Saturday, August 01, 2020


(This follows on from Puzzle.)

A few mathematicians demurred. Some thought that there were no infinite numbers, because there never are all of the counting numbers. Just as theoretical entities are created by theorists theorising, so these mathematicians thought of numbers as being created by mathematicians doing mathematics. But that view did little to undermine the idea of redefining the whole numbers. A view that would have undermined that idea was the view that all infinities lie far beyond human comprehension. But that view had become a lot less plausible following the creation of the calculus by the British mathematician Isaac Newton (1643–1727) and others.

There was, in short, a consensus for redefining the whole numbers, amongst early twentieth century mathematicians. And in order to make the new whole numbers cohere with the rest of mathematics, mathematicians decided to give everything in mathematics a new definition, in a new language (an axiomatic set theory). Mathematicians have since designed other languages, which are to some extent analogous. For example, when a computer is used to do various calculations, the calculations are first translated into the computer’s programming language. The machine can then do all the hard work, without any worries about human error. The computer code for, say, a problem in Newtonian mechanics is much more complicated than the original problem. But it facilitates a much faster and more reliably accurate solution.

A computer model of a mathematical model of a mechanical system is just a different mathematical model of that system, one that is just as effective but much more efficient. And in a similar way, the new definitions did not adversely affect applications of any mathematical model or method. They just allowed academic mathematicians to carry on working, effectively as normal (although actually within one new bit of mathematics), without having to worry about inconsistencies. As I shall shortly describe, the inconsistencies were just beyond the periphery of the new mathematics. The new whole numbers would produce consistent results.

Other mathematical properties and relations translated into the new language essentially unchanged, though. The new language was all about saving existing mathematics from the puzzling, but fortunately peripheral, inconsistencies. So, to translate mathematics into the new language, all that was needed was to put the new definitions in front of the first chapters of existing textbooks. And that did not even have to be done physically. The mere possibility of such a “chapter zero” was usually enough.

New textbooks tended to begin with an initial chapter on set theory, though. And eventually those changes trickled down to school mathematics, in the form of the new math. The new math was quite controversial, fifty years ago. But this book is not about the new math. Nor is it about the difficulty of proving, in the new mathematics, that the new language would definitely produce consistent results—a difficulty discovered by the Austrian mathematician Kurt Gödel (1906–1978). This book is not about the technicalities of academic mathematics. It is about solving the puzzle of the inconsistencies that changed the nature of academic mathematics.

My explanation of those inconsistencies is controversial, to say the least. But conventional thinking has been unable to explain them. Cantor’s discoveries have proved to be extremely puzzling. It is now a hundred years since the new language was designed, and mathematicians are still using those definitions to avoid inconsistencies. And just to call Cantor’s paradox mathematical is controversial nowadays, the word “mathematical” having changed its meaning, for the experts on mathematical matters, as a result of that paradox.

Mathematics has become a mathematical model of mathematics, and a deliberately incomplete model at that. But in itself, that is not too odd. More than two thousand years earlier, the Greek mathematician Euclid replaced geometry with a similarly axiomatic model of geometry, and that axiomatisation worked out well enough, not least because it turned out to be false. In the nineteenth century, it facilitated the abstract investigation of non-Euclidean geometries, and in 1919 spacetime was found to be non-Euclidean.

You need know nothing of Euclid—or axioms—to understand this book, but I do need to indicate the importance of Cantor’s discoveries. And readers are more likely to have heard of Euclid than Cantor. My explanation of the inconsistencies is so controversial, it might appear unwarranted if you did not know how important was their inexplicability within atheist worldviews. If you find my talk of axioms too opaque, though, simply skim-read or skip this section. It is completely incidental to the main argument of this book.

Originally, Euclidean space was presumed to be an accurate mathematical model of physical space. But the axioms of Euclidean geometry include the famous parallel postulate—Euclid’s fifth postulate—which turned out to be false of actual spacetime. Nevertheless, it remains a very useful model, because it is a very accurate model of our natural spatial intuitions. And in itself, Euclidean space is a mathematical object, defined by the Euclidean axioms. Euclidean shapes—such as the familiar circles and triangles of elementary geometry—are mathematical models of the shapes of circular and triangular things. But in themselves, they are defined by the Euclidean axioms. And it is in a similar way that the axioms of the new mathematical language—an axiomatic set theory—say what sets are, and hence, by way of the new definitions, what everything in modern mathematics is (including Euclidean space).

Axioms are just elementary descriptions of properties. Euclid’s first postulate, for example, says that between any two points, you could in principle draw a straight line. His second postulate says that you could extend that line beyond those points, to get an infinitely long straight line. The two axioms together mean that, in Euclidean space, any two points lie on a straight line. Mathematical axioms are usually written in mathematical symbols, which makes them look much more mysterious than they really are.

I will be avoiding the use of symbols as much as possible. But not even symbols are intrinsically mysterious, it is just that unfamiliar symbols are like hieroglyphs and graffiti. Unfamiliar symbols are strange and intimidating, but those for the counting numbers—1, 2, 3 and so forth—are so familiar, we tend to think of them as being the numbers themselves. In fact, the numbers themselves are merely referred to by those symbols, just as they are by the words one, two and three. You will have learnt the meanings of those words when you learnt English. And when you learnt to read and write, you will probably have learnt the symbols 1, 2 and 3. Those three symbols have exactly the same meanings as those three words. There is nothing mysterious about them.

For another example that will probably be familiar to you, the symbol = means is equal to. The two lines in that symbol are of equal length, which is why that symbol was chosen to represent mathematical equality. In an equation, the two quantities on either side of it are supposed to be the same (for example, 2 + 2 = 4). Other symbols are also indispensable in this book, but I will be explaining them as simply and comprehensively as I can as I get to them—as a rule. The following symbols, from set theory, are an exception to that rule. It is just that some of you may find it helpful to see some simple examples of the new definitions (if not, then just skim-read or skip the rest of this section).

To begin with, the whole number 0 is now defined to be the empty set, Ø, in a standard set theory. Sets contain mathematical objects much as bags contain things, or classes contain members. And empty sets are sets with nothing in them, so they are a bit like empty bags, in a story about bags. Fictional bags are howsoever their stories say they are—if a story describes a bag as bottomless, then it is bottomless, in that story—and similarly, axiomatic sets have whatever properties their axioms say they have. One of the axioms of the standard set theory asserts the existence of a set with no members, for example. And another says that different sets have different members. Those two axioms mean that there is one empty set in the standard set theory. And other axioms ensure that the empty set behaves just like the whole number zero, when it interacts with the sets that are defined to be those other numbers.

Repeatedly adding one generates the counting numbers. And nowadays, the process of adding one to a whole number is the putting of the set that is that number into the set that is that number. The whole number 1 is 0 + 1, and if you put Ø into the empty set then you get a set containing Ø. So, the whole number 1 is now defined to be {Ø}. It is like a bag that contains an empty bag and nothing else, in a story about bags.

Similarly, the whole number 2 is 1 + 1, and if you put {Ø} into {Ø} then you get {Ø, {Ø}}. And 3 is similarly defined to be {Ø, {Ø}, {Ø, {Ø}}}. Another way of looking at that “whole number” is that it is {0, 1, 2}, where 2 is {0, 1}, and 1 is {0}. Similarly, the new 4 is {0, 1, 2, 3}. Arithmetical facts about these “numbers” are deduced from the axioms of the standard set theory.

To prove that 2 + 2 = 4, for example, the set-theoretical adding of 1 is used in these equations: 2 + 2 = (2 + 1) + 1 = 3 + 1 = 4. That is a huge improvement on the earliest twentieth century proofs. To prove that 1 + 1 = 2, for example, took the British mathematician and philosopher Alfred North Whitehead (1861–1947) and Russell hundreds of pages of the first volume of their Principia Mathematica (published by Cambridge University Press in 1910). In English, 1 + 1 = 2 just says that one thing and another thing make, in total, two things, which is essentially the definition of “two” in English. Why did Whitehead and Russell not take 2 to be 1 + 1 by definition? Because such common sense had been leading mathematicians into inconsistency. Whitehead and Russell’s 2 exists (insofar as such things can be said to exist) within the axiomatic structure of their Principia. It is 2 no more than any model is the thing modelled.

Axiomatic sets are mathematical models of classes. But classes are a rather philosophical concept, best left until chapter 2. We can also think of sets as models of collections, which are very simple things. When referring to a lot of things, it may well be impractical or even impossible for us to refer to them all individually. There is therefore a need to refer to things collectively. A collection of things is simply all those things, being referred to collectively. There is no collection of one thing, there is just that thing. And there is no empty collection. A number of things, in the common sense, is just a collection of things.

Following Cantor’s discoveries, collections of mathematical objects were replaced with sets of redefined mathematical objects—that is, with sets of sets—with one exception. Because the redefinitions were designed to protect mathematics, mathematical properties and relations are essentially the same in the new mathematics as they always were. So, the assumption that the new whole numbers are collectively a set leads to an inconsistency. But this inconsistency is within set theory, so it simply shows that it would be false to assume that the new whole numbers are collectively a set.

By redefining whole numbers and collections within set theory, mathematicians created an assumption for this inconsistency to falsify. The totality of the new whole numbers is not a set. But there are all of them, in the theory, and so there is a collection of them all. Mathematicians call the totality of the set-theoretical whole numbers a proper class. The sets of the standard set theory never, in any of their mathematical uses, have too many things in them. They are like fictional bags that never, in any of their stories, have too many things in them.

The inconsistencies are just beyond the periphery of set theory, so they are just beyond the periphery of the new mathematics. However, the original inconsistencies, associated with ordinary whole numbers, still exist. Mathematicians have redefined mathematics, for academic purposes, but there is still mathematics in the common sense. It is just that it now includes set theory. Analogously, writing a novel adds some books to reality. It does not make reality go away, for all that we can enter the world of the novel by reading it.

(This is the second section of the first chapter of The Way of Things. The next section is Reality.)

No comments: