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trilobite submitted a new PF Insights post
Simple Geometry, Deep Math
Continue reading the Original PF Insights Post.
Simple Geometry, Deep Math
Continue reading the Original PF Insights Post.
Yes, that's a fascinating topic! Thanks for mentioning it.Robert Smart said:We should perhaps mention the boundary between computable numbers (includes e and pi) and non-computable numbers. Since there are only a countable number of computable numbers there are much more non-computable numbers. The Mathematics you get into with this is near the boundary with Philosophy. To me the non-computable numbers don't exist [disagree: show me one]. However they give us a model of the real line that is logically consistent and helps us to think clearly about it.
I believe that a computable number is one that can be computed to any desired number of decimal places by a finite number of operations. Usually, this is expressed in terms of a computer program that accomplishes the required calculation. The interesting thing is that the computable numbers are countable, essentially (I think) because there are only a countable number of computer programs. Since the real numbers are uncountable, this means that not only do non-computable numbers exist, but that "most" numbers are non-computable! However, as with transcendentals, actually exhibiting a non-computable number is not easy.Jimster41 said:care to explain a bit what the def of a non-computable number is?
Assuming LEM and assuming that decimal representation for real numbers is used, whatever you have written should be correct.trilobite said:I believe that a computable number is one that can be computed to any desired number of decimal places by a finite number of operations. Usually, this is expressed in terms of a computer program that accomplishes the required calculation. The interesting thing is that the computable numbers are countable, essentially (I think) because there are only a countable number of computer programs. Since the real numbers are uncountable, this means that not only do non-computable numbers exist, but that "most" numbers are non-computable! However, as with transcendentals, actually exhibiting a non-computable number is not easy.
I guess I meant "not easily understandable." But this is not an area I know a whole lot about. I would be interested to see an easily understandable non-computable number, if you have an example.SSequence said:However, it is not quite clear what you meant with the last sentence though. It is quite easy to demonstrate specific examples of uncomputable real numbers. Of course just given a description it might not be easy at all to show that the given real number is uncomputable (and this is perhaps what you meant).
The standard definition is that a number is algebraic if it is the root of a polynomial with rational coefficients. However, this wording was not very useful for the purposes of my article, so I chose a slightly different description.SSequence said:The category of algebraic/transcendental numbers seems to be an interesting one. What is the simplest (and also precise) definition of algebraic numbers?
Well there are many examples that one can easily give of course. I should perhaps describe a few that are more instructive.trilobite said:I guess I meant "not easily understandable." But this is not an area I know a whole lot about. I would be interested to see an easily understandable non-computable number, if you have an example.
SSequence said:In other words there is no marker as to when the values f(0), f(1), f(2) etc. end. What I mean here is that suppose we had f(0)=10, f(1)=6, f(2)=99, ... then the first few decimal places will be like:
0.10699...
Is the answer of computability of this real number independent of the function F? If it is, then is this real number always computable or always uncomputable?
Well it would seem that it certainly depends how you define the words involved really. But, for example, many mathematicians (though not all) would agree that the definitions of reals (in post#8) described in examples (1), (2) and (3) are "well-specified", "well-defined", "well-described" (whatever you want to call it) but they will say that those real numbers do not have a computable decimal representation.Jimster41 said:How wrong is it to cartoon it this way: A computable number is an example of a specific symbol that is repeatably accessible through some finite set of (repeatable?) symbolic steps. A non-computable number is one that cannot be specified (because specifying it would show it is computable).
w.t.h?
Thanks for describing your interesting (and ambitious!) project. [By the way, for some reason your post does not appear in "Discuss in the Community", at least not for me. Fortunately, I spotted it in Insights by scrolling down below my article, although usually I can't see any of the comments there, at all!]David Reeves said:Problems arise even from simple high school geometry. Here is my example. Keep in mind, this is not my main area of expertise.
I have been working on a physics-related AI project, and part of this project was developing an expert system. An expert system's two main modules are known as the knowledge base and the inference engine.
I decided to work on the inference engine first. I developed an automated theorem prover in LISP. This ATP uses first order predicate logic (FOPL).
After solving a few toy problems as a test, I turned my attention to Euclid's Element of Geometry. Since high school I regarded this as the quintessential book of infallible deductive logic. So I was shocked when I ran into difficulties in the very first proposition of Book 1. In this proposition, Euclid constructs an equilateral triangle in a very simple way. There seems no doubt that his proof is valid. Until, that is, one demands strict use of FOPL, with an acceptable set of axioms. By acceptable axioms I mean "self-evident truths" concerning what we now call Euclidean space.
Unfortunately, Euclid makes some steps in his first proof which are not justified, based on his initial set of axioms. It's very clear that what he is doing is mixing deductive logic, with statements based on looking at a diagram, and applying our common-sense notions concerning geometrical objects in 2D space. Yet I would say his proof is valid. I can perceive its truth. This tells me that at least for some cases, proof in the sense of strict formal logic is not required. So I say Euclid was doing the right thing by taking a practical approach. Now my belief in logic as the ultimate verifier of mathematical truth begins to fade a bit.
I knew beforehand that some of the earlier geometry theorem proving programs ran into difficulties, but I wanted to see for myself. I assumed that there must be some way of proving all of Euclid using FOPL, provided one had a sufficient set of axioms for each proof. Perhaps the old timers did not have powerful enough computers to handle this task? Perhaps their funding ran out? Or their expertise was demanded for more important problems?
My conclusion was not what some logicians would accept. I decided that Euclid was right, and that trying to fit every one of his proofs into the narrow confines of FOPL is perhaps not a useful approach. But I was still looking for the automated proof. Should I use a higher order logic? That raises other issues, such as the lack of any generally accepted automated proof method, comparable to proof by resolution for FOPL.
I looked into other techniques, such as found in the old PLANNER system. But PLANNER was never fully developed, in the sense of Hewitt's original proposal. I also tried PROLOG and ran into various problems. So I developed my own variation of backward chaining that works well. It runs in acceptable time using a LISP interpreter running on my PC. I decided to go ahead with FOPL because I saw no practical alternative.
Now the problem was the axioms I needed for each proof. I do not like my axioms. They are too complicated.
I looked in as many places as I could for someone who had actually published a complete automated proof, using FOPL, of Euclid, beginning with a relatively small number of simple axioms. Does it exist?
People talk about axioms sets of Hilbert, Tarski, and Birkhoff. But has anyone actually used one of these axiom sets and proved every one of the propositions of Euclid? Or have they come up with a published set of axioms and proofs of their own? A paper that only presents a small number of proofs does not impress me, because I can do the same thing quite easily. Also, I want a print-out from the program, not the human transcription.
I found out recently that in 2003 Meikle and Fleurot looked into Hilbert's geometry, as presented in his Grundlagen, and discovered some flaws. I'm not surprised. Even geniuses make mistakes. That is one reason we try to develop automated theorem provers and proof verifiers.
Far from giving up on this issue, it's something that still fascinates me. If nothing else, I have a FOPL theorem prover that works, and indeed without relying on recursion with backtracking. I won't say it's the fastest.
The other module is the knowledge base. When I look into how to represent math and physics knowledge, it's a more difficult problem than developing a FOPL prover. Of course it's often stated that we need second order logic. But as I said, I was beginning with what could be done using just FOPL.
On a more positive note, my favorite moment in this work was when my theorem prover made the constructions needed for Book I Proposition I. People might ask a human who solves this, "how did you come up with that idea?" Some would say it's human creativity or imagination. Or is it? In fact, it's just logic. Working only from the original axiom set, which of course excludes all the results of later proofs, what method do you have to prove that two lines are equal? The method my theorem prover came up with was that two lines are equal if they are both radii of the same circle. That is exactly what Euclid does.
But originally in the problem there is only a line, and no circles. I needed to add a construction axiom. Now the prover can constructs the circles. But it clearly has no "imagination" or "insight" or whatever other humanistic term you may use. It's just searching through many possibilities until it finds one that works.
My AI project has turned out to be much harder than I thought it would be. It's not like Chess programming, which I have studied. It was realized decades ago that creating a Chess program which could beat the world champion was only a matter of a faster processor and more memory. This could be seen by extrapolating a graph that showed the Chess rating vs. the hardware factors. I think this is because Chess is actually a simple problem, compared to geometry.
In conclusion, I think it's remarkable that for thousands of years humans taught Euclid's geometry as the best example of strict deductive reasoning. His reasoning method seems simple, so that even a young student can understand it. Euclid uses simple logic, and computers are logic machines. Yet creating a programmatic "Euclid in a box" turns out to be a challenging problem.
trilobite said:Thanks for describing your interesting (and ambitious!) project. [By the way, for some reason your post does not appear in "Discuss in the Community", at least not for me. Fortunately, I spotted it in Insights by scrolling down below my article, although usually I can't see any of the comments there, at all!]
Again, visit "Gödel, Escher, Bach" by Douglas Hofstadter. He discusses these concepts at length.SSequence said:First define any partial recursive/computable function f:N→N (where N={0,1,2,3,...}) as whatever a simple C-like program (to use a concrete example) of the following type could calculate:
int f (int x) {
// insert some program code
}
To define any partial recursive/computable function f:N x N→N whatever a simple C-like program of the following type could calculate:
int f (int x, int y) {
// insert some program code
}
and so on. Whenever the program is stuck in a loop and doesn't get to the "return" command, it is assumed that the function f isn't returning any value for that particular input (and hence the function f is considered partial in that case).
I do not think that is enough. Have you seen the chapter where he introduces the "programming languages" FLooP, BLooP and GLooP?SSequence said:Well I am aware of it and I have sort of flipped through it (reading a few paragraphs/pages here and there).
David Reeves said:You are so right when you mention how our school math class glosses over certain things. Your example of the ellipse is very good. It seemed so simple when I learned the formula. It's even easy to construct an ellipse, using two pins and a string. I can measure the circumference by straightening out the string and using a ruler. So why is the math difficult?
That used to happen to me, too (not that long ago.) For some reason, the feature works for me now. I don't think it has to do with membership level, but I have no idea why this happens.David Reeves said:I enjoyed this article very much. I would like to give it 5 stars. But when I try to do so, I am asked to "login in" even though I am already logged in. Do I lack the necessary membership level?
That's an excellent question, but I don't have an answer. It's one thing to live in a physical universe that is hard to understand, but as my article points out, "mysteries" lurk even in the ideal world of rudimentary geometry (squares, circles, ellipses!) It would be "nice" if the square root of 2 and pi were good old rational numbers. (Nice, but maybe boring!)David Reeves said:It's even easy to construct an ellipse, using two pins and a string. Then I can take another string and make it follow the curve of the ellipse. I can measure the circumference by straightening out the string and using a ruler. So why is the math difficult?
Thanks for this interesting remark. My personal bias, for which I have no evidence, is that the "correct" math is continuous, although we use it to model a universe that is likely discrete.David Reeves said:But in the back of my mind is the digital physics of Zuse, in which the most correct math is discrete, because our universe is a process carried out on some kind of cosmic grid. But as Zuse pointed out that raises its own problems, including a clash with relativity theory.
Big fleas have little fleas,trilobite said:Thanks for this interesting remark. My personal bias, for which I have no evidence, is that the "correct" math is continuous, although we use it to model a universe that is likely discrete.
Svein said:Big fleas have little fleas,
Upon their backs to bite 'em,
And little fleas have lesser fleas,
and so, ad infinitum.
Is there a proof that intergral of perimeter of an ellipse is can't not be written in terms of elementary functions or is it that nobody has found the solutions and it is a open problem ?trilobite said:Thanks for this interesting remark. My personal bias, for which I have no evidence, is that the "correct" math is continuous, although we use it to model a universe that is likely discrete.
Great question! Unfortunately, I don't know the answer. Every reference I can find simply asserts that there is no elementary formula without giving any justification. If someone can comment with insight into this question, I would definitely be interested to hear it.Buffu said:Is there a proof that the perimeter of an ellipse cannot be written in terms of elementary functions or is it that nobody has found the solution and it is an open problem?
Simple geometry is the study of basic shapes and their properties, such as points, lines, angles, and polygons. It is often taught in elementary and middle school as an introduction to more complex mathematical concepts.
Deep math refers to the more advanced and abstract concepts of mathematics, typically studied in high school and beyond. It involves complex mathematical structures, theories, and proofs.
Simple geometry provides the foundation for understanding more complex mathematical concepts in deep math. It introduces students to fundamental principles and shapes that are used in more advanced math courses.
Simple geometry is useful in everyday life, such as measuring angles and distances, while deep math has applications in fields like engineering, physics, and computer science. It is also essential for problem-solving and critical thinking skills.
Practice and repetition are key to improving your understanding of simple geometry and deep math. It is also helpful to seek out additional resources, such as textbooks, online tutorials, and working with a tutor or study group.