[Paleopsych] EDGE: Verena Huber-Dyson: On the Nature of Mathematical Concepts
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EDGE: Verena Huber-Dyson: On the Nature of Mathematical Concepts
http://www.edge.org/3rd_culture/huberdyson/index.html et seq.
[Links omitted. Best, perhaps, to click on the URLs.]
ON THE NATURE OF MATHEMATICAL CONCEPTS: WHY AND HOW DO MATHEMATICIANS
JUMP TO CONCLUSIONS? by Verena Huber-Dyson
Verena Huber-Dyson Photo
(In early December, I received the following email message from Verena
Huber-Dyson - JB)
"Quite a while ago my son George Dyson handed me a batch of comments dated
Oct. 29 - EDGE 29 (What Are Numbers, Really? A Cerebral Basis For Number
Sense", by Stanislas Deheane and Nov. 7 - EDGE 30 (the subsequent Reality
Club discussion) by your group of EDGE researchers. I read it all with
great interest, and then my head started spinning.
Anyone interested in the psychology (or even psycho-pathology) of
mathematical activity could have had fun watching me these last weeks.
And now here I am with an octopus of inconclusive ramblings on the
Foundations bulging my in "essays" file and a proliferation of
hieroglyphs in the one entitled "doodles". It is so much easier to do
mathematics than to philosophize about it. My group theoretic musings,
the doodles, have been a refuge all my life.
Although I am a mathematician, did research in group theory and have
taught in various mathematics departments (Berkeley, U of Ill in
Chicago and others), my move to Canada landed me in the philosophy
department of the University of Calgary. That is where I got exposed
to the philosophy of Mathematics and of the Sciences and even taught
in these realms, although my main job was teaching logic which led to
a book on Gödel's theorems. To my mind the pure philosophers, those who
believe there are problems that they can get to grips with by pure
thinking, are the worst.
If I am right you Reality people all have a definite subject of
research and a down to earth approach to it. That is great.
There are two issues in your group's commentary that I would like to
address and possibly clarify.
For a refutation of Platonism George Lakoff appeals to non standard
phenomena on the one hand and to the deductive incomplenetess of
geometry and of set theory on the other. First of all these two are
totally different situations, to each of which the Platonist would
have an easy retort. The first one is simply a matter of the
limitation inherent in first order languages: they are not capable of
fully characterizing the "intended Models", the models that the
symbolisms are meant to describe. The Platonist will of course
exclaim: "If you do not believe in the objective existence of those
standard models, how can you tell what is standard and what is not ?".
The deductive incompleteness of a theory such as geometry or set
theory, however, simply means that the theory leaves some sentences
undecided. Here the Platonist will point out that your knowledge of
the object envisaged is incomplete and encourage you to forge ahead
looking for more axioms, i.e., basic truths !
Incidentally I consider myself an Intuitionist not a Platonist.
I wonder whether it is appropriate for me to send you my rather
lengthy discourse on non standard phenomena. You may find it tedious.
Yet I believe that the question how it is possible for us to form
ideas so definite that we can make distinctions transcending the reach
of formal languages is pertinent to your topic "what are numbers
really?" It is very difficult to put these phenomena into a correct
perspective without explaining at least a little bit how they come
The other contribution is a simple illustration of the naive
mathematical mind at work on the number 1729! And a remark about a
VERENA HUBER-DYSON is a mathematician who received her PhD. from the
University of Zurich in 1947. She has published research in group
theory, and taught in various mathematics departments such as UC
Berkeley and University of Illinois at Chicago.She is now emeritus
professor from the philosophy department of the University of Calgary
where she taught logic and philosophy of the sciences and of
mathematics which led to a book on Goedel's theorems published in
by Verena Huber-Dyson [2.16.98]
Notation: x for products: 2 x 3 =6, ^3 for cubes: 2^3 = 8, ^exponent:
2^11 = 2048.]
While engaged in the mathematical endeavor we simply jump, hardly ever
asking "why" or "how". It is the only way we know of grappling with
the mathematical problem that we are out to understand, to articulate
as a question and to answer by a theorem or a whole theory. What
drives our curiosity is a question for psychologists. Only after the
jump has landed us on a viable branch the labor of proving the theorem
or constructing a coherent theory can set in. The record of the end
result, usually a presentation at a conference, a paper in a learned
journal or a chapter in a book, is laid out in a sequence of rational
deductions from clearly stated premisses and rarely conveys the
process by which it has been arrived at.
The question why we have no other choice but to jump has received a
remarkably precise answer through Gödel's Proof of Incompleteness in
1931 and Tarski's analysis of the concept of Truth in the thirties in
Poland. Since then the development of a rigorous concept of an
algorithm has led to a proliferation of so-called undecidability and
inseparability results underscoring the limitations of the formal
The question how we jump has many aspects. First: what does the
jumping consist of, what are we doing when we jump, what is going on
in our minds when we are hunting down a mathematical phenomenon? And
then: what is guiding us, how come we jump to CORRECT conclusions?
Even if the guess was not quite correct, it usually was a good hunch
that, properly adjusted, will open up new territory. Where do these
hunches come from? Probably the simplest recorded answer to that
question goes back to Plato and has spawned a school of thought in the
Foundations of Mathematics that bears his name. It puts those hunches
on a par with our spontaneous reactions to physical messages "smell
that? someone must be roasting a lamb in the next clearing", "there is
a storm brewing in the South West, I can feel it in my bones".
According to Plato's view mathematical objects exist eternally and
immutably in a realm of ideas, an abstract reality accessible, if only
dimly, to pure reasoning. That is how we discover them and their
properties. By now, what with 2000 years of escalating experience with
mathematics and painstaking critical analyses of its tenets, Platonism
is no longer the accepted view in the Foundations. But, if nothing
else, it is a wonderful allegory and an extremely useful working
To put it bluntly, while at work a mathematician is too busy
concentrating on deciphering the hints he can gather from the trail he
is following to stop and bother asking how the trail got here. It is
enough for him to have a good hunch that the trail will lead to the
The following is a slightly polished version of my spontaneous
response to the assortment of EDGE-comments on Stanislas Dehaene's
question "What Are Numbers, Really? A Cerebral Basis For Number Sense"
and the subsequent discussion at The Reality Club. After a simple
illustration of how we ponder, jump and then fill in the steps I
address some general considerations raised on EDGE, which leads me to
an exposition of the limitation phenomena.
Although keeping technicalities to a minimum, both conceptually and
typographically, I am careful to be precise and correct. In our field
the smallest inaccuracy can have disastrous consequences leading head
on into contradictions.
1729 AN EXAMPLE OF MATHEMATICAL REASONING
Stanislas Dehaene brings up the Ramanujan-G.H.Hardy anecdote
concerning the number 1729. The idea of running through the cubes of
all integers from 1 to 12 in order to arrive at Ramanujan's
spontaneous recognition of 1729 as the smallest positive integer that
can be written in two distinct ways as the sum of two integral cubes
is inappropriate and obscures the workings of the naive mathematical
mind. To be sure, a computer-mind could come up with that list at a
wink. But what would induce it to pop it up when faced with the number
1729 if not prompted by some hunch? Here is a more likely account:
Confronted with 1729 you will recognize at a glance that:
i) 1729 = 1000 + (810-81) = 10^3 + 81 x (10-1)
= 10^3 + 9^2 x 9
= 10^3 + 9^3 = (1 + 9)^3 + 9^3
= 1 + 3 x 9 + 3 x 9^2 + 9^3 + 9^3
= 1 + (3^3 + 3 x 3^2 x 9 + 3 x 3 x 9^2 + 9^3)
= 1 + (3 + 9)^3
= 1^3 + 12^3 in view of the pattern
ii) (a + b)^3 = a^3 + 3 x a^2 x b + 3 x a x b^2 + b^3.
Now all those 3's in the above expressions spring to attention, you
fleetingly call up THE EQUATIONS
iii) (a + b)^3 + d^3 = a^3 + (c + d)^3
a^3 + (3 x a^2 x b + 3 x a x b^2 + b^3)
= a^3 + (c^3 + 3 x c^2 x d + 3 x c x d^2)
and JUMP to the conclusion that the choice of (1,9; 3,9) for a,b; c,d
will give you the smallest positive integer that can be written as the
sum the cubes of two integers (a+b) and d and also of a different pair
a and (c+d). You have a well trained instinct. But, if called upon, it
will be a simple matter to fill in that jump by a proof, the fixed
coefficients 3 ruling out smaller choices for b,c,d, once the minimal
possible value 1 is chosen for a.
ANALYSIS OF A TRAIN OF THOUGHT
The best way to understand the process encoded above in technical
shorthand is via a metaphor, which should be spun out at leisure. Say
you are driving into a strange town, and, for some reason or other, a
building complex catches your attention. It does not just pop into
your field of vision; at first glance you see it as a museum, a villa,
a church or whatever. And then, depending on your particular interests
and background, you may recognize its shape, size and purpose, muse
over its style, venture a guess as to its vintage, and so forth.
Upon meeting 1729, your first reaction will probably be to break it up
into the sum of 1000 and 729, because of our habit of counting in
decimal notation. Stop for a moment to consider what would have been
facing Ramanujan if Taxi cab companies were favoring binary notation!
[11011000001 = 11011000000 + 1 = 11^3 x 100^3 + 1^3 = 101^3 x 10^3 +
11^3 x 11^3 = 1111101000 + 1011011001]. On the other hand, if you are
one of those people obsessed with prime factorization you'll "see" the
product 7 x 13 x 19 when somebody says "1729" to you while a
before-Thompson-and-Feit but after-Burnside group theorist will say
"Aha that is an interesting number, all groups of order 1729 are
solvable" and anyone with engineering experience immediately thinks of
the 1728 cubic inches contained in a cubic foot . But a historian
of Mathematics will see 1729 as the year of Euler's friend and
benefactress Catherine the Great's birth.
Next you decide, more or less deliberately, how to investigate the
phenomenon. Do you drive to the nearest kiosk, buy a "Baedecker",
search for that building and read through all you can find in there
about it, before you make up your mind about what you want to know, in
other words, assuming you have a kiosk full of lists handy in your own
mind, do you run through all the integral cubes smaller than 1729? If
so, why cubes?
If you have that kind of mind you probably would first run through the
squares before getting to the cubes. The less methodical tourist,
eager to enjoy rather than out to complete his (or her) knowledge, may
choose to investigate in a haphazard way, spurred on by curiosity,
guided by experience, using skills automatically while impulsively
following hunches, prowling, sniffing, looking behind bushes, and then
jump to rational conclusions.
Now return to Ramanujan and see how the first thing that springs to
the naive eye beholding the number 729 is that adding 81 = 9^2 turns
it into 810, whereupon 10 drops its disguise, shows one of its true
natures as the sum of 1 and 9 and, lo and behold, all those powers of
3 start tumbling in. All the while you are aware of the pattern ii),
just below the threshold of consciousness, exactly as a driver is
aware of the traffic laws and of the coordinated efforts of his body
and his jeep. That is how you find your way through the maze of
mathematical possibilities to the "interesting" breakdown of 1729 into
two distinct sums of integral cubes.
When you stop to ask yourself what is so great about that, something
clicks in your mind: you are facing a positive integer with a certain
property, you know that
iv) every collection of positive integers has a least member
(in terms of its natural ordering).
That knowledge, always hovering below the threshold of consciousness,
prompts the question whether 1729 might in fact be the LEAST positive
integer expressible in distinct ways as the sum of two cubes. Having
another look at the representation of 1729 as a sum of various powers
of 3 as held in your mind's eye and exhibited in the third line of i)
above, the more or less conscious awareness of ii) invites you to
break up those sums of cubes according to the pattern iii) where you
assume "without loss of generality" that a < d = a + b, and hence c <
b. At this point the solution a = 1, b = c^2 = d and c = 3 surfaces by
inspection as "obviously" yielding the minimal value for (a + b)^3 +
ABOUT MATHEMATICAL ACTIVITY
I have gone through this simple illustrative example at such length in
order to underscore a few of my pet contentions:
What we sorely need is a phenomenological study of mathematical
practice. Polya and Lakatos had independently started out on that
path, I do not know to what extent it has been followed up.
Mathematicians are well aware of how they work, whether by themselves
or in teams. But their goals are results that must be presented in a
conclusive and "clean" form that makes them publicly accessible, at
least within the profession, a form that necessarily obscures the path
that led to them, just as the most beautiful tombstone will sum up a
life but give no inkling of how it really has been lived, to use an
observation by Claude Chevalley .
a) Much mathematical reasoning is done subconsciously, just as we
automatically obey traffic rules and handle our cars, whether we know
why and how they work or not. Symbolic notation is an "artificial aid"
used to secure a hold like a piton, to survey a situation like a
geological map and to encode general patterns for repeated
application. But it is not mathematics. Mathematics can be done
without symbols by a particularly "gifted" individual, like, e.g.,
Ramanujan. What that gift consists of is one of the questions raised
in the EDGE piece. Obviously we are not all of us born with it. Nor do
I believe that all people born as potential mathematicians become
actual ones. Tenacity of motivation, an uncluttered and receptive
mind, an unerring ability to concentrate the mind's focus on long
intricate chains of reasoning and relational structures, the self
discipline needed for snatching such a mind out of vicious circles,
these are only a few characteristics that spring to mind. They can be
cultivated. Experience will train the judgment to distinguish between
blind alleys and sound trails and to divine hidden animal paths
through the wilderness.
b) Free association plays an important role, an agility of mind that
allows reasoning to jump ahead with a sure touch, after which comes
the dogged toil of constructing proofs.
c) Conceptual visualization is an indispensable attendant to
mathematical thinking. Formalization is only a tool and may encourage
lazy thinking! Look at the freshmen who enroll in math because they
assume they won't be expected to produce coherent arguments or to
write grammatical text, that bureaucratic neatness in "plugging in
numbers and turning the crank" will suffice to pass the course.
It is fascinating to browse through some of the essays on the
Foundations of Mathematics by the topologist and logician L.E.J.
Brouwer, the father of Intuitionism. You will find very few formulas
in them, and yet they are rigorously reasoned, tightly and succinctly,
more so than many formal texts. 
d) That practice, familiarity, experience and experimentation are
important prerequisites for successful mathematical activity goes
without saying. But less obvious and just as important is a tendency
to "day dream", an ability to immerse oneself in contemplation
oblivious of all surroundings, the way a very small child will abandon
himself to his blocks. Anecdotes bearing witness to the enhancement of
creative concentration by total relaxation abound, ranging from
Archimedes' inspiration in a bath tub to Alfred Tarski's tales of
theorems proved in a dental chair.
The tenet that MATHEMATICAL OBJECTS ARE MENTAL CONSTRUCTS conceived by
the human species for the purpose of forging its way through life and
environment is compelling. How could we orient ourselves in space
without discerning dimensions and estimating distances, how could we
keep track of possessions and offsprings without a sense for numbers
(cardinals), groupings and hierarchies (ordinals)?
Maybe the prototypical shepherd just kept a heap of pebbles handy by
his cave, one for each sheep, to make sure by MATCHING that at the end
of the day he had his whole flock together ÷ the first occurrence of
the mathematical arrow. The next guy paid attention to the pecking
order among his charges and chose his pebbles accordingly. And then ÷
much later ÷ one with a poetic twist of mind gave individual names to
his sheep and picked pebbles to match their personalities in looks,
color, shape and mood so that, if one went missing, he could tell by
looking at the leftover pebble which one of his flock to search for
where, according to the culprit's specific idiosyncrasies. Finally,
with all that time on their hands, some of the shepherds started
creating poetry or inventing music, others projected and extrapolated
their minds into higher realms of mathematics ÷ and started wondering.
Here is the beginning of mathematics, not only arithmetic, the whole
works, structures (you start grouping your flock, and those groups
will interact), mappings and probably even the concept of infinity,
"what if those ewes keep lambing and lambing till I run out of
pebbles...". Pretty soon these concepts become PHENOMENA and begin
evolving in interaction with their creators and with the uses they are
When the ANTHROPOLOGIST has told his story and the PHENOMENO- LOGIST
has had a look at how a mathematician's mind works it is for the NEURO
PHYSIOLOGISTS to figure out what is going on in the brain of those
shepherds and their descendants. The PSYCHOLOGY of mathematical
activity ÷ and obsession ÷ also deserves attention and is bound to
shed light on the mystery of the prodigy.
The view of mathematical "objects" as mental constructs forever caught
up in a dynamic process of evolution was succinctly articulated by
L.E.J. Brouwer, the Dutch topologist who, during the first quarter of
this century, founded the school of INTUITIONISM as the most
compelling alternative to PLATONISM. Occasionally Intuitionism is
accused of leading into solipsism. But the understanding of
mathematical intuition as a sense for charting one's way around an
environment including fellow creatures implies that its tools, the
concepts, must be evolving by joint and competing efforts of a
community. Very much in keeping with what I understand is Stanislas
Dehaene's view. With Brouwer I believe in preverbal mathematical
perception, where by perception I mean an activity, a process of
"seeing as", picking out of patterns and imposing frames of reference.
Friedrich Wilhelm Nietzsche (1844-1900) had a keen understanding of
the anthropological evolution of mathematics and rational thinking.
His Der Wille zur Macht (The Will to Power, 1887) contains poignantly
expressed insights into the genesis of the laws of Logic, many of them
George Lakoff's stress on image schemes and conceptual metaphors is
compelling, especially his suggestion of "expansion to abstract
mathematics by metaphorical projections from our sensory-motor
experience". Yes we do have mathematical bodies! On a primordially
homogenous environment we impose a grid commensurate in size and
compatible in shape with our bodies as we know them from direct
experience. One step further, we project our bodies beyond what is
immediately perceivable, spurred on by a tenacious intention "to make
sense of it all". Have you ever noticed how many mathematicians are
rock climbers? The process of mulling over a mathematical problem
displays a striking similarity to that of surveying a cliff before the
ascent; of visualizing and comparing alternate routes, from the big
lines of ridges, ledges and chimneys down to the details of toe and
finger holds, and then weighing possibilities of what might be
encountered beyond the visible; all in perfectly focused
concentration, projecting ahead, extrapolating, performing so-called
"Gedankenexperimente" (thought experiments) and sensing them
throughout one's bones and muscles. And finally setting off to break
trail through the folds of a brain!
Already in 1623 Blaise Pascal articulated in his Penses (thoughts) the
observation that the abstract schemata we impose on the world in order
to interact meaningfully with it are shaped by the experience of our
During the last half century the evolution of so-called CATEGORY
THEORY out of algebraic topology has developed a dynamic language of
diagrams in which the abstract concepts of universal algebra find
their natural habitat.  "Diagram chasing" ÷ a systematic form of
hand waving ÷ is a way of making sense of the abstract structural and
conceptual under-pinnings of mathematics, including Arithmetic and
Geometry, Logic and set theory, as well as of the juxtaposition
between discrete and continuous phenomena. It turns out that Topoi, a
particularly prolific species of categories, have the structure of
intuitionistic Logic ÷ an amazing corroboration of INTUITIONISM. F. W.
Lawvere at SUNY Buffalo, a pioneer in the field since the early
sixties, and his associates are beginning to make significant
contributions to cognitive science.
As to PLATONISM, whether deliberately or inadvertently, most
mathematicians still act and talk as if they were dealing with objects
that are part and parcel of the furniture of their Universe. I do it
myself, and so does George Lakoff when he refers to the straight line
and the reals. It is such a convenient make-believe stance, not to be
confounded, however, with the deep allegorical truths revealed in the
poetry of Plato's dialogues.
But there is more to be said when we stop to contemplate what we call
REALITY. Think how often a writer will create characters only to find
them taking on a life of their own, doing things or getting into
trouble that their creator had not intended for them at all. So, the
positive integers are mental constructs. They are tools shaped by the
use they are intended for. And through that use they take on a patina
of reality! Nor do they rattle about in isolation. They interrelate,
they pick up individual personalities through interaction, by their
position in the natural ordering, by splitting into primes, by what
they are good for, in what contexts they play what roles.
And before we know it we have a problem on our hands like Fermat's
Last Theorem! Its statement can be explained to every child, using a
bit of hand waving and the ever handy dots. Through generations the
belief in its truth had grown for ever more entrenched. No counter
example was found, but no proof was in sight either until Andrew Wiles
 succeeded in blazing the final trail to the goal through abstract
territory, rugged and disconnected in places and prepared by the toil
of his peers in others. To the experts the proof is illuminating, but
not to the ordinary mathematician in the street. By now our tools are
so highly developed that they bring us information about our own
creations that we cannot fathom with the unaided mathematical senses,
even though it may concern situations whose meaning we can understand
perfectly well. In physics and astronomy we are used to similar
situations: our instruments can reach physical phenomena way beyond
the reach of our physical bodies. The interpretations of these
messages from beyond are encoded in theories of our own construction.
The method of FORMALIZATION is by now widely accepted, used and
discussed. But it has limitations and is trailing some baffling
"non-standard" phenomena in its wake. In order to put these into
proper perspective a technical digression is needed.
While mathematics is forging mental tools for charting our way through
the world, our brains playing very much the part of our senses, things
become so intricate that we need artifacts for keeping track of those
constructs. That is where symbols come in ÷ algebraic notation,
diagrams, technical languages and so forth ÷ as mechanisms for storing
and surveying insights and for communicating about them. Extension of
this method to the analysis of mathematical reasoning itself leads to
so-called meta mathematics and symbolic logic.
Allowing the articulation of "axioms" and of rules of deduction
governing their use, the systematic construction of formal languages
leads to FORMALIZED THEORIES consisting of theorems, i.e., wellformed
sentences (wfs' for short) obtained from axioms by chains of
deductions according to those rules.
A formal proof is a finite sequence of wfs' starting with axioms,
hanging together by the formal rules and ending with the theorem
proved by it. The formalized theory itself becomes a topic for
theoretical investigation since it is bound to have properties that go
beyond what we put into it. Will it be formally consistent in the
sense that the negation of a theorem will never show up as a theorem
too? Is it formally complete ,i.e., does every sentence have a proof
unless its negation has one? These are typical problems for the
The choice of axioms is not arbitrary. We are guided by common sense
of mathematical perception, by criteria that deserve investigations to
which the EDGE group seems to be making valuable contributions. As we
acquire and develop intuitive concepts of sets, spaces, geometries,
algebraic structures and all the rest, we try to grasp them by
characteristic properties and are led to basic postulates.
Occasionally sustained experience reveals that the original
construction was not fully determinate, that the axioms are not
complete. They don't suffice to pin down the intended concept
uniquely. Some sentence A ÷ Euclid's fifth postulate for instance ÷ is
left undecided by what was considered an axiomatic characterization of
the concept ÷ of, say, a geometry. Both A and its negation not-A are
formally consistent with the axioms. Well, for some purposes it is
useful to assume Euclid's parallel axiom for geometry, or
well-foundedness for sets, at other times it may be handy to deal with
bottomless sets or crooked squares. The tools are evolving as we are
using, refining and adjusting them. Such experiences that at first
look like failures deepen conceptual understanding and expand
The situation of the arithmetic N over the natural numbers 0,1,2,3,...
and that of the ordered field R of the reals are more subtle. In both
cases we "know exactly" what structure we have in mind, there is no
question of bifurcation of concepts. Yet in the case of N a complete
axiomatization founders on the requirement of effectiveness while,
even though completely formalizable, the elementary theory of R, has
so-called non-standard models, as does every theory of an infinite
These phenomena are a manifestation of the precarious balance between
algorithmic precision and expressive power inherent in every formal
language and its logic. The most popular, widely taught formalization
is the first order predicate calculus, also called elementary logic, a
formalization of reasoning in so-called first order predicate
languages. That apparatus leads from "elementary axioms" to
The important requirement for any formalization is the existence of
both a "mechanism" (algorithm) for deciding, given any well formed
sentence (wfs) of the language concerned, whether or not that wfs is
an axiom, and one for deciding of any given configuration of wfs's
whether or not it is an instance of one of the rules. The resulting
concept of a formal proof is decidable, i.e., there exists an
algorithm, which, when fed any finite sequence of wfs', will come up
with the "answer yes" (0) or the "answer no" (1) according as that
sequence is a formal proof in the system or not. The resulting
axiomatizable theory will in general only be effective in the sense
that there exists an algorithmic procedure for listing all and only
those wfs' that are theorems. That does by no means guarantee a
decision procedure for theoremhood. In fact most common theories have
been proven undecidable.
To start with, a familiar structure like N or R will serve as the
so-called STANDARD MODEL or INTENDED INTERPRETATION for the elementary
theory meant to describe it. Observe that the notion of a standard
model presupposes some basic concept of mathematical reality and
truth. Gödel talks of "inhaltliches Denken" (formal thinking) in
juxtaposition to "formales Denken" (formal thinking). His translators
use the term 'contentual'. 'Intentional' might be just as good a
Of course one might dodge the need for a metaphysical position by
using terms like "preverbal" or "informal".
But that does not make the problem go away. If we want to talk about
standard models, if we want our theories to describe something ÷
approximately and formally ÷ what is it that we want them to describe?
A question that would not disturb a Platonist like Gödel. The
formalist's way out is to throw away the ladder once he has arrived at
his construction and to concentrate on the questions of formal
consistency and formal completeness, purely syntactic notions.
A theory is formally consistent if and only if for no wfs A both A and
not-A are theorems and formally complete if and only if for every wfs
A either A or not-A is a theorem.
To an extreme formalist the existence of an abstract object coincides
with the formal consistency of the properties describing it. If at
all, he will draw his models from yet another theory, most likely
some, necessarily incomplete, formalization or other of elementary set
theory, presumed ÷ but only presumed ÷ to be consistent. An
We, however, are left with the conundrum of Mathematical Truth and the
semantic notions that depend on a "meaning" attached to the theory.
With respect to an interpretation of its language over a structure S a
theory, formalized or not, is
sound (semantically consistent) if and only if only wfs' true in S
semantically complete if and only if all wfs' true in S are theorems.
Granted a clear and distinct idea of the structures N and R we talk of
the sets of all wfs' that are true under the intended interpretations
on N and on R as True Elementary Arithmetic, TN, and as the True
Theory TR of the Reals.
Consider an EXAMPLE: Leaving aside the question where N comes from, I
should think that we all know what we mean by the wfs
(F^3) for ALL positive integers x, y and z: the sum of the cubes of x
and y is not equal to the cube of z.
F is short for FERMAT. To explain it to a naive computer mind, we
would say: "Make two lists as follows; in the left one, L, write down
successively the results of adding the cubes of two positive integers,
1^3 + 1^3, 1^3 + 2^3, 2^3 +2^3, 1^3 + 3^3, 2^3 + 3^3, 3^3 + 3^3,...,
and into the right one, R, put all the cubes 1^3 (1), 2^3 (8), 3^3
(27), 4^3 (64), 5^3 (125) and so on. Now run through both lists
comparing the entries. (F^3) claims that you will never find the same
number showing up both on the left and on the right". A computer can
easily compile these lists in so orderly a fashion and run through
them so systematically that, for each bound N, it will, after a
computable number of steps, say f(N) of them, have calculated and
compared all pairs of numbers in L and in R smaller than N. You will
probably agree that this tedious explanation makes it sufficiently
clear what we mean here by ALL. You may want to use nicer language
like talking about NEVER finding a matching pair. The purpose of
symbolization, however, is not only orderliness, but clarification.
The dual to the so-called UNIVERSAL QUANTIFIER (for all) is the
EXISTENTIAL QUANTIFIER. Just think for a moment, assuming (F^3) were
false, how easy it would be for your patient computer to prove that.
It would only have to go on long enough until it found a COUNTER
EXAMPLE, i.e., a positive integer that shows up in both lists, R and
L. Having done so it would have proved
NOT(F^3) there EXIST positive integers x,
y and z, such that the sum of the cubes of
x and y is equal to the cube of z.
In 1753, using clever transformations of the problem, Euler succeeded
in proving the restriction (F^3) of Fermat's theorem to cubes. But
Fermat's general Conjecture
(F) for ALL positive integers x, y, z and n,
n greater than 2,
the sum of the n-th powers of x and y
is not equal to the n-th power of z
has only been proved conclusively a few years ago by means of
techniques way beyond elementary arithmetic. It should be noted here
that variable exponentiation is not part of the language of N but can
be paraphrased in it. In the above procedure you will have to organize
your left list according to an enumeration of triples (x,y,n) and the
right one according to pairs (z,n).
The capacity to visualize an ongoing sequence of calculations and
comparisons leads to an understanding of what is meant by the truth of
(F). Yet, in spite of many efforts, its proof had to wait till
algebraic geometry and number theory had achieved the maturity
necessary to allow its construction.
We have a pretty good understanding of what we mean when we claim that
ALL integers ÷ or all pairs or triples of them ÷ have a certain
property, provided that we understand the property itself. The most
manageable kind of properties that integers may have are what we call
recursive or computable. They are susceptible to a decision procedure
as illustrated by the example of checking for fixed n and any given
triple of integers x,y,z whether or not the sum of the n-th powers of
the first two is equal to that of the third.
A property P of triples of numbers is called recursive if and only if
its so-called characteristic function that takes on value 0 at the
triple (m,n,q) if that triple has the property and value 1 otherwise
(its decision function) is computable by an algorithm like a Turing
machine, or, equivalently, is recursive.
The amazing ÷ often elusive ÷ power of the universal quantifier
brought home by Gödel's incompleteness proof, discussed in the next
section, is again manifest in the intrinsic difficulties with which
the conclusive proof of Fermat's theorem is fraught.
Based on a naive concept of Truth, every true theory of a definite
structure is complete and consistent in both senses, a pretty useless
observation. For, a byproduct of Gödel's Incompleteness proof of 1931
 is the non-formalizability of elementary arithmetic, TN, and with
it of many other theories.
EVERY SOUND AXIOMATIZATION OF ELEMENTARY ARITHMETIC IS INCOMPLETE.
The most natural candidate for axiomatizing TN goes back to Giuseppe
Peano (1895) and consists of the recursive rules for addition,
multiplication and the natural ordering on the set N of non negative
integers built up from 0 by the successor operation that leads from n
to n+1, together with the Principle P of Mathematical Induction, which
postulates that every set of numbers containing 0 and closed under the
successor operation exhausts all of N, or, equivalently, that every
property enjoyed by 0 and inherited by successors is universal. P is a
principle that adults may consider a definition of the set N, while
children will ÷ in my experience ÷ take it for granted. But, if you
want to articulate it in the language of the first order predicate
calculus you run into trouble. As illustrated in example (F)
elementary languages can quantify over individuals. But quantification
over so-called HIGHER ORDER items like properties is beyond its scope.
P is a typical sentence of second order logic.
PEANO ARITHMETIC, PA, is the first order approximation to second order
arithmetic obtained by replacing P with the following schema of
infinitely many axioms
(PW) If 0 has the property expressed by the
wff W and,
whenever a number x has that property,
then so does x+1,
then all natural numbers have property W.
one for each wff (well formed formula) W of the elementary language of
Reformulating (PW) in terms of proofs rather than truth sheds light on
it and illustrates how one might want to go about replacing the basic
concept of truth in mathematics by a primitive notion of proof.
Writing W(x) for "x has property W" the Principle of Proof by
Mathematical Induction reads
(PPW) Given 1) a proof of W(0) and 2) a
method for turning any proof of W(x)
into a proof of W(x+1)
THEN there exists a proof of
"for all x: W(x)".
Note how appealing this formulation is: Given any number n, you only
have to start with the proof given by 1) and then apply the method of
2) n times to obtain a proof of the sentence W(n). But there is a
subtlety here. So understood, the principle only guarantees that, for
every number n, a proof of W(n) can be found, a typical "for all ÷
there exists ÷" claim. Its power lies, however, in the "there
exists-for all ÷" form of the conclusion as exhibited above.
These may sound like nit-picking distinctions, but they are of great
proof-theoretic significance. For instance, from the consistency of
PA, proved by means transcending PA, follows:
If g is the Gödel number of the Gödel sentence G then:
for each natural number n, the sentence
"n is not the Gödel number of a proof
of the sentence with Gödel
is a theorem of PA.
However G itself, namely the sentence
"for all x: x is not the Gödel number of a
proof of the sentence with Gödel number g"
is not a theorem of PA.
G is the sentence that truthfully claims its own unprovability. Much
deep work is required to establish this result rigorously.
Occasionally proofs by mathematical induction are confused with
arguments based on so-called 'inductive reasoning', a term used in
philosophical discussions of logic and the sciences ÷ yet another
reason for all these elaborations.
Axiomatized but undecidable theories are a fortiori incomplete. In
1939 Tarski proved that
THERE IS A FINITELY AXIOMATIZABLE FRAGMENT OF PA ALL OF WHOSE
CONSISTENT EXTENSIONS ARE UNDECIDABLE. 
And yet, if we accept an intentional concept of truth, we seem to
obtain a complete theory from Peano's mere handful of axioms together
with that one marvelous second order tool P on which so much of our
mathematical thinking hinges. For:
ALL MODELS OF PEANO'S SECOND ORDER AXIOM SYSTEM ARE ISOMORPHIC.
Still, any attempt to formalize second order arithmetic is again
doomed to founder on the cliffs identified by Gödel and Tarski. The
juxtaposition of these claims is and ought to be baffling. In fact
they bring home the discrepancy between the naive and the formalist
concept of a model. From the naive point of view they mean that higher
order logic cannot be completely formalized. Even so completeness
proofs for it are widely hailed ÷ at the price of allowing all sorts
of non-isomorphic models even for second order Peano Arithmetic.
Enough of that for now. Fermat's last theorem may well be beyond the
scope of elementary Peano Arithmetic. In other words, (F) is
presumably left undecided by PA. A few mathematically interesting
theorems expressible in the language of PA with that property are
already known. They are embeddable in stronger but still convincing
first order theories, some elementary set theory or other.
After all that we are faced with the question where new axioms come
from, in other words with THE PROBLEM OF THE NATURE OF MATHEMATICAL
TRUTH. To declare "OK, as of October 27, 1995, the day that Wiles was
awarded the Prix Fermat by the town of Toulouse, (F) shall be added to
the list of axioms for elementary arithmetic" would seem quite
inappropriate. We want more intuitively obvious first principles.
NON STANDARD MODELS
In amazing contrast to TN the first order theory TR of the ordered
field of the REALS has been successfully and completely formalized ÷
starting with Euclid's axioms, improved by Hilbert just before the
turn of the century and completed as well as proved complete by Tarski
about the time of the Second World War. My immediate reaction when I
first heard of this feat was shock and distrust of those Berkeley
logicians. "How could that be? The reals are so much more complicated
than the integers. Aren't the natural numbers defined as the non
negative integral reals?" Well, the solution of that conundrum lies in
LIMITATION OF EXPRESSIVE POWER INHERENT IN FORMAL LANGUAGES.
As a matter of fact, the natural numbers are not "elementarily
definable" among the reals; there is no wff of the language of R that
picks out the natural numbers among the reals.
Moreover, in spite of its completeness, TR has non-isomorphic models!
It has countable models, uncountable ones, Archimedean as well as
Non-Archimedean ones; some harbor hyperreals, others only standard
reals... What is going on? First the chicken-or-egg question must be
faced: what comes first, the model or the theory? Ever since the
elaborations by Tarski in 1934 and by Mal'cev in 1936 of the results
by Lwenheim of 1915, and by Skolem of 1920 (a brief exposition will
follow below) we understand that first order chickens are prone to lay
a medley of eggs, some "real" in the Platonic sense of being standard
and others weird, artificial, substitutes, freaks, in short
non-standard. The Ur-hen, the axiomatization, originated from a
standard egg, the "intended interpretation", a natural mathematical
construct like our everyday arithmetic of the positive integers, or,
more sophisticated, the real number system of the 19th century. After
the chicken has grown to maturity it starts laying models, and,
roaming through the virtual reality of model theory instead of free
ranging in Platonic realms, it comes up with non-standard eggs. The
only constraint on those is consistency and the verification of the
axioms, i.e., the genetic chicken code. These models are hatched
within the confines of some entrenched formalization of set theory.
What really lies at the basis of non standard objects like hyper reals
is ÷ again ÷ the limitation inherent in first order languages. In the
elementary language of real number theory we cannot distinguish
between Archimedean and non Archimedean orderings and that opens the
door to constructions that were scorned by my teachers although they
might use infinitesimals as a handy figure of speech the way we still
talk Platonically. We thought that Cauchy and Weierstrass'
arithmetization of analysis had done away with that alleged abuse of
language, but now it is back en vogue again and very useful too (see
NON STANDARD PHENOMENA are closely connected with the SEMANTIC
COMPLETENESS OF ELEMENTARY LOGIC, first proved by Gödel in 1930  and
extended in many ways since, in particular by Henkin who also dealt
with formalizations of higher order logic. The underlying meta theorem
rests on two facts, one inherent in the finitary nature of a formal
deduction, the second involving non- constructive instructions for
building a model
1. WHENEVER ALL FINITE SUBSETS OF A SET OF WFS' ARE CONSISTENT THEN
SO IS THE ENTIRE SET and
2. EVERY CONSISTENT SET OF WFS' HAS A MODEL.
By definition Semantic Completeness of a formal calculus means
EQUIVALENCE BETWEEN FORMAL DERIVABILITY AND SEMANTIC VALIDITY where
validity stands for truth under all interpretations, i.e. in all
At first this looks like an amazing result especially in view of
currently rampant incompleteness. It is unfortunate that popular
literature so often fails to make a clear distinction between the two
concepts of semantic and of syntactic completeness (pp.10,11). Only
the experienced reader will automatically know from the context which
notion is at stake.
As a matter of fact the completeness of first order logic is achieved
at a price: the expressive poverty of the formal language.
Completeness proofs for higher order logic are ensnared in the same
kind of bargain. They are based on a concept of model that to the
naive mind seems contrived. Elementary languages are incapable of
distinguishing between arbitrarily large finiteness and infinity, and
so are forced to tolerate the infinitely small. Consider the infinite
set of wfs' 0 < a < 1, a + a < 1, a + a + a < 1,..,a + a + a +...+ a <
1,... and let U be its union with TR, the set of all wfs' that are
true in the field R of the reals. Every finite subset V of U has a
model: just take R and interpret a by 1/n, where n is the number of
symbols occurring in that finite set V. By 1) then the whole set U is
consistent and so, by 2) it and with it the elementary theory of the
reals has a model which harbors an element satisfying all these
inequalities, i.e., a non- Archimedean, non-standard, or hyper, real
a. It is positive and yet smaller than any fraction 1/n, n a positive
Ruled by its logic, the language cannot prohibit such anomalies. But
there is a silver lining to this shortcoming: Because of the
consistency of infinitesimals with TR every truth about the reals that
can be expressed in the elementary language of R holds for all reals ÷
standard or not ÷ and so, by Gödel's completeness theorem, it has a
formal proof. And if the approach via infinitesimals is smoother that
is just great. One cannot help but marvel at the native instinct with
which the seventeenth century mathematicians went about their work
Similarly, any first order theory of N, including TN, has models that
contain infinitely large integers. The elementary theory of finite
groups has infinite models and so in fact does every first order
theory of arbitrarily large finite models.
All this is meant to explain that these non standard phenomena have no
bearing on the question whether Platonism is an appropriate view of
the origin of Mathematics. I am deliberately not using the word
"correct". Whether Platonism is "true" seems an ill posed question,
luring into vicious circles. How can we contemplate the truth of this,
that or the other "ism" before we have a clear and distinct idea of
what ÷ if anything ÷ we mean by the Truth of a theory?
The existence of non standard models should NOT be confounded with the
occurrence of incomplete concepts like that of a geometry or that of a
set. In the case of hyperreals we are running into limitations of the
formal language while dealing with complete theories, in the second
case we are simply facing the fact that the intuitive concept, say of
a geometry or a set, that we had in mind when setting up the
formalization is not completely fathomed yet, in both senses of
completeness. Of course the easiest reaction is to say, "that concept
is out there, let us go look more closely and we shall eventually find
its complete characterization". In this frame of mind Gödel is reputed
to have been convinced that we shall eventually understand enough
about sets to come up with new axioms that will decide the continuum
hypothesis. But in other cases the expedient policy will allow a
concept to bifurcate ÷ sailors have no trouble with non-Euclidean
The big question is where our standard concepts come from, how do we
all know what we mean by the Standard Reals? How can we distinguish
between Archimedean orderings and non Archimedean ones, when we cannot
make the distinction in first order language? Well we can always
resort to hand waving when words fail. We can indeed communicate about
them beyond the confines of formalism. They are conceptions,
constructions, structures, figments of our imagination, of the human
mind that is our common heritage. Other creatures may have other ways
of making sense of and finding their way in a Universe that we are
sharing with them.
This century has seen the development of a powerful tool, that of
formalization, in commerce and daily life as well as in the sciences
and mathematics. But we must not forget that it is only a tool. An
indiscriminate demand for fool proof rules and dogmatic adherence to
universal policies must lead to impasses. The other night, watching a
program about the American Civil Liberties Union I was repeatedly
reminded of Gödel's Theorem: every system is bound to encounter cases
which it cannot decide, snags that will confront its user with a
choice between either running into a contradiction or jumping out of
the system . That is when, with moral issues at stake, cases of
precedence are decided by thoughtful judgment going back to first
principles of ethics, in the sciences alternate hypotheses are formed
and in mathematics new axioms crop up.
Returning to my question, think of mathematics as a jungle in which we
are trying to find our way. We scramble up trees for lookouts, we jump
from one branch to another guided by a good sense of what to expect
until we are ready to span tight ropes (proofs) between out posts
(axioms) chosen judiciously. And when we stop to ask what guides us so
remarkably well, the most convincing answer is that the whole jungle
is of our own collective making ÷ in the sense of being a selection
out of a primeval soup of possibilities. Monkeys are making of their
habitat something quite different from what a pedestrian experiences
as a jungle.
To sum it all up I see mathematical activity as a jumping ahead and
then plodding along to chart a path by rational toil.
The process of plodding is being analyzed by proof theory, a prolific
branch of meta mathematics. Still riddled with questions is the
 cf. Charles Simonyi on EDGE quoting Richard Feynman
 cf. D. Guedj: Nicholas Bourbaki, collective mathematician, an
interview with Claude Chevalley, The Mathematical Intelligencer vol.
7, no. 2, 1985.  Cf. the two anthologies Philosophy of Mathematics:
selected readings edited by P. Benacerraff and H.Putnam, Cambridge
University press 1964 and reprinted with some deletions 1985, and J.
van Heijenoort's From Frege to Gödel; a source book in mathematical
logic 1879-1931 Harvard University Press 1971.
 In a fine essay, entitled "Bemerkungen zu zwei wenig beachteten
'Gedanken' Pascals" ("Remarks on two of Pascal's 'thoughts' that have
hitherto received little attention") in his collection Ausgewþhlte
Vortrþge und Aufsþtze (selected talks and essays), Bern 1955 pp.
226-234, the Swiss psychoanalyst and phenomenologist Ludwig Binswanger
elaborates on Pascal's "penses" (thoughts or reflections) on this
topic and substantiates them with clinical experiences pertaining
especially to the role played by symmetries in Rorschach tests.
 cf. chapter III of my Gödel's Theorems; a Workbook on
Formalization, Teubner Texte zur Mathematik, Stuttgart-Leipzig 1991.
 Modular elliptic curves and Fermat's Last Theorem, Ann. Math. 141,
1995, 443 551.
 ber formal unentscheidbare Sþtze der Principia Mathematica und
verwandter Systeme Monatshefte fr Mathematik und Physik 37,
1931,173-198. For a translation see Kurt Gödel, Collected Works, Volume
1, ed. S. Feferman et al, Oxford U. Press 1986.
 cf. p. 39 ff. of the excellent and very readable monograph
Undecidable Theories by Tarski, Mostowski and Robinson, North Holland
 cf. the collected works cited before.
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