Does infinity exist?

Everyone says “as x approaches infinity”, but what does that mean?

To approach means to get closer to, but regardless how close we get to infinity, we’re still infinitely far away.

Near-infinite doesn’t make sense.

You made a lot of specific points and I’ll try to handle them one by one rather than attempt a big bang reply. Hopefully we can bring some focus. First, Wildberger is a crank; and I’m not the only one who says so. He’s an interesting case because he has also done serious mathematical work. It’s only his ideas on infinity that are regarded (by a lot of people, not just me) as cranky.

reddit.com/r/math/comments/ … my_school/

scienceblogs.com/goodmath/2007/ … e-sets-and

math-frolic.blogspot.com/2012/0 … erger.html

reddit.com/r/math/comments/ … _goldbach/

physicsforums.com/threads/n … ra.772409/

You can Google “Norman Wildberger crank” for yourself and get a wide variety of articles on the topic.

It doesn’t reflect badly on me to label someone a crank who is (a) widely labelled a crank; and (b) happens to actually be a crank.

Please read someone other than Wildberger on the topic of infinity. It doesn’t help your cause because his ideas on infinity are generally regarded as cranky.

It’s not calling cranks cranks that affects my credibility, any more than if I called the Pope a Catholic. Wildberger’s a known mathematical crank. Rather, your quoting known cranks hurts your own credibility.

By the way whether or not Wildberger is a crank is not all that important to us. What’s important is that you’re wasting your time quoting him to me. Also I’ve known about him for several years, if I called him “some guy” in some post it’s probably because someone posted a link to a video and I didn’t bother to watch the video.

It means x gets arbitrarily large. That’s ALL it means. Didn’t they explain that in your 1200 page calculus text? No matter. “x goes to infinity” or “x approaches infinity” means that x gets arbitrarily large. It’s not bounded. It’s just a figure of speech. Although we can formalize it using the extended real numbers, in which we add a pair of symbols (+ \infty) and (- \infty) and assign them formal properties that let us use them as we need to. I’m pretty sure that’s in your calculus text too. But it’s ok if these fine points aren’t clear. Nobody is expected to learn anything in calculus beyond the basic techniques. The fine points of getting everything logically correct are taught in a subject called real analysis, taken by math majors.

In any event the use of limits at infinity in calculus is completely different than the transfinite ordinals and cardinals studied in set theory. But if you even believe in the familiar real number line taught in high school, that’s an example of an infinitely long mathematical object that’s indispensable in physical science and even social science. The familiar Gaussian probability curve, or “bell curve,” is defined over the entire real line and is one of the most important concepts in probability and statistics.

I also wanted to mention that your point about the scientist’s quote about infinity in physics is a good one and I have something substantive to say about it, but not tonight. So I hope we won’t get sidetracked on these two minor issues (Wildberger’s crankitude and the meaning of “x goes to infinity”) before I get to what I consider the more substantive and important point about infinity in physics.

Just dropped in to say I’m hard at work writing down my thoughts on the meaning of infinity as applied to physics. I’m trying to make it brief and clear. That may take a little more time. I completely take your point regarding physical infinities and I’m drafting a response that I hope will shed light.

Whether or not other people think he is a crank is completely irrelevant. It’s an appeal to popularity fallacy committed for the purpose of supporting an ad hominem fallacy. A bunch of people say he’s a crank (appeal to popularity), so he must be a crank, and because he’s a crank, his ideas on infinity are wrong (ad hominem redirection from the topic to the person).

Despite various publications of results where hand washing reduced mortality to below 1%, Semmelweis’s observations conflicted with the established scientific and medical opinions of the time and his ideas were rejected by the medical community. Semmelweis could offer no acceptable scientific explanation for his findings, and some doctors were offended at the suggestion that they should wash their hands and mocked him for it. In 1865, Semmelweis suffered a nervous breakdown and was committed to an asylum, where he died at age 47 of pyaemia, after being beaten by the guards, only 14 days after he was committed. en.wikipedia.org/wiki/Ignaz_Semmelweis

The guy who suggested hand washing was also labeled a crank and that is the argument you’re appealing to. Some people on the net who like the idea of infinity are calling Wildberger a crank specifically because he doesn’t and that is supposed to mean something? Do they also think Gauss and Hilbert are cranks??? Should it matter if they did?

From this I can learn more about you and the people calling Wildberger a crank than I can learn about Wildberger from all the slandering. And I’m probably more inclined to believe Wildberger is not a crank specifically because people insist that he is. That philosophy would have served me well in Galileo vs The Church concerning geocentrism and from my perspective it’s at least equally likely, if not more likely, that the cranks wind up being right in the end.

“All truth passes through three stages: First, it is ridiculed. Second, it is violently opposed. Third, it is accepted as being self-evident.” - Arthur Schopenhauer

The only reason to vilify people is if they’re a threat and if they’re a threat, it’s probably for good reason.

That’s why I say it does more damage to your credibility to insult his.

“Slander is the tool of the loser” - Socrates

“When you’re out of ammo, throw mud.” - Me

Mudslingers sling mud because that’s all the have to sling.

Anyway, even if you generally discredit Wildberger, it wouldn’t mean anything unless you could show that he is incapable of being correct on anything. Even the biggest fool isn’t wrong about everything.

And lastly, I don’t quote these people because I need their references, but because I’m trying to consolidate various quotes and arguments/ideas about infinity into one place.

Is it also a figure of speech to say “near-infinite”?

I’m ok with figures of speech, but Michio Kaku said “as density approaches infinity…” and it dawned on me at that moment that what he said doesn’t make sense because there is no way to approach infinity; regardless how close you get, you’re still just as far away. So the statement means absolutely nothing when taken literally, but figuratively, ok, I get it.

Take your time. It’s no problem as I’m kinda wrapped up in political threads anyway.

Can non existence exist in the absolute sense ? No it cannot which means existence has always existed [ and always will ]
This means that it exists infinitely in the past and will exist infinitely into the future

We are here discussing the existence of infinity then how can it not exist in the mind ?
It only needs to exist as a mental concept in the same way that any other thought does

He is absolutely right : the singularity as traditionally defined does not exist in nature as zero dimension and infinite density are mutually incompatible
But what is not mutually incompatible is infinitesimal dimension combined with finite density and this is therefore the true definition of the singularity

This definition also allows for time to have existed before the Big Bang because the singularity did not experience it as it [ time ] is asymptotic

Something has always existed, but I wouldn’t call that something “existence” since I define existence as a relationship between subject and object. How can something be said to exist if it doesn’t exist: in something, as a function of something, in relation to something, etc? If we talk about objective existence where the object exists only in relation to actuality, then the standpoint of actuality becomes the subject.

Like James said, that which has no affect, does not exist. So if the grand totality of everything has no affect on anything else (because there is nothing else), then we can’t talk about it as existing.

Where does the absolute exist? There is no “where”. Location is only defined inside the absolute.
What does it exist in? There is no “what” because all “what” is inside the absolute.

There is no way to talk about the absolute and any concept we think we have of it simply has to be wrong.

I think it means that time, like location, is only defined inside the thing we’re trying to measure. Time is an emergent property and a consequence of relationships between moving bodies. For instance, I could drive to the next town in 1/24 revolution of the earth… or so many billion vibrations of a certain atom. Time itself does not exist and eternity is not infinite time, but absence of time.

Well, let’s talk about squared circles. Or maybe we can discuss what the universe looks like from the outside even though there is no such thing as “look” outside the universe. This goes to show that we can talk about things without conceptualizing them. We can discuss nonsense without having a concept of the nonsense. I once had a dream where a cat had its head in its mouth. Likewise with infinity: there has never been a person who could properly conceptualize what he fools himself into believing he has. All we can do is imagine the biggest thing we can and we call that “good enough, close enough” and pretend we’ve conceptualized infinity, but we’ve only approached it and our approach is still infinitely far away.

I don’t understand. Can you unpack that a little? The way I understand it is physicists generally regard time before the big bang as north of the north pole: there is no such thing as before the big bang. Time cannot exist before there is something in existence that is moving in relation to something else that is either not moving or moving with a different velocity.

Time is internal to the universe, subjective to it, and not an objective thing existing independent of the universe which could preside over the formation of the universe and record what happened before. Whatever happened before, has no affect on anything, so it doesn’t exist :wink:

Nothing north of the north pole is only true in relation to the Earth as an isolated body but not so in relation to its position within the Universe
So this is where it fails as an analogy in trying to explain why time did not exist before the Big Bang as it assumes nothing existed before it did

It is not known what did or did not exist before the Big Bang because that is only as far as back as physics can currently go
This is demonstrably not the same as saying it cannot go any further back because the BB is the absolute beginning of time

If the singularity was a space of zero volume and infinite density then nothing could have existed before it including time

This definition is wrong because both zero volume and infinite density cannot exist in actuality
As the former would have no dimension or property and the latter can only exist in finite form

A singularity less absolute in physicality however would allow for time to exist before it as it would not be the totality of all that existed

Also if time did begin at the Big Bang it would mean absolute nothing existed before it but this is actually invalidated by quantum mechanics
As absolute nothing can only exist infinitesimally not infinitely because of the existence of quantum fluctuations which disturb vacuum states

So quantum mechanics absolutely forbids the existence of a singularity as traditionally defined

Also the Big Bang was not the beginning of the Universe as such but only local cosmic expansion

I totally agree. I totally agree. I hope saying it twice will convince you that I mean it.

When infinities arise in physics equations, it doesn’t mean there’s a physical infinity. It means that our physics has broken down. Our equations don’t apply. I totally get that. In fact even our friend Max gets that.

blogs.discovermagazine.com/crux/ … g-physics/

The point I am making is something different. I am pointing out that:

All of our modern theories of physics rely ultimately on highly abstract infinitary mathematics

That doesn’t mean that they necessarily do; only that so far, that’s how the history has worked out. There is at the moment no credible alternative. There are attempts to build physics on constructive foundations (there are infinite objects but they can be constructed by algorithms). But not finitary principles, because to do physics you need the real numbers; and to construct the real numbers we need infinite sets.

I collected some examples of the infinitary math underlying physics. I tried to be brief. Each example could be expanded to a book or the work of a lifetime. I’ll do my best to answer specific questions. As with Fubini I regret that it’s beyond me to explain any of these examples fully and in detail with perfect clarity and without requiring effort on the part of the reader. That’s what TED talks are for. /s

  1. The rigorization of Newton’s calculus culminated with infinitary set theory.

Newton discovered his theory of gravity using calculus, which he invented for that purpose. However, it’s well-known that Newton’s formulation of calculus made no logical sense at all. If (\Delta y) and (\Delta x) are nonzero, then (\frac{\Delta y}{\Delta x}) isn’t the derivative. And if they’re both zero, then the expression makes no mathematical sense! But if we pretend that it does, then we can write down a simple law that explains apples falling to earth and the planets endlessly falling around the sun.

It took another 200 years for mathematicians to develop a rigorous account of calculus from first principles; and those first principles are infinitary set theory. No set theory, no real numbers, no calculus, no gravity.

encyclopediaofmath.org/inde … f_analysis

  1. Einstein’s gneral relativity uses Riemann’s differential geometry.

In the 1840’s Bernhard Riemann developed a general theory of surfaces that could be Euclidean or very far from Euclidean. As long as they were “locally” Euclidean. Like spheres, and torii, and far weirder non-visualizable shapes. Riemann showed how to do calculus on those surfaces. 60 years later, Einstein had these crazy ideas about the nature of the universe, and the mathematician Minkowski saw that Einstein’s ideas made the most mathematical sense in Riemann’s framework. This is all abstract infinitary mathematics.

en.wikipedia.org/wiki/Differential_geometry

en.wikipedia.org/wiki/Introduct … relativity

  1. Fourier series link the physics of heat to the physics of the Internet; via infinite trigonometric series.

In 1807 Joseph Fourier analyzed the mathematics of the distribution of heat through an iron bar. He discovered that any continuous function can be expressed as an infinite trigonometric series, which looks like this:

$$f(x) = \sum_{n=0}^\infty a_n \cos(nx) + \sum_{n=1}^\infty b_n \sin(nx)$$

I only posted that because if you managed to survive high school trigonometry, it’s not that hard to unpack. You’re composing any motion into a sum of periodic sine and cosine waves, one wave for each whole number frequency. And this is an infinite series of real numbers, which we cannot make sense of without using infinitary math.

Fast forward to present time. Fourier series underlie the propagation of digital signals over the Internet. They allow us to converse in this very moment.

en.wikipedia.org/wiki/Fourier_series

  1. Quantum theory is functional analysis.

If you took linear algebra, then functional analysis can be thought of as infinite-dimensional linear algebra combined with calculus. Functional analysis studies spaces whose points are actually functions; so you can apply geometric ideas like length and angle to wild collections of functions. In that sense functional analysis actually generalizes Fourier series.

Quantum mechanics is expressed in the mathematical framework of functional analysis. QM takes place in an infinite-dimensional Hilbert space. To explain Hilbert space requires a deep dive into modern infinitary math. In particular, Hilbert space is complete, meaning that it has no holes in it. It’s like the real numbers and not like the rational numbers.

QM rests on the mathematics of uncountable sets, in an essential way.

ps – There’s our buddy Hilbert again. He did many great things. William Lane Craig misuses and abuses Hilbert’s popularized example of the infinite hotel to make disingenuous points about theology and in particular to argue for the existence of God. That’s what I’ve got against Craig.

  1. Cantor was led to set theory from Fourier series.

In every online overview of Georg Cantor’s magnificent creation of set theory, nobody ever mentions how he came upon his ideas. It’s as if he woke up one day and decided to revolutionize the foundations of math and piss off his teacher and mentor Kronecker. Nothing could be further from the truth.

Cantor was in fact studing Fourier’s trigonometric series! One of the questions of that era was whether a given function could have more than one distinct Fourier series. To investigate this problem, Cantor had to consider the various types of sets of points on which two series could agree; or equivalently, the various sets of points on which a trigonometric series could be zero. He was thereby led to the problem of classifying various infinite sets of real numbers; and that led him to the discovery of transfinite ordinal and cardinal numbers. (Ordinals are about order in the same way that cardinals are about quantity).

In other words, and this is a fact that you probably will not find stated as clearly as I’m stating it here:

If you begin by studying the flow of heat through an iron rod; you will inexorably discover transfinite set theory.

I do not know what that means in the ultimate scheme of things. But I submit that even the most ardent finitist must at least give consideration to this historical reality.

ias.ac.in/article/fulltext/ … /0977-0999

Conclusion

I hope I’ve been able to explain why I completely agree with your point that infinities in physical equations don’t imply the actual existence of infinities. Yet at the same time, I am pointing out that our best THEORIES of physics are invariably founded on highly infinitary math. As to what that means … for my own part, I can’t help but feel that mathematical infinity is telling us something about the world. We just don’t know yet what that is.

Most definitely, and one that I would never personally use. Nothing is “near infinite,” I agree with you about that. Physicists and others use it to mean “really big.”

When physicists talk about infinity they often have NO IDEA what they’re saying in terms of math. Physicists misuse the word infinity terribly; and of all the physicists who do that, the celebrity physicists do it the worst.

You’re reading way too much into words people are using very informally.

I almost don’t feel like this needs saying…

To be accurate about infinite sets, it’s proper to say, “the sequence approaches 2”. Rather than, “the sequence is 2”

If there is no time, then what does “before” mean?

If time had a beginning, then there is no before. If there were a before, then after the before would not be the beginning of time.

Oh I see. But in order to have time, we need things in motion through a spacial construct. How can there be things in motion if the universe is so small?

But time doesn’t exist relative to light, yet there is not nothing.

Yes that makes sense.

That could be true.

That was an excellent post and qualifies as a treasure to be found on this site! :obscene-drinkingcheers:

Thanks for the link and I would have showcased it all on its own had I seen it first :slight_smile:

I see what you mean, but as Max pointed out when describing air as seeming continuous while actually being discrete, it’s easier to model a continuum than a bazillion molecules, each with functional probabilistic movements of their own. Essentially, it’s taking an average and it turns out that it’s pretty accurate.

But what I was saying previously is that we work with the presumed ramifications of infinity, “as if” this or that were infinite, without actually ever using infinity itself. For instance, y = 1/x as x approaches infinity, then y approaches 0, but we don’t actually USE infinity in any calculations, but we extrapolate.

Hilbert pointed out there is a difference between boundless and infinite. For instance space is boundless as far as we can tell, but it isn’t infinite in size and never will be until eternity arrives. Why can’t we use the boundless assumption instead of full-blown infinity?

I didn’t know he developed calculus specifically to investigate gravity. Cool! It does make sense now that you mention it.

I’m going to need some help with this one. If dx = 0, then it contains no information about the change in x, so how can anything result from it? I’ve always taken dx to mean a differential that is smaller than can be discerned, but still able to convey information. It seems to me that calculus couldn’t work if it were based on division by zero, and that if it works, it must not be. What is it I am failing to see? I mean, it’s not an issue of 0/0 making no mathematical sense, it’s a philosophical issue of the nonexistence of significance because there is nothing in zero to be significant.

Isn’t this the same problem as previous? dx=0?

I can’t make sense of it WITH infinitary math lol! What’s the cosine of infinity? What’s the infnite-th ‘a’?

Well, thanks to Hilbert, I’ve already conceded that the boundless is not the same as the infinite and if it were true that QM required infinity, then no machine nor human mind could model it. It simply must be true that open-ended finites are actually employed and underpin QM rather than true infinite spaces.

Like 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.”

We can claim physics is based on infinity, but I think it’s more accurate to say pretend or fool ourselves into thinking such.

Max continued with, “Our challenge as physicists is to discover this elegant way and the infinity-free equations describing it—the true laws of physics. To start this search in earnest, we need to question infinity. I’m betting that we also need to let go of it.”

He said, “let go of it” like we’re clinging to it for some reason external to what is true. I think the reason is to be rid of god, but that’s my personal opinion. Because if we can’t have infinite time, then there must be a creator and yada yada. So if we cling to infinity, then we don’t need the creator. Hence why Craig quotes Hilbert because his first order of business is to dispel infinity and substitute god.

I applaud your effort, I really do, and I’ve learned a lot of history because of it, but I still cannot concede that infinity underpins anything and I’d be lying if I said I could see it. I’m not being stubborn and feel like I’m walking on eggshells being as amicable and conciliatory as possible in trying not to offend and I’m certainly ready to say “Ooooohhh… I see now”, but I just don’t see it.

Craig is no friend of mine and I was simply listening to a debate on youtube (I often let youtube autoplay like a radio) when I heard him quote Hilbert, so I dug into it and posted what I found. I’m not endorsing Craig lol

I still can’t understand how one infinity can be bigger than another since, to be so, the smaller infinity would need to have limits which would then make it not infinity.

Right, because of what Max said about the continuum model vs the actual discrete. Heat flow is actually IR light flow which is radiation from one molecule to another: a charged particle vibrates and vibrations include accelerations which cause EM radiation that emanates out in all directions; then the EM wave encounters another charged particle which causes vibration and the cycle continues until all the energy is radiated out. It’s a discrete process from molecule to molecule, but is modeled as continuous for simplicity’s sake.

I’ve long taken issue with the 3 modes of heat transmission (conduction, convention, radiation) because there is only radiation. Atoms do not touch, so they can’t conduct, but the van der waals force simply transfers the vibrations more quickly when atoms are sufficiently close. Convection is simply vibrating atoms in linear motion that are radiating IR light. I have many issues with physics and have often described it as more of an art than a science (hence why it’s so difficult). I mean, there are pages and pages on the internet devoted to simply trying to define heat.

quora.com/What-is-heat-1
quora.com/What-is-meant-by-heat
quora.com/What-is-heat-in-physics
quora.com/What-is-the-definition-of-heat
quora.com/What-distinguishes-work-and-heat

Physics is a mess. What gamma rays are, depends who you ask. They could be high-frequency light or any radiation of any frequency that originated from a nucleus. But I’m digressing…

It just means we’re using averages rather than discrete actualities and it’s close enough.

I think it means there are really no separate things and when an aspect of the universe attempts to inspect itself in order to find its fundamentals or universal truths, it will find infinity like a camera looking at its own monitor. Infinity is evidence of the continuity of the singular universe rather than an existing truly boundless thing. Infinity simply means you’re looking at yourself.

Anyway, great post! Please don’t be mad. Everyone here values your presence and are intimidated by your obvious mathematical prowess :sunglasses: Don’t take my pushback too seriously :slight_smile: I’d prefer if we could collaborate as colleagues rather than competing.

Small is a relative term and from a classical perspective the quantum is incredibly so but this does not mean motion is not actually possible at this level
Three of the four fundamental forces operate at the quantum level inside of an atom in relation to the motion of its protons and neutrons and electrons
Also motion is a universal constant so any space no matter how small will feature motion even if it is just quantum fluctuations disturbing vacuum states

This statement violates the Law Of Non Contradiction as it is impossible for an infinity to also be not infinity
The problem here is that in relation to small infinity you are focusing on the small rather than the infinity

All infinite sets contain an infinite number of members but some of them will have more members than others and some will have less
For example the infinite set of primes is smaller than the infinite set of reals because primes occur less frequently on the number line

But primes are no less infinite than the reals despite this [ and yes it is incredibly counter intuitive but more importantly it is also true ]

I think we first need to answer the question of if there is a smallest point in the universe.

If there is a smallest point, then the extents of the universe would have to be bigger than that smallest point in order to have concepts of moving from point to point.

If there is not a smallest point, then the universe has no size relative to its smallest point, and regardless how small the universe became, it would still be infinitely larger than its own points.

The evidence so far suggests that the smaller we go, the less certain we can be of location and the more probabilistic location becomes. I am under the impression that space is not empty space with things moving about independent of the space-substance, but a collection of fields analogous to a computer screen where things cannot be smaller than the pixel (a quantum) that generates them. So if the universe were smaller than a pixel, then how could movement exist?

Also, neither time nor space exists relative to a photon. Every photon emission and reception from the beginning of time to the end of time, took no time and covered no space from its own perspective, which seems to imply there is no objective time standard by which to measure spacetime itself, but that time is an artifact of spacetime and contained only therein. Time is subject to and emergent from spacetime and not objective to it.

It seems obvious that something existed before, but I can’t understand what “before” means. From the photon’s perspective, where was it yesterday? “Yesterday” only applies to me and not the photon which itself caused a sense of “yesterday” to manifest. Time cannot apply to the thing that causes time. It’s like trying to use a knife to cut the same knife.

Primes are infinite only because reals are infinite. If reals were not, then primes could not be, so the cardinality of primes is dependent upon the cardinality of reals. This is like saying the age of a son will always be less than the age of his father, except if they live forever, in which case they will be the same age on the day that forever arrives (which can never happen). It’s the same with primes and reals: there will always be more reals than primes until the day eternity arrives.

If a completed infinity of reals are ALL numbers in actual existence and potential existence and nonexistence and any other existence you can conceive, then from where will you find more numbers to have a larger infinity? And if you do find more numbers to make a larger infinity, then it wasn’t ALL the numbers to start with.

Ordinals assume an infinity of reals can be completed and then some new number comes in order right after that completed infinity. Like Gauss, I can’t make sense of a completed infinity. Further, the first infinity should include all of the second or else it wasn’t infinity.

[youtube]https://www.youtube.com/watch?v=SrU9YDoXE88[/youtube]