'math' is theory, 'science' is theory, all/any theory is rooted in philosophy. This is historically so, for 'science' in the modern material sense of the word, is best understood historically as a development of theory within an already long established tradition of 'philosophy of nature'. The confusion here arises from the current scientific paradigm being largely philosophically illiterate while unknowingly representing a very specific philosophical doctrine known generally as 'positivism'. Positivism being a theory of science which rejects metaphysics, another much longer established body of theory within the school of philosophy that is somewhat foundational to the concept of nature which manifested modern science to begin with. The problem being the fact that positivism was/is a very much contested and unproven theory within philosophy that has been adopted nearly universally by all scientific praxis towards large material reward. These are exceedingly important issues facing the academy, tantamount to a crisis in my opinion.

Science Definition: The word science comes from the Latin "scientia," meaning knowledge.

How do we define science? According to Webster's New Collegiate Dictionary, the definition of science is "knowledge attained through study or practice," or "knowledge covering general truths of the operation of general laws, especially as obtained and tested through scientific method and concerned with the physical world."

What does that really mean? Science refers to a system of acquiring knowledge. This system uses observation and experimentation to describe and explain natural phenomena. The term science also refers to the organized body of knowledge people have gained using that system. Less formally, the word science often describes any systematic field of study or the knowledge gained from it.

What is the purpose of science? Perhaps the most general description is that the purpose of science is to produce useful models of reality. Most scientific investigations use some form of the scientific method.

Science as defined above is sometimes called pure science to differentiate it from applied science, which is the application of research to human needs. Fields of science are commonly classified along two major lines:

◾ Natural sciences, the study of the natural world, and

◾ Social sciences, the systematic study of human behavior and society.

Definitions of mathematics

Aristotle defined mathematics as "the science of quantity", and this definition prevailed until the 18th century. Starting in the 19th century, when the study of mathematics increased in rigor and began to address abstract topics such as group theory and projective geometry, which have no clear-cut relation to quantity and measurement, mathematicians and philosophers began to propose a variety of new definitions. Some of these definitions emphasize the deductive character of much of mathematics, some emphasize its abstractness, some emphasize certain topics within mathematics. Today, no consensus on the definition of mathematics prevails, even among professionals. It is important to highlight that a great numbers of mathematicians take no interest in a precise definition of mathematics, or consider it undefinable. Some just say, "Mathematics is what mathematicians do.

Three leading types of definition of mathematics are called logicist, intuitionist, and formalist, each reflecting a different philosophical school of thought. All have severe problems, none has widespread acceptance, and no reconciliation seems possible.

An early definition of mathematics in terms of logic was Benjamin Peirce's "the science that draws necessary conclusions" (1870). In the Principia Mathematica, Bertrand Russell and Alfred North Whitehead advanced the philosophical program known as logicism, and attempted to prove that all mathematical concepts, statements, and principles can be defined and proven entirely in terms of symbolic logic. A logicist definition of mathematics is Russell's "All Mathematics is Symbolic Logic" (1903).

Intuitionist definitions, developing from the philosophy of mathematician L.E.J. Brouwer, identify mathematics with certain mental phenomena. An example of an intuitionist definition is "Mathematics is the mental activity which consists in carrying out constructs one after the other.” A peculiarity of intuitionism is that it rejects some mathematical ideas considered valid according to other definitions. In particular, while other philosophies of mathematics allow objects that can be proven to exist even though they cannot be constructed, intuitionism allows only mathematical objects that one can actually construct.

Formalist definitions identify mathematics with its symbols and the rules for operating on them. Haskell Curry defined mathematics simply as "the science of formal systems". A formal system is a set of symbols, or tokens, and some rules telling how the tokens may be combined into formulas. In formal systems, the word axiom has a special meaning, different from the ordinary meaning of "a self-evident truth". In formal systems, an axiom is a combination of tokens that is included in a given formal system without needing to be derived using the rules of the system.

However, for many mathematicians, mathematics is the science of numbers and their operations, interrelations, combinations, generalizations, and abstractions and of space configurations and their structure, measurement, transformations, and generalizations.

Definitions of Philosophy

Philosophy is the study of the fundamental nature of knowledge, reality, and existence, especially when considered as an academic discipline. In other words, is the use of reason in understanding such things as the nature of the real world and existence, the use and limits of knowledge, and the principles of moral judgment.

Based on the above definitions, I consider the Philosophy as the mother of all sciences and matemathics also a nature science

Maths are in the sciences.  It belongs to a group of sciences called Formal/Mathematical Sciences (and are the bedrock of science).  Fields like Computer Science and Statistics are formal sciences as well.  They all play by the rules of science, but don't necessarily explore the physical world (in contrast to empirical/natural sciences).  Though one can make an argument about maths (even the nature of computing) are rooted in nature.  We can't do science without some bit of the formal sciences (we need models after all).  Unlike these, philosophy doesn't adopt the scientific method.  So no, philosophy isn't a science.  Philosophy is precursor to Science.  

Let me first quote the words of Richard Feymann "Nature speaks to us in Mathematics".

Definitely Mathematics is science in the sense that it is a systematic, formulated, logical, and verifiable knowledge. In fact, Mathematics provides language (or mathematics is the language) for all sciences to explain, analyse and verify the findings.  That is why Mathematics is called the queen of all sciences.

For me, Philosophy is not science, but every science has its own philosophy and  the connections of philosophy with sciences are deep and influential.

Dear friends and colleagues, thank you very much for your wonderful ideas from different aspects on mathematics, philosophy and science.

Some people argue that knowledge and science are close related but not exactly the same things. For example, some one may have the knowledge of curing certain diseases, but such knowledge may not be called science: anything we know can be called knowledge but only some of them can be called science.

As both mathematics and philosophy are very “abstract” (one can not see and touch “infinite”,… ), this may be one of the reasons that people think they are not science.

Are mathematics and philosophy the knowledge which can be called science?

Mathematics is one of of the major branch of Science. All sciences reveal themselves in the form of mathematics..!!! Mathematics is the logical backbone of all sciences.

Philosophy is not a science as it related to foundations, methods, and implications of a subject. 

Dear Chithra and Stefan, thank you.

So, there are many subjects in our knowledge mansion, but some are called science some are not.

1, how can we call Philosophy which is not called science? 

2, why a subject related to foundations, methods, and implications is not science---------- What is science?

Sincerely yours, Geng

If a field doesn't perform the scientific method, it isn't a science.  On top of that, if it provides no explanatory power, it isn't a formal science or an empirical science.  So that rules out most social sciences (which is fine, social science is great at archiving things).

Dear Mr. Daniel Page,

I am sorry, can we say philosophy doesn't perform the scientific method?

Sincerely yours, Geng

Dear Mr. Stefan Gruner , thank you for your recommendation.

For my follow-up questions (1) above is easy to answer. But for the follow-up questions (2) is not that easy to answer because all branches of science relate to foundations, methods, and implications in their own ways and produce different scientific contents.

Does philosophy perform the scientific methods in its own way or it doesn't perform the scientific methods?

Sincerely yours, Geng

Dear Mr. Ellis D. Cooper，you are right that hundreds of carefully written books over at least two centuries have addressed more carefully articulated questions on what science, mathematics, and philosophy are.

But what we are doing in this thread is totally different from the previous. Driven and powered by the newly discovered infinite related Harmonic Series Paradox, we have very clear target which is different from previous talks and writings------hoping to solve some of the infinite related fundamental defects in our mathematics (science, philosophy) through the discussions here.

Sincerely yours, Geng

Facing the newly discovered family member of 2500-year old Zeon’s Paradox------Harmonic Series Paradox, something should be done.

We have now more than a dozen of “philosophy of mathematics”; they are not being for “after dinner or tea time talks” but for the healthy foundation of mathematics.

Dear Mr. Stefan Gruner , thank you very much for your recommendation of a good book.

RE: “ The "method" of philosophy is mainly to "discuss" things with "wordy" arguments”. This is what philosophy does but not how philosophy does. It is pity that people sometimes confuse on “what philosophy and mathematic do-- how philosophy and mathematic do” so they doubt whether philosophy and mathematics are science.

Sincerely yours, Geng

Dear Mr. Stefan Gruner , thank you again for your recommendation of some  good scholarly books or essays on Science versus Philosophy.

I think there are really a lot to do in the field of philosophy of mathematics to solve some fundamental defects in infinite related area of our mathematics.

Sincerely yours, Geng

            I believe, the best and objective way to answer this questions is, we must define the boundaries and contours of each of the terms “science”, “philosophy” and “mathematics”, for example, based on our shared understanding and widely accepted conventional wisdom. Philosophy is a science, if the contours of “philosophy” fit with in the boundaries of “science”.

            Unfortunately the meanings of the terms are not same at the different times in the history. That is, the meanings of these terms have been evolving over time. Also each term may have different meaning in different context or each person may interpret slightly differently.

Let me quote a great insightful quote by Galileo: “Measure what can be measured, and make measurable what cannot be measured.”

The real scientific knowledge comprises of objective facts, where objective facts can be measures and subjective concepts can’t be measured. In other words, scientific process requires trying to find a way to transform things that can’t be measured. That is, “measure what is objective, and if things are subjective, try to make the things objective.”

Many experts told me, it is foolish to say software engineering is real engineering. I have no problem with that, but my contention is it is possible to transform software engineering into real engineering.

I am sure everyone agrees that, engineering (i.e. design and development) of one of a kind physical products such as an experimental jet fighter or prototype of next generation spacecraft is real engineering. Why software engineering is not a real engineering (if we invent real-software-components for achieving real CBD for software), and if there is virtually no difference between the engineering (i.e. designing and development) of complex software product and the real engineering (i.e. assembling individually designed and tested real functional components for building one-of-a-kind experimental jet fighter or spacecraft)?

In this case, the contours of CBD of software engineering fit well with the boundaries of widely accepted real engineering. Similar reasoning can be given to various sub fields of computer science. For example, if we believe various sub-scientific disciplines such as virology, mycology, parasitology, and bacteriology are real sciences, then componentology (e.g. for achieving real CBSD) and neuronology (e.g. for achieving real artificial intelligence) can be made real science, by making them virtually no different from the known real sciences.

My point is, we need to define (based on conventional wisdom and widely accepted shared understanding) counters of ‘’what is science” and “what is mathematics” to see if the contours of mathematics fall within the boundaries of our shared understanding of the term “science”. Our conventional wisdom and shared understanding could change over time.

However, today my understanding of philosophy do not fall within the boundaries for my understanding of the term “science”. So I don’t believe, they are sciences but they are more useful than most scientific disciplines. So I don’t believe, branding each of them as a science don’t increase their usefulness not decrease their usefulness. This is just my personal perspective.

I believe, we use mathematics to deal with things like measurement of objectivity of scientific knowledge, while philosophy deals with things like defining the rules, guidelines and understanding our accuracy, context and nature of our knowledge of science. This is my understanding, correct me if I am wrong.

Best Regards,

Raju

Dear Mr. Raju Chiluvuri ,

Thank you for the experienced and insightful idea. You are right, (1) the boundaries and contours of each of the terms “science”, “philosophy” and “mathematics” are very important, (2) the transformations between  objective and subjective is very important, though both are difficult. I share with your ideas:

1, Science is our human’s, it is undoubtedly that we really can own (create) our common “scientific concepts and terms” for our science through studies, discussions…. This is one of the reasons that we are here and we cherish this destiny that ties many of us together.

2, I also believe how to unify “objective world” and “subjective world”, how to cognize and translate “things in human science and things in natural world”, “sets in mathematics” and “sets in nature”, “infinitesimals in mathematics” and “infinitesimals in nature”,…, have been the essential, important and toughest works for scientists.

3, I have been working very hard on the cognitions of “concepts and terms” because I think they exert great impact on logics and fundamental theories-------if the concepts and terms are unscientific, the relationships

(logics) between and among them are unscientific.

I also present a difficult case of defining the term of “science”: some people even argue that knowledge and science are close related but not exactly the same things. For example, some one may have the knowledge of curing certain diseases, but such knowledge may not be called science: anything we know can be called knowledge but only some of them can be called science. So, what is science?

Driven and powered by the newly discovered infinite related Harmonic Series Paradox------one of the family members of 2500 years old Zeno’s Paradoxes, many “concepts and terms” such as “finite”, “finite”, “infinite”, “potential infinite”, “actual infinite”, “infinite related numbers”, “infinite related mathematical theories”, “mathematics”, “philosophy of mathematics”, “philosophy”, “science”,…. This is a huge field and our works conform us that there are really some suspended fundamental defects in present infinite related science theory system ever since Zeno’s time.

Sincerely yours, Geng

Dear Mr. Ouyang,

You said:

I have been working very hard on the cognitions of “concepts and terms” because I think they exert great impact on logics and fundamental theories-------if the concepts and terms are unscientific, the relationships (logics) between and among them are unscientific.

You absolutely right. Terminology is like cultural differences, where a term could mean certain thing to a group of people while the same term means different to another group of people.

I am facing so much trouble with the terminology. For example, in computer science, each kind of software components is defined as either a kind of software parts either having certain useful properties (e.g. reusable or standardized) or software conform to a so called component model. That is, term component is used as a synonym to useful part. Then CBSE (Component Based Software Engineering) is defined as using so called software components. It is kind of saying flopping fake wings is flying.

This has no basis in reality or fact. The fact is, we all know physical functional components are a very special kind of physical parts having very unique characteristics. And an ideal CBD for physical products require implementing over 95% of the features and functionality in such functional physical components (e.g. as an hierarchy of replaceable components logically equivalent to http://real-software-components.com/CBD/CBD-structure.html and a process logically similar to http://real-software-components.com/CBD/CBD-process.html).

It is not necessary that even a single large functional component in the CBD structure of hierarchy need to be reusable or have any of the properties erroneously attributed to the so called software components.

It is kind of like defining the term “horse” is synonym to the word “animal”, then definition for each kind of CBSE is like defining that using cat is a kind of cavalry and using pigs is another kind of cavalry, and so on. This paradigm and conventional wisdom was so deeply entrenched over 45 years, it is became impossible to use term ‘component’ to explain the reality of real CBD. One can find a dozen widely accepted core concepts of the existing paradigm to discredit each of the core concepts of new CBD paradigm, so it is impossible to pick any core concept to start explaining.

I would not have faced this kind of problem, if the researchers 50 years ago were to use any term such as ‘widget’ (instead of the term component) and widget based software engineering (instead of CBSE) 45 years ago. But the terminology is so confusing, any statement I made in the context of new meaning using terms, would have different and conflicting meaning in the old paradigm.

So I agree that, if contours and boundaries of each term has different meanings in different groups, it would become impossible to communicate between the groups. It is extremely hard to explain, unless one has personal experience in this kind of two conflicting perceptions of realities.

Best Regards,

Raju

Dear Mr.  Raju Chiluvuri ,

Happy to share with you the research experiences on terminology. I am working in the infinite related area; some suspended problems are exactly the same as those of that in Zeno’s time 2500 years ago (unimaginable?). Ok, I will try my best in my remaining life.

Best Regards, Geng

Dear Mr. Ouyang,

You are not alone, the US Supreme Court and many experts have been perplexed about the boundaries of scientific discoveries and concepts in applied sciences (e.g. engineering inventions) in countless cases. One can get patent for concepts in applied science (e.g. engineering inventions) but can’t get patent for scientific facts.

If you create a virus, I think, you can get a patent form the USPTO for the process of creating the living thing. But you can’t get a patent on the genetic structure of a virus. I am not an expert, but that was my understanding. There was so much case history related to patenting of inventions on the blurry boundary separating inventions and discoveries, which might be helpful. The US Supreme Court and CAFC have been struggling for decades to draw a bright light for separating applied sciences from the basic sciences.

My personal understanding of the ‘term’ science is knowledge of facts only about the physical beings and physical phenomena (where many physical beings interact) such as big bang (if it is true). Not any knowledge constitutes science, such as invention of useful things. Inventions of useful things fall under applied science are engineering. Often engineering inventions are derived by relying on the scientific knowledge.

However, some times accidental inventions lead to scientific discoveries: For example, you gave an example: “some one may have the knowledge of curing certain diseases, but such knowledge may not be called science.” . This falls under applied science or engineering, as per the law for patenting.

In this case, we all agree there must exist a valid scientific reasoning (a set of facts and/or explanation of phenomena), if the cure is working (even no one knows the reasoning). For example, I accidentally invented real CBD for software more than a decade ago, by repeatedly creating hundreds of CBD structures (i.e. hierarchy of replaceable components more than a decade ago) to confirm it. It took me more than a decade of passionate research to find valid reasoning.

Best Regards,

Raju

 Thank you dear Mr. Raju Chiluvuri for your encouragement, I think we are all working hard in our pioneering research fields.

Very happy to know you and your experienced ideas are really insightful and helpful.

Sincerely yours, Geng

Dear Mr. Stefan Gruner

 Thank you very much for your recommendation.

Sincerely yours, Geng

Dear Mr. Luciano da F. Costa, I totally agree with you.

Dear Mr. Hijaz Ahmad, thank you for your ideas.

Sure, mathematics is a science and it makes the base of science.

Five of my published papers have been up loaded onto RG to answer such questions:

1，On the Quantitative Cognitions to “Infinite Things” (I)

https://www.researchgate.net/publication/295912318_On_the_Quantitative_Cognitions_to_Infinite_Things_I

2，On the Quantitative Cognitions to “Infinite Things” (II)

https://www.researchgate.net/publication/305537578_On_the_Quantitative_Cognitions_to_Infinite_Things_II

3，On the Quantitative Cognitions to “Infinite Things” (III)

https://www.researchgate.net/publication/313121403_On_the_Quantitative_Cognitions_to_Infinite_Things_III

4 On the Quantitative Cognitions to “Infinite Things” (IV)

https://www.researchgate.net/publication/319135528_On_the_Quantitative_Cognitions_to_Infinite_Things_IV

5 On the Quantitative Cognitions to “Infinite Things” (V)

https://www.researchgate.net/publication/323994921_On_the_Quantitative_Cognitions_to_Infinite_Things_V