Hermes
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« on: March 13, 2012, 16:01:22 PM » 

In a scientific approach it is practice to allow for a margin of uncertainty. This is appropriate, since the scientific experiment is subject to observations where a margin of error cannot be ruled out. Furthermore universality cannot be proven with examples. The mathematical theorem has been held up as an example of absolute and universal truth, but the theorem is based on postulates which serve as points of departure for the mathematical system within which the theorem is valid. If "universal" is meant to mean all encompassing and unique, a mathematical theorem may therefore also fail to comply with universal validity. This still leaves one possibility: the axiom or selfevident truth. An example from Euclid's Elements: If two things area equal to another thing, they are also equal to each other. This might constitute an anomaly in traditional skeptical reasoning: an axiom cannot be proven to be universal. Wikipedia does not distinguish between an axiom and a postulate. This has not always been the case. Is there a distinction? It is hard to fathom a mathematical system where the above Euclidian axiom does not apply.



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GCG


« Reply #1 on: March 13, 2012, 16:16:02 PM » 

........



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Tweefo


« Reply #2 on: March 13, 2012, 16:49:32 PM » 

........ I agree. It's been a long hard day..



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Superman
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« Reply #3 on: March 13, 2012, 17:07:47 PM » 

Well GCG I can tell you what I can understand of it but perhaps it is not 100% In a scientific approach it is practice to allow for a margin of uncertainty Here is a quote from Natalie Angier, a science journalist, from her book The Canon, "Science is uncertain because scientists really can't prove anything, irrefutably and beyond a neutrino of a doubt, and they don't even try. Instead, they try to rule out competing hypotheses, until the hypothesis they're entertaining is the likeliest explanation, within a very, very small margin of error – the tinier, the better.” You cannot prove anything in science 100% This is appropriate, since the scientific experiment is subject to observations where a margin of error cannot be ruled out. Supports the above. Based on the premise that we are biased. Furthermore universality cannot be proven with examples. The mathematical theorem has been held up as an example of absolute and universal truth, but the theorem is based on postulates which serve as points of departure for the mathematical system within which the theorem is valid. Mathematical theorem is sometimes used as evidence support something that is universally true. But Hermes is saying that the mathematical theorem is based on postulate. A postulate is when we assume something as correct. If "universal" is meant to mean all encompassing and unique, a mathematical theorem may therefore also fail to comply with universal validity. Hermes is defining universal as all encompassing and unique. That mathematical theorems fail to support the fact of universality I don’t know enough about mathematical postulates to comment. This still leaves one possibility: the axiom or selfevident truth. An axiom presupposes proof. That mean an axiom cannot be proven. You need to rely on an axiom to prove something. An axiom is assumed to be true e.g. A thing is itself. An example from Euclid's Elements: If two things area equal to another thing, they are also equal to each other. Again ‘A thing is itself’ or a cat is a cat. This might constitute an anomaly in traditional skeptical reasoning: an axiom cannot be proven to be universal. Statement that I need to research further. Wikipedia does not distinguish between an axiom and a postulate. Axiom = Postulate. This has not always been the case. Is there a distinction? It is hard to fathom a mathematical system where the above Euclidian axiom does not apply.
As far as I understand Hermes is asking is there a distinction between an Axiom and a Postulate. He find it hard to believe. Is this correct Hermes am I understanding it correct?



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LJGraey
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« Reply #4 on: March 13, 2012, 17:15:06 PM » 

I'm starting to wonder if that hissing in my ears isn't coming from my brain... This is going to keep me busy for a while...



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BoogieMonster
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« Reply #5 on: March 13, 2012, 17:28:32 PM » 

I heard this cool poem once: "Lines that are parallel meet at Infinity!" Euclid repeatedly, heatedly, urged.
Until he died, and so reached that vicinity: in it he found that the damned things diverged.



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Mefiante
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« Reply #6 on: March 13, 2012, 17:32:56 PM » 

An axiom is not the same thing as a postulate. An axiom is applicable to formal systems (chiefly, logic and mathematics) and is a selfevident truth, whereas a postulate applies to natural sciences and is something that is a foundational principle that is held to be true because no reliable observations have ever shown it to be false. Before Russell and Whitehead proved that “1+1 = 2” using propositional logic calculus, this would have been an axiom of arithmetic. In contrast, Einstein’s idea that the speed of light is the same in all inertial frames constitutes a postulate. We cannot conceive of a fuzzy, imprecise situation in which “1+1 ≠ 2” without either being contradictory or perverting the meaning of the symbols that are used, or both, while it is hardly challenging to imagine inertial frames in which Einstein’s postulate doesn’t hold (in fact, our intuitive Galilean conception requires it). In this way, “1+1 = 2” becomes a universal truth with zero room for error, as derived ultimately from laws of logic that are themselves axiomatic. If these laws of logic are subverted, it is not possible to construct a consistent arithmetic, so these axioms bootstrap themselves as universal truths as well because their falsity leads inevitably to absurdities that we cannot experience.
'Luthon64



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Superman
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« Reply #7 on: March 13, 2012, 17:52:55 PM » 

Thank you Mefiante. I read understand and then have to read again to make sure I understand it correct. But it will sink in.



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Hermes


« Reply #8 on: March 13, 2012, 21:50:23 PM » 

Wikipedia does not distinguish between an axiom and a postulate. I have misread the article and apologize for the misinformation. As far as I understand Hermes is asking is there a distinction between an Axiom and a Postulate. He find it hard to believe. Is this correct Hermes am I understanding it correct? I find it hard to distinguish. Again ‘A thing is itself’ or a cat is a cat. I have come across literature where "a cat is a cat" is presented as an "axiom of definition". In my opinion it is a circular argument and says nothing about the existence of universal truth. "Lines that are parallel meet at Infinity!" Euclid repeatedly, heatedly, urged.
Until he died, and so reached that vicinity: in it he found that the damned things diverged. Did Piet Hein become a poet because he failed geometry?



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Hermes


« Reply #9 on: March 13, 2012, 22:00:42 PM » 

An axiom is not the same thing as a postulate. An axiom is applicable to formal systems (chiefly, logic and mathematics) and is a selfevident truth, whereas a postulate applies to natural sciences and is something that is a foundational principle that is held to be true because no reliable observations have ever shown it to be false. 'Luthon64 I like this distinction. Although a postulate "applies to natural sciences" it can also serve as a point of departure for a mathematical system?



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Mefiante
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« Reply #10 on: March 13, 2012, 22:36:29 PM » 

Although a postulate "applies to natural sciences" it can also serve as a point of departure for a mathematical system? Yes, absolutely! Remember that all of mathematics is ultimately traceable back to realworld problems that needed solving. It’s in the way mathematics develops: A realworld problem is abstracted and solved. A similarity with or parallel to another, apparently unrelated problem is noticed. Further abstraction ensues to subsume the two problems under the same descriptive framework. And hey presto, new mathematics comes into being. For example, the complexity of knot theory, which derives from the study of knots in string loops plus a healthy dose of graph theory and topology, is hard to overstate. 'Luthon64



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Superman
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« Reply #11 on: March 14, 2012, 17:34:26 PM » 

I have come across literature where "a cat is a cat" is presented as an "axiom of definition". In my opinion it is a circular argument and says nothing about the existence of universal truth.
It has also been called a Tautology, Bromide etc ad infinitum



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st0nes


« Reply #12 on: March 15, 2012, 09:13:04 AM » 

We need to distinguish between inductive and deductive reasoning. Formal logic and mathematics relies on deductive reasoning in which the conclusion is proved from the premises which are axiomatic (all humans are mortal, Socrates is human, therefore Socrates is mortal); natural science relies on inductive reasoning (we've never seen a swan any colour other than white, so we can postulate that all swans are white; but we may be required to reexamine this when some fool goes to Australia and sees a black swan).



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BoogieMonster
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« Reply #13 on: March 15, 2012, 11:09:49 AM » 

You can still be wrong. "The earth is flat" could (has) easily become a "selfevident fact" to someone with little information. This does not make you LOGIC invalid, which is what people struggle to realize time and again. LOGIC is still valid, even if you get to "untrue" results, because your initial assumptions could very well be wrong. Getting people to "reason" with you during a conversation by stating "given X" is a task I often eschew due to lack of belief in average human intelligence.



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Superman
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« Reply #14 on: March 15, 2012, 22:54:38 PM » 

For those outside the operating theater, however, all the quarreling, the hesitation, the emendations and annotations, can make science sound like a pair of summer sandals. Flipflop, flipflop! One minute they tell us to cut the fat, the next minute they are against the grains. Once they told us that the best thing to put on a burn was butter. Then they realized that in fact butter makes a burn spread; better use some ice instead. All women should take hormone replacement therapy from age fifty onward. All women should stop taking hormone therapy right now and never mention the subject again. The Canon, Natalie Angier. This illustrates what I have experienced in my life. It had a really deep impact on my life but I don’t want to go into it here. This has not made me cynical of science but it taught me a very valuable lesson. Science is most probably the most powerful knowledge we have today but it is sometimes hard to make good choices based just on science. That is why I try to develop a method of thinking in my life where I know as much as I can about science(and sometimes it is hard to find any or even recognize good studies relating to a problem) and learn a method of thinking ( critical thinking, logic and philosophy) that help me to make better choices in life. And here Natalie Angier concludes in the chapter of Thinking Scientifically, The Canon: That you have to be willing to make mistakes if you are going to get anywhere is true, and also a truism. Less familiar is the fun that you can have by dissecting the source of your misconceptions, and how, by doing so, you realize the errors are not stupid, that they have a reasonable or at least humorous provenance. Moreover, once you’ve recognized your intuitive constructs, you have a chance of amending, remodeling, or blowtorching them as needed, and replacing them with a closer approximation of science’s approximate truths, now shining round you like freshly pressed coins.


« Last Edit: March 16, 2012, 08:24:58 AM by Superman, Reason: Grammar »

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Hermes


« Reply #15 on: March 16, 2012, 16:18:11 PM » 

"All humans are mortal" or "the earth is flat/spherical" are not held to be true because the converse would result in a collapse of logic or arithmetic, but based on sensory observations. They do therefore not qualify as axioms in compliance with Mefiante's requirements.



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st0nes


« Reply #16 on: March 16, 2012, 16:24:35 PM » 

"All humans are mortal" or "the earth is flat/spherical" are not held to be true because the converse would result in a collapse of logic or arithmetic, but based on sensory observations. They do therefore not qualify as axioms in compliance with Mefiante's requirements.
No, you are right, but that is just a trivial example quoted in almost all logic textbooks of valid logic whose conclusion is true if the premises are.



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Rigil Kent
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« Reply #17 on: March 17, 2012, 06:22:02 AM » 

... due to lack of belief in average human intelligence. I strongly believe in average human intelligence. An axiom must appeal to our intuitive, yet precise, understanding of it's components (the words making up the axiom.) Unless we can flawlessly conceptualize a point and a straight line, we cannot hope to understand that it is always possible to join two of the former with one of the latter. Because of this crucial link between the truth of an axiom, and our full understanding of it's components, an interesting, almost subconscious "reversal" of the logical process takes place. It appears that the axiom also reinforces our understanding of it's components in a backwards direction! For example, if I have problems logically grasping the axiomatic statement " concentric circles have a common center", then I know that it is time to reconsider my understanding of center, circle, and perhaps even common. And perhaps, in a similar though slightly more spooky way, a theorem reinforces the understanding of it's axioms too. Rigil


« Last Edit: March 17, 2012, 07:18:23 AM by Rigil Kent »

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Hermes


« Reply #18 on: August 13, 2012, 16:25:26 PM » 

If these laws of logic are subverted, it is not possible to construct a consistent arithmetic, so these axioms bootstrap themselves as universal truths as well because their falsity leads inevitably to absurdities that we cannot experience. 'Luthon64 Can one deduce that the antithesis of a paradox must be an axiom? If one examines a paradoxical statement such as "Truth does not exist," one finds that if the statement were true, at least one truth would exist, namely the statement itself. If the statement were false, truth would also exist. Therefore "Truth exists" must be an axiom. One could apply the same argument to the fact that the concept of "almighty" is paradoxical and therefore the nonexistence of an almighty god must be axiomatic.



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Mefiante
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« Reply #19 on: August 13, 2012, 17:09:01 PM » 

A very interesting question, but no, I don’t think that’s true as a universal rule. Selfreferential propositions, e.g. “This sentence is false”, are always tricky and normally require the use of (a hierarchy of) metalanguages for their resolution. In other words, they seem paradoxical when considered on a naïve level but are sensibly apprehendable in the correct framework and so it’s neither obvious what the correct antithesis would be, nor how any such possible antithesis would constitute an axiom. In the case of the given example, you might take “This sentence is true” or perhaps “Every sentence except this one is false” as the antithesis. The result is something that is tautological, incoherent and/or unfruitful. If we consider a paradox that does not rest on selfreference, e.g. Zeno’s AchillesandtheTortoise paradox, we note again that it’s only a paradox because there are perceptional assumptions that conspire to mislead our thinking. Zeno’s paradoxes are rigorously resolvable within the correct framework (specifically, infinite convergent series and calculus). The antithesis would be something perhaps like “Achilles will overtake Tortoise at some point”, which again is a truism (if it is true that Achilles can run faster than Tortoise). I can’t think of any way to reformulate an antithesis to this paradox so that it results in something that could rightly be labelled an axiom. 'Luthon64



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cyghost
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« Reply #20 on: August 14, 2012, 07:46:20 AM » 

You have a beautiful mind.



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Hermes


« Reply #21 on: August 14, 2012, 10:53:43 AM » 

Both "antithesis" and "paradox" are problematic here.
Let's try something related: If the falsity of a statement introduces selfcontradiction, the statement must be an axiomatic truth.



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Mefiante
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« Reply #22 on: August 14, 2012, 19:21:56 PM » 

You have a beautiful mind. If that was directed at moi, I’m truly touched that you should not only think so, but also say so. Merci beaucoup, patron. Both "antithesis" and "paradox" are problematic here. Yes, and this was a substantial part of what I meant to convey. If the falsity of a statement introduces selfcontradiction, the statement must be an axiomatic truth. I think this may be true for some statements within the framework of an axiomatic formal system, but not so generally. If we take an axiom of a formal system and simply negate it, we obtain something (technically, it’s called a ‘sentence’ of the system) that is selfcontradictory (i.e. incoherent and thus false) in that system because it directly contradicts one of its axioms. If we now negate this false sentence, the original axiom reemerges, so in this case your contention is valid. However, the system also has sentences that are socalled ‘theorems’ (i.e., sentences that are provably true in that system) which consist of (some of) the system’s axioms put in a particular logical arrangement. The negation of such a theorem, while clearly false (because it contradicts what is valid in that system), is not itself an axiom because the theorem itself (which is the negation of its own negation) requires more than one axiom of the system for its proof. In this case, the contention fails. The short symbolic version is ~(~ P) = P where P can be either an axiom or a theorem. 'Luthon64



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cyghost
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« Reply #23 on: August 15, 2012, 07:44:01 AM » 

If that was directed at moi, I’m truly touched that you should not only think so, but also say so. Merci beaucoup, patron. It was, been thinking it all along, it was just time to say it out loud dammit.



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Hermes


« Reply #25 on: August 15, 2012, 12:29:05 PM » 

I think this may be true for some statements within the framework of an axiomatic formal system, but not so generally. If we take an axiom of a formal system and simply negate it, we obtain something (technically, it’s called a ‘sentence’ of the system) that is selfcontradictory (i.e. incoherent and thus false) in that system because it directly contradicts one of its axioms. If we now negate this false sentence, the original axiom reemerges, so in this case your contention is valid.
However, the system also has sentences that are socalled ‘theorems’ (i.e., sentences that are provably true in that system) which consist of (some of) the system’s axioms put in a particular logical arrangement. The negation of such a theorem, while clearly false (because it contradicts what is valid in that system), is not itself an axiom because the theorem itself (which is the negation of its own negation) requires more than one axiom of the system for its proof. In this case, the contention fails.
The short symbolic version is ~(~P) = P where P can be either an axiom or a theorem.
'Luthon64 I think it is obvious that a selfcontradiction must contain an untruth. What I am testing, is if its negation may lead us to a special category of truth. Surely the negation of a theorem will create a contradiction with the system, but I would like to see an illustration where the negation of a theorem creates a selfcontradiction.


« Last Edit: August 15, 2012, 12:54:32 PM by Hermes »

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Mefiante
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« Reply #26 on: August 16, 2012, 18:49:35 PM » 

Take the following general sentence of propositional logic: S = P → ~ P (In words: “If P is true then P is false.”) This is clearly an archetype of a selfcontradictory statement. The truth of a logical implication S = P → Q can be negated in just one way, namely when you observe a case where P entails ~ Q instead of Q, and so directly contradicts S. That is, there is a complementary instance of S, its negation ~ S = S' = P → ~ Q. If we replace Q with ~ P, we have our original selfcontradictory sentence S, and its negation becomes S' = P → ~(~ P) = P → P, which is always true. (Technically, it’s called a tautology, which is indeed a special category of truth, viz. something trivially true.) However, S' is not an axiom unless P itself is an axiom, in which case S' is merely an awkward restatement of P. So, once again, the negation of a selfcontradictory proposition is not necessarily an axiom. By turning the situation around and starting with the tautology S = P → P where P could be an axiom or a theorem, we can now negate S in the opposite direction as before, and so arrive at the situation where the negation of a theorem creates the selfcontradiction S = P → ~ P. If the above still doesn’t satisfy your requirements then I don’t understand properly what you are driving at, in which case perhaps you need to delve into some formal logic and the arcane field of axiomatic formal systems to crystallise your meaning. 'Luthon64



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Hermes


« Reply #27 on: August 17, 2012, 12:58:45 PM » 

This answers my questions most satisfactorily. Wikipedia describes a tautology in logic as follows: [A] tautology is a formula whose negation is unsatisfiable. Unsatisfiable statements, both through negation and affirmation, are known formally as contradictions. It follows that the negation of a selfcontradiction must lead to an irrefutable truth. Intuitively it seems selfevident, but it would not suffice to rely on intuition when it comes to proving irrefutability. That the negation of a selfcontradiction leads to a tautology rather than (necessarily) an axiom, is of secondary importance. This is a useful tool, because selfcontradictions are much easier to spot and point out than tautologies. Mefiante, thank you for your (widely admired) guidance.



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Mefiante
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« Reply #28 on: August 17, 2012, 13:25:46 PM » 

Always glad to be of service where I can in the pursuit of understanding. And thanks for the compliments — both of them! 'Luthon64



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Hermes


« Reply #29 on: August 17, 2012, 15:14:22 PM » 

However, S' is not an axiom unless P itself is an axiom, in which case S' is merely an awkward restatement of P. Oops, now I have doubts about this. I don't think that the characteristics of S' should be a function of P's veracity at all. Even if P is an axiom, it should not make S' an axiom. S' appears to me to be an "if...then" statement in which P is both the condition and the fulfillment.



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Mefiante
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« Reply #30 on: August 17, 2012, 15:58:49 PM » 

Well yes, S' = P → P is a tautology (i.e. trivially true) irrespective of the truthvalue of P. However, the meaning of S' (as opposed to its logical truthvalue) is, as said, an awkward restatement of P. If, moreover, P is an axiom or a theorem then P is true in addition to S'.
The more general case of S = P → Q is discussed more fully in the post I linked to earlier.
The trick is in being fully aware of the distinctions between truthvalue, logical validity and meaning. A logical argument can be both true and logically valid (by being sufficiently general) but nonetheless without meaning because meaning will usually lie in particular instances of it.
'Luthon64



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Hermes


« Reply #31 on: August 17, 2012, 16:30:03 PM » 

Thanks. This was just an aside. Have a nice weekend.



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