Is Mathematics a science, or a philosophy?

[American]

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.

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]

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 )

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

 

[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]

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. 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]

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 )

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

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]

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.

 

[Chris Haynes]

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.

 

[Chris Haynes]

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]

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]

I'm not really sure the point of your post,

I was thinking the same thing! LOL

 

[Taffer]

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]

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]

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]

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.