A well-known scientist once gave a public lecture on astronomy. He described how the earth orbits around the sun and how the sun, in turn, orbits around the center of a vast collection of stars called our galaxy. At the end of the lecture, a little old lady at the back of the room got up and said:
"What you have told us is rubbish. The world is really a flat plate supported on the back of a giant tortoise."
The scientist gave a superior smile before replying, "What is the tortoise standing on?"
"You're very clever, young man, very clever," said the old lady. "But it's tortoises all the way down!"
What do we mean when we talk about science and knowledge? What exactly do these terms mean - the dialogue? the process? the set of causally linked so-called facts? these ideas realized in technology? These are all important players but more important than these perhaps is the context of these linked facts, of this process and of this dialogue.
From this context we are lead to axioms we are required to take on little more than faith. Take for instance the flat earther from the tale above who asserts that “It's turtles all the way down.” Is this mocked worldview really that different from the foundations behind science and knowledge? Is it not true that there are a large number of these so called turtles we rest upon the foundations of how we think? Is it not so that science rests on turtles, even if perhaps they are carried on the shoulders of giants?
The previous story taken from this a different context depicts an illusion in our language - and a very useful multileveled one at that. On the base level we have the traditional interpretation, mocking the flat earther as a fool. From the secondary context of examination, we can see her view is in essence no different from that of the scientist - she at least admits this openly. We hide our faith behind social and lingual constructs so we can be ignorant of our faith. Then, we have at least a third additional context - that ultimately the flat earther, while correctly noting the infinitely recursive nature of the justification of our knowledge, is damned. We must consider the contexts of secondary and tertiary relevance - say, William James' tell of the story as well or even infinite regression in general. We can further extend this to involve the discussion here as a yet another relationship - that between the other contexts.
Our inevitable choices of axiom and context - these guesses are now so ingrained in our culture and language that they at times will even appear to us as objectively true facts or relations of facts and shape our observational language and thought - in spite of the fact that their real worth and validity remains largely to be shown. If we don't sucuum to an infinite regression of turtles we will have to settle with a demarcation made on what can only amount to faith or convenience.
It has been argued in some circles that Godel's Incompleteness Theorem shows us that we must take statements on faith. As John Barrows quips “If a religion is defined to be a system of thought which requires belief in unprovable truths, then mathematics is the only religion that can prove it is a religion!”
The particular aspect that we look at now is its claim that any well made system capable of expressing elementary arithmetic cannot be both consistent and complete. This is to say that for any consistent formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true, but not provable in the theory; this is known as the Godel Sentence of the system. This would be a quite naive view of the situation; when one looks more into the details of Godel's Theorems one sees that in actuality it just says that one can construct a paradoxical sentence in a sufficiently complex system. This is similar to the case of the Liar's Paradox - we can construct a mathematical statement that is analogous to: This sentence is false.
More subtly, we are talking about the implications of over-defining our formal systems (perhaps even the validity of formal systems in general) and the trade-off with having then the power to construct statements that are beyond our systems capability to deal with. Our system becomes powerful enough that it is able to create unruly situations, namely self-referential paradoxical statements that cannot be proven but are true.
While this does show that we must take statements on faith, at least in principle, it does not do so in a meaningful or useful way and thus is without fruit for us. The fact we can construct paradoxical statements simply shows us we are dealing with an approximation that at some venture that needs to be disillusioned. It is not accurate to say that Godel's incompleteness theorem forces us to take these axioms on faith - more instead that it forces us to view strict formalism with a skeptical view. We don't expect English to be devoid of all such internal paradoxes, and it is unreasonable to assume our mathematical language to be devoid of this type of issue at this time.
This case also tends towards an infinite regression as shown by Turing. If we aim to eliminate this issue through defining within our system these phrases of paradox we instead just push the issue to be dealt with on the next layer of abstraction. For example, if we deal with the issue of this sentence is false we would then have to deal with with the issue of the phrase: (this sentence is false) is false and so on ad infinitum. We must at some point take the matter simply on convention or faith.
Turtles anyone?
The next likely target to examine is Induction. Induction can be said to be present when we take a statement about a singular event or a series of singular events and come to a conclusion about universal truths from them. The unfortunate truth is that we have no logical validation for this methodology. This is known as the Problem of Induction.
The problem may be related as follows: If one has seen twelve black birds, and he were to use induction it would bring him to believe that all birds are black. This is obviously not the case because any day a white bird might be observed. Science relies on there existing some validity to induction as it attempts to turn experience into universal statements about the universe. Or as Ludwig Wittgenstein puts it to example: “It is an hypothesis that the sun will rise tomorrow: and this means that we do not know whether it will rise.”
A principle of induction, which is to say a valid formal logical definition that supports induction as a logically sound methodology, may indeed be outside of our reach.
Thus if we try to regard [the principle of induction's] truth as known from experience, then the very same problems which occasioned its introduction will arise all over again. To justify it, we should have to employ inductive inferences; and to justify these we should have to assume an inductive principle of a higher order; and so on. Thus the attempt to base the principle of induction on experience breaks down, since it must lead to an infinite regress.
And even more turtles begin to show themselves!
The hope for a unifying method of science based around a purely logical inductive basis is hopeless. And so we must turn instead to a demarcation to allow us to work as logically as possible - a demarcation that can only be made on faith or perhaps faith disguised as convention.