wtf wrote:Serendipper wrote:Has it been proven that integration cannot be formalized without infinity?

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 science isn't founded upon infinity and neither is math. That is especially true since no person nor machine can comprehend infinity, yet both people and machines routinely do math.

But is all of modern mathematics, including nonconstructive math and uncountable sets, necessary to found physics?

Physics breaks down when infinity is introduced. Example: black holes. Physics cannot describe what is assumed to be infinitely dense, infinitely small, infinitely large.

I no of nobody who claims to be able to found physics on finitary principles.

Physics has never been able to deal with infinity, so how can it be founded on it?

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.

Existence is the relationship between subject and object. If the subject does not perceive/behold/comprehend/relate to/be affected by the object, then the object doesn't exist.

You may be unhappy with this, because we do have infinite subsets of the naturals

We don't "have an infinite set" of anything. You induce/infer that we do, but you haven't proved that any such sets: 1) could exist. 2) do exist.

So constructivism still needs "a little infinity," but far less infinity than full set theory.

All we need is a finite number that won't fit in the universe, called a "dark number".

For instance, an infinitesimal is a number greater than zero, but smaller than any means of measure, so it's a reciprocal of a dark number, which is less than infinity, but larger than anything we could measure.

Once we exceed the carrying capacity of the universe, nothing is changed by assuming yet bigger numbers.

How are you going to express the differential equations of biology?

Just write them down like we have been.

Why is it so important to you?

You're asking why is it important to me to combat absurd ideas? Why is it important to you to advance them? lol

Serendipper wrote: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.

I quite agree. Nobody thinks the axiom of infinity is "true" in any meaningful sense. Rather, infinite sets are USEFUL to mathematicians,

The **ramifications** of infinite sets are useful to mathematics. We cannot behold infinity, but we can say if infinity were the case, then this conclusion could be drawn. IOW, if x could be infinite, then 1/x would be zero. We suspect that to be true because we extrapolate bigger and bigger numbers while observing the effects on the function, which seem to tend to zero, so we conclude (without going all the way out to infinity) that if x were infinite, then the function would be zero.

A good example is 1+2+3+4+5+... = -1/12. If we stop adding at some finite location, the answer will be a large positive integer, but if we go all the way to infinity, the answer is -1/12. If that is counterintuitive because it violates our extrapolation, then maybe 1/x does too.

If you don't like -1/12, the Achilles heel in the proof is the assumption that 1-1+1-1+1-1+1-1+.... = 1/2. If we ever stop the process of adding and subtracting 1, the answer will always always always be either 1 or 0, but somehow, at infinity, the answer becomes 1/2 and one could either take it or leave it, I guess.

But the -1/12 has some empirical evidence for substantiation, so if we assume it's true, then the 1/2 is also true, which means that things too large/small to measure are in a superposition of states, which is what we see in quantum physics.

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?

As long as everyone knows it's a game, then it doesn't bother me, but when they start on about "existence is necessarily infinite" or "the universe is infinite" and therefore ___________, then I feel like I have to reinvent the wheel for each new interlocutor, so I figured redirection to this thread would substitute.

Serendipper wrote: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."

We don't need the queen to play chess. So what?

We don't need pink elephants, unicorns, leprechauns, teapots either to play chess.

Serendipper wrote: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.

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.

You proved my point. In the 1840s there was no practical use for non-euclidean geometry; therefore it was math for the sake of math. The advanced math of today is likewise math done for the sake of math without any practical use. An engineer, scientist, or anyone who isn't a mathematician would not endeavor to study math that is, maybe, one day 200 years from now, might have a practical use.

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.

Probably because waves are assumed to extend to infinity.

So you're just flat out wrong on the facts here.

Even if that were true, it could be the case that the "facts" are wrong, but I don't think I'm wrong on the facts even if it meant something if I were.

Advanced abstract math is indispensible for modern science.

If that were true, it would not be called "abstract".

Abstract-

adjective

1) thought of apart from concrete realities, specific objects, or actual instances: an abstract idea.

2) expressing a quality or characteristic apart from any specific object or instance, as justice, poverty, and speed.

3) theoretical; not applied or practical: abstract science.

Abstract math is disconnected from reality.

Serendipper wrote: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?

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.

Perhaps, but it's also possible I'm arguing against positions you have expressed, but, for whatever reason, claim you haven't.

Serendipper wrote: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.

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.

If you're not interested in explaining it, then I'm not interested in learning it. My time is finite and I have to choose where to spend it.

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.

Integration is essentially the summation of infinitesimals and I don't see why it would matter from which direction we begin the addition, and if it does matter, then it's likely that some order-of-operation has been violated (or some similar problem).