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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I agree. It’s been a long hard day…

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 [i]Elements:[/i] 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.

[url=http://en.wikipedia.org/wiki/Axiom]Wikipedia[/url] 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?

:-[ 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…

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.

An axiom is not the same thing as a postulate. An axiom is applicable to formal systems (chiefly, logic and mathematics) and is a self-evident 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

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

I have misread the article and apologize for the misinformation.

I find it hard to distinguish.

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.

Did Piet Hein become a poet because he failed geometry? >:D

I like this distinction. 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 real-world problems that needed solving. It’s in the way mathematics develops: A real-world 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

It has also been called a Tautology, Bromide etc ad infinitum

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 re-examine this when some fool goes to Australia and sees a black swan).

You can still be wrong. “The earth is flat” could (has) easily become a “self-evident 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.

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. Flip-flop, flip-flop! 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.

“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.

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

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.

A very interesting question, but no, I don’t think that’s true as a universal rule. Self-referential 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 self-reference, e.g. Zeno’s Achilles-and-the-Tortoise 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