Xerxes

No, this is not the definition of fixation. If I don't know what s, N and P are, how can I comment on what follows? Define terms. As above, this is not what fixation means! Ah. Then I'm done with you my friend. Have a nice day.

Here is my original comment, the one to which you took objection. To which I can only say, if it is not true in mathematics/logic, it's not true in "science"

I doubt this is true. I gave examples earlier of neutral alleles at isozyme and DNA polymorphism loci. Their appearance in a population over time is in no way related to selection (natural selection, that is) In what sense? Chance has ...poor odds? I don't follow. Anyway, as above, we know from evidence that loads and loads of neutral mutations are fixed in effectively H-W populations. How to account for it? Having claimed the right to use my own (defined) terminology, I should be slow to criticize here. But you are getting yourself into a sematic quicksand. Say I am an architect: my job is to "design" buildings. The first building I put up falls down the next day. Was it fit for purpose? No. Was it designed? Yes, but poorly, but it was still designed! Conversely, if I am not an architect, I might well be able, by pure chance i.e. without design, to erect a buildilng that lasts for centuaries. I fully endorse this. Once again, I merely note that selection is only one of many mechanisms by which allele frequencies change over time. This, all along, has been my only, very minor, point. Look, if it means so much to you, or anybody else, I am happy to do that. I am not bogged down with particular terminogies, I am quite flexible here. It's no big deal.

No, this is not right, whoever states it. Directional?????? I have no idea what that means. Pray tell. I have no idea what Mayr thinks he means by "changes in diversity". Do you? Maybe change in allele frequency? Hm. Adaptation is, I remind you once again, is an a posteriori judgement. Good heavens! And what the **** does hereditary similarity mean? It is meaningless. Hey! We call it the change in allele frequence over time! Ha! Every time I see that "I" word I want to throw up. Sorry, but I seriously doubt your source here.

OK, I refuse to argue the finer points of semantics. If you cannot see "useful" embedded if either of these definitions, then fine. I can. But it really isn't substanttive. Good - H-W can be "reversed" in the sense that if their equilibrium doesn't apply, then something else must be going on. But what? Yay, not only drift! H-W is a theorem in population genetics, all sorts of effects can lead to deviation form H-W equilbrium. I invite you to show me the "mathematical tests" that distinguish the various population effects that lead to lack of equilibrium in this sense. I cannot bring any to mind, maybe I'm being slow here. Look, the statement "sqrt(x) is an integer" is true when x = 4. This statement is false for almost all other integer values of x. So the statement cannot be generalized, and I say that sqrt(x) = integer is false in general. Do we really care about this issue? Are we trying to score points (I hope not). I cannot understand the problem (but if you do have one, I am quite willing to use different terminology, it's a trivial thing).

OK, "drives" is not the best word. I merely meant that selection is not the only known mechanism by which evolution (aka change in allele frequency over time) can occur. Yikes! What on earth can you mean by designs? Assuming you're not an adherent of ID, let me try and guess. "An organism is designed to fit its environment perfectly by natural selection" Something along those lines, perhaps? If this a wrong summary of your position, please say. Yes, it is true in general that natural selection operates to match phenotype to environment as closely as possible. This would be an asymmtote if environments were stable, which they are not in general. But allele frequecies change regardless of this "pressure" True, but as I said in an earlier post, if I a) admit I am slightly abusing terminology and b) am willing to explain what I mean, I cannot see the problem. Terminology is, after all, entirely abtitrary. Of course that is the historical definition. But surely you can see that that any barrier that prevents sub-populations from interbreeding leads to precisely the same outcome, so there is no need for multiple definitions. Oh by the way, I note that Mayr's classification uniformly uses (or pre-supposes) some form of isolation. Personally I do not see a functional distinction between them. P.S. I note you have several other posts prior to the one that I just responded to. Let me get to them.

Yup, we all agreed that as a definition of evolution. Then you are insisting that neutral alleles are not maintained at "polymorphic" levels? We know that they are. This is a change in allele frequency, but you are right in a sense you don't possibly realize (aren't I cheeky!). It is not (necessarily) a continuous change (I can show you the calculus if you want) Use the word "improvement" again, and I'll find out where you live and come round and eat your goldfish. Anyway, to Mokele and Dak: We don't really disagree in substance, merely language, which is not that interesting. I have a proposition to put to you clever guys, which you might well find controversial. We'll see. Meantime, Saturday night, I'm going out to get hammered. Care to join me?

Then you and I have different notions of what a priori means. To me, if a venom kills more efficiently, then it is more potent. This is an a posteriori argument. Id est, observe the effect of the venom and conclude it is more potent. Anyway, this is getting silly - look at the conclusion of my respone to Dak, folowing.....

Dak I am sorry to say this, as we had what I thought of as a good rapport on poor old WiSci, but.... This is precisely the sort of loose language I was originally objecting to! Define "advantageous". You cannot do it a priori, but only by looking at the way in which your "traits" are propagated over generations. This in no way implies selection, for the reasons I gave. Of course selection is an agent of evolution, but it is by no means the only one. As to what you mean by "neutral evolution" I can only guess. Not heading towards some perfect future, perhaps? Maybe you meant neutral variation? But we know it exists stably as all sorts of polymorphisms on all sorts of sexually reproducing organisms.

Neither am I any longer! Ah yes, that was it. I do not like, on purely theoretical grounds, the bland assumption that only selection drives evolution. You see, I happen not to agree. But that's cool (no doubt you don't find cricket as interesting as I do?) Yes, well more literally, different father(land). But I use it, I don't think too abusively, to refer to any situation that restricts interbreeding. Not all of these, of course, are geographic. And I would use exactly the same definitions. Maybe I started off being too snooty, for which I apologize. I'm afraid I am a bit of a pedant in these matters. Ho hum.

Yes Forgive my editing your quote (bold and italic), but this is an important point. It's not "obviously" true, but an assumption. But that's OK, as it is a premise, in your words. Malthus? Not, I think, a reliable source, but again a valid premise. Well, I don't like that way of putting it, but fine. How does this imply natural selection? Look, I know this thread is supposed to be about natural selection, so maybe I'm spoiling the party. But it is an important point. Evolution was once described by a man whose name I always have trouble spelling, as the change in allele frequency over time. Which is roughly what you were saying. But....we know of at least three classes of mechanism that explains this phenomenon: natural selection, drift and allopatry. So at least two out of three evolutionary engines are nothing to do with selection. There was in the 1960's, I think, a heated debate as to whether selectively neutral mutations could ever become fixed in a population. We now know (isozyme polymorhisms, DNA sequence polymorphisms in non-coding regions) that they can and do become fixed, and this is therefore evolution in the absence of selection. Oh, and by the way. If others don't like my use of terminology, that's fine with me. However, I reserve my right to use it as long as I am willing to explain it when asked (which I always am).

You are obviously getting cross, but then.... Then give me a definition of "useful" that doesn't entail the definition of "do better" (or v.v.) A syllogism is a logical chain, as you put it, which admits of no possible exceptions. Empirical data, of course, always do. I fail to see what H-W has to do with it. Are you suggesting, perhaps, that as all alleles at a locus are not in H-W equilibrium, selection must have occurred? We know this to be "false in general". Which brings me to this extremely uninteresting point. Sorry, but that is the way I and my mathematically inclined colleagues use the term "in general". You don't need to like it, but that's how it is. x is true in general if there are no exceptions ("always" is not usually used as it implies a time dimension). x is not true in general means that x is not always true. x not-true = x false. How about this, as a slightly less jargonistic compromise? The statement that x is true is not true in general, therefore the statement that x is true in general is false i.e. the statement that x is false in general is true? Any better?

Please don't lecture me on evolutionary genetics. What you say is manifestly true (by observation), but, logically, "useful" is predicated on "do better". It's not a syllogism (but it's still true, empirically) You don't need a source, it's elementary. x is false = x is not true. x is "false in general" means that, whenever x might be true, it may sometimes not be true, what's so radical?

Whatever you say.

No shit!! Look, I'm not going to argue philosophy, but the major premise is observed, the minor is assumed; it is not logically self-evident that this is true, unless one looks at the "conclusion first. Anyway, drop it, it's not important. Then you were confused by my slightly mathematical terminology. One says that x is false in general if one can't find a proof that rigourously exludes not-x. And proofs do not, in general, come from data, these are merely probabilities. But, if you conclude from the data that there are exceptions to x, you are entitled to say "not true (i.e. false) in general" (As it happens, Lewontin is my hero, so I would agree with him, wouldn't I?)

luca I agree with almost everything you said. I am an evolutionary geneticist by training (ain't I grand!), so why wouldn't I? I have, however, constantly to take issue with students who think that a gene "is" DNA, and that genetics is somehow the study of DNA. You have shown that you don't think so - bravo! Where I slightly disagree with you, though it's not a point I would want to press, is that Darwinian theory is syllologistic in the usual sense of the term. I don't know if philosophers would recognize the word "antisyllogism", but that's really what it is: we see the survivors, we know there's a stuggle, we assume the survors are the fittest. The mistake, in my opnion, is that made by people like Des Morris, who went that one step too far: if it exists today, it must have been selected for, which we now know to be false in general

Ya, well you need to be a bit careful about terminology here. A linear operator T acts upon elements of a vector space V, and where T: V-->V it is referred to as a linear transformation. But if T:V-->W, with V and W being different spaces, then T is an operator, not a transformation. But yes, if V and W are of the same dimension, the answer to your question is yes. But they may not be of the same dimension! With that qualification, we can ask: as the maximum number of linearly independent vectors in a vector space V of dimension n is n, is there an operator T:V-->W where W is of dimension n - 1? The answer is yes, there are such operators, so that the n linearly independent vectors in V may not be linearly independent in W under T (because W only admits of n - 1 linearly independent vectors). However, you asked a different question. If the vectors w in W with dim W = n are independent, are the v in V independent if T:V-->W and dim V = n - 1. In general, I suppose, the answer is yes, except that, whereas I can think up an operator which takes an n-space to an n-1-space, I cannot think of one that goes the other way.

Yes that's exactly what I meant! Was there any ambiguity in not bothering with LaTeX? As to substance......?

Say I have a set X and a topology T on X = {X {} A} i.e A is an open subset of T. Then the complement of A is X - A = Ac, which is closed. Now the interior of A, int(A) is the largest open set (or the union of all open sets) contained in A which is A, and the closure of A, cl(A) is the smallest closed set in {X {} Ac} containing A which is X. So if bd(A) = cl(A) - int(A), we have that bd(A) = X - A = Ac. Similarly, the closure of Ac is the smallest closed set containing Ac, which is Ac = X - A. So, by an alternative definition of the boundary of A, cl(A) intersect cl(Ac) = X intersect (X - A) which is X - A = Ac. I've tried it out on a number of arbitrary topologies of my own devising, and the answer is always the same, except where sets in T are both open and closed or neither. Surely it's not right in general, though? Hmm. Does that make sense? (just back from the pub)

I have, and while I hugely enjoyed Harvey Flander's little book, I must admit that Bachman's is "better". It takes an easier pace, and doesn't make too many assumptions about our prior knowledge. Yes, it's excellent (for we beginners)

Trying to delete some nonsense. How?

Yes, DQW gave a nice exposition. I might add that the [math] g_{ij} [/math] assigns a coefficient to the derivatives of the generalised Pythagorean. 1 or zero is usual in Euclidean 3-space (hence the identity with the K. delta). As I recall, Lovelock & Rund have a nice and full definition. See if I can dig it out. Hmm..why didn't my LaTeX work? EDIT Cheers DQW!

Well, well. Got it now, nice explanation. Thanks. It also explains why, if the Reimann integral exists, it is equal to the Lebesgue. Neat.

Thanks for that. I have this book, it's one of those I find less than helpful on this subject. On the whole, though, it is a splendid book, and a really good read, thanks again. I especially liked their treatment of analytic functions.Thanks also for the web sites, but I think I need an interactive session on this, as I am totally baffled. Cheers.