This day, twelve years ago,

this Robin was sunbathing

 

Definitions

In the seventeenth century, Leibniz, a German Lutheran famous for inventing the infinitesimal calculus (following Newton's invention of a similar method), was primarily working on a logical calculus (following Descartes' algebraic innovations). Leibniz thought that a logical calculus would make the world a more peaceful place: if two people disagreed, they would not need to argue, they could just sit down with paper and pencil and compute the true answer. Leibniz hoped, for example, to use his logical calculus to show Locke that 2 + 2 = 4 is not an empirical truth but a logical truth:

Starting with the logical axiom that things are identical to themselves (expressed in symbols as A = A, with A being 2 + 2 in this example), Leibniz applied the logical rule that replacing equal things preserves equality three times (with three definitions, a definition of 2 as 1 + 1, a definition of 3 as 2 + 1, and a definition of 4 as 3 + 1) in order to obtain that arithmetical truth:

          2 + 2  =  2 + 2
          2 + 2  =  2 + 1 + 1
          2 + 2  =  + 1 3 + 1
          2 + 2  =  + 1 4

Laid out like that, the logic of that proof of 2 + 2 = 4 is certainly very clear, and Leibniz wanted to bring such clarity to all of our reasoning, by expressing in symbols all of the logical axioms and rules of inference used in logical thinking.

In the nineteenth century, Frege, another German Lutheran who wanted to make logic more like algebra and to show that arithmetical truths are logical truths, noticed that Leibniz had not shown that 2 + (1 + 1) = (2 + 1) + 1. Still, it is obvious that for small numbers of things, how many things we have does not depend upon which ones we look at first. And if we had to prove every obvious step in a logical proof, then we would never get to prove anything logically.

Logical proofs begin with descriptions that are obviously true (or descriptions that would clearly be true were the assumptions of the proof true) and then, step by logical step (each step being such that from something true it would clearly take us to something true), they take us to conclusions whose meaning and truth are thereby obvious.

Frege thought that logic should be more algebraic, though. So, he did a lot of work on using symbols to describe logical axioms and rules of inference. And he added to Leibniz's definitions by defining 1 to be 0 + 1, and by defining 0 and + 1.

In the twentieth century, Russell was infamous for proving 1 + 1 = 2 with hundreds of pages of obscure symbols. Russell wanted to make logic algebraic and to show that arithmetical truths are logical truths, and Frege's efforts had ended in failure because of a logical paradox that Russell had discovered at the start of the twentieth century (following Cantor's discovery of a mathematical paradox at the end of the nineteenth century). But Russell soon tired of mathematics. And for a hundred years, most mathematicians have implicitly defined 0 and + 1 within an axiomatic set theory. Has that enabled them to prove 2 + 2 = 4 in a rigorously logical way?

To be precise, they can prove in a rigorously logical way that 2 + 2 = 4 where those symbols have set-theoretic meanings.

Many more steps would be needed for a proof of 2 + 2 = 4 where those symbols have their ordinary meanings, and those steps could not all be as rigorously logical because axiomatic sets are algebraic models of collections whose sizes are natural numbers like 2 and 4, and such models are bound, because of such paradoxes, to be at least a little inaccurate in some rather obscure ways. It is in a much more obvious way that the symbols 2, 4, + and = get their ordinary meanings from the meanings of words like one, two, three and so forth; and those words get their meanings in much the same way as the other basic words of English get theirs. Note that while the meanings of such words are described by dictionary definitions, they are not given to them by those definitions (indeed, they could not be, because in order to define a word, other words must be used).

At the end of the nineteenth century, Lewis Carroll used a fictional dialogue to show that if we needed rules of inference to tell us how to use rules of inference, then we would never get to prove anything logically. And it is fairly obvious that if we had to define the terminology of a proof in order for it to be a logical proof, then we would only be able to prove algebraic results logically. And of course, if the logic of a logical proof had to be axiomatic, then we would have had to have had logical axioms before we had any logical proofs. In reality, we hypothesize logical axioms when we have a lot of similar proofs, all of which are clearly logical, and then test those hypotheses as logically as we can. Logical axioms are bound to be less certain than our most logical proofs, so how could deductions from logical axioms and definitions of the algebraic (rather than the dictionary) kind be the most logical kind of proof? In reality, the fact that two things and another two things make four things can be seen by counting them:

There is (to begin with the first two things) a first thing and a second thing, and then there is a third thing (the first of the other two things) and a fourth thing (the second of the other two things).

That is a logical proof because it is obviously the case that counting a small number of things can tell us how many of those things there are, and it is a proof of 2 + 2 = 4 because the ordinary meaning of "2 + 2 = 4" is that two things and another two things make four things. It is because those things can be any logically possible things that 2 + 2 = 4 is a logical truth. Would that proof of 2 + 2 = 4 be more logical if it included an explanation of how counting works? Maybe, and such an explanation could always be added. But would we get a more logical proof of 2 + 2 = 4 if we modelled that proof very accurately? No, because even if that model was a very logical proof, it would be a very logical proof of a model of 2 + 2 = 4.

What's the matter?

What is physical reality? Most obviously, it is a lot of problems, the most important of which we have in common. Physical reality is problematic for us because we are physical and because we have feelings. So, it is pretty obvious that physical things can have feelings. Nevertheless, it might seem strange that physical things could have feelings:

Physical things appear to be composed of physical particles, and it is hard to see how any number of physical particles could possibly add up to something with feelings.

Does the difficulty of seeing how that is possible mean that we are not physical? Of course not. It would make as much sense to conclude that we do not have feelings. What about the theory that ordinary physical objects are composed of atoms, could we doubt that theory in light of that difficulty? Well, we could, but it would make more sense to doubt that we are just a lot of atoms. It would make more sense for us to doubt materialism.

Does the difficulty of seeing how some large number of atoms could possibly add up to something with feelings amount to a reason for doubting the logical possibility of materialism, though?

That might seem unlikely when you consider how little we know about what matter is.

Materialism does not seem to be contradictory, so materialism does seem to be a logical possibility.

Contradictions do describe logical impossibilities:

When we say that something is not a certain way, we mean that it is not the case that it is that way, so it cannot be the case that something is a certain way and that it is not that way, and so contradictions describe impossibilities.

And although contradictions can also describe possibilities (as when we say "it is and it isn't," meaning that in a sense it is, but in another sense it isn't), they describe them much less accurately than they describe impossibilities. So in short, a contradictory hypothesis is not a logical possibility. And for what other reason could we rule something out on logical grounds? Upon reflection, you should be able to see that it makes sense to say that something is a logical possibility when it is not contradictory.

So what?

It is precisely because we do not know much about what matter is that it is, for all we know, possible that this universe was deliberately created out of nothing but its logical possibility. Of course, such a creator would certainly seem unrealistic to atheists. So note that the possibility of such a creator is not the same as the possibility that the picture of a God that atheists pick up from the religions around them is a realistic picture. More importantly, so far as logic is concerned, the concept of such a creator does not seem to be contradictory.

Rational atheists should therefore have as little problem with the logical possibility of theism as they do with the logical possibility of materialism.

However, to remain atheists they must, in light of my proof that atheism is illogical, either show that the logic of that proof is flawed, or deny one of its premises. And insofar as logic is not algebraic, there is nothing wrong with the logic of that proof. And they could hardly deny the existence of things like these words and remain rational. So, they would have to deny the logical possibility of theism, if they want to remain rational atheists.