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Is Mathematics an art or a science? This is a deep question with which I have had many discussions with my friends and teachers.

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    Can you define art and science so we can know what we're dealing with? This sounds like its rife for equivocation. Also, can you explain what specifically you think philosophers can address relative to this problem? – virmaior Mar 11 '14 at 02:39
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    What definitions of art and science are you using? And are those the only possibilities? Carpentry is part art, part science ... but most would call it a craft. As they say in philosophy: Define your terms! – user4894 Mar 11 '14 at 02:57
  • Does it need to be one or the other? I definitely think it's not an Art, but I'm not entirely convinced it's a Science either. – Paul Ross Mar 11 '14 at 18:44
  • @Ross: There are experiments that are done in mathematics via computers - and its likely that this will increase in the future. – Mozibur Ullah Mar 11 '14 at 21:12
  • @MoziburUllah But that's the empirical investigation of computational behaviour, right? If we think Maths is just one branch of the study of computer science, then I'd be more than happy to call it a science, but I'm fairly confident that this isn't how mathematicians see the matter. – Paul Ross Mar 11 '14 at 21:51
  • @Ross: Its more in the sense of finding evidence for conjectures. For example the Birch-Swinnerton-Dyer conjecture. You're right though that isn't how mathematicians see it. Its a complex issue. Another line of enquiry which is still very early are theorem-provers; I suspect that they will prove important in the future in encoding mathematics formally, but will also be used as aides in creating correct proofs and formulating conjectures: The result of a calculation might not just be a number but a proof. – Mozibur Ullah Mar 11 '14 at 22:05
  • For example, one might sketch out a proof of a certain conjecture and leave ask the theorem-prover to fill in the rest. – Mozibur Ullah Mar 11 '14 at 22:10
  • @MoziburUllah Fair enough (I'm all for automated theorem proving), but we're going to have to decide a lot more about our set theory than ZFC before our recursive proof-generation procedures are going to be anywhere near the level of Analytical strength required for wider application. – Paul Ross Mar 11 '14 at 22:18
  • it is definitely both! – vzn Mar 12 '14 at 02:43
  • Isn't science concerned with knowledge about the universe and if so, isn't this mainly the question whether Mathematics was invented or discovered in another form? – Drux Mar 12 '14 at 07:15
  • You need to define 'art' and 'science' for this question to be reasonably answered here, and once you do that it will be readily clear which category math falls under, making this question rather trivial. In general, these definition questions are not a great match for this site, especially this one which will generate a load of unfocused discussion with no real answer. – stoicfury Mar 12 '14 at 21:15
  • Process is ART, result is just fact/science. – Asphir Dom Mar 11 '14 at 14:36

3 Answers3

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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".

But :

(i) scientific ones a "devoted to" knowledge of the world (physical and social) ; arts are not aimed at knowledge

<p><em>(ii)</em> mathematics gives us knowledge, and this is the "essence" of science.</p>
Mauro ALLEGRANZA
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    I don't understand your point. Satanists have a community, bowlers have a community, Unitarians have a community, the Chamber of Commerce has a community. Are all these things therefore sciences? – user4894 Mar 11 '14 at 16:58
  • @user4894 - you must see at least T.S.Kuhn and Scientific community :"The scientific community is a diverse network of interacting scientists working on particular scientific fields, and within particular institutions. Objectivity is expected to be achieved by the scientific method. Peer review, through discussion and debate within journals and conferences, assists in this objectivity by maintaining the quality of research methodology and interpretation of results." – Mauro ALLEGRANZA Mar 11 '14 at 17:07
  • You seem to be confusing A implies B with B implies A. It's true that science is/has a community; but being or having a community doesn't make something a science. – user4894 Mar 11 '14 at 18:48
  • @user4894: Having a community isn't the single and sole condition that makes it a science. One must also enquire after it goals etc. – Mozibur Ullah Mar 11 '14 at 21:08
  • @MoziburUllah - you are right; scientific communities have common goals and methods that are radically different from those of priests or magicians ... – Mauro ALLEGRANZA Mar 12 '14 at 07:46
  • I would disagree. Science is Aristotelian. Mathematics is plutonic https://philosophy.stackexchange.com/questions/8394/is-mathematics-an-art/102130#102130 – Tom Huntington Aug 24 '23 at 23:03
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Definitely an art. Science has a strict definition that I think most scientists and philosophers of science use. Unlike the bi-directional relationship between logic and mathematics, mathematics is a tool used within the scientific method which is not scientific itself.

Laoch
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    Do you really think so ? 2,5 millenia of studying Euclid's Elements, and now we discover that we all wasted our time ... – Mauro ALLEGRANZA Mar 11 '14 at 16:21
  • Could you be more explicit? I don't really understand your statement. Science according to google is the intellectual and practical activity encompassing the systematic study of the structure and behaviour of the physical and natural world through observation and experiment.

    Interestingly they define mathematics as the abstract science of number, quantity, and space, either as abstract concepts ( pure mathematics ), or as applied to other disciplines such as physics and engineering ( applied mathematics ).

    But I still don't see how it can be considering a science by definition.

     – Laoch Mar 11 '14 at 16:51
  • "According to google ..." is not my preferred way to understand philosophy and science. From Wiki Mathematics : "Gauss referred to mathematics as 'the Queen of the Sciences. In the original Latin Regina Scientiarum, as well as in German Königin der Wissenschaften, the word corresponding to science means a 'field of knowledge', and this was the original meaning of 'science' in English, also; mathematics is in this sense a field of knowledge." I personally prefer Gauss to Google ... – Mauro ALLEGRANZA Mar 11 '14 at 17:00
  • Accepting for the moment that it's not a science, why does that make it an art? – Paul Ross Mar 11 '14 at 18:43
  • So would I. You'll notice though he said Queen of science not a science. Mathematics is used in science you can't really do science without it but simply put it isn't empirical. – Laoch Mar 11 '14 at 19:05
  • @Laoch: How is Mathematics not empirical? Isn't geometry empirical? Aren't numbers an abstraction of the idea of quantity? Google is a search-engine not a who. One doesn't usually quote the index of a book, but the person who wrote the book. – Mozibur Ullah Mar 11 '14 at 19:43
  • @Loach - why do you think that science is only empirical ? How we can think about "modern" science - from Galileo and Newton to quantum mechanics - without mathematics ? – Mauro ALLEGRANZA Mar 11 '14 at 20:49
  • Again science uses mathematics that does not imply mathematics and science are congruent. Look science does something and mathematics does something but not the same thing! Are you really trying to tell me math tries to do the following?:
    1. Gather evidence in experiments and observations.
    2. Develop theories about the evidence.
    3. Look for contradicting evidence falsify a theory.

    This is not what mathematics is about. Maths is about deduction, science about observation. Now if you were to say maths and philosophy maybe subsets of each other that might be interesting.

     – Laoch Mar 11 '14 at 21:17
  • All physical scince is about observations ? What about constant speed of light (Einstein) or Heisenberg's principle ? They are very "abstract" theoretical principle assumed for building very complex (mathematically developed) theories that are "checked" against reality regarding some specific empirical consequences. The time of "baconian" science (gathering "facts" through observations, devoided of theory) as long been ended (in fact, it was seldom the driver of scientific progress ...). – Mauro ALLEGRANZA Mar 11 '14 at 21:26
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    @Laoch, although I'm sympathetic to the idea that Maths isn't empirical in the same sense that physics or biology are empirical, there is still a sense that at least in some areas of maths research, what we're doing is proposing some statements hypothetically as axioms in the spirit of experimentation and seeing what follows as theorems when we do. The theorems that follow are data, which although not "in their own right" falsifiable are properties that may be desirable or useful (or not) in a formal model. – Paul Ross Mar 11 '14 at 22:02
  • @Laoch: Mathematics isn't just about deduction. Thats whats generally emphasised. Take the Weil Conjectures: "What was really eye-catching, from the point of view of other mathematical areas, was the proposed connection with algebraic topology. Given that finite fields are discrete in nature, and topology speaks only about the continuous, the detailed formulation of Weil (based on working out some examples) was striking and novel...The analogy with topology suggested that a new homological theory be set up". Is this solely deduction? – Mozibur Ullah Mar 11 '14 at 22:24
  • @Ross - I agree with you that Math is (mainly) not empirical; my point of view is that science is not only empirical: we have a lot of "theoretical" physiscs : see Lagrange, Hilbert, ... – Mauro ALLEGRANZA Mar 12 '14 at 07:44
  • @Mauro This doesn't show why maths should therefore be called science and it's fairly clear that the scientific nature of theoretical physics has caused a lot of controversy. – Laoch Mar 12 '14 at 09:03
  • @Ross I admit I have been conflating art with the arts or humanities department in academia. – Laoch Mar 12 '14 at 09:04
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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.

Mozibur Ullah
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  • I get that this is deliberately archaic, but surely Klein's Erlangen Program (http://en.wikipedia.org/wiki/Erlangen_program) explicitly supplanted this unitarian view of the nature of geometry? – Paul Ross Mar 11 '14 at 22:09