Like most people of my generation- my sensibility has been shaped by science more than anything else. I am using ‘science’ in the broadest sense to describe a systematic schema for thinking, which starts of with a small set of axioms (or self evident truths) and by a logical evaluation of data/information posits and proves hypothesis, which can then have predictive power, by mean of inductive inference.
Provability is hence central to the scientific approach- in fact philosophically that marked the deviation of modern science from religion
(In the pre-industrial revolution period ‘faith’ was as valid as ‘reason’ for advancing a theory. Science was accorded primacy (after a long and fierce struggle) only when its explanatory power clearly oustripped that of religion)
It is thus disconcerting if in the province of science, there exist truths that are not provable.Such truths exist- in that most impregnable of fortresses – mathematics! In fact the discovery of the ‘incompleteness’ of mathematics can be extended, by the same reasoning, to show that any formal system is inherently incomplete ( since all formal systems have mathematical analogues).
This was proposed (and proved dare I say!) in a paper titled On Formally Undecidable Propositions in Principia Mathematica and Related Systems I by the Austrian mathematician Kurt Godel in 1931
The proof is vast and rather abstruse- it took a whole book- Godel, Escher, Bach by Douglas Hofstadter for me to grasp it. But what is more fascinating than the proof, or even the insight, are the implications.
Hofstadter himself brilliantly builds on Godel’s Incompleteness theorem to build the best explanation I have read on the classic question of metaphysics- namely, the mind-body problem.
At a more general level the philosophical implications are mind-boggling. It demonstrates that there exist definite truths, that are definitely not provable! Truth and provability, seen as concomitant if not identical, have effectively been delinked!
The reason this is so disconcerting is that we have always had a sense that with science we were inexorably moving towards a situation when everything could be explained and proved ( possibly in infinite time; the relevant fact is not whether it was practically achievable, but that theoretically we were moving towards that).
What the incompleteness theorem demonstrates is that we are consigned to have atleast some knowledge without demonstrable basis. That is, provability is a weaker concept than truth, the known truths will never be circumscribed by provable hypothesis.
But wait a second- doesn’t that sound suspiciously similar to religion- that something is true without it being proved!
That is actually a specious argument; formal systems have not been undermined to the extent that in terms of validity it is on par with religion. It is only that we now have a more modest expectation of the explanatory power of the methodology of logic, which was hitherto over-estimated.
But in a fundamental sense the certainty of formalism has been challenged- we will always know things that we cannot prove- that is inescapable. In the abstract, at least experientially that does parallel a religious experience- where you know something is true but cannot prove it.
For someone who is scientifically inclined, but has turned from atheism to being a believer only recently, its comforting to know that I am not being inconsistent vis a vis my scientific temper, when I know something without being able to prove it!
5 comments:
I read old GEB, as my friend Taktin called it when he asked me whether I read the book. It was on my shelf, and I must say that I'm sorry I lied to him when I told him that I read it. (Otherwise he would have never shut up about it! My reputation!) Subsequently, I read it, and found myself scratching the old noggin about questions of provability.
Truth is stronger than its ability to be proved, but like you, I find it specious that when faith has been proven to be a cruder tool than reason, it has suddenly been looked upon more kindly. One's failings do not excuse the others'.
But that doesn't mean that there won't be some ridiculous breakthrough in human thought from which we won't develop other points of analysis. (I am beginning to wonder if that world'll become obsolete, even.) And when that happens, maybe we will be able to "prove" a lot of things, or at least understand them in other dimensions.
Anyway, wow. You've hurt my puny brain.
An alternative and, in my opinion, fascinating way of looking at science and truth is critical rationalism, the philosophy of Karl Popper.
A simplified version of Popper's philosophy is: Human understanding (or science) evolves through theories. It is not necessary for a theory to be provable. However, inconsistencies or empirical findings can disprove a theory. When such flaws are discovered in a theory, it is discarded and replaced by a theory with no known flaws. As we discover more flaws, we improve our theories.
Assigning special status to certain "self-evident" truths or axioms seems like an arbitrary process to me.
With critical rationalism, we avoid the cumbersome idea of provability and deal instead with "disprovability."
Faith can probably never be disproved and can therefore claim to be a legitimate part of a theory (in the Popperian sense). If you feel that faith adds to the explicatory power of your theory, you are justified in your stand.
I am uncomfortable with the notion of scientific enquiry focusing on 'disprovability' as opposed to 'provability'- this form of enquiry would lead to the formulation of a whole set of fanciful theories that cannot be 'disproved', but simply not have explanatory power (i.e. the theory is advanced in only a limited sense, ensuring consistency but eschewing comprehensiveness). Also, in the Popperian sense, is it necessary to build causal linkages between observed phenomena and proposed undelying causes with the same degree of rigor that 'classical' mode of enquiry would require ( I don't know about this - I am asking since I get the feeling that disprovability lowers the bar of methodological skepticism which underlines most scientific enquiry as we know it). Would Popperian philosophy be succesful then-in the sense of advancing human understanding, though I do concede that it is as self-consistent as Occam's Razor- the parallel underlying principle to generate hypothesis in classial science.
Let me give you an analogy from Computer Science (what else?) to explain what I believe to be the relation between Popper's philosophy, science and Occam's razor. Let us say you want to answer the "ultimate question" (and you disagree with the other Douglas in that you think the answer is not 42), then in CS terms you are trying to perform a search on a very large search space.
In other words, imagine that there is a space where every point represents a theory or set of theories that can explain the universe completely (not necessarily in a deterministic sense). Typically, science (in the "classical" sense) places positive constraints on this search space. It says things such as: the point representing the solution should have properties X,Y,Z. Occam's razor is more of a heuristic than a constraint. Occam's razor tells you stuff like: Since there is no constaint on the presence of property P in the solution point, start searching for points without property P. Popper's philosophy says that at any given time, you have a temporary solution point (which is the "best" point you have found so far) and set of feasible points. As our understanding improves (i.e. as we carry on the search algorithm), we find that the temporary solution point keeps changing (and improving), while the set of feasible points keeps shrinking.
Think of it in terms of the game of minesweeper (Microsoft is evil). You can prove that some squares have mines and some are clear while you cannot prove anything about other squares. Yet, when you reach an impasse, you continue playing (you open up a random square).
In management-speak, the focus of Popper's philosophy is on the process, not the product. I just had to say that. I registered for the GMAT today. ;-)
Speaking of minesweeper (Microsoft is evil) and computer science, the humble game is in the forefront of a decades-long search for an answer to the "P=NP?" question. Oh, and you get 1 million dollars if you can do it.
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