In his autobiography, the great logician and mathematician, Bertrand Russell presented the decisive moment his older brother, Frank, launched his intellectual career by beginning him on Euclidean Geometry. Bertrand Russell was eleven. As well as being a great logician and mathematician, Russell was probably the most renowned professional atheist of his time*. As befits a budding professional doubter, the younger boy wanted to know why he should accept the axioms of plane geometry without proof. His brother didn't have an answer except to point out that unless he did accept them they couldn't go on.
I had been told that Euclid proved things, and was much disappointed that he started with axioms. At first I refused to accept them unless my brother could offer me some reason for doing so, but he said, "If you don't accept them we cannot go on", and I wished to go on, I reluctantly admitted them pro tem. The doubt as to the premises of mathematics which I felt at that moment remained with me, and determined the course of my subsequent work.
Bertrand Russell: Autobiography, volume 1 page 32
There is, to date, no absolute logical proof of the foundations of geometry, nor, as Russell found to his crushing disappointment in the late 1920s, of mathematics. The proof, famously provided by Kurt Godel, would seem to be that those don't exist within human reason. After more than two millenia of looking for them with some of the finest of human minds, call me skeptical, but I believe that kind of logical closure is as imaginary as other human quests for dreams of perfection.
There was, obviously, a strong emotional desire in young Bertrand for an absolute logical and intellectual certainty that is, simply, not available. If that absolute certainty is unavailable for the artificial forms of geometry and numbers, all of them human creations abstracted from humans' experience of the physical world, why would anyone expect absolute logical and intellectual certainty for less refined aspects of human experience? I think it's exactly that emotional desire for certainty that ruled large parts of Bertrand Russell's thinking and writing. He says that his conditional acceptance of the axioms of geometry was motivated by a desire to get on with the subject. He obviously wanted to have geometry more than his imagined perfection, for the time being.
Euclidean geometry is an application of the methods of logic as well as mathematics. All of mathematics is reliant on the human practice of logic, all mathematical proof is founded in logic. The consequence of that is that if any logical flaw is discovered in an exposition of mathematical propositions, the proposal fails. Mathematics rests on a basis of logic and, again, as was famously demonstrated by Godel, logic has no absolute foundation in itself. It rests on something that is far more disreputable in today's culture, human experience and persuasion. That has been known for far longer. For example, there is no logical reason to believe that because in every case you can conceive of it to follow that if a = b and b = c that a will equal c in each and every possible case. We believe that out of our common experience of the external world.
I'd go farther than that and point out that the first part of that, a equaling b is, clearly, a human construct. a is not b but it is held to be equal to b. a is also not identical to b, even if every single other thing about them is the same. b could be something exactly like a for every other purpose except that b was not a. There is at least that aspect that they don't share, a is not b. Equality would include that aspect of difference. That relationship of equality of two separate objects is the product of human thought, not an intrinsic attribute of a and b. And the statement can be even more problematic. If we say that a equals a and that b also equals a, then we have set up two different meanings for the relationship of equaling, both identity of something to itself and functional similarity of a thing to something other than itself. Obviously, in that case, the relationship of equality would not always be equal to itself. We can determine that from the imaginary entities "a" and "b" which we are capable of giving a similarity that real things don't really have. For real things that have an existence separate from human imagination, those relationships become even more tenuous through the differences between things held to be equal**.
We are accustomed to ignoring problems like that in cases when every single person in the world would maintain that a equals b but that is the case. Like the axioms of plane geometry, ignoring their equality is an invention of human convenience is something that works. If someone denied that common experience, they would be held to be irrational. We create that equivalence out of our abstracted experience of the physical world. We maintain that their equality is a useful idea because it works. It is a creation of human utility.
What can be said of the impossibility of an absolute foundation of logic separable from the far less than absolute foundation of human experience is even more true of higher level intellectual activity. Logic and mathematics are two of the foundations of science. Any scientific proposal that can be refuted by a failure of its logical or mathematical basis fails as science. Or that should be the case if the public promotion of science is honest. Far from being the verification of logic or mathematics, science is their ward. As repugnant as the idea is to many atheists who believe science is the absolute test of everything, science isn't even the proof of itself. Science relies on belief as certainly as mathematics or logic, there is no separable higher level of any of those products of human experience that is independent of that fact. It is extremely rare for a scientist to admit that, especially, in my experience, for an atheist to admit it. One of the few examples I know is found in Joseph Weizenbaum's great and tragically unknown book, Computer Power and Human Reason:
The man in the street surely believes such scientific facts to be as well-established, as well-proven, as his own existence. His certitude is an illusion. Nor is the scientist himself immune to the same illusion. In his praxis, he must, after all, suspend disbelief in order to do or think anything at all. He is rather like a theatergoer who, in order to participate in and understand what is happening on the stage, must for a time pretend to himself that he is witnessing real events. The scientist must believe his working hypothesis, together with its vast underlying structure of theories and assumptions, even if only for the sake of the argument. Often the "argument" extends over his entire lifetime. Gradually he becomes what he at first merely pretended to be; a true believer. I choose the word "argument" thoughtfully, for scientific demonstrations, even mathematical proofs, are fundamentally acts of persuasion.
Scientific statements can never be certain; they can only be more or less credible. And credibility is a term in individual psychology, i.e.. a term that has meaning only with respect to an individual observer. To say that some proposition is credible is, after all to say that it is believed by an agent who is free not to believe it, that is, by an observer who, after exercising judgement and (possibly) intuition, chooses to accept the proposition as worthy of his believing it. How then can science, which itself surely and ultimately rests on vast arrays of human value judgments demonstrate that human value judgments are illusory? It cannot do so without forfeiting its own status as the single legitimate path to understanding man and his world.
Immediately after that rather bracing idea, Weizenbaum placed the myth of scientific certainty in its undoubted and highly dangerous social context:
But no merely logical argument, no matter how cogent or eloquent can undo the reality that science has become the sole legitimate form of understanding in the common wisdom. When I say that science has been gradually converted into a slow-acting poison, I mean that the attribution of certainty to scientific knowledge by the common wisdom, an attribution now made so nearly universally that it has become a commonsense dogma, has virtually delegitimatized all other ways of understanding.
The rest of the book is a brilliantly argued analysis of the problems coming from that most common of faith holdings.
I think Joseph Weizenbaum's confessions of the real nature of, not only science but of logic and mathematics is one of the bravest intellectual acts I've ever encountered. He questions the status which science claims and which is claimed for it and asserts that foremost creed of modern life is dangerous. With it he jeopardized his credibility with less rigorous thinkers, his professional standing and the usual foundations of atheism. I've never encountered another atheist who so basically impeached the intellectual conceit that most of the articulation of atheism is based in. The use of science and logic in atheist polemics against religion requires that their own dependence on belief, on persuasion be ignored or denied in a sputtering display of derision.
* Though, officially, Russell was an agnostic. There's a case that he stole that position from Joseph McCabe who will figure in a later post. Today Russell's position has remained in atheist culture whereas McCabe has been eclipsed. I think there were political reasons in the post-war period cover up the period of atheist history that McCabe represents.
** In nothing else is the problem so undeniable as the frequent holding that people and other living beings are interchangeable objects. What began in the 19th century scientific sexism and racism that made generalized statements about women, men, Black people, "Redskins" etc. is matched today in the ubiquitous faith in the genetic determinism of organisms, the resurgent expression of the same faith. The dependence of that faith on the allegedly scientific study of twins, the assertions of identity based on a shared genome ignores the fact that even identical twins raised together do not share the same identity but are, in fact, individual people. Even when the twins, themselves, deeply want to believe otherwise.
Even clones are not identical to the "parent", not even on the most basic level of physical appearance. Organisms are not abstractions that can be manipulated like angles made by intersecting lines, they cannot be rotated and mentally superimposed on each other, becoming identical. Even real lines in real space can't really be manipulated into identity. How much less can organisms, which exist as independent beings in reality be scientifically squeezed into the same kind of imaginary, geometric non-space. Even those ants I mentioned the other day are individuals. If science can pretend they are for every desired purpose, that doesn't change the fact that the purpose of science is not a comprehensive and exhaustive description and manipulation of the lives of individuals within an ant colony. Yet so many people today are content to pretend that human beings are interchangeable vehicles of DNA, ignoring even the differences in that famous molecule and the variability of its functions in even individual cells.
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