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Reuben
Hersh: What is Mathematics,
Really?
(Jonathan Cape: 1997)
“The working mathematician
is a Platonist on weekdays, a formalist on weekends. On
weekdays, when doing mathematics, he’s a Platonist,
convinced he’s dealing with an objective reality
whose properties he’s trying to determine. On weekends,
if challenged to give a philosophical account of this
reality, it’s easiest to pretend he doesn’t
believe in it. He plays formalist, and pretends mathematics
is a meaningless game.... Does it matter? Yes. Truth and
meaning aren’t recondite technical terms. They concern
anyone who use or teaches mathematics. Ignoring them leaves
you captive to unexamined philosophical preconceptions.
This has practical consequences.... ‘What’s
interesting in mathematics?’ is an urgent question
for anyone doing research, or hiring or promoting researchers,
[yet] there’s no public discussion of this question.
No vehicle for public discussion of it. No language or
viewpoint that could be used for such a discussion. Not
to say there should be agreedon standards of what’s
interesting. Precisely because tastes differ, we need
discussion on taste. We have some common standards. That’s
proved by our identity as one profession, and our agreement
that certain feats in mathematics deserve the highest
rewards. Bringing out those standards for analysis and
controversy would be important philosophical work [but]
our inability to sustain public discussion on values betrays
philosophical unawareness and incompetence.... Ideas have
consequences. What I think mathematics is
affects how I present it...[as was shown by] the unfortunate
importation into primary and secondary schools, during
the 1960s, of settheoretic notation and axiomatics. This
wasn’t an inexplicable aberration. It was a predictable
consequence of a philosophical doctrine: Mathematics is
axiomatic systems expressed in settheoretic language.
Critics of formalism in high school say ‘This is
the wrong thing to teach, and the wrong way to teach’
[but] such criticism leaves unchallenged the dogma that
real mathematics is formal derivations from formally stated
axioms. If this dogma rules, the critic of formalism is
seen as asking for lower quality, to give students something
‘watered down’ rather than the ‘real
thing.’ The fundamental question is ‘What
is mathematics?’ Controversy about high school teaching
can’t be resolved without controversy about the
nature of mathematics.”
(Hersh, pp.3941)
Undoubtedly the greatest division within our intellectual
world lies between those disciplines within which mathematical
methods form an essential component, and those that do
not... And, with people educated in the latter generally
making no serious effort to understand the former  despite
the pretensions to critique of the “postmodern”
Humanities  we are, as so often, left to our own devices
re this truly essential task. Thankfully, however, there
is a range of
widely accessible (and often entertaining) writing by
professional mathematicians today which can serve to show
the rest of us how mathematical thinking works, and thus
can help open up not only this field, but also all those
others dependent upon this approach.
Of such works, I’ve found those of Reuben Hersh
the most useful, addressing as they do the fundamentals
via empirical and historical enquiries...and offering,
as well, an enviably hardnosed approach to social constructivism
 a badlyneeded corrective to the Humanities’ smugly
illinformed relativism. Some samples from his early chapter
on criteria for theoretical judgement should make this
point better than anything I can say for myself:
“Evaluate a body
of thought according to its own goals and presuppositions.
Understand it historically, in the sense of history of
ideas. Pay attention to its consequences, theoretical
and practical. Beneficial consequences don’t verify
a doctrine. Harmful consequences don’t falsify it.
But consequences are as important as plausibility, consistency,
or explanatory power.... [And] the argument, ‘What
I don’t know, ipso facto
doesn’t matter’ isn’t new. Age hasn’t
made it palatable.”
(Hersh, pp.245)
“One wants all
three  truthfulness, precision, and simplicity. But,
one usually can’t maximize at once goal A and goal
B. If you’re not willing to pick one goal and ignore
the others (maximum cash flow, for instance  reputation
and legality be damned!) then you have to do some balancing
or juggling. (Work on cash flow, but don’t actually
go to jail.) [And] precision is easier to achieve in a
simple situation than in a complex one. Some phenomena
are inherently imprecise.... Simplicity goes with singlemindedness.
Where several factors interact to give a complex result,
simplicity can be created by ignoring all factors but
one [and] different scholars may single out different
factors. This kind of simplicity leads to fruitless controversy,
like between Red Sox fans and White Sox fans. For example,
both formalization and construction are central features
of mathematics. But the philosophies
of formalism and constructivism are longstanding rival
schools. It would be more productive to see how formalization
and construction interact than to choose one, and reject
the other.... Putting simplicity and precision ahead of
truthfulness is treating philosophy as a game, art
pour l’art , like
some exotic branch of algebra. Philosophy can be serious
 no less than how and why to live. Being serious means
putting truthfulness first. First get it right, then go
for precision and simplicity. My first assumption about
mathematics is: It’s something people do. An account
of mathematics is unacceptable unless it’s compatible
with what people do, especially what mathematicians do.”
(Hersh, p.30)
“Comprehensibility
is valued by readers, not by all writers. Philosophy students
think that, among professional philosophers, incomprehensibility
gets ‘brownie points’ and comprehensibility
gets demerits. Unworthy suspicions aside, it’s a
question of comprehensibility to whom .
What’s incomprehensible to you may be crystalline
to the Heidegger expert. This book aims to be easily comprehensible
to anyone. If some allusion is obscure, skip it. It’s
inessential.”
(Hersh, p.29)
As should already be evident, Hersh is an insightful thinker,
with an unusually terse yet clear and colloquial style.
In some ways, I suspect, this may reflect the mathematical
mindset, in which processual clarity is particularly evident,
and there is no place for ornament? Irrespective, I find
Hersh’s style a very effective one...and highly
appropriate to his subject matter:
“Should the philosophy
of mathematics be precise? ...Mathematics
is precise; philosophy cannot be. Expecting philosophy
of mathematics to be a branch of mathematics, with definitions
and proofs, is like thinking philosophy of art can be
a branch of art, with landscapes and still lives.... It
happens that the creators of foundationist philosophy
of mathematics were mathematicians (Hilbert, Brouwer)
or mathematically trained (Husserl, Frege, Russell). This
training may explain their bias. They sought to turn philosophical
problems into mathematical problems, to make them
precise . This bias was
fruitful mathematically. Some of today’s mathematical
logic descended from the search for mathematical solutions
to philosophical problems. But, even though mathematically
fruitful, it was philosophically misguided.”
(Hersh, p.29)
“I once wrote
that mathematicians hate contradiction. That’s not
accurate. We love it  like a duck hunter loves ducks.
Nothing draws us to the chase like a contradiction in
a famous theory.... Consistency is important. But, it’s
less important than fruitfulness (inside and outside of
mathematics), imaginative appeal, and linking new mathematical
devices to old, respected problems. A contradiction can
generally be fixed up, one way or another. As Bourbaki
explained, ‘freedom from contradiction is attained
in the process, not guaranteed in advance.’ ...In
practice, we can’t always prove in advance the consistency
of all possible deductions. Instead, we develop a technique
for preserving partial consistency  absence of contradiction
up to the latest set of results. In that way, we continue
to forestall contradiction each time it raises its ugly
head.... Mathematical buildings collapse  lose interest,
are forgotten  not because of contradictions, but because
their questions are no longer interesting, or because
another theory answers them better.”
(Hersh, p.32)
“Teacher thinks
she perceives otherworldly mathematics. Student is convinced
teacher really does perceive otherworldly mathematics.
No way does student believe he’s
about to perceive otherworldly mathematics.”
(Hersh, p.238)
By refusing to accept the overly reified notions of mathematics
that mainly prevail in philosophical circles  and which
have helped ruin early mathematical education  Hersh
allows us to approach mathematics as a human endeavour...in
fact, he goes so far as to label his philosophy of mathematics
“humanistic”, a marvellous dissent from avowedly
“antihumanist” approaches within the Humanities
itself. Equally in contrast with such approaches, too,
is his plainspoken approach to issues of genuine complexity:
“Even without
three years of graduate school, you can get a rough notion
of modern mathematics. Here’s a minisketch of its
method and matter. The method of mathematics is ‘conjecture
and proof’. You come to an inherited network of
concepts and facts, properties and connections, called
a ‘theory.’ ...This presently existing theory
is the result of a historic evolution. It is the cooperative
and competitive work of generations of mathematicians,
associated by friendship and rivalry, by mutual criticism
and correction, as leaders and followers, mentors and
protégés. Starting with the theory as you
find it, you fill in gaps, connect to other theories,
and spin out enlargements and continuations.... But you
[don’t do this] in isolation...[for] no matter how
isolated and selfsufficient a mathematician may be, the
source and verification of his work goes back to the community
of mathematicians.... Mathematical discovery rests on
a validation known as ‘proof,’ the analogue
of experiment in physical science. A proof is a conclusive
argument that a proposed result follows from accepted
theory. ‘Follows’ means the argument convinces
qualified, skeptical mathematicians. Here, I am giving
an overtly social definition of ‘proof.’ Such
a definition is unconventional, yet it is plainly true
to life. In logic texts and modern philosophy, ‘follows’
is often given a much stricter sense, the sense of mechanical
computation. No one says the proofs that mathematicians
write actually are checkable by machine. But it’s
conventional to insist that there be no doubt
they could be
checked that way. Such lofty rigour isn’t found
in all mathematics. From one speciality to another, from
one mathematician to another, there’s variation
in strictness of proof and applicability of results, [and]
mathematics that stresses results above proof is often
called ‘applied mathematics.’ ...A naive nonmathematician
 perhaps a neoFregean analytic philosopher  looks into
Euclid, or a more modern math text of formalist stripe,
and observes that axioms come first. They’re right
on page one. He or she understandably concludes that in
mathematics, axioms come first. First your assumptions,
then your conclusions, no? But anyone who has done mathematics
knows what comes first  a problem. Mathematics is a vast
network of interconnected problems and solutions...[and]
sometimes a solution is a set of axioms!”
(Hersh, pp.56)
“When a piece
of mathematics gets big and complicated, we may want to
systematize and organize it, for esthetics and for convenience.
The way we do that is to axiomatize it. Thus a new type
of problem (or ‘metaproblem’) arises: ‘Given
some specific mathematical subject, find an attractive
set of axioms from which the facts of the subject can
conveniently be derived.’ Any proposed axiom set
is a proposed solution to this problem. The solution will
not be unique. There’s a history of reaxiomatizations
of Euclidean geometry, from Hilbert to Veblen to Birkhoff
the Elder. [And, so] in developing and understanding a
subject, the axioms come late. Then, in the formal presentations,
they come early. Sometimes, someone tries to invent a
new branch of mathematics by making up some axioms and
going from there [but] such efforts rarely achieve recognition,
or permanence. Examples, problems, and solutions come
first. Later come axiom sets, on which the already existing
theory can be ‘based.’ The view that mathematics
is in essence derivations from axioms is backward. In
fact, it’s wrong.”
(Hersh, p.6)
By focusing upon what mathematicians do
 rather than viewing mathematics as an independent world
of its own  Hersh succeeds in showing us that it is not
at all as alien as many of us may think...merely much
more reliable/reproducible than the rest of our social
world. In fact, this  rather than any Platonic existence
 is what defines
a field as mathematical, as modern mathematics is an astonishingly
variegated field today, literally overflowing with bizarre
objects & strange relations of all sorts...
“So far, I’ve
described mathematics by its methods. What about its content?
The dictionary says math is the science of number and
figure (‘figure’ meaning the shapes or figures
of geometry). This definition might have been O.K. 200
years ago. Today, however, math includes the groups, rings,
and fields of abstract algebra, the convergence structures
of pointset topology, the random variables and martingales
of probability and mathematical statistics, and much,
much more. Mathematical Reviews
lists 3,400 subfields of mathematics! No one could attempt
even a brief presentation of all 3,400, let alone a philosophical
investigation of them all. To identify a branch of study
as part of mathematics, one is guided by its method, more
than its content.”
(Hersh, p.7)
“What’s
the nature of mathematical objects? The question is made
difficult by a centuriesold assumption of Western philosophy:
‘There are two kinds of things in the world. What
isn’t physical is mental; what isn’t mental
is physical.’ Mental is individual consciousness...[and]
physical is taking up space.... Frege showed that mathematical
objects are are neither physical nor mental. He labelled
them ‘abstract objects.’ What did he tell
us about abstract objects? Only this: They’re neither
physical nor mental. Are there other things besides numbers
that aren’t mental or physical? Yes! Sonatas. Prices.
Eviction notices. Declarations of war.... Platonist philosophy
masks this social mode of existence, with a myth of ‘abstract
concepts.’ ...Once created and communicated, mathematical
objects are there ....
We learn of them as external objects, with known properties
and unknown properties. Of the unknown properties, there
are some we can discover [and] some we can’t discover,
even though they are our own creations.... Mathematical
objects can have welldetermined properties because mathematical
problems can have welldetermined answers. To explain
this requires investigation, not speculation. The rough
outline is visible to anyone who studies or teaches mathematics.
To acquire the idea of counting, we handle coins or beans
or pebbles. To acquire the idea of an angle, we draw lines
that cross. In higher grades, mental pictures or simple
calculations are reified ...and
become concrete bases for higher concepts. These shared
activities  first physical manipulations, then paper
and pencil calculations  have a common product  shared
concepts.... The observable reality of mathematics is
this: an evolving network of shared ideas with objective
properties. These properties may be ascertained by many
kinds of reasoning and argument.... They have the rigidity,
the reproducibility, of physical science. They yield reproducible
results, independent of particular investigators. Such
kinds of ideas are important enough to have a name. Study
of the lawful, predictable parts of the physical world
has a name: ‘physics.’ Study of the lawful,
predicable parts of the socialconceptual world also has
a name: ‘mathematics.’”
(Hersh, pp.1319)
Personally, I am convinced by this...even though one skilled
mathematician of my acquaintance disliked the concept
so much that he refused to finish reading the book! Yet,
as Hersh compellingly argues, the other alternatives are
genuinely incoherent  whereas this proposal is merely
distasteful to minds predisposed to Platonic “solutions”.
Still...Hersh is no diplomat, and he gives short shrift
to some of the “big” (yet vacuous) questions
which have traditionally occupied philosophers in this
area  albeit he does
offer a fascinating (and rather mathematical) conjecture
as to why philosophy (as opposed to history of ideas)
so often makes a fool of itself with idealism:
“Some questions,
which at first seem meaningful, are futile
to answer them neither possible nor necessary. Why are
there rigid, reproducible concepts , such as number or
circle? Why is there consciousness? Why is there a cosmos?
We need not answer Kant’s question, ‘How is
mathematics possible?’ any more than we need answer
Heidegger’s question, ‘Why should anything
exist?’ I haven’t heard about progress on
either problem.... Ethnology, comparative history, developmental
psychology, the development of nonEuclidean geometry,
and general relativity all show that Euclidean geometry
is not built into everyone’s mind/brain. We think
about space in more than one way. We reject Kant’s
answer. Must we still accept his question? ...This much
is clear: Mathematics is
possible. It’s the old saying, ‘What is
happening can
happen.’ ...Since Dedekind and Frege in the 1870s
and 1880s, philosophy of mathematics has been stuck on
a single problem  to find a solid foundation to which
all mathematics can be reduced, a foundation
to make mathematics indubitable, free of uncertainty,
free of any possible contradiction.... That goal is now
admitted to be unattainable. Yet, with the exception of
a few mavericks, philosophers continue to see ‘foundation’
as the main interesting problem in philosophy of mathematics....
I have two concluding points: Point 1 is that mathematics
is a socialhistoric reality. This is not controversial.
All that Platonists, formalists, intuitionists, and others
can say against it is that it’s irrelevant to their
concept of philosophy. Point 2 is
controversial: There’s no need to look for a hidden
meaning or definition of mathematics beyond its socialhistoriccultural
meaning. Socialhistoric is all it needs to be.”
(Hersh, pp.203)
“Niagara Falls
is the outlet of Lake Ontario. It’s been there for
thousands of years. It’s popular for honeymoons.
To a travel agent, it’s an object. But from the
viewpoint of a droplet passing through, it’s a process....
Seen in the large, an object, felt in the small, a process....
Movies show vividly two opposite transformations:
A: Speeding up time
turns an object into a process.
B: Slowing down time
turns a process into an object
...In the socialculturalhistoric
domain, the continuity between object and process is blatant,
even though some institutions, beliefs, and practices
seem eternal. All institutions change. If they change
slowly, over centuries  slavery, piracy, royalty, private
property, female subjection  they are thought of as objects.
If they change daily  clothing fashions, stock market
prices, opinion polls  they are thought of as processes....
High speed and limelapse photography show that the objectprocess
polarities are ends of a continuum. Any phenomenon is
seen as an object or a process, depending on the scale
of time, the scale of distance, and human purposes....
In brief, an object is a slow process. A process
is a speedy object ....
Only in mathematics we think we have pure objects. There,
it is thought, we find nothing but
pure objects. Infinitely many of them! Could this thinking
come from seeing mathematics in too short a time scale?
Wouldn’t a view that encompassed centuries show
mathematics evolving  a process?”
(Hersh, pp.7880)
As I’ve suggested earlier in this review, Hersh’s
version of social constructivism has little or nothing
to do with those currently in favour in the academic Humanities,
since he so evidently has no time for relativistic excesses,
or jargonclotted prose. However, as a mathematician,
he undoubtedly encounters little of such foolishness amongst
his peers, so he basically ignores it here...preferring
to concentrate upon the versions of nonsense common in
his profession, which he does to devastating effect. Moreover,
these are hardly that alien to the Humanities, with both
philosophical idealism and aesthetic formalism having
v.long (and dubious) pedigrees:
“Two principal
views of the nature of mathematics are prevalent among
mathematicians  Platonism and formalism. Platonism is
dominant, but its hard to talk about in public. Formalism
feels more respectable, philosophically, but it’s
almost impossible for a working mathematician to really
believe it.... The formalist philosophy of mathematics
is often condensed into a short slogan: ‘Mathematics
is a meaningless game.’ ...Wittgenstein and some
others seem to think that, since the making of rules doesn’t
follow rules, then the rules are arbitrary. They could
just as well be any way at all. This is a gross error.
The rules of language, and of mathematics, are historically
determined by the workings of society that evolve under
pressure of the inner workings and interactions of social
groups, and the physical and biological environment of
earth. They are also simultaneously determined by the
biological properties, especially the nervous systems,
of individual humans. These biological properties and
nervous systems have permitted us to evolve and survive
on earth, so of course they reflect somehow the physical
and biological properties of this planet. Complicated,
certainly. Mysterious, no doubt. Arbitrary, no.... [Moreover,]
rulemaking tasks don’t follow rules...[but,] rules
are made for a purpose. To be played, or accepted, or
performed by people, they have to be playable or acceptable
by people. Tradition, taste, judgement, and consensus
matter. Eccentricities of individual rulemakers matter....
Some formalists in philosophy of mathematics say discovery
is lawless  has no logic  while proof or justification
is nothing but logic...[but] in real life, there are no
totally rulegoverned activities. Only more or less rulegoverned
ones, with more or less definite procedures for disputes....
Mathematics is, in part, a rulegoverned game...[but]
the notion of strictly following rules without any need
for judgement is a fiction. It has its use and interest.
It’s misleading to apply it literally to real life.”
(Hersh, pp.79)
“The mystery of
mathematics is its objectivity, its seeming certainty
or nearcertainty, and its nearindependence of persons,
cultures, and historical epochs.... Platonism says mathematical
objects are real, and independent of our knowledge. Spacefilling
curves, uncountably infinite sets, infinitedimensional
manifolds  all the members of the mathematical zoo 
are definite objects, with definite properties,
known or unknown. These objects exist outside physical
space and time. They were never created.... An inarticulate,
halfconscious Platonism is nearly universal among mathematicians
[since] research or problemsolving, even at an elementary
level, generates a naive, uncritical Platonism. In math
class class, everybody has to get the same answer. Except
for a few laggards, they do
all get the same answer. That’s what’s special
about math. There are right answers .
Not because that’s what Teacher wants us to believe.
Right because they are
right.... Yet most of this Platonism is halfhearted,
shamefaced. We don’t ask, How does this immaterial
realm...make contact with flesh and blood mathematicians.
We refuse to face this embarrassment: Ideal entities independent
of human consciousness violate the empiricism of modern
science...yet most mathematicians and philosophers of
mathematics continue to believe in an independent, immaterial
abstract world  a remnant of Plato’s heaven, attenuated,
purified, bleached, with all entities but the mathematical
expelled...like the grin on Lewis Carroll’s Cheshire
cat.”
(Hersh, pp.1112)
“We can understand
the working mathematician’s oscillation between
formalism and Platonism, if we look at her work experiences
and at the philosophical dogmas she inherited  Platonism
and formalism. Both dogmas say mathematical truth must
possess absolute certainty. Her own experience in mathematics,
on the other hand, offers plenty of uncertainty.... The
basis for Platonism is awareness that the problems and
concepts of mathematics are independent of him as an individual.
The roots of a polynomial are where they are, regardless
of what he thinks or knows. It’s easy to imagine
that this objectivity is outside human consciousness as
a whole...[however,] once mysticism is left behind, once
scientific skepticism is focused on it, Platonism is hard
to maintain.... [And] formalism needs its own act of faith.
How do we know that our latest theorem about diffusion
on manifolds is formally deducible from ZermeloFrankel
set theory? No such formal deduction will ever be written
down. If it were, [due to its length,] the likelihood
of error would be greater than in the usual informal or
semiformal mathematical proof. Now another question:
‘How come these examples were known before their
axioms were known? If a theorem is only a conclusion from
axioms, then do you say Cauchy didn’t know Cauchy’s
integral formula? Cantor didn’t know Cantor’s
theorem? Formalism doesn’t work! Back to Platonism.
We don’t quit doing mathematics, of course. Just
quit thinking about it.”
(Hersh, pp.423)
Perhaps most intriguing to me, in this book, is the way
Hersh makes very real sense of such quintessentially human
aspects of mathematics as aesthetics, intuition, and error
 situating mathematics firmly within the human world,
where it (now) seems to belong. For, once we ditch unworkable
idealism and formalism  and insist on seeing mathematics
as a human process  it becomes much less forbidding,
even if its workings (as such) still remain outside my
range of skills. Yet, thanks to Hersh & many others,
I now have some useful understanding of such skills, and
of the viewpoints which structure them.
And this is no small thing...
“There’s
an amazing consensus in mathematics as to what’s
correct or accepted. But, just as important is what’s
interesting, important, deep, or elegant. Unlike correctness,
these criteria vary from person to person, speciality
to speciality, decade to decade. They’re no more
objective than esthetic judgements in art or music. Mathematicians
want to believe in unity, universality, certainty, and
objectivity, as Americans want to believe in the Constitution
and free enterprise, or other nations in their Gracious
Queen, or their Glorious Revolution. But while they believe,
they know better. To become a professional, you must move
from front to back. You get a more sophisticated attitude
to myth. Backstage, the leading lady washes off powder
and blusher. She’s seen with her everyday face.
The frontback codependence makes it hopeless to understand
the front while ignoring the back.... You can’t
understand a restaurant meal if you’re unaware of
the kitchen. Yet you can present yourself as a philosopher
of mathematics, and be aware only of publications washed
and ironed for public consumption.”
(Hersh, p.39)
“Accounting for
intuitive ‘knowledge’ in mathematics is the
basic problem of mathematical epistemology. What do we
believe, and why do we believe it? To answer this question,
we ask another question: what do we teach and how do we
teach it? Or, what do we try to teach, and how do we find
it necessary to teach it? We try to teach mathematical
concepts, not formally (memorizing definitions) but intuitively
 by examples, problems, developing an ability to think,
which is the expression of having successfully internalized
something. What? An intuitive mathematical idea. The fundamental
intuition of the natural numbers is a shared concept,
an idea held in common after manipulating coins, bricks,
buttons, pebbles.... [This] intuition isn’t direct
perception of something external. It’s the effect
in the mind/brain of manipulating concrete objects  at
a later stage, of making marks on paper, and still later,
manipulating mental images.... [And] different people’s
representations are always being rubbed against each other,
to make sure they’re congruent.... The difficulty
in seeing what intuition is arises because of the expectation
that mathematics is infallible.”
(Hersh, pp.656)
“‘The mistakes
of a great mathematician are worth more than the correctness
of a mediocrity.’ I’ve heard those words more
than once. Explicating this thought would tell us something
about the nature of mathematics. For most academic philosophers
of mathematics, this remark has nothing to do with mathematics
or the philosophy of mathematics. Mathematics for them
is indubitable  rigorous deduction from premises. If
you make a mistake, your deduction wasn’t rigorous.
By definition, then, it wasn’t mathematics! So the
brilliant, fruitful mistakes of Newton, Euler, and Reimann,
weren’t mathematics, and needn’t be considered
by the philosopher of mathematics. Reimann’s incorrect
statement of Dirichlet’s principle was corrected,
implemented, and flowered into the calculus of variations.
On the other hand, thousands of correct theorems are published
every week. Most lead nowhere. A famous oversight of Euclid
and his students (don’t call it a mistake) was neglecting
the relation of ‘betweenness’ of points on
a line. This relation was used implicitly by Euclid in
300 B.C. It was recognized explicitly by Moritz Pasch
over 2,000 years later, in 1882. For two millennia, mathematicians
and philosophers accepted reasoning that they later rejected.
Can we be sure that we, unlike our predecessors, are not
overlooking big gaps? We can’t. Our mathematics
can’t be certain.”
(Hersh, pp.445)
“Is mathematics
created or discovered? This old chestnut has been argued
forever. The argument is a front in the eternal battle
between Platonists and antiPlatonists.... Let’s
not replow this welltrodden ground. Instead, let’s
listen impartially for ‘create’ and ‘discover’
in nonphilosophical mathematical conversation. Why do
both words  ‘create’ and ‘discover’
 seem plausible? ...When we solve [any clearly and definitely
formulated] problem, we say the solution is ‘found’
or ‘discovered.’ Not created  because the
solution was already determined by the statement of the
problem, and the known properties of the mathematical
objects on which the solution depends.... But solving
wellstated problems isn’t the only way mathematics
advances. We must also invent concepts and create theories.
Indeed, our greatest praise goes to those like Gauss,
Reimann, Euler, who created new fields of mathematics....[which
are] in part predetermined by existing knowledge, and
in part a free creation.... When several mathematicians
solve a wellstated problem, their answers are identical.
They all discover that answer. But when they create theories
to fulfil some need, their theories aren’t identical.
They create different theories.... But then, after you
invent a new theory, you must discover its properties,
by solving precisely formulated mathematical questions....
[Moreover,] you may have to invent a new trick to discover
the solution.... Is mathematics created or discovered?
Both, in a dialectical interaction and alteration. This
is not a compromise; it is a reinterpretation and synthesis.”
(Hersh, pp.735)
Reuben Hersh’s What
is Mathematics, Really? is, to my mind, the best
introduction to mathematical thinking for the rest of
us...all those who cannot think mathematically and, in
consequence, find many specialist disciplines forbidding,
to say the least. And, as I noted earlier, it also serves
to cast a very useful sidelight on a wealth of other issues
 principally methodological  from a mathematical perspective,
and in doing so demonstrates exactly why mathematics has
always been associated with rigorous thought.
Even more valuably, it makes an extremely strong argument
for seeing mathematics as a fundamentally human activity,
thus aiding those of us who would build bridges between
seemingly disparate disciplines, by demonstrating that
such rigour as mathematics enables is not inhuman, after
all...but, rather, exists on a continuum with other forms
of collective knowledge. Such an awareness is doubly crucial
today for, with the knowledgedriven proliferation of
the sciences, and the selfimposed isolation of the Humanities,
we are all too likely to forget Terence’s ancient
aspiration...to forget that “nothing human is alien
to me.”
“This book offers
a radically different, unconventional answer to [its title
question]. Repudiating Platonism and formalism, while
recognizing the reasons that make them (alternately) seem
plausible, I show that from the viewpoint of philosophy
mathematics must be understood as a human activity, a
social phenomenon, part of human culture, historically
evolved, and intelligible only in a social context. I
call this viewpoint ‘humanist.’ I use ‘humanism’
to include all philosophies that see mathematics as a
human activity, a product, and a characteristic of human
culture and society.... This book is a subversive attack
on traditional philosophies of mathematics. Its radicalism
applies to philosophy of mathematics, not mathematics
itself. Mathematics comes first, then philosophizing about
it, not the other way around. In attacking Platonism and
formalism and neoFregeanism, I’m defending our
right to do mathematics as we do.... Of course, it’s
obvious common knowledge that mathematics is a human activity,
carried out in society and developing historically. These
simple observations are usually considered irrelevant
to the philosophical question, what is mathematics? But
without the social historical context, the problems of
the philosophy of mathematics are intractable. In that
context, they are subject to reasonable description and
analysis.”
(Hersh, pp.xiii)

