Does infinity exist?

[Serendipper]

It's not logically or grammatically incorrect to say a wall is infinitely tall, or infinitely wide, a thing can be infinite in some quality without having to be infinite in all qualities.

Whether it's empirically incorrect is another matter.

Infinitely wide is ok because it extends in both directions, but infinitely tall would be a wall that extends around the universe until it connected with the other side of the earth.

Why would the infinitely tall wall curve around the cosmos and touch its bottom?

Because that would be the only place it hasn't yet touched. To say otherwise would be to put a limit on the wall and make it finite.

[Serendipper]

@Serendipper

Then how do you define infinity?

The same as you, but there can be an absolute infinity, and specific infinities.

What's the difference? Either there are infinite amounts of something or not.

Unlimited in some ways and limited in others is unlimited in quantity and limited in identity/category.

Or unlimited in quantity here/now, but limited there/then, or unlimited in x qualities, but not in y.

Sure I'll go along with having unlimited x qualities, but limited y, such as oranges are unlimited in quantity, but not size, shelf-life, color, flavor, price, etc.

I can't conceptualize it and you can't either. In actuality, where in the universe do you suspect that it may be possible to draw an infinite line in one direction, but not in the other?

Conceptually you can draw a line anywhere, but you can also conceive of a road ending one way, but not the other.

If a road were truly infinite, there would not exist a place void of road. To say there is a place without road is to place a limit on the length of the road.

You can also conceive of an impenetrable wall that keeps everything this side of it from crossing over, not that such a wall is necessary for a road to end one way, but not the other.

Beyond the wall, there might be nothing, not merely empty space, but no MEST at all, or there might be stuff.

What's north of the north pole? Nothing. Not because the wall is impenetrable, but because there is no place to exist north of the north pole; there is no there there.

Then, at this moment, it is not infinite because there exists a place for more road.

It's endless backwardly, and endful forwardly.

How can you propose having an infinite road when clearly we could make it longer?

You can make it longer forwardly, but not backwardly.

A road that is truly infinite would extend around and around the universe many many times until it occupied every planck cube in the universe, completely displacing all matter, and until it eventually connected with itself for lack of having anywhere else to go. To say that isn't so is to say the road has a boundary which would make it not infinite.

It has a boundary backwardly, but not forwardly.

Forwardly, the road would continue through every planck cube in the universe before finally terminating at its beginning for lack of having anywhere else to go. An infinite road would occupy every planck volume in the universe, in the forward direction, even if the universe is infinite in size.

Yes it does matter because if there is a place for another apple, but no apple is there, then we have found a boundary and therefore the number of apples is not infinite.

Apples could unendingly sparsely populate the unending universe, and still be unending in number, which means some infinites could be bigger than others.

Unendingly sparsely? What does that mean?

Size must have a zero like temperature and speed or else it couldn't exist. We can't get infinitely colder, infinitely slower, infinitely smaller and if we could, then temperature, speed, size would have no significance/meaning.

I'm not so sure, for example, if two things are both infinitely divisible, but, finitely multipliable, if you will, than one of them could still be bigger, stronger and so on than the other.

I don't know what that means. Do you have an example?

But even if things are necessarily finitely small, the smallest unit of matter, motion and space might still be centillions of times smaller than quarks.

This is the debate I had with James a year or two ago. He asserted an infinite amount of smaller particles and I asked why we occupy this tranche instead of some other. If there are infinitely smaller particles, then no size of particle has any special significance. Why is an atom the size that it is instead of some other size? If every particle were composed of infinitely smaller particles, then no particle would have less than an infinite amount of particles, regardless how big it is, and no particle would be any different from any other, and size would have no meaning.

It seems weird...asymmetrical to me the universe could be infinitely big, but not also infinitely small, and if a thing could be infinitely big, and not infinitely small, than why couldn't a thing be finitely big, and also infinitely small?

Well, no worries, the universe is finitely big and small

[Serendipper]

I'll try to remember that I'm talking to a bunch of 5 year olds. That actually explains a lot.

Lots of good things have been said about being like a child. Jesus said it was conditional to get to heaven

Being childish is generally regarded as being petty, vindictive, and perhaps stupid, but children are open-minded and every thought is outside the box because they haven't formed a box yet. Only a child can learn perfect pitch, which is to say that only a child can learn to accurately perceive aspects of our world. https://www.youtube.com/watch?v=816VLQNdPMM

I can integrate an area over a height to yield a volume without using infinity.

That is a very interesting remark. Of course if you took freshman calculus, you can do that using a rote procedure, say by taking an antiderivative of the kind of elementary functions you see in calculus class. Integrand is \(x^2\) so antiderivative is \(\frac{1}{3} x^3\) kind of thing.

But if you studied the subject more deeply, you would realize that in order to form a logically rigorous definition of an integral, you require modern infinitary set theory. In calculus they don't show you that. Perhaps you remember that when they defined the Riemann integral, they defined lower and upper sums relative to a partition, and then you took the LIMIT over all possible partitions. To formalize that requires the full apparatus of ZF set theory, including the axiom of infinity.

Has it been proven that integration cannot be formalized without infinity?

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.

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."

So to me, the fact that you DO believe in Riemann integration (aka freshman calculus integration) tells me that you've seen the rote procedures, but not the underlying theory nor all the weird counterexamples and corner cases that made 19th century mathematicians realize they needed a rigorous theory. Infinitary math is essential to define an integral and do freshman calculus. They just don't tell you about this until you take a more advanced course in real analysis.

Because in advanced math you're studying applications only to math instead of the real world. 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.

No infinitary math, no logical foundation for freshman calculus. No axiom of infinity, no Riemann integral.

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?

ps -- Let me give a concrete example. You mentioned integrating an area over a height. How about if you have a rectangular metal plate with a temperature at each point and you want to integrate the temperature over the area of the rectangle to determine the average temperature. You could integrate the vertical slices then the horizontal ones or vice versa. This is multiple integration as in second year calculus. But how do you know when the order of integration matters and when it doesn't? How do you know whether it makes a difference if you integrate the x's and then the y's, or first the y's and then the x's? This can be a very tricky business, especially with a weird or pathological integrand or temperature function. This is when you have to drill down to the rigorous, set-theoretic definition of the integral to prove theorems on reversing the order of integration. In other words the moment you go beyond the simplest examples you need some theory; and the theory of integration requires infinitary set theory, or my name's not Guido Fubini!

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.

This reminds me of those hotly debated arithmetic ordering of operations puzzles that I hope you're familiar with because I can't find a good example at the moment, but some will fill the comments section with debates and ultimately have no concrete resolution.

[Jakob]

"No axiom of infinity, no Riemann integral."

No Jesus, no Sacrament.

[wtf]

Being childish is generally regarded as being petty, vindictive, and perhaps stupid, but children are open-minded and every thought is outside the box because they haven't formed a box yet.

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.

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, 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.

https://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?

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, 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?

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? 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?

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.

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?

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.

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 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.

This reminds me of those hotly debated arithmetic ordering of operations puzzles that I hope you're familiar with because I can't find a good example at the moment, but some will fill the comments section with debates and ultimately have no concrete resolution.

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.

[wtf]

"No axiom of infinity, no Riemann integral."

No Jesus, no Sacrament.

No bacon, no BLT.

[Jakob]

Bacon isn't a hypothesis that is only validated by the sandwich its in. It can be eaten (made sense of, valued, used) for its own properties.

Explicating; consider "no humans, no railroads".
Would the railroad suffice as a justification of the human?
In a philosophic sense, I mean.

[wtf]

Bacon isn't a hypothesis that is only validated by the sandwich its in. It can be eaten (made sense of, valued, used) for its own properties.

Explicating; consider "no humans, no railroads".
Would the railroad suffice as a justification of the human?
In a philosophic sense, I mean.

Oh I see your point. Infinitary set theory has been validated by over a century of mathematical practice. Surely you would at least grant me this historical fact, easily confirmed by a study of the mathematical literature.

I still don't follow your religious analogy, perhaps you can explain it to me. Is the Sacrament a validation of Jesus? How so? I don't know anything about Christian theology past the Lord's prayer, which (at the time I went to school) we were required to say every day, along with the Pledge of a Lesion, and to the republic for Richard Stands.

[Jakob]

Lets say Mohammed then. No Mohammed, no Islam. And Islam is the desirable thing, evidently - to a muslim. Like the Riemann integral is to a mathematician. That is why Mohammed is holy, why infinity is "true". Its not like islam is holy because Mohammed is holy. He was made holy (rendered into a desirable idea) by his service to Allah.

Infinity is an idea required to have certain other ideas possible, as you point out. Thats why it exists.

The same goes for Jesus, it is an idea required to make some moral systems work, moral systems which are the criterium. There is no evidence of Jesus directly. Whats more, the idea is that he exists simply to redeem mankind. I see a strong resemblance with the idea of infinity. It exists to make set theory and some other desirable ideas possible. Both are ideas justified by their making other things possible.

A better argument for you to make here would be "no bacon, no pig".
Bacon tastes perfectly good without the BLT. But does the pig serve without the bacon? Would we keep pigs if we didn't like bacon?

[wtf]

Lets say Mohammed then. No Mohammed, no Islam. And Islam is the desirable thing, evidently - to a muslim. Like the Riemann integral is to a mathematician. That is why Mohammed is holy, why infinity is "true". Its not like islam is holy because Mohammed is holy. He was made holy (rendered into a desirable idea) by his service to Allah.

Infinity is an idea required to have certain other ideas possible, as you point out. Thats why it exists.

The same goes for Jesus, it is an idea required to make some moral systems work, moral systems which are the criterium. There is no evidence of Jesus directly. Whats more, the idea is that he exists simply to redeem mankind. I see a strong resemblance with the idea of infinity. It exists to make set theory and some other desirable ideas possible. Both are ideas justified by their making other things possible.

A better argument for you to make here would be "no bacon, no pig".
Bacon tastes perfectly good without the BLT. But does the pig serve without the bacon? Would we keep pigs if we didn't like bacon?

I have to admit I didn't follow all of that. I am simply making a utilitarian argument for mathematical infinity. I suppose the official name for it is indispensability. https://plato.stanford.edu/entries/mathphil-indis/ However I'm not sufficiently knowledgeable about that set of ideas to say if that's exactly what I'm saying, or just influenced by it. My thoughts arise from the ideas of philosopher of math Penelope Maddy, who makes a similar argument in Believing the Axioms. She refers to a philosophical principle she calls MAXIMIZE, which says that given a choice of axioms which allow us to do less, or that allow us to do more, we choose the axioms that allow us to do more. [My paraphrase, not necessarily Maddy's literal words].

The Jesus and Mohammad analogies are inexact, in the sense that J and M were the founders of their respective religions. But we had mathematical integration before we had a theory of infinity. Infinity is currently (but perhaps not necessarily) essential to the foundation of mathematical integration. Whereas Jesus and Mohammad are necessary to the founding of their respective religions, because they were, after all, the founders.

[Magius the Gadfly]

Infinity is a word we devote to things which we do not see nor understand limits of. So for example, the word was first associated with the heavens or space, or the universe because we couldn't imagine calculating or measuring space or the distance of the stars. As Serendipper states in his first post, the word infinite means "immeasurably great". However, what is measurable is changing with time. What was considered infinite 3,000 years ago is no longer considered infinite.

Other ideas in the definitions of "infinite" are problematic. For example "unlimited" and "not finite". These too, change with time. We thought the distance between us and the stars was unlimited and not finite but later we found a way to measure them. Words/ideas like "infinite" are less and less applicable to everyday life and are pushed further and further into abstract corners of fields such as mathematics (abstract). Math is nothing more than a system of generalization, organization and grouping. Something being "infinite" can only make sense in such systems and even then in only limited applications.

What's your take?

~Magius

[Serendipper]

I'd like to single-out this topic despite the digression because I want to proselytize this point of view as much as possible.

Being childish is generally regarded as being petty, vindictive, and perhaps stupid, but children are open-minded and every thought is outside the box because they haven't formed a box yet.

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.

I think that, within philosophical arenas, an open mind is preferable to indoctrination which is what "education" must be taken to mean in the context where childishness is juxtaposed with "been to school". IOW, I'd rather have to bring someone up to speed in order to have a discussion than to debate someone who was already versed in a topic because such education would invariably contain biases that are nearly impossible to overcome.

The only certain barrier to truth is the conviction you already have it. The innocent have no barriers since they don't know anything which makes them ideal candidates to to find wisdom. Those who already "know", can never know, for once they build upon a seemingly solid foundation, it becomes increasingly harder to move.

To me, the colloquial abstract idea of what it means to be intelligent is a function of one's propensity to admit error quickly and move on. Those who dig in will always be stuck and could never find truth... even in infinite time. That is true regardless of my own hypocrisy on the subject (I'm human, afterall, and have demons to slay just like anyone else).

An excerpt from http://www.bbc.com/future/story/2015041 ... ing-clever

Learnt wisdom

In the future, employers may well begin to start testing these abilities in place of IQ; Google has already announced that it plans to screen candidates for qualities like intellectual humility, rather than sheer cognitive prowess.

The challenge will be getting people to admit their own foibles. If you’ve been able to rest on the laurels of your intelligence all your life, it could be very hard to accept that it has been blinding your judgement. As Socrates had it: the wisest person really may be the one who can admit he knows nothing.

[Serendipper]

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

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.

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.

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.

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.

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).

[Serendipper]

Pledge of a Lesion, and to the republic for Richard Stands.

lol

I pledge allegiance to the flag
of the corporate states of America.
And to the conglomeration,
for which it stands,
one nation, under many CEOs,
always divisible,
with liberty and privileges for some.

[Serendipper]

Infinity is a word we devote to things which we do not see nor understand limits of. So for example, the word was first associated with the heavens or space, or the universe because we couldn't imagine calculating or measuring space or the distance of the stars. As Serendipper states in his first post, the word infinite means "immeasurably great". However, what is measurable is changing with time. What was considered infinite 3,000 years ago is no longer considered infinite.

Yup, infinity is essentially any finite that is bigger than our universe. Once we exceed our capacity to measure, then adding more doesn't change anything.

Other ideas in the definitions of "infinite" are problematic. For example "unlimited" and "not finite". These too, change with time. We thought the distance between us and the stars was unlimited and not finite but later we found a way to measure them. Words/ideas like "infinite" are less and less applicable to everyday life and are pushed further and further into abstract corners of fields such as mathematics (abstract). Math is nothing more than a system of generalization, organization and grouping. Something being "infinite" can only make sense in such systems and even then in only limited applications.

What's your take?

Sounds good to me!

Magius

Welcome to the site

[wtf]

To me, the colloquial abstract idea of what it means to be intelligent is a function of one's propensity to admit error quickly and move on.

You better get to it then.

[Jakob]

What flaccid academic pedantry has overtaken this thread since the horde left it.

[wtf]

Yup, infinity is essentially any finite that is bigger than our universe.

Infinity is a subclass of finite? I think we're done here. If you can't even stipulate that finite and infinite are opposites -- that infinite means not finite, and finite means not infinite -- then it's difficult to imagine what further dialog could look like.

[Magius the Gadfly]

Hello Serendiper,

Trust you are well. Thank you for your response and for your welcoming me. It has been a long time and it is good to be back. I was a member here for many many years going back as far as 2002. I went through two names/avatars/nicknames, specifically "Magius" and "The Gadfly of ILP". You can see my posts by checking those names. None of the veterans appear to be here anymore and it will take some time to get my grounding. I feel more at ease with your hospitality. Thank you.

Having said that, allow me to get to the thick of things.

You said:

Yup, infinity is essentially any finite that is bigger than our universe.

I'm not sure I understand your full meaning here but if I understand your words, you are saying that infinity that which is beyond or exceeds beyond or is bigger than our universe. My point is that the word "infinity" and it's meaning don't point to anything real except for our ignorance. Some would say, much like the word "God". Either there is something beyond our universe or there isn't. It either has physical reality or not. Regardless of those conditions, the word infinity doesn't express anything about actual reality outside of our relative perspective. Because we say A or B is infinite, doesn't mean it is, it just mean's we haven't found a way to measure it yet or to comprehend the thing properly. I wanted to clarify that because you appeared to agree with me though you were agreeing with a straw man fallacy (unintentionally). Infinity is relative to ourselves and not connected to physical reality. Put another way, there is nothing you can point to or show and say "Aha, THAT is infinity!"

You said:

Once we exceed our capacity to measure, then adding more doesn't change anything.

But we do add more and that is why we are advancing, slowly, not just in what we consider to be infinite (space, stars, light, etc). What was once limited for us to count or measure is now measurable, like the distance from the Earth to the Moon.

Regards,
~Magius

[Serendipper]

Yup, infinity is essentially any finite that is bigger than our universe.

Infinity is a subclass of finite? I think we're done here. If you can't even stipulate that finite and infinite are opposites -- that infinite means not finite, and finite means not infinite -- then it's difficult to imagine what further dialog could look like.

Yes of course they are opposites.

But I'm suggesting that any finite number that is bigger than our universe is absolutely indistinguishable from any other concepts of infinity. Once the limits of the universe have been exceeded, it makes no difference to anything within the universe to exceed them more.

The immeasurably great has all the same implications and ramifications as a true infinite. If there is a finite number that can never be represented in any way within this universe, then why do we need a bigger number and how will we know when we have it?

[Serendipper]

Hello Serendiper,

Trust you are well. Thank you for your response and for your welcoming me. It has been a long time and it is good to be back. I was a member here for many many years going back as far as 2002. I went through two names/avatars/nicknames, specifically "Magius" and "The Gadfly of ILP". You can see my posts by checking those names. None of the veterans appear to be here anymore and it will take some time to get my grounding. I feel more at ease with your hospitality. Thank you.

What a turn of events... I'm the newbie Welcome back! It's a bit slow around here now, but perhaps it will pickup soon.

Yup, infinity is essentially any finite that is bigger than our universe.

I'm not sure I understand your full meaning here but if I understand your words, you are saying that infinity that which is beyond or exceeds beyond or is bigger than our universe. My point is that the word "infinity" and it's meaning don't point to anything real except for our ignorance. Some would say, much like the word "God". Either there is something beyond our universe or there isn't. It either has physical reality or not. Regardless of those conditions, the word infinity doesn't express anything about actual reality outside of our relative perspective. Because we say A or B is infinite, doesn't mean it is, it just mean's we haven't found a way to measure it yet or to comprehend the thing properly. I wanted to clarify that because you appeared to agree with me though you were agreeing with a straw man fallacy (unintentionally). Infinity is relative to ourselves and not connected to physical reality. Put another way, there is nothing you can point to or show and say "Aha, THAT is infinity!"

Yes, like Gauss said, "I protest against the use of infinite magnitude as something completed, which is never permissible in mathematics. Infinity is merely a way of speaking, the true meaning being a limit which certain ratios approach indefinitely close, while others are permitted to increase without restriction."

Infinity is never anything you can point to because it can never be anything "completed" by definition which is why I say it can't be thought of as existing.

Once we exceed our capacity to measure, then adding more doesn't change anything.

But we do add more and that is why we are advancing, slowly, not just in what we consider to be infinite (space, stars, light, etc). What was once limited for us to count or measure is now measurable, like the distance from the Earth to the Moon.

I see what you're saying about precision, but I think there are hard limits to the universe. For instance, we can't measure anything smaller than the spacetime fabric since there is no such thing smaller than the thing that determines what size means. It's like measuring the distance south of the south pole. Once we measure down to a certain size, location loses meaning. And what does it mean to be bigger than the universe if the universe is the only thing that determines what size means? And what does it mean to be bigger than that? Once we exceed the limits of the thing that determines meaning, exceeding it more doesn't change anything.

[Serendipper]

On the infinite

by

DAVID HILBERT

https://www.math.dartmouth.edu//~matc/R ... sophy.html

Some highlights:

- But a still more general perspective is relevant for clarifying the concept of the infinite. A careful reader will find that the literature of mathematics is glutted with inanities and absurdities which have had their source in the infinite. For example, we find writers insisting, as though it were a restrictive condition, that in rigorous mathematics only a finite number of deductions are admissible in a proof — as if someone had succeeded in making an infinite number of them.

- Before turning to the task of clarifying the nature of the infinite, we should first note briefly what meaning is actually given to the infinite. First let us see what we can learn from physics. One's first naïve impression of natural events and of matter is one of permanency, of continuity. When we consider a piece of metal or a volume of liquid, we get the impression that they are unlimitedly divisible, that their smallest parts exhibit the same properties that the whole does. But wherever the methods of investigating the physics of matter have been sufficiently refined, scientists have met divisibility boundaries which do not result from the shortcomings of their efforts but from the very nature of things. Consequently we could even interpret the tendency of modern science as emancipation from the infinitely small. Instead of the old principle natura non facit saltus, we might even assert the opposite, viz., "nature makes jumps."

- In addition to matter and electricity, there is one other entity in physics for which the law of conservation holds, viz., energy. But it has been established that even energy does not unconditionally admit of infinite divisibility. Planck has discovered quanta of energy.

- Hence, a homogeneous continuum which admits of the sort of divisibility needed to realize the infinitely small is nowhere to be found in reality. The infinite divisibility of a continuum is an operation which exists only in thought. It is merely an idea which is in fact impugned by the results of our observations of nature and of our physical and chemical experiments.

- We have already seen that the infinite is nowhere to be found in reality, no matter what experiences, observations, and knowledge are appealed to. Can thought about things be so much different from things? Can thinking processes be so unlike the actual processes of things? In short, can thought be so far removed from reality? Rather is it not clear that, when we think that we have encountered the infinite in some real sense, we have merely been seduced into thinking so by the fact that we often encounter extremely large and extremely small dimensions in reality?

Does material logical deduction somehow deceive us or leave us in the lurch when we apply it to real things or events? No! Material logical deduction is indispensable. It deceives us only when we form arbitrary abstract definitions, especially those which involve infinitely many objects. In such cases we have illegitimately used material logical deduction; i.e., we have not paid sufficient attention to the preconditions necessary for its valid use. In recognizing that there are such preconditions that must be taken into account, we find ourselves in agreement with the philosophers, notably with Kant. Kant taught — and it is an integral part of his doctrine — that mathematics treats a subject matter which is given independently of logic. Mathematics, therefore, can never be grounded solely on logic. Consequently, Frege's and Dedekind's attempts to so ground it were doomed to failure.

- In summary, let us return to our main theme and draw some conclusions from all our thinking about the infinite. Our principal result is that the infinite is nowhere to be found in reality. It neither exists in nature nor provides a legitimate basis for rational thought — a remarkable harmony between being and thought. In contrast to the earlier efforts of Frege and Dedekind, we are convinced that certain intuitive concepts and insights are necessary conditions of scientific knowledge, and logic alone is not sufficient. Operating with the infinite can be made certain only by the finitary.

The role that remains for the infinite to play is solely that of an idea
________________________________________________________________________

So that's that.

But...

There is one quote that gave me pause:

Although euclidean geometry is indeed a consistent conceptual system, it does not thereby follow that euclidean geometry actually holds in reality. Whether or not real space is euclidean can be determined only through observation and experiment. The attempt to prove the infinity of space by pure speculation contains gross errors. From the fact that outside a certain portion of space there is always more space, it follows only that space is unbounded, not that it is infinite. Unboundedness and finiteness are compatible.

He draws distinction between infinity and the unbounded which undermines the definition I first proposed.

An analogy is the money supply, which is unbounded, but always finite.

So the definition of the infinite must be stipulated that it is not unbounded in potentiality, but actuality. The money supply may be unbounded in theoretical potential, but it can never be actually unbounded, meaning that an infinite amount of money would already exist.

This is similarly so with numbers: the quantity of numbers may be theoretically unbounded given the inference that we could always add more, but to therefore claim infinite numbers must already exist is a fallacy of speculation no different than saying an infinite amount of money exists right now merely because it has an untested and theoretical potential to exist.

Just like new money is issued when needed, new numbers are created when needed and new space is created when needed, but none of these things are actually infinite.

David notes that infinity does not exist in reality, but I maintain infinity cannot exist because that which has no boundaries is not a thing and the unattainable potential can never be existent as an actuality.

[Jakob]

I think you're honing in on the full explanation.

Key is that all human conceptions of the universe are naturally less than complete, qualitatively speaking, and thus for any conception to even make some claim to pertinence it must explicitly be open ended.

So positing infinity as a concept is an excuse for falling short in concrete terms. Any concrete terms fall short of the whole. Infinity isn't a concrete term.

The part can't encompass the whole and a thought, an idea, is a part of the universe.

[wtf]

David notes that infinity does not exist in reality ...

I think we're all (at least you and I) in agreement that (as far as we know) no actual infinity is instantiated in nature. That is consistent with our current understanding of physics.

What I don't understand about your point of view is why you reject purely mathematical infinity. After all, mathematical infinity is "just an idea," as you say. It's an abstraction like the game of chess. Unlike chess, math is useful in the world. That doesn't make it real, just a useful abstraction like law and money and property and all the other abstractions on which civilization is built.

I do believe that Wittgenstein may have said something along the lines that words and ideas gain meaning from their use. So that the number 3 is an abstraction, but every time we use it in the world, as in 3 books or 3 planets, we give meaning to the abstraction.

Can you explain to me what is your objection to mathematical infinity regarded as a pure abstraction? And even perhaps as an abstraction that gains meaning from its use in modern physical science? My understanding is that you object to infinity even as a mathematical abstraction, please clarify if I've got that wrong.

[Serendipper]

I think you're honing in on the full explanation.

I'm honing in on where I cannot go

Key is that all human conceptions of the universe are naturally less than complete, qualitatively speaking, and thus for any conception to even make some claim to pertinence it must explicitly be open ended.

So positing infinity as a concept is an excuse for falling short in concrete terms. Any concrete terms fall short of the whole. Infinity isn't a concrete term.

Maybe we must realize a marriage between precision and vagary.

The part can't encompass the whole and a thought, an idea, is a part of the universe.

Yes I think so. Thoughts are part of the universe and therefore thoughts cannot embrace the universe. Trying to embrace the universe with an aspect of the universe results in infinite regression which is evidence that the initial assumption was false.

As David said, "We have already seen that the infinite is nowhere to be found in reality, no matter what experiences, observations, and knowledge are appealed to. Can thought about things be so much different from things? Can thinking processes be so unlike the actual processes of things? In short, can thought be so far removed from reality? Rather is it not clear that, when we think that we have encountered the infinite in some real sense, we have merely been seduced into thinking so by the fact that we often encounter extremely large and extremely small dimensions in reality?

Does material logical deduction somehow deceive us or leave us in the lurch when we apply it to real things or events? No! Material logical deduction is indispensable. It deceives us only when we form arbitrary abstract definitions, especially those which involve infinitely many objects. In such cases we have illegitimately used material logical deduction; i.e., we have not paid sufficient attention to the preconditions necessary for its valid use."

This arrived in the news yesterday:

Matter Sucked in by Black Holes May Travel into the Future, Get Spit Back Out

Astronomers and physicists have long held that the idea of a singularity simply must be wrong. If an object with mass has no size, then it has infinite density. And, as much as researchers throw around the word "infinity," infinities of that kind don't exist in nature. Instead, when you encounter an infinity in a real, physical, science situation, what it really means is that you've pushed your mathematics beyond the realm where they apply. You need new math.

It's easy to give a familiar example of this. Newton's law of gravity says that the strength of the gravitational attraction changes as one over the distance squared between two objects. So if you took a ball located far from Earth, it would experience a certain weight. Then, as you brought it closer to Earth, the weight would increase. Taking that equation to the extreme, as you brought the object near to the center of Earth, it would experience an infinite force. But it doesn't.

Instead, as you bring the object close to the surface of Earth, Newton's simple law of gravity no longer applies. You have to take into account the actual distribution of Earth’s mass, and this means that you need to use different and more complex equations that predict different behavior. Similarly, while Einstein's theory of general relativity predicts that a singularity of infinite density exists at the center of black holes, this can't be true. At very small sizes, a new theory of gravity must come into play. We have a generic name for this new theory: It's called quantum gravity.

All this is just to say that when we encounter infinity, it means we've done something wrong.