Mathematics and Science

[Surlethe]

Tonight at dinner, my father contended mathematics is just another science. I begged to differ, citing the fact mathematics is based on pure deductive logic, rather than empirical evidence. However, I got to thinking: at its root, mathematics requires a set of postulates assumed to be true, which in turn take their basis from observations. So, in light of that fact, can mathematics be considered a science?

[Akhlut]

While I'm getting training to be a scientist, math hasn't been my strongest point, however, I think things can be proven, more or less, through graphing. I'm not sure if that can be constituted as proof, though.

[Zadius]

Science wouldn't accept empirical evidence either if it were possible to prove something absolutely, just as you can in mathematics.

[General Zod]

Mathematics is more like a language of numbers that's used to explain abstract concepts, as opposed to explaining any specific function of the natural world in and of itself, which is what science is supposed to do. Least that's my grasp on the matter.

[Lord Sabre Ace]

Math is used to quantify science. Most, if not all, things in science have some kind of equation that goes with them.

[Kuroneko]

Tonight at dinner, my father contended mathematics is just another science. I begged to differ, citing the fact mathematics is based on pure deductive logic, rather than empirical evidence. However, I got to thinking: at its root, mathematics requires a set of postulates assumed to be true, ...

Quite right.

... which in turn take their basis from observations. So, in light of that fact, can mathematics be considered a science?

Well, that depends. Loosely speaking, there are observations in mathematics of how structures behave, but the key difference is that those structures are not of nature, but abstractions created by people--there is absolutely no requrement in mathematics that what is investigated has any correspondence to physical reality. That still, however, is a frequent motivator for actual mathematics, so one can draw the distinction between "pure mathematics" and "applied mathematics." The former is definetely not a science [1] in the strict definition of the term, but the latter can be construed as a kind of theoretical science. The key question is then how to define "science". Is it limited to investigation of the natural world? If so, mathematics does not automatically qualify, although certain portions of it may. Is it any sort of systematic investigation, with no limitation to the topic of study? If so, mathematics does qualify. Personally, I prefer the stricter definition.

[1] Although fields of pure mathematics often become of great practical interest after some time. My current avatar, G.H.Hardy, is a very ironic example of that--living through World War I, he took great pride in that his work contributed absolutely nothing to the new technological horrors of that age. Later on, his field of specialization--number theory--became of a great military interest for applications such as encryption.

[Lucifer]

Math, or Calculus is classified under "science" at my university, but I can see the key difference in proving in Math, and use of disproofs and evidence instead of proof in science. And you're all right; math is used in science, but it's usually theoretical. You use the numbers and concepts from math to try to understand the results you get in science; it's a way of relating quantitative measurements.

[Davis 51]

I had a mathematics teacher once who claimed that Math was "pure science." Beyond that, I cannot say.

[wolveraptor]

...so....
What are these postulates assumed to be true?

[Kuroneko]

...so.... What are these postulates assumed to be true?

Hilbert axioms, Peano axioms, Zermelo-Fraenkel axioms... the list goes on. Which ones are assumed to be true depends on the particular mathematical system in consideration. It would be more proper to think of them as definitions rather than assumptions, although there is an significant undercurrent of Platonism among mathematicians.

[drachefly]

In mathematics, axioms are unjustified simple statements which have consequences.
Someone who applies the mathematics choses which axioms apply to their situation.

For example, you can take Euclid's parallel axiom (given a line and a point not on the line, there is exactly one line which goes through the point and is parallel to the line), in which case you get flat space; or you can take a different axiom (there are exactly two such lines) and get a space of uniform negative curvature; or you can take a different axiom still (there are exactly zero such lines) and get a space of uniform positive curvature.

So, you see, mathematics works off of axioms that can be true or not, and it's irrelevant to the mathematics whether they are or not; but if the axioms reflect the reality of some situation somewhere, then the mathematics can be applied usefully to that situation.

[Kuroneko]

For example, you can take Euclid's parallel axiom (given a line and a point not on the line, there is exactly one line which goes through the point and is parallel to the line), in which case you get flat space; or you can take a different axiom (there are exactly two such lines) and get a space of uniform negative curvature; ...

Not quite. That sort of axiom does not give a geometry; moreover, it is inconsistent with the rest of the Hilbert axioms assuming standard logic. If there are two such lines, it is always possible to construct a third distinct from them, and so on.

...or you can take a different axiom still (there are exactly zero such lines) and get a space of uniform positive curvature.

Yes.

So, you see, mathematics works off of axioms that can be true or not, and it's irrelevant to the mathematics whether they are or not; but if the axioms reflect the reality of some situation somewhere, then the mathematics can be applied usefully to that situation.

That's the sort of Platonic attitude that I was referring to earlier. It assumes the things that the given axioms refer exist to prior to the fact, so that the axioms "could be false." Such an intepretation might perhaps be appropriate for applied mathematics, where one seeks to model some phenomena, but it is definitely false in general. The axioms define their referents--a set is any object that behaves in this manner; points and lines are any objects that are related in that manner; and so on. Metaphysical nature of those objects is irrelevant; all that matters are the relationships between them--the structure. For that matter, any set of axioms is as valid as any other--the only relevant tests of the resulting systems is whether they are interesting or useful, and neither of those two properties has any necessary connection with truth.

[Surlethe]

So, you see, mathematics works off of axioms that can be true or not, and it's irrelevant to the mathematics whether they are or not; but if the axioms reflect the reality of some situation somewhere, then the mathematics can be applied usefully to that situation.

For that matter, any set of axioms is as valid as any other--the only relevant tests of the resulting systems is whether they are interesting or useful, and neither of those two properties has any necessary connection with truth.

I thought that was what he was trying to communicate: you can work the mathematics up from different axioms without trouble (like discarding some axioms in favor of others). At least, it struck me his use of "true" and "false" related to whether the axiom was chosen as a base for the mathematics, not in some absolute sense.

[brianeyci]

I thought about this for awhile too, not just for thinking sake but because if (assuming I don't flunk out) and when I get my double major in English and Mathematics, what will I say to people who ask me what I got my degree in? Because I might fulfill both requirements for a science and an arts degree, what should I choose to be on my diploma? (Those who are asking the possibility of this, it is not unless you want to take 5 years, which is what I am probably going to end up doing, and University of Toronto considers mathematics as "science" courses).

As a major I am only taking a partial load and not nearly the difficulty of courses a specialist would be taking. I am also taking no applied mathematics courses. As well science is not mathematics, and I would be sorely misrepresenting myself if I said I was a "scientist" or had significant training in science after I got my degree. I have no significant training in chemistry, physics, biology, geology, or any other science discipline save computer science which I have only a passing familiarity with after a few first year courses.

So no, in my view mathematics is not science, and when I come out I will gladly put the Bachelor of Arts moniker on my diploma rather than insist on a Bachelor of Science, even if it somehow ends up being detrimental for my job prospects. And, I will tell people I have an arts degree, but always mention my majors with a sort of snide pride, because let's face it a major in Film Theory and Psychology is not the same as double majors in English and Mathematics.

Anyway that is assuming I don't flunk out and I can survive a couple more years .

Brian

[Kuroneko]

I thought that was what he was trying to communicate: you can work the mathematics up from different axioms without trouble (like discarding some axioms in favor of others). At least, it struck me his use of "true" and "false" related to whether the axiom was chosen as a base for the mathematics, not in some absolute sense.

Perhaps, but if that was the intent, then the statement was very poorly worded. Consider the statement "bachelors could be unmarried or married, and we don't care whether they are married on unmarried." If this is a statement of alternative possible states for bachelors, as implied by the second clause, then it is clearly false, because bachelors are incapable of being married. His statement has the exact same problem--if an expression was "not chosen as a base," then it is simply not an axiom.

As a major I am only taking a partial load and not nearly the difficulty of courses a specialist would be taking. I am also taking no applied mathematics courses. As well science is not mathematics, and I would be sorely misrepresenting myself if I said I was a "scientist" or had significant training in science after I got my degree.

A bachelor of science does not a scientist make; that takes more effort. The fact that mathematics falls under B.Sci. is a well-known matter of convention. Therefore, you would not be misrepresenting yourself on either of those grounds. Ultimately, of course, the choice is your own.

[brianeyci]

A bachelor of science does not a scientist make; that takes more effort. The fact that mathematics falls under B.Sci. is a well-known matter of convention. Therefore, you would not be misrepresenting yourself on either of those grounds. Ultimately, of course, the choice is your own.

If I was a mathematics specialist (14 out of 20) I wouldn't hesitate at all, but I'm not and having more than half my courses come from other disciplines, declaring myself to have a B.Sci. is ridiculous at least from my view.

Brian

[drachefly]

For example, you can take Euclid's parallel axiom (given a line and a point not on the line, there is exactly one line which goes through the point and is parallel to the line), in which case you get flat space; or you can take a different axiom (there are exactly two such lines) and get a space of uniform negative curvature; ...

Not quite. That sort of axiom does not give a geometry; moreover, it is inconsistent with the rest of the Hilbert axioms assuming standard logic. If there are two such lines, it is always possible to construct a third distinct from them, and so on.

What do you mean? If you take the first four Euclidean axioms and then add the fifth, you are constrained from having a curved space.

Or if you are complaining about the last thing I said, about the assumption there are exactly two, that depends on your definition of parallel. In gaussian negative-curvature geometry, the typical statement of parallellism is that parallel lines are asymptotic. This is another example of how in 3 dimensions, say, skew lines are not considered parallel.

So, you see, mathematics works off of axioms that can be true or not, and it's irrelevant to the mathematics whether they are or not; but if the axioms reflect the reality of some situation somewhere, then the mathematics can be applied usefully to that situation.

That's the sort of Platonic attitude that I was referring to earlier. It assumes the things that the given axioms refer exist to prior to the fact, so that the axioms "could be false." Such an intepretation might perhaps be appropriate for applied mathematics, where one seeks to model some phenomena, but it is definitely false in general.

Yes, I misspoke somewhat. I meant, "They can be applied to some situation accurately, or not" when I said "True or not".

The axioms define their referents

Are all axioms definitions? The axioms suggested by, say, Goedel incompleteness don't look to me like definitions. Hmm. I guess they could define transfinite numbers...

[Kuroneko]

What do you mean? If you take the first four Euclidean axioms and then add the fifth, you are constrained from having a curved space.

True enough, but that does not mean that your alternative axiom gives a geometry. Replacing the parallel postulate with the statement that for every given line and a point not on the line, there are exactly two lines parallel to the given one passing through the given point gives and inconsistent system. It does not give a geometry; it gives a system in which every proposition is true (assuming standard logic). The mention of Hilbert axioms was because the Euclidean ones do not actually guarantee a geometry either (for a different reason: they are incomplete).

Or if you are complaining about the last thing I said, about the assumption there are exactly two, that depends on your definition of parallel. In gaussian negative-curvature geometry, the typical statement of parallellism is that parallel lines are asymptotic. This is another example of how in 3 dimensions, say, skew lines are not considered parallel.

There are infinitely many parallel lines in negative-curvature geometry. If one has two, one can construct more--exactly what I said above. Proving this is not difficult.

Are all axioms definitions?

Yes, but I should have been more precise: the collection of axioms of a given system defines the referents of those axioms. Lines in parabolic (Euclidean) geometry are different things from lines in elliptic or hyperbolic geometry, despite the usage of the same word. They have different properties, therefore they are different objects.

The axioms suggested by, say, Goedel incompleteness don't look to me like definitions. Hmm. I guess they could define transfinite numbers...

In what sense does the Gödel incompleteness theorem "suggest" any axioms? If G is the Gödel sentence for a given system, is there any reason to prefer G over ¬G?

[Durandal]

Once something is proven in mathematics, it's proven. Period. This alone is more than sufficient to differentiate it from science.

Statistics is a sort of exception though. Pure statistics is simply running operations on numbers, but there is a scientific element in statistical analysis regarding how best to gather data. Since its objective is to accurately model the real world, statistics is something of a science. But algebra, for example, is not really scientific. It's just numbers, and there are no real concerns as to where those numbers come from.

As for mathematics assuming certain postulates to be true, that's not entirely accurate. Humans define what number goes in what spot on the number line. There is no assumption that our number line is "how things work" in nature.

[drachefly]

Replacing the parallel postulate with the statement that for every given line and a point not on the line, there are exactly two lines parallel to the given one passing through the given point gives and inconsistent system.

Please prove this, using the definition of parallel I gave.

My non-euclidean geometry text book (by Bolyai) used that definition and said there were two such lines. One was asymptotic at one end of the given line; the other was asymptotic at the other end of the given line.

Other lines - for example, the line which we would normally consider most parallel locally (it's not getting any closer or further at the given point) - are NOT considered parallel in this scheme.

Are all axioms definitions?

Yes, but I should have been more precise: the collection of axioms of a given system defines the referents of those axioms.

Okay, I'll take that.

Lines in parabolic (Euclidean) geometry are different things from lines in elliptic or hyperbolic geometry, despite the usage of the same word.

We do have more appropriate terms, such as 'geodesic'

The axioms suggested by, say, Goedel incompleteness don't look to me like definitions. Hmm. I guess they could define transfinite numbers...

In what sense does the Gödel incompleteness theorem "suggest" any axioms?

we can take G or ¬G as an axiom and see what comes out. Those would be new axioms.
Of course, adding such an axiom doesn't end this, because the new system also has a Godel sentence

If G is the Gödel sentence for a given system, is there any reason to prefer G over ¬G?

no, and that's my point.

[drachefly]

NOTE:

above, when I say you can take G or (not)G, I merely meant that it was possible to consider the system in which these were so; not that you had to make a choice.

[Wyrm]

Replacing the parallel postulate with the statement that for every given line and a point not on the line, there are exactly two lines parallel to the given one passing through the given point gives and inconsistent system.

Please prove this, using the definition of parallel I gave.

I can provide an informal proof.

Take line AB and point C. Then by your axiom there exists two lines CD and CE that are distinct and do not meet AB at any point. These two lines define to half-planes (call them H and J) that do not contain any of AB. The lines CD and CE are distinct, so are these half-planes, so there must exist points that are in one half-plane but not in another. Let F be one such point. Then there exists a line CF, contianing points from H and J. Any point that is not contained in H must be a member of J, so the entire line CF must be within the union of H and J. Since no points of H or J is also contained in AB, neither is any point in CF. Therefore, if there are two lines parallel to a line, there are an infinite number of them. Although not formal, my proof does provide a guideline for a more formal and rigorous proof.

However, since your postulate says that there are exactly two such lines, the geometry is inconsistent. Leave out the "exactly" part, and you got a consistent geometry.

[drachefly]

Replacing the parallel postulate with the statement that for every given line and a point not on the line, there are exactly two lines parallel to the given one passing through the given point gives and inconsistent system.

Please prove this, using the definition of parallel I gave.

I can provide an informal proof.

Take line AB and point C. Then by your axiom there exists two lines CD and CE that are distinct and do not meet AB at any point. These two lines define to half-planes (call them H and J) that do not contain any of AB. The lines CD and CE are distinct, so are these half-planes, so there must exist points that are in one half-plane but not in another. Let F be one such point. Then there exists a line CF, contianing points from H and J. Any point that is not contained in H must be a member of J, so the entire line CF must be within the union of H and J. Since no points of H or J is also contained in AB, neither is any point in CF.

But CF is not asymptotic to AB, and thus is not a parallel line as I defined it.

[Surlethe]

I can provide an informal proof.

Take line AB and point C. Then by your axiom there exists two lines CD and CE that are distinct and do not meet AB at any point. These two lines define to half-planes (call them H and J) that do not contain any of AB. The lines CD and CE are distinct, so are these half-planes, so there must exist points that are in one half-plane but not in another. Let F be one such point. Then there exists a line CF, contianing points from H and J. Any point that is not contained in H must be a member of J, so the entire line CF must be within the union of H and J. Since no points of H or J is also contained in AB, neither is any point in CF. Therefore, if there are two lines parallel to a line, there are an infinite number of them. Although not formal, my proof does provide a guideline for a more formal and rigorous proof.

Quick question -- I don't see how the bolded part is justified; doesn't the existence of points on AB which are neither in H nor J imply an infinite number of points not in H union J?

[drachefly]

He meant any point which is on CF and not in H must be in J. That's perfectly justifiable. The problem is that CF isn't parallel as I defined it.