Debate #001: Is math a science?

Ricky · 2008-10-16 07:52:50

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.

It seems that we are starting to get to the main idea. When ever you want to group a set of things into different categories, you need criteria. A good choice for criteria is one which is easily decidable and relates to a fundamental similarity between that which you are grouping. Of course there are many other things to consider, and these are typically only relevant to your specific set.

There is one fundamental idea that is shared universally between all things which we all agree are sciences (astronomy, biology, chemistry, physics, geology, paleontology, archaeology, etc). This whole idea of "falsification". It is the way that science works. We can keep an idea until it gets falsified by tests, and the things we say are "true" are merely "not falsified yet". Take any subject which is agreed by virtually everyone to be a science, and I can guarantee you the fundamental principle at the heart of it is falsification.

This idea of falsification is not shared by mathematicians. Indeed, mathematicians prove statements don't hold by counterexamples, but this is very far removed from falsification. When a mathematician says a mathematical statement is true, it does not simply mean "not falsified yet". A mathematical theorem being regarded as true is a much stronger statement then saying a theory in physics is true.

In the end, is this not what we want? A statement in mathematics is indeed stronger than a statement in the sciences. By grouping mathematics into the sciences, you are at the very least making it seem that statements in mathematics carry just as much weight as statements in evolutionary biology, when of course this couldn't be further from the truth. By saying mathematics is a science, you conflate the idea of mathematical truth with scientific truth, when in fact these two are quite far apart.

Perhaps an even more clear criteria is the way in which mathematical knowledge and scientific knowledge grows. The only way a revolution in mathematics can occur is if someone makes a statement which is not true. In order to do this, a mistake must be made, and it must have gotten passed all the people in the mathematical community that said "yea, that's right". That is to say, revolutions (overturning past ideas) are not expected in mathematics. On the other hand, revolutions are not only expected in the sciences, but they are demanded. We fully expect that theories in physics will eventually be overturned, and even more so in biology or archaeology, because that's how science works. It's an approximation to the truth that only gets better and better. Mathematics on the other hand starts with small systems and grows and abstracts to get bigger and bigger. But we don't expect someone to suddenly tell us that Euler's theorem on the existence of Euler paths in graphs is incorrect. Sure, it could happen that a flawed theorem is accepted by the mathematical community. But this is entirely different than the sciences because in the sciences, we know from the start that every theory is flawed.

Edit: When I say scientific "theory", what I mean is an explanation for a large set of data. For example, g = 9.8m/s^2 is not a theory, but Relativity is. That the Earth rotates around the Sun is not a theory, but the evolution of planetary systems is. That gene frequency in populations change over time is not a theory, but the theory of Evolution is.

All_Is_Number · 2008-10-16 09:10:48

Ricky wrote:

Take any subject which is agreed by virtually everyone to be a science, and I can guarantee you the fundamental principle at the heart of it is falsification.
This idea of falsification is not shared by mathematicians. Indeed, mathematicians prove statements don't hold by counterexamples, but this is very far removed from falsification.

It's not far removed from falsification, it is falsification.

When a mathematician says a mathematical statement is true, it does not simply mean "not falsified yet". A mathematical theorem being regarded as true is a much stronger statement then saying a theory in physics is true.

In both disciplines, conditions are set implicitly or explicitly that must hold in order for the statements to be true.

By saying mathematics is a science, you conflate the idea of mathematical truth with scientific truth, when in fact these two are quite far apart.

Not at all. That's like saying biological truths are conflated with geological truths because they are both sciences.

Perhaps an even more clear criteria is the way in which mathematical knowledge and scientific knowledge grows. The only way a revolution in mathematics can occur is if someone makes a statement which is not true. In order to do this, a mistake must be made, and it must have gotten passed all the people in the mathematical community that said "yea, that's right". That is to say, revolutions (overturning past ideas) are not expected in mathematics.

Of course not, because everyone in the mathematical community has said "yea, that's right." Unexpected does not imply unlikely.

On the other hand, revolutions are not only expected in the sciences, but they are demanded.

Demanded by whom? The last revolution of science (restricting my statements to Physics, Cosmology, and Quantum Theory) was the development of quantum theory. Before that, Newton's laws of motion and calculus. Everything else, including relativity, has been more evolutionary than revolutionary.

We fully expect that theories in physics will eventually be overturned, and even more so in biology or archaeology, because that's how science works. It's an approximation to the truth that only gets better and better.

Sounds a lot like numerical methods. Oh, wait

Last edited by All_Is_Number (2008-10-16 09:11:47)

Ricky · 2008-10-16 12:33:31

It's not far removed from falsification, it is falsification.

Not in the least! Falsification is a method under which an idea goes strenuous test after test and we hold it true so long as it continues to pass tests. Mathematicians do not accept ideas which have only resisted tests, mathematicians only accept ideas which have been proven. This is why we have not concluded that the Riemann Hypothesis is true, no matter how many of the zero's we have verified fit.

When a mathematician says a mathematical statement is true, it does not simply mean "not falsified yet". A mathematical theorem being regarded as true is a much stronger statement then saying a theory in physics is true.
In both disciplines, conditions are set implicitly or explicitly that must hold in order for the statements to be true.

You're either not understanding my point, or you're ignoring it. When a mathematician says something is true, he means for it to be proven so in the system he is studying. When a scientist says something is true, he means that it has not been falsified yet. These two ideas are entirely different.

Not at all. That's like saying biological truths are conflated with geological truths because they are both sciences.

Truths in biological and geological sciences are not far apart at all. They both rely on falsification, so it is perfectly fine to conflate the two.

Of course not, because everyone in the mathematical community has said "yea, that's right." Unexpected does not imply unlikely.

Show me the last time a mathematical idea accept by the mathematical community was shown to be incorrect. Not that it matters anyways, I was not arguing that mistakes in mathematics are unlikely. Rather, the entire argument was unexpected. Whether or not they are likely does not hold any significance in my showing the difference between growth of information in the sciences and that in mathematics. Saying it is "likely", or even that it could be likely, is a non sequitur.

Demanded by whom? The last revolution of science (restricting my statements to Physics, Cosmology, and Quantum Theory) was the development of quantum theory. Before that, Newton's laws of motion and calculus. Everything else, including relativity, has been more evolutionary than revolutionary.

Someone hasn't be paying attention to string theory, the theory behind planetary evolution, expansion of the universe, the existence of dark matter, the existence of dark energy, the existence of dark flow. Relativity was certainly revolutionary, it showed for one thing that gravity was a force which altered spacial dimensions rather than a "pulling" force, as Newton thought of it. That turned physics up on it's head! Of course, find all the elementary particles was also revolutionary since before hand the atom was thought to be an indivisible piece of matter.

And those are really just the big things.

We fully expect that theories in physics will eventually be overturned, and even more so in biology or archaeology, because that's how science works. It's an approximation to the truth that only gets better and better.
Sounds a lot like numerical methods. Oh, wait

Your comparison is rather ridiculous. You seem to be attempting to conflate iteration and recursion with the iterative method of science. The theorems and algorithms in numerical analysis don't change, but the theories in the sciences do. Every day.

mikau · 2008-10-16 13:23:07

Is math a science? Isn't that entirely up to Zach?

All_Is_Number · 2008-10-16 14:21:07

Ricky wrote:

Not in the least! Falsification is a method under which an idea goes strenuous test after test and we hold it true so long as it continues to pass tests. Mathematicians do not accept ideas which have only resisted tests, mathematicians only accept ideas which have been proven. This is why we have not concluded that the Riemann Hypothesis is true, no matter how many of the zero's we have verified fit.

Both discard ideas once there is evidence to suggest they are not true. Mathematicians have not abandoned the Riemann hypothesis, because there is no evidence to suggest it is not true. Scientists do not claim relativity to be correct. Scientists do not claim Newton's laws of motion to be correct. They recognize them for what they are, models that approximate reality closely under particular conditions.

You keep making assertions about scientists that aren't true.

You're either not understanding my point, or you're ignoring it. When a mathematician says something is true, he means for it to be proven so in the system he is studying. When a scientist says something is true, he means that it has not been falsified yet. These two ideas are entirely different.

I understand your claim. I just recognize it for what it is: incorrect. Again, you are making unsubstantiated claim about scientists in order to falsely differentiate between them and mathematicians, who are themselves scientists. Scientists don't accept just any idea that hasn't yet been disproved. That's simply not how science works.

Truths in biological and geological sciences are not far apart at all. They both rely on falsification, so it is perfectly fine to conflate the two.

As do mathematicians, so why not conflate scientists and mathematicians? There's far fewer differences between physicists, cosmologists and mathematicians than there are between biologists and physicists or biologists and cosmologists.

Someone hasn't be paying attention to string theory, the theory behind planetary evolution, expansion of the universe, the existence of dark matter, the existence of dark energy, the existence of dark flow. Relativity was certainly revolutionary, it showed for one thing that gravity was a force which altered spacial dimensions rather than a "pulling" force, as Newton thought of it. That turned physics up on it's head! Of course, find all the elementary particles was also revolutionary since before hand the atom was thought to be an indivisible piece of matter.

Those are all evolutionary advances in science. Are you familiar with the events and discoveries that led to them? Einstein, for example, just took Mach's work a small step further with relativity. With regard to gravity, you are confusing the theory with the explanation.

Of course, find all the elementary particles was also revolutionary since before hand the atom was thought to be an indivisible piece of matter.

You'll notice that I mentioned quantum physics as being revolutionary. Thanks for reiterating the point.

Your comparison is rather ridiculous. You seem to be attempting to conflate iteration and recursion with the iterative method of science. The theorems and algorithms in numerical analysis don't change, but the theories in the sciences do. Every day.

Are you saying that numerical analysis has not been refined over the course of its existence? Interesting. And wrong.

What you seem to be missing is that even though a mathematician can, in a relatively short amount of time, check a hypothesis against all known possibilities, while the scientist must conduct experiment after experiment, or make observation after observation, which may take years (or decades, or longer), they are doing the same thing. They are testing their hypotheses for flaws. It is the very act of logically testing hypotheses that makes something a science. It doesn't make any difference whether it takes five minutes, or two centuries. As long as mathematicians continue to rely on logic in proofs, mathematics will remain a science. It's really that simple.

Ricky · 2008-10-16 15:13:04

Scientists do not claim relativity to be correct. Scientists do not claim Newton's laws of motion to be correct. They recognize them for what they are, models that approximate reality closely under particular conditions.
You keep making assertions about scientists that aren't true.

Yes, they do. But they understand when they say such things that they mean scientifically true and the difference between this an absolutely true (of course). But if you think mathematicians practice falsification, then you have a misunderstanding of the entire idea behind falsification in science. "Not yet falsified" is the highest level a scientific theory can get, and the same is not true in mathematics.

You're either not understanding my point, or you're ignoring it. When a mathematician says something is true, he means for it to be proven so in the system he is studying. When a scientist says something is true, he means that it has not been falsified yet. These two ideas are entirely different.
I understand your claim. I just recognize it for what it is: incorrect. Again, you are making unsubstantiated claim about scientists in order to falsely differentiate between them and mathematicians, who are themselves scientists. Scientists don't accept just any idea that hasn't yet been disproved. That's simply not how science works.

I didn't say that they do! You're trying to extrapolate on my words, and you're putting things in that aren't there. Of course for the scientific community to accept something it has to go under rigorous testing. But once this happens it still gets the label "not falsified". When the mathematical community accepts something, it gets the label "proved".

Those are all evolutionary advances in science. Are you familiar with the events and discoveries that led to them? Einstein, for example, just took Mach's work a small step further with relativity. With regard to gravity, you are confusing the theory with the explanation.

No, they aren't. When Einstein first came up with Relativity, he threw in some constants because otherwise his theory would imply that the universe was expanding. When Hubble showed that the universe was actually expanding, he took them out. Now we know the universe is not only expanding, but expanding at an accelerating rate. We must now put some constants back into Relativity equations so that they still hold. Each of these show an old idea being falsified and replaced with a new.

And a theory is an explanation. Relativity is formed around mass acting on spacetime. In Newtonian mechanics, gravity is a pushing force. This is why no one expected that gravity would act on light, because you can't push light. But you can act upon the spacetime that the light travels over, thereby influencing the direction and travel of the light.

Let me add another. The origin of the moon was actually shown to be from a meteor hitting earth, rather than a satellite being trapped in the Earth's orbit. This was an overturning of a previous accepted idea, a revolution.

You'll notice that I mentioned quantum physics as being revolutionary. Thanks for reiterating the point.

While the Standard Model is of course intertwined with quantum physics, the existence of elementary particles is not.

Are you saying that numerical analysis has not been refined over the course of its existence? Interesting. And wrong.

New theorems have been made, but the old theorems (even if no longer used) are just as true.

What you seem to be missing is that even though a mathematician can, in a relatively short amount of time, check a hypothesis against all known possibilities, while the scientist must conduct experiment after experiment, or make observation after observation, which may take years (or decades, or longer), they are doing the same thing.

You think a mathematician can check against all the numbers that only numbers equivalent to 1 modulo 4 can be written as a sum of two squares? Absolutely not. In all cases outside of discrete mathematics, mathematicians can not check a conjecture against all known possibilities. Indeed, once they come up with a proof, they don't check against possibilities as they know it would be fruitless.

All_Is_Number · 2008-10-16 15:44:30

Ricky wrote:

"Not yet falsified" is the highest level a scientific theory can get, and the same is not true in mathematics.

You still aren't getting it. Let's take Special relativity as an example. Scientists know it's not correct, and don't claim otherwise, despite your contrary and incorrect assertions.

Given the postulates Einstein started with, Special Relativity theory is absolutely correct. The uncertainty is in the postulates. That's no different from Euclid's work. It doesn't apply in the real world, but given his postulates, it holds absolutely true. The difference lies in mathematicians having the luxury of defining their working environment. Scientists don't have that luxury. They have to describe the world in the best terms they can. If those terms are wrong, it keeps their theories from being as accurate as they otherwise could be. However, given the environment they described and thought they were working in, their theories remain as valid as anything mathematicians come up with.

I didn't say that they do!

Good. Then you finally recognize that falsifiability is not the single defining characteristic of science. We're making progress.

You're trying to extrapolate on my words, and you're putting things in that aren't there. Of course for the scientific community to accept something it has to go under rigorous testing. But once this happens it still gets the label "not falsified". When the mathematical community accepts something, it gets the label "proved".

It is proved only for the given conditions. If those conditions change, the validity of the proof may or may not hold, just like any other science.

No, they aren't. When Einstein first came up with Relativity, he threw in some constants because otherwise his theory would imply that the universe was expanding. When Hubble showed that the universe was actually expanding, he took them out. Now we know the universe is not only expanding, but expanding at an accelerating rate. We must now put some constants back into Relativity equations so that they still hold. Each of these show an old idea being falsified and replaced with a new.

Each shows a theory being modified to better model reality as our understanding of reality becomes more precise. The conditions changed.

And a theory is an explanation.

Nothing could be farther from the truth. Well, okay, some things could, but you're completely wrong with that assertion.

The origin of the moon was actually shown to be from a meteor hitting earth, rather than a satellite being trapped in the Earth's orbit. This was an overturning of a previous accepted idea, a revolution.

You certainly have low standards for what defines revolution.

New theorems have been made, but the old theorems (even if no longer used) are just as true.

Only some have been replaced with more accurate approximations. Just like any other science.

You think a mathematician can check against all the numbers that only numbers equivalent to 1 modulo 4 can be written as a sum of two squares? Absolutely not.

Are you really being this disingenuous with your argument?

In all cases outside of discrete mathematics, mathematicians can not check a conjecture against all known possibilities. Indeed, once they come up with a proof, they don't check against possibilities as they know it would be fruitless.

Hmmm If we assume you are correct here, then it would imply that an absolute proof would not be possible. Yet you've claimed that such proofs are what differentiate mathematics from the other sciences. I guess we can consider the discussion complete.

Ricky · 2008-10-16 17:03:01

Given the postulates Einstein started with, Special Relativity theory is absolutely correct.

I would like to argue this point in depth because it seems like this is the major divide. Given the postulates that Einstein started with, his work was arguably mathematically correct. However this does not mean that it must carry over to the physical world. Just because I can subtract 5 feet from 3 meters per second does not mean it has a physical meaning.

I would like for you to illuminate exactly what you consider the postulates of his theory to be.

Oh and by the way, from the United States National Academy of Sciences:

Some scientific explanations are so well established that no new evidence is likely to alter them. The explanation becomes a scientific theory.

All_Is_Number · 2008-10-17 04:37:48

Ricky wrote:

Given the postulates Einstein started with, Special Relativity theory is absolutely correct.
I would like to argue this point in depth because it seems like this is the major divide. Given the postulates that Einstein started with, his work was arguably mathematically correct.

Given his postulates, his work is correct, not "arguably correct."

However this does not mean that it must carry over to the physical world.

No, it doesn't. But we can't hold his work to a higher standard than mathematicians' if we're going to compare the validity of science versus the validity of mathematics. If we can legitimately say that Einstein was wrong, then we can just as legitimately say that Euclid was wrong or that the Pythagorean theorem does not hold on a large scale in the natural world, and is therefore wrong.

On the other hand, if we say that Euclid and other mathematicians were correct, given his postulates, then we must extend that same courtesy to scientists' work if we are to compare their relative accuracy.

Just because I can subtract 5 feet from 3 meters per second does not mean it has a physical meaning.

Sorry, you cannot subtract 5 feet from 3 m/s. You may as well try to subtract 5 kilometers from 7 liters.

I would like for you to illuminate exactly what you consider the postulates of his theory to be.

For special relativity, Einstein started with two postulates:

1. The laws of physics are the same in all inertial reference frames.

2. The speed of light in a vacuum is equal to the value of c, independent of the motion of the source.

Oh and by the way, from the United States National Academy of Sciences:
Some scientific explanations are so well established that no new evidence is likely to alter them. The explanation becomes a scientific theory.

A theory, at least with respect to physics, quantum mechanics, cosmology, etc. are the mathematical constructs used to make predictions. Explanations are the words used to describe those theories. If the theory is once removed from reality, then the explanation is twice removed. All that is required of an adequate explanation is consistency with the mathematical theory. A small modification to the theory, such as special relativity provided for Newton's laws of motion, can cause major problems with the explanation.

Scientists are concerned with theories, while laymen and young school children are typically perfectly satisfied with explanations.

Ricky · 2008-10-17 08:30:12

Scientists are concerned with theories, while laymen and young school children are typically perfectly satisfied with explanations.

You cannot preform science by pretending it exists in a mathematical vacuum. The equations in physics all have meanings, and this meaning is separate from their mathematical meaning.

No, it doesn't. But we can't hold his work to a higher standard than mathematicians' if we're going to compare the validity of science versus the validity of mathematics. If we can legitimately say that Einstein was wrong, then we can just as legitimately say that Euclid was wrong or that the Pythagorean theorem does not hold on a large scale in the natural world, and is therefore wrong.

We say a theory in physics is correct if it allows us to make accurate predictions about the physical universe. Do you disagree?

All_Is_Number · 2008-10-17 09:17:42

Ricky wrote:

You cannot preform science by pretending it exists in a mathematical vacuum.

I never said or implied that you can.

The equations in physics all have meanings, and this meaning is separate from their mathematical meaning.

That's ridiculous. The equations mean the same in science as in math. If they didn't, the equations would be useless, and mathematical proofs meaningless. It is precisely because the equations have consistent meaning that makes them useful.

We say a theory in physics is correct if it allows us to make accurate predictions about the physical universe. Do you disagree?

That statement is oversimplified to the extent that I cannot agree or disagree.

George,Y · 2008-10-28 16:27:47

That the Earth rotates around the Sun is not a theory

Well I should point out it is not a fact, but definately only a view point simpler than that the Sun rotates around the Earth, which is true as well. Think about it.

Ricky · 2008-10-29 07:53:32

I want to first say, All_is_Number, that to me, you seem to be of the opinion that mathematics is a science and everyone who thinks differently is wrong. This is not a bad view to hold at all on some topics, indeed, I hold it on many of them. But in this case, I disagree. I find the question of whether or not mathematics is a science to be very much open to debate. I think that each side can make a compelling case to why it is and isn't, and that in the end it comes down to personal choice. I think it also has a lot to do with your background, and whether it is scientific in nature (a pun!) or mathematical. Do you agree with this?

Your view on theoretical physics is a bit odd. Or perhaps maybe it is mine which is odd, but let me explain. If we were to find that a hypothesis (again, let me state for clarity that a hypothesis is what is assume to be true for a conclusion to hold, this is the mathematical usage of the word) to Einstein's Theory of Relativity (ToR) is false, then eventually the ToR would be forgotten, it would be of no use. Einstein's logic would still be correct, but we would still regard the ToR as being false. Because in the end, what is true or false in physics relies on the universe telling us what holds and doesn't hold. The razor in physics that is used is "does it work?". Work of course meaning does it allow us to make accurate predictions. In mathematics, there is no such razor. We don't ask in mathematics whether or not it holds in the real world because the question alone does not make sense.

But it is very easy to get confused with theoretical physics. And indeed, you have caused me to reflect upon theoretical physics, I was asking myself if we could possibly classify theoretical physics as a mathematics. And the conclusion I came to was "no", after I found the error in my thinking. Theoretical physics requires so much mathematics that it is easy to get the two confused. Let me give a quick example.

In quantum physics, one of the most common problems is to find the wave function of a given system using initial conditions. This work is all mathematical in nature, solving a differential equation using various techniques and coming up with a linear combination of different states, and then finding out what the coefficients are by normalization.

The work above is mathematics in theoretical physics, but it is not theoretical physics. Theoretical physics is the work done before and after the equation is solved. If you can hand the work to a mathematician who knows absolutely nothing of physical interpretation of symbols, and he can solve the problem, then I would conclude that his work is mathematical in nature and not theoretical physics.

Theoretical physics on the other hand is discovering things like the Schrodinger equation, or coming up with experiments, or interpreting the mathematical work to it's physical meaning. We can thus separate the mathematics from the theoretical physics. What do you think of this?

bitus · 2008-10-29 10:21:41

Mathematics is not only the Queen of the Sciences, is the ONLY Science. The others are "something like Science". The most "un-scientific" elements are statistics, experiments and -sometimes- theories. In fact, the difference between Mathematics and other Sciences is the difference between a theorem and a theory. Many refer to Maths as "the pure science" to distinguish it from the others. Of course, it's an art too.

All_Is_Number · 2008-10-29 16:45:11

Ricky wrote:

Your view on theoretical physics is a bit odd. Or perhaps maybe it is mine which is odd, but let me explain. If we were to find that a hypothesis (again, let me state for clarity that a hypothesis is what is assume to be true for a conclusion to hold, this is the mathematical usage of the word) to Einstein's Theory of Relativity (ToR) is false, then eventually the ToR would be forgotten, it would be of no use.

Which is exactly why we no longer use Newton's laws of motion. Oh, wait, we still use them more often than relativity theory. Just as Euclidean geometry is "wrong" in the context of the real world, so are Newtonian mechanics. Yet, both are still extremely useful.

The razor in physics that is used is "does it work?". Work of course meaning does it allow us to make accurate predictions.

Whether or not it works has nothing to do with being right or wrong.

In mathematics, there is no such razor. We don't ask in mathematics whether or not it holds in the real world because the question alone does not make sense.

Nor does that question alone make any sense in any of the other sciences. The beginning assumptions can be incorrect, yet the theory can still be very useful.

The work above is mathematics in theoretical physics, but it is not theoretical physics. Theoretical physics is the work done before and after the equation is solved. If you can hand the work to a mathematician who knows absolutely nothing of physical interpretation of symbols, and he can solve the problem, then I would conclude that his work is mathematical in nature and not theoretical physics.
Theoretical physics on the other hand is discovering things like the Schrodinger equation, or coming up with experiments, or interpreting the mathematical work to it's physical meaning. We can thus separate the mathematics from the theoretical physics. What do you think of this?

You make it sound as though the difference between a mathematician and a scientist is like the difference between someone who is an expert at writing and someone who is not only capable of writing, but also knows something worth writing about.

Personally, I think mathematicians deserve more respect than that, but to each their own.

George,Y · 2008-10-31 01:53:47

Ricky, since maths is free of examination of reality, I wonder- is it free of examination of logic as well?

Can mathematics use logic when it needs it, and throw it away when it gets in the way?

then indeed it is the queen, very powerful and very inconstant.

Can a paradox be a legal concept in mathematics? And how should mathematicians regard it? What need they explain to outsiders who just need to use mathematics in reality? Do they say, you see, there is a flaw here, but we like it ,we really love it, it is the only way we think we come up with it, take it or leave it?

If you are a customer, what will you feel when a sales representitive says something like this about a car?

George,Y · 2008-10-31 01:59:12

. Just as Euclidean geometry is "wrong" in the context of the real world, so are Newtonian mechanics. Yet, both are still extremely useful

Well, it doesn't mean we use them without recognizing their faults, and without knowing how they coincidentally discribes things right in some cases, but will definately fail in other cases.

If the assumption is wrong, the theory is doomed to fail in some cases, even if it looks right, funcitions well in other cases. History acknowledges this fact and people's ignorance very often.

George,Y · 2008-10-31 02:01:46

People will take the result of a theory based on wrong assumption for granted and then for a long time accidentally "discover" the fault in reality, how smart!

George,Y · 2008-10-31 02:03:49

Just like the neo-liberalism and market-fundamentalism. We haven't come up with a substitute, so folks just bear with it and listen to our preach it is the truth we know so far even we know its assumptions are ungrounded.

All_Is_Number · 2008-10-31 05:07:52

George,Y wrote:

Well, it doesn't mean we use them without recognizing their faults, and without knowing how they coincidentally discribes things right in some cases, but will definately fail in other cases.

I agree. We still use Newton's equations because in most cases, the difference between them and the more accurate equations associated with relativity theory is less than the precision of the measurements can show (i.e. √(1-v²/c²)≈1).

Ricky · 2008-10-31 11:04:18

All_is_Number, you're right, my wording was completely off. Allow me the chance to restate what I was trying to say.

We consider the theory in science correct if it allows us to make accurate predictions about the universe in which we live. How "accurate" is "accurate" is up to debate, and I'd rather not journey down that road for fear we may not return. But it does need to be accurate, whatever that may mean varying from theory to theory and situation to situation. Do you find this more agreeable?

If you agree, then of course mathematics makes no predictions about the universe in which we live (by itself), so "accuracy" has no meaning. A model makes a prediction, sure enough, but a model is separate from the mathematics that it uses.

You make it sound as though the difference between a mathematician and a scientist is like the difference between someone who is an expert at writing and someone who is not only capable of writing, but also knows something worth writing about.
Personally, I think mathematicians deserve more respect than that, but to each their own.

If that's what you think, then that's your opinion. Personally, I believe that pure mathematics is something worth writing about. And I would state this even ignoring the fact that pure mathematics is used throughout physics and applied mathematics. I find it entirely worth it to relate the equation x^4 - 2 = 0 to the symmetries of the square, or to find a general solution to a combinatorial problem if an equivalence relation forms a group structure by counting orbits. Personally, it matters not to me whether this is "useful" in terms of physics or statistics or economics. It is meaningful to me without having a use.

But, in the end, pure mathematics consistently supplies the places where it's applied with new ideas and new ways to solve problems.

All_Is_Number · 2008-10-31 12:34:14

Ricky wrote:

If you agree, then of course mathematics makes no predictions about the universe in which we live (by itself), so "accuracy" has no meaning.

So you're saying that mathematicians can never be wrong because they make their own rules up as they go along?

A model makes a prediction, sure enough, but a model is separate from the mathematics that it uses.

I'm not sure I agree (or disagree) with that statement. Often, the mathematics are developed specifically to create models for the application.

Personally, I believe that pure mathematics is something worth writing about.

I think you misunderstood the analogy. Understanding maths is like being able to write. Being able to apply the maths is like being able to write and having something to write about.

And I would state this even ignoring the fact that pure mathematics is used throughout physics and applied mathematics.

That would make them applied mathematics.

Personally, it matters not to me whether this is "useful" in terms of physics or statistics or economics. It is meaningful to me without having a use.

I too recognize the allure of mathematics as their own means to an end. It's not the only branch of science that has limited usefulness. I also recognize the attraction to answering the age old question of where the universe came from. In both cases, I don't see any practical usefulness of such studies. And in both cases, it is highly probable that practical discoveries will be made.

But, in the end, pure mathematics consistently supplies the places where it's applied with new ideas and new ways to solve problems.

That statement is equally true the other way.

Ricky · 2008-11-02 05:42:35

So you're saying that mathematicians can never be wrong because they make their own rules up as they go along?

No. What's wrong for a mathematician is making a (logically) false statement. Reaching a contradiction where none is expected, as well as just having something that doesn't follow.

I'm not sure I agree (or disagree) with that statement. Often, the mathematics are developed specifically to create models for the application.

Agreed, but that mathematics can exist with our without the application. It is in this sense that it is separate from the application.

I think you misunderstood the analogy. Understanding maths is like being able to write. Being able to apply the maths is like being able to write and having something to write about.

And I disagree. I believe that understanding mathematics is having something to write about.

And I would state this even ignoring the fact that pure mathematics is used throughout physics and applied mathematics.
That would make them applied mathematics.

No, this is incorrect. Pure mathematics is not just mathematics that has not been applied, I believe this to be rather established. It is true that if a mathematics has not been applied, then it is pure. However, the converse is not true. I don't believe there is any universal criteria, however one large determining factor is "why did it come about"? If it came about to solve a real world problem (Newtonian calculus), then it would be applied. If it came about to solve a mathematical problem, then it would be pure. Again, this isn't the only determining factor. One of the easiest fields to see this is number theory and graph theory. Both of these were conceived before they were ever applied, and they remain as pure mathematics. But each became applied with the advent of computers. Another determining factor is how "mathematically" the question is stated. If a question can be stated without reference to an applied model or problem, then it has a large chance of being pure mathematics. The volume of three intersecting spheres would hence be a pure question, even though it came up during my research in computational biology.

I also recognize the attraction to answering the age old question of where the universe came from. In both cases, I don't see any practical usefulness of such studies.

That's weird. I had this thought come up before when reading a book by Hardy, but I have always thought that understanding the world we live in is useful. You and Hardy both seem to think that useful and economical are synonyms. Is this correct (at least for you...)?

That statement is equally true the other way.

Agreed.

All_Is_Number · 2008-11-02 16:56:51

Ricky wrote:

What's wrong for a mathematician is making a (logically) false statement. Reaching a contradiction where none is expected, as well as just having something that doesn't follow.

Right. When that occurs, it is often, perhaps usually, due to an incorrect assumption, just like in any other branch of science.

And I disagree. I believe that understanding mathematics is having something to write about.

Your disagreeing with a claim that was not made. The analogy does not imply that understanding mathematics is not a worthy topic of writing. I wouldn't say such a thing. I like mathematics, I just recognize that they are a discipline of science.

Pure mathematics is not just mathematics that has not been applied

Yes, by the definition of applied, mathematics that have been applied in the real world are elements of applied mathematics.

That's weird. I had this thought come up before when reading a book by Hardy, but I have always thought that understanding the world we live in is useful. You and Hardy both seem to think that useful and economical are synonyms. Is this correct (at least for you...)?

Understanding the world we live in is useful. The utility of understanding the universe's origin, however, remains to be seen. Nonetheless, I favor the pursuit of more knowledge relevant to that endeavor.

Ricky · 2008-11-03 07:04:28

What's wrong for a mathematician is making a (logically) false statement. Reaching a contradiction where none is expected, as well as just having something that doesn't follow.
Right. When that occurs, it is often, perhaps usually, due to an incorrect assumption, just like in any other branch of science.

In mathematics, there is no concept of a hypothesis (which is what I believe you mean by assumption) being true or false. Whether the hypothesis leads to your conclusion or not depends upon your conclusion. In science however, there is a concept of a hypothesis (sciences call this an assumption) being wrong. That is, whether or not that hypothesis (assumption) holds true in the universe.

Yes, by the definition of applied, mathematics that have been applied in the real world are elements of applied mathematics.

And by the definition of "horse fly", it is a fly that is also a horse

Group theory is used in combinatorics for a counting problem which is relevant to counting the number of possible organic molecules that can be formed from certain atoms, and this is used by chemists. Therefore, group theory is applied. Differential geometry is used to describe solutions to partial differential which arise out of physics, so differential geometry is applied. Analysis is used in statistics to find functions with certain properties to make distributions relative to particular models used by actuaries, so analysis is applied.

Only one problem: Group theory, differential geometry, and analysis are all considered to be part of pure mathematics.

The problem is with the misleading term "applied mathematics". Wikipedia has nice articles on both pure and applied, and the problem that arises when trying to find the precise distinction between the two. I won't repeat their content here, but I do suggest reading them.

Math Is Fun Forum

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