Is Mathematics considered a science?

Question

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?

Answer 1

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.

Answer 2

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.

Answer 3

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.

Answer 4

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).

Answer 5

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.

https://www.wikiwand.com/en/Sieve_of_Eratosthenes

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.

Answer 6

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.

Answer 7

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.

Answer 8

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.

Answer 9

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.

Answer 10

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.

Answer 11

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.