18th Jul, 2021

Metropolitan Autonomous University

Question

Asked 29th Aug, 2018

In classical times there was a concept of pure science that was produced entirely in the intellect.

More recently sciences were developed by testing the intellectual product with empirical data. These sciences are not regarded as being pure.

Mathematics, often regarded as pure science, has for most of history been based on postulates of geometry that could not be proven. Then came relativity and other geometries. In the past century there was considerable effort to reformulate mathematics on a firmer basis of conditional sets. Math is now regarded as being somewhat more pure than before, while producing two generations of graduating students in some countries who are not able to do simple arithmetic.

Fortunately I had some excellent teachers who explained the two systems and why they were both needed. Other teachers displayed the Gödel's incompleteness theorems.

In academic settings there seems to be a difference of opinions about whether or not math is a science, and whether or not it is pure.

Is Mathematics A Science?

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If you agree that mathematics does not base its certainty on experimental or observational results and you share my belief in the factual (experimental) basis of science then mathematics cannot be a science. But this not prevents mathematics from being the language of science.

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Deleted profile

you wrote :

has for most of history been based on postulates of geometry that could not be proven

etc etc

It is uncredible to read this on a sciences forum.

What is a theory ?

What are these two systems ?????

the "pompon" : Gödel's incompleteness ...

The old math and the new math are two systems. Maybe you aren't old enough to remember the transition or the explanation for making a change.

The history is a famous story encountered in the teaching of mathematics. Apparently you haven't found it before, but a great many other researchers did.

Old math was built on geometry. One example is a plane triangle assumed to have 180 degrees total angles. It was not proven, and now under relativity is not in general exactly true. The new math has conditional postulates within which the math is consistent, although the postulates now built upon set theory are defined but not proven.

The Gödel's incompleteness is a set of proofs that postulates of mathematics can never be completely proven. It brings the question on a sciences forum if math is a science, considering that science should eventually be proven.

Before Gödel it was thought that the new math could be proven, but now is realized the math can be arbitrarily defined in a consistent system, but not logically proven.

Mathematic proofs are not based on empirical measurements, only on intellectual processes. For this reason in some context it is said to be pure.

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Yes because it follows rigorous structure of events leading to a tentative conclusion.

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In many ways, math is closely related to science. Mathematics is such a useful tool that science could make few advances without it. However, math and standard sciences, like biology, physics, and chemistry, are distinct in at least one way: how ideas are tested and accepted based on evidence.

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Deleted profile

you are talking of old and new teached maths in high schools. The "new maths" are not so new, it is easy to check it. They bring mainly rigor. In 2018, writing in a sciences forum that postulates are just postulates and asking presumed scientists to comment is something astonishing.

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Dear Jerry,

In addition to the nice answers. I have a little bit to add.

I agree that different axioms produce different geometries but all are consistent. In general, mathematics is everywhere in physics, chemistry, computers, music, golden ratio, nature, arts, astronomy, all engineering sciences, philosophy, psychology, food science, etc...

Ahh, Mathematics is not a science,

it is the queen of all sciences and it's servant.

Best regards

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No, as it clearly does not rely on empiricism or the scientific method to establish results — indeed, if we applied the SM to the Riemann Hypothesis (RH), then the conclusion would be true since all evidence indicates that all nontrivial zeros lie on the critical line.

Mathematics relies on crafting arguments which is obviously an art. Therefore math is an art and not a science.

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It depends on the definition of "Science".

Is it empirical? Yes, to a "certain extent", but the "truth" is not ascertained based on empirical evidence.

Is it based on logic? You bet!

What kind of logic? Sorry I am out of my territory here.

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31st Aug, 2018

I would consider this kind of philosophical questions to be outside of science and/or math. It nice to debate about with some friends on a rainy Sunday afternoon, but otherwise it is totally irrelevant and not adding anything about science and/or math itself. For example, it would not make General relativity any more or less true.

Mathematics is a science of a very different kind. Its principle of inference is deduction; in other sciences it is induction, which is not guaranteed in the same way. The "problem of induction" was said by Bertrand Russell to be the only philosophical problem which does not come down to issues to do with language (using it out of context).

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When the doctors degree is PhD, it seems appropriate for a philosophical discussion to occur about deficiencies in the foundations and about research for future improvements.

If math allowed empirical proofs, then the postulates could be proven within a context, and Gödel's incompleteness would be overturned. Already empirical arguments are allowed to disprove some theorems.

Physical sciences and engineering professions all use mathematics to make predictions, followed by empirical tests to validate the results. In this way the sciences are legitimizing the math as used in the science.

It seems unlikely that math teachers will be convinced to allow empirical proofs. Graphing of the functions is helpful, although it is more useful when some empirical data occurs on the same graph.

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Mathematics, based on the axioms that we can agree about, is a science. Homeopathy is however not.

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Certainly mathematics is a science that has evolved a lot from the idea that came out at the beginning which is the idea of counting.

Nobel Prizewinner Richard Feynman had this to say about mathematics:

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Mathematics is a living, breathing, changing organism with many facets to its personality. It is creative, powerful, and even artistic.

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Mathematics is clearly a Science, and is probably the Science with most applications in other fields (including Biology and Medicine). All "abstract" main results in Math. have found applications, some of them after decades. It is also an Art, comparable with Music.

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No. Fundamental to the sciences is the experimental method, and that method isn't very useful in mathematics. Mathematics is, instead, a tool of thought---one of our only two. (The other is language.)

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Meta-mathematics is concerned with reasoning "about" mathematical theories. Analogously, mathematics can be thought of as "meta-science" which is concerned with reasoning "about" scientific theories.

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Mathematics is a study of geometry, arithmetic and measurement, as well as the study of dimensions, change, structure and space. In other words, mathematics is defined as a science that conducts a broad and comprehensive study of all abstract structures through the use of a number of mathematical proofs, It is defined as a comprehensive study of all numbers and their different patterns.

Mathematics is the basis of all science. No science can do itself without the existence of mathematics; it is the language of communication in the world that any specialist can understand, but scientists and especially mathematical philosophers have not been able to define it. Related to this science.

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Mathematics is considered a science. However, Paul Halmos wrote: "Mathematics is not a deductive science — that's a cliché. When you try to prove a theorem, you don't just list the hypotheses, and then start to reason. What you do is trial and error, experimentation, guesswork. You want to find out what the facts are, and what you do is in that respect similar to what a laboratory technician does."

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In my opinion, it is yes because it involves a lot of theories, laws and proving that makes science a science

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Deleted profile

Scientific 'Knowledge' of a subject is 'Science' and therefore 'Mathematics' is a science.

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Yes, mathematics is a science simply because it is the building block for almost everything we used on a daily basis in different areas of studies, such as sports, materials, architecture, engineering, chemistry, etc.

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9th Sep, 2018

Being a part of Nature, the human mind by definition can not invent anything that, under certain conditions, would not occur anywhere in nature. Therefore, the most abstract and "incredible" mathematical constructions necessarily have a material reference. Therefore, in turn, mathematics is a dialectical removal of all natural (and not only) sciences.

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Jerry Decker,

A very illogical question, for anyone, especially for one engineer. I did not see an engineer of what ?? Why, the technique is mathematics, too, but it is not pure, it is applied mathematics, and without pure mathematics, there is no applied mathematics !
Still now there is a fundamental (= pure) sciences (natural and social, and one of them certainly the most important is mathematics.

Mirjana

I do not claim to be a mathematician but we may not claim mathematics as a totally science in the sense that it is not a subject of matter of laboratory but for any areas whether it is a science or finance we have to take recourse to mathematics .

This is my personal opinion

I believe so....Prof. Jerry Decker

As a science - which it is - it has grown into such a formidable complex "machinery" that no-one knows all math. It has been said that David Hilbert is the last mathematician who grasped all math that there was - and he died in 1943. Nowadays school teaches pupils less than before, which is quite a shame - as the generations after us need all the tools that they can muster in order to reverse the dooms-laden horror of global warming.

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There seems to be a mix of opinions in RG and also in college departments where both Arts and Sciences degrees are awarded in Mathematics, sometimes in the same university. In a long Engineering career, five decades I earned both Arts and Science degrees in Mathematics, at different degree levels and in different colleges. It gave me an opinion that Mathematics is too large to put in a single category.

Mathematics is not only science, but it is also the root of all science. Currently, we can observe the development of novel descriptive mathematical methods and theories capable to describe bio-medical phenomena.

Those who follow it closely are witnessing a birth of the novel discipline that is going beyond all what was so far available in the description of those phenomena.

Complex systems offer to pursue highly individualized and personalized medicine in the future.

Although a significant number of people believe that mathematics is a natural science, it is not. Therefore, mathematics is not a natural science, although it is very strongly applied in natural sciences. Mathematics is a very specific way of principled-philosophical thinking about ideas, concepts and processes with them.

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Mathematics is the science which deals with the logic of shape, quantity and arrangement. It is applicable everywhere..

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The structure of mathematics and science is similar with one significnt difference. The basic results of science are based on observation or experimentation. The analogous results in mathematics are based on logical proofs. Sometimes science uses mathematics to establish hypotheses or transform or interpret results, but there must be some observation or experiment (a designed observation) to be science. In mathematics the theorem replaces the observation.

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It is plausible to think that humans first started considering empirical objects and shapes and then made perfect abstractions of these shapes , creating the first mathematical objects.

They also used counting and made calculations in their daily lives , which helped them understand and elaborate more advanced and abstract methods of calculation , numerical systems and numerical operations.

At times empirical scientific observations instigate the creation of new mathematics , and at other times mathematical theories help explain new empirical data.

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Mathematics is the science that deals with the logic of shape, quantity and arrangement. Math is all around us, in everything we do. It is the building block for everything in our daily lives, including mobile devices, architecture (ancient and modern), art, money, engineering, and even sports.

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There is no science without Mathematics.

Usually, postulates or hypothesis of mathematical structure firstly depend on observation of reality problems, see for example the topological structure which consider as new geometry. Then the philosophy frame appear to investigate invisible knowledge or to predestine to the occurrence of more applications in practical life.

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Mathematics is not a science. Science is based on empiricism; Mathematics is based on theorem proving.

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Certain possible qualifications of my original statement pertain to certain areas of applied mathematics in which a researcher is doing mathematics in a science such as physics, chemistry, biology, etc.. In these cases the worker is in both mathmatics and science. Theorems may be proved but some of the hypotheses can be based on empiricism. Some applied math can therefore be in the intersection of science and mathematics. I say 'some' because there are different views on a definition of applied math. My definition excludes areas such as differentiol equations, numerical analysis, linear algebra, etc. and involves the use of math in discovery and understanding of other subjects.

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Mathematics is a study of structures of logic and reason. When the reasons are natural behaviors, behaviors in which science studies, we have applied mathematics. The dichotomy of pure and applied mathematics is becoming a vast overlap in which more mathematics is becoming applied.

One person is important here to quote:

Nikolai Lobachevsky - “There is no branch of mathematics, however abstract, which may not some day be applied to phenomena of the real world.”

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If mathematics is science, then why do we have two different names for the same fields!

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Deleted profile

Maths are sometimes scientific tools, sometimes the main object of a scientific theory, anyway distinct within the same application.

When doing pure maths or economy, for example constructing a very abstract geometry or an economic theory, the sciences is the geometry or the economy.

But when studying the mathematics in fundamental logic or in philosophy, maths becomes a pure sciences, even if misc maths tools are called.

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The view of science we have today is as laid out in Newton's "Philosophiae Naturais Principia Mathematica." That outline the basic "laws if physics" as perceived by Newton at the time gave rise to the scientific method as we see it today which more or less establishes that there are no "laws" in the natural sciences - only theories to be falsified.

What is science, to quote Richard Feynman, "It doesn't matter how beautiful your theory is, it doesn't matter how smart you are. If it doesn't agree with experiment, it is wrong." So Newton's "Principia: had more to do with science than mathematics independent of its title. What's the difference?

British mathematicians Alfred North Whitehead and Bertrand Russel developed "Principia Mathematica" to try to establish the logical ground rules for mathematics. The Whitehead/Russel "Principia" is to mathematics what Newton's "Principia" is the the natural sciences. Mathematics is not science - although a lot of the problems addressed in mathematics rise in science since mathematics become the precise language necessary to describe the predictions that arise out of science so an experiment can be performed to explore a given theory. Whitehead's and Russel's work arose because of the lack of rigor they saw at the time and the need for more rigor in mathematics. The classic example was in algebraic geometry - the zero sets of polynomials in several complex variables. The Italian school was based on a lot of "geometric intuition and handwaving" and some of their conclusions were wrong! Hilbert, Noether and other recognized this. Riemann and others through the development of complex analysis showed how abstraction to Riemann surfaces provided a way forward in one complex dimension. The Italian school was floundering not because of lack of smarts or intuition but because they had run out of tools to address any algebraic varieties of dimension greater than one.

Chow, Weyl, and Zariski then wrapped the precise language and methods from abstract algebra around the problems. This and embedding the problem into the projective closure allowed them and others that came along to move algebraic geometry onto a firm logical footing and it flourished.

So while mathematics and science are close they are not the same. The questions are different, the methods are different and the goals are different. Of course there is a fuzzy middle - known more or less as "applied mathematics" which is somewhere in between. However, the mathematician is trained to pursue questions that expand mathematical knowledge through a precise logical framework from axiomatic set theory and formal logic. The scientist is more interested in the answering of questions related to the natural world, it just so happens that those questions are stated in the precise language of mathematics. As experimental physicists Leon Lederman, Wolf and Nobel prize winner in Physics, said in his wonderful book, "The God Particle", "There is a pecking order in physics. The experimental physicists answers to the theoretical physicists who answers to the mathematician who only answers to God."

A mathematician will say - "it's nice that my work applied, but that's not why I did it." The scientist will then say - "if you didn't do it for a reason, why do it at all?" The mathematician will reply - "because of the internal beauty of the results."

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All science is based on observation or experiment. Mathematics is based on axioms, definitions and logical proofs.

There was a very interesting article in the Johns Hopkins University Magazine a few months about what is mathematics. Mathematics is not science. That it is the language in which science communicates does not make it science any more than English is a science. I think the operative quote is "Mathematics is the music of reason." The mathematician might be happy that his results can be uses or applied by someone working in the scientist but that is secondary to him. Some mathematicians could care less. Mathematics is about going where your creativity takes you - unconstrained about any "real world issues" like gravity, magnetic force, building a bridge, etc.

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No, it is not. Two necessary conditions to be considered science are observation and controlled procedure. The mathematical activity certainly is characterized by controlled procedure, but not by observation. Thus, it lacks a necessary condition to be science.

There are some similarities between math and science but philosophically they are distinct. Although math and statistics are widely used in science, neither is a science. Science is characterized by observations or experiments; math is characterized by logical proofs; However there are some nice analogies between math and science:

The conjecture of math corresponds to the hypothesis of science.

The theorem in math corresponds to the law in science.

The proof in math corresponds to he experiment in science.

Although math and science are distinct, there are similarities in their distinctions..

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Many negative answers to this question stress on the fact that science needs interactions with real facts. There is no science where no support from any empirical test is requried (whence possible). I agree. Accordingly, mathematics is not science. It is the language of (some parts of) science, but not a science itself, just like a grammar is not a novel or a poem. However, in this perspective (no reasonable hope for any interaction with facts), a large part of contemporary theoretical physics has nothing to do with science. It can be regarded as mathematics, maybe intriguing mathematics, although often too sloppy to meet the standards that most of mathematicians still demand. But let's slightly change

our perspective and follow Popper. According to his celebrated criterion, we cannot really say that mathematics is not a science. Yes, generally it is not such, but occasionally it is: the errors are science, till when they are detected as such. Indeed, what is falsifiable is sience, till when it is not definitely falsifield. So, according to this perspective, only bad mathematics is science, although each bad item for a very short time. No hope for good mathematics to have anything to do with science. But let's forget about Popper. He had right intuitions (but not so different from those that many others had), but he grossly sharpened them too much. Saving the core of those intuitions, science should say something on reality. We say something when, in principle, what we cay could be wrong. A sentence that cannot be wrong says nothing (it embodies no information). A correct mathematical inference cannot be rejected, whatever the world is. Only contradictions oppose tautologies. Correct mathematics cannot be wrong. Hence it is empty, whence no hope for it to be a science. So far so good. However, let's consider a corollary of this (correct) conclusion. Suppose you work in a departement where engineers are the majority and you dare to expose the above point of view. Your colleagues will enthusiastically approve your speech, but

they will go a little step further: what you do has nothing to do with science, as you say. On the other hand, you are not doing literature. So, you do nothing. Then, please, withdraw your application for fundings.

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Sorry, I realize that, apart from strange jokes that my keybord liked to play, the last part of my answer is hard to understand. I wrote: "Suppose you work in a department...". I should have written as follows: "Suppose you are a mathematician and you work in a department...".

I like the expression: "The Art of Mathematics" - it puts Mathematics slightly above Sciences.

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Thanks to all comments.

A search of BS in mathematics finds 291 million responses. BA gets 138 million responses. These are the main choices and some colleges offer both. University of Florida is one example. They recommend the BS for students who intend to do graduate studies in Mathematics and the BA for those who might do graduate study in education, engineering, computer science or business.

Science in modern times prefers to have an experimental proof, although some areas of physics prefer a theoretical consistency. In classical thought science was more like math, established in the intellect.

STEM in recent education regards science, technology, engineering, and math as separate disciplines.

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What is interesting is Newton's "*Philosophiae Naturalis Principia Mathematica" - *has little to do with mathematics but established the scientific method that is the standard for science - that is through validation of theories with experimentation and falsification. In Russel's "*Principia Mathematica" *the logical formalism of modern mathematics is established. Up to the time of Russel's work, mathematics did not have the logical rigor necessary. Issues were arising as recognized in the early 20th century as the non rigor had resulted in some errors to arise which was recognized by Hilbert and others. The birth of modern mathematics came out of this. A classic example was the Italian school of Algebraic Geometry - beautiful theory on algebraic surfaces but a lot of it - particularly the later work incorrect.

Historically it was difficult to separate mathematics from the sciences - particularly physics up until the late 19th and early 20th century. However, the lack of rigor caught up with mathematics and required a much more formalistic approach wrapped around not only intuition but a strong logical foundation provided by Russel and mathematicians such as Hilbert, Chow, Zariski, Noether, etc.

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My work in Orbital Mechanics has been called scientific work. It is basically applied mathematics but because it uses Newton's laws and sometimes atmospheric drag it can be called science. There are borderline areas between physics, engineering, and math that give us some flexibility in our designation.

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Yes, for sure!!!

There is continuous growth in the subject, and therefore is science.

Science is beyond limits!!

Mathematical physics is considered math. Theoretical physics is considered physics. The topics studied are the same. The difference between them is in the way that they are approached.

Borrowing from **Berggren:** Mathematics, **the science** of structure, order, and relation that has evolved from elemental practices of counting, measuring, and describing the shapes of objects. It deals with logical reasoning and quantitative calculation, and its development has involved an increasing degree of idealization and abstraction of its subject matter. Since the 17th century, mathematics has been an indispensable adjunct to the physical sciences and technology, and in more recent times it has assumed a similar role in the quantitative aspects of the life sciences.

Math is all around us, in everything we do. It is the building block for everything in our daily lives, including mobile devices, architecture (ancient and modern), art, money, engineering, and even sports.

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All sciences are based on empiricism. Mathematics is not a science but it is analogous to science in that the proof replaces the observation or experiment.

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Mathematics is human made technique to solve problems which can be related to nature, reality or just theoretical ideas or concepts. It uses logic. But all scientific principles and concepts like predictability, repeatability, evidence, proof etc are followed in Mathematics. Scientists, mainly physicists, have hage contributions in development of Mathematics. So it can be treated as science as per the context it is applied. Given, it has no limits as like science, it can also be associated with other branches of philosophy or knowledge.

"No human investigation can be called real science if it cannot be demonstrated mathematically."

― **Leonardo da Vinci**

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