Is Mathematics an art or a science? This is a deep question with which I have had many discussions with my friends and teachers.
Is Mathematics an art or a science?
Disclaim. First of all, I strongly agree with the OP that
This is a deep question.
My personal understanding is that deep questions do not have definitive answers.
Having said that, I think that mathematics has been also an art in the past : the art of reckoning. Today it is certainly a science.
Like for all sciences, we can identify a community of people (the mathematicians) which share : goals, methods, theories, languages, traditions.
Added March, 11 - about the "definition" of mathematics
I do not believe that we may have an "aristotelan" definition of science by its "essence" (like : human = rational animal) ; this simply does not work for complex human historical practices like science.
The only "definition" we may have is the "trivial one" :
mathematics is ... what mathematicians do.
The best approach - for me - is through the wittgensteinian concep of family resemblances [see Phil Inv, §§65-on] mediated trough the work of Thomas Kuhn, the American physicist, historian, and philosopher of science whose book The Structure of Scientific Revolutions (1962) gave us a deep insight into real historical development of science.
The practice of mathematics is "defined" by the common "value" of the mathematical community :
methods and problems : proof, theories
language : symbolic language
institutions : university, research centers
an evolving tradition form Euclid until today
goal : knowledge.
There are of course other communities not "devoted" to mathematics, nor to science in general, but mathematicians know perfectly well how to "recognize" a theorem from a spell or a bowling game or a religious rite.
Added March, 12 - about the "relationship" with other sciences
About the relationship of math with "empirical" sciences (mainy physics) I stay with Morris Kline, Mathematics and the Search for Knowledge (1985), Preface, page v :
How do we acquire knowledge about our physical world? All of us are obliged to rely on our sense perceptions [...]. Major phenomena of our physical world are not perceived at all by the senses. They do not tell us that the Earth is rotating on its axis and revolving around the sun. [...] our chief concern will be to describe what is known about the realities of our physical world only through the medium of mathematics. [...] I shall describe what mathematics reveals about major phenomena in our modern world. Of course, experience and experimentation play a role in our investigation of nature [...].
In the seventeenth century, Blaise Pascal bemoaned human helplessness. Yet today a tremendously powerful weapon of our own creation — namely, mathematics — has given us knowledge and mastery of major areas of our physical world. In his address in 1900 at the International Congress of Mathematicians, David Hilbert, the foremost mathematician of our era, said:
"Mathematics is the foundation of all exact knowledge of natural phenomena."
One can justifiably add that, for many vital phenomena, mathematics provides the only knowledge we have. In fact, some sciences are made up solely of a collection of mathematical theories adorned with a few physical facts.
Contrary to the impression students acquire in school Contrary to the impression students acquire in school, mathematics is not just a series of techniques. Mathematics tells us what we have never known or even suspected about notable phenomena and in some instances even contradicts perception. It is the essence of our knowledge of the physical world. It not only transcends perception but outclasses it.
I totally agree with it. My personal "connection" with the above views about the "mathematical community" is that - as some comments have said - there a lot of "communities".
(i) scientific ones a "devoted to" knowledge of the world (physical and social) ; arts are not aimed at knowledge
(ii) mathematics gives us knowledge, and this is the "essence" of science.
Geometry was encoded into axioms by Euclid over 2 millenia ago. Roughly at the same time in India Panini encoded Sanskrit grammar into a set of precise grammatical rules.
On the face of it they look alike - both a set of rules which one can use to reason about either Geometry or Grammar.
Geometry is empirical. This is generally how Geometry was understood in Babylonian mathematics. Language too is empirical - after all one does not invent a new language - it is learnt from one social habitus. But there is of course a difference: geometry relates to the physical world which is not contingent whereas the social habitus is.
In this sense Geometry and Grammar are sciences or scientia - forms of knowledge.
Is Geometry an art? It is certainly used in Architecture, it is also used in perspectival painting of the Renaissance, it was referenced by Da Vinci in his drawing of Man in the Vitruvian Man, by Francis Bacon in some of his tryptychs. By Bridgit Rily in her op-art and Durer in his drawing of Melancholia. But in all this it is not mathematics as itself but something other that is displayed - either its practical sense or as symbol for reason or the perversion of reason.
Certainly Geometry requires craftsmanship - it is art-isinal. To do it well requires a long apprenticeship. It requires imagination. But unlike art, it does not inquire into the tragic or the comedic, it disdains DaDa and the the surreal, it is orthogonal to the fantastic, the monstrous and the horrifying. It does not understand pataphysics or the mad-antics of pere Ubu. The trials and sorrows of King Lear leave it unmoved as does the mad and unrequited love of Majnun.
There is a religous dimension to Geometry. Allah is The-One. The Pythagoreans made a cult of Number and Geometry. Fragments of Euclid was discovered together in the Dead sea scrolls. The Eight-fold way. The seven days in which the world was fashioned. Nothing and zero. Being, substance and One.
But Geometry and Number is not religion, it is the law of neccessity that one sees in Geometry and Number, or that it symbolises that becomes the symbol of neccessity in the self subsisting and neccessary substance of God that Spinoza enquired after in his geometrical method.
Geometry in its essence is a science though it is also a symbol for other things. The mainstream ontology of mathematics is Platonism - it takes the entities it enquires after as real in a different realm from the ordinary physical world. That is its numbers and its spheres are out there, and one discovers them. This is why some mathematicians say that certain constructions are natural or found in Nature.