(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.

*The Way of Things*. The next section is

*Reality*.)

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