There's been a bunch of discussion here at ScienceBlogs about whether or not mathematicians are qualified to talk about evolution, triggered by [an article by ID-guy Casey Luskin][luskin]. So far, [Razib at Gene Expression][gnxp], [Jason at][evblog1][EvolutionBlog][evblog2], and [John at Stranger Fruit][sf] have all commented on the subject. So I thought it was about time for me to toss in my two cents as well, given that I'm a math geek who's done rather a lot of writing about evolution here at this blog.
I don't want to spend a lot of time rehashing what's already been said by others. So I'll start off by just saying that absolutely agree that just being a mathematician gives you absolutely *no* qualifications to talk about evolution, and that an argument about evolution should *not* be considered any more credible because it comes from a PhD in mathematics rather than a plumber. That's not to say that there is no role for mathematics in the discussion of evolution - just that being a mathematician doesn't give you any automatic expertise or credibility about the subject. A mathematician who wants to study the mathematics of evolution needs to *study evolution* - and it's the knowledge of evolution that they gain from studying it that gives them credibility about the topic, not their background in mathematics. Luskin's argument is nothing but an attempt to cover up for the fact that the ID "scientists petition" has a glaring lack of signatories who actually have any qualifications to really discuss evolution.
What I would like to add to the discussion is something about what I do here on this blog with respect to writing about evolution. As I've said plenty of times, I'm a computer scientist. I certainly have no qualifications to talk about evolution: I've never done any formal academic study of evolution; I've certainly never done any professional work involving evolution; I can barely follow [work done by qualified mathematicians who *do* study evolution][gm-good-ev].
But if you look at my writing on this blog, what I've mainly done is critiques of the IDists and creationists who attempt to argue against evolution. And here's the important thing: the math that they do - the kind of arguments coming from the people that Luskin claims are uniquely well suited to argue about evolution - are so utterly, appallingly horrible that it doesn't take a background in evolution to be able to tear them to ribbons.
To give an extreme example, remember the [infamous Woodmorappe paper][woodie] about Noah's ark? You don't need to be a statistician to know that using the *median* is wrong. It's such a shallow and obvious error that anyone who knows any math at all should be able to knock it right down. *Every* mathematical argument that I've seen from IDists and/or creationists has exactly that kind of problems: errors so fundamental and so obvious that even without having to get into the detailed study of evolution, anyone who takes the time to actually *look at the math* can see why it's wrong. It's not always as bad as Woodie, but just look at things like [Dembski's specified complexity][dembski-sc]: anyone who knows information theory can see that it's a self-contradicting definition; you don't need to be an expert in mathematical biology to see the problem - the problem is obvious in the math itself.
That fact in itself should be enough to utterly discredit Luskin's argument: the so-called mathematicians that he's so proud to have on his side aren't even capable of putting together remotely competent mathematical arguments about evolution.
I have seen several attempts to disprove evolution using population genetics computer simulations.
Under those simulations, we'd be either all dead, or we never could possibly have risen past the amoeba stage.
It takes a lot of gall to disprove a natural phenomena on the basis of a 150-line computer program output.
Let me see: either life as we know it cannot exist, OR an unknown entity is steering evolution along OR the computer model is wrong.
Tough judgement call. The guy plowed a whole 20 hours of programming in his model!
That's exactly what I mean: these creationists that want to criticize evolution don't bother to do the work to understand what they're criticizing. The don't understand science; they don't understand biology; they don't understand how the current theory actually says evolution functions. And they don't want to take the time to learn what it says - they just want to tear it down, *now*, *quickly*.
Just because some bozo knows how to throw together a couple of hundred lines of code doesn't mean that *they understand what that code should really be doing*. Hell, that's one of the biggest problems that we have in software development in general: people don't bother to make sure they understand the problem before they code a "solution".
If you want to do real work on evolution using math, you need to know enough to be able to build a realistic mathematical model of evolution - and building mathematical models is *hard*. You need to really, deeply understand the system you're modeling; and you need to know *what aspects* of the real system you're modeling; and you need to be able to validate that the model correctly represents those aspects.
The creationists don't do any of that - they don't understand the theory they're trying to attack; they don't worry about what aspects they're modeling, or whether those aspects capture the features they're interested in; and they don't bother to validate the model in any way.
Look through all of Dembski's papers, and see if you can find *anything* that shows that the mathematical model used by NFL in any way matches any features of real biological systems.
I have been wondering a lot lately why I feel qualified as a computer scientist to say anything at all about evolution. I concur with the standard disclaimer that my views bare absolutely no authority and should obviously not be given any more weight than some other interested layperson.
At the same time, I think I actually have a pretty good intuition about self-organizing systems that has come from studying cellular automata at a fairly detailed level. The idea that complexity cannot arise spontaneously is just wrong--unless you want to invoke the fairly pointless tautology that end results are inherent in the initial conditions. If we have no way to describe the end results without carrying out a process (whether deterministic or stochastic) it is reasonable to say that this process generated the end results.
Since most ID/creationist arguments take the tack of insisting that complexity cannot be produced by simple-looking computational processes (again whether deterministic or stochastic), I feel confident in dismissing them as obviously wrong. I may get in trouble with biologists as overly glib when I say that evolution is plausible as an especially powerful form of self-organization, but that is how I look at it. Nobody should take my word for it, but I think my view is not one that comes from total ignorance either. I also evaluate the evidence for evolution in particular to the best of my ability, time permitting, and this has never contradicted my intuition.
I also realized that I engage in a kind of meta-reasoning when faced with arguments like Dembski's. If evolution were actually impossible, there ought to be some simple argument that works as a "foot in the door" for intuition. It may be that you need very sophisticated mathematics to hammer out the details, but there should be a summary to the effect that if you assume a couple of steps are proven, you would believe the result. There is no such argument against evolution. The short intuitive arguments are easily refuted. The long, detailed arguments require more effort to follow than most people can be expected to muster. I don't believe this is because the long arguments are needed to prove something true. I believe it's because the claim is false and therefore the short arguments do not exist.
We know that self organization works up to a certain point--nature is filled with repeating patterns, fractal patterns, Euclidean minimal surfaces and minimal energy trajectories. None of these things look anything like uniform randomness (e.g. picking red and blue balls out of a bag, flipping coins, watching TV static). If somebody could give me a good reason to consider that the spontaneous emergence of order should all stop at a certain point and prevent the development of self-replicating adaptive agents, then I might have time to read a longer argument. Because to me, that would be an amazing, counterintuitive result. It is rare for a process to work at some level and then just come to a dead stop.
I'm a layman with a fondness for science shows and (especially) blogs. Doesn't take great expertise to take down most IDer arguments. I may not understand all those do-the-hokey-pokey-and-turn-yourself-around gene transfers PZ Myers talks about, but I know enough to recognize an IDer's argument from ignorance when I see one. I recognize a fundamental misunderstanding of evolution when I see one.
It doesn't take much more than I learned from TV and conversations with my 7th grade biology teacher (since we didn't learn much about biology in biology, ya'll) to see that the ID crowd is suffering from verbal diarrhea.
Math is so pure, Life is so dirty !
Here's a recent example from talk.origins
Is the complexity of evolutionary change explainable?
As far as I can tell, the guy follows a population of 100 16-bits individuals during 1000 iterations, and fail to achive its pre-detemined binary target during this duration.
Hence his existential question.
He seems unaware that there is rougly 1 billion bacteria in a shovel-ful of manure, each with 1 million genes, duplicating every 20 minutes or so. What's the odds that one bacteria mutation produce an useful antibacterial peptide (any pepitde will do) in the whole heap of manure in a week ? Now try a million years.
While evolution as a biological process may not be in the realm of mathematicians, I think it is obvious that the analysing the process by which small changes in a population, filtered through a selection mechanism, can create sophisticated results (even patentable ones: see Koza) certainly is. Understanding these mathematics shoudl be seen as a tool for biologists to leverage as much as chemistry or physics. Mathematicians (and I lump computer scientists in here as specialist mathematicians) will be necessary to hone this understanding. Applying it to biology is for biologists.
I think we're in total agreement. Things like the creationist complexity argument are in some sense so shallow that you don't really need to know anything about evolution to knock them down. If an anti-evolution argument is based on the idea that "randomness can't create information", or "complexity can't emerge from simplicity", then a math person is perfectly qualified to knock down that argument.
On the other hand, if a creationist were to put together a study similar to the goodmath paper I referenced last week, arguing, say that oppressor-operator pairs couldn't co-evolve because of some simultaneity constraint - then the kinds of rebuttals that we generally use against creationists wouldn't work. We'd need to actually understand the biochemical systems that were being discussed in order to understand whether or not the simultaneity constraint was a reasonable assumption. (The paper I linked last friday does show that there is no such restriction.)
He seems unaware that there is rougly 1 billion bacteria in a shovel-ful of manure
This page says it's over a billion in a teaspoon of topsoil (which sounds like it would support a lower density population than manure).
A teaspoon is about 1/6 ounce, so a pint is almost 100 teaspoons. If a shovelful weighs a bit over 5 pounds at the density of water, then it might be as much as 500 billion.
This works in favor of your point, but I think it's good to try to get these numbers right. A factor of 500 is not a nitpick even when the numbers are already very big.
I also happen to remember that probiotic capsules claim multiple billions of live bacteria, but I wasn't sure if this could be interchanged with the density of living bacteria. However, 1 billion/shovelful sounded like an underestimate to me.
BTW, Behe also seems to have trouble with the sheer magnitude of bacterial populations as was clear at the Kitzmiller trial. He has a paper supposedly showing the difficulty of evolving new disulfide bonds, but using his analysis on realistic numbers would suggest this is a comment event. E.g. http://www.stcynic.com/blog/archives/2005/10/behe_disproves_irreducible…
Absolutely; I hope my original post didn't suggest otherwise!
Evolution is definitely a process which is amenable to mathematical modeling; and doing that modeling requires serious mathematical skills. But you can't do that modeling without understanding *what it is* that you're modeling.
There's two ways for math guys to really contribute to evolution. One is to do the usual mathematical abstraction thing: take the basic concepts of evolution, and strip them down to a bare abstraction, and then work with that abstraction to see what it's implications are. The other is to get directly involved in mathematical biology. The former requires at least a moderate amount of knowledge about the evolutionary process; and the latter requires as much knowledge of biology and biochemistry as mathematics.
In the further interest of trying to pin down these numbers, the Kitzmiller link says there are over 10^16 prokaryotes in a ton of soil (which I think we can treat as synonymous to bacteria). That's 2000 lb (BTW, I know there is this wonderful thing called the metric system, but nobody else seems to be using it in this context either). So there are more than 5x10^12 bacteria per pound. Assuming about 100 teaspoons/pound, that's 5x10^10 or 50 billion bacteria in a teaspoon of soil.
Anyone know for sure if this is a good estimate? This is a number I'd like to have at my fingertips (hopefully not under my fingernails though--yeech!), and I am still seeing a divergence of nearly 2 orders of magnitude between sources. It probably depends upon the kind of soil, moisture, temperature, and so forth, but I would like to have as good an estimate as possible.
Typically soils will have in the neighbourhood of 1E8-1E9 bacteria per teaspoon. That could be more or less depending on conditions (moisture, temperature, pH, mineral content, organic matter content, depth, presence of nearby vegetation). Anything less than 1E7 would be considered practically sterile as far as soil is concerned. Soils rich in organic matter and water will have a higher figure, and drier less rich soils lower. Bacteria are aquatic organisms, and so water is an important limiting factor.
Not only these crude population genetics models are off on the scale a natural population can expect to be (millions of zebras, trillions of bacteria), but they often fail to take into account some important mechanisms that enhance the basic effect of Natural Selection.
For example they will not modelize the effect of mate selection. In many if not most natural species, the most fit individuals of a local subgroup are more likely to breed with the most fit (with the usual caveats, skipped here), and thus the less fit with the less fit, with the net effect of increasing the speed of convergence toward a (computerized) fitness goal by a significant factor.
Their point is rarely to suggest there could be yet-undiscovered *natural* mechanisms involved in natural selection. Any failure of their comuter model will be interpreted as invalidating the Theory of Evolution as a whole, although none of their models can conclusively establish that gremlins are involved in new genes creation.
I was at ISIT-2006 last week; no ID-ist was in sight. Instead we were treated to a real information theorist doing biology.
Nobody there gives those ID-ers any shred of credibility.
people don't bother to make sure they understand the problem before they code a "solution"
Mark, in your copious free time, I would love to read your thoughts on this topic in greater depth, preferably in a series as you've been doing with category theory. It is the one thing that makes me tear out my hair and bay at the moon when I deal with colleagues who don't understand how computers work.
Like most of biology, evolution is really complicated. Neutral mutations, genetic drift, large and small populations, genetic bottlenecks, gene duplication, parasitism, recombination, kin selection, reciprocal altruism, etc., etc. The chief virtue of mathematical modeling in biology is in abstraction and simplification--you can can leave out some of the messy details and see if they are crucial. If your simulations don't work without them, you can add back the complexity a little bit at a time and see which aspects really are important and which are not. This is extremely valuable, because you can almost never do this experimentally. In real experiments, the closest you can get is to try to to keep all the variables but one or two the same, but a lot of the time it turns out there was an important variable that you didn't think of or couldn't properly control.
So mathematical modeling is very important to understanding evolution, but its strengths make it essentially worthless for what the ID'ers would like to do. You can't disprove evolution with a mathematical model, because no mathematical model captures the complexity of biology. The closest you can get is to say, "A model with these particular features won't do it, so something else is likely needed."
What the ID'ers would like to be able to say is that there is some fundamental mathematical rule that makes any kind of evolution impossible--something like "random processes can never produce information." Unfortunately, that is obviously untrue, because information can be acquired by a pure random brute force search, and at its worst (mutation without selection) evolution reduces to a random search strategy. So they are stuck trying to make a quantitative argument--that it can't produce information fast enough to account for the speed of evolution. But now they are back into the mess of all the complexities that might enhance the efficiency of evolution.
What ultimately condemns Dembski to obscurity is that his zeal to disprove evolution forces him to ask an uninteresting, sterile question: "What are the conditions where evolutionary algorithms are inefficient?" instead of the interesting question: "What are the conditions required for evolutionary algorithms to be as efficient as they would have to be to account for the complexity and diversity of life?"
tgibbs, I agree with your assessment of Dembski. One comment:
What ultimately condemns Dembski to obscurity is that his zeal to disprove evolution forces him to ask an uninteresting, sterile question: "What are the conditions where evolutionary algorithms are inefficient?"
Actually, it is a mathematically interesting question to ask what are the limitations of certain classes of algorithms given time or memory constraints. The main problem is that these questions are notoriously difficult. For instance, it's obviously true that given enough time, an undirected random search strategy could find a Hamiltonian path in any graph that has one. It's also true that nobody knows how to do it for every graph in a reasonable amount of time, nor has anyone been able to show that no efficient algorithm exists. If you could do the latter, you'd have proved that P!=NP.
Since Dembski's supposed fundamental mathematical barriers to evolution ultimately come down to efficiency, he would almost have to be the "Isaac Newton of complexity theory" if he had resolved any such issues in a rigorous mathematical way. So I do have some additional meta-reasoning about Dembski. Namely, if he's that good, why doesn't he work on (seemingly) simpler complexity questions such as whether P!=NP. The fame ensuing from such a result would be a springboard to his credibility (deserved or not) in other areas. Again, my meta-reasoning says that the most likely answer is that he does not have any breakthroughs in reasoning that would help us to answer the notoriously difficult question of what kinds of problems are feasible in certain time bounds.
I've posted comments on this before. Basically, most people attack Dembski from the standpoint that he doesn't understand evolution. That's true, but I would add that he does not have much insight into how intelligent agents actually design things either. Unless we are talking about omniscient intelligences, the efficiency question comes up in that area as well. Humans aren't smart enough to optimize all objective functions efficiently either, and our progress also involves a good bit of trial and error. Thus, my conclusion is that complexity, both as we see in living things and in human-designed inventions is not the solution of intractable problems, but rather consists of approximate solutions to tractable problems that can be usually be found through some combination of random probing and hill climbing.
To those commenting on the microbes in a teaspoon of soil (seriously, consider the metric system...). The typical numbers given (and I've done these counts myself in various settings) are on the order of 10e8-10e9 per gram of soil. The most abundant biomass is fungal (eukaryotic, but there are generally more bacterial cells (~100x smaller volume, give or take). There is tremendous heterogeneity, ranging from very rich ag soils and rhizosphere (near the plant root) to the poorer arid soils and interstitials. The number of cells can vary by several orders of magnitude.
Of even more interest, from an ev bio pt of view, is the diversity- there are at least 1000s of species in every teaspoonful, and the top teaspoon is very different from one 6 inches down. Not only that, but we can only culture about 0.1% (although we're trying!).
The soil is tremendously densely populated with mind-bogglingly diverse bugs, most of which have only been recognized recently using molecular tech.
Anyone who thinks they understand this system, and can model it's life in a computer is ridiculously ignorant.
john derbyshire told me that reading dembski's work convinced it was a load of crap. the main reason being the maths suck, derb has a math degree and he could see through it.
also, mathematicians do play an essential role as jason and mark are doing, they can slap aside flanking maneuvers.
"No selection can occur unless the protein has functionality."
Anyone who thinks they understand this system, and can model it's life in a computer is ridiculously ignorant.
Missing the point entirely. You might as well complain that we can't model a spaceship launch because spaceships involve lots of atoms with electrons whizzing around.
The model is a simplification: A proper model uses an abstraction of the processes we wish to learn about, to see how they operate in certain conditions.
If you want to dispute a model's findings, show a significant shortcoming in the framework, rather than its inability to be an exact copy of reality. Some details just aren't important enough to the experimental procedure. If you want to say otherwise, please tell us why those details are too important to leave out.
Bronze Dog: I disagree. You can model significant properties of bacterial soil ecology in a way that is tractable. But this is a far cry from understanding the system in the sense of grasping its limitations (which creationists would like to do, but are hopelessly ill-prepared for the task). I think in this particular case, it is safe to say that we don't understand the system fully and that any tractable model will potentially miss something important. It is hard to imagine a model that would allow one to conclude that the bacteria cannot evolve some new (but plausible) characteristic within a sufficiently long (but realistic) time frame.
There are cases in which a many-variable system might have a model that can be predicted without simulating all the individual parts and interactions. For instance, the system could be linear. E.g., suppose you have a cellular automata in which the next state of a cell is simply the XOR of its neighbors. In that case, you can make long range predictions that a particular cell will be 0 or 1 far into the future without doing all the work (e.g., represent the tranformation as a matrix multiplication and use iterated squaring).
In the case of a soil ecosystem, there is no such shortcut. That doesn't mean it is entirely unpredictable. One can predict that no "perpetual motion machine" will evolve, since it is ruled out by physics. Maybe there are good reasons to believe the system reaches some kind of equilibrium that holds into the far future. However, barring any deep insight about its behavior, one would have little basis for drawing such a conclusion.
I deleted one of your comments in the moderation queue; I considered deleting both.
I don't mind anything that someone wants to say in comments here. But I do not *not* consider it acceptable to post a comment that consists of nothing but a link to another page. You're welcome to link to other things, but you need to actually *say something* about what you're linking to, why it's relevant to the discussion. Comments that are nothing but links without any explanation or other content will be deleted.
I left your second comment because it arguably has *some* explanation of why you were linking. But now that you've been warned, next time, I won't approve a post from the mod queue that is nothing but a link with so little context.
> That's not to say that there is
> no role for mathematics in the
> discussion of evolution - just
> that being a mathematician doesn't
> give you any automatic expertise or
> credibility about the subject.
A short comment on this one, but from a mathematicians point of view. One of the most powerful aspects of mathematics and the way mathematicians approach a problem is from a level of abstraction that is alien to most other scientists. A mathematician can understand fluid dynamics by simply understanding the properties of the relevant equations -- even if he/she has never seen what water flow looks like. An algebraist can show you how you can solve a Rubix cube by examining the group theoretic properties of the puzzle, but may have never touched the actual cube him/herself.
What I'm saying is what differentiates a mathematician from another person has always been their ability to see the bare necessities of a problem and analyse it in a logical, deductive, and neutral manner.
This is why Applied Mathematicians who work in certain Physical topics can do excellent research even if he/she has only a rudimentary knowledge of the actual Physics.
I'm not saying that one should take a mathematicians view of Evolution as the be all and end all. But I'm saying that one should keep in mind that mathematicians may be able to lend a different perspective on the issue -- and they may be able to do so with less background knowledge.
Arthur mentioned my question at talk.origins about the "explainability" of evolutionary change. That discussion is completely unscientific I just wanted to give some very very rough estimate of the numbers to get some "hunch" about the changes by mutation, not in bacteria but in higher organisms. To me the probability of mutation seems inadequate to drive the evolution of higher organisms. This process is researched by scientists for 50 years or so and to me it seems that making some basic rough estimate about the feasibility of the process (in higher organisms) should no problem. I am not an creationist (frankly), I would like just to see the numbers add up (or not add up). Is there a rough "right" mathematical model?
I mean, this is not quantum theory, this is the code, effects of changes in that code on the phenotype are roughly known, diversity is known, selection criteria is roughly known, probability of mutation is known and - is 1+1=2 or 1+1<>2? Or is it quantum physics?
I read some of evolutionist-creationist discussions and it always happens that creationist tries to attack evolution and the evolutionist finds an error in his math and laughs at him. The battle of ideologies.
Mark says in this article:
"And here's the important thing: the math that they do - the kind of arguments coming from the people that Luskin claims are uniquely well suited to argue about evolution - are so utterly, appallingly horrible that it doesn't take a background in evolution to be able to tear them to ribbons."
I don't want to see somebody "torn to ribbons" or debunked. I would only very sincerely want to see the answer to the question "are we the result of mutations (selections etc.)?" Evolution is obvious but is the process obvious too?
It is commendable of you to try to find out how biology works. Perhaps I can provide some pointers to give you a start. (Note: Layman alert. A biologist will provide better help. I recommend The Panda's Thumb or Pharyngula blogs to meet some professional biologists.)
It seems the best order to answer your specific questions is from the end up.
Evolution is obvious but is the process obvious too?
Seems your starting point is that you know what "common descent" is. I see that you are familiar with the Talk Origins site, which gives a good description on this.
I am not sure about its material on evolutionary mechanisms, but I will in any case give references later.
The battle of ideologies.
Here you should try to put yourself in the mindset of scientists. Empirical science is a rather modern invention. (It started to become used, and useful, around the 17th century.) But it has rapidly become obvious that it is superior to dogma or philosophy to get at the truth.
So if we accept that science can get at observational fact and tested theories that describes and predict them, we rather here see accepted science (evolutionary biology) against an ideologically motivated idea (creationism).
creationist tries to attack evolution
This is btw a strong hint that creationism is ideology and not science. Trying to find shortcomings with in an existing tested theory is meaningless without suggesting a viable alternative. The alternative must be tested and stand on its own - and that hasn't happened here.
Without an alternative what creationists are involved in is the fallacy of false choice - finding shortcomings with one alternative doesn't show that another alternative is correct. There are more options.
this is the code
To see the genome as code isn't entirely fruitful. The DNA code and the biological machinery that express and regulates it is so tightly interwoven that it is hard separate out the code and study evolution that way. The code is translated and regulated, and non-coding DNA has a lot of functions as well.
What biologists primarily do is study populations of individuals and their alleles which express characteristics.
Is there a rough "right" mathematical model?
As I understand it, there are two ways to model biological populations.
Biologists can go bottom up, looking at the dynamics of allele frequencies, ie how characteristics are selected and fixed (i.e. becomes a permanent occurrence). This is population genetics.
They can also go top down, looking at the dynamics of characteristics directly. This is quantitative genetics.
If basic evolutionary mechanisms (population genetics) agree with what we see in the field (quantitative genetics), these two descriptions should agree. And they do, quantitative genetics simply represents one extreme in regards to genetic architecture (many loci of genes).
As you can see from the links, especially the second one, even simple results like the breeders equation gives quite complex effects in modeled populations. Biological models and effects can be much more complex than physics, IMHO.
I'm not sure that this complexity is what you wanted, but at least you can see that fairly realistic numbers give fairly realistic results. There are papers that use and verify these models on real populations if you are interested. Good luck!
The argument is specious. Before Newton, as we discuss on our website in greater depth, everybody believed that angels pushed the planets around in their orbits, irregular orbits when observed from the earth, a moving body itself. Math made the difference. Once it was precise and clear enough,the angel's path turned out to be ellipses, orbits with such a simple geometry they needed no angels. Hence, as the poet, Alexander Pope, said so eloquently, Newton kicked the angels out of the Heavens. In this case, the math mattered a lot. That is history, not something arguable.
We have a mathematical explication of evolution, a quite original and brilliant take on it, as an extension of the logisitcal (Verhulst) equation. Its math got two thumbs up from no less than Dennis Sullivan, A. Einstein Prof. of Math at Stony Brook, (a university),and winner of the National Medal of Science, 2004, no subversive award as it is given directly by the White House.
Who ever heard of an argument that tries to disprove the efficacy of mathematics as a means to scientific proof without checking out the specifics of the mathematical proof first!!?? In any field!
Dr. Peter V. Calabria, PhD (Biophysics)