jamest wrote:Xamonas Chegwé wrote:You did not win the point, LI.
You merely claimed that the existence of mathematical proofs somehow validated your point. But mathematical proofs are built upon earlier proofs, etc. (it's proofs all the way down) until, in the final analysis, they are built upon axioms which cannot be proven and are merely taken as being self-evident.
Here's the stinger.
Those axioms are accepted as self-evident because they agree with our observation of empirical data.
I don't believe that this is true. That is, I don't believe that whatever has come out of the mouths of mathematicians, or logicians, has necessarily been a consequence of 'observation'.
You are, of course, entitled to believe whatever you will. However, for me to be convinced that your belief system is based upon anything other than blind faith, I would appreciate evidence of how any of Euclid's postulates and axioms diverge from observation. (The exception being the parallel postulate, the negation of which leads to some very useful results in what is known as non-Euclidean geometry - ie. the geometry of non-flat surfaces. The postulate holds perfectly well for the geometry of flat planes however.)
I only have to cite Zeno of Elea - who lived about 2500 years-ago - to lend weight to my point. He was famous for his logical parodoxes, derived purely from logic, which made claims contrary to observation!
Actually, Zeno's paradoxes were derived from (faulty) logic
applied to observation, not purely from logic. But continue.
Logical conclusions are not necessarily a subsequent claim of observation. That much is evident in the works of many philosophers, not just Zeno. The same applies to mathematicians.
People make mistakes and use faulty logic, in other words? I doubt anyone would dispute that.
And, btw, Euclidean logic is not necessarily 'wrong' - it's just at-odds with how 'the world' is perceived.
Is it? How please? And are you referring to Euclid's postulates here? Or to the logic which builds upon them to produce mathematical proofs?
People are so short-sighted.
So this is an eyesight issue? Don't prescription lenses help?
They garner information, for instance, that proves that Euclid's axioms are not absolute - and instantly attack Euclid's axioms.
Which axioms have been proven not to be absolute? Your evidence please? And why would they be wrong to attack a provably untrue premise?
Why don't they fixate upon something else instead, such as the 'absoluteness' of the world that he was talking about?
Why attack that? What has it ever done to you? And what exactly are you referring to when you talk about the absoluteness of Euclid's world?
Nobody has ever proved that Zeno, or Euclid, were 'wrong'.
Aristotle went a great way towards proving Zeno wrong. Archimedes improved upon his work. More recent advances in mathematics, especially calculus, completed the job. All of the common paradoxes come down to summing infinite sequences, the methods for which have been rigorously proven logically.
As for Euclid, exactly what did Euclid ever claim that is at odds with observable reality or logical thought?
All they ever did was show that observation was [sometimes] at-odds with logical thought!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
Well,
Zeno did (not Euclid - I have no idea where you got that notion from) but only because his logic was flawed (although it took several hundred years to pinpoint exactly how with rigour.)
There is plenty about Zeno's paradoxes in
wikipedia, including sketchy explanations of various solutions. Far fuller demonstrations exist if you care to google 'Zeno paradox solution' and can follow the maths.
To sum up. You seem to be saying two things here (correct me if I am wrong.)
(a) That Euclid's axioms are at odds with observation in some way.
(b) That logic built on these axioms can contradict observation, ie. lead to false results.
(a) may well be true but nobody has demonstrated this in the 2,300 since the
Elements were first published. Axioms, by definition, cannot be proven, merely held to be self-evident, but can be shown to be false with a
single counter-example.
(b) may also be true, as every possible result that can be derived from Euclid's axioms is not known and never will be. However, nothing based upon the axioms and constructed via
sound logic springs to mind - I am sure that it would have made quite a splash in mathematical circles had a false result based purely upon Euclid's axioms been proven, so I think I would have heard of it.