Reason and Religion: Irremediably Incompatible Bedfellows?
We find an example of this in Newtonian mechanics. It held the reins of scientific knowledge for about two and a half centuries. Classical physics was a success story the likes of which have occurred few times in human history, if at all. However, a thorn in the side of Newtonian mechanics remained in the form of Mercury’s perihelion. The planet should move in smooth ellipses like the other planets. But it didn’t. It wobbled slightly, creating an unbalancing effect. Thus predictions regarding Mercury’s behavior were less than satisfactorily accurate. And science was stuck with a contradiction. Einstein’s relativity seemed to resolve the issue. At least it resolved this particular issue. But others quandaries remained. For instance, within Einstein’s universe we have the ‘twin paradox.’ One of the twins enjoys a trip around the universe at a velocity approaching the speed of light, while the other twin stays home. When the adventurer returns, many years have transpired on Earth, but as far as she is concerned, her trip lasted only a few weeks. She has aged little, but to her consternation, her sister is now old and infirm. So to the question ‘Can our scientific theories be entirely free of contradictions and hence inconsistencies?’ an ultimate answer can hardly be forthcoming, at least in terms of classical logical terms.
Aristotle is often considered the father of classical logic. Aristotle set out the Identity Principle according to which something is what it is and nothing else, for if it were something else it would not be what it is, hence nothing else can be what that something is either. The Principle of Non-Contradiction is dependent upon the Identity Principle. It says that if something is what it is, then it cannot be anything else. Then there is the Principle of the Excluded-Middle according to which if A is what it is and nothing else, then whatever there is must be A, or it must be Not-A, and there can’t be anything else. There can be no third option between something and that which it isn’t. In a nutshell, that’s about it. Rather disappointing, one might think. Nevertheless, it guided much Western thought for a little over a couple of millennia.
Actually, the condition of classical logic is not as severe as it might appear. When we look at the fine points, we come to understand that A cannot be Not-A at the same time and in the same respect. This greases the wheel a little so it can move in one direction or the other and even make a few wide turns when it must in order to avoid catastrophe. It confines the Principles of Identity and Non-Contradiction to legitimate identities and contradictions and excludes the apparent ones. There is no identity, contradiction, falsity, or unintelligibility in a statement about something that holds at one time and place and in one respect but not at another time and place. To say ‘All people are mortal’ is apparently unquestionable. To say ‘The earth rests on an elephant’s back’ might appear patently absurd, and perhaps even unintelligible for today’s worldly inhabitants. Yet according to certain accounts it was found acceptable by a certain group of people from a past culture. The idea that ‘The universe is (like a) a machine’ predominated Western scientific thinking for a few centuries and came to pervade its general mind-set for generation after generation. Today, however, it has by and large fallen from grace——though there are some well-meaning citizens of the world who apparently don’t know it yet. A sentence is considered contradictory in the strict logical sense only in the event that it is simultaneously subject to affirmation and negation. In such case it is inconsistent, and one might wish to say, unintelligible——though children, poets, and mystics would likely disagree on this point.
In this sense ‘A and not Not-A’ might appear as the sole survivor with respect to iron-clad contradictory sentences. Even such apparently deductive sentences as ‘1 + 1 = 2’ and ‘All people are mortal’ are not absolutely immune to revision under certain circumstances (philosopher Charles S. Peirce himself has said as much). In quantum theory the addition of certain subnuclear particles does not yield a product equal to ordinary arithmetic addition. Rather, it leaves us with less of what we started with. In this manner we might conjecture that we have the equivalent of ‘1 + 1 = 1’. In Boolean logic also, as well as Spencer-Brown’s (1979) ‘Laws of Forms’, if an annotation is made, and then made again, it is the same as if it were made once. Hence also ‘1 + 1 = 1’. Disconcerting.
Even more disconcerting, in quantum theory and relativity, enigmatic numbers called ‘imaginary numbers’ (or √√-1) occasionally pop up. How can we rationalize these numbers? If we say ‘*√-1 = +1’, it’s a strike against us. If we say ‘*√-1 = -1’, we’re victims of another strike. One more strike and we’re out. What are we to do? Say the answer’s both of the possibilities, and it’s neither of the possibilities, which leaves us with a paradoxical situation? There seems to be no rational answer to the problem. Nonetheless, scientists and computer engineers customarily use ‘imaginary numbers’ in their mathematical description of what is presumably ‘real’. Biologist and physicist Howard H. Pattee (1969, 1972, 1979, 1986) goes so far as to speculate that the equivalent of ‘imaginary numbers’ exists at the heart of life processes.
In our age when prosthesis, transplants, and cloning are coming into their own, who knows how and to what extent life can be prolonged a century from now——if by that time we haven’t all destroyed ourselves and our planet Earth to boot. With respect to presumably inductive truths, the sentence ‘All crows are black’ is not absolutely fool proof. We can have no absolute proof, inductively speaking that is, until we have observed each and every crow, past, present, and future, which is to say that we have to be well nigh immortal, thus refuting the deductively or synthetically derived sentence ‘All people are mortal’.
Pouring more salt in our inductive wounds, Carl Hempel’s once demonstrated that in order to obtain absolute proof that ‘All crows are black’, we have to confirm the contrapositive sentence, ‘All nonblack things are noncrows’. This for sure lands us in interminable task. After all, if we spot a nonblack thing, how do we know it is not a crow until we have verified its noncrowness? For all we know it could be an orange crow we overlooked when we were obsessed with our premise that ‘All crows are black’ and we overlook that white thing that happened to be a bird that happened to be a crow. One could perhaps say that by deduction we can try our damnedest to demonstrate that a sentence is not inconsistent, and that by induction we can try our damnedest to demonstrate that a sentence is not consistent, and in both cases we will either end up with an inconsistency, or our search will suffer from incompleteness, given our finite capacities.
The inconsistency or incompleteness of all sufficiently sophisticated formal systems is due to the enterprising work of mathematical logician Kurt Göödel (Nagel and Newman 1964). His two incompleteness theorems reveal the impossibility of our deciding on the truth of a logical system from within that selfsame system. Some scholars have suggested basically the same with respect to ordinary language use, and indeed, with respect to all sign making and taking. The layperson’s example of Göödelian undecidability is a variation of the infamous Cretan paradox in the form of ‘This sentence is not true’. Is it true or is it false? If we say it’s true, then it says of itself that it is not true, so it is false. O.K. So we assume it is false. But that is what it says of itself anyway, so what it says of itself is true, so it is true. We simply can’t win. The sentence refers to itself, it contradicts itself in its self-reference, and thus it sets up an infinite regress. We might wish to say that the Cretan sentence is not true and false simultaneously. We read the sentence, pretty much like we read most sentences, by charitably suspending disbelief and giving it its say. Then, after the fact, we think about what it says of itself, and we balk. Why, the sentence is absurd, nonsensical, and perhaps unintelligible! And we are in a certain sense correct: the sentence actually doesn’t affirm and deny itself in the same breath, in the same instant.