Does infinity exist?

“No axiom of infinity, no Riemann integral.”

No Jesus, no Sacrament.

I understand the openmindedness of children, but the subject we are discussing is better served by assuming the participants are intelligent adults who have perhaps been to school or maybe read and thought a little bit about things.

Now that is a very good question! What is true that not only calculus but all of physical science is currently founded on infinitary mathematics. But, it is this a necessary or a contingent fact? I’m pretty sure it’s contingent. Foundations go in and out of favor. Netwon got results using math that’s not regarded as rigorous today, and in fact required another 200 years to logically formalize.

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

There are researchers trying to find weaker logical structures in which to do math and physics. Finitism (No axiom of infinity, but still with mathematical induction); and ultrafinitism (not even induction); are far too radical and I no of nobody who claims to be able to found physics on finitary principles.

However, constructive foundations are a subject of great interest. In constructive math and physics, an object is said to exist only if it is the output of a Turing machine. I discussed this earlier. So there is no axiom of choice, no uncountable sets, no noncomputable real numbers.

en.wikipedia.org/wiki/Construct … athematics

In set theory we have the full powerset axiom, that says that all of the subsets of a given set exist. In constructive math, only the contructible sets exist. These are the sets whose elements can be cranked out by a computer program (as exemplified by a TM). So the even numbers exist as a subset of the naturals. But most sets in standard math no longer exist because their elements can’t be computed.

You may be unhappy with this, because we do have infinite subsets of the naturals and for that matter we have the full set of naturals, infinitely many of them. So constructivism still needs “a little infinity,” but far less infinity than full set theory.

That is the state of the art today. If you wish to hold out hope of a glorious future in which all of physical science can be founded on ultrafinite or finite principles, that is your right. But why? How are you going to express the differential equations of biology? Why is it so important to you?

I quite agree. Nobody thinks the axiom of infinity is “true” in any meaningful sense. Rather, infinite sets are USEFUL to mathematicians, and infinitary math is useful to physics. Whether it’s necessary, we don’t know.

Put it this way. We could play chess without the queen. The game would be very different and much more dull. So we keep the rules the way they’ve evolved.

The axiom of infinity is like that. It’s a more fun and usesful rule so we keep it in the game. Why does that bother you?

We don’t need the queen to play chess. So what? But you’re wrong on the facts. Without the axiom of infinity, at the very least the constructible sets, you can’t develop the theory of the real numbers sufficiently well to do modern physics. Sure someday someone MIGHT find a way, but in the meantime are you throwing out all of science back to before Newton?

But no, this is quite false on the facts. Differential geometry and non-Eucidean geometry were mathematical curiousities in the 1840’s, and became the mathematical foundation of relativity aftter Einstein.

And quantum physics lives in the mathematical framework of Hilbert spaces, a highly abstract infinite-dimensional vector space studied in a field called functional analysis.

So you’re just flat out wrong on the facts here. Advanced abstract math is indispensible for modern science. Not all of advanced math, but much of it. Sure there is math that’s “out there” today, but who is to say it won’t be essential to the study of the real world a century from now?

Not more logical. More useful. If you’d discuss what I write and not the words YOU put in my mouth, this would be more productive. You are constantly arguing against positions I’ve never expressed.

I don’t say the axiom of foundation is more logical than its negation. On the contrary, they are both equally logical, each being consistent with the rest of the axioms. The axiom of infinity has proven itself more useful so most mathematicians adopt it. There are constructivists, finitists, and ultra-finitists among mathematicans. Especially in the past few years, there’s renewed interest in constructivism due to the influence of computers and automated proof checking.

I linked a Wiki article that contained counterexamples, and I explicitly called out that fact. The Wiki article on Fubini’s theorem contains examples of functions whose integral depends on the order of integration.

Those puzzles only demonstrate the poor teaching of the order of precedence of the arithmetic operators. And the poor understanding of this topic even among elementary school teachers. They don’t hire elementary school teachers for their math acumen. God knows I wouldn’t spend my days among a bunch of ten or twelve year olds.

I don’t see how you can make this comparison. If you think those silly puzzles are anything like the discussion of the axiom of infinity, I don’t think you’ve given the matter enough thought.

No bacon, no BLT.

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.

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. 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.

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

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

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 bbc.com/future/story/2015041 … ing-clever

[i]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.[/i]

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.

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

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

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.

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.

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.

Just write them down like we have been.

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

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.

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.

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

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.

Probably because waves are assumed to extend to infinity.

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.

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.

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

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.

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).

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.

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.

Sounds good to me!

Welcome to the site :slight_smile:

You better get to it then.

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

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.

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:

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:

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

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?

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

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.

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.

On the infinite

by

DAVID HILBERT

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.

  • [i]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.[/i]

  • [i]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 [/i]


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.

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.