No clue, I could not find it, I only know that he works on it — Lionino

I'll believe that he has anything when I see it. Especially, does he purport to offer an axiomatic system? I don't recall, but perhaps he rejects the axiomatic method. That would be fine. But there is no comparing, on one hand, an ostensive treatment of mathematics in which one can leave a lot unexplained, unsupported and without the ultimate objectivity of access to mechanical means of checking proofs with, one the other hand, an axiomatization that submits itself to the constraints and discipline required for that ultimate objectivity.

And sometimes people post questions about mathematical subjects that have bearing on philosophy, such as about infinities, incompleteness and computability. — TonesInDeepFreeze

You have this inverted. These are actually philosophical issues which have a bearing on mathematics. The way that a particular mathematician deals with these issues exposes their philosophical inclinations, or lack thereof.

Works both ways. It would be better if I had said that. The philosophy of mathematics needs for there to be mathematics to philosophize about and developments in mathematics do inform philosophy; and mathematics is liable to being philosophized about by philosophers.

I'll believe that he has anything when I see it. Especially, does he purport to offer an axiomatic system? I don't recall, but perhaps he rejects the axiomatic method. — TonesInDeepFreeze

Honestly, I don't know. I only brought up Wildberg because it is the first name that came to mind when it comes to non-standard mathematics. The last time I heard of him was years ago in some science board where discussions about him were common.
In hindsight I should have brought up intuitionistic mathematics but the connection to infinity is not as straightforward.

Yes, he is prominent, therefore natural to refer to him.

The point I was making is that concepts like infinity, incompleteness, and even computability, extend beyond mathematics. So, the mathematical approach is only one approach to such concepts. The philosophical approach, specifically the dialectical approach, is to consider the way that such concepts appear in all the different fields. The way that each field deals with the concepts demonstrates how that field fits, or does not fit, within a consistent whole philosophy. To say that such concepts are the domain of mathematics, therefore mathematicians ought to define them, is to make a statement not consistent with the world we live in.

concepts like infinity, incompleteness, and even computability, extend beyond mathematics. So, the mathematical approach is only one approach to such concepts. — Metaphysician Undercover

Of course.

Except incompleteness (in the sense of the incompleteness theorem).

To say that such concepts are the domain of mathematics, therefore mathematicians ought to define them — Metaphysician Undercover

I never said such a thing. Maybe other people have.

Except incompleteness (in the sense of the incompleteness theorem). — TonesInDeepFreeze

That's a specific, restricted definition of "incompleteness". The term is slightly different in physics for example. So this is an example of what I am talking about. Mathematics also uses a specific, restricted definition of "infinite", a meaning exclusive to mathematics, determined by the axioms. The mathematicians designing the axioms tailor the meaning of the term, to suit their purposes.

The philosophy of mathematics is a rich area. — TonesInDeepFreeze

Not only that, but when we think about almost anything we are quantifying. Does this action bring about more welfare than the other? Is a pantheist god encompassing of the whole universe? Do our beliefs have different percentages of certainty?

Mathematics spans our thoughts just like language, those are perhaps worthier of investigation than metaphysics — dependent on the former two.

That's a specific, restricted definition of "incompleteness". The term is slightly different in physics for example. So this is an example of what I am talking about. — Metaphysician Undercover

That is silly. Mathematicians don't claim that the mathematical sense of incompleteness trumps all other senses of that rubric in other fields. Just as a biologist talks about cells in an organism does not begrudge a penologist talking about cells in a prison and vice versa.

Mathematics also uses a specific, restricted definition of "infinite" — Metaphysician Undercover

Yes, and no fair minded mathematician would thereby begrudge people in other areas of thought from using other senses of the words.

I've said it over and over: Philosophers, scientists, theologists, et. al should be permitted to use terminology as suits them, and to express concepts as suits them. But when someone says the mathematics is wrong for using terminology in its way too, then that is quite unreasonable, and even worse when cranks claim that the mathematics is thereby wrong and worse, premises that claim on horribly misconstruing the mathematics and outright fabricating that makes it says things it actually does not say.

Words have different meanings in different fields of study. The point is that reasonable people allow that. What is unreasonable is when the crank has his own ways of using words and their related notions and then dictates that mathematicians are wrong for their specialized sense not conforming to the crank's own sense.

The fact that you can stack a property onto a substance to make an object does not mean that that object is instantiated in real life, — Lionino

Yes, for example as in "infinite number" where "infinite" is a property of "number".

Yes, for example as in "infinite number" where "infinite" is a property of "number". — RussellA

There is no such a thing as "infinite" number. See this is an illusion, and source of the confusion.
Infinity is a property of motion or action, nothing to do with numbers. Infinite number means that you keep adding (or counting whatever) what you have been adding (or counting) to the existing number until halted by break signal (as can be demonstrated in computer programming).

A set containing 3 numbers can be made infinite, when it is in the counting Loop 1, 2, 3, 1, 2, 3 .... ∞ Therefore a term "infinite number" is a misnomer. I bet my bottom dollar that you will never find a number which is infinite, because it doesn't exist. If it did exist, then it is not an infinite number.

I said math and philosophy have different way of doing things
— Corvus

They certainly do, which is why I’m wondering what a thread on mathematics is doing on a philosophy forum. — Joshs

Mathematics must have been believing in Philosophy's assistance in clarifying the tricky concepts. :snicker:

Infinity is a property of motion or action — Corvus

I agree. "Infinite" is a property attached to an object, such as "large house" or "infinite number".

As "large" doesn't exist as an object, "infinite" doesn't exist as an object.

As I wrote before: ""infinity" as an adjective means something along the lines "any known set of real numbers can be added to"".

I agree. "Infinite" is a property attached to an object, such as "large house" or "infinite number".

As "large" doesn't exist as an object, "infinite" doesn't exist as an object. — RussellA

If that is the case, then it seems barmy to talk about different size of the infinite sets.

There is no such a thing as "infinite" number. See this is an illusion, and source of the confusion.
Infinity is a property of motion or action, nothing to do with numbers. Infinite number means that you keep adding (or counting whatever) what you have been adding (or counting) to the existing number until halted by break signal (as can be demonstrated in computer programming).

A set containing 3 numbers can be made infinite, when it is in the counting Loop 1, 2, 3, 1, 2, 3 .... ∞ Therefore a term "infinite number" is a misnomer. I bet my bottom dollar that you will never find a number which is infinite, because it doesn't exist. If it did exist, then it is not an infinite number. — Corvus

Extended real number line

In mathematics, the extended real number system is obtained from the real number system ${R}$ by adding two infinity elements: +∞ and −∞, where the infinities are treated as actual numbers.

Extended real number line — Michael

"What is the number line to infinity?
For instance the number line has arrows at the end to represent this idea of having no bounds. The symbol used to represent infinity is ∞. On the left side of the number line is −∞ and on the right side of the number line is ∞ to describe the boundless behavior of the number line.11 Sept 2021" - Google

The philosophy of mathematics is a rich area.

(1) Unfortunately, cranks, who are ignorant and confused about the mathematics post incorrect criticisms of the mathematics, from either a crudely conceived philosophical or a crudely imagined mathematical perspective. That calls for correcting their misinformation about the mathematics itself. — TonesInDeepFreeze

I agree with your points. I just meant that this particular thread doesn’t seem to be getting beyond the correcting of wayward mathematical assumptions in order to deal with the philosophy. I might add that even at the level of securing consensus concerning ‘standard’ mathematics there is likely to be more disagreement than many might expect, perhaps due to the inseparability of philosophical presuppositions and mathematical principles.

"infinity" as an adjective — RussellA

'infinity' is not an adjective.

it seems barmy to talk about different size of the infinite sets — Corvus

No set has different sizes. But there are infinite sets that have sizes different from one another. That follows from the axioms.

One is free to reject those axioms, but then we may ask, "Then what axioms do you propose instead?"

One is free to reject the axiomatic method itself, but then we may ask, "Then by what means do you propose by which anyone can check with utter objectivity whether a purported mathematical proof is correct?"

One is free to respond that we check by comparing to reality or facts or something like that. But then we may point out, "People may reasonably disagree about such things as what is or is not the case in whatever exactly is meant by 'reality' or in what the facts are, so we cannot be assured utter objectivity that way."

One is free to say that we don't need utter objectivity, but then we may say, "Fair enough. So your desideratum is different from those using the axiomatic method."

except it can be used as an adjective, so stop being a dumb cunt who only seems to know "maf." — Vaskane

'infinity pool' for example. But I'm talking about the context here.

If that is the case, then it seems barmy to talk about different size of the infinite sets. — Corvus

There cannot be different sizes of infinite sets

As you say: "Infinity is a property of motion or action..............Infinite number means that you keep adding (or counting whatever) what you have been adding (or counting) to the existing number"

What does "infinite set" refer to?

It cannot refer to an object, an infinite set, as comprehending an infinite set is beyond the ability of a finite mind. It can only refer to the process of being able to add to an existing set.

In other words, "infinite set" refers to "a set that can be added to", where "that can be added to" qualifies the object "a set".

As a "set" is an object it can have a size, and therefore there can be different sizes of sets.

However, as the qualifier "that can be added to" is not an aspect of the size of the set, whilst the expression "different sizes of sets" is grammatical, the expression "different sizes of infinite sets" is ungrammatical.

What is infinity

On the one hand we have the concept of infinity within the symbol ∞, but on the other hand a finite mind cannot comprehend an object of infinite size. So what does our concept of infinity refer to?

As the Wikipedia article Extended Real Number Line notes, the infinities are "treated" as actual numbers, not that the infinities are actual numbers.

As our concept of infinity cannot refer to an object, as comprehending an infinite number of things is beyond the ability of a finite mind, it can only refer to the process of adding to an existing number of things until it is not possible to add any more, which can be comprehended by a finite mind.

IE, "infinity" refers to a process not an object.

if there are infinite whole numbers, and there are infinite decimals between 0 and 1, and there are infinite decimals between 0.1 and 0.12, and there are infinite decimals between 0.1111111 and 0.1111112 [...] — an-salad

Ordinarily, one would take that to be referencing mathematics, as have posters in this thread, not just me.

Which is the exact same boneheaded mistake you made on the other post about infinity. — Vaskane

Actually, in that instance I responded to the poster who has written:

As I wrote before: ""infinity" as an adjective means something along the lines "any known set of real numbers can be added to"". — RussellA

That is a mathematical context.

doesn't equate to math — Vaskane

I didn't say anything about 'equating to math'.

Rather, the context includes mathematics, as also other posters have taken it.

One can do whatever one wants with numbers. That doesn't vitiate that it is reasonable that I and others have commented on the mathematics.

Language is bendable, and often done so for artistic effect. — Vaskane

Hear hear.

'infinity' is not an adjective. — TonesInDeepFreeze

True, infinite is an adjective and infinity is a noun

But it can get complicated.
Music fills the infinite between the two souls - Rabindranath Tagore
Infinity pencil with eraser - Amazon

Is that why you felt the need to correct me when I said
To me it's just silly to argue the point of how big an infinity is when infinity is a concept considering continuity, not size.
— Vaskane
Because it was UNREASONABLE for me to not assume mathematics simply because numbers were involved? That's the real Dunning-Kruger here. — Vaskane

It’s the reverse.

In recent threads, the notion infinity has been raised with reference to mathematics - in the original posts and in replies. And I have not said that therefore the subject must be contained to mathematics. But the mathematical aspects should not be mangled, so I have commented to correct and articulate points about the mathematics that is being referenced. And still that does not even insist that one may not have an alternative mathematics; rather that if, for example, one claims to disprove that there are sets of higher infinite cardinalities, then the context in which there are higher infinite cardinalities should not be misconstrued or misrepresented. And ordinarily, the context of higher cardinalities would be classical set theory. Or for example, if the context begins with intervals on the real number line, then the ordinary context is classical real analysis and it should not be misconstrued or misrepresented. And, again, if one wants to discuss it in some other context, mathematical or otherwise, then that is fine, but that doesn’t preclude that we also discuss it in context of ordinary mathematics.

On the other hand, you post to say that you view it as “silly” to consider the notion of infinity regarding size when it is not regarding size but rather continuity. Then it is reasonable for one to say, “No, this discussion is not silly for talking about size, as indeed size is central to the ordinary context of sets in mathematics as indeed the definition leads right into size rather than continuity.”

Yes, continuity is an important topic related to the infinitude of certain sets. But the notion of infinitude applies even where the topic of continuity is not involved. So it is not silly to talk about the sizes of infinite sets.

So I am not insisting that any discussion be confined to mathematics. But I do say that such a discussion may include mathematics, especially when it starts out with reference to notions that are usually regarded as pertaining to mathematics, especially as the notion of greater infinitudes is ordinarily in context of classical mathematics and especially where the original poster mentions it in connection with Russell’s paradox. But, in stark contrast, meanwhile you are the one who is telling other people that it is silly to talk about the subject in terms of a certain aspect when you incorrectly claim it not concerned with that aspect.

Of course, but I'm saying that in context of sets in mathematics, 'infinity' as a noun invites misunderstanding, especially as it suggests there is an object named 'infinity' that has different sizes.