Is Mathematics a science, or a philosophy?

[Z]

Interesting Ian claims that Math is a philosophy. I claim it is a science. I acknowledge that there are such things as mathematical philosophies, but mathematics, I think, is a science.

Surely, looking up the definition across the internet, it appears that it is a science; and the texts I have available classify it as a science. Yet Ian is unconvinced. He insists it cannot be a science.

SO, to the rest of you - what are the arguments on either side of the case? Science, or philosophy?

Thanks for your time.

:D

[Hydrogen Cyanide]

Neither. It is a way to describe things, it could be a kind of language.

I am an engineer... I use math to describe the motion of objects I am dealing with. Nothing more, nothing less. The science I use provides the variables, constants and relationships (such as Newton's second law of motion and force).

Some folks get into philosophical discussions on the properties of infinity. But I just ignore that and figure that anything devided by something with the limit of infinity is just zero.

[Z]

I could accept that claim - that mathematics is a language. However, it seems the evidence supports it as a science, rather than a language or tool, or a philosophy.

It WOULD be more logical to define Mathematics as 'a language used primarily in commerce, finance, science, etc.' But is there any support to Ian's philosophical claim?

[Mendor]

I don't think maths is a science. To me, "science" implies a degree of empiricism -- observing phenomena, collecting data, formulating hypotheses, applying the scientific method. That's not done in maths (or logic or so on). Maths is a human construct -- it's useful as a tool in science, but it itself is not a science, I don't think.

Whether it's "a philosophy" or not, I couldn't say without better definition of the term "a philosophy".

I like the idea of its being "a language", but I think it's more than that. It isn't merely a "way to describe things" -- it can be used to describe things, but you can also construct mathematical systems and results that only make sense within a mathematical framework -- they don't have any relation to "real world" phenomena like, say, how many apples you have. (Which isn't to say that these results don't have applications -- imaginary numbers don't correlate to anything in the "real world" -- you can't have i apples -- but I understand that if you want to model certain wave patterns, particularly in electronics, it's useful to use them.)

I don't think I've quite expressed my point correctly in the above paragraph, 'cos IANA mathematician (yet). What I mean to say is that maths can be an end in itself, and not necessarily concerned with describing things. If I'm misguided here, please let me know.

[Interesting Ian]

Originally posted by zaayrdragon
Interesting Ian claims that Math is a philosophy.

{sighs} I said it comes under the broad heading of philosophy rather than science.

[Eleatic Stranger]

I don't particularly think it's either a subset of science or philosophy proper (ignoring of course that historically it was, as was science, and so on). However, if I had to choose one I think it's pretty reasonable to say that it's a part of philosophy.

After all, the distinctions between mathematics and formal logic, which is a part of philosophy, are tenuous at best. While this doesn't mean that mathematics is necessarily a part of philosophy (after all, many disciplines overlap) it would tend to indicate the best place of the two to put it if you had to.

[Z]

Interesting...

One has to wonder what, then, is the fundamental thinking error in the people who write these many definitions.

Oh, mind you, I don't always agree that 50 million Elvis fans can't be wrong - but it would seem to me that science still covers mathematics better than philosophy - simply because philosophy is often wishy-washy, changes frequently, and is very subjective, where mathematics is pretty solid and rarely changes.

It would seem, though, that the best place to put Math is firmly by itself - that it is a tool/language which, as Math, is a part of science, and as Logic, is a part of both philosophy and science.

Of course, philosophy can be seen as a subset of science, and science can be seen as a subset of philosophy...

The whole thing is making me dizzy!

Still, according to the so-called experts, Mathematics is a science. Why would they say that?

[Paul C. Anagnostopoulos]

Mathematics is philosophy if the definition of "philosophy" includes mathematics. Does it? If not, then mathematics is its own thing.

Now, care to tell me what difference it makes?

~~ Paul

[Dancing David]

Neither, I agree that it is a language, it is a self referencing symbolic set of representation, it can be use as either philosophy or science.

It has it's roots in counting and philosophy, but until it has empirical roots it is not science.

[Z]

Mathematics is a language, then. So is language a science? A philosophy? An art?

Is Mathematics art?

(I've always thought so...)

And, if Mathematics IS an art, does that suggest there is such a thing as objective art?

(Gawd, I'm starting to pull an Iacchus, aren't I?)

[Z]

Here is a pretty good, brief article that underlies the problem about classifying Mathematics. (http://www.bydesign.com/powervision/Mathematics_Philosophy_Science/PVPAGE12.html)

[American]

Since it is whatever you define, it must be a philosophy. I think what you are wondering is whether there is some platonic mathematical system that runs the universe... it has never worked that way. We observe a system first (as in physics), then find numbers and relationships to describe it. We can further confirm a scientific observation by playing with an equation (deriving whatever from it, doing calculus, factoring, etc), then seeing if expected results follow in reality. If they don't, then you can question whether it's your math or your theory that is wrong. One example was the breakdown of classical physics 100 years ago, where they had to conclude that the math they were using didn't fit with the reality of light speed.

By the way, I made all of that up. Sorry for wasting your time, but I sure enjoyed it.

[uruk]

And, if Mathematics IS an art, does that suggest there is such a thing as objective art?
You know those "caution - wet floor" signs with the guy falling down? That graphic design could be considered "objective art".

[T'ai Chi]

Originally posted by Dancing David
It has it's roots in counting and philosophy, but until it has empirical roots it is not science.

Probability theory, for example, has empirical roots (in the gambling halls of the 17th century), so would that make it a science?

[Z]

Maybe we've stumbled on a philosophical science? Or a scientific philosophy?

Or maybe we're lumping too much under the banner of 'mathematics'?

That is, after all, a damned broad category.

Maybe... maybe some parts of Math are sciences, and other parts are philosophies?

:shrug:

[Taffer]

I think that Mathematics is Logic defined. Maths is simply a way of representing logic, and allowing us to do very complicated logical processes without overheating our own brain.

Even things like Physics formulae, having been defined for the most part through observation, are logical given said observations.

Even abstract ideas like Imaginary numbers, and infinity, are logical tools used to come to a logical conclusion of a problem. I guess I would call it a language, but only the numbers etc. The idea of maths I would classify as logic defined.

Some things, however, I'm not too sure about. For example, we have found through experiment, that one apple plus another apple always equals two apples.

[Badly Shaved Monkey]

Originally posted by zaayrdragon
Interesting...

One has to wonder what, then, is the fundamental thinking error in the people who write these many definitions.

Oh, mind you, I don't always agree that 50 million Elvis fans can't be wrong - but it would seem to me that science still covers mathematics better than philosophy - simply because philosophy is often wishy-washy, changes frequently, and is very subjective, where mathematics is pretty solid and rarely changes.

It would seem, though, that the best place to put Math is firmly by itself - that it is a tool/language which, as Math, is a part of science, and as Logic, is a part of both philosophy and science.

Of course, philosophy can be seen as a subset of science, and science can be seen as a subset of philosophy...

The whole thing is making me dizzy!

Still, according to the so-called experts, Mathematics is a science. Why would they say that?

I'd say maths lies within philosophy. I think science has empiricism inherent in it, which is why no one is quite sure what to do about beautiful, but untestable theories in physics because they are lying on the boundary.

I can see a motivation to put maths in science's camp to the extent that it is a 'discovery' about the world, which is a function also of science. However, strictly, the formal arms of philosophy, I think, would have that same property.

Reading what you have written, I think you are prejudiced against philosophy "because philosophy is often wishy-washy, changes frequently, and is very subjective" so see allying maths with it as pejorative, but that is to confuse the formal field of human intellectual endeavour known technically as philosophy with the completely woolly use of the term, "My philosophy on team selection against the Chicago Bears is...", "Homeopathy is better than allopathy because it is based on an underlying philosophy...", "The bank foreclosed on the mortgage, but I'm philosophical about it" In these cases, the P-word is used to mean an underlying principle or set of governing rules or even jsut a willingness to accept that there is a 'bigger picture' without making any attempt to paint that picture.

[Z]

In other words, it's all a semantical juggling match, right?

Still, out of the first five definitions for 'philosophy' on Google, only two are even vaguely suitable for lumping Math under - but, these two also are vaguely suitable for lumping any and all sciences under them, as well. The first three are completely unsuitable for inclusion of mathematics!

And none of the definitions for 'mathematics' on Google offer 'philosophy' as a governing body.

Curious state of affairs...

[Mendor]

Originally posted by Taffer
Some things, however, I'm not too sure about. For example, we have found through experiment, that one apple plus another apple always equals two apples. Did we find that by experimentation, exactly? Is that not a consequence of our definitions of "one", "two" and the addition operator?

[Taffer]

Originally posted by Mendor
Did we find that by experimentation, exactly? Is that not a consequence of our definitions of "one", "two" and the addition operator?

Well, yes ok, I guess. But that doesn't take away from the fact that we have to learn that one and one makes two. For all we know, when we are born that is, one and one makes five. In fact, there could be a universe where this is true. Not only that, but a major problem with the quantum computer that 'they' are developing is that one and one do not always equal two. Sometimes they do, sometimes they equal seven, other times they equal smiley face, and yet others they equal 'N/A'. It is a big problem with the otherwise great quantum computer, and is one that I'm fairly sure that physicists havn't quite worked out yet.

[Mendor]

Originally posted by Taffer
For all we know, when we are born that is, one and one makes five. In fact, there could be a universe where this is true.How?

[Taffer]

That is exactly what I used to think, but I learned to open my mind to it a bit.

How, for example, can one plus one equal 'b'? Or 'N/A'? We can't concieve it, because our brains work in a universe in which one plus one equals two. But if in the universe one plus one equals three, and there were beings there as well, I'm sure they could understand perfectly how it works, and wonder if there could be a universe where one plus one equals two.

I can't see how it can equal three either, but the point is we don't know that it couldn't, somewhere. We are biased in thinking that it has to equal two simply because that is the way our brain works.

EDIT: This is exactly the same problem as trying to concieve a universe with more then three dimensions (not counting time, for those who do). To our brains, there can only be three; up/down, left/right, forward/backwards. But it is possible that there is a universe that has five dimensions, or twelve, or a hundred and thirty five. Just because we can't imagine it doesn't mean it couldn't exist ;).

[Interesting Ian]

Originally posted by Taffer
That is exactly what I used to think, but I learned to open my mind to it a bit.

How, for example, can one plus one equal 'b'? Or 'N/A'? We can't concieve it, because our brains work in a universe in which one plus one equals two. But if in the universe one plus one equals three, and there were beings there as well, I'm sure they could understand perfectly how it works, and wonder if there could be a universe where one plus one equals two.

I can't see how it can equal three either, but the point is we don't know that it couldn't, somewhere. We are biased in thinking that it has to equal two simply because that is the way our brain works.

EDIT: This is exactly the same problem as trying to concieve a universe with more then three dimensions (not counting time, for those who do). To our brains, there can only be three; up/down, left/right, forward/backwards. But it is possible that there is a universe that has five dimensions, or twelve, or a hundred and thirty five. Just because we can't imagine it doesn't mean it couldn't exist ;).

There's a bit of confusion here. 1 + 1 necessarily equals 2. That is to say 1 + 1 = 2 in all logically possible Universes. But it is different for particulars. It is possible that in a Universe 1 apple + 1 apple = 3 apples. Just imagine putting an apple in an empty opaque container, putting another apple in, then tilting the opaque container and seeing 3 apples roll out!

[Hydrogen Cyanide]

Originally posted by Interesting Ian
There's a bit of confusion here. 1 + 1 necessarily equals 2. That is to say 1 + 1 = 2 in all logically possible Universes. ...

No... it is that way because that is how we have defined it for Base 10 (or any Base above 3). In binary (Base 2 system) 1+1 would equal 10.

As a tool and a language to be useful we often use Base 10 more than others... but for other reasons we could use something else like binary or octal. The 360 degrees around a circle is based on a Base 60 number system from ancient Babylonia:
http://www.mediatinker.com/whirl/zero/babylonian.html

I find that I understand mathematics more when I use it... it is easier to understand the relationships as you work with them. I also think discussions of making math more than what it are usually by those who do not understand it very much (but then again, I once chided a math professor for not knowing much about differential equations).

[Eleatic Stranger]

Oh, mind you, I don't always agree that 50 million Elvis fans can't be wrong - but it would seem to me that science still covers mathematics better than philosophy - simply because philosophy is often wishy-washy, changes frequently, and is very subjective, where mathematics is pretty solid and rarely changes.

ZaayrDragon - Badly Shaved Monkey has already responded fairly definitively to this, but as a serious student of Philosophy (the academic discipline, not the fruity "here's my belief system") I should add that this is just plain untrue. First off it's untrue because mathematics changes with an interesting frequency at the higher levels, and secondly because philosophy is not in the slightest subjective (it's based around the concept of an argument for crying out loud, that's intersubjective at the very least), and because it's about as far from wishy-washy as you can get. Also I should add that googling for the definition of Philosophy isn't really a very good method of argumentation here, if only because the question "what is philosophy?" is one of the central questions of.... philosophy. Any definition of philosophy that isn't controversial is going to be fairly strictly methodological, and that sort of definition is going to be one that mathematics tends to fall under reasonably easily.

Taffer:

I think that Mathematics is Logic defined. Maths is simply a way of representing logic, and allowing us to do very complicated logical processes without overheating our own brain.

Technically, the way of representing logic is called Formal Logic, and while it's the area of Philosophy that is the most mathematical (in fact, there's really little ground on which to divide mathematics and philosophy strictly due to this massive overlap), it's still technically a part of Philosophy and not Mathematics. (In, at the very least, in the sense that if you take Logic in a university it'll be under the purview of the Philosophy department more often than not.)

Also, under the philosophic usage of 'possible worlds' there wouldn't be one where 1 +1 didn't equal 2. 1, +, and 2 are considered rigid designators which apply to the same objects across all possible worlds, and it's logically necessary that 1+1=2. Now, there could be a possible world where in that world's version of English differed enough that the sentence spoken out loud meant something different, but thats an entirely different matter from saying that in that world 1+1 =/= 2.

(Also this is far from the same thing as a possible world in which there are more than 3 spatial dimensions - as it isn't a logically necessary truth that there are 3 spatial dimensions. (Or, at the very least, it's a fairly controversial one that requires some serious argumentation to establish.))

[Hydrogen Cyanide]

Originally posted by Interesting Ian
.... It is possible that in a Universe 1 apple + 1 apple = 3 apples. Just imagine putting an apple in an empty opaque container, putting another apple in, then tilting the opaque container and seeing 3 apples roll out!

When you find that universe where the Conservation of Mass does not conform to THIS universe... then you can go find the means to define quantities there. Until then... apples do not appear unless they have been put in.

Also... about adding in 3 dimensions... be sure to designate if you are using scalars or vectors. It makes a difference.

[Taffer]

Why must 1+1=2? It is only logical to us, because that is the way our universe works. Again, just because we can't concieve it doesn't mean it can't exist.

[Mendor]

Originally posted by Taffer
Why must 1+1=2?Because we define 1, 2, + and = such that it is so. Maths is a human construct.

[crimresearch]

I thought it was a subset of the philosophy of science...


http://www.searchbeat.com/Society/Philosophy/PhilosophyofScience/PhilosophyofMathematics/

[Taffer]

*sigh*

Then why must one apple plus another apple equal two apples? I know maths is a human construct, but is firstly based on observed facts (1 apple +1 apple = 2 apples).

EDIT: And before you start yelling that we define that too, how about if we say (as II said) you have an opaque container. You put two apples in, and three roll out. If that were the case, in their universe, the equivilant of 1 + 1 would = 3. (To our maths). To them, (in their universe) awer + hgodf = utqer.

[Eleatic Stranger]

But Taffer, I can come up with real world examples where 1 plus 1 doesn't equal two.

For instance, take 1 liter of pure water and add 1 liter of pure ethyl alcohol. How many liters will you get as a result? (Hint - it's not 2).

Face it - the fact that mathematics has empirical uses doesn't make it foundationally empirical like the sciences. For instance, the sentence "All bachelors are unmarried men." is a sentence about bachelors and unmarried men. However, the sentence is true in all possible worlds, just like mathematical statements. Do you see how this starts to work?

[Esther]

Interesting discussion.

The idea that Mathematics is a language is called by the mathematicians " Mathematical Formalism" and those who disagree with this theory-- and it's the majority-- enjoy to say" Formalism on Sundays" meaning that we can fool around on Sunday but when Monday comes we must return to real mathematics that ( for the horror of the platonic philosophers ) is called among mathematicians as " Platonic Realism" LOL!!

I do not remember if it was in Cosmopolitian or in sci.maths or sci.logic where in endless threads the choleric Matthew P Wiener argued on the matter, defending the platonic realism that was kind of represented by Goedel's theories on maths and supporting the idea that Mathematics is not a language and is not a physical science either but it is a logical "experience."

[American]

Originally posted by Taffer
Why must 1+1=2? It is only logical to us, because that is the way our universe works. Again, just because we can't concieve it doesn't mean it can't exist.

1+1 = 1+1.

2=2.

That's how I see it.

[Taffer]

And of course that is how I see it also. That is how it is in our universe. But that doesn't take away from the fact that we cannot say it is impossible to have a universe where that doesn't hold true.

[Hydrogen Cyanide]

Originally posted by Taffer
*sigh*

Then why must one apple plus another apple equal two apples? I know maths is a human construct, but is firstly based on observed facts (1 apple +1 apple = 2 apples)....

Because it would violate the Law of Conservation of Mass... which has been observed over several millenia in this universe.

Using Conservation of Mass (or Energy in other cases) makes it easier to predict outcomes (like how many apples will in a container) in this universe.

Should you happen to come across another universe where mass is NOT conserved... then you will have to find a way to predict an outcome. THEN perhaps the notion of 1(apple) + 1(apple) NotEqual 2(apple) may be valid.

Until then... mathematics is a useful tool for predicting outcomes defined by repeated observations.

[Interesting Ian]

Originally posted by Mendor
Because we define 1, 2, + and = such that it is so. Maths is a human construct.

No, I think mathamatics has a real existence. But obviously I agree with you about 1+1 = 2

[Interesting Ian]

Originally posted by Hydrogen Cyanide
When you find that universe where the Conservation of Mass does not conform to THIS universe... then you can go find the means to define quantities there. Until then... apples do not appear unless they have been put in.

Also... about adding in 3 dimensions... be sure to designate if you are using scalars or vectors. It makes a difference.

I was talking about a logically possible Universe, not this Universe.

[Interesting Ian]

Originally posted by Esther
[B]Interesting discussion.

The idea that Mathematics is a language is called by the mathematicians " Mathematical Formalism"

I first came across this idea that maths is a language as a child on reading a Jenning's book by Anthony Buckeridge. Jenning's friend Darbishire said that his father said that mathematics is actually a language. This was greeting with scornful derision by Jennings and co (they were 11 year old boys at "public" school where they actually lived). So they asked a Teacher called Mr Wilkins. He said something like yes it could be called a language. So Jenning's said something like "Wow! How would you say "Good morning, how are you doing today"". Mr Wilkins said "I . .I . .doh!!! you silly little boy!! :mad:"

Anyway, just thought I'd mention that.

[rppa]

Originally posted by American
1+1 = 1+1.

2=2.

That's how I see it.

You are assuming what you're trying to prove. Who says there is any connection between the first line and the second? Until you have accepted that 1+1 = 2, you don't know whether the second line should read
2 = 2
or
3 = 3
or
aardvark = aardvark.

[Eleatic Stranger]

And of course that is how I see it also. That is how it is in our universe. But that doesn't take away from the fact that we cannot say it is impossible to have a universe where that doesn't hold true.

As far as we can say of anything that it is impossible we can say that it is impossible to have a universe where 1+1 does not equal 2 - it is a logical necessity that it do so based on the meanings of the terms.

Would you say that it would be possible to have a universe where "all bachelors are unmarried men" is false? In other words - could there be a universe where some bachelors are not unmarried men?

Since bachelor has the same set of referents as unmarried man, the sentence can be rewritten as "All bachelors are bachelors", or more logically, "(n)(Bn -> Bn)". (Where B is the predicate 'is a bachelor'). In a possible world where this was false, the true sentence (its negation) would read "-(n)(Bn-> Bn)". This is equivalent to saying that "(En)(Bn & -Bn)", which is a straightforward logical contradiction. (There is something, n, that is both a bachelor and not a bachelor.) In other words, a world in which this sentence is false is not a possible world.

Since 1+1=2 is true by virtue, as well, of the meanings of the terms 1, + and 2 it is likewise impossible that there could be a possible world in which 1+1 did not equal 2.

[American]

Originally posted by Taffer
And of course that is how I see it also. That is how it is in our universe. But that doesn't take away from the fact that we cannot say it is impossible to have a universe where that doesn't hold true.

My point was more that 1+1 might not equal 2, rather it only equals "1+1" and nothing else. But I'm willing to accept that 1+1 may not even equal 1+1. The statement 1+1=1+1 may be false in some way, by some matter of perception or maldefinition.

Originally posted by rppa
You are assuming what you're trying to prove. Who says there is any connection between the first line and the second? Until you have accepted that 1+1 = 2, you don't know whether the second line should read
2 = 2
or
3 = 3
or
aardvark = aardvark.

'zactly.

You guys win. I am ducking out now. This discussion is incredibly dorky, and I am embarrassed I was ever a part of it. Further, any musings we may have are pathetic compared to the insanely high level of math that is being achieved in published papers by men whose intelligence is higher than 99.9999% of the world. I certainly have no business dabbling in this field.

[Taffer]

Eleatic Stranger, that can only be said for our universe, because it is the only universe we have ever experienced. And I am not talking about 1+1=2 in the 'language' sense. Of course 1+1=2, beacuse 2, by definition, is 1+1. What I am saying is that we cannot say it is impossible to have a universe where one apple with another apple gives three such apples, because we have never experienced every possible universe. Just this one, where 1 apple + 1 apple = 2 apples. As a skeptic, you must realise that to say that "1+1=2 is knowledge, and cannot be anything else anywhere" cannot be true because there is no way to prove that it isn't different anywhere else.

However, the statement "1+1=2 is true in this universe" is fine, IMHO.

American: You may be right, it may be 'dorky', but then so is arguing what the chance of being born is, or any other strange topics that come up on this board.

[Mendor]

Originally posted by Taffer
Eleatic Stranger, that can only be said for our universe, because it is the only universe we have ever experienced. And I am not talking about 1+1=2 in the 'language' sense. Of course 1+1=2, beacuse 2, by definition, is 1+1. What I am saying is that we cannot say it is impossible to have a universe where one apple with another apple gives three such apples, because we have never experienced every possible universe.Ah, I think I see where we misunderstood one another.

When we say "1 + 1 = 2", that to me is maths.
When we say "One apple plus another apple is two apples", we then (to my mind) start to move in to the application of mathematics to the real world, and empirical sciences. This was my idea of maths being a tool to understand science and not a science in and of itself.

The thing is, though, that if we find that the maths doesn't reflect the real-world situation, we can invent a new system of maths. For example, in a scalar sum, one kilometre plus one kilometre equals two kilometres. But in a vector sum, one kilometre north plus one kilometre west equals sqrt(2) kilometres north-west. The reason I think maths doesn't have a "real" existence (to address Interesting Ian) is that we can do this -- we can start with new axioms and construct different mathematical systems which are perfectly consistent in and of themselves, and which may (or may not) have a relationship with the real world.

Now, to be honest, I can't conceive of a universe where one apple plus one apple equals three apples, but I'm prepared to accept that as a lack of imagination on my part. I'm similarly unable to conceive, say, 11 dimensions, but that doesn't mean it's not a useful or possible concept. I'm sure it wouldn't be beyond mathematicians to come up with systems to deal with addition of apples in that universe (though it's certainly beyond me :D)

But, as Eleatic Stranger points out, "1 +1 = 2" -- given our definitions of 1, 2, + and = and the axioms governing them -- is true for all possible universes, unless we change the axioms. If it turned out that one apple plus one apple gave three apples, we could change the axioms such that "1 + 1 = 3" (though I don't know how we'd accomplish that), but that wouldn't invalidate the old axioms and the statement "1 + 1 = 2 under the old axioms" -- it would just be a different system of maths.

I'm starting to be of the opinion that it's neither science nor philosophy and that it deserves a nice little box all of its own :D

[Eleatic Stranger]

What I am saying is that we cannot say it is impossible to have a universe where one apple with another apple gives three such apples, because we have never experienced every possible universe. Just this one, where 1 apple + 1 apple = 2 apples. As a skeptic, you must realise that to say that "1+1=2 is knowledge, and cannot be anything else anywhere" cannot be true because there is no way to prove that it isn't different anywhere else.

I'm not sure what you mean by 'possible' world if you accept that there could be a possible world wherein a logical contradiction comes out true... If 'possible' doesn't mean 'logically possible' (which, by the way, is about the loosest standard of possibility out there) then I'm completely baffled as to what exactly it means.

It's perfectly possible to come up with situations where things labeled as 1 and 1 are added together to yield something other than 2 - in fact I earlier gave such an example (the result of adding 1 liter of ethyl alcohol to 1 liter of water is approximately (and I'm working from a vaguely faulty memory of chemistry here) 1 and 1/2 liters). The point, as made excellently above, is that this isn't a problem for the truth of 1+1=2, but rather the application of that truth to the empirical situation at hand.

[rppa]

Originally posted by Eleatic Stranger
As far as we can say of anything that it is impossible we can say that it is impossible to have a universe where 1+1 does not equal 2 - it is a logical necessity that it do so based on the meanings of the terms.

Would you say that it would be possible to have a universe where "all bachelors are unmarried men" is false? In other words - could there be a universe where some bachelors are not unmarried men?

You are exactly correct. Statements about 1+1 not equalling 2 in other universes are confusing a mathematical statement (involving definition of "1", "2", and "+") and an empirical one (involving seeing what happens when you put an object next to another object).

I hate to agree with Uninteresting Ian, but he's correct that mathematics is not a science. It's a logical discipline in which theorems are built up from starting assumptions, but there is no need for those assumptions to correspond to a physical universe. Nor is there any such thing as validating or invalidating the postulates. The postulates constitute the universe in which you are working. Mathematics is the discipline of forming those logical constructs, and individual branches concern themselves with particular sets of axioms and theorems. I suppose that makes it a branch of philosophy, but I'm no expert in philosophy.

In mathematics, it is not taken as an assumption that 1+1 = 2. These days the starting point for exploration of the natural numbers is usually the Peano postulates (which define only "1" and "successor"). Since "1+1 = 2" is not a postulate in that system, it must be a provable theorem. And so it is. Here is a proof. (http://mathforum.org/library/drmath/view/51551.html) It starts with the Peano postulates, then defines "+" in terms of "1" and "successor" and defines "2" as "successor of 1".

Discussions of the abstract properties of the symbols "1", "2" and "+" are purely exercises in logic. It's true that the motivation started long ago with physical objects, but the numbers aren't physical objects themselves. In order for this proof to work out differently, you'd have to imagine a universe where logic worked differently.

[hammegk]

"Weak" anthropy notes that sentience would not be available to discuss these questions if this universe didn't obey the logic of math. "Strong" anthropy states sentience must occur in a universe which follows the logic of math as we know it.

How does one accept the 'pre-discovery' physical predictions of math (e.g. Dirac & the positron, many general relativity predictions, the fact that momentum, angular momentum, and energy must be conserved due to symmetries of location, direction & time, QED/QCD guage symmetries, etc.) unless by accepting the math itself to be neither science nor philosophy, but "real"?

[Eleatic Stranger]

Even if math is real, it would still be an open question whether it belongs to philosophy, science, or it's own catagory.

[hammegk]

I'd rather know if math only describes our universe, of if it constrains and defines it. ;)

The latter is my suspicion. :)

[Taffer]

Originally posted by Mendor
Ah, I think I see where we misunderstood one another.

When we say "1 + 1 = 2", that to me is maths.
When we say "One apple plus another apple is two apples", we then (to my mind) start to move in to the application of mathematics to the real world, and empirical sciences. This was my idea of maths being a tool to understand science and not a science in and of itself.

The thing is, though, that if we find that the maths doesn't reflect the real-world situation, we can invent a new system of maths. For example, in a scalar sum, one kilometre plus one kilometre equals two kilometres. But in a vector sum, one kilometre north plus one kilometre west equals sqrt(2) kilometres north-west. The reason I think maths doesn't have a "real" existence (to address Interesting Ian) is that we can do this -- we can start with new axioms and construct different mathematical systems which are perfectly consistent in and of themselves, and which may (or may not) have a relationship with the real world.

Now, to be honest, I can't conceive of a universe where one apple plus one apple equals three apples, but I'm prepared to accept that as a lack of imagination on my part. I'm similarly unable to conceive, say, 11 dimensions, but that doesn't mean it's not a useful or possible concept. I'm sure it wouldn't be beyond mathematicians to come up with systems to deal with addition of apples in that universe (though it's certainly beyond me :D)

But, as Eleatic Stranger points out, "1 +1 = 2" -- given our definitions of 1, 2, + and = and the axioms governing them -- is true for all possible universes, unless we change the axioms. If it turned out that one apple plus one apple gave three apples, we could change the axioms such that "1 + 1 = 3" (though I don't know how we'd accomplish that), but that wouldn't invalidate the old axioms and the statement "1 + 1 = 2 under the old axioms" -- it would just be a different system of maths.

I'm starting to be of the opinion that it's neither science nor philosophy and that it deserves a nice little box all of its own :D

Well said. I, too, cannot imagine an universe where said laws exist, but I just put it down to poor imagination, just as you have.;)

[rppa]

Originally posted by hammegk
How does one accept the 'pre-discovery' physical predictions of math (e.g. Dirac & the positron, many general relativity predictions, the fact that momentum, angular momentum, and energy must be conserved due to symmetries of location, direction & time, QED/QCD guage symmetries, etc.) unless by accepting the math itself to be neither science nor philosophy, but "real"?

These are not "predictions of math". They do not follow from any system of mathematical postulates. They are theoretical predictions based on PHYSICS postulates, postulates which might ultimately be replaced by other postulates. The GR predictions for instance came from the the postulate called the Equivalence Principle. These are assumptions about how the universe behaves. Unlike in mathematics, there is a universe against which such postulates and their predictions can be tested.

[Hydrogen Cyanide]

Originally posted by Taffer
Well said. I, too, cannot imagine an universe where said laws exist, but I just put it down to poor imagination, just as you have.;)

That would be the universe where writers of fantasy, cartoon, sci-fi, adventure,fairy tale and other fiction genres live (including writers of blechy romance novels).

This is the universe where shape-shifters can turn from a huge lion into a little mouse negating any Conservation of Mass... or where someone jumping after a falling person can actually catch up (negating the fact that the acceleration due to gravity is the SAME for both persons)... or that two things can be created from one ("Angels and Demons" episode of Red Dwarf)... or that a supposedly smart woman would pine for years over a wimp when she could have had Clark Gable (I hate that movie).

See more at http://www.badastronomy.com/bad/movies/index.html and http://intuitor.com/moviephysics/index.html .

It is the universe where the dead don't really die, but just go somewhere with a bad communication system (much like my ISP this week)... and one where plain water cures all if you call it "homeopathy", or where vaccines are worse than the actual diseases.

[Taffer]

I'm not really sure the point of your post, but all of that is possible. As I've said before, and will say again. We do not think it could happen because it does not in our universe. It is possible that there are other universes. It is possible that in these universes all these things happen. In fact, you cannot proove to 100% certainty that those things do not happen in our universe. It's just that all our data suggests that it is 'woo woo' bollocs.

[Hydrogen Cyanide]

If you think some of the stuff in BAD movies and books are really possible... then perhaps you need to brush up on your physics.

IF you do find a universe than can create an extra apple from nothing, then be sure to observe and record everything so that you can create the new mathematical models of physics for THAT universe.

Oh, and I would really like to thank Phil Plait for the link to the Physics website that tears apart many movies (including some I actually liked --- like "Abyss"). It looks like a family affair... and I like that they called engineers "applied physicists".

And tonight I was annoyed with an "Inuyasha" cartoon episode.

Okay, I really do try to suspend disbelief, but sometimes it is really really hard. My poor hubby had the misfortune of attending the movie "Firefox" with me (aerospace engineer) and a friend who was a Navy submarine officer --- he got to hear from both of us what was goofy movie nonsense.

[Interesting Ian]

Originally posted by Hydrogen Cyanide
or that a supposedly smart woman would pine for years over a wimp when she could have had Clark Gable (I hate that movie).



Gone with the wind? It's a brilliant book. Never watched the movie. Never could be bothered.

[Interesting Ian]

Originally posted by Taffer
I'm not really sure the point of your post,

I was thinking the same thing! LOL

[Taffer]

Originally posted by Hydrogen Cyanide
If you think some of the stuff in BAD movies and books are really possible... then perhaps you need to brush up on your physics.

IF you do find a universe than can create an extra apple from nothing, then be sure to observe and record everything so that you can create the new mathematical models of physics for THAT universe.

Oh, and I would really like to thank Phil Plait for the link to the Physics website that tears apart many movies (including some I actually liked --- like "Abyss"). It looks like a family affair... and I like that they called engineers "applied physicists".

And tonight I was annoyed with an "Inuyasha" cartoon episode.

Okay, I really do try to suspend disbelief, but sometimes it is really really hard. My poor hubby had the misfortune of attending the movie "Firefox" with me (aerospace engineer) and a friend who was a Navy submarine officer --- he got to hear from both of us what was goofy movie nonsense.

Ahh fair enough, I'll just let it drop. It's not really all that important anyway ;).

[Interesting Ian]

Originally posted by Hydrogen Cyanide
[B]If you think some of the stuff in BAD movies and books are really possible...

So what happened in groundhog day is not possible?

[Taffer]

(Or I guess not...)

Anything is possible, Mr. Ian, with some things simply being very unlikely.

[hammegk]

Originally posted by rppa
These are not "predictions of math". ...

Interesting contention. Is there a chance you didn't understand what I said?

Or do you actually state that the positron was not pre-discovered by Dirac's Equation?

[rppa]

Originally posted by hammegk
These are not predictions of math...
Interesting contention. Is there a chance you didn't understand what I said?

Nope. I understood what you said.

Or do you actually state that the positron was not pre-discovered by Dirac's Equation?

Nope. Had you read beyond the first five words you would have seen what I'm stating, specifically this:
They do not follow from any system of mathematical postulates. They are theoretical predictions based on PHYSICS postulates, postulates which might ultimately be replaced by other postulates.

Mathematics builds from a set of axioms which are arbitrary, beyond being self-consistent. It builds conclusions on those axioms.

What you are talking about is starting with a set of postulates which are not at all arbitrary but which are hypotheses about the universe.
The predictions are logical deductions from PHYSICS postulates, from guesses about nature. Mathematics was the tool used to make those deductions, but the fact that the starting point is a physical hypothesis makes the process fundamentally different from a mathematical conclusion from mathematical axioms.

Why? Because the postulate, and hence the conclusion, might be wrong. Mathematical conclusions (i.e. theorems) don't have that property. The axioms can't be wrong. There's no "reality" to test them against.

So no, I didn't say the positron wasn't predicted prior to its discovery. I said that the process of making that prediction, though it involves calculation and the tools of mathematics, would not be classifed as "mathematics". It doesn't start from the same place as mathematics.

[hammegk]

Oh, goody, we disagree. The basic disagreement is your understanding -- or lack thereof -- of what I said.

I'm happy to agree to disagree on this point.