# How can numbers be infinite? [closed]

Abstract or not, the number concept originates from the world, so it seems to me there are only as many numbers as things, and there comes a point where what we call numbers are only hollow representations, pointing at concepts which are false in the edge cases, and so false in themselves.

If there were 2 things in the universe, then 3 wouldn't be a number, even abstractly, because the abstract is a representation of reality, isn't it? Even if not, it still caries truth/false in the abstract.

Yes, to mathematicians, infinity is countable, because you can theoretically start at 0, count upwards 1, 2, 3, 4, 5... and "reach" ℵ₀ according to the axiom of induction.

But counting is a concept which originates from grouping things. So even in theory, there comes a point where we can't count, if we can't count in reality by adding an item to such number of things.

There is counting-counting, then there is playing with notation.

I don't think the induction follows. Do we think theory has validity just because the notation plays nice?

It seems to me, Zeno's paradox could even be seen as a proof of the finiteness of numbers, because even in the abstract we reach the paradox of the true abstract (there are arrows which can be shot) against the false abstract (it can be divided in an infinite set of motions). Something has to give in.

I know we gave in in thinking that we can sum infinities... Well, I don't give in there. We can't sum infinite sums, because there are no infinite sums to sum.

A representation is true in the extent of the truthness of what it represents. If I add 1 unicorn to 1 unicorn, I get 2 unicorns, because the false representing unicorns are represented by real 1 thing and 1 thing counting... So the abstract numbers are only true representations in the extent of the truthness of the real things they represent. Then they become unicorns which fail in the edge cases, like being.

If it is not being, then it is a false concept, and the infinity of number is a false concept, so it is false.

Even in the abstract, numbers are finite. I can't imagine a new color, can I? The limit of representation. There is no real unicorn concept, only the concepts of horse, corn, and the shapes/sounds. It is concepted artificially, partially, so it is not even a concept, but a meshing of concepts (themselves wholly concepted from reality, so wholly true), through the mesh concept, into a false, void 'concept'. In the same way, there is no concept, for some numbers, and for infinity. It's only playing with notation.

Infinity exists only as much as we can put "in" in front of "finity", and wonder why it doesn't mean anything we can visualize. We can't represent it, because it doesn't exist, even abstractly. It is false.

Well, that's just, like, my opinion

• Are truth and falsity not also abstract concepts which both exist only in the mind? – Moritz Oct 31 '15 at 13:16
• "If there were 2 things in the universe" - say A and B, one could have the sets {A}, {B} and {A, B}, so 3 could be a number. – nir Oct 31 '15 at 13:22
• @nir that reminds me of Russell's comment about his shoes. If I have two shoes does that mean that there are three things: the left shoe, the right shoe and the pair? – Bumble Oct 31 '15 at 13:37
• There is also the idea found in Husserl, I believe, or earlier, that we cannot have a concept "1" without a concept of "2" so that "1" thing can only be a backformation from the idea of "2" things, which we only infer from "3" things, etc. No cognitions without "recognition." – Nelson Alexander Oct 31 '15 at 17:30
• This is not a question, it is a manifesto. – Conifold Oct 31 '15 at 23:38

It seems that you are taking a rather extreme form of mathematical nominalism and denying the existence of distantly mind-independent objects. There is no existent "counter" who can keep on "counting" to infinity, thus it is a mere word, or rather the fanciful negation of the word "finite." Reason exceeds its remit.

This starts off sounding like good, hard-headed common sense, but can soon lead into the radical empiricism of Bishop Berkeley. If nothing is granted "real" existence apart from what can be "counted" or observed, how do you distinguish between your own observing mind and the so-called "existing" world, including other minds in that world? Are they all really there when you aren't looking or counting?

Moreover, what seems most certain by direct experience is that you cannot observe absolute "finitude." Have you ever seen it, chewed it, experienced it? Where is it? As far as each of us can think in all directions and back into our past, don't we discover only a kind of infinite regress of consciousness? Never some clear boundary where "finitude" begins. In terms of strict empiricism, there seems to more evidence for an infinite continuum than for its opposite.

So now we are in the reverse antinomy. We want to kick the stone and say "things exist!" Yet we do not grant existence to all those things that cannot be observed or counted, like some infinite series of numbers or anything those numbers might refer to. Must we then assume, with Berkeley, that all these things exist because they are being observed and counted.... by some possibly infinite Accountant?

• But Kant does not grant "real" existence even to what can be "counted" and observed, those are only forms of perception, and he was not even a nominalist. Infinite, infinitely divisible continuum is clearly an idealization regardless of one's philosophy, so there can be no evidence for it beyond it being a convenient model, Kant showed what happens when mental constructs are ascribed to things in themselves. Hilbert was a nominalist about mathematical objects, so was Quine at one point, that doesn't take one anywhere near solipsism. – Conifold Oct 31 '15 at 23:54
• @Conifold Is it really an obvious idealization? In principle we can represent any real number with a string of digits, so we can describe them all. Why should they be any more of an idealization than the integers? – Matt Samuel Nov 1 '15 at 0:04
• @MattSamuel We can not, by diagonal argument no more than countably many real numbers can ever be represented in principle, even with the usual inductive completion powers we grant ourselves in physics and arithmetic. – Conifold Nov 1 '15 at 1:34
• @Conifold ah but there's also transfinite induction. – Matt Samuel Nov 1 '15 at 1:38
• @Matt Samuel One is probably better off trying to make an empirical case for continuum than for transfinite induction :) None of empirically applicable mathematics depends on the axiom of choice for continuum, and Godel's constructible universe explicitly shows that all uncountable infinities (at least set theoretically understood) along with transfinite induction are at best figures of speech concerning countable infinities. – Conifold Nov 1 '15 at 2:31

A huge numbers of mathematical objects arise from "real" objects in our world like circles, angles, lines, two trees or two persons etc. But also these mathematical objects are not images but abstractions. The mathematical circle is not just an image, i.e. a copy of a real circle.

An abstraction of an object takes some properties of the object and let's other go. It takes the former properties and creates out of them a new object, a concept or an idea. The idea belongs to quite a different category than the original object. Selecting those properties, which are to be built into the concept, often means generalization, ommitting restrictions. And one generalization in mathematics is to allow the concepts of arbitrary big numbers, not just the numbers one has needed to count real objects.

The next generalization concerning the number concept was creating the number "infinite", named "Aleph Zero" by the mathematician Georg Cantor in the 19th century. Cantor also investigated how to extend elementary calculation to include Aleph Zero. From a mathematical point of view it is not the question whether a newly created mathematical object corresponds to real objects. The primary question is whether the concept is free from contradictions and how it connects to the hitherto existing concepts.

The story went on and Cantor created many different, biger and bigger numbers infinity. That's part of set theory, a fascinating mathematical domain.

Summing up and to put it bluntly: Mathematical objects live in a different domain than real objects, they live in the virtual realm of a game. The mathematical game has only one rule: No contradictions!

• @Aleph Zero is the smallest infinite cardinal number. It is the cardinal number of the set of all natural numbers. It is the cardinal number which captures the traditional concept of "actual infinity". That's the crucial point: Aleph Zero does not capture the concept of "potential infinity". It is a much stronger concept. - Cantor showed that several infinities exist, i.e. there are infinite sets with different cardinalities. For details please look up "infinite cardinalities". E.g. Aleph One is the cardinality of the set of real numbers, which is strictly bigger than Aleph Zero. – Jo Wehler Oct 31 '15 at 17:54
• You wrote: "The next generalization concerning the number concept was creating the number 'infinite', named 'Aleph Zero' by the mathematician Georg Cantor in the 19th century." - maybe I misunderstood what you meant, but according to the wikipedia infinity was used in math since the greeks, and "European mathematicians started using infinite numbers in a systematic fashion in the 17th century." – nir Oct 31 '15 at 20:33
• @nir I am not aware that already antique Greek mathematics dealt with infinity. People struggled for centuries to capture infinity into a precise concept. Of course philosophers and theologians used the term, but mostly either undefinded or in the meaning "without borders". – Jo Wehler Oct 31 '15 at 22:00
• Yes, Newton and Leibniz developed calculus introducing the limit concept and l'Hospital found a formula to deal with some fractions of the form infinity divided by infinity. Here infinity was to be taken as the limit of bigger and bigger real numbers. - But Cantor introduced Aleph Zero not as a limit, but as a fixed, well-determined number, hereby extending the domain of finite numbers. – Jo Wehler Oct 31 '15 at 22:01
• Yes, the generalized continuum hypothesis (GCH) states that the powerset of a set with cardinality Aleph N has cardinality Aleph N+1. GCH is consistent with but independent from set theory without GCH (Goedel, Cohen). This is one of the most famous results from mathematics of 20th century. – Jo Wehler Nov 1 '15 at 2:02

The number concept originates from the world.

Yes: it originates from the act of counting : we can easily assume that it starts with "counting things", but at some point it becomes "counting numbers themselves" :

one, two, three, ...

without necessarily matching the number-words with "things".

This "process" lead itself to infinity, at least the so-called "potential" one, i.e. the possibility of an endless repetition of the basic "iterative step" of :

You can see it in the simple game where player A ask player B to state the "greatest number he can think to", and when player B name it, player A immediately reply with "... plus one".

Natural numbers are infinite simply because we cannot stop the process of counting.

• But if a system is a subsystem of truth (like having non-thing numbers), it is necessarily false in regards, like maybe in the existence of what it portrays. So in truth, there are X things, but in this system, there is a X+1 thing... But in truth, there is no such X+1 thing. So in truth, there is no X+1 number. If you are within the system (in the false), there is, and it allows to do things with the notation, which become true again after the process, maybe, but the number doesn't exist, even conceptually. It is only a bridge, a representation for other real concepts. – Pierre Nov 3 '15 at 15:30
• So if you rely too much on it, it breaks. We can pretend there is such thing, because we can play with the notation to represent it, but what good does it do? Your game was never more than sounds and hollow agreements. It is like drawing lines, and saying it represents such non-existing thing. It doesn't mean anything, because we can't use what is represented, if it doesn't exist. – Pierre Nov 3 '15 at 15:30

Well if you don't believe in actual infinities, such as infinite numbers, you are in good company. Gauss, Poincare and Kronecker, to name a few, thought of infinity is nothing more than a kind of useful fiction that allows us to do things with limits. Mathematicians who reject the Cantorian approach to the construction of transfinite numbers are sometimes called finitists.

This is a minority view, however. Playing around with transfinite numbers is interesting and useful. There are, for example, more irrational numbers than rational numbers, even within a finite interval such as 0 to 1. How would you express this without recourse to transfinite number theory?

• You wrote "There are, for example, more irrational numbers than rational numbers, even within a finite interval such as 0 to 1" - here is what Wittgenstein writes about it: "From Cantor's proof, however, set theorists erroneously conclude that 'the set of irrational numbers' is greater in multiplicity than any enumeration of irrationals (or the set of rationals), when the only conclusion to draw is that there is no such thing as the set of all the irrational numbers." – nir Oct 31 '15 at 20:39
• @nir I do not understand Wittgenstein's reasoning. Could you please explain how you understand his text, thanks. – Jo Wehler Oct 31 '15 at 22:10
• @JoWehler: I'm getting a finitist vibe from that text. It sounds like he's arguing that if you can't enumerate it, it doesn't exist as a set (but maybe it does as a proper class?). – Kevin Nov 1 '15 at 2:09

As Bumble points, the position you're taking or insisting on has been taken by some very good mathematicians; On the whole they can be largely divided into camps: one called finitism, who rule out actual infinities, but not potential ones.

Thus, the sequence 1,2,3, ... is allowed; since it's potentially infinite but doesn't require us to believe we can actually reach the infinite.

A more radical position, and one in line with your opening paragraph are the ultra-finitists - who rule out very large numbers; as you say they are not represented...

These are minority positions, amongst mathematicians - however; but were physicists to think on this, they themselves may find that they are at least finitists, if not ultra-finitists; for the result of a measurement must be a finite number to count as a measurement; even if they pass through abstractions that hinge on the mathematically concieved infinite - their tools being manufactured elsewhere, on a Cantorian forge.

This is a really fascinating area, and you are in good company. Some philosophies/schools of mathematics -- like finitism, reject infinity. Others -- like intuitionism -- have nuanced positions on infinity.

With that said, mathematical objects are not "real" nor do they necessarily represent anything "real". Yes, they most likely arose from interaction with the real world, but after that, they were abstracted and became a logical system in their own right. Once that happened, they became capable of having all sorts of properties divorced from the real world, including infinity.

Now you should think of mathematical objects as purely logical constructs forming a closed system, and instead of trying to ensure that these objects accord with intuition, ask instead if they are consistent within this system.

For instance, one definition of Natural Numbers is the Peano Axioms. The key part of this system is that it starts with a number -- 0 -- then defines a successor function that yields a distinct number for any number. Since there's no upper bound defined on this function, it's clear that this system yields infinite numbers.

If by "number" you mean "quantity", then what you say is true - there is a limited quantity in the universe (unless the universe is of an infinite age, ie had no beginning). But if you mean the abstract concept of numbers, then what you say isn't really true. Numbers don't exist to begin with, and from the start are nothing more than a symbol of an idea, whether it is one or whether it a number we have no name for. And if you mean that the concept of "infinity" can't be a number in of itself, but can only be a representation of a series of numbers - that is the definition of any number.

The post presumes that infinity is some abstract concept that is a result of mathematicians faffing about. That it doesn't have any application to the real world. That's not true. In fact, without infinity math does not work.

There's a bunch of leaps and logical fallacies in the original post, but I think this is the big one.

If it is not being, then it is a false concept.

Even in the abstract, numbers are finite.

Let's put these two together. The first seems assert that every concept which cannot be physically represented is false. The second builds on that by saying if you don't actually count to infinity, infinity is false.

Ok, let's imagine a world without infinity. That means there has to be a biggest number. What's the biggest number? Here, I'll pick one. 1,000,000,000,000. There. I have just asserted the biggest number. What happens if I add one to it? 1,000,000,000,001. A new biggest number!

Do it again: 1,000,000,000,002. A new biggest number!

Do it again: 1,000,000,000,003. A new biggest number!

Do it again: 1,000,000,000,004. A new biggest number!

For any biggest number you can imagine, you can add one to it to get a bigger number.

I don't actually need to keep at this task, it's enough to prove the potential of their existence were you to keep at the task. Same as I can add 10 + 20 without actually moving little groups of objects around. This is the great utility of mathematics, I don't need to move physical things around to do math.

Concepts which don't have a physical form are not false, they just don't have physical form. If I have an idea to build a house slightly different from all other houses it's not a false concept on a house. We know houses work. This particular house just doesn't have physical form yet.

But wait! It gets worse! (Or better depending on how you look at it)

I can't imagine a new color, can I?

Sure you can! Infinitely! What happens when you mix red and green? Yellow! What if you mix yellow and red? And the result of that mixing with red again? And that with red again? And again? And again? And again? Each will be a slightly different color.

This is similar to what you get when you try to find the smallest fraction. What's halfway between 1 and 2? 1.5. Between 1 and 1.5? 1.25. Between 1 and 1.25? 1.125. And so on.

The difference is paint is made up of discrete molecules and eventually you'll have nothing but red left. Numbers don't have an end and you can just keep on dividing.

Not only is there an infinity of numbers, there are infinities between every number! The infinite set of positive integers is countably infinite meaning you can potentially count them without leaving any gaps. The infinity between the real numbers is uncountably infinite meaning you can never list them all out (I realize this is not the formal definition). Every time you try to count them you create more gaps to fill in.

This is the infinity of the very small.

There is counting-counting, then there is playing with notation.

Infinity isn't "playing with notation". Without infinity basic math does not work. It would be inconsistent. Without infinity there must be a "biggest number" and that breaks math.

For example, for the set of positive integers (1, 2, 3, ...) it can be said that if `M = N + 1` then `M > N`. But what if N is the biggest number? If M is greater than N, then M must be the biggest number. Contradiction! This might not seem like a big deal, but in mathematics a system which contains contradictions is false. Without infinity integer addition does not work.

Similar problem with the infinitely small and division of the real numbers. If `M = N/2` then `M < N` because `2M = N`. But what if N is the smallest number? If M is less than N, then M must be the smallest number. Contradiction!

Without infinity there are no irrational numbers, numbers which have infinite, non-repeating decimals. Pi is an irrational number. Without Pi, we can't work with circles.

No infinity, no mathematics. (You can construct mathematical systems without infinity, but they're not nearly as useful.)

I'd encourage you to watch these Numberphile videos where various charismatic mathematicians talk about infinity. They're delightful and will provide you with all sorts of insight into how important and practical infinity is. Also how bad most people's understand of infinity is.

In particular...

• I like what you said about colors and irrational numbers. There's food for thought! But maybe the process of computing the irrational is flawed by the idea of an infinitely-round circle being possible. Maybe when we calculate the number, there is a bound to it, which is the bound of the maximum roundness of a circle with the degree of precision of the universe, and that a pi more precise doesnt exist, because a more precise circle doesn't exist, and we're there again playing with the notation of a circle, instead of the exact concept of the roundness of a thing. – Pierre Nov 3 '15 at 14:27
• @Pierre Pi is irrational and it shows up in many equations because circles and spheres are important. In the physical world we can't make a perfect circle, reality is too unpredictable, but you can get pretty damn round. Math is a tool used by physics to create a model of the universe, it's not the universe. We know it's an approximation of reality. – Schwern Nov 3 '15 at 19:16