"I protest against the use of infinite magnitude as something completed, which
is never permissible in mathematics. Infinity is merely a way of speaking, the
true meaning being a limit which certain ratios approach indefinitely close,
while others are permitted to increase without restriction." (Gauss)
"I don't know what predominates in Cantor's theory - philosophy or theology,
but I am sure that there is no mathematics there." (Kronecker)
"...classical logic was abstracted from the mathematics of finite sets and
their subsets...Forgetful of this limited origin, one afterwards mistook that
logic for something above and prior to all mathematics, and finally applied
it, without justification, to the mathematics of infinite sets. This is the
Fall and original sin of [Cantor's] set theory ..." (Weyl)
Modern mathematics as religion
Modern mathematics doesn't make complete sense.
The unfortunate consequences include difficulty in deciding
what to teach and how to teach it, many papers that are logically
flawed, the challenge of recruiting young people to the subject,
and an unfortunate teetering on the brink of irrelevance.
If mathematics made complete sense it would
be a lot easier to teach, and a lot easier to learn. Using
flawed and ambiguous concepts, hiding confusions and circular
reasoning, pulling theorems out of thin air to be justified
`later' (i.e. never) and relying on appeals to authority don't
help young people, they make things more difficult for them.
If mathematics made complete sense there would
be higher standards of rigour, with fewer but better books
and papers published. That might make it easier for ordinary
researchers to be confident of a small but meaningful contribution.
If mathematics made complete sense then the physicists wouldn't
have to thrash around quite so wildly for the right mathematical
theories for quantum field theory and string theory. Mathematics
that makes complete sense tends to parallel the real world
and be highly relevant to it, while mathematics that doesn't
make complete sense rarely ever hits the nail right on the
head, although it can still be very useful.
So where exactly are the logical problems?
The troubles stem from the consistent refusal
by the Academy to get serious about the foundational aspects
of the subject, and are augmented by the twentieth
centuries' whole hearted and largely uncritical
embrace of Set Theory.
Most of the problems with the foundational
aspects arise from mathematicians' erroneous belief that they
properly understand the content of public school and high
school mathematics, and that further clarification and codification
is largely unnecessary. Most (but not all) of the difficulties
of Set Theory arise from the insistence that there exist `infinite
sets', and that it is the job of mathematics to study them
and use them.
In perpetuating these notions, modern mathematics
takes on many of the aspects of a religion.
It has its essential creed---namely Set Theory, and its unquestioned
assumptions, namely that mathematics is based on `Axioms',
in particular the Zermelo-Fraenkel `Axioms of Set Theory'.
It has its anointed priesthood, the logicians,
who specialize in studying the foundations
of mathematics, a supposedly deep and difficult subject
that requires years of devotion to master. Other mathematicians
learn to invoke the official mantras when questioned by outsiders,
but have only a hazy view about how the elementary aspects
of the subject hang together logically.
Training of the young is like that in secret
societies---immersion in the cult involves intensive undergraduate
memorization of the standard thoughts before they are properly
understood, so that comprehension often follows belief instead
of the other (more healthy) way around. A long and often painful
graduate school apprenticeship keeps the cadet busy jumping
through the many required hoops, discourages critical thought
about the foundations of the subject, but then gradually yields
to the gentle acceptance and support of the brotherhood. The
ever-present demons of inadequacy, failure and banishment
are however never far from view, ensuring that most stay on
the well-trodden path.
The large international conferences let the
fellowship gather together and congratulate themselves on
the uniformity and sanity of their world view, though to the
rare outsider that sneaks into such events the proceedings
no doubt seem characterized by jargon, mutual incomprehensibility
and irrelevance to the outside world. The official doctrine
is that all views and opinions are valued if they contain
truth, and that ultimately only elegance and utility decide
what gets studied. The reality is less ennobling---the usual
hierarchical structures reward allegiance, conformity and
technical mastery of the doctrines, elevate the interests
of the powerful, and discourage dissent.
There is no evil intent or ugly conspiracy
here---the practice is held in place by a mixture of well-meaning
effort, inertia and self-interest. We humans have a fondness
for believing what those around us do, and a willingness to
mold our intellectual constructs to support those hypotheses
which justify our habits and make us feel good.
The problem with foundations
The reason that mathematics doesn't make complete
sense is quite easy to explain when we look at it from the
educational side. Mathematicians, like everyone else, begin
learning mathematics before kindergarten, with counting and
basic shapes. Throughout the public and high school years
(K-12) they are exposed to a mishmash of subjects and approaches,
which in the better schools or with the better teachers involves
learning about numbers, fractions, arithmetic, points, lines,
triangles, circles, decimals, percentages, congruences, sets,
functions, algebra, polynomials, parabolas, ellipses, hyperbolas,
trigonometry, rates of change, probabilities, logarithms,
exponentials, quadrilaterals, areas, volumes, vectors and
perhaps some calculus. The treatment is non-rigorous, inconsistent
and even sloppy. The aim is to get the average student through
the material with a few procedures under their belts, not
to provide a proper logical framework for those who might
have an interest in a scientific or mathematical career.
In the first year of university the student
encounters calculus more seriously and some linear algebra,
perhaps with some discrete mathematics thrown in. Sometime
in their second or third year, a dramatic change happens in
the training of aspiring pure mathematicians. They start being
introduced to the idea of rigorous thinking
and proofs, and gradually become
aware that they are not at the peak of intellectual achievement,
but just at the foothills of a very onerous climb. Group theory,
differential equations, fields, rings, topological spaces,
measure theory, operators, complex analysis, special functions,
manifolds, Hilbert spaces, posets and lattices---it all piles
up quickly. They learn to think about mathematics less as
a jumble of facts to be memorized and algorithms to be mastered,
but as a coherent logical structure. Assignment problems increasingly
require serious thinking, and soon all but the very best are
brain-tired and confused.
Do you suppose the curriculum at this point
has time or inclination to return to the material they learnt
in public school and high school, and finally organize it
properly? When we start to get really picky about logical
correctness, doesn't it make sense to go back and ensure that
all those subjects that up to now have only been taught in
a loose and cavalier fashion get a proper rigorous treatment?
Isn't this the appropriate time to finally learn what a number
in fact is, why exactly the laws of
arithmetic hold, what the correct
definitions of a line and a circle are, what we mean
by a vector, a function,
an area and all the rest? You might
think so, but there are two very good reasons why this is
nowhere done.
The first reason is that even the professors
mostly don't know! They too have gone through a similar indoctrination,
and never had to prove that multiplication
is associative, for example, or learnt what is the right order
of topics in trigonometry. Of course they know how to solve
all the problems in elementary school texts, but this is quite
different from being able to correct all the logical defects
contained there, and give a complete and proper exposition
of the material.
The modern mathematician walks around with
her head full of the tight logical relationships of the specialized
theories she researches, with only a rudimentary understanding
of the logical foundations underpinning the entire subject.
But the worst part is, she is largely unaware of this inadequacy
in her training. She and her colleagues really
do believe they profoundly understand elementary mathematics.
But a few well-chosen questions reveal that this is not so.
Ask them just what a fraction is,
or how to properly define an angle,
or whether a polynomial is really
a function or not, and see what kind
of non-uniform rambling emerges! The more elementary the question,
the more likely the answer involves a lot of philosophizing
and bluster. The issue of the correct approach to the definition
of a fraction is a particularly crucial one to public school
education.
Mathematicians like to reassure themselves
that foundational questions are resolved by some mumbo-jumbo
about `Axioms' (more on that later) but in reality successful
mathematics requires familiarity with a large collection of
`elementary' concepts and underlying linguistic and notational
conventions. These are often unwritten, but are part of the
training of young people in the subject. For example, an entire
essay could be written on the use, implicit and explicit,
of ordering and brackets
in mathematical statements and equations. Codifying this kind
of implicit syntax is a job professional
mathematicians are not particularly interested in.
The second reason is that any attempt to lay
out elementary mathematics properly would be resisted by both
students and educators as not going forward, but backwards.
Who wants to spend time worrying about the correct approach
to polynomials when Measure theory
and the Residue calculus beckon instead?
The consequence is that a large amount of elementary mathematics
is never properly taught anywhere.
But there are two foundational topics that
are introduced in the early undergraduate years: infinite
set theory and real numbers. Historically
these are very controversial topics, fraught with logical
difficulties which embroiled mathematicians for decades. The
presentation these days is matter of fact---`an infinite set
is a collection of mathematical objects which isn't finite'
and `a real number is an equivalence class of Cauchy sequences
of rational numbers'.
Or some such nonsense. Set theory as presented
to young people simplydoesn't
make sense, and the resultant approach to real numbers
is in fact a joke! You heard it correctly---and
I will try to explain shortly. The point here is that these
logically dubious topics are slipped into the curriculum in
an off-hand way when students are already overworked and awed
by all the other material before them. There is not the time
to ruminate and discuss the uncertainties of generations gone
by. With a slick enough presentation, the whole thing goes
down just like any other of the subjects they are struggling
to learn. From then on till their retirement years, mathematicians
have a busy schedule ahead of them, ensuring that few get
around to critically examining the subject matter of their
student days.
Infinite sets
I think we can agree that (finite) set theory
is understandable. There are many examples of (finite) sets,
we know how to manipulate them effectively, and the theory
is useful and powerful (although not as useful and powerful
as it should be, but that's a different story).
So what about an `infinite
set'? Well, to begin with, you should say precisely
what the term means. Okay, if you
don't, at least someone should. Putting
an adjective in front of a noun does not in itself make a
mathematical concept. Cantor declared that an `infinite
set' is a set which is not finite.
Surely that is unsatisfactory, as Cantor no doubt suspected
himself. It's like declaring that an `all-seeing
Leprechaun' is a Leprechaun which can see everything.
Or an `unstoppable mouse' is a mouse
which cannot be stopped. These grammatical constructions do
not create concepts, except perhaps in a literary or poetic
sense. It is not clear that there are
any sets that are not finite, just as it is not clear that
there are any Leprechauns which can
see everything, or that there are
mice that cannot be stopped. Certainly in science there is
no reason to suppose that `infinite sets' exist. Are there
an infinite number of quarks or electrons in the universe?
If physicists had to hazard a guess, I am confident the majority
would say: No. But even if there were
an infinite number of electrons, it is unreasonable to suppose
that you can get an infinite number of them all together as
a single `data object'.
The dubious nature of Cantor's definition was
spectacularly demonstrated by the contradictions in `infinite
set theory' discovered by Russell and others around the turn
of the twentieth century. Allowing any old `infinite set'
à la Cantor allows you to consider the `infinite set'
of `all infinite sets', and this leads to a self-referential
contradiction. How about the `infinite sets' of `all finite
sets', or `all finite groups', or perhaps `all topological
spaces which are homeomorphic to the sphere'? The paradoxes
showed that unless you are very particular about the exact
meaning of the concept of `infinite set', the theory collapses.
Russell and Whitehead spent decades trying to formulate a
clear and sufficiently comprehensive framework for the subject.
Let me remind you that mathematical theories
are not in the habit of collapsing. We do not routinely say,
"Did you hear that Pseudo-convex cohomology theory collapsed
last week? What a shame! Such nice people too."
So did analysts retreat from Cantor's theory
in embarrassment? Only for a few years, till Hilbert rallied
the troops with his battle-cry "No one shall expel us from
the paradise Cantor has created for us!" To which Wittgenstein
responded "If one person can see it as a paradise for mathematicians,
why should not another see it as a joke?"
Do modern texts on set theory bend over backwards
to say precisely what is and what is not
an infinite set? Check it out for yourself---I cannot
say that I have found much evidence of such an attitude, and
I have looked. Do those students learning `infinite set theory'
for the first time wade through The Principia?
Of course not, that would be too much work for them and their
teachers, and would dull that pleasant sense of superiority
they feel from having finally `understood the infinite'.
The bulwark against such criticisms, we are
told, is having the appropriate collection of `Axioms'! It
turns out, completely against the insights and deepest intuitions
of the greatest mathematicians over thousands of years, that
it all comes down to what you believe.
Fortunately what we as good modern mathematicians believe
has now been encoded and deeply entrenched in the `Axioms
of Zermelo--Fraenkel'. Although there was quite a bit
of squabbling about this in the early decades of the last
century, nowadays there are only a few skeptics. We mostly
attend the same church, dutifully repeat the same incantations,
and insure our students do the same.
Let us have a look at these `Axioms', these
bastions of modern mathematics. In what follows,
and
are `sets'.
1. Axiom of Extensionality: If
and
have the same elements, then .
2. Axiom of the Unordered Pair: For
any
and
there exists a set
that contains exactly
and .
3. Axiom of Subsets: If
is a property (with parameter ),
then for any
and
there exists a set
that contains all those
in
that have the property .
4. Axiom of Union: For any
there exists a set ,
the union of all elements of .
5. Axiom of the Power Set: For any
there exists a set ,
the set of all subsets of .
6. Axiom of Infinity: There exists
an infinite set.
7. Axiom of Replacement: If
is a function, then for any
there exists a set .
8. Axiom of Foundation: Every nonempty
set has a minimal element, that is one which does not contain
another in the set.
9. Axiom of Choice: Every family of
nonempty sets has a choice function, namely a function which
assigns to each of the sets one of its elements.
All completely clear? This sorry
list of assertions is, according to the majority of
mathematicians, the proper foundation for set theory and modern
mathematics! Incredible!
The `Axioms' are first of all unintelligible
unless you are already a trained mathematician. Perhaps you
disagree? Then I suggest an experiment---inflict this list
on a random sample of educated non-mathematicians and see
if they buy---or even understand---any of it. However even
to a mathematician it should be obvious that these statements
are awash with difficulties. What is a property?
What is a parameter? What is a function?
What is a family of sets? Where is
the explanation of what all the symbols mean, if indeed they
have any meaning? How many further assumptions are hidden
behind the syntax and logical conventions assumed by these
postulates?
And Axiom 6: There is an infinite
set!? How in heavens did this one sneak in here? One
of the whole points of Russell's critique is that one must
be extremely careful about what the words `infinite set' denote.
One might as well declare that: There is
an all-seeing Leprechaun! or There
is an unstoppable mouse!
Just to get you thinking about whether in fact
you understand the
` Axioms',consider
the set As
we do. Please stop reading for a moment, and just consider
this set.
Thanks for considering it. Ah, but someone
has a question! Yes? You would like to know what
is? Very well, I will tell you. I am not sure if I am legally
obligated to (am I?), but I will tell you anyway---
is itself a set, also a very simple one, with just two elements,
called
and
ThusCan
we move on now? Wait, someone insists on knowing: what are
and
They are also sets, also with two elements each, so thatAnd,
before you ask, each of the elements ,
,
and
is itself a set, with also exactly two elements. Does the
pattern continue? Suppose it does,
would that make
legitimate? But suppose it doesn't,
and that I refuse to reveal a pattern, perhaps because non
exists. In modern mathematics we are allowed to consider patterns
that do not have any pattern to them. In such a case does
still exist? Does it exist if I invoke some appropriate new
`Axiom'?
The Zermelo-Fraenkel `Axioms' are but the merry
beginning of a zoo of possible starting points for mathematics,
according to modern practitioners. The `Axiom of Choice' has
numerous variants. There is the `Axiom of Countable Choice'.
`The Axiom of Dependent Choice'. There is the `Axiom that
all subsets of
are Lebesgue measurable' (which contradicts the `Axiom of
Choice'). Not to mention all the higher possible axioms concerned
with large cardinals. You can mix and match as you please.
I have been a working mathematician for more
than twenty years, and none of this resembles in any way,
shape, or form the subject as I have come to experience it.
In my studies of Lie theory, hypergroups and geometry, there
has never been a point at which I have pondered---should I
assume this postulate about the mathematical
world, or that postulate? Of course
one makes decisions all the time about which definitions
to focus on, but the nature of the mathematical world that
I investigate appears to me to be absolutely fixed. Either
has an eleven dimensional irreducible representation or it
doesn't (in fact it doesn't). My religious/ philosophical/Axiomatic
position has nothing to do with it. So I am confident that
a view of mathematics as swimming ambiguously on a sea of
potential Axiomatic systems strongly misrepresents the practical
reality of the subject.
Does mathematics require axioms?
Occasionally logicians inquire as to whether the current `Axioms'
need to be changed further, or augmented. The more fundamental
question---whether mathematics requires any
Axioms ---is not up for discussion. That would be like
trying to get the high priests on the island of Okineyab to
consider not whether the Divine Ompah's Holy Phoenix has twelve
or thirteen colours in her tail (a fascinating question on
which entire tomes have been written), but rather whether
the Divine Ompah exists at all. Ask that
question, and icy stares are what you have to expect, then
it's off to the dungeons, mate, for a bit of retraining.
Mathematics does not require `Axioms'. The
job of a pure mathematician is not to build some elaborate
castle in the sky, and to proclaim that it stands up on the
strength of some arbitrarily chosen assumptions. The job is
to investigate the mathematical reality of
the world in which we live. For this, no assumptions
are necessary. Careful observation is necessary, clear definitions
are necessary, and correct use of language and logic are necessary.
But at no point does one need to start invoking the existence
of objects or procedures that we cannot see, specify, or implement.
The difficulty with the current reliance on
`Axioms' arises from a grammatical confusion, along with the
perceived need to have some (any) way to continue certain
ambiguous practices that analysts historically have liked
to make. People use the term `Axiom' when often they really
mean definition. Thus the `axioms'
of group theory are in fact just definitions. We say exactly
what we mean by a group, that's all. There are no assumptions
anywhere. At no point do we or should we say, `Now that we
have defined an abstract group, let's assume they exist'.
Either we can demonstrate they exist by constructing some,
or the theory becomes vacuous. Similarly there is no need
for `Axioms of Field Theory', or `Axioms of Set theory', or
`Axioms' for any other branch of mathematics---or for mathematics
itself!
Euclid may have called certain of his initial
statements Axioms, but he had something else in mind. Euclid
had a lot of geometrical facts which he wanted to organize
as best as he could into a logical framework. Many decisions
had to be made as to a convenient order of presentation. He
rightfully decided that simpler and more basic facts should
appear before complicated and difficult ones. So he contrived
to organize things in a linear way, with most Propositions
following from previous ones by logical reasoning alone, with
the exception of certain initial statements
that were taken to be self-evident. To Euclid, an
Axiom was a fact that was sufficiently obvious to not require
a proof. This is a quite different meaning to the use
of the term today. Those formalists who claim that they are
following in Euclid's illustrious footsteps by casting mathematics
as a game played with symbols which are not given meaning
are misrepresenting the situation.
We have politely swallowed the standard gobble
dee gook of modern set theory from our student days---around
the same time that we agreed that there most certainly are
a whole host of `uncomputable real numbers', even if you or
I will never get to meet one, and yes, there no doubt is
a non-measurable function, despite the fact that no one can
tell us what it is, and yes, there surely are
non-separable Hilbert spaces, only we can't specify
them all that well, and it surely is
possible to dissect a solid unit ball into five pieces, and
rearrange them to form a solid ball of radius two.
And yes, all right, the Continuum hypothesis
doesn't really need to be true or false, but is allowed to
hover in some no-man's land, falling one way or the other
depending on what you believe. Cohen's
proof of the independence of the Continuum hypothesis from
the `Axioms' should have been the long overdue wake-up call.
In ordinary mathematics, statements are either true, false,
or they don't make sense. If you have an elaborate theory
of `hierarchies upon hierarchies of infinite sets', in which
you cannot even in principle decide
whether there is anything between the first and second `infinity'
on your list, then it's time to admit that
you are no longer doing mathematics.
Whenever discussions about the foundations
of mathematics arise, we pay lip service to the `Axioms' of
Zermelo-Fraenkel, but do we every use them? Hardly ever. With
the notable exception of the `Axiom of Choice', I bet that
fewer than 5% of mathematicians have ever employed even one
of these `Axioms' explicitly in their published work. The
average mathematician probably can't even remember the `Axioms'.
I think I am typical---in two weeks time I'll have retired
them to their usual spot in some distant ballpark of my memory,
mostly beyond recall.
In practise, working mathematicians are quite
aware of the lurking contradictions with `infinite set theory'.
We have learnt to keep the demons at bay, not by relying on
`Axioms' but rather by developing conventions and intuition
that allow us to seemingly avoid the most obvious traps. Whenever
it smells like there may be an `infinite set' around that
is problematic, we quickly use the term `class'. For example:
A topology is an `equivalence class of atlases'. Of course
most of us could not spell out exactly what does and what
does not constitute a `class', and we learn to not bring up
such questions in company.
There is also the useful term `category'. Consider
the `category of all finite groups'. Given any set
whatsoever, I can create a one element set
whose single element is
Then I can define
to be a group, by defining
Thus for every set
there is a group with one element which determines .
So if you believe that the `set of all sets' doesn't make
good sense, then how can the `category of all finite groups'
be any better? Do category theorists begin their lectures
to the rest of us with a quick primer as to what the term
`category' might precisely mean? Does the audience get nervous
not knowing? Back in the good old nineteenth century they
probably did, but nowadays those who attend research seminars
regularly are quite used to taking for granted abstractions
that they feel incapable of understanding.
Another good example arises from the usual
definition of a function. Although
the official doctrine is that a function is prescribed by
a domain (a set) and a codomain (a set) as well as a rule
that tells us what do with an element of the former to get
an element of the latter, we know that in practice the domain
and codomain can be dispensed with in shady circumstances,
or the term can be replaced by the somewhat more flexible
`functor', particularly in category theory. To illustrate---when
we define the fundamental group
of a topological space ,
we instinctively know that it is better not to writebecause
chances are Top and Group are not `properly defined infinite
sets'. We just employ the everyday understanding of a function,
namely that it suffices to say what kind
of an object it inputs, what kind of
an object it outputs, and what it does
precisely to an input to get an output. No need to have all
the possible inputs and outputs arranged in front of us neatly
as two sets. This kind of understanding can be usefully extended
to many more mundane situations. Do you really think you need
to have all the natural numbers together in a set to define
the function
on natural numbers? Of course not---the rule
itself, together with the specification of the kinds
of objects it inputs and outputs is enough. As computer
scientists already know.
Why real numbers are a joke
According to the status quo, the continuum
is properly modelled by the `real numbers'. What is a real
number? Let's start with an easier question: What is a rational
number? Here comes set theory to our aid. It is, according
to some accounts, nothing but an equivalence
class of ordered pairs of integers. Thus when my six
year old daughter uses the fraction
what she is really doing is using
the `equivalence class'
Good
grief. But let's carry on. A Cauchy sequence
of rational numbers is a sequencewhere
each of the
is a rational number (of the kind just mentioned) with the
property that for all
there exists a natural number
such that if
and
are bigger than
thenBut
here is a very important point: we
are not obliged, in modern mathematics,
to actually have a rule or algorithm that
specifies the sequence
In other words, `arbitrary' sequences are allowed, as long
as they have the Cauchy convergence property. This removes
the obligation to specify concretely the objects which you
are talking about. Sequences generated by algorithms can be
specified by those algorithms, but what possibly could it
mean to discuss a `sequence' which is not generated by such
a finite rule? Such an object would contain an `infinite amount'
of information, and there are no concrete examples of such
things in the known universe. This is metaphysics masquerading
as mathematics.
To get you used to the modern magic of Cauchy
sequences, here is one I just made up:Anyone
want to guess what the limit is? Oh, you want some more information
first? The initial billion terms are all
Now would you like to guess? No, you want more information.
All right, the billion and first term is
Now would you like to guess? No, you want more information.
Fine, the next trillion terms are all
You are getting tired of asking for more information, so you
want me to tell you the pattern once and for all? Ha Ha! Modern
mathematics doesn't require it! There doesn't need
to be a pattern, and in this case, there isn't, because I
say so. You are getting tired of this game, so you guess
Good effort, but sadly you are wrong. The actual answer is
That's right, after the first trillion and billion and one
terms, the entries start doing really
wild and crazy things, which I don't need to describe to you,
and then `eventually' they start heading towards
but how they do so and at what rate is not known by anyone.
Isn't modern religion fun?
So now what is a real number? It is an equivalence
class of Cauchy sequences! That's right, not just one, not
just two, but an entire equivalence class of them. We can't
even list the elements of such a `class', since each and every
one of them contains an `uncountable' number of Cauchy sequences.
So of course we have already absorbed the `infinite set theory'
to make sense of these statements, and we still ought to `explain'
the equivalence relation. Let's forego that, and just present
a representative example. Here is a real number, where I have
saved considerable space by not presenting rational numbers
in their full glory: Like
to guess what real number this is? You're right! It is
However did you know?
Now that you are comfortable with the definition
of real numbers, perhaps you would like to know how to do
arithmetic with them? How to add them, and multiply them?
And perhaps you might want to check that once you have defined
these operations, they obey the properties you would like,
such as associativity etc. Well, all I can say is---good luck.
If you write this all down coherently, you will certainly
be the first to have done so. On top of the manifold ugliness
and complexity of the situation, you will be continually dogged
by the difficulty that in all these sequences there does not
have to be a pattern---they are allowed to be completely
`arbitrary'. That means you are unable to say when two given
real numbers are the same, or when a particular arithmetical
statement involving real numbers is correct. Even a simple
statement like
will cause you consternation, since you have to phrase everything
in terms of unending Cauchy sequences, and in the absence
of solid conventions for specifying infinite sequences, you
will wrestle with the question of whether the Cauchy sequence
really does represent
or perhaps just appears to from this end of things.
Perhaps you would like to consult the usual
`Constructing the Real Numbers' section in your favourite
calculus text instead. Have a look, and see what passes for
logical thinking in modern mathematics education. Then to
really sink your spirits, open up a `rigorous' analysis text,
and thumb through to the critical section where they explain
the continuum ---exactly what a real
number is and how one operates with them. This is the
heart of the matter---the bedrock on which modern analysis
is built. And in all such books, waffling and ambiguity is
what you find, unless the subject is passed over altogether.
Some of them are honest about it. Others cleverly confuse
the issue by allowing talk about `sets' of rational numbers
without any mention of how you actually specify
them. It is in this gap that the logical difficulties lurk.
A set of rational numbers is essentially a sequence of zeros
and ones, and such a sequence is specified properly when you
have a finite function or computer program which generates
it. Otherwise `it' is not accessible in a finite universe.
This critical issue of describing the points
on the continuum should have a strong
connection with notions of computability,
but it turns out, according to the standard dogma, that computable
real numbers are just a `measure zero slice' of `all real
numbers'. Despite the fact that neither you nor anyone else
has been able to write down a single `non-computable real
number' and the undeniable fact that they never play the slightest
role in any actual scientific, engineering or applied mathematical
calculation.
Even the `computable real numbers' are quite
misunderstood. Most mathematicians reading this paper suffer
from the impression that the `computable real numbers' are
countable, and that they are not
complete. As I mention in my recent book, this is quite
wrong. Think clearly about the subject for a few days, and
you will see that the computable real numbers are not
countable , and are complete.
Think for a few more days, and you will be able to see how
to make these statements without any reference to `infinite
sets', and that this suffices for Cantor's proof that not
all irrational numbers are algebraic.
When it comes to foundational issues, modern
analysis is off in la-la land.
But what about the natural
numbers?
Okay, you say, perhaps you have a bit of a
point here, but surely you are going too far in denouncing
infinite sets altogether. After all, there is one infinite
set that we can be absolutely sure of,
one that is so familiar, so cut and dried, it is beyond reproach.
What about---the set of all natural numbers you ask?? Have
a look, here it is in its glorious entirety:Well,
perhaps not in its entirety, but we all know what those three
little dots represent, don't we? All the rest of those numbers,
squeezed in between the
and the right bracket!
The ancient Greeks believed that the natural
numbers are not finite, but that didn't mean they agreed that
you could put them all together to form a
well-defined mathematical object. A finite set we can
describe explicitly and specify completely---we can list all
its elements so there is no possible ambiguity. But the question
is---are we allowed to state that all
of the natural numbers are collectible into one big set?
Some will argue that a mathematician can do
whatever she likes, as long as a logical contradiction doesn't
result. But things are not so simple. Are we allowed to introduce
all-seeing Leprechauns into mathematics
as long as they seem to behave themselves and not cause contradictions?
A far better approach to create beautiful and useful mathematics
is to ensure that all basic concepts are entirelyclear and straightforward right from the
start. The onus is on us to demonstrate that our notions
make sense, instead of challenging someone else to find a
contradiction.
We'll now see that the concept of the `set
of natural numbers' is neither clear nor straightforward,
but immersed in complexity. The difficulties start when we
leave the familiar and comfortable domain of microscopic natural
numbers, and start pushing on through the sequence in an effort
to write down bigger and bigger numbers. Pretty soon expressing
numbers in decimal form like gets
uneconomical, and it is easier to use exponential notation.
Iteration allows us to write a tower of three tens: Let's
keep on going, and write down the number where
the tower of exponents on the left has altogether
tens. My guess is that
is already bigger than any number ever used (meaningfully)
in mathematics or science, but I could be wrong. In any case,
it's still early days in our exploration of
as we've only been at it for five minutes. How about where
the number of brackets is
Please think about this number for a few minutes. This should
not be too much of a burden on you, since you routinely bandy
the set
of all natural numbers about.
Next we could introduce
then
then some suitable iterate of iterates, say
then
then eventually
and so on, and so on, constrained only by the limits of our
imaginations, and the amount of writing paper at our disposal.
Assuming our imaginations are not a problem, there is the
issue of space, for as we keep going and keep going, we are
going to start running out of memory space to write down our
increasingly large numbers. First they will fill a page, then
a book, then our hard drives. Of course we can make our computers
bigger and our coding more efficient, start dismantling stars
and spreading our memory banks across galaxies. But... the
universe is almost certainly finite. Eventually, you
and I may have vaporized and rearranged all the stars, furniture
and other creatures in our quest to write down yet bigger
numbers, and now we are starting to run out of particles with
which to extend our galactic hard drive. Suppose you reduce
me to atoms in the interests of science, and perhaps your
outer extremities too. At some point, you are going to write
down a number so vast that it requires all the particles of
the universe (except for some minimal amount of what's left
of you). May I humbly suggest you call this number
in honour of the last person you vaporized to create it?
Now here is a dilemma. Once you have written
down and marvelled at
in all its glory, where are you going to find From
this end of things---the working end---the endless sequence
of natural numbers does not appear either natural nor endless.
And where is the infinite set
The answer is---nowhere. It doesn't exist.
It is a convenient metaphysical fiction that allows mathematicians
to be sloppy in formulating various questions and arguments.
It allows us to avoid issues of specification and replace
concrete understandings with woolly abstractions. What seems
to be a happy and well behaved sequence when viewed from the
beginning is more like an enormous fractal when viewed from
the other end.
Unlike
the numbers
and
are dramatic anomalies in the zoo of natural numbers, because
they can be written down using so little space. Their complexity,
or informational content, is much smaller than they are themselves.
Most numbers are not like this at all. To emphasize this point,
let's make a crude calculation to bound the number of possible
numbers we could write down by treating the entire universe
as an enormous hard-drive, packed row upon row with elementary
particles to encode some gigantic number. Suppose that in
one dimension the universe is at most
metres wide, that there are perhaps
dimensions (to make room for future versions of string theory),
that the smallest possible particle size is
metres, and that there are say
different particles that we could place at any one point in
the universe. So the number of possible configurations of
particles filling up all of the universe is at mostAlthough
this is a respectable number, it pales to insignificance when
compared to .
Conclusion: The vast majority of numbers less than
cannot be written down in our universe. These numbers are
completely inaccessible to us, and always
will be. But
can be written down in one line. Numbers `close' to
in the sense of having expressions that are not all that different
from that of
form little `islands of simplicity' in a sea of overwhelming
complexity.
It follows that long before you get to
you are going to reach numbers whose prime
factorizations are impossible, since some of the factors,
if they existed, would require more room to write down than
For example
is almost surely such a number---I claim it has
no prime factorization. Neither you nor I nor anyone
ever living in this universe will ever be able to factor this
number, since most of its `prime factors' are almost surely
so huge as to be inexpressible, which means they don't
exist.
Perhaps you believe that even though you cannot
write down numbers bigger than
you can still abstractly contemplate them!
This is a metaphysical claim. What does a
number bigger than
mean, if there is nothing that it counts, and it can't even
be written down? Believing you can `visualize' an all-seeing
Leprechaun or an unstoppable mouse in your mind, by some melange
of images, descriptive phrases and vague feelings, does not
mean they exist. By all means, write plays and poems about
all those numbers beyond ,
but don't imagine you are doing mathematics. Twentieth century
physicists have learnt to disregard `concepts' which are not
measurable or observable in some form or another, and we mathematicians
ought to be equally skeptical.
Elementary mathematics needs to be understood
in the right way, and the entire subject
needs to be rebuilt so that it makes complete sense right
from the beginning, without any use of dubious philosophical
assumptions about infinite sets or procedures. Show me one
fact about the real world (i.e. applied maths, physics,
chemistry, biology, economics etc.) that trulyrequires mathematics involving `infinite
sets'! Mathematics was always really about, and always will
be about, finite collections, patterns and algorithms. All
those theories, arguments and daydreams involving `infinite
sets' need to be recast into a precise finite framework or
relegated to philosophy. Sure it's more work, just as developing
Schwartz's theory of distributions is more work than talking
about the delta function as `a function with total integral
one that is zero everywhere except at one point where it is
infinite'. But Schwartz's clarification inevitably led to
important new applications and insights.
If such an approach had been taken in the twentieth
century, then (at the very least) two important consequences
would have ensued. First of all, mathematicians would by now
have arrived at a reasonable consensus of how to formulate
elementary and high school mathematics in
the right way. The benefits to mathematics education
would have been profound. We would have strong positions and
reasoned arguments from which to encourage educators to adopt
certain approaches and avoid others, and students would have
a much more sensible, uniform and digestible subject.
The second benefit would have been that our
ties to computer science would be much stronger than they
currently are. If we are ever going to get serious about understanding
the continuum---and I strongly feel we should---then we must
address the critical issue of how to specify and handle the
computational procedures that determine points (i.e. decimal
expansions). There is no avoiding the development
of an appropriate theory of algorithms. How sad that
mathematics lost the interesting and important subdiscipline
of computer science largely because we preferred convenience
to precision!
But let's not cry overlong about missed opportunities.
Instead, let's get out of our dreamy feather beds, smell the
coffee, and make complete sense of mathematics.
