Limitless Possibilities

Freedom is happily functioning under whatever restrictions exist without the will to dispute them. Freedom is respecting and even appreciating the limitations imposed on you, be they from human authorities or nature or physics, there are always rules imposed.

Unrestricted freedom doesn’t exist – to be free of politically/legally imposed restrictions is to be anarchic, which is equivalent to being subjugated to nature itself instead. Living in fear of disease and animal attacks and weather and deprived of the comforts of society is hardly any more free than living under the rule of a king, paying taxes, so forth.

Fact is, we give up the freedoms of anarchy for the protection of society. Society also protects freedoms and grants us privileges in addition that we wouldnt have in nature or anarchy.

The concept that we could live in a happy, protected, comfortable, technological, progressive, and moral society… without the need for law, government… is idyllic but foolish. For this to work, we would all need to recognize and adhere to an objective/absolute and good moral and ethical code (restrictions in themselves), and do so willingly without the threat of persecution or prosecution for failure. It is an impossibility, and indeed the notion bares self-contradictory attributes.

http://answers.yahoo.com/question/index?qid=20110609021248AAnlrOu

What would it take to live in such a “good moral code” since we can not agree across cultures on many things?

“Everything can be taken from a man but one thing: the last of human freedoms – to choose one’s attitude in any given set of circumstances, to choose one’s own way.”

Viktor E. Frankl

Love & Mathematics (Photo credit: Lost Archetype)

Subject: Mathematics, is there really freedom?I have some questions just about math in general that i consider thought prevoking.First, Can you ever truly think differently in math? Ultimately everyone is bound by a basis of rules set down by the creators of the most basic of math.Also, can math TRULY be proven as each proof is all traceable to assumptions made by the earliest of math’s creators? And those are based on a system of numbers we assume to be correct.Ultimately i see math as all operations in that they branch all from a single seed and that seed be one big assumption we could never really concecrate with rock hard fact based on no assumptions. Now i am no mathematician by any means and most likly have the smallest of mathematical knowledge compared to everyone else here. Perhaps you could clear up a gross misconception i am understanding to be truth or at the very least state your opinion. I however understand the need for mathematics as without it we would crumble as a society.
English: Square root of x formula. Symbol of mathematics. (Photo credit: Wikipedia)	Ultimately i see math as all operations in that they branch all from a single seed and that seed be one big assumption we could never really concecrate with rock hard fact based on no assumptions. That’s actually a reasonable, though incomplete, description of the nature of mathematics. There have been attempts to derive all mathematics from the basic principles of logic, but as far as I know none have ever been deemed successful. What we are left with, then, are basic axioms of mathematic on which the system is built, exactly the same as with geometry. Just as there are non-Euclidean geometries, there are also non-standard mathematics formed by removing or changing the basic axioms. If you read up on Godel’s Incompleteness Theorum, you can find out some more about what those systems look like. We can have different systems of mathematics. The question is which most accurately models, or acts as a metaphor for, the world in which we live (the world of experience). Once we understand that mathematics is an abstract system, which is not part of the world but rather is a metaphor of part of the world, then we must choose between systems pragmatically.
American Regions Mathematics League (Photo credit: Wikipedia)	Why are numbers beautiful? It’s like asking why is Beethoven’s Ninth Symphony beautiful. If you don’t see why, someone can’t tell you. I know numbers are beautiful. If they aren’t beautiful, nothing is. – Paul Erdös
$Mathematical square knot (basic depiction).$ Mathematical square knot (basic depiction). (Photo credit: Wikipedia)	Also, can math TRULY be proven as each proof is all tracable to assumptions made by the earliest of math’s creators? Forgot this question. Mathematical statements can be proven true within the system of mathematics. Whether or not they can be proven true for the world is another question, and the answer to it depends on your own conception of truth. This is not the place to discuss that now, but you may find exploration fo what truth means through this question very fruitful.
The essence of mathematics is its freedom Cantor.	Godel, Turing, and Chaitin have all disposed of the idea of a single monolithic structure of mathematics. Any formal axiomatic system (FAS) which includes the four arithmetic operations is incomplete, ie. has true statements that cannot be proved.So, our only recourse is to regard each branch of mathematics as a structure on its own and build the bridges that enable us to use them together, eg. analytic geometry. And, of course, each of these has its own FAS which induces still more incompleteness.So Lodestones statement “Mahematical statements can be proven true within the system of mathematics.” should really say “its own system of mathematics.”
English: Math is prerequisite to Ontology. Math and Ontology are prerequisite to Physics. Math, Ontology and Physics are prerequisite to Chemistry (Photo credit: Wikipedia)	It should also be noted that there are true mathematical statements that cannot be proven within its system, as was shown by Gödel. While some mathematical statement can be proven from axioms, the question remains, what does their truth mean? Is math a science? Kant argued that mathematics is founded on intuitions of space and time, and thus contain a priori synthetic truths, which means that although it is thought of as necessarily true, it is not merely tautologous but extends our knowledge, and thus is a science about the conditions of experience.
Dansk: Dedikeret til matematik (Photo credit: Wikipedia)	“Math” does not exist, except as a collective of “mathematics”. Each branch of mathematics can be studied independently so each is a discipline in its own right. That’s one issue.However, what do we mean by “true but unprovable”? I was taught that “true” = “provable”. If we can’t prove it, it remains “unproven”.I think what we mean is “unprovable” and “noncontradictory”. So, there are mathematical statements which appear isolated, unrelated to the remainder of a discipline. We can not prove them and we cannot disprove them.This is what Chaitin means when he says that there is randomness in the heart of mathematics, “facts” which are logically independent of the rest. You can affirm them as axioms and build a new system or you can deny them and build a different new system.
English: Mathletes at a Texas Math and Science Coaches Association (TMSCA) Mathematics Tournament (Photo credit: Wikipedia)	Well you were taught wrong, because Gödel proved that provability is a weaker notion than truth. Unproven statements can be true or false nonetheless, but more than that, there are unprovable statements that are true in all consistent mathematical systems.Chaitin has some interesting ideas, and he is right about constructive nature of mathematics, but if I remember correctly he argued that its quasi-empirical, whatever that is suppose to mean, or even biological, where I think he is getting into a whole different realm. Also his idea of “randomness” seem to be of information theoretic character, pertaining to entropy or algoritmic complexity or something of that odd sort.
some math (Photo credit: Wikipedia) Differences between Euclidean and non-Eucl. geometries (Photo credit: Wikipedia) Wykres math (Photo credit: Wikipedia)	I have some questions just about math in general that i consider thought prevoking. First, Can you ever truly think differently in math? There are many different kinds of mathematical systems, and there are many different ways of thinking about math. Ultimately everyone is bound by a basis of rules set down by the creators of the most basic of math. I’ve never heard of this. I don’t know who these creators are, what the basic math is or what the basic rules are. Also, can math TRULY be proven as each proof is all tracable to assumptions made by the earliest of math’s creators? I never heard of every proof depending on assumptions made by math’s creators or of math’s creators. And those are based on a system of numbers we assume to be correct. Some mathematics makes no reference to numbers at all. For the most part, though, in many contexts the natural numbers and certain properties of them are, in some sense, taken as a given. Ultimately i see math as all operations in that they branch all from a single seed and that seed be one big assumption we could never really consecrate with rock hard fact based on no assumptions. If you’re saying we need to make assumptions that are not empirically testable, then (1) I don’t think anyone denies we need to make assumptions, (2) my sense is that usually mathematics does not address whether its assumptions are empirically testable, especially since they’re not empirical assumptions. Now i am no mathematician by any means and most likely have the smallest of mathematical knowledge compared to everyone else here. Perhaps you could clear up a gross misconception i am understanding to be truth or at the very least state your opinion. I however understand the need for mathematics as without it we would crumble as a society. To understand mathematics it helps to understand mathematical logic and set theory. This includes understanding formal systems and model theory, which put your questions into a quite sharply articulated context.
	We can have different systems of mathematics. The question is which most accurately models, or acts as a metaphor for, the world in which we live (the world of experience). Once we understand that mathematics is an abstract system, which is not part of the world but rather is a metaphor of part of the world, then we must choose between systems pragmatically. But some of the most salient questions seem not to concern the world of our experience. Most famously, what in the world of experience would you look at to determine the truth of the continuum hypothesis?. What pragmatic test would you offer for settling one way or the other on the continuum hypothesis?