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How do we label the idea that the likely correctness of an argument or theory decreases with the size of the exposition required to expound it?

Let me make a probabilistic analogy. Suppose we have a chain of length n, made of chain links which under strain of 9000 lb fail with a probability of 0.01. Then even at n = 100 there is a greater chance the chain will fail than not.

Back to philosophy: much of philosophy is based on an archaeology of theory: a very deep extensive archaeology. How often is this depth seen as a problem and how often are re-foundational texts offered?

Edit: Ignore the third paragraph if you'd like. Emphasize the first paragraph and the second only for illustraion.

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    You seem to be asking two questions (1: about argumentation; 2: about archaeological research in philosophy). Generally, posts here work best when they consist of only one question. Which one do you want answered?
    – user2953
    Aug 2 '16 at 21:43
  • In a circuitous way this makes me think of Occam's Razor.
    – commando
    Aug 2 '16 at 22:51
  • Yeah, I feel somehow looped in the question. For the first part, I'd call it an "argument for fundamental complexity".
    – virmaior
    Aug 2 '16 at 23:12
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    For the last paragraph, I don't understand the question per se but as a guess if I'm following the basic idea, I'd say there have been a few different attempts to put philosophy on / under /back on "first principles" / "solid axioms" , etc., but they don't reset the entire discipline. Instead, they get their own following that stands in competition with the older foundations.
    – virmaior
    Aug 2 '16 at 23:13
  • There are correct formal proofs of arbitrarily large finite length.
    – user4894
    Aug 3 '16 at 0:46
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Your first observation sounds like the principle of parsimony, stated cynically, if you are looking for a name. The principle is that the most compact theory is preferred over any more complicated theory that explains the same data. And yours, among other, more aesthetic, considerations is the basic argument supporting the principle.

But the connection implied in the second part is just not true. In general, philosophers have tended to avoid long strings of old arguments as a way of introducing new theories. They try to build positive arguments 'low' and close to the ground of intuition. The long-chain arguments based on centuries of refinement, outside of fairly dead traditions like Scholasticism, appear primarily when criticizing positions and motivating changes or refinements. But avoiding an obstacle does not incorporate the obstacle itself into your theory, even if the statement of the obstacle becomes part of your work, as an excuse for the lack of parsimony.

So what is given by the archaeology is a broad range of counterarguments and test cases one should consider while building. It is not source code, so much as regression tests, and does not produce fragile arguments, but robust ones.

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I don't have an answer to your first question; but an observation that may help bring forward more helpful answers by focusing on a particular discipline where formal arguments are made as a kind of case study.

Take physics, here simplicity is often put forward as a key criteria and thus also a kind of mantra to characterise useful ideas and theories and perhaps almost as a way of thought and life; Einstein said a theory should be simple as possible and no simpler, and this is a thought that needs to be unpacked and to be laid out.

Take Maxwells Equations, not as we now find them but as he first wrote them down. Written out fully, they comprise twenty equations - hardly simple; however, when we look in an undergraduate textbook, we find they have been simplified to four equations and when we look at a more advanced textbook, one for graduates, we find only two. (On this evidence one might think that the next step and the last step is to reduce it to a single equation).

What we have here is not simplicity but simplification, and this has been ground out by inspiration and hard work by several generations of collaborative work between physicists and mathematicians.

What then of Maxwells work? It cannot be simple in light of its later simplifications, yet there is something simple in it that has been grasped. Some new reality has made itself manifest even if it remains still uncovered.

Because it is new, it cannot be written down easily in the old forms, hence the complexity of its expression, and the arguments that uncover it; arguments here are complex.

Physics yearns towards the tautological; here truth manifests itself solely as no more than the truth of form, and no less as there is nothing more than form. This is the truth of mathematics where 1+1=2 merely by its form - it cannot be otherwise. (It can become more by being less, to reduce it to the truth of 1, simple in itself - the Pythagorean truth - but this strp is a step towards truth on a symbolic level).

Thus physics, when it looks for its own essences, looks for that expression that comes closest to mathematics, and it's in this uncovering, clearing and motion that simplicity displays itself: to come as close to the tautological without being merely and only tautological as physics partakes of the earth and air in a way that mathematics cannot.

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One might call it the Regulae problem from Descartes' Regulae ad directionem ingenii. Harry G. Frankfurt brings this problem out in discussing the Meditations. He refers to cases:

...such as the proof of a proposition t from a self-evident principle p by seeing that p entailed q, q entailed r, r entailed s, and finally s entailed t.... There could be no doubt in Descartes' mind about the first principle, when he clearly and distinctly perceived it, nor could he be in doubt about any one of the steps as it was taken. But, when he reached t, he had travelled such a distance that he might no longer perceive either his first principle or the first steps in the proof. He might have to remember, for instance, that he had already proved s.Since his memory was fallible, he could not be absolutely certain of this.... The proof of t, since it depended on memory, could not be certain so long as Descartes entertained doubts about his memory. . . The proof of t could be made certain by God's certification of the use of memory. ( Harry G. Frankfurt, 'Memory and the Cartesian Circle', The Philosophical Review, Vol. 71, No. 4 (Oct., 1962), 327-328.)

If we delete God from the picture, then there are good probabilistic grounds for assuming that the longer the argument, the greater the problem of recall of all the premises and of the relations between them - i.e. the higher the probability of mistaken recall.

Descartes realises the greater the length of argument, the greater the chance of error through incorrect recall in Rules for the Direction of the Mind, Rule III (The Philosophical Writings of Descartes, I, ed. J. Cottingham & others, Cambridge : CUP, 1985, 15.)

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