"Do we need math to understand nature or Do we need to understand nature to (create) math?" This was the topic I've been debating with a guy for a long time. The whole point he says is that "we cannot UNDERSTAND nature without mathematics" How could that be? Isn't that in other way? How can we create maths without understanding nature? Didn't we know objects fall to earth without the laws of gravity? Didn't we know that every action has equal and opposite reaction without newtons 3rd law of motion? These laws are already there in nature, and we just used math to write it down. For example 1+1=2 is the way nature shows us. We used numbers to express it. The laws of addition and subtraction etc aren't made by math itself, it's shown by nature. Or am i wrong? Don't we understand nature without maths?
I think we don't need math, animals can be said to understand nature without math. An orangutan for example has a pretty good understanding of nature, surely.
I definitely think we need math to understand electricity. Who could understand electricity with no math? Who could build a radio circuit or something like that with no math?
We invented the laws of addition and subtraction because they are a convenient set of symbolic manipulations that match up to the way some real-world things behave. But there is nothing that prevents us from inventing other sets of symbolic manipulations that do not match up to anything in the real world. That's what "pure" mathematics is. If somebody comes along later and finds a mapping to things in the real world, that's great. But it's not essential.
We can certainly write down math that has nothing to do with nature, but at least for me, only the math that can be used to understand nature better is interesting. I think the point is that we can understand it better with mathematics. There are many questions about nature that can only be answered by a theory of physics. A theory of physics consists of a piece of mathematics and a set of correspondence rules that tells us how to interpret the mathematics as predictions about results of experiments. Sure, but did you know that the path taken by a thrown object is (neglecting air resistance and the small variations in the direction of the gravitational force over a small region of the surface of the Earth) a parabola? Did you know that the path taken by a planet around the sun is an ellipse? Even if you know these things, don't think you understand them better when you see that there's a single formula that explains both? Without that formula, can you even be sure that the same force is responsible for both? How about light (including the invisible kind, like radio waves). How can you be reasonably sure that light is electromagnetic without finding Maxwell's equations and that they have solutions that describe waves? Newton must have had a pretty good idea about it. Real numbers were invented to mimic certain aspects of space and time. But the definition is based on our intuitive understanding of space and time, which has been proven wrong by special relativity, and then even more wrong by general relativity. General relativity still agrees with our intuition about short distances along a line, but the fact that we haven't been able to fully combine general relativity and quantum mechanics into a new theory strongly suggests that in reality, that intuition is wrong too, probably extremely wrong (at extremely short distances). This doesn't mean that we should change the definition of real numbers. The fact that we have discovered that they're not as "real" as people probably used to think they were doesn't make them any less useful.
I'm not sure how numbers are affected by special or general relativity. If I observed a mathematical calculation carried out by someone moving relative to me, then would I disagree with the result of that mathematics?
I don't buy the notion that mathematics mimic nature. In my opinion, elementary propositions such as 1+1=2 is a rule applied to nature (via counting), not a rule extracted from nature. This applies to all of mathematics.
Numbers are not affected at all by any physics properties or theorems. What makes you think that they might be?
SR and GR both include phenomena like time dilation and length contraction, which show that our intuition about space and time is wrong. GR shows that there's a better model of space and time than ##\mathbb R^4##.
Are you saying that the only mathematics that is valid is that which directly describes the physical universe? In SR, the Lorentz transformation and the energy-momentum four-vector are direct applications of standard "pre-SR" mathematics using ##\mathbb R^4##. I would say that SR and GR left mathematics itself unchanged. Much of the non-Euclidean geometry required by GR had already been devised by 19th century mathematicians. Moreover, mathematics can happily deal with higher, even infinite dimensional vector spaces and manifolds that have no direct correlation to the physical universe. As the universe does not, as far as we know, have infinite dimensions, does that invalidate the mathematics of infinite dimensional vector spaces?
No. I answered that question before you asked it by saying that even though the definition of the real numbers was inspired by naive intuition about space and time, the fact that space and time have turned out to be different from what we thought at first is not a reason to change the definition of the real numbers. It must have at least sped up the development of differential geometry, so no, it didn't leave mathematics unchanged. Those things are extremely useful in quantum mechanics. To most mathematicians, that must have been a very good reason to study them instead of things that appear to have no relevance to physics. Of course not.
I agree there's no reason to change the definition of Real numbers. But, using your knowledge of SR and GR, how would you redefine them to be more useful to physics? Can you give me an example of one idea that might form the basis of "relativistic number theory"?
Why would I redefine them? They're already very useful, so I guess we're talking about defining something new. GR has given us some new things, in particular Lorentzian manifolds, but why would it give us new numbers?
Don't you think we can understand electricity without math? It's all ideas that helped us to develop AC current or DC current. We analyzed those currents and then later developed maths for it
An orangutan knows if you jump, you fall down. He also knows that if he pushes against a tree, it pushes back.
How do we create maths? Show me picture of bear you saw. (Shown is crude picture, bear and bear), Later, "I saw bears". Like how? Show me picture of bear you saw. (Shown is a crude picture, bear and bear and smaller bear and smaller bear) DIFFERENT bear between sitings. WE MAKE PICTURES AND THEN MAKE NUMBERS.