It was how to get that looking like the gaussian bell centered at x=0 in a coordinates system and add some background as the "undefinable" future of mathematics discoveries. Such discoveries are all so "useless" when they are made, but on some day millions of people use cameras and computers which could never been made without those "useless" inventions in their own times.
And this is one of the most beautiful things in mathematics. It is not "useless", it are "purposeless" for itself, because it is not done for any other "purpose" than for examining the consequences of the ideas in mind just for finding the truth. And look there! It is exactly this that enables everyday life some centuries later.
Wow, 97/100!!! Why dodn't you go to physics, Saad? You seem to have to necessary appetite for that.
Anyway, a big, a huge pity that most of the works aren't translated from russian to english language. Especially maths... the guys there are really incredible!
Nick,my first ever scientific book to read,at 12 or 11,I do not remember was titled the entertaining physics,a Russian one that translated to Arabic,by MIR publishing house,at the times of the previous,USSR,it was given to me for free,at one book show that held in Mosul,during the sixties, that book influence me regarding the subject of science,and I have got the highest mark in physics at the final in the secondary school,and that was 97/100 cheers, Saad.
That link was one of Podnieks' contributions to the foundations of mathematics. He writes very very well and very very understandable about such subjects right in the very foundations of mathematics. A very lucid, extremely sharp mind. Really, sometimes when you read his works, you start questioning what you thought as "given" unitl now.
Like... if the barber shaves all people that don't shave themselves, then... who shaves the barber? ;-) A typical, clear and simple... undecidable question, though it is still possible to formulate it.
thank you Nick, that link was a student text book, it is enough for me now ,I have introduced to Gödel's Theorem in a good way,I know for now,now something about mathematics university at Greece,also some facts about mathematical world,and how it enclose everything,and how the mathematician looks at the surrounding worlds,and that is fair and enough for me I think,thank you very much for the time ,effort,and revision of my inputs about your Godel. my regards,and Merry Christmas, Saad.
Algorithm 1. Given the axioms of a fundamental formal theory T this algorithm produces a closed PA-formula RT. As a closed PA-formula, RT asserts some property of the natural number system.
Algorithm 2. Given a T-proof of the formula Tr(RT) or the formula ~Tr(RT), this algorithm produces a T-proof of a contradiction.
Therefore, if T is a fundamental theory, then either T is inconsistent, or it can neither to prove, nor to refute the hypothesis RT. A theory that is able neither to prove, nor to refute some closed formula in its language, is called incomplete. Hence, Goedel and Rosser have proved that each fundamental theory is either inconsistent, or incomplete.
Which also tells you that *any* such system is also formalizable by... natural numbers!!
Now, to your emphasis on the word "within". Exactly this is the very pwoer of that theorem. You always are "within" some formal system. But only mathematics proves to you something about that from "within". It doesn't need to go "out" for proving that. Actually there is no "out" at all. And at the end the theorem also says that no system "knows" exactly about its own self. It only contains a rather "diffuse" representation of the own self.
For a good and very understandable work on that, consider reading http://www.ltn.lv/~podnieks/gt.html (Scroll down until you come at the table of contents and read it from the start to the end.)
But I start by asking you, what *is* a "rigidly logical mathematical system"? Is there any system at all, that isn't "rigidly logical mathematical?" Any system, in order to be a system at all, has its axioms, its own logic and inference rules, out of which its "truts" (e.g. theorems) can be reached. Mathematics is not only "numbers" and "shapes" and "functions". Anything can be mathematized.
So we return to the uncompleteness theorem, which you didn't really dealt with. You did read some things about it but form what yoi write to me, I must assume that you didn't read mathematics about it. Goedel didn't state it at all for "rigidly logical mathematical systems". (Simply because this would be a tautology.) He proved it for *funamental* systems, and this word "fundamental" doesn't have to do with what one might "feel" about it. It has a mathematical definition that doesn't care about what one might "think" of it.
A formal theory T is called a fundamental formal theory, if and only if there is a translation algorithm Tr from PA into T such that, for all closed PA-formulas F, G:
1) If PA proves F, then T proves Tr(F).
2) T proves Tr(~F), if and only if T proves ~Tr(F).
3) If T proves Tr(F), and T proves Tr(F->G), then T proves Tr(G).
One good formulation of the theorem is then: If T is a fundamental formal theory (only conditions 1, 2 are necessary), then one can build a closed PA-formula RT (i.e. a formula asserting some property of natural numbers) such that if T proves Tr(RT) or T proves ~Tr(RT), then T is inconsistent.
Now, we reduced it back to the question about: What is a formal system?
And a formal system is each and every system the "fundaments" can be formalized. For example, any religion is a formal system, since it does have its dogmas (axioms), its own logic (which can be also not standard-logic), out of which then we derive other "truts" (its theorems).
So, you want me to prove you here and now that any statement about an almighty (for "complete") and at the same time perfect (for "consistent") "deity" is simply completely impossible?
BTW, mathematics is everywhere because it is in our minds, even if we think it isn't.
Very well Nick,I think this the only site on the planets that designed and purposed for photography that contains such a nice logical mathematical synopsis of some pure scientific theory,I would like to add one thing here that this fine Goedel,stated that within any rigidly logical mathematical system there are propositions (or questions) that cannot be proved or disproved on the basis of the axioms within that system and that, therefore, it is uncertain that the basic axioms of arithmetic will not give rise to contradictions. look to his statement,WITHIN ANY RIGIDLY MATHEMATICAL,I mean he put condition to his theory by the word WITHIN, Any how thank you for the effort of writing,the time taken,and your mathematical enthusiasm and devotion,and it seems to me this is in some way is related to engineering,my very best wishes and regards, Saad.
First of all, there are no stupid quastions. (But there *are* indeed stupid answers! ;-))
I don't have any theory, Saad. What makes you think that I do? Whatever I said up to know is proved not by me but by many other people. So, one can't speak about "my" theory.
Any theory is inconsistent or incomplete (if it satisfies some requierments) - this is a theorem by Goedel. (Actually most theories do satisfy those requierments.) A theorem itself is not "inconsistent" or else it wouldn't be a theorem. If I understoof you right, you mean that the argument "each theory is either inconsistent or incomplete" could be used against the argument itself. But the term "theory" means something else than the term "therem". It is not the same.
A theorem is a statement about something and can be either treu or false. A theory is a collection of theorems, axioms, objects, etc, ect. So, Goedels theorem proves that a theory will be either:
inconsistent, which means that it can generate a statement "A is true" and at the same time "A is not true". This is named inconsistency, contradiction.
or it will be:
incomplete in the sense that there will be true statements about which it is provable that they can't be proved neither true nor wrong.
Notice that the latter doesn't say that there will be things that we can't prove if they are right or wrong. It says that there are thuths about which we *know* that they are true but we can't *prove* that they are true. So the theorem says that trith a a stronger notion than the proof, which no other science is able to prove too. Mathematics is the only science that also allows much space for truth that can't be proved but it is simply known.
And they they say tome that mathematics is not "human" and the like. ;-) Which other science is so well-behaved in order to accept and prove the own limitations?
Dear Nick, first,as a mathematician do not say what is this stupid question, could we judge that who made the human mind products inconsistant, by his theory,and tell him your theory is inconsistant ,according to your theory ? and I know for sure he was told this critique. Saad.
And so... my beloved Pallas Athena in front of the university of Athens will not welcome you with the arrogance of the "wise", or the power of the "kings". She will welcome you and everybody else with the free, pure, limitless and humble spirit of what she represents! It is the same spirit that leads to humanism, to real deep respect! It is at the end the same language that nature itself speaks: Mathematics! BTW, count up the leaves as they appear on concentrical circles on your image "Lilium", and find... the Fibonacci numbers! Did you care counting them also on each an every other structure in nature? ;-) Why is that? ;-)
Well, this is why mathematics is the humble sweet queen of sciences in a low level dress, and this is why even in our days of absolute political power, nobody, absolutely nobody enters her place carrying guns at the universities in Greece. Even the most powerful units of army are not allowed to enter the place of *mental* power in their property as "powers tha be". And this you will still only find in Greece. Nowhere else!
Or face that human smile of my Athena, my Euclides, my Pythagoras, my Archimedes, and dare then enter that place with violence. Athema will look at you and say; "But kid, we are studying theorems here. You think it's necessary to enter this place like that?"
To the really international community of thinking people!
Now, you may say, what is the use of that? To develop a whole theory of numbers that are based on the... imaginary unit? Does that number exist? The root of -1... ????
Still, some decades later physicists were dealing with electric current. (The same thing that brings light where you live.) And see there! The whole technology that is needed in order to understand and use alternating current/voltage is nothing else than the complex numbers, based on the imaginary unit, that was so "academic" when it was discovered. Gauss didn't have any idea about our electric energy supply - he did that just for knowing. And after some centuries it was found to have also practical relevance. ;-)
I could go on for ever with such examples, but the point remains the same. Mathematics is not bound to any "purpose" at all. It is free, it hugs the searching spirit, and its own spirit says not: "Discover the useful". It says: "Go discover, whatever that might be!" And this is its very own delicate kind of beauty - it is willing to accept and celebrate any kind of "useless" discovery, provided it is logically right. In mathematics you don't look for possible "industrial" or other "usefullness". You look simply for the nacked truth, only the truth, and nothing else. And some centuries later they see that you made the tool before it was even needed. ;-)
That's why everybody knows Einstein, but almost nobody knows Goedel, Dedekind, Mandelbrot. Didn't I tell you about? Well, Goedel was the one to *prove* that any kind of system, be it religion, ethics, government, etc, that "was made by human thought" has to be either imperfect or to be inconsistent. The consequence of that: There exists no god at all - and this is not philosophy! This is mathematics!
In other words, as it can be stated strictly mathematically, mathematics says to you with this "theorem of incompleteness" by Goedel, that there are indeed things.. which are only to be found by your intuition. You are the one to believe or not believe any kind of god at all - it is up to you and not to philosophers and theologists of any kind.
Watch out here! Other sciences may say that to you too by "feeling", but only mathematics *proves* that! It is a difference to say that by belief or to say that by a proof! Mathematics is the one and only science that states and proves its own imperfection! It says to you: "You wanted me to be so clear and transparent, that I might as well humbly disappear."
Well, when you do pure mathematics you don't do that for some kind of "real, material" profit. I will use a very typical example.
When Pythagoras formulated the wellknown theorem about the square of the hypothenuse being equal to the sum of squares of the other two perpendicular sides of any right-angled triangle, there was not much "practical" reason or practical "usage" of that. It was a rather "purely academic delight" to know that. It was developed and discovered without any "purpose" waiting to be statisfied and "solved" by the theorem. Nonetheless, some thousands of yeras later another mental power, named Einstein, develops first the special relativity theory, an then the general relativity. Now, this very development at its start is based on... Pythagoras! It is the same old god crystal clear theorem that made special relativity and then also general relativity possible. And even if we don't remember this today, the very function of our common GPS cannot be guarantied without the theorems of relativity, and thus: without the theorems of Pythagoras that (at those times) were made without any kind of "practical" reason. Think of that! The ships that nowadays approach safely some harbour use a GPS that works inly with knoeldege of relativity, which works only with the knowledge of that old good theorem that was discovered... just for being discovered.
Another example: The CD-ROM in your OPC works just because we know about quantum mechanics. In quantum mechanics some differential equations (and their solutions) are of fundamental importance. Now guess what? The very family of such differential equations was examined and also solved literally centuries before we knew anything at all about the begavior of matter in subatomic scales. It was pure interest to know what is the behavior of such solutions, and then... some centuries after that work of pure mathematics the physicians saw that the necessary tools for understanding the subatomic world were already available, since mathematicians made them before their "practical" usage was recognizable.
An even simpler example that has to do with this image: Gauss delivered limitbreaking works about complex numbers. When he die that it was rather "academic" what a complex number might represent. These numbers have to do with the root of -1, the imaginary unit that we call i.
being not my specialty ,I found the about is difficult for me to understand in its first part, Nick. I tried to like the second part of your about with the photo,it appears also difficult, your love for mathematics is not a secret for me,may I ask you kindly to clarify it for the commoners,pleas ? Saad.
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