Lots of good things have been said about being like a child. Jesus said it was conditional to get to heaven
Being childish is generally regarded as being petty, vindictive, and perhaps stupid, but children are open-minded and every thought is outside the box because they haven’t formed a box yet. Only a child can learn perfect pitch, which is to say that only a child can learn to accurately perceive aspects of our world. youtube.com/watch?v=816VLQNdPMM
Has it been proven that integration cannot be formalized without infinity?
How can we assume an axiom and then claim anything is proven because of that axiom? Since infinity is an axiom, then it can substantiate nothing.
The proof is in the pudding: it gives the right answers consistently and doesn’t require notions of infinity to implement, which is my point: we do not need infinity to “do math.”
So to me, the fact that you DO believe in Riemann integration (aka freshman calculus integration) tells me that you’ve seen the rote procedures, but not the underlying theory nor all the weird counterexamples and corner cases that made 19th century mathematicians realize they needed a rigorous theory. Infinitary math is essential to define an integral and do freshman calculus. They just don’t tell you about this until you take a more advanced course in real analysis.
Because in advanced math you’re studying applications only to math instead of the real world. Advanced math is for people who have exhausted the practical uses of math and have graduated to the study of math for the sake of math.
No infinitary math, no logical foundation for freshman calculus. No axiom of infinity, no Riemann integral.
So I have no logical foundation, but you simply assume the axiom of infinity and use that assumption to claim your foundation is more logical than mine?
ps – Let me give a concrete example. You mentioned integrating an area over a height. How about if you have a rectangular metal plate with a temperature at each point and you want to integrate the temperature over the area of the rectangle to determine the average temperature. You could integrate the vertical slices then the horizontal ones or vice versa. This is multiple integration as in second year calculus. But how do you know when the order of integration matters and when it doesn’t? How do you know whether it makes a difference if you integrate the x’s and then the y’s, or first the y’s and then the x’s? This can be a very tricky business, especially with a weird or pathological integrand or temperature function. This is when you have to drill down to the rigorous, set-theoretic definition of the integral to prove theorems on reversing the order of integration. In other words the moment you go beyond the simplest examples you need some theory; and the theory of integration requires infinitary set theory, or my name’s not Guido Fubini!
I don’t see why it would make a difference whether we integrate in the x or y first so long as the function accurately describes the temperature variation. If the right answer is only coming out in one direction, then some more-fundamental assumption is probably flawed. Perhaps you could explain the temperature problem in more detail and show why it matters in one direction vs the other, then maybe we can see why.
This reminds me of those hotly debated arithmetic ordering of operations puzzles that I hope you’re familiar with because I can’t find a good example at the moment, but some will fill the comments section with debates and ultimately have no concrete resolution.