Physicist chose to ignore the infinite nature of universe while modeling it as mathematics but it is a fact that we can never model the universe completely in terms of mathematics. Still all the physics we make is in terms of mathematics and our computation machines are also in the same form. Real universe contains many systems we cannot be modeled accurately as perfect or probabilistic input-output using finite responses. Comparing with this, let us consider another paradigm of physics computers. Let us consider a physical system, which cannot be modeled in more details because of the sheer un-predictability of its responses. But given two exactly same systems, their responses to an input are > 99% in agreement. Wouldn't it be better to create a replica of the remote system in the lab and predict the response of a remote system by giving the input to the system in lab and then model a large more complex system as a combination of the physical results?
My question is wouldn't this be a better model than pure mathematical models?
Thank you Shalender.
I agree that physicists ignore infinities, but especially, they ignore the difficulties presented by Gödel's Theorems or Turing's Halting problem. In my view, confronting these problems is going to lead to answers to very big questions in physics, such as what is behind the Measurement Problem and why is 3-space not 2-space or 4-space?
As to how to model. I believe in isomorphism. That is to say, algebraic isomorphism plus logical isomorphism. I don't mind whether its mathematics or a machine; but I want to see it replicate Nature, both, what is predictable and what is random.
Aleksei> I agree partially. There are some extremely high entropy systems where no approximation, linear piece wise or even a memory map of input output works because the system never terminates (or terminates in a extremely long time) and keeps producing newer results (for example if you want model matter density along a straight line in universe). A partial sample of the system and a model based on it does not produce correct results because it is unable to capture the inner entropy of the system. As you rightly said that we do real wind tunnel simulation for aircraft design but there is no formal paradigm of modeling physics systems as a combination of such black boxes. Given a set of high entropy black boxes can we define what complex systems can be modeled as a combination of results of these systems? If we can do that we can make a physics computer to simulate such specific problem, which not only runs mathematics but also gives input to the "basis systems", measures the output and then uses the data to simulate the big system? Imagine it as a computer with attached jars of systems, which respond to the input in a specific way.
Steve> This is on the similar line of thought of your question -- can a system model the who universe including itself? It may if the universe is fractal but I think fractal model may need some more thinking.
The question of analytical representation.
I don't know anything about numerical analysis. Well, I know computers can't represent analytical mathematics. But, I don't believe analytical stuff is going on in Nature either. Maybe for the same reason that computers don't like exact analytic maths. I don't really believe there is any difference between a computer and any other physical thing. The same physical and logical problems are present in both. Everything is a machine of some sort.
I believe all measurements are rational, not real. I don't believe any irrational information can be communicated, even if it is present in the ontology. And so, for instance, although we understand the notion of the exponential function pretty well, I believe it cannot be exactly communicated in Nature.
I have always been very entertained by a story I heard some years ago. A student told his supervisor: "Irrational numbers don't actually exist". The supevisor told the student, "I'll show you one with a triangle", and proceeded to fold a square of paper to demonstrate a hypotenuse of length root-2. The student answered by saying, yes but that's not a triangle. And it wasn't, was it.
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