The question in another thread about whether or not mathematics is a science was an interesting one to contemplate. Beyond the basic answer I gave in that earlier thread, thinking about the question got me to thinking about geometry, but I thought sharing those thoughts in the other thread would be too much off topic, so I decided it might be better to start another thread rather than have even the small risk of spinning too much of the discussion off away from the primary theme of that thread.
There are many kinds of geometries. Each kind has its own unique set of axioms (though there may be a large number of subsets of axioms that are the same or very similar amongst various geometries). Many, if not all, geometries are logically inconsistent with each other because they have some different fundamental axioms. Mathematicians have no problem with the inconsistencies amongst geometries because they see mathematics as a field of understanding that is about the logical application of axioms related to quantities and is, in its most dogmatic form, "agnostic" about any metaphysical truth or falsity being affixed to one set of axioms just because it is logically inconsistent with another set of axioms. Pure mathematics treats all systems of axioms as equal in the application of basic rules of logic; e.g., the rules of any axiomatic system must be consistent with the law of non-contradiction or it will be deemed mathematically illogical.
Significantly for purposes of this discussion, mathematics has no necessary empirical factor against which to test a system of axioms, because pure mathematics is "agnostic" with respect to any relationship between mathematical axioms and physical reality (events).
Science, on the other hand, is anything but "agnostic" about its application of logic to the hunt for patterns in observations of physical events. Science is empirical. To make a scientific hypothesis is to make a claim about the presence of a pattern in physical events. The hypothesis must not only be logically valid (which is the minimal requirement for a mathematical system), it must also be physically tested to see if it properly predicts relevant physical events.
So...
there are many geometries, but if Einstein is right, there is only one geometry for space-time. Mathematics and science are fairly distinct spheres of thought, but there is a deep conceptual connection between the two due to the natures of mathematics being about quantities and science being about measuring quantities when observing to try to verify a hypothesized pattern in physical events. Certain axiomatic systems about quantities in mathematics work to predict patterns in physical events (and other mathematical axiomatic systems do not).Oftentimes, the hunt for patterns in physical events inherently involves a hunt for mathematical equations derived from axiomatic systems that "match" observed patterns.
I think it's interesting to note, though, that in mathematics you can have an "ironclad" proof, but not in science. In science, any conclusion about what is most likely to be the correct identification of a pattern in physical events is a conclusion that is tentative. Nothing in science is proven in the "ironclad" sense. I think that is a reflection of the fundamentally different natures of mathematics and science. Mathematical axiomatic systems are closed loops of logic - though the loopz can expand and contort in the most intriguing ways. When something is mathematically proven, a logical loop is closed. Until there stops being the experience of new events, though, the logic of science can never be closed - it must always be open to the possibility of more accurately finding patterns in physical events. Science can't prove anything (in the "ironclad" sense); the loop is always open.
Good post, and I agree that no theory can be proved and is always tentative. In my opinion mathematics is a tool. Solitary
I wouldn't call math a science, but a lot of sciences, maybe all of them, need math. Without the math, much of science would just be some guy pontificating. Religion doesn't use much math beyond basic counting two of each "kind" sort of things. It would be irrelevant, perhaps even harmful to the cause.
Good post. I'm not very familiar with metaphysics relating to math, so I don't tend to get into those discussions but cool.
I think mathematics is an extension of language. Mathematics has rules much like the rules of syntax, and symbols much like written language. Some people claim that math is the language of nature, but I think that's just poetic phrasing. Electrons don't have language; people do.
Mathematics and language are even somewhat interchangeable. No matter how complex a computer program it all reduces to mathematical instructions coded into semiconductors, and yet we use languages to program our computers.
As far as the relation between science and math, I think that scientists tend to use the more convenient language to record or communicate a particular idea. If a scientist is writing about the social behavior of a band of gorillas, English would probably be the language of choice for most or all of the paper. For papers in Physics, it's typical to see a mixture of English and mathematics.
Quote from: "entropy"When something is mathematically proven, a logical loop is closed. Until there stops being the experience of new events, though, the logic of science can never be closed - it must always be open to the possibility of more accurately finding patterns in physical events. Science can't prove anything (in the "ironclad" sense); the loop is always open.
Science doesn't tell us what's true. Science can only tell us what isn't true and we are then left to assume what is true after we have eliminated all other possibilities as false. And of course there are always other possibilities so the loop is always open.
Quote from: "SGOS"I wouldn't call math a science, but a lot of sciences, maybe all of them, need math. Without the math, much of science would just be some guy pontificating. Religion doesn't use much math beyond basic counting two of each "kind" sort of things. It would be irrelevant, perhaps even harmful to the cause.
Mathematics provides a language for quantitative measurement - as aileron was talking about - that is objective in the sense that anyone should be able to duplicate a test of a scientific hypothesis and get the same results from the measurements. This is a large part of how science gets around the problem of subjective impressions - or as you suggest, "just some guy pontificating".
I think "mathematics as language" is a pretty accurate description of what it does. It's a language designed to be understandable by everyone, that leaves no real margin for error, and has no subjectivity.
It's no wonder that we operate computers using math. All of the english in programming is just window-dressing for raw mathematics that operate the programs.
Quote from: "GurrenLagann"Good post. I'm not very familiar with metaphysics relating to math, so I don't tend to get into those discussions but cool.
Yeah, I'm not sure about the ontology of mathematics. But I think we can talk about the relationship of mathematics and science without assuming a particular ontological status for mathematical "entities". I suppose, though, that if a person assumes that mathematical entities exist in physical reality like matter and energy, then that could lead to some fascinating conjectures about the relationship of mathematics to science. Maybe a synesthetic who sees colors or hears musical notes when he contemplates certain numbers is apprehending the connection between mathematics and physical reality in a profound way. :-D
It's strange that rather simple mathematical expressions should precisely describe natural events (E=mc2). And why does the square and square root show up so much in nature?
Take the speed of a falling body. Know the duration of the fall, the distance a body falls in that duration, multiply some numbers and then square the thing, and presto. You know how fast it's falling at any given second. It's just gravity, but to figure it out, you need squares and square roots. Why is that? What's so special about squares and square roots? They keep showing up as if nature can't do anything without using some arbitrary mathematical language with a square root here and there to do it.
My intuition tells me math just describes nature, yet on the surface it seems like nature takes it's orders from math. I've always wondered why it seems like that.
Quote from: "Johan"Science doesn't tell us what's true. Science can only tell us what isn't true and we are then left to assume what is true after we have eliminated all other possibilities as false. And of course there are always other possibilities so the loop is always open.
That's true. :) In a practical sense, what we are left with in science is a best guess about what is likely to be true. In the functioning world that seems to be extremely useful. That's what engineering is all about.
A square root is just representing a 2-dimensional attribute, usually. Actually they're deceptive; they're not as meaningful as they look. It's not so much E = m*c^2, but it's E = m*c*c. These are, of course, equivalent, but it looks much more mathematical if you use the square root sign.
If you replace all the square roots with what they represent (ie; a square is not 6m^2 of surface but rather 6*m*m meters) it'll look very different, but represent the same. Except now all the "math" is gone, it's just a bunch of physical constants.
Then if you remember that the meter is just an arbitrary piece of length decided on by some french emperor, you get to the point where the surface of a square is just the multiplication of two distance vectors that have a unit conversion in there to make it clear to readers that we're talking distance, and roughly how much distance.
And then multiplication can be described as taking a distance vector, making a number of copies, and then putting them side by side, pointing in the same direction, with one unity-vector distance between them, and you realise that ultimately surface area is just expressed in a number on a numbers-line, and all the mathematical constants like meter or second is just stuff we add for our own clarity, so that we can easily see what the real world effect is.
This probably stopped making sense a while ago.
For the hell of it, because this kind of thing was an eye-opener to me as a kid, I'm gonna do some juggling.
E = mc^2
---
E = m * c * c
---
c = 299 792 458 * m / s
(Replacing one m with kg, because e=mc^2 uses m for mass but c uses it for meters)
---
E = kg * 299 792 458 * m / s * 299 792 458 * m / s
---
E = 8.9875518e+16 * kg * m * m / ( s * s )
---
E = 8.9875518e+16 * kg * m^2 / s^2
This is a perfectly valid way to write the amount of energy E present in matter.
Now we can make it even funnier... we're arrived at the unit kg * m^2 / s^2... and that's actually a really well known unit, namely the Joule. So I guess E is 8.9875518e+16 Joules.
Is that useful? I don't know. Probably not. But not a lot of people seem to realise you can do this kind of stuff. The math is just a descriptor, and you just get taught a few and then usually lose track of what they mean.
Quote from: "Plu"This probably stopped making sense a while ago.
No worries. That's the same thing I thought about my own question. Sometimes it's like that when trying to express yourself.
I can see why formulas work when working with simple geometrical objects. That's more cut and dried. But nature? Specifically, things like behavior of gravity, energy as a function of the speed of light squared? Whoa! Those things seem mysterious to me. It's like a mathematician designed the physical laws that our universe is dependent on. It's like he asked, "How strong a force should I make gravity?" And then he arbitrarily started with an equation that included a square or square root, did the math, and declared, "That's how strong a force I will give to gravity!"
After he had done the math, he could have said, "Just to make it complicated, I'll thrown in a +14 for the Hell of it, and use that for the amount of energy in mass, but he didn't. :-D
I think you are approaching it from the wrong side. This is really hard to explain, especially in English, but I'll try anyway...
The meter is an arbitrary distance. (You probably know that Napoleon first introduced it because he was sick of all the different distance units.)
However, the concept of distance is not arbitrary. We understand that there is such a thing as distance, and we needed to measure it. So we decided to pick our numbers scale, pick a random distance, and then said "1 distance unit".
Then, when we came across other kinds of abstract concepts, we picked a unit for them as well. We took a random point in time and said "1 time unit".
And thus enters the formula... because we have an arbitrary unit for distance and an arbitrary unit for time, we can now create a formula for the speed of an object. (Because speed is a measurement of distance over time). So we create the formula "1 distance unit per 1 time unit", to determine the abstract concept of how quickly something is moving.
This also brings into light the seperation between numbers of units... numbers are meaningless. Only when a number (like 3) is combined with a unit (like the meter) does a number represent something useful. This is why it would be impossible to have a formula that has a "+14" in it... 14 does not mean anything. In order to be added to the formule, it would need a unit. And if it has a unit... we'll need an arbitrary conversion unit for it. And hey, we might as well pick one that simplifies the formula. Like, if we add some constant force to all of these... we might as well pick our unit as "equal to that constant force" and simplify our formula from (randomly) G=vkr+14z to G=vkr+X, and suddenly have something that looks elegant. Just because we picked a different number, or introduced a new constant.
--
Writing this I came up with what is probably the perfect example of "picking your units wisely".... the formula to calculate the circumference of a circle. It's really messy:
Circumference = length * 3.14159265359. That doesn't look elegant at all.
Hey, I know... lets convert the complicated number into a constant.
Circumference = length * ?.
Boom. Instant elegance. Not a property of the universe, just a property of "picking your units wisely since they are arbitrary anyway"
Another example. If your lamp said: "60kg?m^2?s^-3", that would not look very elegant, would it? On the other hand, if we just make it elegant by defining a new unit called the Watt which is defined to be equal to 1kg?m^2?s^-3 we can say a lamp is 60Watt and suddenly it looks elegant and simple again, even though it is not really so.
Quote from: "Plu"This is really hard to explain, especially in English, but I'll try anyway...
Hmmm, for some reason I thought you were British. Where are you from?
Quote from: "Plu"I think you are approaching it from the wrong side.
Yeah, that same thought is occurring to me. I think the discussion about units of measure being arbitrary helps clear up some of my confusion. I may have to just think about this some more. I still feel confused, but about what, I'm not entirely sure. Does that sound weird?
Quote from: "Plu"The meter is an arbitrary distance. (You probably know that Napoleon first introduced it because he was sick of all the different distance units.)
Is he like the father of the metric system?
Quote from: "Plu"However, the concept of distance is not arbitrary. We understand that there is such a thing as distance, and we needed to measure it. So we decided to pick our numbers scale, pick a random distance, and then said "1 distance unit".
Then, when we came across other kinds of abstract concepts, we picked a unit for them as well. We took a random point in time and said "1 time unit".
This may be important. I'm not sure. Question: Does gravity have a unit of it's own, or is that something we just calculate from other arbitrary units?
There's no "inconsistencies" between the different geometries. They can be classified by a single factor of curvature with k = 0 for flat space, +1 for spherical, and -1 for hyperbolic. It turns out that empirically, our 3-D world is flat, while 4-D spacetime (in GR) is hyperbolically curved.
Quote from: "Johan"Science doesn't tell us what's true. Science can only tell us what isn't true and we are then left to assume what is true after we have eliminated all other possibilities as false. And of course there are always other possibilities so the loop is always open.
I'm not sure where you're coming from here. I think perhaps you mean scientific generalizations cannot be proven true, but certainly there is a role of confirmation in science that enables us to call observations of some phenomenon true.
Some microorganism infect cells; helium is a superfluid at or below 2.17 K; crows are in the phylum chordata. Without resorting to brain in a vat arguments, what other possibilities do we need to eliminate in order for these statement to be true?
Quote from: "SGOS"This may be important. I'm not sure. Question: Does gravity have a unit of it's own, or is that something we just calculate from other arbitrary units?
Historically, our units have been quite arbitrary, however we have found that nature has three fundamental units: c, the speed of light,
? , Planck reduced constant and G, Newton's gravitational constant. Out of these three, we can build any units whether it's for time, space, or mass.
Quote from: "josephpalazzo"There's no "inconsistencies" between the different geometries. They can be classified by a single factor of curvature with k = 0 for flat space, +1 for spherical, and -1 for hyperbolic. It turns out that empirically, our 3-D world is flat, while 4-D spacetime (in GR) is hyperbolically curved.
I was referring to consistency in axioms. Non-Euclidean geometries substitute different axioms for the parallel postulate in Euclidean geometry.
I will admit that somewhere I got the impression that there are subsystems of each type of geometry (i.e., geometries with more dimensions) that contained axioms that were inconsistent with other subsystems of that type of geometry, but with some quick research I see that is likely not to be true.
another thread without pictures? This sucks..
:D/
Quote from: "aitm"another thread without pictures? This sucks..
:D/
(//http://i243.photobucket.com/albums/ff277/josephpalazzo/7a04139e468ab3370ef6f7406e404309.png) (//http://s243.photobucket.com/user/josephpalazzo/media/7a04139e468ab3370ef6f7406e404309.png.html)
(//http://i.imgur.com/ucCwfwx.jpg)
(//http://i.imgur.com/jayWl1q.jpg)
Does this help? Solitary
The more you know... whatever... :-D
(//http://i243.photobucket.com/albums/ff277/josephpalazzo/Curvedspace.jpg) (//http://s243.photobucket.com/user/josephpalazzo/media/Curvedspace.jpg.html)
It has been many years since I took geometry in college, so I had remembered incorrectly - I had thought that the axioms of Euclidean geometry need to be modified as you go through different numbers of dimensions (and the same thing with the other basic types of geometry). I see now that the basic axioms of Euclidean geometry (or non-Euclidian geometries) don't change as the number of dimensions increases.
It's the axioms which are the core of what I'm talking about, not the conclusions that are drawn about geometric shapes based on those axioms. Even though I was off about the effect of more dimensions, it doesn't change the underlying point I was making - that mathematicians don't consider one geometry to be more true or false than another geometry just because it is axiomatically different from another. And there are axiomatic differences between the three different kinds of geometry. The differences are:
//http://threes.com/index.php?option=com_content&view=article&id=2199:geometry-3-basic-types&catid=72:mathematics&Itemid=50
Quote1.) Euclidean: Given a line L and a point P not on L, there is exactly one line passing through P, parallel to L.
2.) Hyperbolic: Given a line L and a point P not on L, there are at least two lines passing through P, parallel to L.
3.) Elliptic: Given a line L and a point P not on L, there are no lines passing through P, parallel to L.
It is the difference in those axioms in each of the three basic types of geometry that differentiate the three geometries. But mathematicians don't consider one type of geometry "true" and the other two "false". They are just different geometric systems. That's what I meant when I said the mathematicians are "agnostic" about one valid axiomatic system relative to another valid axiomatic system.
Quote from: "entropy"It has been many years since I took geometry in college, so I had remembered incorrectly - I had thought that the axioms of Euclidean geometry need to be modified as you go through different numbers of dimensions (and the same thing with the other basic types of geometry). I see now that the basic axioms of Euclidean geometry (or non-Euclidian geometries) don't change as the number of dimensions increases.
It's the axioms which are the core of what I'm talking about, not the conclusions that are drawn about geometric shapes based on those axioms. Even though I was off about the effect of more dimensions, it doesn't change the underlying point I was making - that mathematicians don't consider one geometry to be more true or false than another geometry just because it is axiomatically different from another. And there are axiomatic differences between the three different kinds of geometry. The differences are:
//http://threes.com/index.php?option=com_content&view=article&id=2199:geometry-3-basic-types&catid=72:mathematics&Itemid=50
Quote1.) Euclidean: Given a line L and a point P not on L, there is exactly one line passing through P, parallel to L.
2.) Hyperbolic: Given a line L and a point P not on L, there are at least two lines passing through P, parallel to L.
3.) Elliptic: Given a line L and a point P not on L, there are no lines passing through P, parallel to L.
It is the difference in those axioms in each of the three basic types of geometry that differentiate the three geometries. But mathematicians don't consider one type of geometry "true" and the other two "false". They are just different geometric systems. That's what I meant when I said the mathematicians are "agnostic" about one valid axiomatic system relative to another valid axiomatic system.
To add to that, mathematicians work with the axioms and see where that can lead. However, geometry of the real world is determined by matter. This is the fundamental gist of General Relativity. In Einstein's field equations,
R[sub:2ji54nw3]??[/sub:2ji54nw3] - ½g[sub:2ji54nw3]??[/sub:2ji54nw3]R + g[sub:2ji54nw3]??[/sub:2ji54nw3]?= 8?G/c[sup:2ji54nw3]4[/sup:2ji54nw3] T[sub:2ji54nw3]??[/sub:2ji54nw3]
The right-hand side (T[sub:2ji54nw3]??[/sub:2ji54nw3])is determined by the presence of matter in a given spacetime. This in turns determines the geometry on the left-hand side (g[sub:2ji54nw3]??[/sub:2ji54nw3]).
Quote from: "josephpalazzo"To add to that, mathematicians work with the axioms and see where that can lead. However, geometry of the real world is determined by matter. This is the fundamental gist of General Relativity.
This is why I chose geometry to talk about when trying to make the case that mathematics is not a science; rather mathematics and science are fairly distinct intellectual endeavors even though mathematics is key to the functioning of much of science. Science is about ferreting out the rules of the patterns of physical events in nature. Mathematics is about assuming rules (axioms) and, as you say, "see where that can lead" - and where it leads to is patterns of relationships of mathematical entities. They are both fundamentally about patterns, but science is about finding the patterns in nature and mathematics is about assuming axioms and seeing what patterns develop. It turns out that some of the mathematic patterns work to predict the patterns of physical events in nature, like the General Relativity equation that you note, but that does not imply that mathematics is a science. If anything, it appears to imply that science is a branch of mathematics - except for that pesky "empirical" thing about science. :)