An interesting question I've been thinking quite a bit about recently. Do you think human mathematics are invented to explain the world around us, or do you think we are really discovering aspects of the universe when we discover new mathematical laws? Or something else entirely?
...Good question, stuntddude. I think it's a little bit of both. We discover more aspects of the world with new mathmatical laws, but they were created to explain the world. It's sort of like the chicken-or-the-egg question, I think. What are your thoughts?
As of yet, I'm personally undecided, and may remain so for a while. It seems absurd to me that a number would be a real physical things, and so how can you discover something that isn't real to begin with? On the other hand, are numbers not modeled after properties of real things? If there are two trees in my yard, isn't it a property of the trees in my yard that there are two of them? And if mathematics is a human invention not originally intended for, say, complex physics... how can it fit so well if it isn't a discovery? And how can it have been invented if some things that are discovered mathematically aren't understood non-mathematically until years or decades later?
I think its definitely a human concept. You can't find some "mathematics" or a "number" in the universe. You can find things that can be represented by mathematics, but not the mathematics itself. Essentially math is just a modeling tool - a way to construct a system that we can use to understand the universe. But it is never reality - it can be very darn close though.
"Are we really discovering aspects of the universe when we discover new mathematical laws?"
No. There have lots of beautiful, awesome mathematical theories that don't have anything to do with the way the universe really works. Math is very, very important to physics, but we must never forget that reality governs and shapes the equations we create, not the other way around. String theory is a prime example of a theory that is beautiful mathematically, but (might) have nothing to do with reality.
Mathematics are some of the fundamental laws that the universe is built on. There are different ways to interperet them (for example, binary vs. base ten) but the behavior of the numbers does not change. 10+10= 100 (binary) is true, and so is 2+2=4, and so is 'one apple put with another apple makes a pair of apples'. Humans can choose different ways of modeling these laws, but they are fundmental.
One of the strongest pieces of proof for this is the fact that math always holds true. We do not have 'chemistry math' and 'physics math' and 'accounting math' and 'cooking math'- we use the same set of numbers and laws for every application.
Imaginedangerous: I think you have it backwards. You don't model mathematical laws - you model physical laws with mathematics. Just because they don't change doesn't mean they're "real"
"You can find things that can be represented by mathematics, but not the mathematics itself."
You can't find math or any number as an object or a physical thing, but it seems clear that it is an intrinsic property of a pair of trees that there is, in fact, two of them. The same way it's an intrinsic property of these particular trees that they have branches, their quantities are also properties of them. (quantum physics, anyone?
And so, if these quantities are observable traits, then does it not make sense that numbers are real traits in the universe? That they're real properties of things in said universe? If so, maybe it makes sense that the universe which is so full of numbers as properties can be discribed mathematically. After all, even the most complex of number-based mathematics (to my knowledge) is based on the four basic operators -- addition, subtraction, multiplication, division -- that clearly affect the numerical properties of real things in the universe (I can add or take away or multiply or divide the number of trees and it will always give me the same result from the same operations).
And even if most math is inaccurate at describing the universe, even if I can make up a set of meaningless numbers and operations, that doesn't mean I can't also discover real mathematical laws and relationships. To use a simple analogy, I can make up whatever kind of animal I want -- a bat with an eagle's wings? Done, I'm not imagining it -- but does that mean I couldn't also go out and discover a new, real animal species? I don't think so.
...There are some things that make my brain hurt when I try to comprehend them. Chemistry is one of those things. This thread is another. ;-)
stuntdude: So I see your point about a pair of trees, but the fact that there are two of them is just a human way to describe the system and its properties. The distinction between this is that I think the number 2 models or describes the physical system. Where as you think that it is part of the physical system.
I stand by my earlier position, but I understand what you're saying and I could be convinced to change my mind - I'll have to think about this some more.
I realized I've been arguing the side that they're discovered in this thread, but I'm still not sure. It seems intuitively very wrong that mathematics would be anything but a human invention.
Maybe humans "discovered" mathematics in the same way that we "discovered" how to build tools. I dunno.
Quantum: "Just because they don't change doesn't mean they're 'real'."
Um, forgive me if I'm wrong, but isn't that how, well, science works? One of the reasons we know gravity is a law instead of an odd apple-tree-related quirk is that we can see that it works every single time. The earth has a gravitational pull on everything- everywhere. Every other object in the universe also had a gravitational field- from the sun to the moon to my laptop. We know that it is a law because it applies every single time.
Same goes for most other scientific concepts. The idea that 'matter is made of atoms' applies every single time, which is how we know that 'atoms' are real and not just a human construct used to model extremely small amounts. The idea that 'parental organisms pass genes to offspring' applies every single time, which is how we know that genes are real and not just a handy analogy used to describe why I look like my dad. Likewise, because math holds true in every case, we know that numbers are an integral property of a system instead of just a fancy human way of trying to understand things.
(Of course, I'm assuming that by 'real' you meant 'an integral quality of a system'. If you meant 'having a physical representation' that's different.)
1. Yes. That's how science works, but it doesn't mean that everything that's repeatable is scientific fact.
2. Mathematics isn't an integral quality of something. It describes things that are an integral quality of something. Its a subtle (and meaningless in terms of actually calculating stuff) distinction. But it is a distinction.
How do you know there's a distinction?
One can't find mathematics in something - only what math describes. For example, 2 trees. We use the number 2 to represent the idea that at point A there is an object that looks like a tree and at point B there is another of the same object. We explain that idea using the concept of 2 because its really awkward otherwise. I don't know if that's very clear or not...
I still don't think I get it. Couldn't the same argument be applied to any property of a system?
Yeah...I think so. But basically any way we describe a system is just a model of the real thing, right? I don't know. I'll have to think about this some more. It still seems very wrong to me that math is real. I'm pretty sure its just a human concept, but I'll have to think a bit more.
'Any way we describe a system is just a model of the real thing'. If that's true, then all science is invented by humans and we don't really 'discover' any of it. The Copernican view of the solar system is only a model. The theory of evolution is only a model. Our knowledge of the atom is only a model. None of it actually holds in reality.
No, not everything that's repeatable is a scientific fact- but everything that repeats every time without exception is. 'I go on Teen Ink in the afternoons' is a repeated event, but not a scientific law. There are often exceptions. 'Matter is made of atoms' is repeated and always consistant. This makes it a law. '1+1=2' is repeated and always consistant. That makes it a scientific law.
And maybe it 'seems' wrong, but that doens't mean it isn't true. It seems wrong to me that time can curve, or that over 90% of our DNA can be junk, or that some liquids can be less dense in thier frozen state- but it happens anyway.
Imaginedangerous: Of course. Even if something seems wrong doesn't mean its not true. That's obviously not a good argument one way or the other. It doesn't mean we should totally rule out what "seems" wrong though. Sometimes you're right.
Also, in regard to science being a model. Yes. That I can say with confidence as I took a Research in Physics class this year and we spent about a week talking about that. All science is a model invented by humans to model what is "real". Of course, those models can be very, very accurate. So in that sense we are "discovering" things about the real world based off of models we "invented". But no model, and no science, is 100% accurate with no uncertainty. All scientific measurements have uncertainties of some sort and thus so do all theories/models. As a very simple example take a paperback book. That book is obviously real. Now, how do you determine its properties. One such property is its volume. You can measure each side length and the thickness to calculate this volume, but the calculated volume isn't the true volume of the book (not a true property). It is a very good approximation based off of a model of the book as a perfect rectangular prism. The model is limited by the jaggedness of the book and the precision of your measurement. So your model is a good description of the book that takes into consideration one of the properties - its volume - but your model is not, and never will be reality. That doesn't make it useless, though.
One more thing. You say 1+1=2 is a scientific law. That's not true and that's what I meant by not everything that's repeatable is scientific fact. I'll clarify. Everything that's repeatable every single time without exception is not necessarily scientific fact. 1+1=2 is not science. It doesn't explain anything about the real world. It doesn't model anything. It is just a statement of mathematical fact.
"1+1=2 is not science. It doesn't explain anything about the real world. It doesn't model anything."
I'll disagree with this on the grounds that not only have you said before that mathematics does model the real world, but because you were (sort of) right the first time. In fact, a huge portion of scientific law is purely mathematical, and I daresay the point of such laws is to model the real world.