I understand the openmindedness of children, but the subject we are discussing is better served by assuming the participants are intelligent adults who have perhaps been to school or maybe read and thought a little bit about things.
Now that is a very good question! What is true that not only calculus but all of physical science is currently founded on infinitary mathematics. But, it is this a necessary or a contingent fact? I’m pretty sure it’s contingent. Foundations go in and out of favor. Netwon got results using math that’s not regarded as rigorous today, and in fact required another 200 years to logically formalize.
But is all of modern mathematics, including nonconstructive math and uncountable sets, necessary to found physics?
There are researchers trying to find weaker logical structures in which to do math and physics. Finitism (No axiom of infinity, but still with mathematical induction); and ultrafinitism (not even induction); are far too radical and I no of nobody who claims to be able to found physics on finitary principles.
However, constructive foundations are a subject of great interest. In constructive math and physics, an object is said to exist only if it is the output of a Turing machine. I discussed this earlier. So there is no axiom of choice, no uncountable sets, no noncomputable real numbers.
en.wikipedia.org/wiki/Construct … athematics
In set theory we have the full powerset axiom, that says that all of the subsets of a given set exist. In constructive math, only the contructible sets exist. These are the sets whose elements can be cranked out by a computer program (as exemplified by a TM). So the even numbers exist as a subset of the naturals. But most sets in standard math no longer exist because their elements can’t be computed.
You may be unhappy with this, because we do have infinite subsets of the naturals and for that matter we have the full set of naturals, infinitely many of them. So constructivism still needs “a little infinity,” but far less infinity than full set theory.
That is the state of the art today. If you wish to hold out hope of a glorious future in which all of physical science can be founded on ultrafinite or finite principles, that is your right. But why? How are you going to express the differential equations of biology? Why is it so important to you?
I quite agree. Nobody thinks the axiom of infinity is “true” in any meaningful sense. Rather, infinite sets are USEFUL to mathematicians, and infinitary math is useful to physics. Whether it’s necessary, we don’t know.
Put it this way. We could play chess without the queen. The game would be very different and much more dull. So we keep the rules the way they’ve evolved.
The axiom of infinity is like that. It’s a more fun and usesful rule so we keep it in the game. Why does that bother you?
We don’t need the queen to play chess. So what? But you’re wrong on the facts. Without the axiom of infinity, at the very least the constructible sets, you can’t develop the theory of the real numbers sufficiently well to do modern physics. Sure someday someone MIGHT find a way, but in the meantime are you throwing out all of science back to before Newton?
But no, this is quite false on the facts. Differential geometry and non-Eucidean geometry were mathematical curiousities in the 1840’s, and became the mathematical foundation of relativity aftter Einstein.
And quantum physics lives in the mathematical framework of Hilbert spaces, a highly abstract infinite-dimensional vector space studied in a field called functional analysis.
So you’re just flat out wrong on the facts here. Advanced abstract math is indispensible for modern science. Not all of advanced math, but much of it. Sure there is math that’s “out there” today, but who is to say it won’t be essential to the study of the real world a century from now?
Not more logical. More useful. If you’d discuss what I write and not the words YOU put in my mouth, this would be more productive. You are constantly arguing against positions I’ve never expressed.
I don’t say the axiom of foundation is more logical than its negation. On the contrary, they are both equally logical, each being consistent with the rest of the axioms. The axiom of infinity has proven itself more useful so most mathematicians adopt it. There are constructivists, finitists, and ultra-finitists among mathematicans. Especially in the past few years, there’s renewed interest in constructivism due to the influence of computers and automated proof checking.
I linked a Wiki article that contained counterexamples, and I explicitly called out that fact. The Wiki article on Fubini’s theorem contains examples of functions whose integral depends on the order of integration.
Those puzzles only demonstrate the poor teaching of the order of precedence of the arithmetic operators. And the poor understanding of this topic even among elementary school teachers. They don’t hire elementary school teachers for their math acumen. God knows I wouldn’t spend my days among a bunch of ten or twelve year olds.
I don’t see how you can make this comparison. If you think those silly puzzles are anything like the discussion of the axiom of infinity, I don’t think you’ve given the matter enough thought.