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What is a proof?
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| trancaholic |
Hi
I'm currently writing a small piece of text on what it constitutes to prove something to somebody, and then I thought that alternatives to what I've come up with myself would be interesting. And where else to turn than here?
Assumptions should be that we all live in a real world and that we can trust our senses. What would it then take for somebody to prove something to you? E.g. for your best friend to prove that he/she has bought a loaf of bread an hour ago? That he/she is in love with somebody? For me to prove that I'm the legal owner of the computer I'm typing this on? That Antarctica exists? ... |
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| Pjotr G |
To prove something means that using some valid rules on some premises, you get to what you wanted to proof, and there is no possible way that the premises are true and the result false.
An example of a valid informal proof is
premises:
- If A then B
- A
result
- B
or this
premises:
- If A then B
- Not B
result
- Not A |
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| Azareal |
Interesting question - and although I haven't taken inductive / discreet mathematics in over 4 years, I should be able to come up with _something_ useful here.
A successful proof occurs when you take a problem, (ie: did I buy a load of bread an hour ago?) and prepose it to either be true or false. Then, you split up the preposition using simple and logical facts that are either a) common irrefutable knowledge or b) have been proven in the past.
Of course, real life differs from mathematics : the language is completely different (semantically anyhow). Suppose your friend is trying to prove to you that he is in love. First, he has to define all the emotions / thoughts/ feelings / states of love. He has to prove that he exhibits each of those, and that each is in correlation to the person to whom he feels he is in love.
Love in this case is a signifier - it stands in for what he is actually feeling for another person. Semantically, it means different things to different people, but he doesn't have to prove that you are in love with his significant other, he has to prove that he is in love with his significant other.
I think my head is going to explode - it would be a lot eaiser to proof something tangible - ie that you are the legal owner of your computer - matching product numbers from your computer to bills of sale, matching the bill of sale to yourself is a simple chain that constitutes ownership. Of course, anyone could doctor/manufacture such information, but a good computer techie could examine your machine, find your 'fingerprints' all over the hard drive (files / programs/ set up -- everyone leaves their styles on the computer) and then go on to prove that the drive is the same drive installed in the computer upon manufacturing.
I guess what I am saying is that the easiest way to prove something is to break it up into small parts - and then proove each small part to be true. The sum of the parts equals the proof of the whole.
Confusing neh? :)
~Azareal |
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| Renegade |
Alright, what we're really talking about here is epistemology, and the problem is that there are so many different theories and ideas about this subject that we could spend all day discussing it. Perhaps if you could be a little bit more specific about what you're trying to find out it may be a bit easier for us to help you? Is this for university/school or something?
Anyway, on with the post.
| quote: | premises:
- If A then B
- A
result
- B |
The only problem with this logic is that it only really applies to mathematics and the "logical positivist" train of thought. I'm sure you all have some idea about mathematical knowledge so I'm not going to bother getting into it, but as for positivism, if we advocate that train of thought then we are left with some fairly serious repurcussions.
I won't get into the complex specifics of this logic (unless you really want me to) but essentially logical positivism was born in Austria in the 1930's. All philosophical thought before this point was speculative, and generally fairly metaphysical: that is, philosophy was its own unique train of thought, and the views propounded by philosophers were made independantly of all other systems of thought (i.e. Leibniz's monad theory - that everything in the universe was comprised of alligned particles called "monads" that could account for any occurance in the universe - was born independantly of whatever was known back then of physics and chemistry). What the positivists set out to do was firstly was establish what the aim of philosophy was and then, secondly, to implement a system of thought - a revolutionary new form of philosophy - that succeeded in providing the results we required from philosophical thought.
They decided that "semantics" (the definition of words) was all - after several millenia of philosophical enquiry - that philosophy amounted to (and sadly philosophy has been mired in this rigid definition ever since). What semantics would allow for was the creation of a unified field of thought - that is, the merging together of all doctrines of human knowledge - and that that, in turn, was the only way to "true" knowledge. According to the positivists, the only things we could possibly know were strict empricism (the sciences - but not any theoretical science, i.e. astrophysics, psychoanalysis etc.) and strict rationalism (pure logic - i.e. tautologies, basic mathetmatics etc.).
So where am I going with all this? Well, basically, with regards to "proof" or epistemology - the topic at hand here - if you want to believe the positivistic train of thought, any metaphysical concept - love, god etc - can be dismissed straight out hand as meaningless and devoid of truth. How so? Well, seeing as all philosophy amounted to, according to their logic, was semantics, anything that couldn't be precisely defined became meaningless and thus, devoid of any element of truth. So, if you were to get someone to prove "that he/she is in love with somebody", the simple answer is, by this train of thought followed to its logical conclusions, either they aren't in love at all or they may as well not be. Not a pretty picture though, eh?
Anyway, I still say that they were on the right track when all said and done. Semantics is the beginning of philosophy and they were right to point that out, but they were wrong to suggest, in essence, that it was all it amounted to. Thus, as Azareal pointed out, before your friend were to prove that he was "in love" his first step would be to define what love is (using semantics) and then to point to all empirical evidence - pertinent to his behaviour, emotions etc. - that correlates with his own definition of love. Thus if his definition of love is having a certain feeling inside of him, and those feelings do exist, then, for all intents and purposes, he is in love. He may not be able to justify it to you (until there is such a time, perhaps, when we have the scientific knowledge available to better quantify what love is - biologically, maybe) but proof, in this sense, must be entirely subjective. If you demand objectification of his feelings, then you'll only end up disappointed. So far as I'm concerned, love - like a great many things - is in the eye of the beholder.
In fact, just reading back, I think me and Azareal are saying pretty much the same thing, it's just that it's taken me 10 times longer to say it. ;)
As for all the other things you've suggested there, the verification lies entirely in empirical proof - so we should be able to find some common, objective standards that make the proof of all those situations attainable.
Without having really thought this through, the bread and butter of empirical thought is to be able to make circumstantial predictions based on any given contention or stance, and whether these predictions are accurate or not should go a long way to determining the veracity of the original contention. Confused?
For instance, if you want proof that "Antarctica exists", the best place to start would be to make a number of predictions that must come true should the contention - that Antarctica exists - be true. So, if Antarctica does exist we should rightfully expect a number of things: that it has been discovered and mapped, that it has been viewed from space, that we can speak to people who have been there and so on. These predictions simply form the basis of what we should expect given the claim being made, and, uniformly, we can suggest that these things ring true. We can assume that Antarctica exists simply because almost any prediction we make about its existence (within reason - i.e. assuming we have full faculty of our senses as you suggested above and so on) rings true.
Of course, you could quite easily point out that all we have in terms of proof of Antractica's existence is hear-say: for instance, there may be a massive NASA conspiracy whereby they've photoshopped that massive continent into all the images they've ever taken of planet Earth, and that anyone who has claimed to be to this continent has been brainwashed into thinking that, and that they've only been taken to some other frozen land so that the tour guide can make an easy buck. Firstly, these claims are, in a way valid. I think that this is the point you're trying to make actually - without actually ever having been there, how can we be sure that Antarctica exists?
Well to quote Neitzsche, "Another reality is entirely undemonstrable". Either the world is as we see it - and we haven't been mislead for the entirity of our lives regarding the geography of our planet - or the world isn't as we see it and we have been mislead. The first conclusion is demonstrable and can be proven using an empirical method such as the one I set out before, or even, possibly, via the use of a logical method (such a conspiracy is untenable, what reason would they have to lie/how could we have been misled for so long etc.) where as the latter conclusion is entirely undemonstrable and devoid of any considerable merit. There is only one reality, not several "possible modes" of reality.
And, if you want a slightly more complicated method of proof (i.e. if you oculd be bothered trying to apply this logic) I suggest you read up on the Hegelian Dialectic (which is form of logic that Hegel nicked off Plato and Karl Marx, in turn, nicked off Hegel for his socialist theories). Basically, without wishing to go into great detail, for every thesis there is an anti-thesis: i.e. there is the thesis "that Antarctica exists" and the thesis "that Antarctica doesn't exist". The Hegelian method then, would suggest that we spell out each thesis in turn, and ascertain the truth - or verifiability - involved in each statement, and by taking the truthful - or most truthful - parts of the thesis and antithesis we can combine them into an entirely new thesis - a synthesis. Then, with this new synthesis there will be a new antithesis and we repeat the process again.
I should point out that this method was meant to apply only to political theory and so on (which is why Marx found it so appealing) but, if you can be arsed spelling out the basic tennets of every theory you come across, it is a very commendable form of logic. Theoretically, if you continue this process ad infinitum - and do it properly and thoroughly every time - you should eventually be left with a statement that is entirely true. Well, that's the theory anyway, but as with all things philosophical it rarely ends up that way.
*sigh*
Anyway. Hope that goes some way to answering your question? |
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| Renegade |
Oh, I forgot to point out what Pjotr G's post had to do with logical positivism.
In short, basically the positivists used a form of mathematical logic similar to that in their "probability tables" (which I'm not going to explain here). Essentially, any statement that couldn't be verified using that sort mathematical "A given B" type logic was dismissed as either meaningless or untrue. That was vaguely the point I was trying to make.
Anyway, enough. I'm going to bed. |
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| drizzt81 |
also noteworthy:
if A then B
does _NOT_ imply
NOT A then NOT B
you can still have B without A |
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| Orbax |
1. {[(A.B) > A] . C . (B>A)}
2. A.B
3. LvC
4. L>A
5. C>(A.B)
6. ~L /...~(A.B) |
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| Orbax |
1. This world is real
2. senses are trustworty
3. If something goes against our senses it isnt trustworthy
4. If it is outside the world is is either trustworthy or not
5. I went to a store on this world.
6. I felt, saw, and smelled the money in my hand go to the cashier
7. I felt, saw, tasted, and smelled the bread./...The bread being purchased was a real event.
1. (W. R)
2. (S. T)
3. (~S>~T)
4. ~W>(Rv~R)
5. (W. O)
6. (S. M)
7. (S. B)
8. (E)
I think hehe been a while.
The definition of a valid argument is that if all the premises are true then the conclusion Must be true. The truth of the argument is the tough part. The best way to set it up is to say You want to prove P. Well if P is true then ABC. if P is false then ~A, ~B, and ~C. So if any premise line has a counter example then its false. So you ahve to have if antarctica existed there would be a land mass. If it didnt exist there owuldnt be one. If I loved you I would want the best for you. If I didnt love I wouldnt want the best. So then you go through and show one or the other of the premises as true thus proving the validity of P or ~P |
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| trancaholic |
| quote: | Originally posted by Orbax
1. This world is real
2. senses are trustworty
3. If something goes against our senses it isnt trustworthy
4. If it is outside the world is is either trustworthy or not
5. I went to a store on this world.
6. I felt, saw, and smelled the money in my hand go to the cashier
7. I felt, saw, tasted, and smelled the bread./...The bread being purchased was a real event.
1. (W. R)
2. (S. T)
3. (~S>~T)
4. ~W>(Rv~R)
5. (W. O)
6. (S. M)
7. (S. B)
8. (E)
I think hehe been a while.
The definition of a valid argument is that if all the premises are true then the conclusion Must be true. The truth of the argument is the tough part. The best way to set it up is to say You want to prove P. Well if P is true then ABC. if P is false then ~A, ~B, and ~C. So if any premise line has a counter example then its false. So you ahve to have if antarctica existed there would be a land mass. If it didnt exist there owuldnt be one. If I loved you I would want the best for you. If I didnt love I wouldnt want the best. So then you go through and show one or the other of the premises as true thus proving the validity of P or ~P |
I really loved these proofs, yet I have no tangible evidence that there is a land mass, that you smelled the money and the bread and that you would do the best for me. Everything that you (or anyone else) inform me about on the net is based on trust. Which will be further explained in the next post... |
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| trancaholic |
| quote: | Originally posted by Renegade
Alright, what we're really talking about here is epistemology, and the problem is that there are so many different theories and ideas about this subject that we could spend all day discussing it. Perhaps if you could be a little bit more specific about what you're trying to find out it may be a bit easier for us to help you? Is this for university/school or something?
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You're absolutely right. This is epistemology which is why I think it is both relevant and necessary to ask people of different social and geographical origin about these matters before drawing any conclusions.
The thing I am going to use it for is in conjunction with a project we're doing at university. We have proven that a multi-operator version of the axioomatic system of modal logic S4 is both sound and complete with regards to a semantic model we call a multi-relational Kripke frame. With this result we can reason about security protocols in a formal manner, and thus see if there are any holes in them. E.g. if we have premises such as paid_phone_bill=>have_made_call etc. we can determine whether it is possible to end up in situations where it is not the case that phone_bill_received=>phonecompany can_prove calls_have_been_made and similar.
However, our semantic model rests heavily on some assumptions on what a proof is (which gets extremely complicated model-wise when you consider things such as contflicting interests and assumptions of the reasoning capabilities of the third party to which things are to be demonstrated). So far we have come to the lukewarm interpretation, that there's no difference between being convinced of something, and have that something proven to you.
Consider my knowledge that "Mauro Picotto" is an italian man who produces electronica. This is something I am convinced of, yet I've never met him myself, only bought albums, read about him heard his voice on live-sets and so forth. On the contrary I have seen a direct proof of the Bolzano-Weierstrass Theorem. However, if someday I found out that there were no such "Mauro Picotto" (Mili Vanilli anyone?) I would be just as surprised/chocked as if I found out the Bolzano-Weierstrass Theorem were wrong. That is, my convictions (I use no "strong" because I don't believe you can have levels of conviction) are as real to me as what I've seen proof of.
But as this is epistemology and our work focus on what *any* third person might consider a proof, I thought opinions from fellow TAs would (although the samplesize is small) be great for backing up and/or contrasting our result.
| quote: | Originally posted by Renegade
For instance, if you want proof that "Antarctica exists", the best place to start would be to make a number of predictions that must come true should the contention - that Antarctica exists - be true. So, if Antarctica does exist we should rightfully expect a number of things: that it has been discovered and mapped, that it has been viewed from space, that we can speak to people who have been there and so on. These predictions simply form the basis of what we should expect given the claim being made, and, uniformly, we can suggest that these things ring true. We can assume that Antarctica exists simply because almost any prediction we make about its existence (within reason - i.e. assuming we have full faculty of our senses as you suggested above and so on) rings true.
Of course, you could quite easily point out that all we have in terms of proof of Antractica's existence is hear-say: for instance, there may be a massive NASA conspiracy whereby they've photoshopped that massive continent into all the images they've ever taken of planet Earth, and that anyone who has claimed to be to this continent has been brainwashed into thinking that, and that they've only been taken to some other frozen land so that the tour guide can make an easy buck. Firstly, these claims are, in a way valid. I think that this is the point you're trying to make actually - without actually ever having been there, how can we be sure that Antarctica exists?
...
Anyway. Hope that goes some way to answering your question? |
It does. Indeed. Maybe because your entire post does not contradict our work :), but then again maybe you're just persuasive in your argument. |
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| trancaholic |
| quote: | Originally posted by Pjotr G
To prove something means that using some valid rules on some premises, you get to what you wanted to proof, and there is no possible way that the premises are true and the result false.
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Thanks for the reply. The proof/induction rules are OK, but those I can (and have) get from any old book, but they only move the question: How do you know a premise is really true? If you mean strictly true in the sense of "having it in my hand - examining it" then I guess others wouldn't be able to prove much to you?
This is probably because I have failed to phrase my question in an clear manner (I didn't clarify too much as not to influence the replies), but the proof techniques and such are really not relevant to me (although they make great discussion material) only opinions/reflections on what makes you say "Oh, then it must be like that".
btw. an interesting paradox lies in that (in logical proofs) you can't have both correctness and interpretation. Either the entire truth is contained in the premises (in which case what you are trying to prove is trivialy true) or it is not (which means the proof is not correct). Makes your head hurt doesn't it?;) |
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| trancaholic |
| quote: | Originally posted by Azareal
A successful proof occurs when you take a problem, (ie: did I buy a load of bread an hour ago?) and prepose it to either be true or false. Then, you split up the preposition using simple and logical facts that are either a) common irrefutable knowledge or b) have been proven in the past.
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What is "common irrefutable knowledge" but a circular definition? (Sorry had to ask).
| quote: | Originally posted by Azareal
I think my head is going to explode - it would be a lot eaiser to proof something tangible - ie that you are the legal owner of your computer - matching product numbers from your computer to bills of sale, matching the bill of sale to yourself is a simple chain that constitutes ownership. Of course, anyone could doctor/manufacture such information, but a good computer techie could examine your machine, find your 'fingerprints' all over the hard drive (files / programs/ set up -- everyone leaves their styles on the computer) and then go on to prove that the drive is the same drive installed in the computer upon manufacturing.
I guess what I am saying is that the easiest way to prove something is to break it up into small parts - and then proove each small part to be true. The sum of the parts equals the proof of the whole.
Confusing neh? :)
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Like Renegade's post I like yours as well as I don't see direct opposition to te work I'm contributing to. Can I conlude from your two last paragraphs that you find something proven beyond any doubt when the alternative (e.g. me having tampered bills, threatened individuals and so forth) is extremely unlikely in view of your experience? |
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