Understanding MathematicsDate: 07/25/99 at 22:02:09 From: Kiki Nwasokwa Subject: Understanding Mathematics Dear Dr. Math, I have a problem. I want to analyze and understand every aspect of mathematics from the simple to the most complex - but I feel compelled to ask questions like "how can we prove that the commutative property is true?" Then I set out to prove it - and I do. I start with a set of my own, as well as accepted, postulates and then build on them to create a larger axiomatic system of definitions, proofs, and theorems... I cannot accept what I have been taught. I can learn it, but I must know WHY and HOW until I'm splitting hairs, and wondering whether it all really matters anyway. This all started when I asked why/how the product of two negative numbers is positive, and actually proved it. ("Proved" as in I showed my proof to my high school math teacher and he was impressed.) What followed was my belief that everything in math must be proven in order for it to be true. In other words, for me, everything must be a definition, a postulate, or a theorem. (I'm not sure whether I believe in properties yet.) The problem, though, with all of this is: How do I know that MY postulates and theorems are true, or are even being done correctly? Is anything really true in math? Is there a name for the type of scrutiny and analysis that I do and enjoy? Is this what they call "number theory"? (Please say yes, because then I can sleep better at night knowing I can study this all in graduate school.) Date: 07/26/99 at 21:26:20 From: Doctor Ian Subject: Re: Understanding Mathematics Hi Kiki, One way to define 'mathematics' is to say that it is the practice of deriving theorems from axioms -- nothing more, nothing less. Some of the theorems that have been derived in the past have turned out to be useful for building and designing things, for explaining empirical phenomena, and for balancing one's checkbook -- but those are applications of mathematics, not mathematics itself. So, to answer your question, the name for what you are doing is not 'number theory', unless you are restricting your inquiries to the properties of integers. The name for what you are doing is 'mathematics'. Mathematics can also be defined as the construction and exploration of formal systems. The advantage of this definition is that it emphasizes the somewhat subtle point that mathematics IS formal -- which is just another way of saying that it has no NECESSARY relation to anything in 'the real world'. What all this means is that what is 'true' in mathematics depends entirely on what axioms you start with, and what rules you use to combine them. To take your example of 'proving' that the product of two negatives is positive -- your result is true given the system you set up, but it may be false in other systems. Kurt Godel showed that any formal system that you can set up can have the property of completeness, or the property of consistency, but not both. (Doug Hofstadter's book _Godel, Escher, Bach_ provides a wonderful introduction to this idea, and many others -- if I were you, I would go out and buy a copy as soon as you finish reading this message.) What does this mean? A system is complete if any statement that is true can be derived. A system is consistent if any statement that can be derived is true. Do you see the difference? In a complete system, you can derive anything that is true. But what Godel showed was that if you can derive everything that is true, you will also be able to derive things that are false. In a consistent system, you can't derive anything that is false. But what Godel showed was that if you can only derive things that are true, then there will always be things that are true that you won't be able to derive. Quite a pickle, isn't it? In 1901 Bertrand Russell (who once defined mathematics as "the subject in which we do not know what we are talking about nor whether what we say is true") discovered the following paradox: Some sets, such as the set of teacups, are not members of themselves. Other sets, such as the set of all non-teacups, are members of themselves. Call the set of all sets which are not members of themselves S. If S is a member of itself, then by definition it must not be a member of itself. Similarly, if S is not a member of itself, then by definition it must be a member of itself. This little bump in the road prompted a complete re-examination of the foundations of mathematics. Godel's discovery was in part a result of that re-examination. When Warren McCulloch, one of the great minds in the theory of computation, was asked upon entering Haverford College what he wanted to do. He replied, "I have no idea, but there is one question I would like to answer: What is a number, that a man may know it, and a man, that he may know a number?" The point is, these are the kinds of things that mathematicians think about, which means that you appear to already be well on your way to becoming a mathematician. Thomas Huxley once said that the happiest man was the one who had found his work, and had only to do it. Now I have a question for you: Why wait until graduate school to study number theory, or anything else, if that's what you want to do? The books you'll be told to read when you get there have already been written. If you read the biographies of great mathematicians (and I think you should), you'll find that one thing they all had in common is that they didn't wait around for other people to teach them mathematics. They went out and learned it from books, or made it up as they went along. I suggest that you take the same approach. You can start here: http://csmaclab-www.uchicago.edu/philosophyProject/sellars/carroll/tortoise.html Don't hesitate to write back if you want to talk more about this, or anything else. - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ Date: 07/27/99 at 19:22:57 From: Kiki Nwasokwa Subject: Re: Understanding Mathematics Thank you very much. I feel more confident about how to deal with my "problem." Your response to my first question prompted even deeper questions on the nature of mathematics: Are the "rules" governing mathematics arbitrary? Is a system of mathematics "true" or "valid" just as long as these rules are followed consistently? What is the measure of such consistency? Date: 07/31/99 at 11:57:51 From: Doctor Ian Subject: Re: Understanding Mathematics Hi Kiki! Nice to hear from you again. Your questions are deep ones. That's good. Those questions turn out to be the most fun. To answer your last question first, the 'measure of consistency', so to speak, of any system is whether you can use the rules of the system to derive a contradiction. Once you can derive a contradiction, you can derive anything at all -- which is how you know that your system is complete, and therefore not consistent. It turns out that the answer to your first question is 'yes and no'. The rules are arbitrary, in the sense that you can choose any rules you want. But the rules are not arbitrary, in the sense that most possible sets of rules that you could choose turn out to be not very useful. I recommend that you read the article "Communication with Alien Intelligence," by Marvin Minsky, which was published in the April 1985 issue of _Byte_ magazine. It's probably not clear from the title what it has to do with your question, so I'll excerpt a little of it here, so that you can see why you'd want to read the rest of the article: Once, while I was still a child in school, I heard that 'minus times minus is plus'. How strange it seemed that negatives could cancel out -- as though two wrongs could make a right. I wondered if there could be something else, still like arithmetic, but having yet another sign. Why not make up some number things, I thought, that go not just two ways, but three? I searched for days, making up new little multiplication tables. Alas, each system ended either with impossible arithmetic (e.g., with one and two the same), with no signs at all, or with an extra sign. Eventually, I gave up. If I had had the courage to persist, as Gauss did, I might have discovered the arithmetic of complex numbers, or, as Pauli did, the arithmetic of spin matrices. But no one ever finds a three-signed imitation of arithmetic because, it seems, it simply doesn't exist. Try, for example, to make a new number system that's like the ordinary one except that it skips some number -- say, 4. It just won't work. Everything will go wrong. You'll have to decide what 2 plus 2 is. If you say that this is 5, then 5 will have to be an even number, and so also must 7 and 9. Then, what's 5 plus 5? Is it 8, or 9, or 10? You'll find that to make the new system at all like arithmetic you'll have to change the properties of all the other numbers. Then, when you're done, you'll find that you have changed only those numbers' names and not their properties at all. Similarly, you could try to make two different numbers be the same -- say, 139 and 145. But then, to make subtraction work, you'll have to make 6 the same as 0 and 4 plus 5 equal to 3. Suddenly, you'll find that the sum of two positive numbers is smaller than either of them--and that scarcely resembles arithmetic at all. (In fact, this leads to modular arithmetic, which has a certain usefulness in abstract mathematics but is worse than useless for keeping track of real things.) And so it goes. A version of this article can be found on the Web at MIT: http://www.ai.mit.edu/people/minsky/papers/AlienIntelligence.html . So 'arithmetic', which is one particular system for generating and combining numbers, appears to be useful, while many 'similar' systems appear to have all kinds of complications that make them useless. To use an imperfect analogy -- don't try to push this too far -- when you have an attractive melody, there are lots of other 'similar' melodies that aren't attractive at all. It appears that attractive melodies -- and useful mathematical systems -- are little islands surrounded by seas of unattractive -- and useless -- ones. The question that Minsky is addressing in his article is: Why should this be the case? In a sense, then, a mathematician is like an explorer looking for islands in the Pacific, back in the days before airplanes and satellites. There's no way to know where they'll be, or just what they'll be like, but -- to an explorer, at least -- there's nothing quite as exciting as finding one. One other point: Note that Minsky mentions that modular arithmetic is useful for some kinds of tasks, and not for others. In general, mathematicians don't go out looking for systems because they hope that the systems will be useful for any particular purpose -- except perhaps when that purpose is to generate a valid proof in another system. (For example, entire fields of mathematics were generated by mathematicians trying to prove Fermat's last theorem.) Mathematicians generally look for systems because they find such systems interesting, and for no other reason. Having discovered various systems and worked out interesting results within them, they leave it for other people to determine whether the systems are useful -- as when the quantum physicists in the 1930's found a use for linear (matrix) algebra, and as when the cryptologists of recent decades found a use for number theory. One exception to this is the mathematics that will be needed to work out the new String Theory of physics. Usually in physics, theorists think up a theory, then look around for the mathematical tools that they will be able to use to make predictions with the theory. Always in the past, they've been able to find such tools. But for the first time in the history of physics, the theories are ahead of the mathematics. It should be interesting to see how this plays out. I hope you'll keep asking questions like these, and even if you find that Dr. Math can't answer them, read as much as you can about mathematics and mathematicians. A couple of good places to start would be with Richard Courant's book, _What is Mathematics?_, and a series called _The World of Mathematics_, edited by James R. Newman. - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ |
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