Good Math, Bad Math

As you’ve probably heard from other ScienceBlogger’s, that paragon of
arrogant ignorance, Dr. Michael Egnor, is back at it again – and he’s abusing
the language of logic in a way that really frustrates me. I’ve written
about this before, but the general topic recently came up in comments, so
I thought I’d bump it up to the top, along with another slap aimed at Egnor.

For those who don’t know, Dr. Egnor is a brain surgeon at SUNY Stonybrook – an excellent school, and Dr. Egnor is, from all information I’ve heard, an outstanding surgeon. In his free time, he blogs for the Discovery Institute, using his
status as an accomplished brain-surgeon to try to boost the bullshit spewing out of DI.

One of Dr. Egnor’s favorite attacks in his anti-evolution screeds always makes me think of a line from one of my favorite movies: “Hello, my name is Inigo Montoya, you killed my father, prepare to die”. Oops, no, not that one. (Sorry, couldn’t resist.) The real line is “You keep using that word. I do not think it means what you think it means”.

You see, what Egnor keeps doing, over and over again, is arguing that
evolution is just a tautology, and that therefore it’s meaningless. He
defines evolution as the statement “that which survives, survives”. He almost never gets through one of his posts without that accusation in one form or another: evolution is a tautology, and that implies that it’s meaningless and worthless as an explanation of anything.

Leave aside for the moment the fact that he mis-states the key premise of evolution. That’s a huge, obvious, and deliberate mistake, but let’s just ignore it for now. Instead, I’d like to just look at the problem with his statement about tautologies. What exactly is a tautology? And does
criticizing something as “just a tautology” actually make any sense?

In logic, a tautology is a statement which is inevitably true solely by virtue of its structure. Alternatively, it’s a logical statement which is true for any binding of its variables – that is, the structure of the statement means that it’s true, regardless of the meanings assigned to its basic elements.

For example, “A ⇒ A” – that is, “If A is true, then A is true.” It doesn’t matter what meaning you assign to A, the statement “If A is true, then A is true” will always, inevitably, be true.

For a slightly more complex example, “A ∧ (A ⇒ B) ⇒ B” – in english, “If A is true, and it’s true that if A is true, then B is true, then B must be true.” It doesn’t matter what A and B are; it doesn’t matter whether A and B themselves are true or false. The statements above will always be true. It cannot possibly be anything but true, by the definitions of the basic elements of the propositional logic in which it’s written.

Now, for what appears to be a change in course: What is a proof?

A proof in a particular logic L is a set of basic statements (axioms), a
sequence of inferred steps, and a conclusion, such that the conclusion can produced from the axioms by the application of a series of inferrence rules defined by L, where each inferrred rule produces a new fact using one of the inferrence rules of L.

That sounds a bit hairy, so let’s pick it apart a bit. A logic is a
system which (loosely speaking) consists of three parts. First, it has
a syntax – that is, a system of rules for describing how you can form
valid statements in that logic. Second, it has a set of inferrence rules – rules
that describe how to use true statements to produce other true statements; each
production of a new true statement from a set of known statements is called
an inference. Finally, the logic has semantics – that is, a
way of assigning meaning to statements within the language.

What does an inference rule look like? Put extremely simply, an inference rule
consists of a structural pattern which says given a set of true
statements that match the patterns, you can produce a new true statement.
For example, a classic inference rule of propositional logic is (in an unfortunately awkward syntax, due to the limitations of HTML):

  • Given: “A” and “A ⇒ B”
  • Infer: “B”

That is, for any value of A and B, if you have the statements that
match the patterns “A” and “A ⇒ B”, then you can infer “B”.

An inference rule is entirely syntactic (which is to say, structural). It doesn’t
rely on meaning at all. Without knowing what any of the statements in an
inference step mean, you can apply the inference, so long as you have the correct
structural elements.

So – a proof is a series of inference steps which lead from a set of axioms to a
conclusion. Hopefully, you should be starting to see where this is going: a proof is a
series of applications of structural rules that lead from its axioms to its

If the common logics that we use for most scientific studies (that is, propositional logic and first-order predicate logic), you can take any proof, and turn it into a single, massive logical statement. How? Take the axioms, and join them with logical ands. For each application of an inference rule, write a version of that inference as a logical implication, and join that implication to the proof with a logical and. Then add the result of the implication as a new statement, again joined
with logical and.

Let’s take an example – one of the old classics. Our axioms:

  1. All men are mortal: “∀m: IsAMan(M) ⇒ IsMortal(M)”.
  2. Socrates is a man: “IsAMan(Socrates)”.

Our desired conclusion is “Socrates is mortal”; in logical syntax, “IsMortal(Socrates)”. We’ll do the proof in first order predicate logic.

There’s one inference in the proof – the application of logical implication. Written as an implication itself, that works out to “IsAMan(Socrates) ∧ (IsAMan(Socrates) ⇒ IsMortal(Socrates)) ⇒ IsMortal(Socrates)”.

So let’s join it all up:

(∀m: IsAMan(M) ⇒ IsMortal(M)) &land; (IsAMan(Socrates)) ∧ (the axioms)

(IsAMan(Socrates) ∧ (IsAMan(Socrates) ⇒ IsMortal(Socrates)) ⇒ IsMortal(Socrates)) ⇒ IsMortal(Socrates) ∧ (the inferrence)

IsMortal(Socrates) (The result of the inference, and also the conclusion).

There it is – the full proof as a single logical statement? What did we do by translating the proof into a statement this way? We produced a tautology. Every proof can be stated as a tautology. And therefore, every provably true statement can be stated as a tautology.

All of math is, ultimately, nothing but a set of tautologies.

The theory of relativity is nothing but a tautology.

The entire practice of science-based medicine is nothing more than the application
of a mass of tautologies.

“Just a tautology” isn’t a meaningful criticism of an idea – because all provable ideas are “just” tautologies.

If he wants to claim that evolution is vacuous, then he should say that; but he deliberately appropriates and abuses the language of logic in order to
make his argument sound more serious, just like he uses his
medical credentials to make his bogus arguments sound more credible.

Finally, a brief mention of what’s wrong with Dr. Egnor’s statement of the
tautology of evolution: it leaves out a crucial element of the theory – a part
without which the theory loses all of its explanatory value.

You see, tt’s true that evolution says that those that survive, survive. But
evolution says more than that. Evolution says that children aren’t exactly the same as
their parents, and those changes can be inherited by their children.
A properly complete (if reductionist) statement of the theory of evolution is:
“Children are heritably different from their parents, and those that survive and
reproduce, survive and reproduce children which are heritably different from themselves”. This statement of evolution is still tautological in the same way as Dr. Egnor’s statement, but it’s is a vastly different and more meaningful
statement that Dr. Egnor’s vacuous one. Why, do you suppose, does Dr. Egnor constantly
leave out that key bit, about children being heritably different from their

Do I even need to answer that question? I think not.

A personal note here; if you’re not interested in personal ramblings, feel
free to skip this. I’ve been asked why I give Egnor and friends so much attention. There is a good reason for it. I’ve got a serious emotional stake in the effects of
the way that people like Egnor – a teacher at a medical school – deny the
importance of evolution in medical education.

My father died a year and a half ago. What finally killed him was pneumonia. But what caused his death was the stupidity and ignorance of an asshole
doctor. My father died of an antibiotic resistant infection. His doctor was,
unfortunately, a fundamentalist christian, but for some reason, my dad trusted him.
This doctor watched as a series of infections ravaged my father’s body, and
at pretty much every step, he did the wrong damned thing. The reasoning
behind his errors relates directly to the kind of argument Egnor makes: antibiotic
resistance isn’t the production of new traits; it’s merely the selection
of existing traits in a population. So he prescribed antibiotics in a way that
anyone with a damned clue about how bacteria evolve would have predicted would increase the antibiotic resistance of the bacteria.

What’s going to happen in you take a staph infection, and give it penicillin? There’s a good chance you’ll kill the infection. What if the penicillin doesn’t? Then you know you’re dealing with a resistant infection. What’s the right thing to do next? My dad’s doctor gave him more beta-lactam antibiotics with the addition of clavulanic acid, which is an agent that defeats the most common mechanism of penicillin resistance. When that didn’t work, he gradually increased the dose of clavulanic acid – the perfect thing to do to help the bacteria evolve increased resistance. Then he put him into a room with a patient with antibiotic resistant pneumonia. After all, they both had antibiotic resistant infections.

The guy’s pig-ignorance of how bacteria evolve led him to follow a
treatment plan that could almost have been designed to create deadly
strains of resistant bacteria. (And that same doctor prescribes antibiotics
like candy. Got a sniffle? Here, have some antibiotics. They probably won’t do anything, since it’s probably a viral infection, but what’s the harm in being sure? Dumb bastard.)

It’s incredibly important that doctors understand this stuff. Not just
understand that antibiotic resistance exists, but understand
how it develops, and how that development can be enabled
by inappropriate treatment decisions. Egnor argues vehemently that
discussions of evolution absolutely do not belong in medical
education – that any discussion of the process of evolution is, at best,
a waste of time for medical students. Attitudes like that cost lives. And
to me, that cost isn’t abstract at all.


  1. #1 bob koepp
    December 15, 2008

    Mark – While it’s true that logical proofs can be restated as tautologies, it’s simply not true that the theory of relativity is nothing but a tautology. The theory of relativity is not simply a logical structure — it includes a particular interpretation which seems to involve reference to a non-symbolic “world”.

  2. #2 Paul W. Homer
    December 15, 2008

    Nice post, re-expressing everything as a tautology is an interesting consequence (that never really dawned on me :-)

    I’m sorry to hear about your Dad. What often scares me as I dig into various different industries is the degree in which we ‘believe’ we know far more than we actually know. Math is rigorous, most other disciplines however rely heavily on more guess work than they are probably willing to admit.

    BTW: I’m fond of saying that software programs are similar (identical?) to mathematical proofs. In your definition of a proof you can perilously close to a reasonable definition of code (syntax, semantics, etc). Could you tie the two a little closer together in a post someday? For myself, I see one as a variation on the other, but it would be nice to have someone with a stronger background clarify it a bit.


  3. #3 Mark C. Chu-Carroll
    December 15, 2008


    You’re absolutely right about that – but that’s equally true for any scientific theory, including evolution.

    You can take bits and pieces of relativity, and turn them into a really silly-sounding tautologies. (“Mass warps space, therefore space is warped”.) Of course there’s deep semantics (meaning) underlying relativity – there’s deep semantics behind all real scientific theories.

    But the fact that all or part of a theory can be stated as a tautology isn’t a criticism of a theory – it’s nothing less than the statement that the theory is logically valid.

    Egnor claims that the tautological nature of a statement of the theory of evolution means that evolution is trivial,
    lacking in meaning, and not worthy of any serious consideration. But in fact, it’s nothing of the sort: the tautological statement of evolution is actually incredibly profound, and describes a hugely significant, observable phenomenon with effects on the world that almost can’t be overstated.

    “Offspring are different than their parents, and the ones that survive to reproduce, survive to reproduce offspring different than themselves” is a tautology. But it’s also an
    (incomplete but useful) description of an astonishing theory that explains so much of the world we observe.

  4. #4 Blake Stacey
    December 15, 2008

    Egnor’s “tautological” statement of evolution leaves out an important piece. It would be substantially improved by a simple addition: “That which survives, survives to make babies.

  5. #5 James Barlow
    December 15, 2008

    No argument with the maths, but I thought the word “tautology” was originally a reference to a weak argument in classical rhetoric, and was only subsequently pinched for formal logic.

    Wikipedia seems to agree with me, although that doesn’t necessarily mean anything.

    Does Egnor intend to use the rhetorical meaning of tautology rather than the formal logical meaning? He’d still be missing the point, but for different reasons.

  6. #6 Barry
    December 15, 2008

    “Math is rigorous, most other disciplines however rely heavily on more guess work than they are probably willing to admit. ”

    Another way to say this is that one can be very rigorous when dealing with pure logic; once one has to deal with the outside world, one is no longer dealing with only pure logic.

  7. #7 Walker
    December 15, 2008

    The guy’s pig-ignorance of how bacteria evolve led him to follow a treatment plan that could almost have been designed to create deadly strains of resistant bacteria.

    Sounds to me like solid grounds for a malpractice lawsuit.

    Though I could understand if you didn’t do that; not everyone wants to fight those nasty battles when it isn’t going to change the outcome.

  8. #8 Jonathan Vos Post
    December 15, 2008

    Egnor is almost entirely wrong. Mark Chu-Carroll is almost entirely right. Walker is probably right that Egnor is exposed to the risk of a Medical Malpracatice Suit. He should be very careful to always make his Med Mal insurance payments on time. What would be more interesting would be a trial by a med school student of his, complaining that Egnor was intentionally teaching a combination of nonsense and religious doctrine disguised as science.

    Technically, there are Mathematical statements which are neither tautologies, nor are their negations tautologies. These fall into 2 classes: (1) those whose truth is contingent on something to which they refer; (2) those which are undecidable.

    The movie line close to what Dr. Egnor preaches might be:

    “There is no good or evil. There is only power, and those too weak to seek it.”
    – Lord Voldemort

    If one waxes theological, then I believe that Dr. Egnor is doing evil. He is hiding behind his social power as a doctor and a med school professor.

    Which puts Mark Chu-Carroll in the role of — Harry Potter? Dumbledore? We never see the Hogwarts students doing Math, which I’d think would be a prerequisite for Astrology and Potions.

  9. #9 Anonymous
    December 15, 2008

    Your final paragraphs are heartbreaking.

  10. #10 Mark C. Chu-Carroll
    December 15, 2008

    Re #7:

    Unfortunately, that’s not up to me. Personally, I would sue the guy, just to prevent him from doing this to anyone else. (I don’t need the money, and I don’t want to make any money from my father’s death.) But the person who’d need to support a suit would be my mom, his widow. She would never go along with it.

    Re #8:

    The doctor in question isn’t Egnor. My father’s problem had nothing to do with brain surgery; it dated back to a very tricky cancer surgery in his leg, which left a lot of vascular damage. The reason that I mentioned this in a post about Egnor is that Egnor is an example of the kind of teacher who produces doctors like the guy who killed my father.

    Doctors need to understand evolution, at least as it applies to infectious agents, because it
    affects how they should practice medicine. Someone like Egnor who doesn’t believe in evolution, who doesn’t believe
    that it’s possible to create new variants of a bacteria
    or virus, simply isn’t going to stress the importance of practicing in a way that minimizes the risk of producing new, resistant variants. The fact that new variants
    that didn’t exist before can be produced by
    using the wrong drugs or the wrong isolation procedures is
    absolutely crucial. And it doesn’t matter a bit if someone like Egnor understands that resistance is a problem, if he doesn’t appreciate how his actions can contribute to that problem.

  11. #11 jhn
    December 15, 2008

    “Proof” exists only in logic and math.

    Relativity and evolution, and the fact that we’re not brains in jars, can never be proved. Certainty about the content of the external world is simply not available to us. Science only allows us to get as close as humanly possible.

    It is not possible to emphasize this enough.

  12. #12 D. C. Sessions
    December 15, 2008

    Mark — sorry to hear about your father. I still miss mine.

  13. #13 Pocket Nerd
    December 15, 2008

    He prescribed antibiotics in a way that anyone with a damned clue about how bacteria evolve would have predicted would increase the antibiotic resistance of the bacteria.

    First and foremost: I’m terribly sorry about your father. (And I wish I could offer something beyond the standard expression of grief and sympathy, which always sounds so sadly impotent to me.)

    Second: The story of your father’s death is a concrete example of why the “evolution controversy” matters. It’s not merely an abstract debate between ivory-tower eggheads, with no impact on the “real world.” A doctor must understand evolution because it is the unifying principle of biology; a doctor who denies evolution is de facto incompetent, just as much as a pilot who denies the influence of gravity, and he risks the lives of those who trust themselves to his care.

    “Professional” creationists like Egnor are not merely well-meaning nuts spouting a harmlessly wrongheaded idea. Their counterfactual, anti-science garbage visibly damages human lives.

  14. #14 Mark C. Chu-Carroll
    December 15, 2008

    Re #13:

    That’s exactly my point. I get a lot of gripes along the lines of “Why keep droning on about the same stupidity?”. Writing about the things these people promote isn’t just an amusing “Gee, look at the idiot” thing, like a lot of my bad-math posts. It’s important to call these people out. It doesn’t matter how repetitive it is – it’s important to keep hammering on the point that this isn’t just stupidity, this is a matter of life and death. People like Egnor are responsible for promoting lies that kill people. And that’s no joke.

  15. #15 Doug Knox
    December 15, 2008

    Let me see if I understand the situation. Dr. Chu-Carroll’s father died because, instead of being prescribed a regimen of antibiotics strong enough to defeat the already antibiotic-resistant infection, the creationist physician only aided the evolutionary process by increasing the dose of clavulanic acid by increments. Forgive my ignorance, but if the hospital docs go for broke, is the evolutionary process supposed to lie down and wait for the oncoming train? Seems to me like the process just pushes the envelope a little harder. By making a really harsh environment for the pneumonia bacteria, we create a microscopic K-T boundary. The surviving bacteria get an evolutionary kick-start, and pretty soon someone else’s father dies.

  16. #16 BouncingBosons
    December 15, 2008

    #15 – The idea is, I believe, to up the dose/change the drug to be potent enough to destroy enough bacteria that the patients own immune system can destroy the leftover resistant ones, and prevent them from ever leaving that host.

    However, I claim no expertise in this field, and that might be wildly wrong.

  17. #17 notedscholar
    December 15, 2008


    I’m going to have to say you’re quite right about this one. In fact, even the formulation which the good Doctor gives is arguably not a tautology. “What survives, survives” is not an identity proposition!

    Of all the criticisms of Evolution, many are bad, but perhaps Michael’s is the worst!


  18. #18 Tercel
    December 15, 2008

    @ #15

    No, that’s not quite how it works. If an infection is appropriately treated, it is either completely wiped out, or weakened enough such that the immune system can wipe it out.

    Remember that, in general, antibiotics are not intended to eliminate the infection on their own. They work by assisting the natural immune response. Our immune system adapts to new pathogens, so nothing can become resistant to it. Of course, an infection can still be strong enough that our immune system alone is not enough, which is when you need antibiotics.

    As an interesting note, consider that an infection is more likely to overcome the immune system if it attacks immune cells, rather than some other type of cell in our body. This is why HIV is infectious. Unfortunately, it is viral and thus unaffected by antibiotics. Hence the problem…

  19. #19 Mark C. Chu-Carroll
    December 15, 2008

    Re #15:

    When it comes to antibiotic resistance, there’s a lot we know, and a lot we don’t.

    One thing that’s reasonably clear is that gradual approaches, like increasing the clavulanate, don’t work.

    With respect to other things – the idea is twofold.

    Most antibiotic resistant bacteria are less fit than the non-resistant bacteria in an antibiotic free environment. (For example, the most common way of protecting against β-lactam antibiotics is by producing
    a substance called β-lactamase. Resistant bacteria waste energy producing the β-lactamase.)

    So – if between the antibiotics and the patients immune system, you can wipe out enough of the bacteria that only a very small number of resistant ones can escape, you can prevent the resistant strain from getting a sufficient foothold to become dominant in any environment.

    Then you combine that with rotating antibiotics. For example, my pediatrician, when there’s a proven bacterial illness, rotates any particular patient through different antibiotics, which work by different mechanisms. There’s a good chance that an infection which is resistant to penicillin won’t be resistant to, say, sulfa.

    When you carefully use these things in combination – with some drugs held in reserve for emergency use only – you can do a pretty good job of preventing powerful resistant strains from becoming dominant.

    It’s not a perfect strategy. But there is no perfect strategy when it comes to evolving things. When you’ve got billions upon billions of novel variations being tried every day, there’s no way around the fact that you’re trapped in an arms race – there are so many ways around your defenses being tried every day that it’s inevitable that a successful one will turn up sooner or later. So you need to vigilant about monitoring, and be very careful about only using the drugs when absolutely necessary, and continually be developing new drugs that work by different mechanisms.

  20. #20 Dave Robinson
    December 15, 2008

    I’m sorry to hear about your father.

    It’s the kind of thing that just reinforces the idea that Creationists (especially young-Earth Creationists) should be permanently barred from any position that requires them to make decisions affecting other people: especially if those decisions are to be based on the evidence before them.

  21. #21 bob koepp
    December 15, 2008

    I think we can all agree that Egnor’s statement is crap. I doubt that he knows what a tautology is. However, it is true that students of evolutionary theory have had to address what some have construed as a “tautology problem”.

    It’s not the theory per se that has been challenged as tautological, but a carelessly formulated empirical claim that some have proposed as an explication of natural selection; i.e., “other things being equal, fitter members of a population are more likely to survive and reproduce than less fit members.” (I’m sure you’ve all heard claims of that sort.) The problem arises when ‘fitness’ is defined as a function of survival and reproduction, rather than the other way around. But this “problem” has been thoroughly explored and (most of us think) resolved.(Eliott Sober devoted a chapter to this in his 1984 book, “The Nature of Selection”.)

  22. #22 Paul Murray
    December 15, 2008

    Can’t agree. Sorry. The theory of relativity states “the universe we live in is shaped like *this*”. That’s a contingent fact, not a nessesary one.

    The people who identify “survival of the fittest” as being a tautology are making an error of equivocation, assisted by dropping a few words out of that phrase.

    The full version would be something like “survival of the genotype carried by the individuals that are fittest”. A genotype, a pattern, is not the same things as individual organisms that are built according to that pattern.

    Or maybe you could express it as a falacy of division/composition: a population is not the same things as the individuals ini it, any more than a house is the same thing as the bricks that make it up.

  23. #23 Paul Murray
    December 15, 2008

    That is: Lionkind is not the same thing as individual lions. Thus the “tautology” is “lionkindness survives insofar as being a lion enables an organism to survive and breed”.

    Now, that’s not a tautology at all. And when we understand that “lionkindness” is a fuzzy concept – there are bigger and smaller lions, more and less ferocious ones – and that we must qualify the above with “in some particular environment” then we can begion to appreciate that change over time in categories of living things is possible.

    Of course, now that we all know about DNA and so on, this quaint 19th century formulation becomes about genes: particular sequences of nucleotides in a particular long and complicated replicating molecule. We are on firmer and less airy-fairy philospohical ground.

  24. #24 B-Con
    December 15, 2008

    Being a math person and by no means very proficient in biology, I went to HowStuffWorks and, sure enough, they have some information. Excerpt from :
    Antibiotics stop working because bacteria come up with various ways of countering these actions, such as:
    * Preventing the antibiotic from getting to its target: [...]
    * Changing the target: [...]
    * Destroying the antibiotic: [...]

    How do bacteria pick up these drug-fighting habits? In some cases, they don’t. Some bacteria are simply making use of their own inherent capabilities. However, there are many bacteria that didn’t start out resistant to a particular antibiotic. Bacteria can acquire resistance by getting a copy of a gene encoding an altered protein or an enzyme like beta lactamase from other bacteria, even from those of a different species. There are a number of ways to get a resistance gene:

    * During transformation [...]
    * On a small, circular, extrachromosomal piece of DNA [...]
    * Through a transposon [...]
    * By scavenging DNA remnants from degraded, dead bacteria.
    This seems to be a classic case of a life form encountering a problem and then taking steps to counter-act the problem. Some of the counter-methods involve obtaining information about the threat and then changing to become resistant to that threat — ie, “micro-evolution”. I don’t see why anyone, including those of any religion, would have a problem accepting that fact.

  25. #25 PSC
    December 15, 2008

    First I agree with the thrust of your article, and I think the point is well made, but let me quibble with your view on what is a “tautology”. Generally a collection of axioms in a proof system are broken down into “logical axioms” and “non-logical axioms”. The logical axioms can be seen as a technical artefact of the the proof theory, and can often be replaced with some other entirely different group of axioms. For instance a Hilbert system has “A => (B => A)” as a logical axiom, amongst others, but a natural deduction system does not contain this axiom; however the consequences of the two systems themselves are identical.

    There are then “non-logical axioms”, for instance “the successor of any number is greater than 0″ would be a non-logical axiom for natural-number arithmetic.

    I’d consider a “tautology” to be the consequences of the proof system itself, without the additional non-logical axioms.

    Another argument is to write out the whole of the evolutionary theory; if we have:

    - a population of reproducing entities
    - imperfect reproduction, so a child entity is not identical to a parent entity
    - differential rates of survival of members of the population

    it is inevitable that the population will diversify.

    This clearly isn’t a tautology (in any sense of the word).

    Formalizing this in an axiomatic system would of course be a challenge!

  26. #26 The Science Pundit
    December 15, 2008

    I guess “It’s just a tautology.” is the new “It’s just a theory.”

    I know you’ve talked about your father before, and I don’t quite remember if you’ve told the whole story like you did in this post, but I’m very sorry to hear about it. I agree that people like Egnor are scum that actually cause a great deal of harm with their stupidity.

  27. #27 Jonathan Vos Post
    December 15, 2008

    #25: if A and B and C then “it is inevitable that the population will diversify” is not all that needs to be said. The “better” genes spread through the population. The survival of the fittest, at the gene level, needs to be expressed as a differential equation (which may be beyond Dr. Egnor’s ability, but well within that of Mark CC and many of the readers of this blog).

    I’d add the gene transport equations, Fisher’s equation, which appeared in his seminal paper R. A. Fisher, “The wave of advance of advantageous genes”, Ann. Eugenics 7 (1937) 335-369.

    The partial derivative of p with respect to t =
    k times (the second partial derivative of p with respect to x) + mp(1-p)

    where where p is the “frequency of the mutant gene” and m is “intensity of selection in favour of the mutant gene.” [I left the British spelling in for historical flavour].

    Fisher considered a population “distributed in a linear habitat which it occupies with uniform density.” The case of two possible preexisting alleles occupying the locus of the mutant gene was simplified to that of a “parent allelomorph, which we shall suppose to be the only allelomorph present.” The physiological or anatomical characteristic governed by the mutant gene was assumed to be recessive, in keeping with the “common recessiveness of observed mutations.”

    For the full background and derivation, see:

    Or, in this simulation-rich day and age, the fundamental theorem of evolution as proven by the then-Chair of Computer Science at the University of Michigan, Dr. John Holland.

    Egnor can play word games disguised as logic games, but Math does not lie.

  28. #28 Jud
    December 15, 2008

    My reading of Egnor (which is not as extensive as Mark’s, for reasons made evident by the coda to his post) is a bit different than “there can’t be any new kinds.” Rather, I interpret him to be saying “God rather than evolution creates kinds (new or old).”

    It’s difficult for me to understand why these IDiots want to give a supposedly loving God “credit” for MRSA, etc. (So God the micromanager tweaks resistance genes and plugs flagella into bacteria’s butts so they can get a change of scenery, but Hitler He can’t be bothered with, eh? Oh, ‘scuse me, Hitler’s supposed to be Darwin’s fault, right? So does that make Darwin more powerful than God, or just how’s all this supposed to work?)

    Whether Egnor’s read as discounting the possibility of new kinds of resistant bacteria, or as giving God “credit” for them, I’m not sure it makes much difference regarding the eventual result. I doubt a medical school teacher who holds either premise can do a decent job conveying the importance of the subject of bacterial antibiotic resistance or inculcating a thirst in his/her students to learn more about it.

  29. #29 Tyler DiPietro
    December 15, 2008

    “You see, what Egnor keeps doing, over and over again, is arguing that evolution is just a tautology, and that therefore it’s meaningless.”

    Then his claim is literally false, at least. Tautologies always possess semantic content. I think it’s his use of the phrase “meaningless” that is misguided, in this case conflated with “trivial”.

    Never send a surgeon to do a logicians work, I guess. :P

  30. #30 OilIsMastery
    December 16, 2008

    Everyone defines evolution defferently. Some people define evolution as simply “change,” but again, everyone defines change differently. I’ll give you some examples. The Sphenodon aka Tuatara is alleged to be the fastest “evolving” animal but it hasn’t changed in over 140 million years. Similarly echinoids and sharks. They haven’t changed in the past 400 million years so I’m not sure what it means to say they’ve evolved.

  31. #31 Christophe Thill
    December 16, 2008

    The first time I saw the “tautology argument” was in one of Gould’s essays in “Ever since Darwin”. He was replying to an article by Tom Bethell, who in 1976 was already an ignorant anti-evolution fighter.

  32. #32 Stephen Wells
    December 16, 2008

    The theory of evolution, and the theory of general relativity, are logical tautologies, which as Mark says means they’re logically valid; NOT being a tautology is a BAD thing in logic :)

    The contingent fact- the bit where we leave pure logic and encounter empirical facts- is that those theories seem to describe the way the world really behaves.

    Newtonian gravitation is also logically valid and can be phrased as a tautology; that makes it a valid theory; it just turned out to not work.

  33. #33 Chris Lomont
    December 16, 2008

    Mark, you really should look into a better definition of tautology if you’re basing your attack on Egnor’s argument on a single word. Although I agree he is off base, you are on (almost) equally incorrect footing with your usage of tautology. Read the two wikipedia definitions to start, as others have posted.

    First of all, almost all equations in science are *not* tautologies since they relate physically derived items up to our current level of understanding. Wikipedia lists E=mc^2 as an example of an equation that is *not* a tautology, since energy, mass, and speed of light are defined elsewhere and this equation shows a relationship between them. Similarly many equations in math are not merely tautologies – they relate definitions. Different equations are true/false under different axiom sets, so which are tautologies? Godel’s Incompleteness theorems make it clear you cannot axiomatize the integers – for any finite axiom set there will always be a true statement not provable from the axioms. So even in math claiming all equations are tautologies is incorrect – each equation’s truth depends on definitions and which axiom set you start with.

    If you’re attacking someone’s argument based on claimed misuse of a word you should look up a precise definition of the word and understand it first. Otherwise you’re misleading others in the same vein as Dr. Egnor.

  34. #34 trrll
    December 16, 2008

    Thank you, Mark. It really grates on me, for exactly the same reason you describe, every time I hear that moronic “tautology” criticism. And they never seem to realize that claiming that evolution is a tautology is equivalent to stating that is it provably true.

  35. #35 bob koepp
    December 16, 2008

    trrll –
    “claiming that evolution is a tautology is equivalent to stating that is it provably true.”

    I think you might be overlooking what’s problematic here. If a statement is “provably true” then it’s empirical content is zilch. Presumably, evolutionary theory aspires to greater empircal content than that.

    I’m quite sure that Egnor is wrong. But not for the reasons and in the ways some commentators think.

  36. #36 Jud
    December 16, 2008

    For the folks who’ve posted about the Wikipedia definition, and/or said Egnor means something like the Wikipedia definition when he speaks of natural selection/evolutionary theory as a “mere” tautology: Yes, but that only shows an *additional* sense in which Egnor willfully refuses to “get it” where evolution is concerned. It doesn’t at all mean that Mark is incorrect when he points out that as a matter of logic, an explanation that can be reduced to a tautology is a good thing.

    Regarding the rhetorical rather than logical sense of tautology, Egnor’s formulation of “that which survives, survives” deliberately omits (1) the mechanism of or reason for survival (fitness for the environment produced by natural selection, genetic drift, etc. vs. Goddidit – there’s obviously a huge gap between those two mechanisms); and (2) exactly what we mean by “that” – are we considering survival only on the level of the individual (where things can look rather arbitrary – think of the Far Side “God at His Keyboard” cartoon, where He’s looking at a nerd walking along a sidewalk as a crane lifts a piano overhead, His finger poised above a key labeled “Smite”), or at the level of groups, particularly genotypes/phenotypes, where we can derive the sorts of mathematical principles Wright, Haldane, Fisher, etc., came up with?

  37. #37 trrll
    December 16, 2008

    I think you might be overlooking what’s problematic here. If a statement is “provably true” then it’s empirical content is zilch.

    So all provably true statements have zero empirical content? The Pythagorean theorem? The quadratic equation? Fourier decomposition? Euler’s identity? Goedel’s Theorem? The Binomial Theorem? If that’s what “zero empirical content” means, then evolution has a lot of company, including many of the greatest achievements of human thought.

  38. #38 Valhar2000
    December 17, 2008

    #15: The idea is to make the infectious agent go extinct. Even the most superficial study of the history of life on Earth should convince you that this is not only possible, but common.

  39. #39 bob koepp
    December 17, 2008

    trrll – You apparently think mathematics is an empirical science. I don’t.

  40. #40 Jonathan Vos Post
    December 17, 2008

    There is a recent and computer-driven branch of Math which has empirical content: Experimental Mathematics. Google it, and the journal of the same name: “a journal devoted to experimental aspects of mathematics research. It publishes original papers featuring formal results inspired by experimentation, conjectures suggested by experiments, and data supporting significant hypotheses.”

    In opposition to those who pick absurdly stupid mis-definitions of Evolution (i.e. OilIsMastery in #30), and those who fatally mix Religion and Evolution (i.e. Egnor), see the NABT [National Association of Biology Teachers] Statement on Teaching Evolution.

    As stated in The American Biology Teacher by the eminent scientist Theodosius Dobzhansky (1973), “Nothing in biology makes sense except in the light of evolution.” This often-quoted declaration accurately reflects the central, unifying role of evolution in biology. The theory of evolution provides a framework that explains both the history of life and the ongoing adaptation of organisms to environmental challenges and changes.

    While modern biologists constantly study and deliberate the patterns, mechanisms, and pace of evolution, they agree that all living things share common ancestors. The fossil record and the diversity of extant organisms, combined with modern techniques of molecular biology, taxonomy, and geology, provide exhaustive examples of and powerful evidence for current evolutionary theory. Genetic variation, natural selection, speciation, and extinction are well-established components of modern evolutionary theory. Explanations are constantly modified and refined as warranted by new scientific evidence that accumulates over time, which demonstrates the integrity and validity of the field.

    Scientists have firmly established evolution as an important natural process. The nature of science, experimentation, logical analysis, and evidence-based revision based on detectable and measurable data are procedures that clearly differentiate and separate science from other ways of knowing. Explanations or ways of knowing that invoke metaphysical, non-naturalistic or supernatural mechanisms, whether called “creation science,” “scientific creationism,” “intelligent design theory,” “young earth theory,” or similar designations, are outside the scope of science and therefore are not part of a valid science curriculum.

    The selection of topics covered in a biology curriculum should accurately reflect the principles of biological science. Teaching biology in an effective and scientifically honest manner requires that evolution be taught in a standards-based instructional framework with effective classroom discussions and laboratory experiences.

    Adopted by the NABT Board of Directors,1995. Revised 1997, 2000, May 2004, and 2008. Endorsed by: The Society for the Study of Evolution,1998; The American Association of Physical Anthropologists, 1998.

  41. #41 bob koepp
    December 17, 2008

    jvp – I’m aware of the use of computers to “test” such things as the four color theorem. But I don’t see how that endows such a theorem with empirical content. Can you cite an empirical (i.e., non-formal) consequence of any mathematical statement?

  42. #42 Jonathan Vos Post
    December 17, 2008

    Bob Koepp: this is not about testing for solutions to theorems. It is abou DISCOVERING theorems by deep computing. That is, in a sense, using computers in interesting ways to systematically search part of the mathematical subset of what Fritz Zwikcy called “the ideocosm” — the space of all possible ideas.

    This is (in my humble opinion) empirical, even if it is not the Scientific Method applied to the physical (chemical, biological) world. The distinguished editorial board of that journal has established a number of important things.

  43. #43 bob koepp
    December 17, 2008

    jvp – Thanks for the link to an interesting journal. I’m not sure I’d agree that this shows maths to be an empirical discipline, though it does show how an experimental approach can be fruitful even in the formal sciences.

  44. #44 Jud
    December 18, 2008

    “…though it does show how an experimental approach can be fruitful even in the formal sciences.”

    Eh? I thought the “scientific method” (a/k/a “methodological naturalism”) encompassed experimentation, and that this method was central to any science worthy of the name.

    It’s trivially simple to show propositions that can be proved empirically true can also be expressed as tautologies. Put 4 objects in front of you, take two of them away. You’ve just shown empirically that 4-2=2, which reduces to 2=2. Both expressions are logical tautologies. Now for purposes of certain kinds of “content,” e.g. teaching someone math, I’d say 4-2=2 has more educational value. In other words, some tautologies have “more content” than others in terms of fitness for particular uses, such as science or math education, or rhetoric.

    That was never in dispute. In fact it’s one of the points that’s been made throughout the post and comments that Egnor is pulling a sleazy rhetorical trick by taking something instructive and filled with content, evolutionary theory as applied specifically to the development of bacterial antibiotic resistance, and expressing it as a “mere” tautology, nearly content-free.

  45. #45 bob koepp
    December 18, 2008

    Mark – As the host of this blog, perhaps you would consider writing a post about salient differnces between formal sciences like maths and empirical sciences like biology.

  46. #46 Stephen Wells
    December 18, 2008

    I just read Chaitin’s Meta Maths. He winds up arguing that almost all mathematical truths are “true for no reason” and must be found experimentally.

  47. #47 chaos_engineer
    December 18, 2008

    There’s an easy way to figure out Creationists mean by “tautology”. Creationist arguments like this are usually simple projection: In the mists of time, someone said that a Creationist argument was tautological and therefore invalid, and Creationists naturally responded with, “No, your arguments are the tautological ones!” Once they’d said that, they had to be able to present a tautological argument for evolution, and “What survives, survives” was the best they could come up with.

    Now, what did that long-forgotten person mean when he said that Creationism was tautological? I think he was trying to say they were making circular arguments that had no predictive value.

    Take the question: “All bats have one arrangement of wing bones, and all birds have a completely different arrangement. Why?” A biologist could look at the arrangements and say, “Bats and birds must have evolved wings independently; they couldn’t have evolved from a common winged ancestor. We should be able to confirm this by looking at the fossil record.” But a Creationist could only say, “Bats and birds have different bone arrangements because God created them that way.”

  48. #48 John_in_Oz
    December 18, 2008

    I’m sorry to be pedantic. There are truth-seeking children raised in the Creationist camp, and its important that they get told the truth clearly so they can see it despite the intentional obfuscations of their elders, without the confusion caused by innocent errors.
    You said:
    ‘A properly complete (if reductionist) statement of the theory of evolution is: “Children are heritably different from their parents, and those that survive and reproduce, survive and reproduce children which are heritably different from themselves”‘
    I would change this to read:
    1) “[SOME] Children…
    2) …heritably different from [those parents].
    Alternatively of course, it’s I who have got it wrong, and I have much to learn, which would be exciting; either way I’d be grateful if you’d address this comment.

  49. #49 Evan Murphy
    December 19, 2008

    Paul W. Homer, #2:

    Yes! Types are propositions, and programs that satisfy those types are proofs! This is the Curry-Howard correspondence, and it’s up there with ice cream and world peace as a contender for The Best Thing Ever.

    Now, a type is only a useful statement if the type system is sound, exactly like a logic, and type systems may have additional considerations about computability and completeness. And a program (or function) is only a proof of its type if the type system is strong.

    This is pretty cool, but then you look at how logical and, logical or, and implication map in intuitionistic logic directly onto tuples, disjoint unions, and function application in a simply typed lambda calculus analogue, and it gets all kinds of awesome. And you can go much, much farther than that.

    Mark CC did a post on it a while ago: Programs Are Proofs. It’s a good introduction, but if you want deeper treatment, I thought Pierce’s Types and Programming Languages was very good.

  50. #50 rpsms
    December 19, 2008

    #48, can you cite or even fabricate a specific instance where a child is NOT heritably different from BOTH biological parents?

  51. #51 Jonathan Vos Post
    December 19, 2008


    “can you cite or even fabricate a specific instance where a child is NOT heritably different from BOTH biological parents?”

    Yes. When there is only one biological parent, i.e. a human reproductive clone.

  52. #52 Mark C. Chu-Carroll
    December 19, 2008

    Re John_in_Oz #48:

    In reality, children are always different from parents. DNA replication is sufficiently error-prone that the chances of seeing a child whose genes contain zero new mutations is vanishingly small. It’s not impossible – but it’s unlikely enough that in an ultra-reductionist statement like the one in this post, there’s no need to add weasel-words.

    In particular, there’s no reason for weasel-words, because in real evolution, we’re talking about populations, not individuals. So you’ve got a generation of parents, which produce a generation of offspring. The offspring generation will be heritably different from the parent generation, even if there are individuals within the offspring that aren’t.

  53. #53 Anonymous
    December 19, 2008

    “All of math is, ultimately, nothing but a set of tautologies.

    The theory of relativity is nothing but a tautology.

    The entire practice of science-based medicine is nothing more than the application of a mass of tautologies.”
    ^^ Simply not true.
    I think Mark is confusing inductive and deductive logic.

  54. #54 Jonathan Vos Post
    December 20, 2008

    Absolutely Regular
    Monday, March 26, 2007
    Is math tautological?

    Scott throws the following teaser:

    One example is the surprisingly common view that “all mathematical propositions are tautologies,” and therefore can’t convey any new information

    and of course I can’t help but take the bait.

  55. #55 Jonathan Vos Post
    December 20, 2008

    As Eric W. Weisstein at MathWorld points out, the term has both a strictly formal and a common usage definition.

    “A tautology is a logical statement in which the conclusion is equivalent to the premise. More colloquially, it is formula in propositional calculus which is always true (Simpson 1992, p. 2015; D’Angelo and West 2000, p. 33; Bronshtein 2004, p. 288).”


    Bronshtein, I. N. and Semendyayev, K. A. Handbook of Mathematics, 4th ed. New York: Springer, 2004.

    Carnap, R. Introduction to Symbolic Logic and Its Applications. New York: Dover, p. 13, 1958.

    D’Angelo, J. P. and West, D. B. Mathematical Thinking: Problem-Solving and Proofs, 2nd ed. Upper Saddle River, NJ: Prentice-Hall, 2000.

    Mendelson, E. “Tautology.” ยง1.2 in Introduction to Mathematical Logic, 4th ed. London: Chapman & Hall, pp. 17-24, 1997.

    Simpson, J. A. and Weiner, E. S. C. (Preparers). The Compact Oxford English Dictionary, 2nd ed. Oxford, England: Clarendon Press, 1992.

    Egnor and his dangerous lying ilk never specify which definition they use. Funny, how weak they are on definitions. Almost as if they’re hiding something.

  56. #56 bob koepp
    December 20, 2008

    I appreciate the efforts to reinforce the point that tautologies are not meaningless, and the point that there are tautological aspects of scientific theories. This is inevitable given that the terms in which theories are formulated require definitions. But the fact remains that tautologies don’t have empirical consequences until they are paired with non-tautological empirical posits.

  57. #57 trrll
    December 20, 2008

    You apparently think mathematics is an empirical science

    This is semantic hair-splitting. Whether or not you consider mathematics to be an empirical science, the Pythagorean Theorem can tell you how much distance you will save by cutting across a diagonal–an empirical fact.

    Tautological statements relate a set of premises to a conclusion that necessarily follows from those premises. If those premises are empirically found to true in the real world, then the conclusions is necessarily an empirical fact.

    In the case of evolution a tautological formulation of natural selection would be as follows:

    IF organisms exhibit heritable traits that affect propagation of those traits to future generations THEN the population will evolve over time in such a way as to favor those traits that directly or indirectly enhance their own propagation.

    Since it is empirically verified that organisms do exhibit heritable traits that affect reproductive fitness, evolution is logically inevitable, and this is thus an empirical fact. This is a big problem for ID/creationists, because logically they have to accept that evolution by natural selection occurs (due to the tautological, i.e. necessarily true, nature of evolution), but have to argue that the observed changes over time are not due to this known mechanism of change.

    The usual strategy is to suggest that there is some invisible boundary, sort of like the way flat-earthers imagined the edge of the world as being somewhere just out of sight. Evolution is allowed to produce change up to this point, but no further. They call this “microevolution,” and use it as a pretext to dismiss the many observed cases of evolution and speciation (“Yes, but these are only ‘microevolution.’ That much, and only that much, is allowed to happen. But anything bigger than that is ‘macroevolution,’ and can only occur by magical intervention”).

    The problem is that we now have the genome, and we know that at the level of the gene, there is absolutely no difference between the kinds of alterations that underlie big differences (say the difference between cats and dogs) and little ones (say the difference between a siamese cat and a Manx cat)–big differences are just the result of a large number of small changes.

  58. #58 Bad
    December 21, 2008

    Evolution isn’t even about what survives anyway: it’s about what successfully reproduces. A tree that lives for thousands of years but never reproduces (because of, say, some genetic anomaly) is just as much an evolutionary dead end as a creature that is killed before it can reproduce.

    And the issue isn’t even simply that things survive to reproduce. It’s that’s they do so _non-randomly_ based on their ability to meet the demands of their environment.

  59. #59 Dick C. Freeware
    December 22, 2008

    Fail blog is fail.

    Prove me wrong.

    (Oh wait, you can’t.)

  60. #60 J. Barkenhagen
    December 24, 2008

    I met a catholic priest many years ago who claimed that evolution was just more evidence that whatever began our existence had a tremendous creative imagination. He said that it only reveals the limited mind of a man or woman that cannot give this ‘creator’ the credit for the fantastic plan that is evolution and that science is our way of trying to understand how it was done.

    My mother was also killed by foolish ‘medicine’. Maybe your father is dancing to her piano playing in heaven while waiting for his beautiful wife. Perhaps our science we will find this ‘heaven’ one day and we can dance with them.

  61. #61 Jonathan Vos Post
    December 27, 2008

    If I may re-post a comment of mine from another science blog:

    Smackdown, please (yes, Egnor, I’m talking to you)

    IANAL (I Am Not A Lawyer).
    TINLA (This Is Not Legal Advice).

    In my opinion, Dr. Michael Egnor is a very dangerous man. I consider him a Medical Malpractice suit waiting to happen. Because he teaches at a Medical School, I suspect that he could even be sued by the family of someone who dies because of certain types of Medical Malpractice by his former students. Remember that when you get called upon as an Expert Witness, Mark Hoofnagle or “PalMD.”

    Please, Chris Hoofnagle, correct me if I’m wrong on the Law.