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?
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
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.
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.
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.
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.
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 .
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
"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
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.
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.
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)
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?
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?
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)
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
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?
why stop at cartesian notions of mind-body etc. first of all there's kant. and u have to at least acknowledge heidegger.
the problem with this question is that it already brings a notion with it, the notion of "actual proof" or "real proof," already working with a notion of objective reality that we can get to and then "proof" meaning that we are "sure" we "got to it." how do we justify any of these other than that they "work in general" with others?
because our existence, our reason, our language, etc. all fall in line with "whatever has worked" to propogate the species, the answer for "what is sufficient proof?" would simply be "whatever convinces sufficient people."
Bigballs, I love you! I have so many non-mathematician friends/acquaintees that do not know the meaning of word "proof". The use it like we use "is likely to be". The humanitarian/social sciences have prooved so many false statements... About Godel theorem. I think the theorem you are talking about was formulated for axiomatic theory called "formal algebra". I remember we were prooving(FORMALLY!) that 2*2=4. Took us 2 hours. The most conviencing result for Godel's theory is certainly not the completeness of formal algebra. He has another remarkable result, that says that, say W is a statement that cannot be prooved or disprooved in formal algebra S. Let's add W(or not W) to the axioms. Then, there exists another statement that cannot prooved or sidproved...and so on. The result is: if system is rich enough, then it is incomplete. While formal mathematical systems are strong enough to produce some real proofs, the non-formal sciences do not have such oppurtinity. Mostly they use stastical methods to proove/disprove something. While they are unable to take into account every factor that impacts the experiments, some sciences have really remarkable results. Biology is the example. Others, like, say, psycology(I believe) are really lacking methods. I believe that it is not only due to the very complex nature of the objects they deal with, it is due to their methods(please psychology students don't turn this into a discussion, this is my very subjective opinion). I believe that ALL sciences will gain if they put some math into.
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.
Did you just try to resolve Russell's paradox? =) Think again
P.S. BigBalls, you screwed up the definition a little...
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?
On November 13 2005 08:57 PlayJunior wrote: I have so many non-mathematician friends/acquaintees that do not know the meaning of word "proof". The use it like we use "is likely to be". The humanitarian/social sciences have prooved so many false statements...
I would like to see an example of this. Sounds like absolute BS to me.
I have not read all of the responses to this thread, so I may be repeating what has already been stated.
The simplest answer to this question is "it depends." The question depends on the area of study in which one is seeking proof. In most analytical contexts (i.e. mathematics, logic, etc.) one can deductively prove something based on deductively valid proofs. These a priori proofs rely on syntactical patterns in logic that always yield a certain result. There are some complications here, but let’s just leave it at that for now. In the sciences, deductive proof is usually impossible. Proofs in the sciences are rather inductive proofs, or more accurately abductive proofs- proofs to the best explanation. These proofs rely on a finite set of data, and the building up of this data to support some hypothesis. As such, very few scientific proofs reach the status of deductive proofs.
Thank you for posting this topic. I see this issue confused so many times due to simply misunderstandings of these issues. The intelligent design thread is a perfect example of people misunderstanding that scientific proofs or theories are not deductively valid proofs as one would see in logic problems or mathematics.
On November 12 2005 19:17 LTT wrote: Proof outside of an axiomatic system does not exist. Knowledge certainly does.
What is knowledge outside of an axiomatic/logical system?
Experience? The senses? Perceptions? Most knowledge isn't even in an axiomatic/logical system, it simply can be interpretted that way post hoc, but that doesn't mean that's how it is. describing is not explaining.
On November 13 2005 07:36 mitsy wrote: why stop at cartesian notions of mind-body etc. first of all there's kant. and u have to at least acknowledge heidegger.
the problem with this question is that it already brings a notion with it, the notion of "actual proof" or "real proof," already working with a notion of objective reality that we can get to and then "proof" meaning that we are "sure" we "got to it." how do we justify any of these other than that they "work in general" with others?
because our existence, our reason, our language, etc. all fall in line with "whatever has worked" to propogate the species, the answer for "what is sufficient proof?" would simply be "whatever convinces sufficient people."
Of course we could, at any time, say this for any argument. Since anything we observe is filtered through our senses and anything we think is just the behavior of our minds, then how do we know anything that we are convinced of is nothing more than a mass delusion? How is any restriction on a proof anymore valid than any other, since those restrictions are just human creations? Well... you could always take the minimalist approach, and restrict it to the axioms you define and the "common sensed belief" of logic
On November 12 2005 19:17 LTT wrote: Proof outside of an axiomatic system does not exist. Knowledge certainly does.
What is knowledge outside of an axiomatic/logical system?
Experience? The senses? Perceptions? Most knowledge isn't even in an axiomatic/logical system, it simply can be interpretted that way post hoc, but that doesn't mean that's how it is. describing is not explaining.
What happens when two people percieve different things, or know two contradictory statements?
On November 13 2005 07:36 mitsy wrote: why stop at cartesian notions of mind-body etc. first of all there's kant. and u have to at least acknowledge heidegger.
the problem with this question is that it already brings a notion with it, the notion of "actual proof" or "real proof," already working with a notion of objective reality that we can get to and then "proof" meaning that we are "sure" we "got to it." how do we justify any of these other than that they "work in general" with others?
because our existence, our reason, our language, etc. all fall in line with "whatever has worked" to propogate the species, the answer for "what is sufficient proof?" would simply be "whatever convinces sufficient people."
Of course we could, at any time, say this for any argument. Since anything we observe is filtered through our senses and anything we think is just the behavior of our minds, then how do we know anything that we are convinced of is nothing more than a mass delusion? How is any restriction on a proof anymore valid than any other, since those restrictions are just human creations? Well... you could always take the minimalist approach, and restrict it to the axioms you define and the "common sensed belief" of logic
Edit: I also think that the main question lies in the validity of the belief in logic and the definition of axioms, as well as the concept of what is true vs. not true. Truth in axiomatic systems has the luxury of not based on any perceived reality outside the functionality of logic. We can always just describe it as some abstract property. Now... if someone was to reject logic and the whole axiomatic system thing, then that would be their right, imo. But, then they couldn't really use it to prove anything to themselves. Additionally, pretty much everyone will agree with the founding statements of logic as being absolutely true, all the time. (ie A->B B->C means A->C, whenever A,B,C are true or not true, and with the accepted definition of implies. So if A is true, then B is true, as is C. Or equivalently, if A->B means "if A has some property, then so does B", then most people would also agree we can deduce if A has some property, then so does C). But if they accept this, then they already accept the mathematical notion of proof in an axiomatic system, since that is all you can do to prove something. So in short, you'll either accept logic and axioms and therefore the ultimate standard for proofs, or you won't ;]
On November 13 2005 08:57 PlayJunior wrote: Bigballs, I love you! I have so many non-mathematician friends/acquaintees that do not know the meaning of word "proof". The use it like we use "is likely to be". The humanitarian/social sciences have prooved so many false statements... About Godel theorem. I think the theorem you are talking about was formulated for axiomatic theory called "formal algebra". I remember we were prooving(FORMALLY!) that 2*2=4. Took us 2 hours. The most conviencing result for Godel's theory is certainly not the completeness of formal algebra. He has another remarkable result, that says that, say W is a statement that cannot be prooved or disprooved in formal algebra S. Let's add W(or not W) to the axioms. Then, there exists another statement that cannot prooved or sidproved...and so on. The result is: if system is rich enough, then it is incomplete. While formal mathematical systems are strong enough to produce some real proofs, the non-formal sciences do not have such oppurtinity. Mostly they use stastical methods to proove/disprove something. While they are unable to take into account every factor that impacts the experiments, some sciences have really remarkable results. Biology is the example. Others, like, say, psycology(I believe) are really lacking methods. I believe that it is not only due to the very complex nature of the objects they deal with, it is due to their methods(please psychology students don't turn this into a discussion, this is my very subjective opinion). I believe that ALL sciences will gain if they put some math into.
Could you give me a link to a not too difficult formal proof? I'm certainly interested in something like this, i want to know what it looks like and how difficult it is to prove anything using it. also, what axioms does it use.
The humanitarian/social sciences have prooved so many false statements...
I think this is true, although by logic is it hard to find one which is false, otherwise it hasn't been proven. But we can watch the tendencies of the past (newton for example claiming that speed at which objects fall down was dependant of weight). Many past 'proofs' have been found to be wrong, and there are undoubtedly many more...
hmm i love these threads, i'm only 14 and just got done with geometery last year so i can't understand too much of whats being said, but i try to learn and look stuff up, by the way, how old are you guys that you know all of this?
On November 12 2005 19:43 BigBalls wrote: 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.
Somehow it reminds me the halting problem (is that the name?) that was developed by Alan Turing regarding his machine.
what's interesting to me is what angles people approach the question from, based on their different backgrounds.
let me sort of rephrase the question. since there is no formal proof outside of an axiomatic/logical system, what is accepted as a proof? I suppose an overwhelming amount of evidence and data supporting a theory that cannot possibly be a statistical anomoly could be considered a loose proof. any way we can generalize proofs in different areas of study?
On November 12 2005 19:17 LTT wrote: Proof outside of an axiomatic system does not exist. Knowledge certainly does.
What is knowledge outside of an axiomatic/logical system?
In order to assert "I know that p", four conditions must be met: 1.) That p be true. 2.) That I believe that p. 3.) That I have good reasons, or sufficient evidence, for my belief that p. 4.) That I have no other evidence that might undermine my belief.
(Taken from Abel's Man is the Measure) I'd highly recommend this book for you BigBalls. It goes into quite a bit more detail and does a fairly good job of presenting contrasting views on a variety of philosphical subjects.
I think this is true, although by logic is it hard to find one which is false, otherwise it hasn't been proven. But we can watch the tendencies of the past (newton for example claiming that speed at which objects fall down was dependant of weight). Many past 'proofs' have been found to be wrong, and there are undoubtedly many more...
Again, can you give us examples of what has been "prooved" wrongly by the humanities and why we have no clue? I am going to defend my field of interest from ignorant remarks.
I think this is true, although by logic is it hard to find one which is false, otherwise it hasn't been proven. But we can watch the tendencies of the past (newton for example claiming that speed at which objects fall down was dependant of weight). Many past 'proofs' have been found to be wrong, and there are undoubtedly many more...
Again, can you give us examples of what has been "prooved" wrongly by the humanities and why we have no clue? I am going to defend my field of interest from ignorant remarks.
Dunno if you count ancient philosophy as part of your field of interest or not, but... Xeno's achilles and tortoise paradox. Just a flawed inductive proof ;D
what's interesting to me is what angles people approach the question from, based on their different backgrounds.
let me sort of rephrase the question. since there is no formal proof outside of an axiomatic/logical system, what is accepted as a proof? I suppose an overwhelming amount of evidence and data supporting a theory that cannot possibly be a statistical anomoly could be considered a loose proof. any way we can generalize proofs in different areas of study?
Me 22, become 23 soon. Major in computer science and mathematics. PHD student, 1st year. I liked your formulation very much. Anyhow, as I pointed before, statistical evidence can prove wrong when the factors that have impact on experiment aren't quite clear. 2 moltke: Let me answer you with an anecdote. A writer, a phisic and a mathematician are travelling in a treain through Scotland. They see sheep grazing in a field. Writer: "Watch! How interesting...In fact, all the sheep in Scotland are black!" Phisic: "Hmm...I'd say that in this part of Scotland most of the sheep are black" Mathematician: "All I can state is that there are some sheep in Scotland which are black on at least one side...".
A proof will always require a leap of faith somewhere (a premise/postulate/assumption) that you must take as truth. Then you deduce new statements from your set of premises using basic logical inference rules. One of these new statements might be the statement you are looking to prove. The tough part is actual being convinced that your premise is "correct". Once that occurs, you can immediately prove a statement when you have the inference trail figured out.
Without the leap of faith, all you have is a definition - an arbitrary declaration made by a human stating "this is MY truth". Sure, if you want, look at a definition as the simplest albeit degenerate form of a proof.
proof is the act of bringing into appearance convinced-being. you can't take it out of context. if you're talking about one individual, making them convinced is proof. if you're talking a nation, what convinces "them", or rather, "your them," is different but still clear cut. and for most educated people, it comes down to convincing those endoctrinated in science; but doesn't that leap over the whole question? it sure does. how do you convince someone that something is a scientific fact? it is still with text and for the most part they never can verify it fully but rather trust their specialized "society," all the same as we commoners do in a different, maybe less consistent form, but trust and convinced-being nonetheless!
On November 14 2005 01:32 mitsy wrote: proof is the act of bringing into appearance convinced-being. you can't take it out of context. if you're talking about one individual, making them convinced is proof. if you're talking a nation, what convinces "them", or rather, "your them," is different but still clear cut. and for most educated people, it comes down to convincing those endoctrinated in science; but doesn't that leap over the whole question? it sure does. how do you convince someone that something is a scientific fact? it is still with text and for the most part they never can verify it fully but rather trust their specialized "society," all the same as we commoners do in a different, maybe less consistent form, but trust and convinced-being nonetheless!
Rather pathetic but interesting. Anyhow, having matematical education, I'd say that proof is something that is not dependent on individual being convinced. That is, every formal proof in a formal system can be FORMALLY verified by a donkey, because the proof can be reduced to application of some rules to axioms or preconditions(English is my third language, so sorry ). Let's mix these 2 concepts. We will distinguish proofs in formal and non-formal systems. With proof in a formal system well defined, let's try to define a proof in a non-formal system. "We will call a proof in non formal system evidence of facts/stastical information/experiments or whatsoever if they suffice to convince anyone with sufficient knowledge in the area." How do you like this guys?
whats your metaphysics, mr. math? ever get past the mind-body problem? got a formula for that?
Sorry, didn't get the point. I just tried to assemble your and Bigballs' defiitions of proof.
Mitsy was being either immature or just desiring to throw out random ideas he/she learned in a philosophy course. For whatever reason, mitsy threw out a classical philosphical problem about the relation between the mind and body, or more specifically how something immaterial can affect something material and vis versa.
The mind influences the body through such things as thoughts of food producing saliva, listening to scary stories producing goosebumbs, and thoughts of sex having obvious effects. The body influences the mind most noticably through things like caffeine, alcohol, or any sorts of drugs.
Science has come close to bridging the gap, but even if we could map out every electronic pulse that caused a certain state of mind, or every thought that caused a certain pulse to travel through us, would we be able to say that those pulses are identical to that state?
On November 15 2005 01:02 baal wrote: these threads are boring even for me, and im a pseudo intelectual sophistic biatch who loves to argue about philosophy.
I am sure you haven't read the thread but feel obliged to post some random stuff about discussion being boring. Or it is a must to post in every thread even if you have nothing to say? P.S. The discussion is not about philosophy.
All you need to do is ask: "What comes to mind upon hearing the word proof?" When something is proved, the only manifestations (the reality of it) are beliefs and predictions. People assume the proved thing is true and always will be, and make predictions based on that assumption. If we hold nothing as proved, we cannot make any assumptions and thus cannot deduce anything. We then live in an unstable world with no idea what's going on. Our desire to use assumptions actively in order to move through life with confidence is the chief motivator in proclaiming something as proved. The more abstracted from time and space and culture you want your beliefs to be, the more you struggle to get consistent proofs, or apparently consistent proofs (which come down to being the same thing, only the former is glorifying and the latter is negative).
I know rationalists will be disappointed and reject this view, but the thing is it actually describes how we live, and how we will live. It shows which factor (our desire to use assumptions actively in order to move through life with confidence) determines how "deep" or consistent upon analysis a series of statements called "proof" must be for a given person.
On November 15 2005 01:02 baal wrote: these threads are boring even for me, and im a pseudo intelectual sophistic biatch who loves to argue about philosophy.
I am sure you haven't read the thread but feel obliged to post some random stuff about discussion being boring. Or it is a must to post in every thread even if you have nothing to say? P.S. The discussion is not about philosophy.
On November 15 2005 10:03 Pseudo_Utopia wrote: All you need to do is ask: "What comes to mind upon hearing the word proof?" When something is proved, the only manifestations (the reality of it) are beliefs and predictions. People assume the proved thing is true and always will be, and make predictions based on that assumption. If we hold nothing as proved, we cannot make any assumptions and thus cannot deduce anything. We then live in an unstable world with no idea what's going on. Our desire to use assumptions actively in order to move through life with confidence is the chief motivator in proclaiming something as proved. The more abstracted from time and space and culture you want your beliefs to be, the more you struggle to get consistent proofs, or apparently consistent proofs (which come down to being the same thing, only the former is glorifying and the latter is negative).
I know rationalists will be disappointed and reject this view, but the thing is it actually describes how we live, and how we will live. It shows which factor (our desire to use assumptions actively in order to move through life with confidence) determines how "deep" or consistent upon analysis a series of statements called "proof" must be for a given person.
I think you are social/humanitarian. Clearly you mixed up a bunch of things.
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
Oh god i'm too tired to read if this has come up yet.
I think "Bible" has the truth and proof. If you wanna know something, just read there! It's all there if you make a few hundred assumptions, hit a few baby's head to the rock and make a crusade.
well i may have not answered the right question earlier in this thread.
the verb is _constitutes_. i would say that for something to be proven it need be consistent with other already proven things. but is "being proven" ever more than a perception, and is our set of "already proven things", even if it is the largest possible set of things that are consistent with eachother, necessarily "truer" than a smaller set?
"what do we mean by proof" is a slightly different question, kind of like "what convinces us that something is proven?"
On November 16 2005 01:10 mitsy wrote: well i may have not answered the right question earlier in this thread.
the verb is _constitutes_. i would say that for something to be proven it need be consistent with other already proven things. but is "being proven" ever more than a perception, and is our set of "already proven things", even if it is the largest possible set of things that are consistent with eachother, necessarily "truer" than a smaller set?
"what do we mean by proof" is a slightly different question, kind of like "what convinces us that something is proven?"
I didn't understand the middle paragraph. Once more pls
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)
I think that's freakin weird and it has no actual application that I can think of.
Oh my that's great intution you have here ^^ in fact recursively it's the formal creation of the natural numbers ! (math freak ftw) 0 = Ø 1 = Ø, {Ø} 2 = Ø, {Ø}, {Ø,{Ø}} 3 = Ø, {Ø}, {Ø,{Ø}}, {{Ø}} ..... if you like it you could read some math-theory books and if you don't already read those you might like it (no joke)
Godel and Hilbert worked on this kind of theories, Hilbert whos was very skilled with proof by absurd.
exemple : let's have x1,x2,x3,x4,x5 (like1a2a3a4a5a) and you know that the total sum isn't null prove that there are 3 xi ( "i" in {1,2,3,4,5} ) that their sum isn't null either.
Everything is set upon a basic principle : "This world has a definite number of rules. " People seek these rules(like axioms) and want to determine the world around us on the base of deductions and proofs. But what if this world is based on indefinite number of rules? Then we will never know how it works since humans seem to only work with definite knowledge. We will only know the piece of cake, but never taste the real truth.
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.
That's why we have system of sets and not set of sets.
How it work in physic: Someone come up with some new theory as a answers to some experimentali showed contradiction of current theory.Then he compute results of some experiments using his new theory (usually it's expanded current theory).Then if numbers are equal with numbers from experiment and it survive every other attempt(read as a some insane experiment) of another smart-ass it's considered as "we don't have any evidence why we should consider it as an another bullshit usless theory so we can say (DISCLAIMER : we don't provide warranty of any kind) it's true and we will give Nobel prize to this lucky bastard".
That's the problem with super-strings theory b/c it works in cases when we are able to make computation but there are many computations that nobody can currently do.But where we can it works super fine.
But my English is super newbie so i really don't know if it's even uderstable.I can explain it in slovak but in english it's very hard for my and that's why i don't post in general forum.
EPT
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