The common perception of mathematics

I have from time to time brought up the question of the nature of mathematics in discussions with people from other disciplines. The common perception of mathematics by non-mathematicians is that mathematics is simply the language of the quantitative, a formalism developed to solve real-life problems. It follows from this view that mathematical objects have about the same ontological status as the words of normal language - at best they point to something that's real, namely, physical objects, but they are not themselves real. It is difficult to explain to non-mathematicians why this view is an inadequate description of mathematics. In the June 2000 issue of the EMS Newsletter, Jean-Pierre Bourguignon writes:

Some fallacies about mathematics. For
some people, mathematics is just the language
of the quantitative. They base their judge-
ment on the fact that mathematics enter-
tains a special relationship with language;
this opinion is shared by some of our fellow
scientists. We mathematicians know how
wrong this is, and how much effort goes
into building concepts, establishing facts,
and following avenues we once thought
plausible but turn out to be dead ends.
This widespread belief forces us to consid-
er more carefully how mathematics inter-
acts with other disciplines and what is the
exact nature of the mathematical models
that appear ever more frequently.

The challenge of explaining to non-mathematicians the perception of mathematics obtained from years of deep study of the subject is made difficult for several reasons. There seems to be a general mistrust, in academic or philosophical discussions, of the judgement of mathematicians regarding their own subject. After all, wouldn't a long and intense study of the characters of the Lord of the Rings eventually engender the view that these characters have an existence of their own?
There is a cognitive difficulty in understanding mathematics the way working mathematicians understand it. Not only does one need a tremendous effort of abstraction, it is also not enough to just read about the results of mathematics in reviews or books, because these presentations are usually mathematics after it has crystallized into formalism. Mathematics as a living development is a completely different thing (as mentioned in the above quote), and this can only be understood by actually engaging in research, or by listening to what mathematics have to say about their work. Here is Bourguignon again:

A special link with truth: Confronted with
a mathematical statement, a mathemati-
cian's goal is to prove that it is "true". What
does this mean? Of course, now that math-
ematicians agree to work in the context of
a potentially completely formalised theory,
ts meaning can be none other than 'the
statement can be deduced from the basic
axioms agreed upon'. In fact, if we are to
address this question in the context of the
relations of mathematics to society, we are
forced to take it in a broader, more philo-
sophical, perspective because we have to
confront it with reality. This amounts to
deciding whether there is a mathematical
reality of which this statement is a part,
and of the status of this reality in relation
with ordinary 'sensible' reality. On this
broader issue, mathematicians have differ-
ent views: from the 'Platonists' at one end
of the spectrum who believe that they 'dis-
cover' new territories and new facts of the
mathematical world, to the 'intuitionists' at
the other end who view mathematical con-
structions as purely human scaffoldings
based on consensus opinions among a
restricted community - in other words,
that mathematics is 'invented'. I have no
doubt that the vast majority of mathemati-
cians are closer to the Platonist viewpoint
than to the other one, or at least that they
spend a good portion of their professional
time behaving as if they were, as André
Weil once put it.

One of the symptoms of non-mathematicians perception of mathematics is the view that mathematics is not a science. If maths is a formal language that deals with logical relations between fictional entities or mental constructions, then how can it be a science? Many scientists view science as an empirical activity, first and foremost. For most working mathematicians this is an unfamiliar view which does not resonate with their everyday experience of mathematics as an activity of uncovering synthetic and deeply meaningful truths. In his conclusion Bourguignon writes:

From all the above challenges concerning
the image of mathematics, the most serious
seems to me the need to make it evident that
mathematics is a science and alive. The rest
should follow.