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1 Mathematics and Physics Have the Same Foundations. 2 The Underlying Structure of Mathematics and Physics. 3 The Metamodeling of Axiomatic Mathematics. 4 Simple Examples with Mathematical Interpretations. 15 Axiom Systems of Present-Day Mathematics. 21 What Can Human Mathematics Be Like? Graphical Key.
And—it should be said at the outset—we’re still only at the very beginning of nailing down those technical details and setting up the difficult mathematics and formalism they involve.) For integers, the obvious notion of equivalence is numerical equality. For hypergraphs, it’s isomorphism. Experiencing the Ruliad.
But by the end of the 1800s, with the existence of molecules increasingly firmly established, the Second Law began to often be treated as an almost-mathematically-proven necessary law of physics. There were still mathematical loose ends, as well as issues such as its application to living systems and to systems involving gravity.
Fast numbers-based ways to do particular computations are often viewed as representing “ exact solutions ” to corresponding mathematical problems. Still, there is in a sense one other kind of computational reducibility that we do know about, and that’s been very widely used in mathematicalscience: the phenomenon of continuity.
Focusing on STEM (science, technology, engineering, and mathematics) activities can be an excellent strategy to keep students engaged in winter. As the air rushes out of the balloon, it propels the balloon forward, vividly illustrating Newton’s Third Law of Motion – for every action, there is an equal and opposite reaction.
And indeed it increasingly seems as if the “secret” that nature uses to make the complexity it so often shows is exactly to operate according to the rules of simple programs. And indeed over the past three centuries there’s been lots of success in doing this, mainly by using mathematical equations.
But in the rare cases its been used in mathematics its typically been to confirm things that were already believed to be true. It is, I think, an interesting challengethat gets at the heart of what one can (and cant) expect to achieve with formalized mathematics. But what about something more like a theory in naturalscience?
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