The uncorrected simplifications/oversimplifications of Hamilton's Rule (Re: Removing Lewontin's Fallacy From Hamilton's Rule)
- From: "John Edser" <edser@xxxxxxxxxx>
- Date: Tue, 15 Nov 2005 15:29:33 -0500 (EST)
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.
> 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. 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.
> 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. 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.
> > 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.
> > > 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.
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.
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.
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.
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.
>
> > 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.
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.
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.
> snip invalid argument from authority<
> > 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.
> > 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.
> > 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.
> > 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.
> > 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.
This is because each epistatic probability has to be multiplied because each
gene is independently assorted within one organism. 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.
Quite clearly r^e where e>1 has two major negative effects for HR:
(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.
> > 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?
Regards,
John Edser
Independent Researcher
edser@xxxxxxxxxx
.
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