Thanks to Karl Popper, there is a clearcut distinction between mathematics and science.

Science depends on hypotheses that are falsifiable. For example, the hypothesis that a heavier object falls to the ground faster than any lighter object. Galileo devised experiments that proved that the hypothesis about falling objects was false.

Mathematics depends on theorems that are provable. Experiments do come into picture in proving a theorem. Mathematics also lives on conjectures. A conjecture is falsified by giving a proof that the conjecture can be contradicted.

Without any doubt Mathematics is a science and not a symbolic prose.

Mathematics is a science because first mathematics is defined as a science (or group of related sciences) dealing with the logic of quantity and shape and arrangement and second because science is defined as a branch of knowledge or study dealing with a body of facts or truths systematically arranged and showing the operation of general laws; or a systematic knowledge of the physical or material world gained through observation and experimentation, or any of the branches of natural or physical science; or a systematized knowledge in general; or knowledge, as of facts or principles; knowledge gained by systematic study; or a particular branch of knowledge; or any skill or technique that reflects a precise application of facts or principles.

Thanks to Karl Popper, there is a clearcut distinction between mathematics and science.

Science depends on hypotheses that are falsifiable. For example, the hypothesis that a heavier object falls to the ground faster than any lighter object. Galileo devised experiments that proved that the hypothesis about falling objects was false.

Mathematics depends on theorems that are provable. Experiments do come into picture in proving a theorem. Mathematics also lives on conjectures. A conjecture is falsified by giving a proof that the conjecture can be contradicted.

For the second part of the question, there is a clearcut distinction between poetry and mathematics.

A poem expresses some perception about the world or about something that someone has written of spoken. Poems typically are written in mellifluous language without symbols. A test of a good poem is whether or not the poem can be sung. As far as I know, theorems in mathematics have never been set to music.

Results in mathematics are not really 'discovered'. That is why I feel, mathematics is perhaps different from science. Then again, I feel, if one can catch the rhythm, mathematics can be poetry of a high class. But yes, of course mathematical results can not be sung like poetry!

I think mathematics is both an art and a science. It is in art as the mathematician can engineer new mathematical objects the same way as engineer builds a bridge or an artist creates a masterpiece. Then he can use scientific methods to discover the properties of these objects and relations to the 'real' world.

I disagree that the mathematical results are not really 'discovered'. In my opinion the properties of mathematical objects are usually discovered by careful observations and experiments. So before starting to prove something the mathematician usually has a pretty good idea what he wants to achieve. So the only significant difference with other sciences is that objects under consideration are artificially constructed mathematical abstractions.

On another topic it was asked if economics is a science. I copy here my comment as it applies well here too.

Interesting question, and interesting comments on it. What is "Science" ? Basically, it is human constructed knowledge which never tells humans "truth" but consistent hypothesis and testing about what truth seems to be. I was an astronomy (which is essentially physics) passionate before becoming an economist, and I do not see any difference in two fields, from this point of view.

In my opinion, what is really peculiar to so-called "social" sciences is that the observer and the observed are the same. Here we all fall in the Goëdel case of a science which is structurally incomplete. Though, physics is incomplete too, as incompleteness apply to all mathematical system which is complex enough.

Thus, from a philosophical perspective, i would say there is no "pure" science in any field, just the neverending fight against our neverlasting infinite ignorance.

We could say that Maths is a meta-science, ie a procedure that can generate, classify, develop, implement ... and many other verbs, every scientific discipline.

@Sergey Salishev:...I disagree that the mathematical results are not really 'discovered'. In my opinion the properties of mathematical objects are usually discovered by careful observations and experiments. So before starting to prove something the mathematician usually has a pretty good idea what he wants to achieve. So the only significant difference with other sciences is that objects under consideration are artificially constructed mathematical abstractions.

The notion of discovery by mathematicians deserves a closer look. For a Platonist, mathematics pre-exists in nature. Then the discovery process entails finding out what is already in nature. Perhaps you will agree that this does not square with the discovery process that you described.

Also, for a Platonis, ideas are absolute forms. The way to discover an absolute form such as the category SET (objects of the category are sets and the morphsims are mapping between the sets). The empty set is an initial object in SET. The empty set is also an example of an absolute form for a Platonist. Another example of an absolute form is the product (catesian product of sets) in SET. And SET is a prototype of a concrete category, provide a particular concrete category resembles SET.

So, based on what you have written, a question to ask is this: Are you discovering concrete categories during the discovery process that typifies the work of a mathematician? In other words, do mathematicians become aware of absolute forms such as the category SET and then, if there is interest in a particular category, start looking via experiments (in nature) for instances of concrete categories.

For more about the category SET, see the attached pdf file.

Dear James,

In another link, once when I used the term 'discovery' with reference to mathematics, there were objections raised. Finally, I started to accept that perhaps I was wrong!

Now I can see that you too have the same idea that mathematical results are indeed discoveries. Thank you.

Dear Hemanta,

I am in favour of a mixed (hybrid) view of mathematics. It is fairly obvious that lots of what we know about mathematics from well-known mathematicians stems from a combination of discoveries from our experience (our walking about in the natural word) and our seeing the world through the "eye glasses" supplied by set theory, geometry, infinitesimal geometry (Efremovic's name for proximity spaces), category theory, and on and on.

But then the nagging question is this. Was Plato correct when he postulated the existence of perfect forms (ideas such as beauty, even SET) that objects in the world imperfectly resemble? Plato's idea was that we should look and study more and more perfect forms of something that represents an ideal. If we following Plato's model for discovering ideal forms, then we will work our way toward a knowledge of a perfect form by cultivating the habit of finding and studying objets that more and more come close to a perfect form.

Just look at the difference between the general idea of the category SET (its objects are sets and its morphisms are mappings between sets) and concrete categories that resemble SET. The well-known distinction between a category such as SET and corresponding concrete categories is reminiscent of Plato's idea of absolute forms and concrete objects that resemble the absolute forms.

See the attached paper, for more about the category SET.

Dear James,

I shall go through the article. Thank you for attaching it. I hope, everyone following this question will find it very meaningful.

Dear Professor Khan,

Indeed, you have pointed out the actual reason why I have posted this question. In India, the Universities offer both M.A. and M. Sc. degrees in Mathematics. Those who study Mathematics at the undergraduate level with subjects such as Political Science are awarded the degree of Master of Arts, while those who study Mathematics at the undergraduate level with subjects such as Physics are awarded the degree of Master of Science! Two such students might actually study together for two different degrees! In fact, this was the root reason why I wanted to discuss this point.

@James Peters:...For a Platonist, mathematics pre-exists in nature. Then the discovery process entails finding out what is already in nature. Perhaps you will agree that this does not square with the discovery process that you described.

In my opinion these two points of view do not contradict each other. First of all it is not that clear that 'ideal object' is. For me 'ideal object' is a solution of some optimization problem. So different though closely related problems can yield different 'ideal' solutions. E.g. for a continuous system we get one optimal regulator and if we consider the same system in discrete time the continuous regulator may become unstable or suboptimal. And even if the exact implementation of the 'ideal' solution is possible mathematics usually provide multiple equivalent formalizations of it.

Then the Platonic theory was presented in school the example used was Table, we know what table is, we can imagine the idea of table, but before the first table was invented numerous generations of people lived without it. And for some people the 'ideal table' is not actually a table but a 'dastarkhan'.

For me a Platonic universe of ideas is like a dark cluttered attic. Is it not a discovery when you occasionally find your grandmother's jewelry here?

Sergey,

We agree! It is definitely not clear what an ideal object is. Plato's notion of idea objects is, perhaps, a characterisation of something that is never found in mathematics.

What Plato does suggest that is interesting for us is this. The discovery process in mathematics is in stages. Typically, the discovery process starts with a preliminary result. And then later the preliminary results is extended, enriched and improved to obtain a new result. Apparently, there is end to the discovery process. It just keeps getting better and better.

Dear Hemanta,

I am really honoured that you invited me last day to join this lively discussion. Perhaps I cannot agree more with Professor Peters. He is really an expert in this matter. However, since you have asked me to contribute my thoughts, so some of my incoherent thoughts are as in the following:

PROPOSITIONAL KNOWLEDGE (science) is true knowledge different from doxa (common belief and opinions, not known to be true). Scientific Knowledge would not be possible without belief justification (or something very much like it). Since, true knowledge is science, for that matter alone, mathematics to be designated as a science it should be true knowledge or knowledge justified. Propositional knowledge can be justified by philosophical analysis, on the basis of a priori knowledge, perception, by empirical method as well as realisation of results by its application. Since, from Plato till now, every mathematical school of thought, classical, non-Euclidean, naturalist intuitionalist (constructionalist) all justify mathematics as propositional knowledge (science) in one or other way.All the more to Its credit it also has universalism as its characteristics, an acid test of science

For your analogy of classical mathematical results “look more like poetry in symbolic prose than mathematics”. Consider the question that why do mathematicians continue doing Mathematics, in spite of a certain failure to secure its very foundations or to remove contradictions? On the prima facie of multiplicity of schools of mathematical thought, it is but natural that contradictions cropped in and foundations changed (something like paradigm shift). One may think of transition from Euclidean to non-Euclidean mathematics or foundation of mathematics laid down by intuitionalists in contrast to all other schools of thought. It is not that there is not a common ground of the mathematics (mathematical objects), it has always been there. But, too many minds have gone in different directions. I do not think a plurality of cosmologies in physics consistent in their disciplinary context (for that they have survived) are as poetry in physics terms.

Confusion whether mathematics is science or not arises from the fact that it relies on both logic and creativity (pure imagination of mathematical objects on the basis of axioms as pure mental images of a poet, painters or fiction writer or to some extent in history based on scanty evidence around which a historian weaves history from his/her perspective or ideology). It is why in almost all universities of India the Department of Mathematics come under both faculties of arts (humanities, like philosophy) and science.

Dear MFK,

Thank you for your analysis.

Martin, Vitaly: I very much enjoy reading "mathematics definitely is a science more than the other 'sciences', despite the opinions of philosophers", and I mostly agree with you except for the possible literal understandings of the ranking part ("math ... is more than other[s]...).

I tend to find young Popper's refutationist doctrine of science both a great advancement - when contrasted to verificationism - and a very poor demarcation principle; in contrast, Dr. Mohammad Khan's understanding above that science "can be justified by philosophical analysis, on the basis of a priori knowledge, perception, by empirical method as well as realisation of results by its application" certainly does not risk of running into metaphysics (one of K.R.P. major preocupations which tended to sadly prescribe, in my view, the study of most domains of knowledge to some form of crude, if piecemeal, engineering), and, to me, seems quite promising. I think Dr. Khan outlines quite a broad program which does not inhibit the systematic study of some subjects (I really liked that in your post, Martin!) which is not the same as attributing the status of science to the subjects themselves (it's like someone - perhaps Robert Merton - ~ said: what interests us is not whether witches exist or not, it is that there are people who believe they exist). Beyond a crude instrumentalist view of math - to which I may tend to very sadly and crudely subscribe, for practical reasons - I think there might be quite a promising role for math if we admit such a program; I just do not know how we could convene and start it as the very definition of math is part of such a huge fragmentation of knowledge in the scientific and philosophic fields.

Sergio,

Good post. Your observation is appropriate for experimental mathematics but not for pure mathematics. From the book that you cited, experimental mathematics is defined to be the methodology of doing mathematics that induces the use of computations for

1. Gaining insight and intuition.

2. Discovering new patterns and relationships.

3. Using graphical displays to suggest underlying mathematical principles.

4. Testing and especially falsifying conjectures.

5. Exploring a possible result to see if it is worth formal proof.

6. Suggesting approaches for formal proof.

7. Replacing lengthy hand derivations with computer-based derivations.

8. Confirming analyically derived results.

Jonathan Borwein and David Bailey are not talking about non-experimetal or traditional pure mathematics such as that found in Hausdorff's Set Theory or Bourbaki or even Euclid.

Sergio,

Another good post! However, notice that you are using an ad hominem argument in citing in writing "according to Borwein and Bailey, Gauss was an experimental ("impure" ?) mathematician". In fact, it is possible for a mathematician to do pure as well as applied mathematics. The term "experimental mathematics" comes from Borwein and Bailey, which looks like the introduction of a new type of mathematics that appears to have had its connection with traditional pure mathematics truncated. But then because Borwein and Bailey wrote about this, does not make it so.

@Sergio,

Very good post! I will try to find a copy of Lindley's book. For whatever reason, Borwein and Bailey do stretch things a bit in their characterisation of mathematics. There are countless beautiful theorems (Euclid's prime number theorem among them) that have proofs that are independent of experimentation. Take a look at Smirnov's 1952 paper on proximity spaces to see what I mean. One of Smirnov's great contributions was his finding and proving theorems based on either Efremovic's axioms or variations of Efremovic's axioms (introduced by Smirnov in his 1952 paper (in Russian), later translated into English in 1964 by the American Mathematical Society.