Notices
Results 1 to 12 of 12

Thread: What do you thnk?

  1. #1 What do you thnk? 
    Forum Ph.D.
    Join Date
    Apr 2008
    Posts
    956
    What do you think of the following “analysis” of mathematics, written by an economist with only an amateur knowledge of mathematics? I’ll tell you what I think – but first tell me what you think.

    In regard to whether real analysis makes "water-tight" proofs possible, I would say that even that statement is highly debatable. In any scientific endeavor, what is considered as "water-tight" today often becomes not so after some time. In fact, Henry Poincare made that observation in one of his speeches. How do we know for sure that we have not made some kind of implicit assumptions based on what we currently consider as "self-evident"?

    Generally speaking, what is considered as a valid proof is driven more by the fame of the one who produces the alleged proof than by mere merit. Take, for example, Any Wile's "proof" of Fermat's Last Theorem. Have you seen it and understood it yourself? If not, why do you accept that it has been proved? It seems to me that, just like other things in the human world, what is "true" is a matter of social consensus. In this case, it is more like the lack of dissent opinion from a very small elite group of experts. In the history of math, there are many math geniuses who did not water-tight proofs or produced proofs with errors. For example, Riemann's proof of his mapping theorem was known to be defective. It was not cleaned up until many years later. Yet, the idea was correct. In addition, it is commonly known the Euclid's work is full of holes. Yet, that fact has not prevented him from making an important contribution to mathematics. Thus, it seems that the importance of formality in general and of real analysis in particular has been oversold.

    It seems that our basic disagreement has something to do with your unfamiliarity with the history and philosophy of mathematics. Many people think of math as the last bulwark of certainty. It is not. Math has lost its certainty since the emergence of non-Euclidean geometry. I tend to agree with you that if you have an axiomatic system and you deduce your conclusion through careful logical reasoning in every step, then you can achieve a high degree of certainty. But what about the assumptions of the system? By definition, the axioms are assumed to be true. They are not subject to tests.


    Reply With Quote  
     

  2.  
     

  3. #2 Re: What do you thnk? 
    . DrRocket's Avatar
    Join Date
    Aug 2008
    Posts
    5,486
    Quote Originally Posted by JaneBennet
    What do you think of the following “analysis” of mathematics, written by an economist with only an amateur knowledge of mathematics? I’ll tell you what I think – but first tell me what you think.

    In regard to whether real analysis makes "water-tight" proofs possible, I would say that even that statement is highly debatable. In any scientific endeavor, what is considered as "water-tight" today often becomes not so after some time. In fact, Henry Poincare made that observation in one of his speeches. How do we know for sure that we have not made some kind of implicit assumptions based on what we currently consider as "self-evident"?

    Generally speaking, what is considered as a valid proof is driven more by the fame of the one who produces the alleged proof than by mere merit. Take, for example, Any Wile's "proof" of Fermat's Last Theorem. Have you seen it and understood it yourself? If not, why do you accept that it has been proved? It seems to me that, just like other things in the human world, what is "true" is a matter of social consensus. In this case, it is more like the lack of dissent opinion from a very small elite group of experts. In the history of math, there are many math geniuses who did not water-tight proofs or produced proofs with errors. For example, Riemann's proof of his mapping theorem was known to be defective. It was not cleaned up until many years later. Yet, the idea was correct. In addition, it is commonly known the Euclid's work is full of holes. Yet, that fact has not prevented him from making an important contribution to mathematics. Thus, it seems that the importance of formality in general and of real analysis in particular has been oversold.

    It seems that our basic disagreement has something to do with your unfamiliarity with the history and philosophy of mathematics. Many people think of math as the last bulwark of certainty. It is not. Math has lost its certainty since the emergence of non-Euclidean geometry. I tend to agree with you that if you have an axiomatic system and you deduce your conclusion through careful logical reasoning in every step, then you can achieve a high degree of certainty. But what about the assumptions of the system? By definition, the axioms are assumed to be true. They are not subject to tests.
    This guy does not really understand mathematics. He has a point, but he does not really understand that point.

    It is quite true that the level of rigor that is expected in modern mathematical proofs is not evident in much older mathematics. The modern definition of the function and real proofs did exist much prior to Cauchy.

    It is also true that modern mathematical proofs generally use shortcuts and allusions for some of the simpler steps that can be filled in by any good mathematician if necessary, but that are tedious and detract from the overall presentation of a new result. This is not really a weakness.

    The most recent issue of the Notices of the American Mathematical Society contains some interesting discussions of work that has been done to reduce many modern proofs to pure symbolic logic traceable directly to the Zermelo Frankel axioms plus Choice. It has a direct bearing on the otion of the rigor and and reliability of modern proofs.


    Reply With Quote  
     

  4. #3  
    Forum Ph.D.
    Join Date
    Apr 2008
    Posts
    956
    That’s just what I think.

    But I think I had better elaborate on this. After all, I asked for an opinion, and once an opinon (with which I agree) has been given, it is all too easy for me to say, “That’s just what I think.” So I had better elaborate on what I think.

    One of the things I did was try to point out to my friend that mathematical statements are different from scientific theories. The former is deductive, whereas the latter is inductive. Since the statements in a mathematical proof are deductive in nature, the truth of a valid mathematical proof is absolute. This cannot be said of a scientifc theory that has been “verified” by experiment. The methods which are employed to test a scientific theory are at best inductive in nature. An inductive procedure can never be relied upon with the 100%-certainty of a valid deductive process.

    My friend then claims that statistics is a branch of mathematics, and statistics is in a large part inductive in nature. Which leaves me at a loss on how to proceed – except that I still know that my friend is still confused about what’s deductive and what’s inductive.

    Well, this gives you an idea of what my friend thinks. My friend claims to be a scientist of some sort. Unfortunately, it saddens me that his behaviour in this does not seem scientific at all. No, it’s not because his notions of mathematics are mistaken. Lots of scientists hold mistaken views all the time, if only out of ignorance, but this does not make them unscientific. What is unscientific about my friend is that he seems bent on clinging to his mistaken views despite evidence to the contrary. This is what greatly saddens me indeed.
    Reply With Quote  
     

  5. #4 Re: What do you thnk? 
    Forum Freshman
    Join Date
    Oct 2007
    Posts
    39
    Quote Originally Posted by DrRocket
    Quote Originally Posted by JaneBennet
    What do you think of the following “analysis” of mathematics, written by an economist with only an amateur knowledge of mathematics? I’ll tell you what I think – but first tell me what you think.

    In regard to whether real analysis makes "water-tight" proofs possible, I would say that even that statement is highly debatable. In any scientific endeavor, what is considered as "water-tight" today often becomes not so after some time. In fact, Henry Poincare made that observation in one of his speeches. How do we know for sure that we have not made some kind of implicit assumptions based on what we currently consider as "self-evident"?

    Generally speaking, what is considered as a valid proof is driven more by the fame of the one who produces the alleged proof than by mere merit. Take, for example, Any Wile's "proof" of Fermat's Last Theorem. Have you seen it and understood it yourself? If not, why do you accept that it has been proved? It seems to me that, just like other things in the human world, what is "true" is a matter of social consensus. In this case, it is more like the lack of dissent opinion from a very small elite group of experts. In the history of math, there are many math geniuses who did not water-tight proofs or produced proofs with errors. For example, Riemann's proof of his mapping theorem was known to be defective. It was not cleaned up until many years later. Yet, the idea was correct. In addition, it is commonly known the Euclid's work is full of holes. Yet, that fact has not prevented him from making an important contribution to mathematics. Thus, it seems that the importance of formality in general and of real analysis in particular has been oversold.

    It seems that our basic disagreement has something to do with your unfamiliarity with the history and philosophy of mathematics. Many people think of math as the last bulwark of certainty. It is not. Math has lost its certainty since the emergence of non-Euclidean geometry. I tend to agree with you that if you have an axiomatic system and you deduce your conclusion through careful logical reasoning in every step, then you can achieve a high degree of certainty. But what about the assumptions of the system? By definition, the axioms are assumed to be true. They are not subject to tests.
    This guy does not really understand mathematics. He has a point, but he does not really understand that point.

    It is quite true that the level of rigor that is expected in modern mathematical proofs is not evident in much older mathematics. The modern definition of the function and real proofs did exist much prior to Cauchy.

    It is also true that modern mathematical proofs generally use shortcuts and allusions for some of the simpler steps that can be filled in by any good mathematician if necessary, but that are tedious and detract from the overall presentation of a new result. This is not really a weakness.

    The most recent issue of the Notices of the American Mathematical Society contains some interesting discussions of work that has been done to reduce many modern proofs to pure symbolic logic traceable directly to the Zermelo Frankel axioms plus Choice. It has a direct bearing on the otion of the rigor and and reliability of modern proofs.
    I do not understand why you say the guy does not know what he is talking about, then you give two examples in which you say that he is actually right and then in the end you say that he got it all wrong because it is not a weakness.
    Reply With Quote  
     

  6. #5 Re: What do you thnk? 
    . DrRocket's Avatar
    Join Date
    Aug 2008
    Posts
    5,486
    Quote Originally Posted by evariste.galois
    Quote Originally Posted by DrRocket
    Quote Originally Posted by JaneBennet
    What do you think of the following “analysis” of mathematics, written by an economist with only an amateur knowledge of mathematics? I’ll tell you what I think – but first tell me what you think.

    In regard to whether real analysis makes "water-tight" proofs possible, I would say that even that statement is highly debatable. In any scientific endeavor, what is considered as "water-tight" today often becomes not so after some time. In fact, Henry Poincare made that observation in one of his speeches. How do we know for sure that we have not made some kind of implicit assumptions based on what we currently consider as "self-evident"?

    Generally speaking, what is considered as a valid proof is driven more by the fame of the one who produces the alleged proof than by mere merit. Take, for example, Any Wile's "proof" of Fermat's Last Theorem. Have you seen it and understood it yourself? If not, why do you accept that it has been proved? It seems to me that, just like other things in the human world, what is "true" is a matter of social consensus. In this case, it is more like the lack of dissent opinion from a very small elite group of experts. In the history of math, there are many math geniuses who did not water-tight proofs or produced proofs with errors. For example, Riemann's proof of his mapping theorem was known to be defective. It was not cleaned up until many years later. Yet, the idea was correct. In addition, it is commonly known the Euclid's work is full of holes. Yet, that fact has not prevented him from making an important contribution to mathematics. Thus, it seems that the importance of formality in general and of real analysis in particular has been oversold.

    It seems that our basic disagreement has something to do with your unfamiliarity with the history and philosophy of mathematics. Many people think of math as the last bulwark of certainty. It is not. Math has lost its certainty since the emergence of non-Euclidean geometry. I tend to agree with you that if you have an axiomatic system and you deduce your conclusion through careful logical reasoning in every step, then you can achieve a high degree of certainty. But what about the assumptions of the system? By definition, the axioms are assumed to be true. They are not subject to tests.
    This guy does not really understand mathematics. He has a point, but he does not really understand that point.

    It is quite true that the level of rigor that is expected in modern mathematical proofs is not evident in much older mathematics. The modern definition of the function and real proofs did exist much prior to Cauchy.

    It is also true that modern mathematical proofs generally use shortcuts and allusions for some of the simpler steps that can be filled in by any good mathematician if necessary, but that are tedious and detract from the overall presentation of a new result. This is not really a weakness.

    The most recent issue of the Notices of the American Mathematical Society contains some interesting discussions of work that has been done to reduce many modern proofs to pure symbolic logic traceable directly to the Zermelo Frankel axioms plus Choice. It has a direct bearing on the otion of the rigor and and reliability of modern proofs.
    I do not understand why you say the guy does not know what he is talking about, then you give two examples in which you say that he is actually right and then in the end you say that he got it all wrong because it is not a weakness.
    I had assumed that the reader had some understanding of the nature of mathematics, which Jane does. Let me elaborate for you.

    Modern mathematical proofs are in principle traceable through the rules of formal logic to the basic axioms of mathematics -- nominally the Zermelo Frankel axioms plyus choice. Proofs from long ago may not be so traceable, but the modern formulations of the results are. These results are absoslute in nature, and show the logical connection to the axioms.

    Of course one cannot prove the axioms. That is why they are axioms. Mathematics has never claimed that the axioms are true or can be proved. They are simply an accepted starting point and intuitively plausible. To make an issue of this is to misunderstand mathematics entirely.

    Mathematical theorems are statements of conclusions that MUST follow if one accepts the axioms. Nothing more and nothing less. The style with which proofs are presented by professionals does not make this traceablility clear to amateurs, and that is necessary in order that the proofs not be overly pendantic and also so that the intuitive notions that led to the discovery of the result are not completely obliterated. That is a strength and not a weakness.

    His other statements are nothing more than "proofs are constructed by humans so how do we know that there are no mistakes?". That question can be asked of almost anything and the answer is always the same -- you cannot be absolutely certain of anything. So what ? That's life. But you can be absolutely certain that if one has not made a mistake in the logic then mathematical proofs are correct and the result does indeed follow from the premise. You can also be absolutely certain that the major results in mathematics have been reviewed by a horde of experts many times from many angles and the chances of a real error are small. Those errors that creep in, and some do, do not go unnoticed for very long, and when they are found things get cleaned up.
    Reply With Quote  
     

  7. #6  
    . DrRocket's Avatar
    Join Date
    Aug 2008
    Posts
    5,486
    [quote="JaneBennet"]
    My friend then claims that statistics is a branch of mathematics, and statistics is in a large part inductive in nature. Which leaves me at a loss on how to proceed – except that I still know that my friend is still confused about what’s deductive and what’s inductive.
    quote]

    Statistics if often practiced more as a branch of science than as a branch of mathematics. While there are many statisticians with a firm grasp of the relevant mathematics it is also true that there are those who have done rather well in statistics without a firm grasp of the foundations of probability.

    I have seen, for instance, elementary treatments of statistics that start with the assumption that there are infinitely many independent equally events. If you think about that for a minute you can conclude that they must all be of probability zero.

    It is easy to become confused about what is deductive and what is inductive. Mathematical research proceeds inductively to discover new results. Once you have guessed correctly then the deductive tools come out to prove that the guess was correct. Rarely is the inductive step revealed, so the non-participant never sees that aspect of mathematics.

    Compounding the confusion is the recent emphasis on "applied mathematics". There are many "applied mathematicians" who never prove anything. They DO NOT prove theorems. That inductive approach can confuse non-mathematicians.

    Mathematics and particularly modern mathematics at the research level is extremely hard, perhaps impossible, to explain to non-mathematicians, even advanced scientists and engineers. That is because it requires a great deal of training to even understand the problems let alone the solutions. I have no idea how to remedy this situation. It is likely that you will never be able to explain to your friend the difference between science and mathematics, and in particular the nature of mathematical proof as opposed to the validation of scientific theories.
    Reply With Quote  
     

  8. #7  
    Forum Professor sunshinewarrior's Avatar
    Join Date
    Sep 2007
    Location
    London
    Posts
    1,525
    I'd go with the points made by Jane and Dr Rocket but add a leetle something in defence of the economist:

    1. This might be an argument about apples and oranges. He is making philosophical points about maths - effectively meta-mathematical speculations. These are, without question, fair game, and a degree in mathematics is not necessary to make these comments. Hofstader, in Godel Escher and Bach, makes similar points, and, as he himself pointed out, Euclid's "parallel lines" axiom is a bit of a sore point, philosophically speaking.

    2. He is, however, making a mistake in trying to cross over from meta-mathematics to pure maths and use these meta-mathematical speculations as evidence that there is anything 'wrong' with the proofs of pure mathematics (like Andrew Wiley's proof of Fermat's Last Theorem).

    3. There is also, however, a glimmer of truth in some of his claims about modern proofs. I believe that the Four Colour Theorem has only been proven thanks to the efforts of large amounts of computer work. That is, we trust that it is a correct proof, but humans by themselves have no way of generating it, or even checking it: we have to place some degree of faith in the computer's having got it right. To that extent, even pure maths these days is becoming, in the scientific or inductive sense, a touch experimental!

    I don't know if the two of you will agree with this, but then I'm coming at it not from the point of view of a mathematician (even though I love the subject) but as a (wannabe) philosopher!
    Reply With Quote  
     

  9. #8  
    . DrRocket's Avatar
    Join Date
    Aug 2008
    Posts
    5,486
    Quote Originally Posted by sunshinewarrior
    ...

    3. There is also, however, a glimmer of truth in some of his claims about modern proofs. I believe that the Four Colour Theorem has only been proven thanks to the efforts of large amounts of computer work. That is, we trust that it is a correct proof, but humans by themselves have no way of generating it, or even checking it: we have to place some degree of faith in the computer's having got it right. To that extent, even pure maths these days is becoming, in the scientific or inductive sense, a touch experimental! I
    There is no more than a very faint glimmer. a dying ember.

    It is true that the 4 color theorem involved a complex computer program. The heart of the program however is the ability to reduce the problem to a finite number of cases that can be generated and checked by computer. It is also true that other independent computer programs have been used to reduce the proof of the 4 color theorem to the formal manipulations of symbolic logic within Zermelo Frankel, plus Choice, axiomatic set theory and formal logic. So the proof has been checked quite thoroughly.

    Mathematics is most certainly not experimental, and that is what separates mathematics from science. There is a place for experiments in helping to check some exmples, but they don't replace rigorous proofs. Computers have been used, and used profitably, to do long an laborious calculations. A recent example is the calculation of the character table of E8, a particularly complex exceptional Lie group. But there is a great deal of traditinal theory that underlies that long computer calculation. And it was a calculation, not an experiment.
    Reply With Quote  
     

  10. #9  
    Forum Professor sunshinewarrior's Avatar
    Join Date
    Sep 2007
    Location
    London
    Posts
    1,525
    Quote Originally Posted by DrRocket
    Quote Originally Posted by sunshinewarrior
    ...

    3. There is also, however, a glimmer of truth in some of his claims about modern proofs.
    There is no more than a very faint glimmer. a dying ember.
    I quite agree. I can get beside myself with frustration when someone with only a little post-modernist learning decides to claim that mathematics (or science) is run by an elite akin to a priesthood. This is nonsense on stilts as anybody with the slightest inclination can check a number of the proofs herself without a degree in mathematics. Most elementary proofs are ones even I can check and I stopped any formal mathemtical studies by the time I turned 19 (many, many moons ago). This is not the sort of stuff a priesthood allows.
    Reply With Quote  
     

  11. #10  
    Forum Junior
    Join Date
    Apr 2007
    Location
    Somewhere near Beetlegeuse
    Posts
    205
    If I may, I think all three of you have misunderstood the point the economist was trying to make. Dr. Rocket said that proofs are checked quite rigorously; he said of the 4-colour theorem:

    Quote Originally Posted by Dr. Rocket
    It is also true that other independent computer programs have been used to reduce the proof of the 4 color theorem to the formal manipulations of symbolic logic within Zermelo Frankel, plus Choice, axiomatic set theory and formal logic. So the proof has been checked quite thoroughly.
    (emphasis added by me)

    The economist's point was that you personally have not done this checking. You personally do not know in a formal sense that the proof is correct, you are simply taking it on trust that the checking has been done correctly, rigorously, and accords with your axioms.

    In this sense you are taking it somewhat on trust that the proofs are correct. He is not claiming any knowledge that a particular proof is not correct, he is merely pointing out that there is some faith involved.

    At least, that is how I read his words as posted by JaneBennet.
    Everything the laws of the universe do not prohibit must finally happen.
    Reply With Quote  
     

  12. #11  
    . DrRocket's Avatar
    Join Date
    Aug 2008
    Posts
    5,486
    Quote Originally Posted by numbers
    If I may, I think all three of you have misunderstood the point the economist was trying to make. Dr. Rocket said that proofs are checked quite rigorously; he said of the 4-colour theorem:

    Quote Originally Posted by Dr. Rocket
    It is also true that other independent computer programs have been used to reduce the proof of the 4 color theorem to the formal manipulations of symbolic logic within Zermelo Frankel, plus Choice, axiomatic set theory and formal logic. So the proof has been checked quite thoroughly.
    (emphasis added by me)

    The economist's point was that you personally have not done this checking. You personally do not know in a formal sense that the proof is correct, you are simply taking it on trust that the checking has been done correctly, rigorously, and accords with your axioms.

    In this sense you are taking it somewhat on trust that the proofs are correct. He is not claiming any knowledge that a particular proof is not correct, he is merely pointing out that there is some faith involved.

    At least, that is how I read his words as posted by JaneBennet.
    I don't think that was the intent of the economist.

    It might surprise you to know that in fadt EVERY theorem that I have used in research I have PERSONALLY checked and can trace the proof back to the Peano axioms. Now I have not reduced the proof to just formal manipulations within symbolic logic, but I have checked the proof in the more conventional manner of making sure that each step in the proof was valid. This is of course subject to human error, and I have not used anything like every theorem ever proved, but I have personally checked every theorem that I have used. I have even made mistakes myself and had to correct them later, after a gap in logic was found by someone else -- but the results were correct, and the gap was closed before they were published.
    Reply With Quote  
     

  13. #12  
    . DrRocket's Avatar
    Join Date
    Aug 2008
    Posts
    5,486
    Quote Originally Posted by numbers
    If I may, I think all three of you have misunderstood the point the economist was trying to make. Dr. Rocket said that proofs are checked quite rigorously; he said of the 4-colour theorem:

    Quote Originally Posted by Dr. Rocket
    It is also true that other independent computer programs have been used to reduce the proof of the 4 color theorem to the formal manipulations of symbolic logic within Zermelo Frankel, plus Choice, axiomatic set theory and formal logic. So the proof has been checked quite thoroughly.
    (emphasis added by me)

    The economist's point was that you personally have not done this checking. You personally do not know in a formal sense that the proof is correct, you are simply taking it on trust that the checking has been done correctly, rigorously, and accords with your axioms.

    In this sense you are taking it somewhat on trust that the proofs are correct. He is not claiming any knowledge that a particular proof is not correct, he is merely pointing out that there is some faith involved.

    At least, that is how I read his words as posted by JaneBennet.
    I don't really think that is what the economist meant.

    In any case it might surprise you to know that in fact I have PERSONALLY checked EVERY theorem that I have used in research. Now, I have not reduced those proofs to just manipulation of symbols in formal logic, but I have checked each step and each assertion in the proofs in the conventional manner. I have not used and checked anything like every theorem ever proved in mathematics, but I have checked a lot of the more common theorems -- that is what is done in graduate mathematics classes and in research seminars. Proofs are checked all the time, very carefully, and by individuals and groups with a lot of insight. Before a mathematician uses a result, he checks it. And if the theoremis a big result he checks it from several angles, and consults colleagues to get second, third, fourth, etc. opinions -- which is what happens in seminars. That is how mathematics research is done.
    Reply With Quote  
     

Bookmarks
Bookmarks
Posting Permissions
  • You may not post new threads
  • You may not post replies
  • You may not post attachments
  • You may not edit your posts
  •