Blogging Dawkins, Chapter One

The other day I sallied forth to the local Barnes and Noble to pick up my copy of Richard Dawkins’s new book The Greatest Show on Earth: The Evidence for Evolution. As I walked into the store I noticed a person stacking books on the main kiosk. She asked me if I was looking for something in particular.

Now, ordinarily I would have said something like, “No thanks, I’m good.” I spend so much time hanging out at Barnes and Noble, you see, that I’m pretty certain I know the layout of the store better than most of the people who work there. Besides, half the fun of browsing is all the must-have books you notice while searching for the one you are specifically looking for.

But since I am living in one of the most conservative counties in Virginia I wanted it to be heard far and wide that I was excited by the prospect of a new Dawkins book. So I replied, “Yes. I am looking for the new book by Richard Dawkins. It is supposed to come out today.”

She furrowed her brow and said something like, “I just saw it a moment ago, but now I can’t remember where. What is the title?” I replied, “The Greatest Show on Earth.” While we were talking I had scanned the main kiosk. Dan Brown’s new book was hard to miss, but there was no sign of Dawkins. I began to grow worried. My employee friend still could not quite seem to remember where she had seen the book. I began to fear it might be buried deep within the science ghetto. (The “Christian Inspiration” section is about three times larger. Groan.)

That was when she moved to the side and I noticed, almost directly behind her, a huge display given over entirely to Dawkins’s book. I smiled and said, “I think I know where you saw it.”


I was planning simply to read the book and then post a review of it. But then it dawned on me that, while the official purpose of this blog is to discuss issues related to evolution and creationism, I had never given any systematic consideration to the evidence for evolution. So how about a chapter by chapter consideration of the case for evolution as presented by Richard Dawkins? Here we go.

My intorduction to Dawkins’s writing came when I was in graduate school. I had just started to get interested in evolutionary biology, and Stepehn Jay Gould was the only contemporary evolutionary biologist with whom I was familiar. I had read his first three essay collections, and though I loved all three of them I decided it was about time to get someone else’s view of the subject. I browsed through the evolution section of the local public library and noticed a book called, The Blind Watchmaker: How the Evidence for Evolution Reveals a World Without Design. Intriguing! I checked out the book and read it. My first time through I was disappointed, since the book did not really present any evidence for evolution. To me it seemed the point of the book was to clear up certain common misconceptions about evolution, as opposed to making a case from the ground up.

So I was gratified to see, in the preface of the present book, a forthright statement that I had successfully ascertained his intent.

The relatively short Chapter One is really just an introduction. Dawkins notes that the term “theory” means different things in different contexts. In everyday language it tends to mean something pretty close to “hypothesis,” that is, a proposed explanation that is only slightly above an outright guess. But in scientific language it tends to refer to an explanatory scheme that successfully accounts for a wide array of observed facts. The germ theory of disease, for example, is surely something far more worthy of general agreement than a mere hypothesis.

As others have noted, this dichotomy can cause confusion when people discuss “the theory of evolution.” People tend to think the everyday usage is intended when it is really the scientific sense that is meant. Dawkins explains that to avoid the confusion he likes to use the mathematician’s word “theorem” to describe evolution. Just to make it clear, though, that he is not really talking about a mathematical theorem, he alters the word to, “theorum”

At first I balked at this. First he coopts our word, then he misspells it! Outrageous! But I started to soften my view when Dawkins suggested it should be pronounced to rhyme with “decorum.” That is kind of a cool sounding word. And the fact is that mathematicians do sometimes talk about “evolution equations” (roughly, they describe systems that unfold in time). If we can use his word, I suppose I can not begrudge him the use of one of our words.

There is also the fact that the real difference between “theory” and “ theorem” is that the latter carries with it the notion of proof. Theorems are proved, theories not so much. Dawkins wishes to convey that evolution ought to be granted a level of certainty that is not well-captured by the word “theory.” Okay, fair enough.

I do think, though, that Dawkins sometimes overdoes the level of certainty we ought to accord to evolution. It is very well-established indeed, and I would agree we can repose rather a lot of confidence in it. But is it really as certain as the idea that Paris is in the Northern Hemisphere? That seems a bit much.

Dawkins also includes an important discussion on the nature of inferential evidence. It is sometimes thought that eye-witness testimony is somehow the gold standard of evidence, while the circumstantial version is more suspect. Since evolution is based entirely on circumstantial evidence, some people use this as a reason to doubt it. I have had more than one creationist suggest to me that since I didn’t actually see our ape-like ancestors evolve into modern humans, I really have no basis for being confident of the idea. Dawkins quite rightly squashes that bit of silliness. He uses the familiar analogy of a detective trying to solve a crime without the benefit of an eye-witness. No one thinks the detective’s efforts will inevitably be in vain.

Finally, I can’t let this one go by:

Pythagoras’ Theorem is necessarily true, provided only that we assume Euclidean axioms, such as the axiom that parallel straight lines never meet. You are wasting your time measuring thousands of right-angled triangles, trying to find one that falsifies Pythagoras’ Theorem. The Pyhtagoreans proved it, anybody can work through the proof, it’s just true and that’s that. (p. 11)

Attributing the proof of the Pythagorean theorem to the Pythagoreans is rather like attributing the authorship of the Torah to Moses. Traditional, but historically dubious. There is the evidence that the Babylonians were familiar with it a full thousand years before the Pythagoreans arrived, and contemporaneous Chinese mathematicians were also aware of the theorem. There is also the issue of “parallel lines never meet” being one of Euclid’s axioms. That is a bit of a simplification, to put it kindly.

Of course, were I now to follow the example of some of The God Delusion‘s more hyperbolic reviewers, I would at this point dismiss the book out of hand, declare that Dawkins is comepletely out of his depth, and protest that he has no business discussing the history of mathematics without first reading a few dozen texts on the subject.

I will resist the temptation, however, and move gamely on to Chapter Two.

Comments

  1. #1 Brian English
    September 24, 2009

    I read Stewart’s Taming the infinite recently (I think that was the title in any case) and isn’t the parallel lines never meeting a postulate that many attempted to prove from the 4 axioms and failed? Something like that, which helped lead to non-Euclidean geometries. Also, regarding mathematical proof, Stewart says that since Goedel, there is not mathematical truth, only proof, or some such. It seems Dawkins is working with a pre-Goedelian view of logic and maths…..

    Still, I’m about 200 pages into the book, and though there’s been no surprises, it’s good to think about evolution from Dawkins’ angle.

  2. #2 John Danley
    September 24, 2009

    This could be a side effect of the academic, informational cascade “theorum.” Nonetheless, there will be no refunds due to mathematical oversimplifications. The hypothetical “fish to be fried” will be much less numerically inclined.

  3. #3 G.D.
    September 24, 2009

    I haven’t read Stewart, but with respect to Gödel it must surely be the other way round (or something similar). After all, what Gödel proved was that there are unprovable truths of arithmetics (and in any system with anything remotely like the expressive power of arithmetics); that’s what it means to say that arithmetics is incomplete. He also showed that you cannot prove the consistency of arithmetics.

    Those results entailed the immediate demise of the Hilbert programme and was a serious blow to logicism (the view that mathematics can be reduced to logic) and a real problem to any kind of formalism. Sure, there are ways to avoid the result (and completeness and consistency can be proven “from outside” of arithmetics), but you have to adopt a pretty different kind of logical and/or arithmetical axioms.

  4. #4 Bernard Kirzner, M.D.
    September 24, 2009

    September 24, 2009

    I ordered the book on Amazon last week, to be delivered when available. I received it the next day.

    Too soon to say, but it seemed a mix of old and new with special attention to documenting the evidence backing Evolution as an inductive conclusion. That’s were circumstantial evidence comes in.

    Jason. That was the point of bringing in mathematical proofs, as a contrast between deductive and inductive reasoning.

    Looking forward to a good read.

    By the way, I also couldn’t resist getting the Frank Schaeffer book, “Crazy for God: How I grew up as One of the Elect, Helped Found the Religious Right, and Lived to Take All (or Almost All) of It Back. Rachel Maddow had a brief discussion with him about radical right ways of thinking, he pitched this book, and he recommended that we just move on by the radical right, they aren’t going to change, but they are outnumbered. I think Obama should listen to him, he needs to forget about bipartisanship.

    Bernard Kirzner, M.D.

  5. #5 Brian English
    September 24, 2009

    Fair enough G.D. I’ll need to read that section again. I probably misunderstood it.

  6. #6 Blake Stacey
    September 24, 2009

    “Parallel straight lines never meet” sounds more like a definition than an axiom — “parallel lines are those which never meet”. The axiom would then be, to use Playfair’s formulation, that given a line L and a point P not on L, one and only one line may be drawn through P which is parallel to L.

  7. #7 Zack Lewis
    September 25, 2009

    “Theorum” is a bit on the silly side. I wonder why “principle of evolution” doesn’t catch on.

  8. #8 RBH
    September 25, 2009

    Yeah, “theorum” is one of his weaker neologisms. I suspect that it is a meme that will rapidly go extinct.

  9. #9 Valhar2000
    September 25, 2009

    Bernard Kirzner, M.D. wrote:

    [...]they aren’t going to change, but they are outnumbered.

    That is easier said than done, I’m afraid. They seem to account for something like 10%, or by some accounts as much as 20% of the population. Even the most optimistic numbers I’ve seen place them at 5% of the population. That is more than enough people to create serious disruptions if they feel that there is no recourse but outright hostility, and there is no shortage of conservative leaders advocating just that.

  10. #10 Brian English
    September 25, 2009

    From the Stewart book:

    After Goedel, mathematical truth turned out to be an illusion. What existed were mathematical proofs, the internal logic of which might well be faultless, but which existed in a wider context – foundational mathematics – where there could be no guarantee that the entire game had any meaning at all. Goedel did not just assert this: he proved it. In fact, he did two things, which together left Hilbert’s careful, optimistic programme in ruins.
    (p. 244)

    I guess Stewart isn’t a platonist like Goedel and doesn’t think mathematical truths exist outside in some platonic realm.

  11. #11 IanW
    September 25, 2009

    I’m looking forward to reading it. PZ’s “review” was more like a sales pitch!

  12. #12 NewEnglandBob
    September 25, 2009

    I am a couple of chapters ahead of you. I am taking the tact of reading it for general knowledge and principles and not trying to analyze every sentence. I can look here for that :)

  13. #13 heddle
    September 25, 2009

    Not to mention that the Chinese proof of the Pythagorean Theorem contained no words. Simply beautiful.

  14. #14 heddle
    September 25, 2009
  15. #15 Brian English
    September 25, 2009

    David, is the link to a proof that uses a patchwork of two sized square arranged like parquetry with triangles overlain? Your link doesn’t work, so I ask.

  16. #16 Brian English
    September 25, 2009

    Something like this:

  17. #17 Brian English
    September 25, 2009

    Try again. If this is that type of proof. I’d never make a good mathematician. Because it’s not intuitive, it relies on assumptions of larger squares being the sum of smaller squares, etc.

  18. #18 Brian English
    September 25, 2009

    I quit, your comments thingy lies. It says I can use simple html tags, yet when I do, it doesn’t work with them. I tried to post an image twice and failed that many times. I reckon this is why David’s link failed as well.

    Oh well, C’est la vie.

  19. #19 heddle
    September 25, 2009

    Brian that is the proof, maybe this will work

    or the cut and paste method:

    http://www.geom.uiuc.edu/~demo5337/Group3/chinese.gif

    It is a valid proof relying only on computing the area of the red square directly and comparing it to the sum of the areas of its composite pieces:

    1) big red square (side c): A = c^2

    2a) triangles long side a, short side b, area = 0.5*a*b (four of them)
    2b) inner red square, area = (a-b)^2
    2c) total composite are = 4*(0.5*a*b)) + (a-b)^2 = a^2 + b^2

    thus c^2 = a^2 + b^2

    At least I believe it is legitimate–maybe Jason can comment.

  20. #20 Thony C.
    September 25, 2009

    On the subject of the theorem of Pythagorus; although Chinese, Babylonian and Indian mathematics all have knowledge of the theorem probably/possibly earlier than the Greeks (the dating of both Indian and Chinese mathematics is highly contentious) the first real proof of the theorem is in Euclid who attributes the theorem to the Pythagoreans.

  21. #21 Thony C.
    September 25, 2009

    The Chinese proof that heddle is touting is from the Chiu-chang suanshu of which the oldest surving text dates from 263 CE, that is more than 500 years later than Euclid’s Elements. The Chinese text is supposedly based on an early text but when it was original composed is impossible to establish. As to its validity as a proof, the Chinese author only actually ‘proves’ (that is demonstrates) the theorem for the case of a three, four, five triangle and it is not clear if he intends his ‘proof’ to be extended to the general right angle triangle.

  22. #22 Lettuce
    September 25, 2009

    >> But is it really as certain as the idea that Paris is in the Northern Hemisphere? That seems a bit much.>>

    Doesn’t seem a bit much to me. I know where Paris is, just as I assume evolution is true by reading everything I can on the subject. Like I know Jesus wasn’t the son of God, although I think his precepts were on solid ground, even if he never said them.

  23. #23 Thony C.
    September 25, 2009

    On Euclid and parallel line:

    In Euclid’s Elements definition 23 is as follows:

    Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet another in either direction

    Postulate five (the so called parallel axiom) is as follows:

    That if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than two right angles.

    Both taken from Euclid, The Thirteen Books of The Elements Translated with introduction and commentary by Sir Thomas L. Heath. Vol. I (Books I and II) p. 190 definition p.202 postulate.

  24. #24 Jason Rosenhouse
    September 25, 2009

    From David Burton’s textbook The History of Mathematics: An Introduction

    Although tradition is unanimous in ascribing the so-called Pythagorean theorem to the great teacher himself, we have seen that the Babylonians knew the result for specific triangles at least a millenium earlier. …Because none of the various Greek writers who attributed the theorem to Pythagoras lived within five centuries of him, there is little convincing evidence to corroborate the general belief that the master, or even one of his immediate disciples, gave the first rigorous proof of this characteristic property of right triangles. … What is certain is that the school Pythagoras founded did much to increase the interest in problems directly connected with the celebrated result that bears his name.

    Concerning the Chinese proof, it is technically true that the diagram itself only illustrates the theorem for the case of a 3-4-5 triangle. It does, however, so strongly suggest a general proof, based on a rearrangement of areas, that I think we can cut the author some slack!

  25. #25 Gingerbaker
    September 25, 2009

    Why not simply capitalize the scientific Theory, and let the small case theory remain as a mere conjecture?

  26. #26 Thony C.
    September 25, 2009

    Jason I don’t in anyway dispute that the Pythagorean theory was know and used before the time of the Pythagoreans however there is no evidence that any of the user offered anything that would be accepted as a proof for the theory before Euclid who certainly lived within less than five centuries of Pythagorus (about two). Heath argues that the whole of Book II of the Elements is based on Pythagorean mathematics

  27. #27 Jason F.
    September 25, 2009

    Jason,

    Since evolution is based entirely on circumstantial evidence…

    Oooofff! That’s exactly the sort of phrase that gets quote-mined. Evolution is not based “entirely on circumstantial evidence”. We see populations evolve all the time, right before our eyes. In most decent undergrad bio classes, you get to evolve a strain of E. coli that has some unique trait (ours was resistance to ampicillin).

    Your statement would better read: “Since common descent of major taxa is based entirely on circumstantial evidence…”

  28. #28 Jason Rosenhouse
    September 25, 2009

    Thony C –

    I was not contradicting you. I just thought it was a useful quote in the context of this discussion.

  29. #29 Jason Rosenhouse
    September 25, 2009

    Jason F –

    Point taken. I’ll be more careful in the future.

  30. #30 Tyler DiPietro
    September 25, 2009

    “After Goedel, mathematical truth turned out to be an illusion. What existed were mathematical proofs, the internal logic of which might well be faultless, but which existed in a wider context – foundational mathematics – where there could be no guarantee that the entire game had any meaning at all.”

    I haven’t read the book in question, but the reasoning here is risible. What Goedel proved was that one could construct undecidable propositions from any formal theory powerful enough to encode arithmetic on the natural numbers. Turing proved a similar result for computer programs when he showed that the halting problem was undecidable. That doesn’t undermine the fact that the overwhelming majority of computational are decidable, and thus doesn’t undermine any notion of “computational truth”.

  31. #31 Brian English
    September 25, 2009

    Tyler, Stewart goes on to state exactly what you state about decideability and Turing’s halting problem. I hope I haven’t given an unfair impression of Stewart. I didn’t want to post several paragraphs so only posted the paragraph I thought pertinent and seemed to say that mathematical truth wasn’t some universal thing, but applied only to specific axioms and systems.

  32. #32 Tyler DiPietro
    September 26, 2009

    I still wouldn’t agree with his exposition in that case. I would accept it if he simply said that Goedel proved we couldn’t construct one system of axioms from which we could prove all assertions formally. That this somehow proves that mathematical truth is an “illusion” is frankly a ridiculous claim that I would argue perpetuates misunderstanding.

    It is of course entirely possible that I’m missing context, but I can’t imagine a context in which I wouldn’t find that paragraph objectionable.

  33. #33 GuLi
    September 26, 2009

    Tyler@29

    I haven’t read the book in question, but the reasoning here is risible.

    Yes. Gödel’s work did not mean anything like “the end of
    Mathematical Truth”, it was only (only!) the end of the hopes
    that a given systemcould both be complete and consistent.

    That doesn’t undermine the fact that the overwhelming majority of computational are decidable,

    I can’t be sure because a crucial word is missing, but I have

    a nagging feeling I could be disagreeing with that. Could you
    please elaborate?

  34. #34 Brian English
    September 26, 2009

    Tyler, that quote is accurate. I’m an ignoramous regarding mathematics. I just regurgitate what I read. It’s difficult because one expert says this, and another says that. ……

  35. #35 Thony C.
    September 26, 2009

    Tyler is basically right, the Stewart quote is way off, which somewhat surprises me as Ian Stewart is normally very much on the ball in his popular presentations of mathematics. In fact the whole thing worries me so much that I have ordered Stewarts book, it’s one I haven’t read up till now.

    Before the work of people such as Tarski and Gödel in the 1930 provability and truth in mathematics were considered to be synonymous in the strict sense; that is within a formal system all true statements are provable and all provable statements are true, the sets of true and provable statements are isomorphic. Tarski’s truth theorem and Gödel’s incompleteness theorem, which are in fact logically equivalent, showed that the two sets are in fact not isomorphic and that truth and provability are for a large number of highly significant formal systems not synonymous. This however does not mean that the concept mathematical truth becomes meaningless, as Stewart seems to be claiming here.

  36. #36 Tyler DiPietro
    September 26, 2009

    “I can’t be sure because a crucial word is missing, but I have a nagging feeling I could be disagreeing with that.”

    I meant most computational problems are decidable (by deterministic Turing machines, with no restrictions on time complexity). Sorry for the word gap.

  37. #37 Tyler DiPietro
    September 26, 2009

    Actually, a little bit of research suggests that I probably jumped the gun on that statement. It looks as though a diagonalization argument shows that the vast majority of possible problems are undecidable by deterministic Turing machines. I don’t know for sure so I’ll just back off of the statement.

  38. #38 heddle
    September 26, 2009

    Why is the Chinese proof limited to 3-4-5 triangles? The triangles in the diagram, for example, do not appear to be 3-4-5 triangles. What am I missing here?

  39. #39 Pseudonym
    September 26, 2009

    I do think, though, that Dawkins sometimes overdoes the level of certainty we ought to accord to evolution. It is very well-established indeed, and I would agree we can repose rather a lot of confidence in it. But is it really as certain as the idea that Paris is in the Northern Hemisphere?
    Probably not, but it’s as certain as many things that we routinely entrust our lives to.

    We design machine safety systems with the built-in assumption that at the non-relativistic speeds that they will encounter, they will not break the laws of Newtonian mechanics. Evolution is just as certain as that.

  40. #40 Pseudonym
    September 26, 2009

    Tyler:

    It looks as though a diagonalization argument shows that the vast majority of possible problems are undecidable by deterministic Turing machines.

    That’s correct. But if it helps, think of it this way.

    Most real numbers are uncomputable. However, pretty much all real numbers that we care about are computable. Why? Because the very act of specifying them usually makes them computable. Most of the uncomputable numbers that we know about have their definitions based, in one way or another, on the notion of Turing computability. If it wasn’t for that, we wouldn’t even be able to express an uncomputable real.

    Having said that, more problems that we care about in computer science do seem to be undecidable than, say, in mathematics, and the reason is that computability dominates the field more. Compiler researchers, for example, routinely run up against the halting problem solved, precisely because compilers manipulate programs.

  41. #41 Johan
    September 27, 2009

    “the first real proof of the theorem is in Euclid who attributes the theorem to the Pythagoreans.”

    Well Euclid gives the first known proofs. (He gives two as a matter of fact, the second being the most elegant, though it requires the theory of proportion.)

    However he does not attribute to anyone.

  42. #42 Thony C.
    September 28, 2009

    Johan @ #40

    You are of course correct in that Euclid doesn’t attribute the proof to anyone but he only gives one proof.

    heddle @ #37

    because the accompanying text says so

    @ Brian, Tyler et al.

    Ian Stewart really does say that about mathematical truth without any further explanation, which sucks!

    @Tyler, Pseudonym

    real numbers being uncomputable has nothing to do with undecidability. a real number is not a true mathematical statement within a formal system!

  43. #43 dave souza
    September 29, 2009

    Off topic for maths, but relevant to the history of evolutionary thought… interesting to see the reference to Playfair’s formulation which I did not know about. However, the same chap famously visited Siccar Point with Hutton and Sir James Hall, and later came up with the formulation “the mind seemed to grow giddy by looking so far into the abyss of time”…

  44. #44 heddle
    September 29, 2009

    ThornyC,

    because the accompanying text says so

    There was no accompanying text to the picture I linked to here (which was sort of the point.) That picture is not from wikipedia, which may be the one to which you refer. The wiki picture was constructed for 3-4-5 triangles. The one in the picture I linked to was generic.

    I dispute that the proof arising from the figure is limited to 3-4-5 triangles. I’d be more than delighted to be shown wrong–but show me the maths!

  45. #45 Johan
    September 29, 2009

    “You are of course correct in that Euclid doesn’t attribute the proof to anyone but he only gives one proof.”

    No, he gives two proofs. In proposition 47 in book 1 he gives the first statement and proof of the theorem.

    And in book 6, proposition 31, he states the theorem again (actually a slightly more general version) and proves it with a quite different proof. The second proof is more elegant.

  46. #46 Takis Konstantopoulos
    September 29, 2009

    I don’t like the word “theorum” for two reasons: (i) it’s a mixture of greek+latin (OK, not the only one, television is another example…), and (ii) there is nothing wrong with the word theory (θεωρία). For many lay persons, “theory” is something fictitious. But we know well what a scientific theory means: it is a well-accepted term with all its implications and consequences. Theorem (θεώρημα) is also well-established term. Evolution is a theory in the same sense that Quantum Mechanics is a theory: it describes, in best possible terms, our current knowledge of the subject; it is compatible with observations and experiment; it has the power of deduction and forecasting. But Evolution is not a theorem! In Greek, both theory (θεωρία) and theorem (θεώρημα) derive from the same verb: “to see”. The ending of the second word suggests, in Greek, “that which has been seen”–i.e. with a proof.

    Now, about Pythagoras. First of all when we talk of Pythagoras we should really be talking about the Pythagorean School (which lasted for hundreds of years until the early christians got rid of them). Second, it is probably true that the Pythagorean Theorem was not only known (this is certain: look at tables of Pythagorean triples…), but had also been “proved”. The issue is with what we mean by “proof”. Possibly people (Chinese….) understood the principles of its validity but did not have the means to express it properly. Certainly, there were no symbols. They had to resort to a “proof in a special case”. It’s a bit like what Diophantus did hundreds of years later: he has no way in describing the solution of, say, a system of two quadratic equations in two (integer) unknowns because of lack of notation; but he does give an algorithm by choosing to present a special case, with special values for the coefficients of his equations.

    What we know about Pythagoras (Pythagoreans) is that they were the first not only to systematize (the then limited) mathematical knowledge but also to make it part of their lives: indeed, they considered Mathematics as important as other aspects of life. So much so that, at some point, “god” was numbers and numbers was “god”. They had no god. They had an ethical system based on numbers and mutual fraternity.

    I often dream of churches (mosques, synagogues, etc.) were people would gather to learn something about Mathematics. About prime numbers, about geometry, about probability…. And discuss; and think; and contemplate…. How much more useful these places would be!

  47. #47 Thony C.
    September 29, 2009

    Johan @ 45

    Book VI proposition 31 is not the Theorem of Pythagorus but the more generalised theorem on similar figures constructed on the sides of right angle triangles.

    Heddle @ 44

    The diagram is taken from the Chiu-chang suan-shu or Nine Books which is accompanied by a text proving the theorem for a 3-4-5 triangle. That it has been posted somewhere in the internet without the text is simply a form of quote mining. The claim on the site you link to that the Chinese ‘proof’ is known since 1000 BC is pure crap and totally unsubstantiated.

    Takis @ 46

    It is the concept of formal proof that distinguishes Greek mathematics from all its predecessors. Understanding the validity of something and producing a formal proof for the same thing is very different.

  48. #48 heddle
    September 29, 2009

    ThonyC,

    That it has been posted somewhere in the internet without the text is simply a form of quote mining. The claim on the site you link to that the Chinese ‘proof’ is known since 1000 BC is pure crap and totally unsubstantiated.

    Huh? What kind of jackass response is that?

    You may be right but are you only saying so? What was wrong with the analysis I outlined in #19? Where is the error?

    Please provide something substantive rather than saying: “oh, you are wrong.”

  49. #49 Tyler DiPietro
    September 29, 2009

    Thony,

    “real numbers being uncomputable has nothing to do with undecidability. a real number is not a true mathematical statement within a formal system!”

    It’s not within a formal axiomatic system, but something like “compute real-number X to an arbitrary degree of accuracy” is a decision problem in computer science. Chaitin’s halting probability is probably the most famous example of a real that can’t be computed, since no partial recursive function can coincide with it on its entire domain of definition.

  50. #50 Thony C.
    September 29, 2009

    heddle in #13 you write:

    Not to mention that the Chinese proof of the Pythagorean Theorem contained no words. Simply beautiful.

    Then in #19 you require 20 words and a whole heap of mathematical symbols to explain your wordless proof!

    The diagram in the internet is taken from a verbal Chinese proof and was by its originators never intended to be wordless, and if you don’t believe me go and read Needham Science and Civilisation in China Vol 3.

  51. #51 Jason Rosenhouse
    September 29, 2009

    heddle –

    The diagram you linked to with the blue and red triangles is not the proof that appears in the oldest Chinese documents on this subject. Your diagram presents a valid, general proof of the Pythagorean theorem by the rearrangement of area that you described. It is essentially what I had in mind when I said in an earlier comment that the Chinese proof so strongly suggests a general proof that I think we can consider it as such.

    Here is David Burton again, from his textbook The History of Mathematics. This comes right after a description oef Euclid’s methods for proving the theorem:

    Such proofs by addition of areas are so simple that they may have been made earlier and independently by other cultures (no record of the Pythagorean theorem appears, however, in any of the surviving documents from ancient Egypt). In fact, the contemporary Chinese civilization, which had grown up in effective isolation from both the Greek and Babylonian civilizations, had a neater and possibly much earlier proof than the one just cited. This is found in the oldest extant Chinese text containing formal mathematical theories, the Arithmetic Classic of the Gnomon and the Circular Paths of Heaven. Assigning the date of this work is difficult. Astronomical evidence suggests that the oldest parts go back to 600 BC, but there is reason to believe that it has undergone considerable change since first written. The first firm dates that we can connect with it are over a century later than the dates for Nine Chapters on the Mathematical Art. A diagram in the Arithmetic Classic represents the oldest known proof of the Pythagorean Theorem.

    There follows the diagram that also appears in the Wikipedia entry. Burton goes on to discuss “the proof inspired by this figure” which, again, was what I was talking about previously. The diagram itself, however, clearly depicts 3-4-5 triangles.

    Since Euclid’s Elements are typically dated to around 300 BC. The dating of the Chinese doucments is a bit nebulous, as Burton suggests. The basic picture, though, seems clear enough. Awareness of the theorem, if not a clear statement and rigorous proof of it, goes back at least to the Babylonians, roughly between 1800 and 1600 BC, and examples of Pythagorean triples go back even further. Things recognizable as modern proofs appeared roughly simultaneously and independently in Greece and China around 300 BC. It seems likely that Euclid and the Chinese writers were basing themselves on earlier works, so that proofs of the theorem are probably older than we think.

    All of which is to say, as I noted in my post, that attributing the theorem to the Pythagoreans is traditional, but historically dubious.

  52. #52 heddle
    September 29, 2009

    Thony C,

    The diagram in the internet is taken from a verbal Chinese proof and was by its originators never intended to be wordless, and if you don’t believe me go and read Needham Science and Civilisation in China Vol 3.

    I’ll take that as “no I cannot say why the analysis is wrong but I’ll continue to provide invitations to read (what is to me) an arcane text.”

    I was introduced to that diagram (that I linked to) years ago by another physics professor as “the Chinese proof”. It was presented w/o words. That does not mean to say that it is never helpful to use words–but that the picture carries the weight of the proof. But words or not, I continue to contend that the diagram I linked is valid for all right triangles, not just 3-4-5 triangles.

    Again, maybe I’m wrong–but I am beginning to sense that you cannot tell me why. Hopefully Jason will comment.

  53. #53 heddle
    September 29, 2009

    Oops he did comment, thanks Jason, our comments crossed in the mail.

  54. #54 Thony C.
    September 29, 2009

    Jason, the two diagrams are graphically different but geometrically identical; heddles linked diagram is simple a modern presentation of the Chinese diagram shown in Wikipedia.

    heddle your proof only becomes a proof with the indeed correct analysis that you give which means that it is in no way a wordless proof! If you think that Needham is arcane then don’t discuss ancient Chinese mathematics because Needham is the greatest expert on the subject!

  55. #55 Jason Rosenhouse
    September 29, 2009

    Thony C –

    That depends what you mean by “identical.” The Chinese diagram clearly depicts 3-4-5 triangles. To get a general proof out of it you must replace the scale provided in the diagram with a more abstract algebraic calculation. That is not difficult to do, certainly, which is why I have been saying all along that the diagram should be considered a general proof. As far as I know the first person to present a general proof based on the Chinese diagram was Bhaskara in 1114.

  56. #56 Thony C.
    September 30, 2009

    Jason: On the whole I agree with you my only objection is the attempt, not by yourself, to say that this proof is 1) non-verbal and 2) older that Euclid. The first is simply not true no matter how suggestive the diagram is one still has to show how the areas are calculated and then algebraically manipulated to arrive at Pythagorus and secondly any pre CE dating for the Chinese proof is purely speculative.

  57. #57 Johan
    September 30, 2009

    “Book VI proposition 31 is not the Theorem of Pythagorus but the more generalised theorem on similar figures constructed on the sides of right angle triangles.”

    As I said it is a generalization. If you take the case of squares you get back the usual Pythagorean theorem.

    Since it is a generalization its proof also proves the ordinary theorem, and in fact there isn’t much simplification if you just want to prove the square case.

    I do not understand why you do not want to count the proof of VI.31 as proof of the Pythagorean theorem. It is a mathematical argument that demonstrates the truth of the theorem. What more do you want?

  58. #58 Thony C.
    September 30, 2009

    In a purely mathematical context you are of course correct but this discussion started in a historical context about possible predecessors of the Pythagoreans who had, or knew of, the so-called theorem of Pythagorus. The proof of the more general form with the theory of proportions is dependent on that theory, which was neither known to the early Pythagoreans nor any of their predecessors. A very minor quibble but from the historical viewpoint very important.

  59. #59 acne information
    October 2, 2009

    I was introduced to that diagram (that I linked to) years ago by another physics professor as “the Chinese proof”. It was presented w/o words.

  60. #60 acne information
    October 2, 2009

    Those results entailed the immediate demise of the Hilbert programme and was a serious blow to logicism (the view that mathematics can be reduced to logic) and a real problem to any kind of formalism

  61. #61 Christophe Thill
    October 2, 2009

    “It is sometimes thought that eye-witness testimony is somehow the gold standard of evidence, while the circumstantial version is more suspect.”

    Every day, for 44 years now, I’ve seen (with my own eyes) the sun rise in the morning and set in the evening. So it pretty much proves that it revolves around the earth, doesn’t it ?

    Also let’s not forget the work that psychologists such as Elisabeth Loftus have done about eyewitness testimony and its lack of reliability.

    We should believe our brain rather than our eyes (especially mine, which are myopic).

  62. #62 JimR
    October 2, 2009

    Just got far enough into Chapter 1 to see the point that genes are immutable packages (more or less). For some reason I always seemed to think there was a reversion to the mean if a small population reproduced within itself. NOT TRUE. The genes remain intact and do not become averaged. To me this is a very important point in understanding genetics. In a small gene pool, the only problem is the chance of recessive genes wiping out the whole population, I think.

  63. #63 Ian Whittaker
    October 3, 2009

    Having read a lot (although not all) of this thread, I have to say that people do seem to be going off on a bit of a tangent. Darwin’s ‘theory’ is as sound as Newton’s or Einstein’s. Problem is, they are called theories. The great ‘unwashed’ do not understand the scientific usage of this term, and so (especially in the U.S. for some reason), tend to equate it with any other ‘theory': i.e. fairies might live at the bottom of ones garden. Which, along with ‘god did it’, is little more than a silly hypothesis, with no evidence, and doesn’t stand scrutiny amongst people with even a modicum of intelligence/ education. I fear that the fight you have over ‘there’ (i.e America), is that your country was initially populated by people who’s views were too extreme in Europe even 200 – 300 years ago. That’s why they were kicked out /shunned, and that’s why they found solace in what became the United States. Having settled there, they had a huge area to populate with people of a similar (unsettled) mind. Meanwhile, in Europe, Australia, New Zealand, Canada etc., people were just getting on with figuring out what Darwin’s theory actually meant. As opposed to Benjamin Franklin (or one of the early presidents) who sent out search parties for the American Mastodon, given that its bones had been dug up in the U.S., and as god (deliberate non-use of capital letter) couldn’t possibly design something that went extinct, due to its being unfit for its environment, must therefore still exist! Can you imagine that anywhere else in the ‘civilised world’, even then? I forgive him that; he was living in a time when people knew no better in that country; absolutely no excuse now. In essence, I fail to see the argument. One is based on scientific investigation, discovery and sheer common sense; the other is based on no evidence whatsoever, and what amounts to uneducated wishful thinking. I can see why Dawkins gets so absolutely pi**ed off with idiots claiming they know better, due to some non-existent deity appearing to them in a ‘dream’. Take George Bush for instance…………..

  64. #64 matt panechelli
    October 9, 2009

    Hey JimR, I believe what you’re referring to is called inbreeding depression. Though population genetics can get confusing the concept is pretty simple. There is an increased possibility that recessive alleles can become fixed(close to 100% frequency of that allele in the population, meaning an individual is very likely to have two recessive alleles) in a small population by simple genetic drift(which is a random event that requires no forcing or selection agent), in a normal population or a large population recessive alleles can persist but they are kept in check by selection forces(basically any individual with two recessive alleles is very likely to die and not contribute those alleles to future generations which in turn keeps the frequency of that recessive allele low).

    A very interesting topic I’ve recently been exposed to, and I haven’t been able to confirm this yet but I’ve heard that cheetahs don’t elicit a host-donor immune response. Apparently scientist have successfully transplanted skin and maybe even organs from cheetah to cheetah and there is no immune response like we see in humans who can reject donor organs. The idea is that there is a small number cheetahs left on this planet and that population is so homogeneous(genetically similar between individuals) that their immune systems don’t respond to donor organs. Now, I’m not sure if this is directly linked to the fact that they are so homogeneous or if it’s possible that some mammals simply don’t have a host-donor immune response(one reason I can think of is that perhaps cheetahs don’t have multiple blood types like humans do and perhaps that’s a prerequisite for host-donor immune response but that would be null and void if there are mammals without multiple blood types that DO exhibit a host-donor response?), a good book to read that offers some insight into the human immune system is “The Mermaids Tail”, really good evo-bio read. I’m really interested in finding out just what is happening with those awesome big cats.

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