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Was Gödel the first person to pose and solve this question in mathematics? In the larger philosophical debate, has this question been posed before? Say by Plato or Aristotle?

One could interpret for example the famous statement by Horatio to Hamlet in this vein:

There are more things in heaven and earth than is dreamt of in your philosophy

That is there are more true things than can be grasped by philosophy.

But I'm looking for similar statements or positions within the Western Philosophical Tradition. The Eastern obviously has the Dao and Zen which has been epistemologically understood in this way.

  • Awareness of the difference between proof and truth is at least as ancient as the first person who attempted to provide a proof. Otherwise why bother with the proof? – David H May 20 '13 at 14:51
  • Good point. I mean that truth exceeds proof. – Mozibur Ullah May 20 '13 at 15:25
  • Since the last edit (#4), I wonder if you caught Gödel right. As far as I'm aware, he didn't state that "truth always exceeds the grasp of proof", unless you mean the whole truth. His first incompleteness result is about some truths being unprovable. (And his second is about systems not being able to prove their own consistency.) – user3164 May 20 '13 at 16:22
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    Always remember that Goedel's result was a technical one about provability within a given formal system. Whenever anyone says Goedel's theorem states that "There are some truths that are not provable" always repeat it back "In formal systems that are consistent and include arithmetic, there are some truths of that system that are not provable in that system." – Mitch Jan 13 '14 at 2:22
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Agrippa the Skeptic (a Greek philosopher from ~1st century AD) is known for supposedly establishing the impossibility of certain knowledge on the basis of five possible grounds for doubt:

  1. Dissent – The uncertainty of the rules of common life, and of the opinions of philosophers.
  2. Progress ad infinitum – All proof requires some further proof, and so on to infinity.
  3. Relation – All things are changed as their relations become changed, or, as we look upon them from different points of view.
  4. Assumption – The truth asserted is merely a hypothesis.
  5. Circularity – The truth asserted involves a vicious circle (see regress argument, known in scholasticism as diallelus).

One can clearly see the influence of Euclid's Elements, which straightforwardly provides a recipe for the enunciating the five reasons for doubt listed above by working the axiomatic method in reverse.

This is most likely the first sophisticated philosophical understanding of the separateness between truth and provability. The next big step forward won't come until the 19th century with the advent of non-Euclidean geometry which necessitated a complete overhaul in the mathematician's notion of what it means for an axiom to be true.

  • I can't see Euclid infuence here. Can you explain more? – Mozibur Ullah May 20 '13 at 17:15
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Not to know of what things one should demand demonstration, and of what one should not, argues for want of education. For it is impossible that there should be demonstration of absolutely everything (there would be an infinite regress, so that there would still be no demonstration).

Aristotle, Metaphysics 1006a 7-9

  • Context: In book Γ Aristotle is defending the principle of non-contradiction by showing that tentative denials of the PNC do presuppose it. – DBK May 20 '13 at 15:57
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Aristotle holds that first principles are undemonstrated truths (Posterior Analytics 72b18):

Our own doctrine is that not all knowledge is demonstrative: on the contrary, knowledge of the immediate premisses is independent of demonstration. (The necessity of this is obvious; for since we must know the prior premisses from which the demonstration is drawn, and since the regress must end in immediate truths, those truths must be indemonstrable.) Such, then, is our doctrine, and in addition we maintain that besides scientific knowledge there is its originative source which enables us to recognize the definitions.

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