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A math book on introducing proofs(?)

  1. Mar 5, 2005 #1
    Hey guys… and girls! I was just wondering if anyone knew of any good books that introduce the concepts and reasoning behind mathematical proofs, starting from the beginners level. (In high school my teachers did not emphasize proofs.) I would like this specifically to help me for first year linear algebra, and Calculus. Thanks a lot everyone, take care.
     
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  3. Mar 5, 2005 #2
    A book that many have recommended that is intended for the individual student that was writen by a world-class expert is, What is Mathematics? by Courant. It is an old book and can be had second hand.
     
    Last edited: Mar 5, 2005
  4. Mar 6, 2005 #3

    matt grime

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    Polya's How to Prove It.
     
  5. Mar 6, 2005 #4
    matt grime: Polya's How to Prove it.

    Excellent suggestion, and a relatively easy book to read.
     
  6. Mar 6, 2005 #5

    matt grime

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    And the best way to learn how to prove something is to read proofs and mimic them, work out the reasons involved in the deductions. There is a book with the same name as Polya's that appears to think explaining what truth tables are is the best way to teach people how to prove things, which is just utter nonsense.
     
  7. Mar 6, 2005 #6

    mathwonk

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    you might be right. i wonder though whether polyas book dates from a time when it was assumed that many people knew how to understand and read a sentence, formulate a negation, a contrapositive, and a converse.

    now when euclidean geometry is no longer taught as it was, it may be that many students need to learn even what it means for a statement to be true or false. this is what truth tables are used for now, not to make a proof but simply to understand one.

    i believe polya assumed that his reader understood that to prove that not every number solved a given equation, it sufficed to find one that did not.

    but maybe we worry to much about such things, and should go back to the substance of proofs, which is simplifying the problem, finding analogies, then trying to use one good idea in a new problem, etc....

    i once gave copies of polya as a present to my graduates of a calculus class for teachers. I wonder if i could get away with using it as a text?

    I know of the book you are talking about of the same name. I was about to adopt it sight unseen, from highly favorable student reviews, when I found from a review that it taught "two column" proofs, and instantly jettisoned it from my list of candidates.

    according to some reviews however the book does encourage the student to reformulate equivalent versions of a problem and look at them for possible inspiration.

    teaching proofs is really hard, even trivial matters such as induction.

    there is a thread on here somewhere in which an endless and depressing discussion is going on and on, over the difference between assuming P and assuming that P implies Q.


    Students are so puzzled by such things that it gets totally in the way of using induction to prove things.

    I was never able to teach my proofs class to understand the logical structure of induction.

    there are also some subtleties that puzzle even some professors, such as the relation with well ordering.

    e.g. it is often claimed that well ordering is equivalent to induction, but it is false for example that a subset of a well ordered set equals the whole set if it contains the least element and the successor of every element it contains.

    e.g. think of the integers re - well ordered so that the even integers are greater than all the odd ones. then an "inductive subset" contains only the odd integers.

    notice that "proofs" of the equivalence assume the special property of the positive integers that there is also a "predecessor" of every element, which is not true in arbitrary well ordered sets.
     
    Last edited: Mar 6, 2005
  8. Mar 7, 2005 #7

    matt grime

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    I think my issue with these attempts to teach proofs by truth tables are that the students get good at showing some logical rules but don't know that in order to prove "not every...." it suffices ot find one counter example, so the whole first 4 weeks of their course are completely wasted. They can prove that if A implies B and B implies C then A implies C but can't then use it to actually prove anything, and no amount of explaining "A if and only if B" seems to get through despite the fact they know the truth table of it is different from A implies B. It might be alright to try and teach a course like this but I think that there needs to be far more examples in it, of a far more complicated nature. But then we're into chicken and egg territory.
     
  9. Mar 7, 2005 #8
    I went looking for this book, but all I found was "How to SOLVE it" as a title. Is "How to Prove It" out of print, or are these two, in fact, one in the same?

    :confused:

    The Rev
     
  10. Mar 7, 2005 #9
    Polya's book is How To Solve It. It's a well known classic.

    How To Prove it is probably the book that Matt was referring to when he warned about a book with a similar sounding title that taught how to do proofs with truth tables. That book appears to be an ordinary logic book and not the same category of book as those that introduce students to basic concepts of mathematics.

    There is another book which is said to have been written (by the author himself in the introduction!) in the same spirit as Polya's book, and that book is How to Read and Do Proofs : An Introduction to Mathematical Thought

    I'm not sure how this one is, I've only flipped through it at a local bookstore, but thought it was worth mentioning.
     
    Last edited: Mar 7, 2005
  11. Mar 7, 2005 #10

    mathwonk

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    in the reviews on amazon, some people liked it, and some think it is pretty simple minded.

    i am worried that "what is mathematics" may be going out of print, as it is available on sale at $15 from amazon or barnes noble, but is out of stock at the publishers.
     
  12. Mar 8, 2005 #11

    matt grime

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    My mistake, How to Solve it is. The important thing is it's Polya's book.
     
  13. Mar 8, 2005 #12

    mathwonk

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    One of the more popular books today on proofs, costing over 100 dollars, is by Bond and Keane, so I picked it up again yesterday to see if I might be willing to impose it on my students.

    The first thing i saw was this: in the chapter on real numbers, they defined "upper bounds" and "lower bounds". Then they gave without comment, as an example of a subset of the reals which is unbounded above, the natural numbers. The said this as if it were obvious.

    THEN they defined a "least upper bound", gave examples, and introduced the least upper bound axiom for reals. Then they proved the important "Archimedean property" for reals using the lub axiom.

    At no time did they ever observe that the property that the natural numbers are unbounded above, is actually equivalent to the Archimedean property. I.e. that the "example" they had given earlier needed the least upper bound property for its verification, and if it were known, then the Archimedean property needed no additional proof!

    This kind of stupidity sends me up the wall. It suggests to me the possibility the authors do not even understand what they are writing about, and I refuse to use such a book, especially when it is priced at 8 times the price of courant and robbins, and roughly the same factor of polya.

    How do we expect students to think intelligently about the material when the authors themselves write in a repetitive, mindless way, merely aping the other standard books on the topic?

    the point of the course is not to memorize the definition of least upper bounds, it is to learn to think logically. the book itself should offer an example of this.
     
  14. Mar 8, 2005 #13
    "How to Prove It: A Structured Approach" by Daniel J. Velleman is brilliant. It starts off with sentential and quantificational logic, which are absolutely necessary in really understanding how to write a proof, and then it gets into building blocks of analysis with relations, and functions. Then it concludes with an in-depth look at mathematical induction and infinite sets. You may think that these topics sound extraneous, but in addition to really understanding the logic behind proofs, you'll get a foot-in on the first topics that you'll find yourself really actually proving while an undergrad. I nearly shed a tear as I recommend this book.

    Polya's book, on the other hand, is really not recommended for what you're looking for, and I believe you'll find it disappointing. It's definitely mandatory reading, but much more from a heuristic point of view than a proof-writing point of view.
     
  15. Mar 8, 2005 #14

    matt grime

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    Hmm, Vellemna? No it's not good. I didn't wish to name names, but Velleman's book is absolutely completely and utterly ******* ****, insert your own preferred terms. It is perhaps the worst book I have ever read, and certainly the worst book about mathematics that has ever been written, that it has been my unfortunate experience to see.

    Velleman's book is only required in the sense of " read a book that is totally pointless..." but then I've got a degree in maths so, plus many more qualifications, so what would I know?
     
  16. Mar 8, 2005 #15

    mathwonk

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    I also find it frustrating that many students seem to like Velleman (from the amazon website reviews), whereas I agree with Matt that it apparently gives the opposite impression of proof writing from the one I want to teach. "2 column" proofs do not make the point I want to make. Proving is not that mechanical, unless the result is tautological.

    On the other hand maybe we underestimate the total lack of familiarity students today have with these tautological mechanics??

    Here is an example of the kind of thing my students find difficult, or nearly impossible:

    Challenge: complete the following definition:

    "A real valued function f defined on the interval I, is said to be 'bounded'
    if and only if..........."

    Less than 10% of my students could complete the previous sentence correctly.

    Many of them simply parroted (incorrectly) a theorem which guarantees that a function is bounded.

    So many of my students do not know the difference between a definition and a theorem, and many could not make a correct statement of a simple concept that we had studied at nearly ridiculous length. (It is not an exaggeration to say that 40 years ago it was considered more than adequate to give this trivial definition once, in a few lines, and expect it to be understood and remembered accurately after that.)

    Perhaps, it is for such students that books like Velleman are intended. But I fear these books still fail to provide the grasp of creative logical reasoning we actually want students to acquire.

    Polya's book is directed at developing a way of thinking that enables the reader to discover proofs. Velleman's is apparently devoted to teaching to the reader what to expect when I say "If I see another such worthless book, I shall scream", and the reader then presents me with a copy of Velleman, or Solow, or Wolf, or Bond and Keane. (And these are among the best of their ilk!)
     
    Last edited: Mar 8, 2005
  17. Mar 8, 2005 #16
    I would possibly agree that Velleman takes a very pragmatic, and seemingly mechanical approach. But it just gives you a way to think about getting started, and how to understand the form of proofs. But seriously..."How to Solve it"? As I recall, that book has nearly nothing relating to the form of proofs. The biggest hurdle isn't in understanding what makes something true or in seeing why something might be true, it's in actually putting it on paper in a way that's rigorous, precise, and clean. People that have already gone passed any hurdle always tend to look back and dismiss the elementary and maybe even slightly dumbed down approaches that got them over that hurdle in the first place. Velleman's book isn't a book one would want to reread many times, as Polya's "might" be, but it's beyond a good book for the original poster who is just looking for a way to get started.

    On a related note..."How to Prove it" using truth tables to prove theorems? That, it does not. I'm wondering if we're referring to different books or if you, matt grime, perhaps only glanced at it? The book begins with sentential logic, and truth tables are for the purposes of understanding logical connectors, not a way to think about proofs. It never once mentions using truth tables as devices in actually writing real proofs.

    Perhaps I misunderstood, but didn't you, mathwonk, say that you read that the book teaches "two column" proofs and that you instantly "jettisoned" it? Does this imply that you haven't read it? If you haven't, I just want to point out that he uses this "two column" proof method as a way to reformulate equivalent versions of a problem, which in your next paragraph you stated was a good thing.

    Let me give a little description of what this "two column" thing is all about. First of all, it's nothing more than a way to organize thoughts, and the only times it's mechanical are when things are trivial. On the left side you put your assumptions, and on the right side you put what you want to prove. Under these you update your givens and your goals using tautologies and/or whatever insights you may have. You do this until you can see how your assumptions lead to your goal. This is something we all do in our heads, and this just gives us a way to keep track of what we're doing. This isn't something he insists that we use forever, it's just an excellent way to visualize progress in your proof, since the average mind can't comprehend a complex proof at one time. You can put your updates into english once you are finished and what you end up with is often a nice proof.
     
  18. Mar 8, 2005 #17

    mathwonk

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    yes i have not read it as i cannot get hold of a copy. i have tried unsuccessfully, but have been forced to go by descriptions on websites.

    The searchable aprt of it on the amazon website only included a few pages and not even a table of contents as i recall. it looked pretty good to me until someone mentioned the dreaded "2 column proof" phrase.

    This has been used by so many people who do not know anything about math that it has a bad connotation for me. thank you for giving me a little more to go on.


    teachers today are forced to pick books often without seeing them since some publishers do not want to send out freebies anymore unless you indicate a preference for their book, and have a large class, etc etc.

    also deadlines make trouble as i have only 4 weeks and some publishers want 6 weeks to send out a book.

    I also went to our science library but they did not have a copy of velleman. I am not willing to spend $35 for a book i will never have any interest in myself.

    OK I just revisited the amazon website and tried to search the book.

    This is a little scary: if mathematical induction is on page 245, what in heaven's name is in the previous 244 pages? Last time I taught this course I started the first day with mathematical induction, and the discovery and proof of formulas for the sum of the nth powers of the first k natural numbers.
     
    Last edited: Mar 8, 2005
  19. Mar 8, 2005 #18
    You're welcome. I can see where you're coming from. I think why a lot of students might be seeing the technique as the holy grail of proof-writing (which obviously leads people of higher stature to scoff at the book) is because it makes all the logic they've learned (which often comes far before real proof-writing) into something they can immediately use to prove theorems. Suddenly the students feel enlightened and elementary proofs go from extremely difficult to something that they actually have a feel for. This causes them to have unfounded confidence in this technique when it's in fact nothing more than a way to organize some thoughts.

    Edit: To answer your last question, a lot of the stuff that comes before mathematical induction is perhaps a little unnecessary for the aim of the book. Every section from relations and after (up until mathematical induction) are mostly exercises in proof-writing mixed with a very basic introduction to some topics that undergrads will soon encounter.
     
    Last edited: Mar 8, 2005
  20. Mar 9, 2005 #19
    Sorry, if it's off topic, but what do you mean by this challenge? Are you saying that you don't want your students to say that "a real valued function [itex]f(x), x \in X[/itex] is bounded on [itex]X_1 \subset X[/itex] if for every [itex]x \in X_1[/itex] there exist [itex]m[/itex] and [itex]M[/itex] such that [itex]m \leq f(x) \leq M[/itex]"? I think this theorem(?) doesn't holds only for monotone functions. So how can we define boundedness for all functions?

    Would a better way to say this be that "A real valued function [itex]f[/itex], defined on the interval [itex]I[/itex], is said to bounded if and only if there exist [itex]m[/itex] and [itex]M[/itex] such that for every [itex]x \in I_1 \subset I[/itex] we have [itex]m \leq x \leq M[/itex]"? EDIT: or maybe not, this sounds a bit circular (and wrong) to me...or is it okay to say that [itex]f[/itex] is bounded iff [itex]I[/itex] is bounded?

    Or maybe the easier way would be to say that [itex]f[/itex] is bounded on [itex]I[/itex] iff there exist [itex]f(m)[/itex] and [itex]f(M)[/itex] ([itex]m,M \in I[/itex]) such that [itex]f[/itex] is defined only for [itex]x \in [m,M][/itex]? This sounds messy too, but what I mean to say, is that [itex]f[/itex] is bounded on an interval if it is defined in that interval and not defined outside of it.

    Don't really know... This may all be wrong, since I'm making this up in my head.

    - Kamataat
     
    Last edited: Mar 9, 2005
  21. Mar 9, 2005 #20

    matt grime

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    For Bifodus. I didn't say Velleman uses truth tables to prove anything (other than results in predicate logic). Perhaps you only glanced at my post? I said it pointlessly starts with truth tables (and things that any student of mathematics at university ought to already know). What it intends the students to understand and what the students actually understand from it are not necessarily related, or are you accusing me of lying about the work my students hand in based upon reading this book?


    It is a complete waste of money, and is the recommended text for a course I teach, so yes, I have looked at it in order to see what my students are beign told to buy. And what I saw wasn't good mathematics. The best way to learn to write proofs is to read a lot of them and write a lot of them and there is no need to have a special book in order to do that.
     
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