they both count genes
> replicated over organism generations. To be able to do this b must be
> divided by n.
Again, you are hopelessly conflating and confusing levels of selection
approaches with kin selection approaches. Done separately, they will
both give you the same answer. But you can't muddle your fitness
accounting in this way and expect to get the right answer.
> > > Exactly the same process must be done
> > > with c where it is divided by n' (the number of actors which is defined
> > to
> > > be 1) and then multiplied by r' which is just the relatedness of the
> > actor
> > > to itself, which of course is 1, so r'c/n' = 1.
>
>
> > if n' = 1 and r' = 1 then r'c/n' = c. This is only equal to 1 when c =
> > 1.
>
> JE:-
> Yes, thanks for the correction. What I meant to write was because r'=1 and
> n'= 1 they have both been ignored and just forgotten. If c>1 then the
> significance to the actors of dividing c by n' becomes obvious.
Uh -- how does dividing by 1 change anything?
> > > It is interesting to not
> > > that if you expand the number of actors to n'> 1 that this logic becomes
> > > much more immediate.
>
> > We are calculating the inclusive fitness for an individual, with that
> > individual defined to be the actor. There is only ever one actor in any
> > inclusive fitness calculation.
>
> JE:-
> Yes, but that one actor is dealing in IBD related genes replicated by proxy
> within just the one group centric recipient which is comprised of n separate
> Darwinian individuals each with their own TDF. The Darwinian proactive actor
> is given two choices: EITHER use x resources itself to normally reproduce c
> organisms OR hand over x resources and allow n recipients as a whole to
> increase their grouped organism fitness by exactly b organisms reducing the
> actor's organism reproduction by exactly c organisms. Donating is argued to
> be the better choice for the proactive actor whenever rb>c. What is missing
> is that it is NOT the better choice if Hamilton's allele has to wait to be
> replicated over organism group generations using rb which is just the gene
> centric gain measured to one group as one unit of selection. The group
> centric b is required to be correctly converted to organism centric since
> genes are only counted within HR over organism generations. The fact remains
> that Hamilton et al never CORRECTLY converted the group centric b to a
> heuristic gene centric measure because they deleted n.
Again, I simply don't follow this line of reasoning.
Also, I am losing track of the extra components we need to squeeze into
HR in order to make it work, according to Mr Edser. So we have "e". And
wasn't there a K or Kmax? And now there is an in.
Why don't I have a go at this 'logic'. Let me see . . .
Okay! Hamilton got terribly confused and forgot most organisms are
diploid. So lets include a ploidy parameter p into Hamilton's rule.
What do we have so far?
Kmax r^e b/n > c
Where should I add in my flashy new parameter. Mmmm . . . how about . .
.. .
Kmax r^e b/n > c^p
here? No, that isn't terribly aesthetically pleasing. I want to raise
the bar, raise it to a new level. Ah ha!
Kmax r^e^p b/n > c
Lovely. I think you will find that that makes all the difference.
> > > 3) The deletion of the variable e within (r^e)b (e = gene fitness
> > epistasis)
> > > by fixing e=1 and just forgetting all about it.
>
> > Out of interest, I have never seen a derivation of this HR with
> > epistasis. Please define e, and properly -- simply calling it gene
> > fitness epistasis is not helpful. And then show me how you move from
> > that definition to a HR where e appears as an exponent of r.
>
> JE:-
> If you assume two alleles at two different loci and not just one allele at
> one locus are required to code for inclusive fitness behavior where each
> allele remains unlinked (on a separate chromosome) then the probability that
> two alleles at two different loci, which are now defined to be required to
> code for just the same inclusive fitness behavior becomes r^e where e = 2.
I see, a two-locus model.
That's not terribly general, John. What about if there is a third
locus? What about the epistasis between this and each of the other two
loci? What about the effect that this third locus has on the epistatic
interaction between the other two loci.
So, back you go, and I don't want to hear from you again until you've
provided the new and improved three-locus Hamilton's rule.
> This is because each epistatic probability has to be multiplied because each
> gene is independently assorted within one organism.
Do you even know what you are saying???
> In the case where
> recipients are related 0.25 and e=2 then relatedness becomes geometrically
> reduced to just 0.0625.
> Since not a single independent gene fitness actually
> exists empirically, after you heuristically define all genes to have an
> independent fitness as Hamilton et al have done, the minimum you can validly
> assume e to be is the haploid chromosome count of the species you are using.
> This remains the case until gene fitness independence becomes empirically
> documented.
HR does not require that gene fitnesses are independent.
> Quite clearly r^e where e>1 has two major negative effects for HR:
Since r^e does not appear in HR, I highly doubt it.
Perhaps you should refer to Edser's rule from now on.
> (a) Forces HR to only allow random mating because genome matching
> becomes out of the question as e increases since the e increase is geometric
> compared to the arithmetic gains from genome matching.
>
> (b) The variable e becomes a useful index of the ability of HR to
> start. The higher e is the less likely it is that inclusive fitness can
> start because the probability that a recipient organism has both genes
> located on two different chromosomes is r^e. Note that if you attempt to get
> around this by assuming the two alleles exist next to each other on the same
> chromosome so they are mostly always inherited together then the chances
> that these two genes mutated next to each other and not apart from each
> other becomes less probable so again, HR cannot start as e increases.
> Because HR is just a tautology "what you gain on the swings you have to lose
> on the roundabouts" otherwise you just end up "getting something for
> nothing". While free lunches are OK within mathematics they are not OK
> within the biological sciences. O'Hara's claim that mutualised expected
> fitnesses cost nothing is utterly wrong as is Felsenstein's proposition that
> the cost of substitution within Haldane's Dilemma can be cost free. Walter
> ReMine's paper on this subject was rejected by Felsenstein e al which IMHO
> opinion constitutes a scandal.
<Yawn>
> > > This act expands Hamilton's "allele" to be 1 haploid chromosome number.
> > > Because HR remains just a tautology, one gene can be expanded or
> > contacted
> > > to anything you like because no biological exceptions can exist when
> > > misusing a tautology as an empirically based theory of nature in its own
> > > right. Whenever e>1 the ability of any "green beard" propositions ceases
> > > because the costs > gains, NO EXCEPTIONS. This condemns the rule to
> > random
> > > mating in all cases. Because HR remains circular no escape from random
> > > mating can ever exist. If you attempt to get around any cost benefit
> > ratio
> > > barrier defining say, the epistatic alleles to be very closer together
> > and
> > > not farther apart so they are more often inherited together then the
> > lesser
> > > the chances that this could happen by random mutation, so you just end
> > up
> > > paying THIS way. ALWAYS, a tautology provides a "what you gain on the
> > swings
> > > you must lose on the roundabouts" argument because you cannot get
> > something
> > > for nothing. Thus e > 1 prohibits all non random mating while also
> > providing
> > > a useful index to how hard it is to start the inclusive fitness process
> > from
> > > scratch. As e increases, the probability that inclusive fitness can even
> > > start geometrically diminishes to zero.
>
> > Gibberish, at least until you define e, and provide the derivation I
> > asked for.
>
> JE:-
> Please see above.
>
> Can we attempt to keep this discourse polite?
Sorry, I take that back. You provideed the definition and the
derivation, and it is STILL gibberish.
.
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