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Inspired by the question Does the term “science” encompass humanities and the social sciences?, I would like to ask a question that has been in my mind for decades.

Does "Science" include Mathematics? Or, put in another way, is Mathematics part of Science?

What I understand about Math are: Math is not humanities. Math is not natural science. Math is not social science. If Math is part of science, what kind of science is it?

When people say STEM, do they mean Math is separate from Science? Or they just say it for the sake of convenience?

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    @DanielR.Collins Sure, but, for instance, the queen of Australia isn't Australian. – Federico Poloni Jun 19 '16 at 17:08
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    @DanielR.Collins interestingly, Elizabeth II is queen of 16 countries but is apparently not legally a citizen of any of them (see here). So being queen of the sciences is perhaps not proof of being a science... – Dan Romik Jun 19 '16 at 17:08
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    @FedericoPoloni beat me to it, but actually she is Australian (as the monarch she is "the embodiment of Australia" or some such nonsense), she is just not an Australian citizen. The legal and philosophical discussions on this question remind me of the question about how many angels can dance on the head of a pin. They are about equally logical and clear, and concern an equally useless question. Same for the current question about math and science, btw. – Dan Romik Jun 19 '16 at 17:12
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    @DanielR.Collins: Given that Gauß was German and I can find German versions of that quote (implying that the English version may be a translation), you should take into account that the German term Wissenschaft commonly encompasses pretty much everything that can be researched, not just "natural sciences". In fact, that difference in meaning between Wissenschaft and science was what the original question this question was motivated by was about. – O. R. Mapper Jun 19 '16 at 19:28
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    @DanRomik I think questions aren't considered duplicates just becase they have been asked on other SE sites. – Kimball Jun 19 '16 at 22:19
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This is a good question, but there is no consensus as to a good answer. Some people think mathematics is obviously a science, some people think it obviously isn't, and some just aren't sure.

It's common to include mathematics as a special case of science in general discussions. For example, universities usually classify mathematics under the sciences, and "scientific publishing" would generally be understood to include mathematics. However, there are exceptions, and abstract discussions of science often don't apply very well to mathematics.

If this distinction matters in a given case, then you'll have to discuss it explicitly, since you can never assume everyone will agree by default.

If Math is part of science, what kind of science is it?

The most compelling answer I've heard is that it is a formal science. However, this terminology is somewhat obscure, and it is certainly not a consensus answer to your question.

When people say STEM, do they mean Math is separate from Science?

I don't think the intention is to assert that they are definitely separate, but rather just to have an inclusive acronym we can all agree on.

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    Super happy to be introduced to the term "formal science", thanks! – Daniel R. Collins Jun 19 '16 at 15:52
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    It's a "mathematical science" is a more tautological, but also common answer. (Statistics is certainly also a mathematical science, and I'd also use that term to refer to much of CS and theoretical physics.) – Noah Snyder Jun 19 '16 at 19:58
  • Though if math is considered a formal science, I would disagree with the statement from the wiki link: the formal sciences do not involve empirical procedures. – Kimball Jun 19 '16 at 22:22
  • Mathematics is also an exact science, so since you can completely go around empirical methods and arrive at definite conclusions with it that are not prone to error, it's safer to take a "fact" in a mathematical textbook and assume that it really is true. – Panzercrisis Jun 20 '16 at 12:37
  • The mistake is in thinking that mathematics is purely an accounting term when in fact it is the language of science. Without numbers, most science can achieve is a hypothesis. In mathematics, the question is asked, a model is made, and a solution is sought to explain the data. This must then be explained. How does it not fit the scientific method? Most science started in math. Newton melded mathematics and physics. Yes, mathematics is a science. – Andrew Scott Evans Jun 20 '16 at 20:14
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The first definition of science that Merriam-Webster gives is:

: knowledge about or study of the natural world based on facts learned through experiments and observation

Math doesn't investigate the natural world and thus isn't a science according to that definition.

In How to Think Straight About Psychology Keith E. Stanovich defines 3 main traits of science.

Three of the most important are that (1) science employs methods of systematic empiricism; (2) it aims for knowledge that is publicly verifiable; and (3) it seeks problems that are empirically solvable and that yield testable theories (the subject of the next chapter).

Math isn't employing empiricism and thus doesn't fulfill criteria (1) and (3).

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    On the other hand, "I have tried to encourage the reader to think of the computer as a physicist would his laboratory - it may be used to check existing ideas about the construction of the world, or as a tool for discovering new phenomena which then demand new ideas for their explanation." -- Tristan Needham, Preface to Visual Complex Analysis (1996). – Daniel R. Collins Jun 19 '16 at 18:11
  • I don't see why people overcomplicate this question. Science is about studying the world we live in. Math isn't (in general). – user37208 Jun 19 '16 at 19:29
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    Empiricism comes up a lot in the practice of mathematics. Study things. Notice a pattern. Formulate a conjecture. Test the conjecture in more circumstances. Loop a few times as you refine the conjecture. Prove the conjecture. The end point is formal, but empirical methods are along the path. (and you don't always make it to that final point) – user13589 Jun 19 '16 at 21:50
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    Math doesn't investigate the natural world - I don't think it's so clear cut. Discrete quantities occur (at least apparently) in the physical world, so counting, and thus combinatorics, investigates the natural world. If you disagree with this logic, then by the same logic, you should also not consider much of theoretical physics a "science." – Kimball Jun 19 '16 at 22:28
  • @Kimball: There is still room for distinction; combinatorics tells how counting works. Combinatorics doesn't tell is that complicated judgments (like "there are three apples in the basket") accurately reflect reality nor that real-world collections really can be quantified by counting. – user13589 Jun 20 '16 at 2:16
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Unfortunately the duplicate in mathematics hasn't got good answers, so I will give it a try.

The answer is: Mu.
It means that the question must be unasked or that neither "yes" nor "no" is right or wrong.

Mathematics is examining the properties of consistent mental models or structures. It started with numbers and geometric figures and it was used for applications (counting, area calculation) from the beginning. It would not be completely wrong to name it "number philosophy" although it has expanded greatly and examines now a innumer...a very great number of concepts.

An example to show the difference to science: Let's say a scientist would try to prove the Pythagorean theoreom without mathematics. He would find out that if we draw squares on the side of a right-angled triangle, the smaller ones look like they have the same area as the big one. He would experiment with it and while they really look very similar, he will never achieve an equal result. The cuts are not completely straight, the material bends, the lines have always some extent. Even if he succeeds in determining that for all tested right triangles the values are mostly equal, he/she can never be sure that it will stay so.

A mathematician can prove that the sides are exactly equal. This is possible because s/he does not use real-world modelling and is therefore not limited by their applications. But while it is not the real-world, it still allows discoveries. It is not self-evident that right triangles have this interesting property.

Mathematics is a necessary part of science because it is a building block for any precise models we need to refine the work. Physical values are models by numbers (or matrices/tensors) and a concept of a dimension. In higher physics countle...a big number of mathematical models are used.

For the reason that mathematicians and other scientists are sharing many mental similarities (curiosity, challenging matters of course and relishing hard, but interesting problems) their faculties are often joined together which results in the STEM field.

So mathematics is "sciency".

But...it is not really a science because it is more fundamental: It does not require knowledge or experience from other scientific fields. If a mathematician travels back in time 20 000 years (the humans were on a comparable intelligence level as today) he would be able to teach a bright kid modern mathematics which is impossible for modern science because there is no infrastructure to replicate experiments. Mathematics also has no room for error or reevaluation. Once the axioms are set (yes, if the axioms change, the result can change), a result is valid for all times. Mathematics is also used massively outside science including the humanities.

So mathematics is not a science.

  • Part of your description of math sounds more like logic than math to me, and I'm not sure about some of your penultimate sentence. Some parts of math don't require understanding other fields, but the same is true in other fields of science. And there has been lot of misunderstandings in math, and consequently reevaluation. – Kimball Jun 20 '16 at 12:28
  • @Kimball "sounds more like logic than math to me". Logic is not only part of mathematics, modern mathematical logic is the base for Zermelo-Fraenkel set theory: the mathematical foundation. Modern logic is much more powerful and stringent than the old syllogistic Aristotelan logic (building sentences from propositions). So yes: Logic is one vital part of mathematics. The reason I use a general "consistent mental models" is the variety of mathematical structures with different propreties which exist: numbers, ,differential forms, groups, nodes, graphs, logical propositions. – Thorsten S. Jun 20 '16 at 19:40
  • @Kimball "Some parts of math don't require understanding other fields". If you mean other fields than math: All parts of what we call math do not require understanding other fields. None. Zero. It is really that brutal, sorry. If you do not believe that, give a counterexample. The misunderstandings and errors depend only on human defectiveness; in natural sciences all the imperfectness of senses, measurement errors and failure to reproduce something lead to failures even if you have made perfect deductions. – Thorsten S. Jun 20 '16 at 20:08
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On my journey through Computer Science, Machine Learning / AI, and Cryptography I've often wondered the same thing. Here's the way I currently see the interactions between math and its related disciplines through the branches of mathematics that I've studied:

Venn diagram of math an its related disciplines

I imagine there's a fair amount of subjectivity here, and that other branches of mathematics will have their own unique place on the Venn diagram.

  • There's experimental moral philosophy. I think that's science, so the nonoverlap between Science and Philosophy doesn't make sense to me. – Christian Jun 21 '16 at 9:57
  • @Christian Having never heard of experimental moral philosophy, I would have a hard time putting it in the diagram :P. I'm not saying that Science and Philosophy don't overlap, just that they haven't overlaped in any of the disciplines I've studied. – Mike Ounsworth Jun 21 '16 at 13:38
  • Weird diagram, I would consider logic strictly mathematical. – YoTengoUnLCD Jul 10 '16 at 4:26
  • @YoTengoUnLCD Interesting. I've heard it the other way; mathematics is a subset of logic - which includes several non-mathematical fields. Wikipedia describes Mathematical Logic (like prepositional logic) as "applications of formal logic to mathematics" ... implying that formal logic is outside mathematics. – Mike Ounsworth Jul 10 '16 at 4:34
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The answer to this question is perhaps not as clear-cut as it might appear, i.e. the issue is not merely definitional.

Some (for example the physicist Roger Penrose, in his book `The Emperor's New Mind') believe that mathematical structures exist in a non-physical, Platonic realm.

Conversely, the quantum physicist David Deutsch has claimed (notably in his book, `The Fabric of Reality') that, since the brains of mathematicians are physical objects, then the structures they can apprehend are constrained by the laws of physics (more specifically, to be computable by a quantum computer).

In `Where Mathematics Comes From', the cognitive linguists Lakoff and Nunez claim that, even if a transcendent Platonic mathematics existed, we would be unable to experience it, since our conceptions are analogised from our physical experience of space, force, motion etc.

Here is a quote from a recent article in the Notices of the American Mathematical Society by AMS President Robert Bryant: People say that mathematics is logical, but the logical aspect is only part of it. Mathematicians usually don’t proceed logically. They make guesses, see patterns, do experiments, develop beliefs. Almost nothing in that process is purely logical.

I'd personally say that this alone puts mathematics on a conceptually equivalent status to physics, but not everyone would agree with this.

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    I guess I don't really understand the relevance of most of what you say to the question at hand. Is the existence of or contemplation of a "non-physical, Platonic realm" part of science? I would think not. – Pete L. Clark Jun 19 '16 at 18:31
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    Pete L. Clark - What I'm getting at is that the question of whether or not mathematics is Platonic is open (but possibly not falsifiable). If mathematics can be reduced to questions about the nature of consistent human pattern perception, then it is within the realm of physics. Certainly, if science can be considered to be "hypothesize, experiment, potentially revise hypothesis, repeat", then Bryant seems to be saying that he thinks mathematics is a science, which I would personally agree with. – NietzscheanAI Jun 19 '16 at 18:37
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    "What I'm getting at is that the question of whether or not mathematics is Platonic is open (but possibly not falsifiable)." Sorry, I didn't get it. I said that the Platonicity of mathematics is not a scientific question, and you hint that it might not be. But if it isn't, why is it relevant to the answer? "If mathematics can be reduced to questions about the nature of consistent human pattern perception, then it is within the realm of physics." This doesn't make a lot of sense to me... – Pete L. Clark Jun 19 '16 at 18:41
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    If, as Deutch claims, all possible mathematics can be simulated by a quantum computer then mathematics is a sub-discipline of physics as a matter of fact rather than being an issue of definition. – NietzscheanAI Jun 19 '16 at 18:44
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    Sorry, your comments are too enigmatic for me to be drawn into a discussion. Just FYI, your practice of writing "more specifically" and then writing something brief that changes the meeting entirely caused me to give up. – Pete L. Clark Jun 19 '16 at 19:12
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Are you a Popperian falsificationist? Then you believe that only those statements that can be disproved by experiment can be classified as science, and as such, mathematics is not a science.

Of course, there are those that believe that mathematics is an experimental science, and the lab is the computer. If you are of this opinion, then math is a science in the Popperian sense, but most people think that software and hardware bugs are so common that this stance is untenable.

If you find Paul Feyerabend's epistemogical anarchism convincing, then you will claim that mathematics is a science, as it produces tangible value.

I personally wouldn't classify mathematics as a science. The best definition I can offer is mathematics is a set of techniques for thinking clearly about quantity and shape. But even that definition can be challenged with little effort.

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    Experiments are performed in the brains of mathematicians, not just `in the lab'. – NietzscheanAI Jun 19 '16 at 18:59
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    While there are certainly some mathematical statements that are not falsifiable (Godel, Turing), surely you are not claiming that this is in general the case? – NietzscheanAI Jun 19 '16 at 19:03
  • "quantity and shape" — math is the study of formal objects, not high school geometry and algebra. – Elliot Gorokhovsky Jun 20 '16 at 23:29
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I'll go for a semi-yes (maybe more).

When one says mathematics is not real, as compared to physics, I generally question the person about the reality of an electron. How does he know an electron is real? Ever seen one? Or guessed through models and measurements?

I long believed mathematics was not a science. However, parts of mathematics have become quite computational. Think about proofs for the 4-color theorem. Mathematics can teach about real-world problems, see the Pentium problem. And I am seduced by Alain Connes views on the existence of an

archaic mathematical reality outside space-time yet as inexhaustible as normal physical reality

discussed in details on SE Archaic mathematical reality:

Take prime numbers, for example, which as far as I'm concerned, constitute a more stable reality than the material reality that surrounds us

So, along with the increasing use of some mathematics in nowadays data engineering (data science/big data buzz words), I am more and more convinced that large parts of mathematics can be considered science... at least no less that string theory (Why String Theory Is Not A Scientific Theory).

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