Science, generally is analyzing information gathered from observing phenomena, and coming up with theories to try and explain the phenomena. Then, attempting to predict a new phenomenon before it happens (when we can do that we usually say that we have discovered "a fundamental law of nature"),and when we can consistently produce the same result, this is regarded as proof of the theory.

Mathematics is different. it does not rely on these experiments in order to claim the discovery of a new truth. Theres a distinction between what Mathematics claims as proof in contrast to science. For a scientist, ten experiments with consistent results might constitute proof, For a mathematician, a million successful experiments is not enough proof. Instead, mathematicians rely on logic. Mathematics is very often inspired by nature, but it is a purely intellectual pursuit. It is just a bunch of ideas in our heads, like philosophy. Pure abstract reasoning.

Mathematics is so intricately related with science, Mathematics being the language used to describe scientific theories, but the difference between methods for arriving at proof appear to make the notion of mathematics as a science inconsistent. Is Mathematics considered a science?

  • I have the feeling that this question has been answered before...
    – iphigenie
    Commented Jul 24, 2014 at 20:28
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    Well, the question asked in the linked answer was whether Maths was an Art or a Science. I personally found that question and the approaches that answers to it took to be rather unsatisfactory in their disconnect with existing literature; a more profitable approach might be to consider the problem of Demarcation in the philosophy of science; see, for instance, plato.stanford.edu/entries/pseudo-science . So please don't close this just because a similar question was closed for being unclear!
    – Paul Ross
    Commented Jul 25, 2014 at 7:48
  • 1
    Also, stoic, part of the purpose of philosophy is trying to establish effective concepts of "math" and "science". It seems silly to expect people seeking answers this question to already come with predefined starting positions. Otherwise, why ask?
    – Paul Ross
    Commented Jul 25, 2014 at 8:01
  • 4
    – CriglCragl
    Commented Feb 23, 2018 at 12:06
  • 1
    Mathematics is NOT a Pure science as mathematics does not require one to perform experiments — nor is mathematics experiential. All sciences requires experiments. So while mathematics does have an empirical side to it it also has components of other things one would liken to the ARTS. So mathematics must be a hybrid subject and not a pure this or that type of topic. It can be more than one thing as an art and a science simultaneously.
    – Logikal
    Commented Aug 12, 2019 at 3:24

11 Answers 11


There seems to be a lot of disagreement concerning this question. Personally, I would argue that although mathematics has many scientific applications, it is not, itself, a science.

Science, as Mauro pointed out in their answer, is devoted to knowledge of the physical world. However, our senses are the only means by which information about the physical world can reach us. Everything else--theories, generalizations, models, formulas, etc.--is just approximation or imaginary, and is only useful or valuable as science inasmuch as it allows us to predict what our senses detect. The essence of science is not theories or models, but observation, experimentation, and prediction, all three of which depend on the senses.

Mathematics, by contrast, is sense-agnostic. It is possible to imagine an intelligent entity, locked in a box with no sensory stimulation whatsoever, happily devising axioms and proving theorems to its heart's content. Given appropriate axioms, such an entity might be perfectly capable of conceiving of concepts such as geometry, numbers, sets, and so on, and could easily reason about them. However, it would be incapable of using those concepts to prove anything about the nature of the box it's kept in, or the world (if any) outside of that box.

Except by analogy, mathematics has no bearing on the physical world and cannot be affected by it. Theories in mathematics, as you've observed, cannot be proven or refuted by experimentation alone, whereas in science that is the only way to do it. Similarly, "facts" in mathematics (axioms) are subjective, imaginary, and subject to change at a whim; whereas facts in science are objective, observed, and can only be supported or contradicted with other facts, never replaced.

I believe that mathematics is an art form, not a science. One of the most famous arguments for this view is the mathematician Paul Lockhart's essay A Mathematician's Lament (which I highly recommend). Specifically, he argues that mathematics is the art of ideas. Ideas are incredibly useful things, and as such mathematics is a powerful tool that has many useful applications. However, I do not think that mathematics itself should be considered a science.

  • 4
    Is that true? If you were born in the box and had no access to the physical world from birth, would you be able to conceive of concepts such as geometry (in particular)? You take it for granted, but that is far from obvious :-) I do agree with the following paragraph though that mathematics says nothing about nature, and I want it to be that way (so that mathematics can do whatever it wants, not just study natural phenomena).
    – Frank
    Commented Apr 5, 2017 at 0:22
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    Mathematics seems to be a language (or many, interrelated), and the Private Language Argument is exactly set at abolishing such Cartesian 'Robinson Crusoeism' as this entity on a box thought experiment. The views on science here don't take in to account views like Popper's that explanations are not necessarily empirical or falsifiable, but rely on other factors like explanatory power. 'Pure' mathematics has again and again been found to describe a field of possibilities in which our world is one specific subset, exactly having a bearing on understanding the world.
    – CriglCragl
    Commented Feb 23, 2018 at 12:27

We have already discussed it here.

My personal belief about the relationship between mathematics and empirical sciences (mainly physics) is expressed in : 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, 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.

The activity of scientists is "devoted to" knowledge of the world (physical and social).

Mathematics gives us knowledge, and this is the "essence" of science.

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    I think there is a body of literature in the debate about Science and Pseudoscience that would disagree with the claim that the production of knowledge is sufficient to constitute science. That would suggest there is no such thing as non-scientific knowledge, which sounds like a difficult claim to support.
    – Paul Ross
    Commented Jul 25, 2014 at 7:56
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    Or, you can reserve the word knowledge for things scientific, and maybe have another word for the rest (extremists could use the word doxa) ;-)
    – Frank
    Commented Apr 2, 2017 at 23:58
  • I don't think mathematicians concern, or should concern, themselves with the natural world. For one thing, it would be boring and limiting. Then, looking at the physical world is the domain of the physicist. Mathematicians normally do not conduct experiments. They are happy with proofs on paper. Mathematics in itself says nothing about nature, and is therefore not a science like physics at all. It happens that the natural science quantify their domains which means mathematics is convenient. That's all.
    – Frank
    Commented Apr 5, 2017 at 0:19

Boethius, following Aristotle, said the "Speculative sciences may be divided into three kinds: physics, mathematics, and metaphysics.":

  1. Physics deals with that which is in motion and is material.
  2. Mathematics deals with that which is material and is not in motion [∵ mathematical objects do not move or change, but they are abstracted from physical objects, which do move or change]
  3. Metascience deals with that which is not in motion nor is material.

(cf. §II of his De Trinitate)

These are known as the Three Degrees of Abstraction.

  • 1
    This is very old, and questionable nowadays, especially the part where mathematics is abstracted from physical objects. This is not needed anymore. You can pick any set of axioms you wish and play the game of mathematics to derive theorems from these axioms using logic. This is very liberating for mathematics, which is now free to deduce without any reference to anything in the physical world.
    – Frank
    Commented Apr 2, 2017 at 23:56
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    @Frank No classification is questionable/unquestionable, because classifications are neither true nor false. It seems you object most to Boethius's definition of mathematics as a speculative science that "deals with that which is material and is not in motion". Certainly today, the definition of mathematics has become very broad, to the point of even being indistinguishable from metaphysics/metascience (e.g., C. S. Peirce thought this when he said in CP 3.428: "Mathematics is the most abstract of all the sciences.").
    – Geremia
    Commented Apr 3, 2017 at 16:27
  • @Frank cf. Peirce's classification of the sciences
    – Geremia
    Commented Apr 3, 2017 at 16:28
  • @Frank C. S. Peirce (CP 1.53): "The most abstract of all the sciences is mathematics. That this is so, has been made manifest in our day; because all mathematicians now see clearly that mathematics is only busied about purely hypothetical questions. As for what the truth of existence may be the mathematician does not (qua mathematician) care a straw. It is true that early mathematicians could not clearly see that this was so. But for all their not seeing it, it was just as true of the mathematics of early days as of our own."
    – Geremia
    Commented Apr 3, 2017 at 16:32
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    Geremia - yes, Kant is around the corner, but I have issues with synthetic a priori.
    – Frank
    Commented Apr 4, 2017 at 21:18

I think it's appropriate to say that mathematics is science in that it yields "reliable" knowledge; however, the knowledge that it yields, as well as the mechanism by which said knowledge is verified, are very different from those of natural sciences, so you cannot apply the same criteria and terms to both (e.g. terms like "proof", "experiment", "evidence").

I don't think it's sufficient to call mathematics an "art form", because mathematics yields (conditional on certain axioms) various objective "facts" (to put it another way, mathematics discovers various useful objective tautologies) -- and in fact this can be said to be its purpose. On the other hand, you'd be very hard-pressed to reconcile "the pursuit of objective facts" with any reasonable definition of an art.

Now, where the "art form" label makes sense is in describing the process by which a mathematician discovers mathematical facts. It's fair to say that this is an art form, because there is no (known) mechanical way in which you can obtain "interesting"/"useful" mathematical facts. It's an undoubtedly creative process, though perhaps more in a sense of putting together incredible puzzles or finding the way through an infinite maze.

However, in the same way, I think any scientific process can be called an "art form" because there is also no mechanical way to formulate great theories or design earth-shattering experiments.

Science earns the "science" label when the creative, "artistic" process of discovery yields reliable (and in this sense objective) facts/laws about certain entities, be they natural (Newton's 2nd law) or mathematical/formal (Euler's identity).

  • 2
    Saying something is science because it produces reliable knowledge is arguing backwards. The reliableness of the (scientific) knowledge is determined by science. There are many other kinds of knowledge than scientific, surely you would have to agree?
    – CriglCragl
    Commented Feb 23, 2018 at 13:25
  • No, IMO, unless you agree with Kuhn, etc, reliability/truthfulness of facts about observations cannot be conditional on the internal criteria of a practice that calls itself "scientific". If you consider "F = m*a" (at least for a certain class events, and specific ways of measuring F, m, a), this is not a fact because "Newton the scientist told us so" -- rather, it is essentially an objectively verifiable, reliable fact about the observable world, which, along with other such facts, makes it reasonable to say that "Newton was a scientist (because he discovered objective facts like this)".
    – Alex Sotka
    Commented Feb 23, 2018 at 23:18
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    But Newton was wrong, at relativistic speeds. Science is always open to new observations, objectively verifiable is an active process rewuiring constant updating & refinement, not Revealed Truth. That's what it means to rest your knowledge base on expetiment.
    – CriglCragl
    Commented Mar 1, 2018 at 11:32
  • I am tying the concept of "scientific truth" to a stmt/law/theory that makes verifiably/objectively accurate predictions, nothing else (also saying nothing about the metaphysics of the terms used in the stmt/theory). I don't know what you mean by "revealed truth", but it's clearly smth else... Newton's laws are still valid because they give exceptionally accurate predictions ("truth" in this specific sense) for a huge class of events... and that's exactly why it's perfectly fine to call them "science" -- and why they're still taught in school, btw.
    – Alex Sotka
    Commented Mar 5, 2018 at 4:05
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    You don't have a good definition of truth, or understand science.
    – CriglCragl
    Commented Mar 5, 2018 at 10:03

This is actually an extremely interesting question because when the word, 'science', comes to mind it's usually chem, bio, physics, EE that pops up, not usually math. I think math for many is seen as a tool that aids all of those natural sciences.

In "The Man Who Loved Only Numbers: The Story of Paul Erdős", a biography of the only man who nearly published more mathematical papers than Pythagoras himself, they actually go into the "sciences" of mathematics.

Most mathematicians believe that deep practitioners of mathematics are actually closer to the heart/soul/essence of the world than physicists and that a good amount of physicists will actually agree.

One of the reasons they gave was about straight lines. Straight lines don't exist in nature, They don't exist on paper, and they don't exist as a circle either, which was defined as an infinite amount of straight lines at an infinite angle or something.

One of the things that makes science, science is the scientific method of hypothesis, testing, theory etc... and mathematics actually has that. Since the days of Pythagoras people believed that through formulas they could come up with every single prime number on the number line. Then several hundred years later, someone did some (aka lots and lots) and found that not to be the case and proved that theory to be wrong.


Here's another one of the the theories/formulations from Ancient Greek which was also proved to be faulty. So if what makes science, science the scientific method, mathematics too has that.

When it comes to pushing boundaries of what IS and IS NOT I feel as though that Mathematics has exponentially less wiggle room for error than any other science. In math everything is precise or it's not and if it's not then it is wrong or at the very least not true.

FUN FACTS: Pythagoras, an obsessor over triangles, believed that the child, the mother, and the father formed a triangle. In the Republic, by Plato, Plato delivers an argument that if that was true, then it would be possible to mathematically come up with the perfect time to have a child. The further you were away from that time, the worse your child would be.

Either way, mathematics is also just another philosophy and in my opinion all science is philosophy.

  • 1
    Also Bertrand Russell and Alfred North Whitehead produced 3 volumes of Principia Mathematica which was seen as one of the most important contributions to mathematics which includes the proof for 1 + 1 = 2 over the course of 80 dense pages. wikiwand.com/en/Principia_Mathematica Which honestly is worth just a quick browse just to scroll down and view the crazy documentation system they used called, "dot notation" and in response to that, which furthers the idea that mathematics does belong as a disciplinary science are all the theories and ideas posted alongside everything els Commented Feb 23, 2018 at 3:14
  • Pythagoras got pi wrong, and denied the possibility of irrational numbers generally existentialcomics.com/comic/189
    – CriglCragl
    Commented Feb 23, 2018 at 13:36
  • Ooooo nice one, I didn't actually know that. I know that he pushed very strongly towards the idea of perfect numbers which are: " a positive integer that is equal to the sum of its proper divisors. The smallest perfect number is 6, which is the sum of 1, 2, and 3. Other perfect numbers are 28, 496, and 8,128." Honestly, humans are pretty weird LOL Commented Feb 23, 2018 at 15:04

I'm actually happy this question popped up in the feed because I'm amazed no one has pointed out the historical changes of the term "science".

Unfortunately I don't have the appropriate amount of time to write a full piece of answer, but I'll attempt to draw some general lines.

If you consider "science", or moreover the scientific community, as dynamic, changeable domain you can see that throughout history the intellectual community has used the term in different ways, which can be claimed to be generally narrowed down to our point in history.

  1. As Geremia points in their answer, Aristotle took science to mean Physics, Mathematics and Metaphysics. This seems like the widest articulation of the scientific domain.

  2. A big leap in history, but the most obvious next "chapter" in the change of the scientific domain's breadth is in the German Idealism time (18-19th centuries), where Kant, Schelling, Hegel and so on treats Physics and Metaphysics and scientific systems, but Mathematics is obvious for them not a science but rather a foundational layer for science.

  3. And nowadays (more particularly since the late 19th century), as most answers here points out, our intellectual community has ruled out Metaphysics as science (the most obvious example is Russell), and Mathematics are formally primarily not treated as a science, albeit some minor objections (which might have been existent throughout history, simply not vocal enough for us to notice).

As to why the change has been made, and what the consequences of those changes have been to the scientific and philosophical communities I unfortunately don't have the required knowledge to provide the objectively correct answer, and unfortunately don't have enough time to suggest my own thoughts, but I will suggest generally that in my opinion the changes were made because of the way we have changed our approach to science and the practical/theoretical distinction (most apparent in early Wittgenstein and Russell); where the changes are probably closely related to our different alternations of the "scientific method".

Again, sorry for zero citation, when I'll have the time I'll put some references.

  • 1
    Nice answer. The fact that we consider X as good science and Y as not doesn't imply that people in the past agreed. Including people we call notable scientists. Newton calculated the age of the earth to be 6000 years. From bible generations. Planck was ashamed of being responsible for quantum theory. So we must beware of presentism : current fashions = truth
    – Rushi
    Commented Aug 9, 2019 at 17:30

One may want to distinguish between pure and applied maths.

Applied maths is a toolkit of techniques and facts for the empirical and social sciences. So it's not a science based on the above criteria, but without these tools there would be very little scientific progress.

Pure mathematics is pursued for its natural and artistic beauty and it's internal logical structure. So it's also not a science based on the above definition. The goal is to further develop and 'flesh out' this logically coherent yet artistically beautiful structure in 'mathematically desirable' and 'interesting' ways, the meaning of which is difficult to convey. (It's not about 'blindly' making up axioms and definitions and logically deducing results or proving theorems. The art is to know what is mathematically meaningful and desirable.)

I would also mention that experimentation has no - zero, zilch - role at all in a correct mathematical proof.

Pure math is also more than an art form - it's a whole lifestyle, culture and community.


Benjamin Peirce defined mathematics as “the science which draws necessary conclusions.” (CP 4.229).

C. S. Peirce's definition (MS [R] 14:4):

Mathematics is the study of the substance of hypotheses with a view to the tracing of necessary conclusions from them.


Some people put mathematics in the "formal sciences" category. However, I'm not sure on the validity of such a category. After all, the methodological differences between mathematics and natural sciences and social sciences are giant. Scientists test their hypotheses and theories based on empirical observations of the natural world. Mathemathicians do not need of empirical obervations.

By the way, I would like to disagree with Frank, who said:
"Or, you can reserve the word knowledge for things scientific, and maybe have another word for the rest (extremists could use the word doxa)".

Frank, are you suggesting that you can't have cooking knowledge, tennis knowledge, knowledge of Mandarin Chinese, history knowledge, knowledge of chess or literature knowledge?

Is it the same a person who knows how to cook than a person who doesn't know it? The first person has acquired some non-scientific knowledge (cooking knowledge) that the second person lacks.

Is it the same a tennis coach than a person who has no idea of tennis? The first person has a non-scientific knowledge (he knows how to play tennis and teachs it) that the second person lacks.

Is it the same a person who talks and writes in Mandarin Chinese than a person who doesn't? The first person has a non-scientific knowledge (Mandarin language) that the second person lacks.

Is it the same a person who has a deep knowledge of history than a person who doesn't? The person who knows a lot about history will have a non-scientific knowledge that the second person will lack.

Is it the same a person who doesn't know how to move the pieces of chess than the current world #1 Magnus Carlsen?

In sum, the idea that there is only scientific knowledge, and no knowledge in other areas of life, is completely wrong.

  • 1
    I you have feedback for another poster, please post them as comments to that post, not as part of your own separate answer. Your answer should be as self-contained as possible - or at least a constructive follow up on another answer. Also please refrain from using derogatory and insulting language in your answers. Commented Oct 24, 2017 at 18:31

What seems to be missing in the replies to your question is the acknowledgement that a mathematical structure (a very complicated one, presumably) can be devised so that it encodes all the observed features of the natural world. There is simply no bound on what can be conceived: the phenomenon of concern may be immeasurable (Hilbert spaces), chaotic (water wave theory), unobservable (random dictionaries for ill-posed problems), meta-mathematical (reverse mathematics) or so complicated as to require more energy to analyze, compute, process, or store than is possible in any physically meaningful way. In this sense, mathematical inquiry somehow seems superior to scientific inquiry.

Oddly, all that the sentient can hope to do, subject to all that the natural world could ever do, is nothing more than a special case than that which can be mathematically conceived. In general, therefore, mathematics is a distraction from the practice of science in much the same way that thinking is a distraction from doing. Our individual and collective consciousness seems to require both in order to flourish.


Philosophy is written in that great book which ever lies before our eyes, I mean the universe, but we cannot understand it if we do not first learn the language and grasp the symbols , in which it is written. This book is written in the mathematical language. (Galileo Galilei).

Is this universal language discovered or invented (or both)? Answering this question also answers your question. I would suggest "Our Mathematical Universe", by Max Tegmark.

So I would carefully lean towards "discovered". Gauss called mathematics "the queen of the sciences" for a reason.

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