Definitions

Leibniz is famous for inventing the infinitesimal calculus (following Newton's invention of the method of fluxions), but his primary goal was always a logical calculus. Philosophical disagreements were relatively fierce in the seventeenth century, and Leibniz thought that a logical calculus would bring peace to philosophy, by enabling disputing philosophers to sit down and calculate the truth. Leibniz disagreed, for example, with Locke's view of the fundamental arithmetical concepts, unity (1) and addition (+ 1). Locke thought that we acquired those concepts by observing a world of things, but Leibniz thought that if we did not already have the concept of an individual thing then we would not be able to see those things as individual things. In short, Leibniz regarded 1 and + 1 as logical concepts. Indeed, he thought that arithmetic is nothing but logic and definitions. For example:

Starting with the logical axiom that things are identical to themselves, which he expressed in symbols as A = A, Leibniz put 2 + 2 in place of A, and in three logical steps, each step using one of three definitions (2 = 1 + 1, 3 = 2 + 1, and 4 = 3 + 1) and the logical rule that replacing equal things preserves equality, he got to an arithmetical truth:

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

With the logical steps of that proof of 2 + 2 = 4 so clearly laid out, it is easy to see why arithmetic might be nothing but logic and definitions (and why 1 and + 1 would not need to be defined in order for such a proof to be rigorously logical). 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.


Euler proved over a hundred theorems about numbers composed of ones added together, in the eighteenth century. And he conjectured that if we multiply such a number by itself five times, then in order to express that fifth power of that number as the sum of other counting numbers raised to the fifth power, we would need at least five of those other numbers. Although a counter-example, 144 (which only needs four other numbers), was discovered in the twentieth century. Now, presumably it goes without saying that in order for 144 to have been a counter-example to Euler's conjecture, that number had to have been a number of the kind that Euler was writing about. But what does it mean for a number to be composed of ones added together? Well, "zwey" was Euler's name for how many things (of a certain kind) he had when he had one thing (of that kind) and another thing (of that kind) and no other things (of that kind), and "2 = 1 + 1" is simply that description written in quotidian symbols. Note that while dictionary definitions describe the meanings of such basic words, they are not how such words get their meanings. Words like zwei and two get their meanings in much the same way as words like blau and blue get theirs (a way with which we are all familiar).

And the reason why 2 + 2 = 4 is that, firstly, 2 + 2 = (1 + 1) + (1 + 1) by definition of words like two, and secondly, (1 + 1) + (1 + 1) = 1 + 1 + 1 + 1 because how many things we have does not depend upon which ones we look at first, and thirdly, 1 + 1 + 1 + 1 = 4 by definition of words like four.

The English name for how many things there are when there is one thing (of a certain kind) and another thing (of that kind) and another thing (of that kind) and another thing (of that kind) and no other things (of that kind) is four, and "1 + 1 + 1 + 1 = 4" is that description written in symbols. Note that while symbols can get algebraic meanings from algebraic definitions and axioms, symbols like 2 and 4 get their quotidian meanings in much the same way as £ and & gets theirs (the familiar way in which words like "pound" and "and" get their meanings). Could that counter-example to Euler's conjecture have been a number of a different kind? Could it, for instance, have been composed of 144 fictional dots collectively called "144" by some twentieth-century mathematicians who had described infinitely many of those dots in order to build arithmetic out of them?


Frege completed a logical calculus in the nineteenth century; and in order to show that arithmetic is nothing but logic and definitions, he tried to use his logical calculus to build arithmetic out of logical axioms and definitions. He added to Leibniz's definitions by defining 1 to be 0 + 1, where 0 and + 1 were defined in terms of concepts that he took to be purely logical. And because associativity had been given an algebraic definition earlier in the century, he noticed that Leibniz, in his proof of 2 + 2 = 4, had not shown that 2 + (1 + 1) = (2 + 1) + 1. Now, it is obvious that 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 proof in order for it to be 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.

However, Frege thought that logical proofs should be more like calculations. Does the logic of a logical proof have to be axiomatic though, in order for it to be a rigorously logical proof? If it did, then we would have had to have had logical axioms before we had any rigorously logical proofs. In reality, logicians hypothesize logical axioms when they have lots of similar proofs that are clearly logical, and then they test those axioms as logically as they can. And because logical axioms are therefore bound to be less certain than a lot of logical proofs (and because logic is all about certainty), rigorously logical proofs do not have to be deductions from axioms and definitions. And Frege's attempt to show that arithmetic is nothing but logic and definitions failed because of a logical paradox discovered by Russell (following Cantor's discovery of a mathematical paradox at the end of the nineteenth century).


Russell designed his own logical calculus to build arithmetic out of, reaching 1 + 1 = 2 after hundreds of pages of algebra, where 1 and 2 got their meanings from some rather algebraic definitions and an axiom that said that there are infinitely many things. Russell wanted to show that arithmetic is nothing but logic and definitions, but how could he possibly have shown that arithmetic is nothing but logic and definitions by reaching such equations as that one after so much algebra? Now, philosophers tend to think that the question is whether his axiom was a logical axiom. However, the question is how he could possibly have shown us anything about arithmetic.

It is obvious that the English name for how many things there are when there is one thing (of a certain kind) and another thing (of that kind) and no other things (of that kind) is two. And there is an obvious way of writing that definition of two in symbols: 1 + 1 = 2.

That equation is a logical truth because those things can be any logically possible things, although that was not logical enough for Russell. How does it fail to be logical, though? If I had a crimson object, and I concluded that I had a red object, that would be logical; and if I also had a scarlet object, and I concluded that I had two red objects, would that not be similarly logical? And after all, what if I had no other red objects, and I defined 2 to be the number of red objects that I have; would I not have given that quotidian symbol another meaning (in addition to its quotidian meaning)?


Hilbert turned Frege's logical calculus into first-order predicate logic (a rather algebraic model of some simple logical reasoning), which became the standard logic in mathematics and philosophy. And for a hundred years, most mathematicians have defined 0 and + 1 within an axiomatic set theory, in order to avoid paradoxes like Russell's. With such definitions, mathematicians can prove in a rigorously logical way that 2 + 2 = 4 where "2" and "4" are names of axiomatic sets. Do "2" and "4" name axiomatic sets? Well, that is the standard meaning of 2 and 4 in rigorously logical mathematics. But consider how male (♂) and female (♀) are biological categories. Even if it became a standard part of the biology curriculum that those two categories are a matter of personal choice, they would not really be a matter of personal choice, because as a matter of fact, they are biological categories. The experts can of course discover new facts about their subjects, but how Orwellian would our society have become if our experts changed the meanings of such basic symbols like that? Axiomatic sets are algebraic, in the sense that they have only the properties described, in symbols, by the axioms of their set theory; axiomatic sets are essentially algebraic models of collections whose sizes are numbers composed of ones. And because such models are bound to be a little inaccurate in some rather obscure ways, because of paradoxes like Russell's, the steps that would be needed to take us from that rather algebraic proof of 2 + 2 = 4, where "2" and "4" are names of axiomatic sets, to a logical proof of 2 + 2 = 4 where those symbols have their quotidian meanings could not all be as rigorously logical as algebra. So, what would a rigorously logical proof of 2 + 2 = 4 look like?

Well, 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 insofar as the meaning of "2 + 2 = 4" is that two things and another two things make four things, which is the quotidian meaning of that equation. Would that proof be more rigorously logical if it included an explanation of how counting works? Maybe, and if it would then such an explanation could always be added. But would we get a more rigorously logical proof of 2 + 2 = 4 if we used first-order predicate logic and set theory to model that proof and then replaced it with that model? That model would be a rigorously logical proof: it would be a rather algebraic proof of an algebraic model of 2 + 2 = 4, and all mathematics is rigorously logical. But would it be a rigorously logical proof of 2 + 2 = 4?