I was inspired to write this entry, mainly as a reminder to myself for future reference, when I learned, yesterday, that a plant geneticist, Dr John C Sanford, wrote a book titled Genetic Entropy & the Mystery of the Genome in which he probably abuses the concept of entropy in order to draw conclusions about his own field: "This book strongly refutes the Darwinian concept that man is just the result of a random and pointless natural process." The sentence, appearing in the book's description is wrong: the Darwinian process is not just random and cannot be described, merely, as a random process. (I also say "probably abuses" because I have not read the book yet.) I'd be interested to see if Jeff Shallit has spotted this, as he often likes to comment on the abuse of Information Theory by Creationists/Intelligent designers.

People transgress boundaries all the time, especially in Academia, and, often, for good reasons. In fact, I am a big supporter of cross-disciplinary research. However, it is not enough to merely have a haphazard knowledge of a secondary field: experience and knowledge are necessary. Otherwise, funny things can happen. Just as the Sanford example I mentioned above. Let me mention a few more:

The domain of biologically-inspired computing: Although I don't condemn the field, there are many silly papers written all the time. Here is one, published in the IEEE Transactions of Evolutionary Computation. This paper says the following, literally: If you have 1 million grains of sand numbered 1, 2, up to 1 million, one of which is red, but you don't know which is red, then there is no algorithm that will perform better than blind search; that is, choose a grain at random, look at the colour, and keep doing it until you find the red one. This theorem, now known as the No Free-Lunch Theorem (NFLT) is proved in a convoluted (and silly) manner in the paper. Many papers of this sort are produced (daily?) by Computer Scientists, Mathematicians and Engineers who decided to use Biological language in their research (without, perhaps, understanding this discipline).

The mathematically-inspired research on religion: Religion, per se, is, of course, not an academic discipline (or should not be), just as Astrology is not. (The study of the religious phenomenon, as a social phenomenon, on the contrary, is an academic discipline, just as the study of mental diseases is.) However, there are many researchers and "researchers" in Academia who have decided to use Mathematics to prove that their religion is correct. I'm saying it bluntly, but it is so even if they pretend to be arguing only about the existence of a deity or about design in the universe. Ultimately, they are only arguing from their own religious viewpoint. Two examples here: William Dembski, Professor of Theology and Science at the Southwestern Baptist Theological Seminary, ex-mathematician, is constantly abusing his prior field, Mathematics, in order to promote the so-called Intelligent Design (ID) ideas, which are versions of the so-called Creationism. In his Jesus Tomb Math paper is not the worst example. But his papers where he "uses" Information Theory for ID purposes are a good laugh. Second example is John Lennox, Professor of Algebra at the more reputable University of Oxford. He is a mathematician and a priest and uses mathematics to "prove" that god exists. I've written about him before.

The physico- mathematically-inspired research on post-modern philosophy is another case, beautifully exposed by Alan Sokal. In his paper, "Transgressing the Boundaries: Towards a Transformative Hermeneutics of Quantum Gravity", he exposes the idiocies of post-modernism, by writing a paper on this field, a paper which he managed to publish! Irene Irigaray is one of the post-modern philosophers who, among other things, argues that equations in Physics are predominantly male.

Examples of this sort abound. I can think of some more. I would be interested in discovering the ones I am not aware of.

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## What measure theory is about

It's about counting, but when things get too large.

Put otherwise, it's about addition of positive numbers, but when these numbers are far too many.

Put otherwise, it's about addition of positive numbers, but when these numbers are far too many.

## The bottom line

Nuestras horas son minutos cuando esperamos saber y siglos cuando sabemos lo que se puede aprender.

(Our hours are minutes when we wait to learn and centuries when we know what is to be learnt.) --António Machado

Αγεωμέτρητος μηδείς εισίτω.

(Those who do not know geometry may not enter.) --Plato

Sapere Aude! Habe Muth, dich deines eigenen Verstandes zu bedienen!

(Dare to know! Have courage to use your own reason!) --Kant

(Our hours are minutes when we wait to learn and centuries when we know what is to be learnt.) --António Machado

Αγεωμέτρητος μηδείς εισίτω.

(Those who do not know geometry may not enter.) --Plato

Sapere Aude! Habe Muth, dich deines eigenen Verstandes zu bedienen!

(Dare to know! Have courage to use your own reason!) --Kant

I don't understand, what's wrong with the NFLT and similar theories? Or are you just talking about proving them in a pointless way by needlessly using pseudo-evolutionary concepts?

ReplyDeleteMichael,

ReplyDeleteThere is nothing wrong with the NFLT in the same sense that there is nothing wrong with the following theorem: If a is smaller than b then 2 times a is smaller than 2 times b.

They are both correct, but also share something else in common: they are both trivial. Despite the fact that the original NFLT paper by Wolpert and Macready makes it look complicated by stating it in a complicated fashion and giving a complicated proof, the essence of the theorem is what I said in my posting. You can find a very nice explanation of its simplicity (triviality, really) in O. Häggström: Intelligent Design and the NFL Theorems, Biology and Philosophy 22 (2007), pp 217-230.

Yes, the use of mathematics and mathematical theories by IDiots/Creationists is laughable/pointless. But, even worse, the NFLT itself is trivial and attempts to use it in more "respectable" fields (like optimization) have yielded nothing much.

From what I remember of undergrad comp sci it didn't seem that trivial. Although based on all the negative results in theoretical comp sci (eg. halting problem, non-computability etc) it should come as no surprise that for these classes of problems there isn't any algorithm that is more efficient across *all* inputs, it would still warrant a proof. And if anything it can do well to deter someone from trying to write an algorithm that they think will solve all cases efficiently (maybe in the same way as the 2nd law will deter the non-cranky from trying to build a perpetual motion machine). But yes, beyond that I don't think there's much use for it so maybe the NFLT has been overblown.

ReplyDeleteMichael,

ReplyDeleteThe negative results of CS are deeper than NFLT. Many people have thought it deeper. But they were mistaken. For example, Ho and Pepyne (2002): Simple explanations of the no free lunch theorem,

Journal of Optimization Theory and its Applications115, 549-570, compare the NFLT to Gödel's incompleteness theorems. (Ho is a, now retired, professor of Applied Maths and Engineering from Harvard.)When I find some time, I'll write a separate post on the NFLT.

ReplyDeletejust a comment.

ReplyDeletethe paper on the NFLT you linked to is co-authored by Wolpert. I do agree that paper is basically trivial, but I would compare it maybe to Russel's 'principia mathematica' or bourbaki's axiomatizations, or similar results in economics.

many of these results are in many ways trivial, but they 'connect the dots'.

sometimes its wise to write down what you wrote (2a>b if a>b) so you can move on to more complex cases (eg ones where that is false, eg where a and b might be nonlinear functions).

(currently in economics, the idea that 'more is better' for example is seen to follow from this commonly, if not from ID.

alternatively I note that Wolpert works at NASA, so 'another world is possible' and more may be better---keep splitting the multiverse, birthing baby black holes (Lee Smolin) Na>b, N@Z ).

wolpert has done some very good work which also 'connects the dots' at a 'higher' level (eg showing the equivalence of game theory and statistical mechanics, essentially).

i gather people like dembski have picked up that theorem to argue for ID, which is obsviously wack. but dembski also picked up a web site, which doesn't follow from the bible (by my minimal reading---maybe i missed the particular book of the bible on the www).

i will also mention that Herbert wilf has a recent paper on arxiv.org showing 'there is plenty of time for evolution' using math pretty way way way beyond me to argue against people like berlinski on why you need someone to encode the evolutionary algorithm.