It has been very hard for me to frame such abstract ideas into words. I generally don't do it. Somehow I managed to pull this off. Enjoy.
Everything you can think of is a solution to 0/0. You multiply anything with 0, you get 0. 1/0 is not defined because you can’t get anything out of nothing even if you reproduce it infinitely. This is where hard math fails to deliver a solution. To satisfy ourselves, we do limit (x tends to 0) 1/x = infinity. But we doubt such ideological ideas. I remember how uneasy my peers felt when we learned about the differential, a quantity as small as possible but not 0. It can't take a definite value because a definite value can be divided further. People get uncomfortable because they think that the Universe is discrete, physically divisible, and material. They kind of distrust dt and dx.
Even though we use approximation, calculus delivers accurate and reliable results. There are no many things that can’t even be calculated without it. We can't ignore that something is innately true about calculus. If we accept calculus as more than just a mathematical fantasy, we are bound to face the philosophical implications of its axioms, one of which says that you can reproduce a differential (let's say dx) infinite times to produce something (x). But how do you reproduce something that doesn't have a discrete value? How do you make a house using an infinite number of bricks that are as small as possible? There is no such thing as the smallest brick if the Universe is discrete and divisible.
Consider atom. It can't be physical because there is nothing stopping us from zooming in and dividing it even further. It must be non-physical and uncountable like energy. But then how do you make physical (what we are biased to call existence) out of non-physical? This question is legit only if we assume physicality. If we allow ourselves to think that our senses might have fooled us, we may not ask this question. Perhaps, hard math & physics fail because physicality is an illusion. If it's true, you don't have to make something out of anything. Perhaps, the universe is nothing, just a void playing with an illusion of physicality.
We don't see magnetic field lines around a magnet, but they are deeply connected with the magnet, in union with it. When we see designs on a piece of fabric, we consider the fabric and the designs on it to be one entity. But we don't see planets, stars, and galaxies as parts of one deeply interconnected infinite multidimensional web. Since our eyes can only see so little, we end up perceiving the Universe as the sum of its parts. Perhaps, we merely identify sharp boundaries and divide the universe into countable things. Perhaps, they merely look countable or approximately countable like random clouds that approximately resemble significant shapes. Perhaps, there is no such thing as countable nouns.
Once we drop the spectacle of discrete countability, we can reasonably propose that hard math & physics are bound to fail. Calculus, on the other hand, doesn’t assume a material world. There is no real/material value to dt. We can reasonably say that atom = dU, where U is the Universe. dU can’t be 0 because we get 1/0. But dU also can’t take any material non-zero value for the universe to be infinite. Thus, dU is atom. The moment it takes a material value, we are no longer talking about atom but about an illusion of atom. One such illusion is what physicists call the atom. The atom, thus, became a discrete unit even though it can be further divided. And the illusion continues.