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## #1 2008-10-10 05:41:23

Ricky
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Registered: 2005-12-04
Posts: 3,791

### Debate #001: Is math a science?

Go.

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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## #2 2008-10-10 06:01:01

G_Einstein
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Registered: 2008-08-30
Posts: 124

### Re: Debate #001: Is math a science?

I think,yes.Math is a science

Se Zoti vete e tha me goje,se kombet shuhen permbi dhe,por SHqiperia do te roje,per te,per te luftojme ne.
God said that all nation exincts on the ground,but Albania will survive,for it,for it we are fighting.

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## #3 2008-10-10 06:12:26

Ricky
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Posts: 3,791

### Re: Debate #001: Is math a science?

The point of debate is to argue.  Why do you think this?  How does mathematics fit in with the other sciences?  What is the definition of science to you?

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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## #4 2008-10-10 07:34:57

mathsyperson
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Registered: 2005-06-22
Posts: 4,900

### Re: Debate #001: Is math a science?

You say that, but his post was still longer than yours. You shouldn't be impartial.

I would say that maths isn't a science.
For me, the process of science is something like:

Make a theory.
Test with experiment.
Does it work?
If yes, relax until someone comes up with another experiment.
In either case, go back to step 2.

One of the major requirements for some statement to be a scientific theory is that it must be falsifiable. If it is false, there must be a way to prove beyond all doubt that that is so.
Scientists love experiments that don't match with their theories, because those cause breakthroughs.

On the other hand, maths requires absolute knowledge.
A statement's proof in mathematics must be completely watertight and once proven, nothing could ever possibly happen to suddenly make that statement false.
In that respect, maths and science are complete opposites.

It's hard to find a general word like 'science' to describe what maths is, but I would say it's closer to being an art.

Why did the vector cross the road?
It wanted to be normal.

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## #5 2008-10-10 07:37:07

integer
Member
Registered: 2008-02-21
Posts: 79

### Re: Debate #001: Is math a science?

By my definition, math is a science.
The word science is derived from the French word scire (to know).

Thus, if you acquire knowledge about something in a systematic way, you have a science.
If the information cannot be verified independently, then you do not have a science.

See this
http://en.wikipedia.org/wiki/Mathematics

This quote is from there:

Carl Friedrich Gauss referred to mathematics as "the Queen of the Sciences".   the word corresponding to science means (field of) knowledge there is no doubt that mathematics is in this sense a science. The specialization restricting the meaning to natural science is of later date. If one considers science to be strictly about the physical world, then mathematics, or at least pure mathematics, is not a science.

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## #6 2008-10-10 08:40:17

Ricky
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Registered: 2005-12-04
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### Re: Debate #001: Is math a science?

You say that, but his post was still longer than yours. tongue You shouldn't be impartial.

I completely agree.  But I don't want to impose my view before I hear what others think.  First off, it might stop some more cautious poster from sharing their ideas.  Also, it allows me to play devil's advocate, making the discussion all that much more fun.

For me, the process of science is something like:

Make a theory.
Test with experiment.
Does it work?
If yes, relax until someone comes up with another experiment.
In either case, go back to step 2.

But what about a similar process with mathematics?

Make a conjecture.
Can you find counter examples?
If yes, come up with a new conjecture.
If no, attempt to prove your conjecture.
If proven, go back to step 1 by weakening hypotheses or generalizing.
If not proven, go back to step 2.

Is this so entirely different from science?

One of the major requirements for some statement to be a scientific theory is that it must be falsifiable.

Would you not say that a conjecture is falsifiable by finding a counter example?

If it is false, there must be a way to prove beyond all doubt that that it so.

I would add "reasonable" in there, somewhere.

A statement's proof in mathematics must be completely watertight and once proven, nothing could ever possibly happen to suddenly make that statement false.

Save a change in the axioms of the system.

Thus, if you acquire knowledge about something in a systematic way, you have a science.

So is my accountant a scientist?  Certainly they acquire knowledge in a systematic way.  And it can most certainly be independently verified.  I suppose you were thinking of a specific type of knowledge, no?

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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## #7 2008-10-10 12:37:33

Chewy
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Registered: 2008-08-07
Posts: 67

### Re: Debate #001: Is math a science?

Math must not be science, because in grammar school we had them as seperate subjects

Seriously, though, a very interesting topic ! I will give this some thought. Good point-counterpoint guys, let's keep it going.

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## #8 2008-10-10 18:27:55

ganesh
Registered: 2005-06-28
Posts: 26,164

### Re: Debate #001: Is math a science?

Chewy,
I agree with mathsyperson in his earlier post in this thread.
Mathematicians work in an entirely different way from scientists.
Although Mathematics is an integral part of any physical science like Physics or Chemistry, and of late, the medical sciences too,
one should understand Mathematics is the highest form of logic.
That apart, mathematical proofs are much different from scientific theories which rely more on observation. Mathematical theories are known to be fool-proof. Come to think of it, some day Archimides principle may proved to be wrong or so Coulumb's law, but can ever to eternity Pythagoras' theorem be proved wrong?
Never, never. This has stood the test of time. This is because of the rigors of a mathematical proof. The number of tests it is subjected to.
Does science have any proof by induction or proof by contradiction? The answer is a big NO!

It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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## #9 2008-10-11 00:01:09

Chewy
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Registered: 2008-08-07
Posts: 67

### Re: Debate #001: Is math a science?

Ganesh,

I agree with mathsyperson as well, and with you in fact. Good points in your last post, Ganesh !

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## #10 2008-10-11 02:01:24

Identity
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Registered: 2007-04-18
Posts: 934

### Re: Debate #001: Is math a science?

I don't think math is a science.

The way I see it is scientists make up their theories according to empirical data and observation.

Mathematicians on the other hand make up their axioms and everything else falls in place. I don't think there is a scientific method to maths.

Scientists are 'forced' to use the scientific method because there exists no other method which will create new and more accurate theories. Hence, there is no proof in science, only evidence, and everything is potentially falsifiable.

With mathematics, you can prove something to be true and there will never be a disproof, as long as you follow the axioms you began with. We shouldn't be focusing on the process of obtaining a solution, but the strength of the solution itself.

Last edited by Identity (2008-10-11 02:05:57)

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## #11 2008-10-11 02:04:27

All_Is_Number
Member
Registered: 2006-07-10
Posts: 258

### Re: Debate #001: Is math a science?

mathsyperson wrote:

For me, the process of science is something like:

Make a theory.
Test with experiment.
Does it work?
If yes, relax until someone comes up with another experiment.
In either case, go back to step 2.

One of the major requirements for some statement to be a scientific theory is that it must be falsifiable. If it is false, there must be a way to prove beyond all doubt that that is so.
Scientists love experiments that don't match with their theories, because those cause breakthroughs.

I think you might be confusing theory with hypothesis.

A hypothesis is what is tested via experimentation.

A theory is a construct, usually mathematical, that utilizes observable scientific laws in such a manner that predictions can be made based on initial conditions. It is often comparable to a mathematical theorem. (Note that a hypothesis could be that a theory is correct/incorrect under certain conditions.)

A hypothesis might be that the amount of time that lapses varies between two reference frames that are not stationary relative to one another. An experiment might be to synchronize two atomic clocks, place one aboard a commercial jet, and leave one on the ground. Then, have the jet take off and fly at its cruising altitude and speed for a few hours, landing at the same airport from which it took off. Compare the times on the clocks. Are they still synchronized? If no, then the experiment has provided evidence supporting the hypothesis. If not, then such evidence was not supported, and the hypothesis needs to be modified, the experiment needs to be refined (e.g. put an atomic clock aboard a space shuttle instead of a commercial jet), or the hypothesis needs to be refined.

Incidentally, relativity theory predicts the amount of time that lapses for the two reference frames will[/b] be different. Einstein derived relativity theory, special and general, mathematically, not experimentally.

After a successful experiment, scientists don't get to "relax." They share their work and data with other scientists, for peer review, to ensure the data actually supports the conclusion, and testing to see if the results can be reproduced independently.

(That's an [i]extremely simplified explanation of the scientific experimental process.)

You can shear a sheep many times but skin him only once.

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## #12 2008-10-11 02:46:55

All_Is_Number
Member
Registered: 2006-07-10
Posts: 258

### Re: Debate #001: Is math a science?

ganesh wrote:

Mathematicians work in an entirely different way from scientists.

They work in very similar ways. Experimental scientists, however, typically don't have the luxury of substituting x, y, and z (i.e. arbitrary variables) in place of constants to demonstrate that something holds for all cases, so they have to do much more testing. Theoretical scientists rely solely on mathematics and imagination.

ganesh wrote:

Although Mathematics is an integral part of any physical science like Physics or Chemistry, and of late, the medical sciences too, one should understand Mathematics is the highest form of logic.

Agreed.

ganesh wrote:

That apart, mathematical proofs are much different from scientific theories which rely more on observation.

This only holds true with experimental science, not theoretical science. Relativity Theory, for example, was established long before its implications were ever observed.

Also, mathematics have relied on observation throughout their development also. The constant relationship between a circle's circumference and it's diameter was observed to be a constant long before anyone understood what or why it was so. The 3-4-5 right triangle was observed to be a right triangle before the Pythagorean theorem was derived.

ganesh wrote:

Mathematical theories are known to be fool-proof.

They, too, have their limitations. For example, we all know that the inside angles of any triangle sum to 180º. Except that only holds true for Euclidian spaces, which only really exist in manmade constructs. It is a convenient tool, since we live in the midst of man made constructs, but it holds no absolute truth. Certain conditions must be assumed, such as parallel lines never intersecting.

ganesh wrote:

Come to think of it, some day Archimides principle may proved to be wrong or so Coulumb's law, but can ever to eternity Pythagoras' theorem be proved wrong? Never, never. This has stood the test of time.

Sure it can. All one needs to do is show that it does not hold true in non-Euclidean spaces, and that non-Euclidean spaces exist. At that point, the Pythagorean theorem has to be modified to specify that it only applies in Euclidian space.

ganesh wrote:

This is because of the rigors of a mathematical proof. The number of tests it is subjected to.

Tests? As in mathematical experiments?

ganesh wrote:

Does science have any proof by induction or proof by contradiction? The answer is a big NO!

Proof by contradiction is how the experimental process is used in attempt to disprove hypotheses. It is the cornerstone of the scientific method. Science does not have the ability to prove things to be true, only the ability to prove things are not true. Nothing is known in science with 100% certainty, except those things that are defined (e.g. we know the speed of light with 100% certainty because we have defined the meter's length in terms of the distance light travels in a vacuum in a fixed amount of time). All that can be known is that a hypothesis has been rigorously tested many, many times, and has never been disproved.

Science and mathematics are tightly interwoven. Neither would exist without the other. They share many methodologies. In many ways, mathematics are science, and in many ways, science is a branch of mathematics.

You can shear a sheep many times but skin him only once.

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## #13 2008-10-12 18:41:10

Tigeree
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Registered: 2005-11-19
Posts: 13,883

### Re: Debate #001: Is math a science?

I think math is a science, but a mathsy sort of science. D'ya get wot I mean?

People don't notice whether it's winter or summer when they're happy.
~ Anton Chekhov
Cheer up, emo kid.

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## #14 2008-10-12 20:39:36

ganesh
Registered: 2005-06-28
Posts: 26,164

### Re: Debate #001: Is math a science?

Thanks, All_Is_Number,
Certainly they were enlightening.
But have you heard of the problem of the 'Mutilated Chess Board'?

It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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## #15 2008-10-13 06:51:07

Ricky
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Registered: 2005-12-04
Posts: 3,791

### Re: Debate #001: Is math a science?

Posted by All_Is_Number:

That apart, mathematical proofs are much different from scientific theories which rely more on observation.

This only holds true with experimental science, not theoretical science. Relativity Theory, for example, was established long before its implications were ever observed.

Relativity theory was established based on the observation that the speed of light has a constant value no matter how fast the object emitting that light is traveling.  The entire mathematical framework rested upon a thought experiment, but that thought experiment could only be played out because of observations.

The 3-4-5 right triangle was observed to be a right triangle before the Pythagorean theorem was derived.

The argument is that sciences such as physics rely more on observations.  The argument is not that mathematics doesn't rely on observations.

They, too, have their limitations. For example, we all know that the inside angles of any triangle sum to 180º. Except that only holds true for Euclidian spaces, which only really exist in manmade constructs. It is a convenient tool, since we live in the midst of man made constructs, but it holds no absolute truth. Certain conditions must be assumed, such as parallel lines never intersecting.

Of course mathematical theorems fail when you remove the hypotheses, there is nothing special in this.  However, it does not bridge the gap between giving evidence for something in science and proving something in mathematics.  The fundamental different is still there: science says "this happens every time we've seen it, so it's always true" where as mathematics says "this is always true".

Science does not have the ability to prove things to be true, only the ability to prove things are not true.

This is argued in depth in the works of Duhem, a major critic of Popper who was the one who originally came up with falsifying as a way to test theories in science.  Duhem argued that at the edge of science, observations almost always rest up underlying theories.  The result is that if you "prove" something is not true that requires an underlying theory, then your "proof that X is not true" is only as sturdy as the theories it relies on.  Therefore, it is not possible do disprove a theory.  Indeed, it may be your disproof that is incorrect.

I don't want to go off the topic, so if you want to talk more about falsification in science, I would ask you to start another thread.

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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## #16 2008-10-13 16:48:48

All_Is_Number
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Registered: 2006-07-10
Posts: 258

### Re: Debate #001: Is math a science?

ganesh wrote:

But have you heard of the problem of the 'Mutilated Chess Board'?

Yes, although I must admit that I don't immediately recognize the relevance.

You can shear a sheep many times but skin him only once.

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## #17 2008-10-13 18:10:56

All_Is_Number
Member
Registered: 2006-07-10
Posts: 258

### Re: Debate #001: Is math a science?

Ricky wrote:

Relativity theory was established based on the observation that the speed of light has a constant value no matter how fast the object emitting that light is traveling.  The entire mathematical framework rested upon a thought experiment, but that thought experiment could only be played out because of observations.

More precisely, one of Einstein's two postulates from which relativity theory was derived was consistent with (but not based upon) the lack of observation of the speed of light varying with respect to the observer's frame of reference relative to the ether. Observation of such a difference had been expected, based on the maths. Let's not forget that the Michelson-Morley experiment was a failure from the perspective of the experimenters.

Of course mathematical theorems fail when you remove the hypotheses, there is nothing special in this.

I'm not sure what you are referring to with hypothesis. A mathematical theorem (as well as a scientific theory) is typically well beyond the hypothesis stage.

However, it does not bridge the gap between giving evidence for something in science and proving something in mathematics.

That gap will always exist, for mathematics exist conceptually, while science is the application of those conceptual tools in the real world. Conceptually, all possibilities can be tested. That's not usually possible in the real world. When working in the real world, it is not often possible to invent a space in which certain problems do not exist. In maths, we can model a particle with a velocity of zero. In reality, a particle in such a state could never be observed. Scientists do not have the luxury of being able to define the environment in which they work. Scientists are continually discovering properties of the environment in which they work.

The fundamental different is still there: science says "this happens every time we've seen it, so it's always true" where as mathematics says "this is always true".

It would be more accurate to say that science claims their models closely resemble reality, based on predictions repeatedly matching subsequent observations. Most scientists are under no illusion of knowing the Truth. Even Einstein recognized that relativity is not absolutely accurate. However, most every scientist recognizes that relativity theory models reality more accurately than Newton's Laws of Motion.

Duhem argued that at the edge of science, observations almost always rest up underlying theories.  The result is that if you "prove" something is not true that requires an underlying theory, then your "proof that X is not true" is only as sturdy as the theories it relies on.  Therefore, it is not possible do disprove a theory.  Indeed, it may be your disproof that is incorrect.

I don't disagree with Duhem's argument as you've expressed it. (It was actually the soundness of the design of the experiment that resulted in the failed Michelson-Morley experiment becoming one of the most famous physics experiments ever.) This is one of the reasons that rigorous peer review is so important in science. By the same token, there are many, many invalid mathematical proofs available, such as those that "prove" 1=0. Of course, the errors may not be so obvious when not made intentionally. This is one of the reasons that rigorous peer review is so important in mathematics.

Comparing mathematics and science is like comparing a toolmaker with a mechanic or craftsman. Without each other, they would both need to find alternate work. Their relationship is of a symbiotic nature.

You can shear a sheep many times but skin him only once.

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## #18 2008-10-13 23:00:45

Ricky
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Registered: 2005-12-04
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### Re: Debate #001: Is math a science?

I'm not sure what you are referring to with hypothesis. A mathematical theorem (as well as a scientific theory) is typically well beyond the hypothesis stage.

I apologize.  In mathematical proofs, it is typical to refer to the states you assume to be true in an implication to be a hypothesis.  For an example, look at the first proof, third line down:

"Since we assumed, by hypothesis"

Likewise, the fact that we are in Euclidean geometry is a hypothesis that goes into the statement of Pythagoras.  It's important to remember that hypotheses can exist implicitly as well as explicitly.

That gap will always exist, for mathematics exist conceptually, while science is the application of those conceptual tools in the real world. Conceptually, all possibilities can be tested. That's not usually possible in the real world. When working in the real world, it is not often possible to invent a space in which certain problems do not exist. In maths, we can model a particle with a velocity of zero. In reality, a particle in such a state could never be observed. Scientists do not have the luxury of being able to define the environment in which they work. Scientists are continually discovering properties of the environment in which they work.

This is a great argument for why mathematics is not science.  Did you intend this?

It would be more accurate to say that science claims their models closely resemble reality, based on predictions repeatedly matching subsequent observations.

That's just a PC version of "it's true".  I could spend an hour with each post rewording all my statements to be technically correct, but I'm not going to do that.  Obviously truth in my previous quote was referring to scientific truth.

I don't disagree with Duhem's argument as you've expressed it. (It was actually the soundness of the design of the experiment that resulted in the failed Michelson-Morley experiment becoming one of the most famous physics experiments ever.) This is one of the reasons that rigorous peer review is so important in science. By the same token, there are many, many invalid mathematical proofs available, such as those that "prove" 1=0. Of course, the errors may not be so obvious when not made intentionally. This is one of the reasons that rigorous peer review is so important in mathematics.

Are you saying that you believe rigorous peer review can solve the problem in falsification as illustrated by Duhem?

Comparing mathematics and science is like comparing a toolmaker with a mechanic or craftsman. Without each other, they would both need to find alternate work. Their relationship is of a symbiotic nature.

Not a bad comparison, but certainly not complete.  Mathematics doesn't just make tools for scientists to use, we do much more than that.  But from this, I would conclude you are of the opinion that science and mathematics are fundamentally different.  Is this correct?

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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## #19 2008-10-15 12:40:18

All_Is_Number
Member
Registered: 2006-07-10
Posts: 258

### Re: Debate #001: Is math a science?

Ricky wrote:

This is a great argument for why mathematics is not science.

Not at all. It just illuminates that mathematician define their working environment to investigate the properties of their tools, while scientists use the known properties of the tools to investigate the characteristics of their working environments.

That's just a PC version of "it's true".

No, it's not the same thing. Scientists are generally very careful to distinguish between the two things.

I could spend an hour with each post rewording all my statements to be technically correct, but I'm not going to do that.  Obviously truth in my previous quote was referring to scientific truth.

It wasn't how you wrote it, it was what you wrote.

Are you saying that you believe rigorous peer review can solve the problem in falsification as illustrated by Duhem?

No, I'm saying that "I don't disagree with Duhem's argument as you've expressed it" (as of your previous post). I've not researched exactly what Duhem's assertions were, so beyond your short description, I haven't enough knowledge to form an opinion about them. Having said that, rigorous peer review is an important component of scientific (including mathematical) research.

Comparing mathematics and science is like comparing a toolmaker with a mechanic or craftsman. Without each other, they would both need to find alternate work. Their relationship is of a symbiotic nature.

Not a bad comparison, but certainly not complete.  Mathematics doesn't just make tools for scientists to use, we do much more than that.  But from this, I would conclude you are of the opinion that science and mathematics are fundamentally different.  Is this correct?

Mathematics is no more different from Science than cosmologists are different from biologists. They all have fundamental differences, but those differences are outnumbered by their fundamental similarities.

Last edited by All_Is_Number (2008-10-15 12:41:39)

You can shear a sheep many times but skin him only once.

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## #20 2008-10-15 13:29:59

MathsIsFun
Registered: 2005-01-21
Posts: 7,685

### Re: Debate #001: Is math a science?

Mathematics transcends what I normally think of as Science, as it investigates things far beyond the physical universe ... in fact it is seldom concerned with the physical universe.

So I guess it all depends how you define Science.

Some definitions limit it to the physical universe:

Wikipedia (not a great source, but handy): Science (from the Latin scientia, meaning "knowledge" or "knowing") is the effort to discover, and increase human understanding of how the physical world works.

ESA (http://www.esa.int/esaMI/Lessons_online/SEMIBLPR4CF_0.html): Science - The systematic study of the structure and behaviour of the physical world, especially by observing, measuring and experimenting, and the development of theories to describe the results of these activities

Others give a much broader definition:

Merriam-Webster http://www.merriam-webster.com/dictionary/science: the state of knowing : knowledge as distinguished from ignorance or misunderstanding

"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

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## #21 2008-10-15 13:57:08

All_Is_Number
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Registered: 2006-07-10
Posts: 258

### Re: Debate #001: Is math a science?

MathsIsFun wrote:

Mathematics transcends what I normally think of as Science, as it investigates things far beyond the physical universe ... in fact it is seldom concerned with the physical universe.

There is no information within the universe that is not a part of the universe, so I'm not sure how mathematics can investigate things beyond the universe.

From Wikipedia's Mathematics entry:

"The mathematician Benjamin Peirce called it 'the science that draws necessary conclusions'. Other practitioners of mathematics maintain that mathematics is the science of pattern, and that mathematicians seek out patterns whether found in numbers, space, science, computers, imaginary abstractions, or elsewhere."

You can shear a sheep many times but skin him only once.

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## #22 2008-10-15 16:45:36

MathsIsFun
Registered: 2005-01-21
Posts: 7,685

### Re: Debate #001: Is math a science?

All_Is_Number wrote:

There is no information within the universe that is not a part of the universe, so I'm not sure how mathematics can investigate things beyond the universe.

I see a distinction between logical and physical.

"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

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## #23 2008-10-16 00:36:25

All_Is_Number
Member
Registered: 2006-07-10
Posts: 258

### Re: Debate #001: Is math a science?

MathsIsFun wrote:

I see a distinction between logical and physical.

You can shear a sheep many times but skin him only once.

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## #24 2008-10-16 05:30:34

mathsyperson
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Registered: 2005-06-22
Posts: 4,900

### Re: Debate #001: Is math a science?

All_Is_Number wrote:

Mathematics is no more different from Science than cosmologists are different from biologists. They all have fundamental differences, but those differences are outnumbered by their fundamental similarities.

I'd have thought that it doesn't matter how many similarities there are, if there is a difference then they are different.
Ducks have a beak like a goose, have feathers like a goose and swim like a goose.
But ducks quack and geese honk, so ducks aren't geese.

Why did the vector cross the road?
It wanted to be normal.

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## #25 2008-10-16 07:18:16

All_Is_Number
Member
Registered: 2006-07-10
Posts: 258

### Re: Debate #001: Is math a science?

mathsyperson wrote:
All_Is_Number wrote:

Mathematics is no more different from Science than cosmologists are different from biologists. They all have fundamental differences, but those differences are outnumbered by their fundamental similarities.

I'd have thought that it doesn't matter how many similarities there are, if there is a difference then they are different.
Ducks have a beak like a goose, have feathers like a goose and swim like a goose.
But ducks quack and geese honk, so ducks aren't geese.

By that logic, physicists and cosmologists can't both be scientists, because they are different.

Your analogy would be more accurate, if we equated scientists to birds, mathematicians to ducks, physicists to geese, cosmologists to quail, etc. Different species of birds are different, yet they are all species of birds.

You can shear a sheep many times but skin him only once.

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