Computer proof ‘blows up’ centuries-old fluid equations

They don't explain this very well, I think.

Take the ordinary differential equation x'(t) = x^2(t), with initial condition x(0)=1. It has the solution 1/(1-t) which blows up to infinity when t tends to 1.

If you try to solve it numerically, using, let's say Euler's method, then this is how you go about it. You pick a step size, let's say 0.1. And iterate this way: you know x(0) = 1, and you also know its derivative x'(0) = x^2(0) = 1. You assume x follows a straight line, so you get x(0.1) = 1 + 0.1 = 1.1. At the next step you add more because x'(0.1) is now 1.21, so x(0.2) = 1.1 + 0.121 = 1.221. You keep going like that.

The numbers will go bigger and bigger, but they will never be infinite. Of course, floating numbers with double precision overflow around 10^308, but if you use a multiple precision library you'll be able to keep going forever and ever.

If you make the time step smaller, the solution will be closer to the actual solution, but still, the algorithm will produce finite values at all times (until it hits overflow).

This project is about real-valued functions which are taken to describe physical reality. Almost all of physical reality has no closed-form analytical description that "traditional mathematics" can operate on. So there arent any relevant symbolic rules of inference yet invented to resolve this problem.

If you want to program a computer to perform these rules on these functions, there arent any -- hence the millenium problem. And if there were some, we wouldnt bother using a computer.

What "using a computer" here means is finding a discrete approximation to this system, searching through that discrete input space until something which looks "infinity-like" occurs in the output space.

Now, a priori, this is never going to constitute a proof of anything. Since the discrete approximation needs, independently, to be analytically shown to be reliable. And, a priori, it's likely to be highly highly unreliable.

It would be trivial to show, for example, an iterated chaotic system is sensitive to an x=pi initial state at "decimal places" that no possible physical computer could provide a discrete approximation of; and hence inferences made via this approximation would be, routinely, false. (This is, for example, why most "climate" models only predict a global mean temperature, and very little else).

So this all comes down to the need to formalise a non-discrete system in discrete terms, and worse in terms that are physically possible implement using electrical switches.

In this case, every output of the system including special designation of bit patterns is, a priori, profoundly suspect.

"the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1/2."

Most of us conflate arithmetic with mathematics. In arithmetic, things start getting 'conceptual' as soon as we no longer can map certain measures and operations to a realizable physical construct. At that precise juncture, math becomes a semantic system and is therefore subject to social consensus.

For example, consider introducing infinity, or even zero, into 'shopkeepers' sense of numbers. Before, you could never add something to a number and end up with the same number, but now 0 + 0 = 0, and a + \infty = \infty . And to the shopkeepers' surprise, some mathematicians may even argue over it.

That depends on if one of the rocks breaks in half as you throw it onto the rock-pile or not. And also if the resulting piece knocked off is large enough to pass your fuzzy and contextual distinction between "rock" and "pebble".

But IMHO, arithmetic such as counting numbers and 2 + 2 = 4 are not part of the natural world. If I have a rock and another rock, I can with minor effort tell them apart: they have different weight, size, shape, density, composition etc. They are each unique individual assemblages of huge numbers of atoms in distinct never-to-be-repeated arrangements. In what way are these 2 unique things "the same" ?

If I have an apple and you give me a frog, I have an apple and a frog. They're not the same. If I have apple A and you give me apple B, do I have 2 apples? I have unique apple A and unique apple B. We can pretend that they're the same if you like, but that category is in our thinking, not in the world, and we also know that we can also notice differences between them.

tl;dr the natural world is not fungible, but behaving as if it is, is a convenient abstraction for mathematics and commerce, not a property of the natural objects.