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Principia Mathematica worth reading ?

  1. May 11, 2005 #1
    "Principia Mathematica" worth "reading"?

    Hi everybody,
    I have read about Principia Mathematica in many books, and sites on the Net, so I would like to ask you if you know anything about it,or even read some parts of it. What is it exactly ? Is this really important? I read that Godel has taken it into account in order to prove his incompleteness theorem. Is it, in its basic concepts, similar as to the mathematical system of the
    ZF-Set Theory?And finally is it worth reading? I have read some reviews in Amazon.com and some say its great, while others that it simply can't be read.
    Any help would be appreciated.

    Note1: I am not familiar at all with mathematical logic
    Note1:I have not posted this in book reviews as I think it's not only a book
  2. jcsd
  3. May 11, 2005 #2


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    Here's a nice article where you can learn more about the purpose, successes, and failures of PM. They were basically trying to establish a logical foundation for math, to show that all of math was actually a subset of logic.
    You wouldn't want to try to learn logic from PM; It's not that kind of book. I can't imagine reading it unless you were already familiar with what it covers. The parts I read were very thorough (read: tedious and repetitive), but that could be for several reasons. I think people read it to gain a greater appreciation for its goals and historical significance, to see how the authors thought, get a better overview of the subject, and so on. If that interests you, it would be worth reading.
    It's one book in three volumes. It does cover set theory, but uses Russell's type theory, not ZF. I'm pretty sure Gödel used the axioms of PM in both incompleteness proofs.
  4. May 11, 2005 #3


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    until i started posting here i could not imagine anyone who would want to read mroe than a few pages of that work. but there are such epopel here, who consider it extremely important.

    I and perhaps most people consider it purely of "historical importance and interest".

    i.e. not only is their stated goal of limited interest, but it is actually impossible of achievement.

    nonetheless, it is silly to ask other people whether it is worthwhile to read a book just read it for christs sake and make up your own mind. you wont melt.
  5. May 11, 2005 #4
    Thanks for your answers
    To mathwonk:I have checked the reviews of PM in amazon and i found a guy under the nickname mathwonk having written some reviews, including one about PM. I guess it's you so I am curious to know why you reject it so much. I have read in the net things like :"Next to Aristotle's Organon, it remains the most influential book on logic ever written. ". So I am wondering: are even maths such a subjective field? And one more thing: why do you write book reviews if you think that it's silly for someone to ask if a book is worth reading?
  6. May 11, 2005 #5


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    to confused: I don't ask for handouts either, but i give to people who do.

    as to Principia, think i stated my views rather clearly and succintly in post 3 above.

    if you don't like my advice feel free to reject it, but why does it bother you that i offer it? too hard to make up your mind if you get conflicting advice?

    be bold, make your own decisions.
    Last edited: May 11, 2005
  7. May 11, 2005 #6
    mathwonk has let it be known previously in this forum that he certainly does not consider Russell to be a Mathematician. I must say Russell shows up on a website: Mathematicians on Stamps, http://jeff560.tripod.com/. However, he seems to receives the most honor for winning the Noble Prize in Literature.

    Anyway, I must point out that Newton wrote: Philosophia Naturalis, Principia Mathematica. Thus, the title under discussion can be confusing. I rather think that Russell and Whitehead were aware of this, and must of had a reason for taking that title, thought I don't know why.
    Last edited: May 11, 2005
  8. May 11, 2005 #7


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    Newton's principia i regard as outstanding.

    but i am just a schlub who gives free advice. nobody needs to take it.

    the pure math advice is objective, but these threads that just evaluate great works are really only gossip.
  9. May 11, 2005 #8


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    Whitehead and Russel's Principia Mathematica was the system that Goedel targeted in his original Incompleteness Theorem. It was others who extended the theorem to ZF and such, I believe.
  10. May 11, 2005 #9
    having just completed a semester doing an independent study in math logic and proving Godel's theorems, I would say it is not worth your time to read PM or even "On Formally Undecidable Propositions of PM and Related Systems" by Godel either. PM is just pages and pages of symbols, it takes Russell and Whitehead around 300+ pages I believe just to prove that 1+1=2. Godel's paper is not worth reading either, people have refined and vastly shortened the incompleteness proofs in a much more concise matter. Anyone who says it is a "great read" is probably lying, anyone who enjoys reading 3 volumes of just symbols is on crack.
  11. May 11, 2005 #10
    Or highly persistent.
  12. May 12, 2005 #11


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    Or is so deep into the receiving end of S&M that he doesn't recognize what pain is anymore.
    Last edited: May 12, 2005
  13. May 12, 2005 #12
    I don't have any problem about your advice. And I am, too, of the opinion that someone has to make his own decisions without listening to what others say. But I specifically asked about PM because I would like to get some information about what is and what knowledge is required in order to understand it. I guess this is objective information. Of course I can check it myself, and I will, but if for example you mentioned that even teachers of mathematics have problems understanding it, then I would probably not bother to look at it at the current time.(there's also a practical problem about getting it-money if you want to buy it, and that it's not easily found in libraries in my country). Anyway, just to clear something once again: I have not any problems with you mathwonk and I just asked about your dislike of Russel. Nothing more.
    Thanks for your answers
  14. May 12, 2005 #13


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    sorry for pulling your chain, confused.

    let me try again;

    1) account of, reason for, a negative impression (mine):

    as a young person i liked logic and math and was always under the impression russell was a great exemplar of both. so i read some of his works but never got into any of them. i guess i thought something must be wrong with me, since if he is so great shouldn't i get something out of them? or else probably i am kind of stupid.

    this insecurity went on a long time. eventually i ran across a book by hausdorff, where he dismissed attempts to "define" numbers by saying he was more interested in how they behaved, and then he began doing fascinating mathematics.

    i realized immediately that i was probably not the only person who found russell uninspiring, and never cared about it again.

    perhaps there is a dichotomy between math types and logic types, and i belong to one and russell to the other. however i did always enjoy greatly the writings of many logicians, and even studied under Willard Van Orman Quine, read Carnap, and many others. so russell was kind of an exception for me, i.e. maybe he is really a philosopher, rather than either a logician or a mathematician?

    at least i never did understand philosophy, in the sense of how many angels can dance on a pinhead, etc....

    at a certain point i even read some parts of PM, just to verfiy for myself that it really took over 100 pages to prove that 1+1 = 2, and found that it did, and that it did not make even that very enlightening to me personally.

    Those unpleasant memories are rekindled whenever anyone mentions taking PM seriously, so you see i am just indulging a personal dislike, rather than giving useful information, but still if you are at all like me, it may be useful to you.

    2) positive impressions: of some people who post here and elsewhere:

    these people regard russell as a genius. they sincerely care about extremely picky and detailed foundational work on the definitions and utterly basic properties of numbers, far below the radar of even most mathematicians, and are able to both appreciate and enjoy russells work.

    conceivably if one is doing or contemplating doing certain types of research, logical or historical, on the foundations of arithmetic, russell and whitehead might be useful.

    still the only to tell is to look at it.

    i woulod feel very bad if you were a budding original researcher in logic or philosophy of arithmetic, and anything i said were to deter you from beginning that career.

    indeed since i am actually slightly more tolerant after 45 more years, i might even get a kick out of it myself, if i had a lot of spare time, and some definite goal in mind.

    i do suggest getting ahold of a copy in a library before laying out over 10 bucks on it however.

    recently i noted that harvard is putting its entire collection online, slowly of cousre, and cornell already has a lot of historical documents available, possibly that one.

    i for example have found an online copy of wirtingers "untersuchungen uber thetafunktionen: at cornells site.

    i'll even get you the address.

    good luck.
    Last edited: May 12, 2005
  15. May 12, 2005 #14


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    well that site has only geometry, but it did provide the following link:

    [I will say however that without PM probably Godels great work would never have been done, so in a sense PM did something wonderful by inspiring the refutation of its own thesis. sort of like watson inspiring holmes by accident.]


    from which the next excerpt is taken:

    Significance of Principia Mathematica
    Achieving Principia's main goal proved to be controversial. Primarily at issue were the kinds of assumptions that Whitehead and Russell needed to complete their project. Although Principia succeeded in providing detailed derivations of many major theorems in set theory, finite and transfinite arithmetic, and elementary measure theory, two axioms in particular were arguably non-logical in character: the axiom of infinity and the axiom of reducibility. The axiom of infinity in effect stated that there exists an infinite number of objects. Thus, it made the kind of assumption that is generally thought to be empirical rather than logical in nature. The axiom of reducibility was introduced as a means of overcoming the not completely satisfactory effects of the theory of types, the theory that Russell and Whitehead used to restrict the notion of a well-formed expression, and so to avoid paradoxes such as Russell's paradox. Although technically feasible, many critics concluded that the axiom of reducibility was simply too ad hoc to be justified philosophically. As a result, the question of whether mathematics could be reduced to logic, or whether it could be reduced only to set theory, remained open.

    Despite these criticisms, Principia Mathematica proved to be remarkably influential in at least three other ways. First, it popularized modern mathematical logic to an extent undreamt of by its authors. By using a notation superior in many ways to that of Frege, Whitehead and Russell managed to convey the remarkable expressive power of modern predicate logic in a way that previous writers had been unable to achieve. Second, by exhibiting so clearly the deductive power of the new logic, Whitehead and Russell were able to show how powerful the modern idea of a formal system could be, thus opening up new work in what was soon to be called metalogic. Third, Principia Mathematica reaffirmed clear and interesting connections between logicism and two main branches of traditional philosophy, namely metaphysics and epistemology, thus initiating new and interesting work in both these and other areas.

    Thus, not only did Principia introduce a wide range of philosophically rich notions (such as propositional function, logical construction, and type theory), it also set the stage for the discovery of classical metatheoretic results (such as those of Kurt Gödel and others) and initiated a tradition of common technical work in fields as diverse as philosophy, mathematics, linguistics, economics and computer science.

    Today there remains controversy over the ultimate substantive contribution of Principia, with some authors holding that, with the appropriate modifications, logicism remains a feasible project. Others hold that the philosophical and technical underpinnings of the Whitehead/Russell project simply remain too weak or confused to be of much use to the logicist. Interested readers are encouraged to consult Hale and Wright (2001), Quine (1966a), Quine (1966b), Landini (1998) and Linsky (1999).

    Contents of

    "Contents of Principia Mathematica
    Principia Mathematica originally appeared in three volumes. Together these three volumes are divided into six parts. Volume 1 begins with a lengthy Introduction containing sections entitled "Preliminary Explanations of Ideas and Notations," "The Theory of Logical Types," and "Incomplete Symbols." It also contains Part I, entitled "Mathematical Logic," which contains sections on "The Theory of Deduction," "Theory of Apparent Variables," "Classes and Relations," "Logic of Relations," and "Products and Sums of Classes"; and Part II, entitled "Prolegomena to Cardinal Arithmetic," which contains sections on "Unit Classes and Couples," "Sub-Classes, Sub-Relations, and Relative Types," "One-Many, Many-One and One-One Relations," "Selections," and "Inductive Relations."

    Volume 2 begins with a "Prefatory Statement of Symbolic Conventions." It then continues with Part III, entitled "Cardinal Arithmetic," which itself contains sections on "Definition and Logical Properties of Cardinal Numbers," "Addition, Multiplication and Exponentiation," and "Finite and Infinite"; Part IV, entitled Relation-Arithmetic," which contains sections on "Ordinal Similarity and Relation-Numbers," "Addition of Relations, and the Product of Two Relations," "The Principle of First Differences, and the Multiplication and Exponentiation of Relations," and "Arithmetic of Relation-Numbers"; and the first half of Part V, entitled "Series," which contains sections on "General Theory of Series," "On Sections, Segments, Stretches, and Derivatives," and "On Convergence, and the Limits of Functions."

    Volume 3 continues Part V with sections on "Well-Ordered Series," "Finite and Infinite Series and Ordinals," and "Compact Series, Rational Series, and Continuous Series." It also contains Part VI, entitled "Quantity," which itself contains sections on "Generalization of Number," "Vector-Families," "Measurement," and "Cyclic Families."

    A fourth volume was planned but never completed.

    Contemporary readers (i.e., those who have learned logic in the second half of the twentieth century or later) will find the book's notation somewhat antiquated and clumsy. Even so, the book remains one of the great scientific documents of the twentieth century."
  16. May 12, 2005 #15


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    just for a text on mathematics and logic in general, i heard some brilliant lectures by paul cohen in about 1965 which appeared later as a benjamin book, called maybe "set theory and the continuum hypothesis". i suggest those, unless you are already past that stage.
  17. May 13, 2005 #16
    Thank you mathwonk for your interest and your help. I have checked in the library of my University and I can find there for sure "Principia Mathematica to *56", which I think includes some sections of the original work. Anyway, I haven't studied logic at all, and have studied ZF-Set Theory only to a really small extent. So, I guess the book you are suggesting may be good for me.

    As for PM, I think I don't have the knowledge to study it yet but some time maybe... The problem is that I am not studying Mathematics (although I am seriously thinking of doing so) but I am in the first year of Electrical and Computer Engineering. So as you can understand we mostly study Analysis and later computability, complexity of algorithms,arithmetical analysis and staff like that. However, I think I am interested in learning more about the foundations of mathematics and the logic,philosophy behind maths. I want to check "Introduction to mathematical philosophy" by Russel. I have read that it's much easier than PM and actually and introduction to it. Is this true?

    Thanks again
  18. May 13, 2005 #17

    matt grime

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    I hadn't actually bothered to read this thread until I did so by accident just now. And that is not an unreasonable synopsis of something that I infer (and it is just my opinion) mathwonk thinks, and is something I believe too: that such foundational parts of mathematics are incredibly uninteresting to mathematicians today. As has been implied this is partly to do with Goedel, which was perhaps inspired by Russell, that no attempt at axiomatization can ever be utterly satisfactory, so why not just get on with it and admit the problems that the theory may possess. Indeed I get more satisfaction from writers who admit their work has major holes. I suppose this might be to do with the fact that my own work would fall apart in the eyes of anyone who thinks that the axiom of choice is fatally wrong.

    In any case, old fashioned mathematical philosophy for many reasons fails to be remotely interesting these days by its sheer lack of importance. I lose track of the platonism versus formalism rubbish that is posted by motivated but misguided souls on such sites as this when all that really matters is if we all agree on the notation, the reasoning and the conclusion. Heck, even if we agree on those it's often hard to decide if someone's argument is correct and has merit. As an analogy we can all agree that the teachings of Jesus Christ have many good points irrespective of whether or not we think he was a real or fictional character. Equally, we may all agree that the existence or otherwise of such a figure is not required to believe in a 'natural law' of morality. And, please, no one start a thread on that. It is a purely illustrative example
    Last edited: May 13, 2005
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