My impression is that we do not prove sqrt(2) is a number, but instead we assume it is a number by means of the completeness axiom. — keystone

(1) In set theory, there is no completeness axiom. Rather, we prove as a theorem that the system of reals is a complete ordered field.

(2) We assume axioms. (Or, in another view, we don't even assume them but rather merely investigate what their consequences are.) The theorems we derive from the axioms are not "assuming by means of". Okay, in a broad loose way of speaking, someone might say that the theorems are essentially just "assumptions" unpacked from the axioms. But that really muddies the matter terribly. Granted, everything we prove is, in a sense, "already in the axioms", but that obscures:

Yes, often we adopt axioms to prove the theorems we already know we want to have. But so what? That is, as they say, a feature not a bug of the axiomatic method.

And any alternative mathematics that is axiomatized is itself going to have that feature. So there's no credit in faulting set theory in particular for that.

And yes one might want for the axioms to be intuitively correct ("true") even if the theorems might be surprising. And with set theory, people's mileages vary. I find the axioms of set theory to be exemplary in sticking to only principles that are in concordance with the intuitive notion of 'sets'.

(3) So getting back to my earlier point: We prove that there is a unique positive real number r such r^2 = 2, and then we prove that r is not the ratio of two integers. Not the other way around as, if I recall, you suggested.

Now you are on a site where you get the chance to learn! — apokrisis

I glean a thing or two here and there. But posting is only a side hobby. I don't have ambitions for philosophy. Sometimes, though, I see things in discussions that I can't resist finding out about more, then I look them up or grab a book.

A scientific mindset means making the creative leap of forming a hypothesis, properly deducing the general constraints of that hypothesis, then inductively confirming the truth or otherwise of that hypothesis in terms of the observed particulars or practical consequences. — apokrisis

Of course.

If you are strong in one of the three aspects of reasoning, why would you be content with leaving the other two weak? — apokrisis

I don't know that I am so terribly relatively weak. And I'm never content. But there's so much else I also need to be doing. Forumcombing is itself a distraction that I probably shouldn't allow myself as it eats so terribly into my time needed for my main pursuits.

So when you hammer on posters, some are indeed just fools or cranks. But also, they might at least be discovering something about how to abductively form hypotheses, or inductively confirm their theories. — apokrisis

That's disastrously overgenerous. I've studied cranks for over 20 years, in forums from here to Timbuktu, and (speaking of inductive inference) one thing is clear: They never learn. They are dogmatic, irrational, intellectually dishonest, and narcissistically self-sure to the core. They persist in their favorite forum for years spewing confusion and disinformation.

reasonable person — apokrisis

I am reasonable in forums. Almost always, my first posts to a crank are without attitude. Merely a statement of the correction. Then, over time, the crank entrenches with even more dishonesty and often with passive put downs and things like that. Eventually, what becomes salient is the crank himself. And eventually I frankly say what is up with them. Believe me, I have so many times practiced restraint hoping that a crank might, miraculously, come to reason. Never happens. And that is not a function of my style. Hundreds and hundreds of other posters in many forums have tried with cranks, and they always fail to get anywhere with it. Not even a millimeter. Always.*

In this thread, you're seeing only recent interchanges. But there is a context with this poster going back over a year(?) or two years(?). I don't know whether you've read much of those threads. If not, then I would understand that you think my approach is arbitrarily harsh. (By the way, this poster is not as overtly dogmatic as usual cranks. Indeed his skill is to deflect by feigning that he is considering the corrections, which I perceive to be disingenuous.)

/

Your comments in your above post are well taken though. Even if I am in countering mode in this post, probably soon later I'll reflect more and benefit more from your point of view.

But now that you've given me advice, I will return the favor:

Your annoyance with me should not permit you to read into my plain words things that are not in them, not even plausibly, not to willfully misconstrue what I say in the worst way, and not to strawman me over and over as you did yesterday. That is beneath you.

/

* Except that one fellow I wrote about recently. The sole counterexample.

Have you noticed how much you assert the negative so as to avoid having to support the positive? — apokrisis

I don't usually support the affirmative because I'm humble enough to admit that I don't have the vision, education, confidence and constancy to arrive at a fixed philosophy. If I were a philosopher, I'd be nowhere in that way. But I'm not a philosopher.

What is life without some form of ontological commitment? — apokrisis

That is a great line.

You yourself have said you have no philosophy to defend on this forum, just a self-appointed need to police it for its mathematical thoughtcrimes and disinformation campaigns. — apokrisis

Having no philosophy is not a disqualifier. Posting is not paintball where you can't participate unless you you are on one of the teams. Not having a philosophy doesn't entail that one doesn't have meaningful things to say. And I find it refreshing when a person doesn't have a philosophical ax to grind.

There are no thought crimes. On the contrary for me. As I don't hew to a particular philosophy, I don't have strong oppositions to other philosophies; and I relish that there are so many tantalizingly different philosophies of mathematics and formal systems; and I believe that freedom to imagine is to be cherished. Spewing of disinformation though is abundant. Moreover, much of my posting is not just making corrections. Your categorical reduction is false. And beneath you, just as your multiple strawmans earlier

I've seen that cartoon, and it is funny.

Of course not. It's not even a formal claim.
— TonesInDeepFreeze

I was teasing. — apokrisis

Okay, you were joking with the 'formal' part. Maybe because you perceive me as asking posters to back up with formal proofs? Or you think I can be charactured that way? I don't know. Anyway, of course I don't ask people to provide formal proofs of informal assertions.

If you gatecrash a comment, you could at least have the courtesy to set out your reasons for your assertions. — apokrisis

Oh come on. I didn't "gatecrash" anything. You posted essentially a one-liner on the subject, itself not an argument. That's fine. And it should be allowed that one may reply in kind. And even if a poster replies tersely to a longer argument, that's not "gatecrashing" or necessarily even rude or whatever.

So you reveal yourself as a pragmatist. — apokrisis

I don't have a philosophy of mathematics; and not one that could be called anything, including 'pragmatism'.

Infinity is a useful idea as far as it goes in the real world of doing things — apokrisis

I am sympathetic to that idea. But I don't personally stake my own understanding of infinitistic mathematics primarily to it.

Truncating pi is practical. — apokrisis

Of course no one expects engineers to write an infinite sequence of digits.

Definitions require a semantic thesis — Cartesian trigger-puppets

Except in formal mathematics, definitions are purely syntactical.

necessary and sufficient conditions — Cartesian trigger-puppets

Indeed.

Of course not. It's not a formal claim.

Anyway, I think the burden of argument is on the side saying that infinitistic mathematics does require a platonist commitment. There are powerful arguments for that side. I very much respect that. But there are two different questions:

(1) Can one have a cogent philosophy of mathematics with infinite sets but that is not platonist? I don't have a firm opinion on that.

(2) Can one work in ZFC without committing oneself to platonism? That's more an empirical question. Such mathematicians as Abraham Robinson do*. And Robinson's own explanation might be pretty good. (but If I recall, I found it not to be entirely glitch-free). For myself, even though I am not a mathematician, I happily study ZFC without having the platonist commitment that the abstract objects of mathematics exist independent of mind.

* A fair number of set theorists do, as does Robinson, say that the notion itself (not even just the existence) of infinite sets is literally nonsense, yet they work in ZFC and recognize its fruitfulness.

Whether by that name or not, the idea goes at least as far back as Aristotle.

I don't believe that infinitistic mathematics requires a platonist commitment.

NASA telling me what's the largest number I can use is like the Department Of Agriculture telling me how many taste buds I can have.

If we agree that there's a largest number we'd ever need to use, then still, what's the harm in having all the rest of the larger numbers in the attic, just in case we ever feel like looking at some larger ones, even if only to play around with them?

I wonder whether the debate lines up with certain kinds of personalities.

One personality type just cannot stand that there are infinite sets. It terribly rankles them that we would allow such a sweeping abstraction. It goes against their way of looking at mathematics as expression of our experiences as a stream of immediate and concrete based perceptions.

The other personality type just cannot stand limiting mathematics to particulars. It bugs them that we would cut off the numbers at some arbitrary point (or a point merely estimated by current cosmological theories) in the succession of numbers. It goes against their way of looking at mathematics as being of greatest abstract generality.

For me, ultrafinitism is ugly that way. I'd rather study mathematics that is not embroiled with a bunch of messy physical stuff like how small an atom is and what is it's rounded-off length, given in a bunch of base ten digits. It's ugly to me to say, "The greatest number is 492^(3327989025)" or whatever. It's so ... choppy.

OP — Real Gone Cat

He's "The Crank With The Friendly Face".

If the infinitistic systemization for mathematics are more powerful, beautiful, and simple — keystone

I don't know definitively that that is the case, but it seems to me to be so.

I do wonder whether our infinitistic systemization can simply be reinterpreted from being based on actual infinity to being based on potential infinity. — keystone

There are systems - such as intuitionistic* ones and others - that are said to embody use of potential infinity. But, personally, I don't know of one that can define within the system 'is potentially infinite', while set theory does define 'is infinite'. So, at least as far as I can tell, saying 'potentially infinite' is not yet, at least, a formalized notion but rather a manner of speaking.

* Though intuitionistic set theory does have an axiom of infinity. The problem though is that we can't directly compare statements in intuitionistic theories with those in classical theories, because the semantics are so radically different.

https://thephilosophyforum.com/discussion/comment/737931

That message is retracted by me.

It's a conceptual issue. He seems to think that, because mathematics is infinitistic, it has a thing that is called 'infinity'. As if the leminscate stands for that thing like the golden arches stand for a hamburger place.

We haven't really got started here — bongo fury

Got started with what? I don't know where you're going with this.

Don't want more! — bongo fury

I don't know what it means to not "want" more than monadic logic. You can't do much mathematics with just monadic logic.

You can reduce n-ary predicate symbols (n>2) to 2-ary predicate symbols. But you have to have at least 2-ary if you want more than the monadic predicate calculus.

He would eliminate n-ary relations (n>1) from the method of models?

But it's only an example of a principle. Presumably, the principle applies to n-place for any natural number n, as indeed it does in model theory.

I mean one-place? — bongo fury

Predicate symbols can be n-place for any natural number n.

Relations can be n-place for any natural number n.

Of course, in the case of the 'snow is white' example, 'is white' is 1-place.

unary predicate — bongo fury

n-ary for any natural number n

unary relations — bongo fury

n-ary for any natural number n

No, I'm just unpacking what's already there.

'Snow is white' is true if and only if snow is white.

I merely unpacked, pedantically really, the right side.

Nothing is missing.

Whatever is meant by 'predicate' and 'property' there, you asked about model theory.

Predicate symbols map to relations on the domain.

So, yes, if I were to render the T-sentence more formally, not so much as an example of a philsophical principle but as a recap of a formal model theoretic formulation, then I wouldn't need to mention 'property'.

Means "Whatever, dude", eh?

I don't know what that means. I'd have to read the rest of the context.

Talk of properties when glossing use of a logical predicate is eliminable? — bongo fury

I don't know what that means.

I actually don’t think definitions need be a requisite, though they are useful insofar as they capture the standard meaning of a term. I try to avoid committing to a definition since it requires a semantic thesis or a theory of public meaning. — Cartesian trigger-puppets

Yet you asked me to comment on your post that includes:

Like, introducing terms without providing a definition or conveying their meaning. I believe when terms are introduced without clear meaning they form a roadblock preventing the conversation to progress. I can’t grant an argument if a premise contains a term that I don’t understand. I can’t even grant that the statement is propositional. — Cartesian trigger-puppets

[emphasis added]

What is more, most of the time no effort is made to define terms or to convey our sense of them. Vagueness and ambiguity often go unchecked, relying instead on the assumption that our interlocutor shares our interpretations. — Cartesian trigger-puppets

It seems necessary to be in agreement on all terms before arguing one way or another on an issue. — Cartesian trigger-puppets

[emphasis added]

If asked whether or not I believe there is a God,I require you provide a definition. — Cartesian trigger-puppets

[emphasis added]

So I took the time to write a post about that. Then you say the opposite, that definitions are not required. .
So I don't understand you.

Does that mean 0 and ∞ aren't numbers — Agent Smith

0 is a number.

I don't understand why you keep skipping my point that using the leminscate as if it stands for an object makes no sense the way you do.

Unless you mean the point of infinity in the extended reals, though I don't think you are even that specific. But even as a member of the extended reals, the point of infinity is not necessarily itself an infinite set. It can be any object that is not a real number. Then we define an ordering using the standard ordering on the reals but setting the point of infinity as greater than all reals.

I don't see any improvement in your revision.

Correct that here 'is' is not for equality but for indicating a predicate.

It seems that we both agree that the T-sentence is missing a necessary condition on the RHS of the biconditional. — RussellA

I didn't say anything like that.

My point is that your description is not an accurate or even reasonable simplification of how set theory proves that there is a complete ordered field and a total ordering of its carrier set. (The carrier set is the set of real numbers and the total ordering is the standard less-than relation on the set of real numbers.)
— TonesInDeepFreeze

I'm not sure what you're referring to. — keystone

I'm referring to the fact that set theory proves there exists a complete ordered field and a total ordering of its carrier set. And then I highlighted that the carrier set is the set of real number and the total ordering is the standard less-than relation on the set of real numbers. I don't know how to put it more simply. Other than perhaps to add that that set together with the ordering is the continuum. I don't recall whether I mentioned it earlier in this thread, though it is utterly basic to discussion of the continuum.

consider an alternate interpretation: in the first step 1 there is a dislodged guest, in the second step there is a dislodged guest, in the third step there is a dislodged guest [etc.] — keystone

The imaginary hotelier can do that also.

What was the 19th century analysis resolution to Zeno's paradox? — keystone

Infinite summation: convergence of an infinite sequence to a limit.

Are you not disquieted that a subset of rooms is equinumerous to the full set of rooms? — keystone

I don't conceive of an infinite set of physical rooms.

As to sets, I already mentioned that I am not bothered that the squares (a proper subset of the naturals) is 1-1 with the naturals.

/

Dartboard paradox. I'm rusty in probability theory. I'd have to go back the books to refresh myself.

Thompson's Lamp and Zeno's paradox. I already addressed those. I don't have more to say about them.

I don't think calculus needs actual infinity to work. — keystone

It does in its common form.

But there are non-infinitistic systems too. I know a little about them, but not enough to say how well they perform.

I suspect (with no evidence to provide) that ZFC doesn't need actual infinity to work either. — keystone

It wouldn't be ZFC then.

We could delete the axiom of infinity, but then we don't get analysis.

Or we could negate the axiom of infinity, but then we don't get analysis but instead a theory inter-interpretable with first order Peano arithmetic.

I have no problem forming a 1-1 relationship between n and the n^2. I just don't think there's an actual set that contains all n (and similarly all n^2). In other words, my qualms are not with the math, they are with the philosophy. — keystone

The axiom of infinity and the results from it are mathematics. If you want a mathematics without the theorems that we read as "there exist infinite sets" and "there exists a set whose members are all and only the natural numbers", etc., then that is not just philosophical but also mathematical

can you imagine the actual endless hotel as a whole? — keystone

Not as a physical object. On the other hand, I know so little of cosmology that I don't know how to dispel my bafflement that the universe could be finite or my bafflement that the universe could be infinite.

whether you can imagine a set of all natural numbers. — keystone

Yes, I do conceive it clearly. I conceive the abstract property of being a natural number. Then I conceive the set of all and only the natural numbers to be the set that pertains to all and only those things having said property. I know what the property is; so it takes only a shift to conceive of a set that corresponds to the property. (Of course, that can't be applied always, lest we get contradiction from unrestricted comprehension. In that case, my naive intuition just needs to adjust to accept restriction.)

That is not at all an argument that there exists such a set. It's only a description of my own intuition.

/

Or another view: If there is any mathematical reasoning that can be considered safe, then it's manipulation of finite strings of objects or symbols (whether concrete sticks on the ground, or abstract tokens). In that regard, I can see the formal derivation in Z set theory of the theorem that we read off as "there exists a set whose members are all and only the natural numbers", though the actual formal theorem doesn't have English words like that. Then, in some a worst case scenario, some crisis where my ability to conceive abstractly is terribly diminished, if I really had to, I could fall back to extreme formalism by taking the theorem to be utterly uninterpreted, but a formula nevertheless to be used in mathematical reckoning.

You asked, here's my answer:

introducing terms without providing a definition or conveying their meaning — Cartesian trigger-puppets

Of course, definitions are crucial.

But how demanding we should be must depend on context.

Since, for example, this thread is about a subject of mathematical logic, different contexts range from just philosophy about mathematics, to a blend of philosophy about mathematics and mathematics itself, to just the mathematics itself. Then there are degrees of formality, from very liberal informality to rigorous formality.

Forum-wide, usually mathematics is not the subject, but still there may be degrees of formality, from liberally speculative philosophy to more rigorous technical aspects of philosophy.

So what context do you have in mind regarding definitions?

Most informally, we know that of course we can't be bogged down by defining every word of English we use, and even if we could, we'd encounter circularity (English is not a formal language in which there are undefined primitives and then a sequence of definitions.)

For philosophy, I would agree that there should be an expectation that a poster should provide definitions for special philosophical terminology where there is a reasonable need to know the specific definitions. But there's still a limit - since we are not posting entire treatises, we don't have the time for everything.

For mathematics, in principle, every mathematical statement should be formalizable (this is called 'Hilbert's thesis'). But that's only in principle; in actual discourse, we have to be allowed informality, as long as we know, in the background of our reasoning, that could formalize it all if we had all the time and patience to do it (I nickname this 'Bourbaki's thesis'). So, yes, mathematicians, at least in principle, must be able to define all terminology down to the primitives. But, again, in a forum we don't have time to define everything down to, say, the sole primitive ('e' for epsilon, i.e. "member of") of set theory.

On the other hand, there are cranks. Cranks often talk as if they are making mathematical statements (not just philosophical statements about mathematics) as they are using mathematical terminology. But their usage is incorrect, usually ludicrously so. And they have no concept even of what a mathematical definition is, or what the specific definitions are of the terminology they use. For me, as far as definitions, that is the worst of a forum such as this; and it's not just this forum, but all over the Internet.

It seems necessary to be in agreement on all terms before arguing one way or another on an issue. Otherwise, how would you know whether or not you agree without a doxastic view of it? — Cartesian trigger-puppets

I know what 'doxastic' means, but I don't know what you mean by "a doxastic view of it" in that context.

therefore cannot grant any statements made by moral realists if they introduce normative terms on a stance-independent construal. — Cartesian trigger-puppets

I know what 'moral realism' and 'normative' mean, and maybe I have a bit of a sense of what 'stance-independence' means, but I don't know what is meant by 'introduce normative terms on a stance-independent construal'.

/

"It is not our business to set up prohibitions, but rather to arrive at conventions." - Rudolph Carnap

Adding to my response about the particular paradoxes. Even if we granted that they indicate flaws in the concept of infinitude, then that is a concept of infinitude extended beyond set theory into imaginary states of affairs for which set theory should not take blame. Those paradoxes don't impugn set theory itself.

In QM we have come to accept a certain level of uncertainty. Why can't we do the same in math? — keystone

I wouldn't argue that we can't. I suppose people already have made logic systems with values such as 'uncertain' that can be be applied to a different mathematics. And I can imagine that certain scientific enquires might be better served by such systems.

But that doesn't erase the rewards meanwhile of classical mathematics.

Consider the proof that sqrt(2) is an irrational number. I would argue that the proof only demonstrates that sqrt(2) is not a rational number and that something beyond rational numbers must exist on the real line. It does not prove that sqrt(2) IS a number. — keystone

We do prove "sqrt(2) is a [real] number".

More exactly:

We prove that there is a unique positive real number r such r^2 = 2, and then we prove that r is not the ratio of two integers.

Naturally.