Underseer
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Wait. Does numerical analysis belong in the natural science forum?
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What category of science is math?
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lpetrich
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Why wouldn't it?
beero1000
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Underseer
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Because unlike science, math is not evidence-based?
OK, you could make the argument that math is ultimately evidence-based because it ultimately comes from counting, which in turn is a consequence of our existence in a universe made up of discrete objects and the need of early languages to describe quantity with greater and greater precision as a result of interacting with the real world, but is that really enough to classify math as a science?
lpetrich
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Mathematics is not an empirical science. It is pure deduction.
Bomb#20
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We already have a forum for pure deduction.
A good a place as any on the site.
It seems like in general speech it is 'math and science'. Math is not exclusive to f0rmal science.
Numerical methods and analysis comes under Applied Mathematics.
Do you have a topic?
I think we had this debate on the old forum.
Math is ultimately validated by real world application.
Applied Math-Numerical Methods
Curve fitting
Interpolation
Solving systems of equations
Solving differential equations
I Started with this book in the 80s. Learning numerical methiods was why I goy my first computer.
If you want to start to dig deeper then generalities it is well worth the used $6.
http://www.amazon.ca/Applied-Numerical-Methods-Microcomputer-Terry/dp/0130414182
Math is ultimately validated by real world application.
Applied Math-Numerical Methods
Curve fitting
Interpolation
Solving systems of equations
Solving differential equations
I Started with this book in the 80s. Learning numerical methiods was why I goy my first computer.
If you want to start to dig deeper then generalities it is well worth the used $6.
http://www.amazon.ca/Applied-Numerical-Methods-Microcomputer-Terry/dp/0130414182
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lpetrich
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How is it validated? The usefulness of math is different from its validity.
lpetrich
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(Me: Mathematics is not an empirical science. It is pure deduction.)
While we are at it, we can also move empirical science back into philosophy, because it used to be called "natural philosophy". "Science" as a general term became common only in the late 19th cy., though references to specific "sciences" are much older than that.
Philosophy?
While we are at it, we can also move empirical science back into philosophy, because it used to be called "natural philosophy". "Science" as a general term became common only in the late 19th cy., though references to specific "sciences" are much older than that.
Underseer
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Math is a useful tool, yes, but ultimately it's just an axiomatic truth: it is true because it is internally consistent with a set of rules that we all agree to follow. It's not at all like proper empiricist/evidentialist disciplines like science. Heck, it's not even as evidentialist as history. The closest we can come to finding an evidential basis for mathematics is counting.
travc
Junior Member
Most mathematics were derived/developed to describe systems in the natural sciences. It isn't science itself, but it is intimately related.
It generally belongs in this forum, though occassionally a thread may get really abstract and be moved off to philosophy or someplace else.
PS: One of the "final lecture" points brought up in one of my applied math classes was pretty damn cool:
For every solvable mathematical equation, there is a (countable) infinity of unsolvable equations. Yet, nearly all the equations we come across describing the physical world are solvable. That isn't an accident, it is a result of how mathematics was developed and what it was developed to do (sort of the environment in which it evolved).
It generally belongs in this forum, though occassionally a thread may get really abstract and be moved off to philosophy or someplace else.
PS: One of the "final lecture" points brought up in one of my applied math classes was pretty damn cool:
For every solvable mathematical equation, there is a (countable) infinity of unsolvable equations. Yet, nearly all the equations we come across describing the physical world are solvable. That isn't an accident, it is a result of how mathematics was developed and what it was developed to do (sort of the environment in which it evolved).
lpetrich
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I've never heard of that one before. Where can I find out more?
Bomb#20
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No, "Logic". Not my fault somebody stuck it in the Philosophy subforum.
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And vice versa.
travc
Junior Member
Solvable in that context is indeed "has an analytical solution", but there are many equations which aren't solvable numerically either. For eg: cos(x) = 2
lpetrich... sorry, I don't really have more info. As I mentioned, it was part of an sort-of "parting lecture" for the last serious math course most of the students in the class would ever take (it was the top level math course required for Engineering degrees at Caltech). I'm also the sort of person who remembers concepts and forgets all the details
The proof involved pretty simple set theory that you'd see in good computer science or computational linquistics... Sadly, I end up making a total hash of it every time I try to explain that stuff.
beero1000
Veteran Member
Solvable in that context is indeed "has an analytical solution", but there are many equations which aren't solvable numerically either. For eg: cos(x) = 2
lpetrich... sorry, I don't really have more info. As I mentioned, it was part of an sort-of "parting lecture" for the last serious math course most of the students in the class would ever take (it was the top level math course required for Engineering degrees at Caltech). I'm also the sort of person who remembers concepts and forgets all the details
The proof involved pretty simple set theory that you'd see in good computer science or computational linguistics... Sadly, I end up making a total hash of it every time I try to explain that stuff.
You can argue something like that using Liouville's theorem - which says that an elementary anti-derivative must have a special form. That implies that many functions, even elementary ones, have no closed-form indefinite integral.
Solvable in that context is indeed "has an analytical solution", but there are many equations which aren't solvable numerically either. For eg: cos(x) = 2
lpetrich... sorry, I don't really have more info. As I mentioned, it was part of an sort-of "parting lecture" for the last serious math course most of the students in the class would ever take (it was the top level math course required for Engineering degrees at Caltech). I'm also the sort of person who remembers concepts and forgets all the details
The proof involved pretty simple set theory that you'd see in good computer science or computational linquistics... Sadly, I end up making a total hash of it every time I try to explain that stuff.
Ok, from algebra all equations or functions do not have infinite ranges and domains. I thought you might have meant something else.
6x = 2 has no solution if x is limited to integers.
lpetrich
Contributor
This theorem?
Liouville's theorem (differential algebra)