Yeah I know, but some here consider themselves fairly proficient in math when measured against their peers and then you come along and wtf lol
True, but is there a smallest number that you could write down? I think that’s the point of all this: where the abstract meets the reality. Like the smallest point, the largest size, the fastest speed, math transcends all of it.
True, but in reality there are smallest quantifiables which makes it discrete, even though the quantums are continuous with other quantums in actuality, but it’s not possible to differentiate a smaller one.
Well: 1) it was really just a shorthand “manner of speaking” like x approaches infinity. 2) To me, physics is the theories. Physics is not the reality just like words are not things. That’s just how I set it up in my mind. 3) You did it yourself: “I know of nobody who claims to be able to found physics on finitary principles.” viewtopic.php?f=4&t=194376&start=175#p2713785
Yes I see, you are equating “physics” with the actuality while I’m equating it with the knowledge of actuality.
When I say “fooling ourselves” what I mean is a mind or machine cannot conceptualize or imagine the infinite. It can’t be modeled, but we imagine the biggest thing we can, then extrapolate by inference and call it good enough. We assume there are bigger numbers without actually empirically verifying it, and the assumption that something exists without proof is what I mean by “fooling ourselves”.
But Max said:
Not only do we lack evidence for the infinite but we don’t need the infinite to do physics. Our best computer simulations, accurately describing everything from the formation of galaxies to tomorrow’s weather to the masses of elementary particles, use only finite computer resources by treating everything as finite. So if we can do without infinity to figure out what happens next, surely nature can, too—in a way that’s more deep and elegant than the hacks we use for our computer simulations.
When you scold me like that, I feel like I have to walk on eggshells.
If reals are not infinite, then primes cannot be infinite, right? So primes are infinite only because reals are infinite and the infinity of primes is just a consequence of the infinity of reals.
Well where is infinity actually used in a calculation?
“Goes to zero” I can handle. “Being zero” makes no sense. And if it’s not zero, then there is no infinity because in order to have infinite slices, each slice would need to be zero-wide.
So how did it work out with Berkeley?
But the limit is just an extrapolation. Instead of saying 1/x =0 where x = infinity, we say 1/x → 0 where x → infinity. Limits aren’t based on infinity, but purposely avoid infinity by extrapolating towards it. We say “if x gets bigger, then 1/x gets closer to zero”.
I can see why lol
How do they model the infinite part of it?
Well somebody must have a finite theory if computers produce answers.
The electromagnetic force has infinite range, but photons cannot be emitted until their destination is found, which means its range cannot be infinite.
Likewise gravity is an interaction that requires a second something that cannot be infinitely far away.
Show me how infinity is used in this y=1/x where x → infinity. Infinity is never used and the math is not based on infinity. That is the point I’m trying to make and the same point is applicable to QM or anything else that “appears” to be based on infinitary math.
I’d concede that 1+2+3+4+… = -1/12 is based on infinitary math because of the assumption that 1-1+1-1+1-1+1-1+… = 1/2. If you reject that is 1/2, then the proof falls apart.
But 1/x where x-> infinity never uses infinity or any strange properties of it.
I need concrete examples to work with, like 1/x or 1-1+1-1+1…
If time stopped then there is no motion because motion is relative to time (ie 50 miles per hour). If there is no change in time, then there is no change in distance. The derivative is the slope of the position plotted against time, mx+b, so the constant velocity would be m. No division by zero necessary.
I’m an asshole. I apologize. It was completely unwarranted for me to go off on you like that. You were asking good questions about what I wrote. I totally apologize. I am way too thin skinned about imagined slights for my own good.
Also … since what’s an apology without a weasel clause … I"m still sensitive about all the beatings I took at the hands of Saint. I should really try to let that go!
I don’t know what that means. The antecedent is false so the implication is valid, but the argument is not sound. It’s based on a false premise. Euclid proved that “there is no largest prime” 2000+ years ago and he had no modern math, barely even the concept of number. He didn’t prove there are infinitely many of anything. Just that there’s no greatest one. Which amounts to the same thing.
Are you familiar with Euclid’s beautiful proof?
Infinity pertains to how we do the math that’s used in the calculations of physics. I’ve agreed to that many times. I make no metaphysical claims about how the world “is” or even if that’s a meaningful question. I only say that the math that the physicists use is based on infinity.
You will have to ask a physicist. Hilbert space is a function space. Think about a continuous function from the reals to the reals, say. Now think about ALL the continuous functions, all at once. That’s like Hilbert space. It’s an infinite-dimensional vector space. (Hilbert space has additional restrictions)
That’s a naive view from the 17th century. Or, it’s a perfectly modern view for a physicist or engineer! And it’s an ongoing philosophical issue.
However in modern standard math, there are no infinitesimals and what you said is not true. They’re not small thingies with discernable qualities.
The formal theory of limits resolves this problem.
Did they teach you about limits? If you didn’t take calculus or if they taught limits badly (very common) one would not necessarily understand that the modern theory of limits has solved this conceptual problem. It took 200 years.
He was a bishop in the Catholic church. Things worked out well for him, he lived a good life and died in 1753, and his name is remembered today. en.wikipedia.org/wiki/George_Berkeley
The modern theory of limits. It solves this ancient problem. It’s one of the greatest intellectual achievements of humanity. I wish they explained all this better to students but they don’t and that’s that. From Newton to Zermelo, the arithmetization of analysis. Putting continuous math completely on the back of logic and the axioms of set theory.
Understand, I’m not saying this is all perfect or that there aren’t philosophical and even mathematical objections. I’m only describing what actually happened, and how most people think today. Tomorrow it could all be different. I am not being dogmatic; but I am being accurate in the history and in describing how mainstream mathematicians see calculus today.
This isn’t a good venue for this discussion, perhaps a different thread? We can talk about the theory of limits. The point though is that it’s not just an extrapolation or gimmick. If that’s all it were you’d be right. The big deal with limits is this:
We can start from the rules of predicate logic; assume a handful of set-theoretic axioms; and develop a comprehensive theory of limits that makes calculus and all its generalizations perfectly rigorous. The generalizations like differential geometry and functional analysis are the mathematical heart of modern physics.
You can’t argue with the actual historically contingent theory. And like I say, nobody has any idea how to do a finitistic theory that works.
I think I explained the bit about the space of all continuous functions earlier. That’s the model to keep in mind. It’s a vector space so it must have a basis. What does a basis look like? That will keep you up nights.
Hey you will have to take that up with Maxwell and Einstein and all those big brains. I myself wonder how a photon can travel forever, but I assume the physicists have worked it out.
Can’t disagree. But does it require a second something? Even if there’s only one massive body in the universe, its gravity distorts the universe, right?
I’m not sure how any of this got directed at me. Am I replying to the wrong post? I am a little confused because you seemed to quote your own post, which I’m replying to. I don’t know anything about forces, I’m strictly talking about the historically contingent development of math and physics.
If by anything you mean the physical world, I don’t disagree. But if you’re talking about the historically contingent theories of physics as they actually developed, you’re wrong.
You shouldn’t confuse the use of infinity in calculus with the infinite sets used in mathematical physics.
You’re looking at some bad Youtube videos. That equation’s not true for the usual definition of +. It’s about a thing called zeta function regularization. No point going off in that nonproductive direction.
Everything in science is based on the real numbers and the real numbers are a very strange infinite set.
If you measured one single instant of time, how would you know that? It’s Zeno’s arrow. If you stop it in time, it’s just sitting there in space. How does it “remember” that it’s going forward with a particular velocity?
Well anyway I hope I responded to the right questions. I might have been confused about some of the quoting.
I wish I knew what he is trying to say because all of it is way way above my head
I should have studied maths when I was younger but now I am too old and stupid
At least you understand him so he is not completely wasting his time being here
I’ll state my thesis as clearly and briefly as possible. No math is required.
Regarding the world “out there,” I take no position and claim no knowledge. There may be a God, there may not be. The world may be discrete or it may be continuous or the question itself may not be meaningful. It may be infinite or finite or the question might not be meaningful. I could be a brain in a vat or a Boltzmann brain: a momentary accidental coherence in a formless and random universe. I take no position, I claim no knowledge, and nothing I say refers to “ultimate reality,” assuming again that the phrase is even meaningful at all.
There are humans, we came out of caves and developed these big brains and our capacity for abstraction. We made up stories about the stars, the seasons, the world. Around the time of the Renaissance our thinking began to become scientific, in the sense that we did repeatable experiments and tried to build mathematical theories about them. This intellectual trend hit its stride with Galileo, and became the dominant way of understanding the world with Newton. It was Galileo who said that the book of nature is written in the language of math. irishtimes.com/news/science … -1.3388465
Since the 17th century, physicists starting with Newton founded their physical theories on infinitary math; that is, math that uses reasoning about infinity. In the next 200 years this infinitary reasoning was placed on a purely logical footing. We can write down some rules of inference, and assume a handful of axioms about sets; and then logically derive all of known mathematics.
The physicists use this infinitary math to do their most advanced theories. General relativity and quantum physics are based on infinitary math. Our basic idea of time is that of a continuous flow modeled by the mathematical real numbers, which are a highly infinite set (a “larger infinity” than that of the plain old counting numbers).
Nobody knows if infinitary math is necessarily the foundation of physical science. It’s the historically contingent work of flawed people. Tomorrow morning someone could come up with a finitary theory of physics. However nobody has done so yet, and there aren’t even any serious efforts to do so. There ARE in fact efforts to base physics on constructive math, which allows infinite objects as long as we can explicitly describe how to construct them. [In standard math there are objects that exist but that we cannot show how to construct].
That’s really it. Of the nature of the world, I know nothing. But the history of math and science has led us to the present day, in which we MODEL the world using mathematics that depends on infinite sets. What that means, I don’t know. I do personally suspect that just as non-Euclidean geometry started out as a mathematical curiosity yet became the foundation of Einstein’s relativity; infinitary mathematics may be telling us something about the world. Something waiting for the next Einstein to discover.
Infinite sets underlie convergence theory, but you can have infinite sets without talking about convergence.
Yes.
Not necessarily. It depends on the distribution. For example there is no way to define a probability for picking one set out of the counting numbers 1, 2, 3, … in such a way that every number has an equal probability. That follows from the axioms for probabilities as set down by Kolmogorov. The requirement of countable additivity implies that no countable set can have a uniform probability measure defined on it.
That doesn’t follow from anything you said. I have no idea what you mean.
No it’s not. That there is more than one person? The fact that I can’t find a parking space because of those pesky other people implies that the mathematical theory of convergence is fatally flawed? No it doesn’t.
That’s just a matter of definition. The sum of 1/2 + 1/4 + 1/8 + 1/16 + … IS exactly 1, because that’s how the sum of an infinite series is defined. This is taught in the second semester of freshman calculus all over the world.
I like this because it is entirely non dogmatic about the nature of actual existence
I try to avoid dogmatism in general and especially on something as complex as this
Convergences add a magic infinitesimal to round off.
When dealing with infinity, the odds of picking the correct number (that an infinity number generator will pick, or bob next door) isn’t 0.0…1. When it converges at infinity, the odds are exactly zero percent.
Convergences work both ways, not only magically adding the infinitesimal, but subtracting it as well.
Untrue. It took 200 years, but mathematicians finally figured out how to banish that kind of imprecise thinking. The theory of limits is perfectly logically rigorous and does away with magic infinitesimals.
Untrue as I pointed out. You can’t put a uniform distribution on a countable set. That’s because of the axiom of countable additivity, explained in the probability link I gave you earlier. If the probability of picking each of 1, 2, 3, … is zero, then the sum of those probabilities is zero. Yet the probability of picking SOME number is 1 as you noted. Therefore we can NOT put a uniform probability distribution on any countably infinite set.
Simply not true. There are no magic infinitesimals in the real numbers.
Let’s say I’m not even doing it as a function of 1/3…
Let’s say I have a paper with the lines.
On each line I start
.3…
.3…
.3…
And I do this forever.
You’re trying to tell me that it equals one without a magical floating point?
I’ll start right now, and live forever just to prove you wrong.
What I think happened after 200 years (and you never gave me a link). Is that mathematicians figured out how to explain how to give hocus pocus legitimacy through bullshit
Inferential proofs are the best we have.
I know if I write those threes forever, that a 4 will never come up, even though I don’t actually do it.
Just like you know that the counting numbers are a well ordered set, knowing that you can’t count them all
Pay particular attention to axiom 3, countable additivity.
By the way, on the general subject of the use of infinity in science … the foundation of probability theory requires infinitary math, as you can see from that link. And probability is the basis of huge amounts of physical and social science. It’s all based on Kolmogorov’s axioms, including countable additivity. I should add that to my list of examples.
Why is that? Someone else might find them of interest. One explains what we mean by the sum of an infinite series. The other describes the basic axioms that underlie modern probability theory. And probability theory underlies everything from weather prediction to the study of heat to quantum physics to the psychology of crowds and modern online advertising and sociology. It’s hard to think of a domain of human activity that doesn’t involve probability.
But why would you not want to at least know that someone on Wikipedia thought these were worthy topics for inclusion? Even if you choose not to read the links, you now at least know that other people find them of interest and importance.
What would make you unhappy about that state of affairs?
