Does Classifying Mathematics as a Science Affect Its Usefulness?

[Dremmer]

What do you think? I'd say it isn't. Dictionaries say it is, but dictionaries aren't always right. They are often wrong.

 

[Mentallic]

Science is a tool that helps us understand the universe through experimental data. The best we can do is to create a law or theory that describes some physical phenomenon and is reproducible. Mathematics however is perfect and a deduction of some basic axioms. Everything we know in mathematics was pretty much derived from 1+1=2.

Anyway, I would say that math isn't a science.

 

[micromass]

The methodology of science and mathematics is completely different. In science everything must be confirmed by experiments. But in mathematics, all things must have a proof, experiments aren't good enough.

So no, I would say mathematics isn't a science. But there is much overlap!

 

[ShayanJ]

So you say every thing which has sth to do with experiments,is a science?

 

[micromass]

No, I say, everything which is a science has to do with experiments.

 

[ShayanJ]

But I think the thing you say is called natural science.
And mathematics is definitely a science(a good one).

 

[micromass]

Then, what is the difference between a natural science and a science?

I believe that all fields like physics, chemistry, biology, psychology,... are sciences. But as sciences they say something about reality. That is the only goal of sciences, to say something about reality.
But mathematics isn't concerned about reality. The real numbers don't exist, the natural numbers don't exist. Mathematics is an abstraction of reality, but is not interested about reality at all. So it isn't a science. With that I don't say that mathematics is not often used in science, but that's another discussion...

 

[HallsofIvy]

Well, of course, the answer depends upon exactly how you define "science"!

If we use the most basic definition- that from the "scientific method", then mathematics would NOT be a science.


But I think there is a more fundamental distinction than that- the basic philosophies behind science and mathematics are different.

In science the truth or falsity of any statement depends entirely on experimentation and observation- is it true of the real world. The underlying philosophy of any sicence is necessarily realist.

In mathematics, however, the truth of falsity of a statement depends upon whether it is consistent with the axioms and postulates of the system. The underlying philosophy of mathematics is necessarily idealist.

(I am, of course, using the words "realist" and "idealist" in their philosophical senses.)

 

[ShayanJ]

MATHEMATICS IS NOT ABOUT REALITY!
That's closing your eyes to the world.You say physics is about reality.physics is expressed by mathematical notions.sometimes a physical notion is defined just because the mathematics of past theories have that notion in front.like potential.And you know the interesting part,most of such notions are understood to be practical or maybe real.
sometimes after many years from the bearing of a mathematical notion,it is shown that it has uses in phyisics like complex numbers and quantum mechanics.
How can you speak as such about mathematics??

 

[micromass]

But it is true. Mathematicians only care about their axioms. As long as everything is consistent, then it's good.

Please tell me how "inaccessible cardinals", "Godels incompleteness theorem", "the Cantor set", "the popcorn function" have any relation to reality.
Even the real numbers have nothing to do with reality! What is true, however, is that the real numbers are used in physics to model reality. But mathematicians don't really care if the real numbers are realistic or not...

 

[micromass]

There are however a few mathematical disciplines that do care about reality. Like statistics or optimization. But for the most mathematicians, reality is a uninteresting concept...

You mention the complex numbers. This is a good example. Mathematicians did not invent complex numbers because they are realistic, they invented them because they were beautiful. It were physicists which used them for realistic purposed like electricity. Mathematicians don't care about the realistic applications of complex numbers...

 

[HallsofIvy]

MATHEMATICS IS NOT ABOUT REALITY!

I said nothing of the sort. That was specifically why I added the statement that I was using the words "realist" and "idealist" in their Philosophical meanings.

That's closing your eyes to the world.You say physics is about reality. physics is expressed by mathematical notions.sometimes a physical notion is defined just because the mathematics of past theories have that notion in front.like potential.And you know the interesting part,most of such notions are understood to be practical or maybe real.
sometimes after many years from the bearing of a mathematical notion,it is shown that it has uses in phyisics like complex numbers and quantum mechanics.
How can you speak as such about mathematics??

Because I, unlike you, distinguish "mathematics" from "applications of mathematics". Yes, mathematics has uses in phyics. That doesn't mean it is the same as physics.

 

[HallsofIvy]

There are however a few mathematical disciplines that do care about reality. Like statistics or optimization. But for the most mathematicians, reality is a uninteresting concept...

And there are some mathematicians who consider "statistics" and "optimization" to be applications of mathematics, not mathematics.

You mention the complex numbers. This is a good example. Mathematicians did not invent complex numbers because they are realistic, they invented them because they were beautiful. It were physicists which used them for realistic purposed like electricity. Mathematicians don't care about the realistic applications of complex numbers...

 

[disregardthat]

You mention the complex numbers. This is a good example. Mathematicians did not invent complex numbers because they are realistic, they invented them because they were beautiful.

The complex numbers were invented because they were useful.

 

[micromass]

The complex numbers were invented because they were useful.

Yes... They were invented because they were useful in solving third-degree polynomial equations. And people solved that polynomials because they were like games, it was for the bragging rights, i.e. "I can solve this equation and you can't". The complex numbers were not invented because they were of any use in reality...

 

[CRGreathouse]

To address the original post: I do not consider mathematics to be a science. I have heard it referred to as a formal science rather than a natural science; that's another valid categorization.

 

[arildno]

I'd rather call maths a form of art (or game), than a science.

 

[paulfr]

Mathematics is a language ... the language of the Natural World ... thus the language of Science.
It is also more than that because Mathematics often preceeds discoveries and applications in the Real World.
So more generally, Mathematics is the language of Abstract Thought.

JMHO

 

[Outlined]

Mathematics is a language ... the language of the Natural World ... thus the language of Science.
It is also more than that because Mathematics often preceeds discoveries and applications in the Real World.
So more generally, Mathematics is the language of Abstract Thought.

JMHO

No definitely not, when I studied applied math I thought Math was a language too but since I moved to pure math (algebra, number theory, discrete math) I learned math is not a language but it is an abstract constructions of the human mind.

It might be the language of physics but I leave that discussion to the physicists.

 

[Tom Gilroy]

I would consider mathematics a science, though perhaps not a natural science. It's certainly a science to at least the same degree that computer science is a science.

Most mathematicians I've met seem to subscribe to the platonist veiwpoint, including myself to some degree. I wouldn't accept that mathematical concepts are dependent of the human mind, I feel it likely that if there is another intelligent lifeform in the universe (superior in intelligence to ourselves), they've reached the mathematical conclusions we have (for example, the smallest faithful representation of the Fischer-Griess Monster has dimension 196883 and any alien intelligence would reach the same conclusion). In some sense, mathematics could be "written into" the universe itself, and then it would indeed both be science and natural science.

Some might protest "but what about these other set theories and the axiom of choice? Surely mathematics is our creation in light of these?" I don't really feel that this argument holds water. The vast, vast majority of mathematics can be formalised using only the axioms for 2nd order arithmetic, which are the bare minimum you need to formalise the arithmetic of the natural numbers, other set theories are largely in place to allow sets themselves to be studied.

I'd almost consider the concepts of the natural numbers and their arithmetic to be necessary for anything I would consider to be "intelligence." So, if there were a superiorly intelligent alien lifeform, I'd imagine they too would have mathematics, and that any conflicting results would be precisely the ones that depend on axioms other than those of 2nd order arithmetic. In some sense, their mathematics might be based on a set theory which is not equivalent to ZFC, however the majority of their mathematics could also be formalised from the axioms of 2nd order arithmetic too.

I've even heard it proposed that mathematics may be "unreasonably effective" in describing the universe for a very reasonable reason; that the universe itself might be isomorphic to some (as yet unknown) mathematical structure. I don't believe it, but it's an interesting idea.

I feel it likely that any definition of an "art form" broad enough to include mathematics could also include the natural sciences, and I don't believe that "mathematics is a philosophy." The idea of mathematics as a "game" to me seems incorrect, as it implies a set of objectives. There are no overall objectives in mathematics as a subject that are set by the mathematics itself, rather the mathematical community sets objectives for itself, and follows the mathematics where ever it may lead them.

Whatever mathematics is, most of the time I'd rather just do it than wonder about what it is or isn't and whether it's "true" or that it should be believed.

 

[g_edgar]

There is a textbook title:
MATHEMATICS: THE SCIENCE OF PATTERNS
Sounds good to me.

 

[robert80]

I believe mathematics is more like art from one point of view. When solving difficult problems in any of the fields, let's say at advanced level on university, sometimes you are really astonished by some proofs of particulair exercise of professors or assistants. I mean, it could be very creative. Anyway, I believe I would put the philosophy and matematics on the very same faculty.

 

[Philosopher_k]

Mathematics most certainly is not a science.

In science we observe and experiment the natural world. In mathematics we are studying something in which we have no idea wether we are totally right or wrong. Mathematics is built upon a set of axioms or assumptions. There is no true way to prove these axioms therefore mathematics will never be complete.

But you argue, are we not unsure of science as well? No. For science as i have defined it is the observation and modelling of reality. If we observe it, correctly without error, consistently then it must be true. We cannot draw up a set of axioms about reality, we can only observe.

 

[mathwonk]

It is depressing to find questions like this on a math forum.

 

[Mentallic]

When there are obviously that many different opinions on the subject? Most certainly not.

 

[HallsofIvy]

It is depressing to find questions like this on a math forum.

Why depressing? Obviously no one conclusion is going to be decided because there are many different ideas as to what "science" or, for that matter, "mathematics" are, but it is always worth while airing differences.

 

[sponsoredwalk]

Mathematics is the language of discovery, it permeates every significant aesthetic and
mental achievement. If we take the word science at face value, i.e. sciere= knowledge,
then mathematics is surely the main tool by which knowledge is achieved.
Theoretical physics is the application of a mathematical model to physical phenomena
aimed at both description & prediction yet mathematics is not a science but a
theoretical physicist relying mainly on mathematics is somehow, I don't get it.

 

[Tom Gilroy]

Mathematics most certainly is not a science.

In science we observe and experiment the natural world. In mathematics we are studying something in which we have no idea wether we are totally right or wrong.

Here you have given the definition of natural science, however the word "science" itself has other meanings, which do encompass mathematics. Moreover, the concepts of right and wrong aren't evenly slightly relevant in mathematics, mathematics is about what is consistent and provable with respect to a given set of axioms.

Mathematics is built upon a set of axioms or assumptions.

And the natural sciences are not? Every scientific theory itself has a set of postulates, which are essentially equivalent to axioms, or assumptions. We only ever have evidence that the postulates are reasonable based on observation, but then we must assume that only what observably manifests itself has any effect.

Moreover, each area of theoretical physics makes many assumptions that there is no evidence for (like the cosmological principle, for example). Further, many of these fields are currently studying systems where there is absolutely no possibility of experimentation or observation, and where there is there is often enormous uncertainty in the observation (cosmology, in particular is plagued by this). As you have defined science, these fields are not sciences.

There is no true way to prove these axioms therefore mathematics will never be complete.

Well, frankly if you believe in the arithmetic of the natural numbers and believe in some of the foundational rules of logic, then you're obliged to believe the vast majority of mathematics, as the vast majority of mathematics can be formalised purely using the axioms of 2nd order arithmetic.

But you argue, are we not unsure of science as well? No. For science as i have defined it is the observation and modelling of reality. If we observe it, correctly without error, consistently then it must be true.

Observation and modelling of reality sure, which is based on many assumptions (including, wherever mathematics is used in the process, that the axioms of mathematics are correct). Also, some degree of error in observation is unavoidable. We cannot perform measurements with arbitrary precision.

We cannot draw up a set of axioms about reality, we can only observe.

It may be so that there is no way to write a list of axioms for science which would never need to be changed or rewritten. However, if you really believe that science is not built upon assumptions, you are very wrong.

 

[micromass]

Here you have given the definition of natural science, however the word "science" itself has other meanings

So, what other meanings do science have?

 

[Tom Gilroy]

So, what other meanings do science have?

Several, the most relevant definition that would encompass mathematics would be a body of knowledge attained through sytematic study, this particular definition being broad enough to include the natural sciences and mathematics.

http://dictionary.reference.com/browse/science

Certainly if computer science is a science, then mathematics must also qualify.