There's a category error that involves thinking that because we can't start at one and write down every subsequent natural number, they don't exist. — Banno

There's an ontology which presumes that numbers exist, it's called Platonism. It's been demonstrated to be a very problematic ontology, and many philosophers claim that it was successfully refuted by Aristotle, as inconsistent with reality.

It is also well-known that those issues do not arise in the same way at the macro scale. — Srap Tasmaner

That's the problem with this type of issue. The supposed universal principles work extremely well in the midrange of the physical domain. Since the midrange is our worldly presence, and that is the vast majority of applications, we tend to get the impression that the principles are infallible, and "true". However, application at the extremes evidently produces problems. Therefore we must take the skeptic's eye to address the real possibility of faults within the supposed ideals.

Logic and mathematics are mental tools or technologies, habits of mind, that we have developed for dealing with things at the macro scale. — Srap Tasmaner

What you call "the macro scale" is really the midrange, the realm of human dealings. Other than the micro scale and the macro scale, we need a third category which might be called the cosmological scale.

This is unsurprising since our mental lives consist, to a quite considerable degree, of making predictions. Logic and mathematics enable us to figure out ahead of time whether the bridge we're building can support six trucks at once or only four. — Srap Tasmaner

It is true, that this midrange scale, what you call the macro scale envelopes pretty much the entirety of our day to day lives. However, as philosophers with the desire to know, we want to extend our principles far beyond the extent of the macro scale. And this is where the issue of incorrectly representing infinity may become a problem.

For example, let's say that the macro scale is in the range of 45-55 in a scale of 0-100. So we might hypothesize and speculate about that part of reality beyond our mundane 45-55 range. If the application of mathematics, to the physical hypotheses leads to infinity in both directions at what is really only 35 and 65, then we have a problem because we place the majority of reality beyond infinity. And, if we close infinity by making it countable, then there is no way for us to know that there is even anything beyond 35 and 65. It appears from our physical hypotheses that we have reached infinity, therefore the extreme boundaries. And, if the mathematics has closed infinity, in the way that it does, then by that principle we actually have reached infinity. Therefore, by that faulty closure of infinity, 35 and 65 are conclude as the true ends of the universe, the true limits to reality, when reality actually extends much further on each side.

Which leads, at last, to my point, such as it is: there is something perverse, right out of the gate, about the insistence on "actually carrying it out". It misses an important point about the value of logic and mathematics, that we can check first, using our minds, before committing to an action, and we can calculate instead of risking a perhaps quite expensive or dangerous "experiment". ("If there is no handrail, people are more likely to fall and be injured or killed" -- and therefore handrail, without waiting for someone to fall.) — Srap Tasmaner

I don't see how this is relevant. The issue is not properly with "actually carrying it out", the problem is with the assumption that it is possible to carry it out. The defining feature of "infinite" renders it impossible to carry it out. So when we say that it is possible to carry out something which is defined as impossible to carry out, this is a problem regardless of "actually carrying it out".

This denigrates the status of "impossible". Now, "impossible" is a very important concept because it is the most reliable source of "necessity". When something is determined to be impossible, this produces a necessity which is much stronger and more reliable than the necessity of inducive reason. So the necessity of what is impossible forms the foundation for the most rigorous logic. For example, the law of noncontradiction, it is impossible for the same thing, at the same time, to both have and have not, a specified property. this impossibility is a very strong necessity. In mathematics, the impossible, and therefore the guiding necessity, is that we could have a count which could include all the natural numbers. if we stipulate that this is actually possible, then we lose that foundational necessity.

The natural numbers turn out to go on forever, and we can prove this without somehow conclusively failing to write them all down. — Srap Tasmaner

So this exposes the problem. We know that the natural numbers go on for ever. Therefore it is impossible to count them, or that there is a bijection of them. They could not have all come into existence therefore it is impossible that there is a bijection of them. This impossibility is a very useful necessity in mathematics. So if we stipulate axiomatically, that it is possible to count them, or have a bijection, then we compromise that very useful necessity, by rendering the impossible as possible.

To see the demonstration that the rational numbers are equinumerous with the natural numbers and complain that it is not conclusive because no one can "actually do them all" is worse than obtuse, it is an affront to human thought. — Srap Tasmaner

This is a misrepresentation of what I am arguing. My claim is that it is definitively impossible to count the numbers. Therefore to represent this as possible is a contradiction. This has nothing to do with whether a human being, computer, or even some sort of god, could "actually do them all". The system is designed so that they cannot be counted. Nothing can do them all, and this is definitional as a fundamental axiom. So, whether or not anything can actually do them all is irrelevant because we are talking about a definition. Therefore, to introduce another axiom which states that it is possible to do them all, is contradictory.

My claim is that it is definitively impossible to count the numbers. Therefore to represent this as possible is a contradiction. — Metaphysician Undercover

This is to spectacularly miss the point.

Because we can prove what the result would be, we do not have to actually carry out the pairing of every rational number with a natural number. Proof is a further refinement of prediction, beyond even calculation. Of course it's impossible to count the elements of an infinite set as you would the elements of a finite set. But for the results we're interested in here, you do not need to. That is the point. We already know what the result would be if it were in fact possible.

(In my old computability textbook, this was described by having Zeus count all the natural numbers: he could finish, by using half as much time to count each successor. But even Zeus could not count the real numbers, no matter how fast he went.)

I can put it another way: what you cannot calculate, you must deduce.

Infinite sets obviously present a barrier to calculation. So we deduce. Having deduced, we label our results, and then calculation becomes available again. We continually cycle between logic and mathematics, not just here but everywhere.

There's an ontology which presumes that numbers exist — Metaphysician Undercover

We don't need much ontology. Quantification will suffice.

We know that the natural numbers go on for ever. Therefore it is impossible to count them, or that there is a bijection of them. — Metaphysician Undercover

How do you know that the natural numbers go on for ever? Have you tried to count them and failed? That doesn't prove that they go on for ever.
But perhaps you have counted some of them. That's easy enough to do.
When we are counting numbers, it is natural to start at the beginning. But we could start at any point in the sequence, and go on as long as we like from there. So we cannot find any numbers that cannot be counted.
So they are countable in the sense that some of them can be counted and we cannot find any numbers in the sequence that cannot be counted.

Admlttedly, the step from there to saying that they go on for ever is an induction.
But it is not an induction like the conclusion that the sun will rise tomorrow morning. We can see from the first few steps that anything that emerges from the function will be a number (because it is the successor to a number) and this is nothing to stop the next number emerging.
So the induction is secure. I know that there is some debate about this, but that is the debate we should be taking up.

They could not have all come into existence therefore it is impossible that there is a bijection of them. — Metaphysician Undercover

Ah, so this is about actual and potential infinities. My problem with that is that I don't see how the idea of a possible abstract object can work. In an Aristotelian system, as I understand it, the concepts of matter and potentiality are linked. But that only applies to physical or material objects. Since abstract objects are not material, I don't see how they can have any potential for anything.
I have to admit that much of the talk about functions, suggests that they produce their results only when we feed in values for the variables. This is misleading. The answers are "always already" defined, before we start calculating. Nothing more is needed for a number to exist.
The philosophical parameters for the debate what it means for a mathematical (abstract) object to exist are well enough defined, so that's the debate we are really involved in.

We don't need much ontology. Quantification will suffice. — Banno

As the popularity of this post shows, we do need clarity on the mathematical object called infinity.

In my view the question comes down to simply just what does it really mean when Cantor showed us that the natural numbers cannot be put into 1-to-1 correspondence with the reals. The standard answer, that the infinity is simply larger, and thus we have larger infinities etc. doesn't really answer everything. It simply lacks the rigorous logic that is so ever present in mathematics. The problem with the Continuum Hypothesis shouldn't come as a surprise.

I loved teaching this stuff to third and fourth grade kids. The Biggest Number game; they say "A hundred ", reply "A hundred and one"; they say "A million million", you reply "A million million and one"; someone says "infinity" and someone says "infinity and one"...

I was surprised, on enlisting in these fora, to find that there are folk who don't get to the stage of understanding that every natural number has a successor, that "...and one" works for any natural number. (not ordinals... another bit of the puzzle.)

And that infinity and one is still infinity. This hazy number play sets up the kid's intuitions. Especially where it doesn't work. Infinity is not part of the structure that lets us play the number game. It needs new rules.

As the popularity of this post shows, we do need clarity on the mathematical object called infinity.

In my view the question comes down to simply just what does it really mean when Cantor showed us that the natural numbers cannot be put into 1-to-1 correspondence with the reals. The standard answer, that the infinity is simply larger, and thus we have larger infinities etc. doesn't really answer everything. It simply lacks the rigorous logic that is so ever present in mathematics. The problem with the Continuum Hypothesis shouldn't come as a surprise. — ssu

Yes, the purely constructive meaning of the diagonal argument, is that any constructable injection from the naturals to the Reals defines the construction of a new real. And all that this constructively implies, is that it is impossible to define a surjection from the natural numbers to the reals. Hence it says nothing about whether or not an injection exists from the reals to the natural numbers, and hence the diagonal argument does not rule out the possibility that the real numbers might in fact be a subset of the natural numbers.

As far as the computable Reals are concerned, all that the diagonalization argument implies is that the computable reals are a subset of the Naturals that cannot be "detached" from the rest of the Naturals by an algorithm. For we know by definition that there aren't more computable reals than naturals, since a computable real refers to a computer program of some sort that has a godel number.

The hypothesis that every real number can be listed by an algorithm, is equivalent to knowing the limiting behaviour of every computer program. So what Cantor actually showed, is an indirect proof that the halting problem cannot be solved, and not that there are "more" real numbers than natural numbers.

Because we can prove what the result would be, we do not have to actually carry out the pairing of every rational number with a natural number. Proof is a further refinement of prediction, beyond even calculation. Of course it's impossible to count the elements of an infinite set as you would the elements of a finite set. But for the results we're interested in here, you do not need to. That is the point. We already know what the result would be if it were in fact possible. — Srap Tasmaner

I demonstrated already, it's not a proper "proof" because it relies on a false premise. This produces your incoherent, unsound conclusion "we can prove what the result would be". The incoherency of proving that the result of an impossible task is anything other than incompletion, is obvious.

I can put it another way: what you cannot calculate, you must deduce. — Srap Tasmaner

Deduction from false premises produces absurdities. That's what Zeno is famous for having demonstrated.

We don't need much ontology. Quantification will suffice. — Banno

Maybe not much, but some. Claiming that numbers "exist" is ontology. If you avoid the ontology, then what are you quantifying?

How do you know that the natural numbers go on for ever? — Ludwig V

Mathematical ideals are produced by definition. People decided that this would be really good, and so the system was designed and maintained that way.

So they are countable in the sense that some of them can be counted and we cannot find any numbers in the sequence that cannot be counted. — Ludwig V

But the issue is whether an infinite quantity is countable. Any finite quantity is, in principle countable. But, since "infinite" is defined as endless, any supposed infinite quantity is not countable.

This problem is strictly confined to Platonism, which treats a number as an object which can be counted. So it inheres within the principal axioms of set theory which premises mathematical objects. Numbering is the means by which we measure a quantity of distinct, individual things. When we assume that a number is a distinct individual thing (Platonism), then we might be inclined to measure the quantity of numbers (cardinality).

The problem which jumps out, is that now we are trying to measure the measurement system with itself. And the ontological issue is that it is fundamentally false to represent a number as a distinct individual thing which can be counted. In reality "a number" is a concept which has its meaning in relation to other numbers (ordinality). Therefore we cannot isolate "a number" to be a distinct object, it would lose its meaning and no longer be able to serve its purpose as the concept it was meant to be. Therefore Platonism, which treats ideas as distinct objects which can be counted is ontologically unacceptable.

Ah, so this is about actual and potential infinities. My problem with that is that I don't see how the idea of a possible abstract object can work. — Ludwig V

I'll give you a brief description why abstract "ideas" are classed as potential by Aristotle. This forms the basis of his claimed refutation of Platonism, and provides the primary premise for his so-called cosmological argument which demonstrates that anything eternal must be actual.

The Pythagorean Platonists, as distinct from Aristotelian Platonists, insisted that ideas, specifically mathematical ideas, had actual existence as eternal objects, eternal truths in themselves. Aristotle premised that it is the geometer's mind which gives actual existence to the ideas. The actual existence of the idea is within the mind. Therefore if we premise that the idea had existence prior to being "discovered" by the mind (as the assumed eternal), we must conclude that it existed potentially. (The cosmological argument then goes on to show that anything eternal must be actual.) So the refutation of Pythagorean Platonism sets up a distinction between an idea within a mind, and the supposed independent idea. Within the mind it is actual, and however it exists in the medium outside of minds, is as potential. So when we assume numbers to be independent objects rather than thoughts within a mind, then they exist potentially, not as actual objects.

The philosophical parameters for the debate what it means for a mathematical (abstract) object to exist are well enough defined, so that's the debate we are really involved in. — Ludwig V

This is probably the heart of the issue. "Exist" is a term which is properly defined by ontology rather than mathematics. Therefore the discipline of ontology is the one which ought to determine whether numbers exist. Notice, @Banno makes some seemingly random claims about the existence of numbers. Since the distinction between what exists and what does not exist forms the basis for our judgements of true and false, we can't simply make an arbitrary, or completely subjective stipulation, or axiom, which defines "exist". That would imply total disregard for truth.

And that infinity and one is still infinity. This hazy number play sets up the kid's intuitions. Especially where it doesn't work. Infinity is not part of the structure that lets us play the number game. It needs new rules. — Banno

Exactly, and we aren't understanding those rules yet. What we see are paradoxes and we simply want to avoid them or assume there's something wrong. There isn't anything wrong, it's that we start from the wrong axioms.

Indeed it's very interesting. I have just later understood when grandfather, a math teacher himself, just shook his head and said it was too difficult, when at first grade at school they started teaching math with set theory. But so progressive and courageous were math-teaching in the 1970's in Finland.

Yet I think mathematics will suprise us some day... once we really understand infinity, it likely is something that can be taught at school for kids. Mathematics is just so beautiful.

I think the real breakthrough will be in when we understand that there is the mathematics that is not computable and thus doesn't start with addition, but is inherently important to mathematics. Yet we (and I think other animals too) have started using mathematics from the practical side of it. "No lions, one lion, two lions, many lions." could be the "mathematical system" for a zebra in the African Savannah, which is very useful for it's survival. It doesn't need calculus, it doesn't ponder infinity. Yet for us when we want to make mathematics a logical system. For the grazer on the Savannah it might be enough, but for us the "0,1,2,many" system isn't enough at all. We need calculus, infinity is obvious in mathematics. Yet if we assume that everything in mathematics starts from counting, from finite natural numbers, we are making a mistake.

Yes., well phrased Although we might differ a bit on the extensibility of maths.

It's just extending the way we talk about numbers. What started with the Biggest Number game gets extended into infinity, both ∞ and ω, the difference being that while ∞+1=∞, ω+1>ω; The first reflecting the teacher's answer "infinity plus one is still infinity", the second, the player's answer "infinity plus one is bigger than infinity". What we have is a division in how we proceeded, in the rules of the game, not in what "exists" in any firm ontological sense. It's chess against checkers, not cats against dogs. Neither set of rules is "true" while the other is "false".

And the great thing about these games is that they are extensible, in that we add more rules as we go, keeping the game coherent, while being able to talk about more and different stuff.

Part of where Meta and Magnus have difficulty is in their insistence that one way of talking is right, the other, they call variously incoherent or inconsistent, both without providing an argument and in the face of demonstrations to the opposition effect. To establish incoherence, they would need to show a violated rule internal to the system, or an explicit contradiction derivable from its axioms.
Mere discomfort with plural rule-sets doesn’t suffice.

Within cardinal arithmetic, ∞+1=∞ is true; within ordinal arithmetic, ω+1>ω is true. Cross-applying the rules is what generates the illusion of contradiction.

I'd also relate this back to my essay Two ways to philosophise, and to the arguments in Logical Nihilism. It's better to have an incomplete theory that is coherent than a complete theory that is inconsistent or artificially restricted. And better to have many differing, incomplete logics than one, monolithic yet restricted logic. These allow for growth. Advocating for new rules, new distinctions, new domains of discourse gives us a normative standard that is neither realist nor relativist.

Critics may conflate pluralism with anything-goes relativism. But only because coherence is doing real work; incoherent extensions are still excluded. Others will insist that without a privileged logic, critique collapses. But critique is local, rules are criticised from within practices or at their interfaces, not from a mythical God’s-eye view.

...Banno makes some seemingly random claims about the existence of numbers. — Metaphysician Undercover

To be is to be the value of a bound variable. ω and ∞ are cases in point. In maths, Quine's rule fits: existence is not discovered by metaphysical intuition but incurred by theory choice. Quantification, ∃(x)f(x), sets out what we can and can't discuss.

I've tried to follow what you are doing here, but scattered inaccuracies and errors make it very difficult. I gather you want to Cantor’s argument into a constructive or even computational lens. It’s valid in that framework, yet you seem to think it can be taken as refuting classical results about cardinality. I musty be misreading you.

The hypothesis that every real number can be listed by an algorithm, is equivalent to knowing the limiting behaviour of every computer program. So what Cantor actually showed, is an indirect proof that the halting problem cannot be solved, and not that there are "more" real numbers than natural numbers. — sime

It's really good that now people are more and more noticing the simple link with Cantor and undecidability resuls of Turing and Gödel. Negative self reference is a very powerful tool in logic.

Part of where Meta and Magnus have difficulty is in their insistence that one way of talking is right, the other, they call variously incoherent or inconsistent, both without providing an argument and in the face of demonstrations to the opposition effect. — Banno

Indeed. And this is why it's actually very informative and interesting to listen to actual finitists as they can make valid criticism of ordinary mathematics. Just like every school in philosophy or economics or whatever, also in mathematics various schools make interesting viewpoints that shouldn't be categorized as being either right or wrong.

Within cardinal arithmetic, ∞+1=∞ is true; within ordinal arithmetic, ω+1>ω is true. Cross-applying the rules is what generates the illusion of contradiction. — Banno

The first uncountable ordinal is the interesting question. What is it, what does it mean and what is the logic then?

I think the main problem is that proving something is inherently close to computation, thus no wonder that we have the undecidability results lurking with the uncountable.

Tough nut to crack, but actually it's great that some large basic questions are still open in math and not everything has been done before us. Because that's the error many do: Russell and Whitehead thought that as everything is already there, they just had to write everything out then in a "small" book.

To be is to be the value of a bound variable. ω and ∞ are cases in point. In maths, Quine's rule fits: existence is not discovered by metaphysical intuition but incurred by theory choice. Quantification, ∃(x)f(x), sets out what we can and can't discuss. — Banno

As I said, Platonism, which is an unacceptable ontology.

As I said, Platonism, which is an unacceptable ontology. — Metaphysician Undercover

Platonism is indeed unacceptable, but quantification is not platonic. Sad you can't see that.

Quantification does not require Platonic commitment; it merely specifies the domain of discourse and what statements about it are true. This is consistent with nominalist or structuralist interpretations.

If, "to be is to be the value of a bound variable", then you are obviously talking Platonism. Anytime a value has being, that's Platonism. I'm amazed that you do not understand this, or deny it, or whatever.

Platonism is indeed unacceptable, but quantification is not platonic. — Banno

Do you recognize that set theory is based in Platonism?

Anytime a value has being, that's Platonism — Metaphysician Undercover

No, Meta. Quantification or assigning a value does not require Platonic commitment. A value can ‘have being’ within a formal system, a constructive framework, or a model, without existing independently as Plato would claim.

Do you recognize that set theory is based in Platonism? — Metaphysician Undercover

Sad. Formally, set theory is just a system of rules. Treating its sets as independently real is a Platonic interpretation, not a necessity.

Guess it's back to ignoring your posts.

No, Meta. Quantification or assigning a value does not require Platonic commitment. A value can ‘have being’ within a formal system, a constructive framework, or a model, without existing independently as Plato would claim. — Banno

You were claiming that numbers "exist", and how to be, is to be a value. Now you've totally changed the subject to "assigning a value".

Formally, set theory is just a system of rules. — Banno

Sure, and those rules are axioms about "mathematical objects". When you were in grade school, were you taught that "1", "2", and "3" are numerals, which represent numbers? Notice, "2" is not a symbol with meaning like the word "notice" is. It's a symbol which represents an object known as a number. In case you haven't been formally educated in metaphysics, that's known as Platonism.

Guess it's back to ignoring your posts. — Banno

And I'll opt to believe that you willfully deny the truth, rather than simply misunderstand.

You were claiming that numbers "exist", and how to be, is to be a value. Now you've totally changed the subject to "assigning a value". — Metaphysician Undercover

Same thing. Again, not my problem that you don't understand this.

Sure, and those rules are axioms about "mathematical objects". When you were in grade school, were you taught that "1", "2", and "3" are numerals, which represent numbers? Notice, "2" is not a symbol with meaning like the word "notice" is. It's a symbol which represents an object known as a number. In case you haven't been formally educated in metaphysics, that's known as Platonism. — Metaphysician Undercover

Very sloppy work. Platonism is not the claim that symbols refer to something, but that mathematical objects exist independently of any theory, language, practice, or mind, and are discovered, not constituted, by mathematics. Nothing here commits to that. You are equivocating between reference and ontological independence.

And I'll opt to believe that you willfully deny the truth, rather than simply misunderstand. — Metaphysician Undercover

You are looking for a rhetorical dodge to get out of the mess you find yourself in.

As I said, denial!

When numbers are assumed to be mathematical objects, these objects simply exist independently of any human mind. The supposed object is not in my mind, nor your mind, because it would be in many different places at the same time. Under the assumption of Platonism, a bijection does not need to be carried out, it can be represented, because it is assumed to already exist independently of any minds, so we just need to reference it. Likewise, the natural numbers can be represented with "{1, 2, 3, ...}" but only if they exist independently of any minds.

Do you understand the difference between the representation of a set of objects, and a formula for the procedure which you called "assigning a value"? Could you read 1, 2, 3, ... as a formula for assigning a succession of values?

1 is a number, and every number has a successor. That's enough to show that the natural numbers exist. — Banno

What you stated her is blatant Platonism.

If however, "1, 2, 3..." signifies to you, a formula for the process of "assigning a value" in a specified sequence, rather than representing an infinity of numbers, this is not Platonism. Can you apprehend the difference? I think you can.

And I think, that's why you switched form "to be is to be the value of a bound variable", to, "assigning a value". right after you stated "Platonism is indeed unacceptable". You know the mathematical principles you argue are thoroughly Platonist, and you feel ashamed of this. So you tried to cover this up. Why the dishonesty? Accept mathematics for what it is, and get on with it. Shameful deception and attempts to disguise your ontology get us nowhere.

Every time that you say 1, 2, 3... represents an infinity of numbers, that is blatant Platonism. It is absolutely necessary that the referenced infinity of numbers must have independent existence because it is absolutely impossible that they could exist within any minds.