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In mathematics, proofs are generally straightforward. Although sometimes hard to proof, they logically make sense. There are a base set of assumptions, from which conclusions can be deduced. For example, suppose a number n is even. Then n = 2j for some integer j. n^2 = (2j)*(2j) = 4*j^2 = 2*(2j^2), which means that n^2 is even.
Paradoxically, this axiomatic style of math is incomplete. what that means is, from a base set of assumptions, there are undecidable propositions. The proof of this was developed by Kurt Godel, and is very short and very sneaky.
Suppose there is a logical machine that can deduce the truth of any statement in a system. Let G be the statement that “This machine will never show that G is true”. Let’s suppose G is true. Then if the machine will never show that G is true, and it fails. Now, let’s suppose G is false. Then G will show the truth of G, but G isn’t true. tricky tricky. so given certain statements in a system, how do we deduce their validity?
For example, in Euclidean geometry, there were originally 4 postulates, or assumptions made from which Euclid tried to deduce everything else about planar geometry. However, there were things he could not prove, so a fifth postulate was added. Further, there are other statements which can not be proven from these 5 postulates. by an inductive way of thinking, given any finite number of postulates in a system, there may still be undecidable statements within that system.
Coming back from that paradoxical tangent, let's extrapolate this from math. What does it mean to prove something in science, in english, in psychology. Science is defined as observation leading to the explanation of phenomena. There is no proof in science. things are observed and conjectures are made, but how is anything ever proven?
Your teacher gives you an english paper. you come up with this thesis then use examples to make arguments which attempt to prove your thesis. but how strict must this proof be? things in language are often left open to interpretation, how do we unambiguiously prove the point we are trying to make?
I decided to make this thread, cause I saw one too many "this further proves that...." statements appearing. how do we really prove anything? Sure we can make arguments and attempt to generalize things based on what we know, but what really constitutes proof? This can be extrapolated to nearly anything. How do we prove the truth of something non mathematical? Hell, even the mathematical methods of proof are flawed because they require a base set of assumptions. How do we know those assumptions even hold?
What do you guys think that are specializing in science, english, psychology, political science, whatever. What kind of things are presented in your field, and are they presented as provable?
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This is far too big a subject for me to write on as a whole, so let me just offer for a starter something I read from Ortega Y Gasset tonight:
That the math student does not really believe in numbers. For instance the negative number makes absolutely no sense to him, for it has no reality (remember the Aristotilean distinction between substance and form, according to whom the former is always superior because the reality of "forms" is completely dependent on the existance of substance)
When we tell him that the number are true only within the propositions of the mathematical system, only then does he understand. Numbers as they work in mathematics and as they appear in reality are not the same phenomena.
Now according to Descartes, who attempted to overthrow the classical line of philosophy; substance is not the superior reality, because our notions of their existance eminate from our own senses. (In his meditations he uses several demonstrations on how our senses can deceive us.) But the superior reality to "The horse exists" is "I think the horse exists" (from which he coined his phrase "I think therefore I am," the realization of which was for him, a confirmable and irrefutable point as to act as the "first principle" for his philosophy.
Gasset interestingly posits that Cartesian skepticism was not radical enough; for Descartes uses the demonstration for example, of an image in a dream as something apparent to the senses, but is not real, whereas when we close our eyes, we fail to see something, but that thing in question remains real. Gasset asserts that there are still the classical prejudices working in the skepticism of this philosopher 
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To answer your question in a more meaningful way however, I hasten to add that there is no certainty in reality, because of the limitations of our own minds. This limitation makes possible the pursuit of understanding, and ultimately truth, a quality which ultimately enriches the mind for all those who do so honestly.
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Proof outside of an axiomatic system does not exist. Knowledge certainly does.
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quote: Suppose there is a logical machine that can deduce the truth of any statement in a system. Let G be the statement that “This machine will never show that G is true”. Let’s suppose G is true. Then if the machine will never show that G is true, and it fails. Now, let’s suppose G is false. Then G will show the truth of G, but G isn’t true. tricky tricky. so given certain statements in a system, how do we deduce their validity?
i read about this before in a book called Godel-Escher-Bach, but i'd already forgotten how it went. This is one of (the one?) the ways showing that our current math system will never be complete -as there will be theorems which cannot be proven nor disproven. Still, this 'proof' sounds a lot to me like the sentence 'i always lie', which would not be true for that sentence itself. That's more like smart-ass layman's philosophy (which annoys the hell outta me). Do you think this single proof is enough to be certain that the current math system isn't valid? Also, ive trouble with this sentence:
Let G be the statement that “This machine will never show that G is true”.
How can G be a statement which includes itself? This would mean you can never define G. It's like creation a dictionary entry for a word which involves the word itself. Im confused and it's 4am, sleep tight.
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You're thinking of the statement: This statement is false. It's a contradiction, it has no true or false value.
It's a recursive defintion but it makes sense. Let G be the statement that "This machine will never show G is true". G is a statement about itself. G is a statement which is a contradiction within the system. It's a weird way to disprove something, but its valid.
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Kek
Bigballs: the mathematician
MoltkeWarding: the philosopher
I don't like philosophy =/
Bigballs: What Godel showed was that for certain axiomatic systems (he was specifically targeting Principia Mathematica's attempted formalization of set theory), not all, have the property that no matter how many axioms you impose you can create a statement which is, via the logical machine, not "printable" and therefore not provable, but is still true.
In reference to your argument about science, and everything else, take the example of string theory in theoretical physics. The physicists define a string, and a space/time of 11 dimensions, to be some object with some properties in this space, much in the same way a mathematician would define a group to be a set of objects with the group properties. So in the strictest sense, they will create an abstract system, and then try to carry out tests that will either show that either this system works in the physical world, or doesn't (although no such tests exist for string theory =/). But demonstrating that such a system will always hold is impossible. How do we show that there will never appear some gate that will issue in hordes of orcs? We can't really, unless we take some statements to be absolute truths, since without them there is nothing to reason with. This applies to all of reality. So... just like in mathematics there is no proof without axioms, there is no proof in this reality without predicated assumptions. This holds for science.
Now take an english paper, or philosophy T.T Much of the time, you'll have logical leaps that are either based on many, many assumptions, (a few of them almost always not "common" and many of them unstated), or just intuition ("common sense") that has no basis in any logical system... Yet people will accept these arguments as "proof", because it has fallen into common usage, and so now you need to distinguish them from context, almost like homophones.
So in short, math proof is pretty much the same as the derivation of the consequences of a scientific theory, (whose validity we test, but never claim to prove), and completely different from the other types of "proofs" which are not real proofs at all.
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Bill307
Canada9103 Posts
I'm actually taking a course on logic for CS, which is culminating with formal verification of program code (weee).
You can indeed prove a lot of things outside of math, but it requires the common acceptance of premises that lead to the conclusion in a logical manner.
For example, suppose I state the following premises:
P1: If any student receives a mark of 50% or less in class X, that student fails the course.
P2: Harvy is a student.
P3: Harvy receives a mark of 50% in class X.
If you agree that these premises are true, then you must logically agree with the conclusion:
C: Harvy fails class X.
Because the premises imply the conclusion, this is a valid proof.
Unfortunately, that's basically as far as we can go with strict formal logic. If you don't believe the premises then you need not believe the conclusion. Perhaps there is good reason to believe the premises, in which case the proof is also sound. But showing the premises are *definitely* true is not something that can always be done formally.
Therefore, I suppose it is true that we can never fully, formally prove anything outside of math or any logical structure. But for many intents and purposes, having a sound proof, with believable premises that imply the conclusion, should be sufficient  .
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Yes in the great richness of life, inherited through the centuries, mankind has developed a collection of common sensed beliefs.
I would take the essays of Dr. Johnson over Einstein as being more true any day.
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On November 12 2005 20:04 MoltkeWarding wrote:
Yes in the great richness of life, inherited through the centuries, mankind has developed a collection of common sensed beliefs.
I would take the essays of Dr. Johnson over Einstein as being more true any day.
I'd rather go by the common sensed belief that results speak for themselves
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On November 12 2005 19:17 LTT wrote:
Proof outside of an axiomatic system does not exist. Knowledge certainly does.
This man speaks the truth.
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"A proof is a proof. What kind of a proof? It's a proof. A proof is a proof.
And when you have a good proof, it's because it's proven." -- Jean Chretien
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On November 12 2005 21:23 Oxygen wrote:
"A proof is a proof. What kind of a proof? It's a proof. A proof is a proof.
And when you have a good proof, it's because it's proven." -- Jean Chretien
Is that an authentic quote? If so, that's hilarious.
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speaking of set theory, this reminds me of extraordinary sets.
a set is extraordinary if it contains itself. For example, A = {1,2,3,A} is extraordinary.
Define S to be the set that contains all ordinary sets and nothing else. Suppose S is ordinary. Then S contains itself because it contains all ordinary sets, which makes it extraordinary, which is a contradiction. Now suppose S is extraordinary. That means it contains itself. But S contains nothing but extraordinary sets, so it cannot contain itself. Another contradiction.
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On November 12 2005 22:02 BigBalls wrote:
speaking of set theory, this reminds me of extraordinary sets.
a set is extraordinary if it contains itself. For example, A = {1,2,3,A} is extraordinary.
Define S to be the set that contains all ordinary sets and nothing else. Suppose S is ordinary. Then S contains itself because it contains all ordinary sets, which makes it extraordinary, which is a contradiction. Now suppose S is extraordinary. That means it contains itself. But S contains nothing but extraordinary sets, so it cannot contain itself. Another contradiction. It simply means that S cannot be ordinary in the first place. It's an extraordinary set that contains all ordinary sets.
This example does illustrate some of the seemingly strange characteristics of set theory. Like that one about successive power sets of the empty set:
A power set of S, P(S), is the set of all subsets of S.
if S = (a, b, c)
P(S) = {a, b, c, {a,b}, {a,c}, {b,c}, {a,b,c}}
(I think I'm missing one element, but you get the point)
The empty set, Ø, contains nothing. It's "null space".
Ø = {}. But...
P(Ø) = {Ø}
P({Ø}) = {Ø, {Ø}}
P({Ø, {Ø}}) = {Ø, {Ø}, {Ø, {Ø}}, {{Ø}}}
etc...
I think that's freakin weird and it has no actual application that I can think of.
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Please, think I'm smart, please, pretty please. I'll show off math problems that I just learned in class. The opinion my fellow TL.netters hold of me is very sacred to me. You guys will love me, won't you?
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On November 13 2005 00:05 TruthBringer wrote:
Please, think I'm smart, please, pretty please. I'll show off math problems that I just learned in class. The opinion my fellow TL.netters hold of me is very sacred to me. You guys will love me, won't you?
shut up plz-_-
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On November 12 2005 23:30 HeadBangaa wrote:Show nested quote +On November 12 2005 22:02 BigBalls wrote:
speaking of set theory, this reminds me of extraordinary sets.
a set is extraordinary if it contains itself. For example, A = {1,2,3,A} is extraordinary.
Define S to be the set that contains all ordinary sets and nothing else. Suppose S is ordinary. Then S contains itself because it contains all ordinary sets, which makes it extraordinary, which is a contradiction. Now suppose S is extraordinary. That means it contains itself. But S contains nothing but extraordinary sets, so it cannot contain itself. Another contradiction. It simply means that S cannot be ordinary in the first place. It's an extraordinary set that contains all ordinary sets. This example does illustrate some of the seemingly strange characteristics of set theory. Like that one about successive power sets of the empty set: A power set of S, P(S), is the set of all subsets of S. if S = (a, b, c) P(S) = {a, b, c, {a,b}, {a,c}, {b,c}, {a,b,c}} (I think I'm missing one element, but you get the point) The empty set, Ø, contains nothing. It's "null space". Ø = {}. But... P(Ø) = {Ø} P({Ø}) = {Ø, {Ø}} P({Ø, {Ø}}) = {Ø, {Ø}, {Ø, {Ø}}, {{Ø}}} etc... I think that's freakin weird and it has no actual application that I can think of.
Actually... it cannot be extraordinary because if it was, it would contain itself, which is an extraordinary set. So all this shows, is that there cannot be an ordinary or extraordinary set with all the ordinary sets and nothing else.
This is one type of problem that the Principia Mathematica tried to resolve, just by adding an axiom, such as "There is no such set S".
Also, the power set includes the null-set, which is the one you're missing Also, it is a bit misleading to right out a, b, c, when it is strictly {a}, {b}, {c}.
If you're interested in the things that can be derived from sets of the null set, read On Numbers and Games by Conway, where he derives all numbers, and a broader category, from this... really good book :D
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On November 13 2005 00:05 TruthBringer wrote:
Please, think I'm smart, please, pretty please. I'll show off math problems that I just learned in class. The opinion my fellow TL.netters hold of me is very sacred to me. You guys will love me, won't you?
You're a moron, and I wasn't talking to you.
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EPT
