I'd be happy if we could focus on your ideas. You clarified what you meant by an identity condition, and I pointed out that it was structural, and seemed to undermine your own thesis. You didn't engage and didn't agree or disagree. I'd find it helpful if we could focus in.

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But ok. An identity condition is a statement that uniquely characterizes a mathematical object. Like "the unique Dedekind-complete linearly ordered set" uniquely characterizes the real numbers.

Ok I'll accept that definition. But note that it's a structural condition, not a Platonic one. — fishfry

You said you wanted to focus on my ideas. Okay, then. Let me explain my motivation for the thread. Please read the following carefully. If I lose you, say where and why. I promise, it's relevant.

So, I am really really interested in whether or not mathematical objects exist in a mind-independent way. Would there be numbers even if we weren't here? I want to say yes. I think that numbers would be here even if we weren't here. I think that brontosaurus had 4 legs long before we counted them. I can't fathom what it would mean to say that it didn't. So that makes me a Platonist, because I don't think that numbers depend on our minds. At least, not in that way.

But there's a problem with Platonism. If I say that something exists, I need to identify it. If I say, "the Blarb exists", then I need to say what the Blarb is. I need an identity condition that picks out the Blarb and only the Blarb.

This is a counter to Platonism, because it confounds the motivation. Would horses have 4 legs if nobody counted them? I, the Platonist, say yes. But if you ask the me for an identity condition, I'm in trouble. See, if say that the legs of a horse are the set {0, 1, 2, 3}, then I've said that the number of horse legs is the natural number four. But is that even true? What if the number of horse legs is the set of all rationals smaller than 4, i.e. the real number 4? How would we know which one it is?

So it comes out to this:

1. To say that a certain thing exists, you need an identity condition for it.
2. You can't always get those identity conditions for mathematical objects.
3. Therefore, we can't say that mathematical objects exist.

Uh-oh!

My hare-brained solution to that was "maybe there's only one mathematical entity". Math is just a single thing. Because then math can be Platonically real without needing identity conditions. You just say that different mathematical objects are what happens when you analyze that one single object in different ways.

I know that sounds weird. Maybe an analogy will help.

So, for example, you know the duckrabbit?



It looks like a duck if you look at it one way. It also looks like a rabbit, though not both at once. However, you can't just see it however you want; it's not a duckrabbitgorilla. It's not a duckrabbithouse. It's not a duckrabbithitler. So there are two valid ways of seeing it, but only one can be used at a time. And some ways of seeing it are invalid.

What if math is kind of like that? There's a single mathematical reality, but it looks different depending on how you analyze it. So if you bring a certain set-theoretic framework, math gives you real numbers, and if you bring a different one, math gives you rational numbers. But the rationals and reals aren't separate, self-identical objects. They're just representations, ways of representing one underlying reality.

Basically, you don't need identity conditions for a representation, because representations don't need to be self-identical in that sense. We can answer the question, "Did horses have 4 legs before anyone counted them?" with "yes" (which is what I wanted). That's because the underlying mathematical reality is Platonic and never changes. But then we have this question: "Was that the rational number 4, or the real number 4, or what?" That's the question that initially flummoxed us. But if there's only one mathematical object, that question is no longer sensible. Represent the number of its legs however you like. You can represent it as rational 4, or real 4, or natural 4. The underlying reality is the same.

I'll come back to that in a moment. Now for structuralism.

The SEP article on structuralism tells me that there's a methodological kind of structuralism, which is basically just a style of doing math. Then there's a metaphysical structuralism, which is an ontology. The former is a style of mathematical praxis and the latter is a philosophical position.

You seem to waver between methodological and metaphysical structuralism, and it confuses me. On the one hand, you take a stance like, "I'm no philosopher. I just find this to be an interesting way of doing math". On the other hand, you do seem interested in the philosophical implications of structuralism, e.g. when you said that modern mathematics tells us new things about the notion of identity. And you're on a philosophy forum discussing it, rather than a math forum.

So let's put the discussion on these questions:

1. Does methodological structuralism imply metaphysical structuralism? Or at least, enable it?
2. Is metaphysical structuralism compatible with Platonism? If so, is it compatible with my idea that there is, in a sense, only one mathematical object?

P.S. the SEP article has it,

Along Benacerraf’s lines, mathematical objects are viewed as “positions” in corresponding patterns; and this is meant to allow us to take mathematical statements “at face value”, in the sense of seeing ‘0’, ‘1’, ‘2’, etc. as singular terms referring to such positions.

...

By focusing more on metaphysical questions and leaving behind hesitations about structures as objects, Shapiro’s goal is to defend a more thoroughly realist version of mathematical structuralism, thus rejecting nominalist and constructivist views (more on that below).

[Shapiro] distinguishes two perspectives on [positions in structures]. According to the first, the positions at issue are treated as “offices”, i.e., as slots that can be filled or occupied by various objects (e.g., the position “0” in the natural number structure is occupied by ∅ in the series of finite von Neumann ordinals).

We ask, "Is the number of horse legs the natural number 4 or the rational number 4?". Well, for Shapiro, the number of horse legs occupies a position. That same position is filled by {1, 2, 3} in the naturals and { x ∈ Q : x < 4 } in the reals. So the answer is, "The number of horse legs is both of those".

The identity condition, then, is "That unique office occupied by {1, 2 3} in the natural numbers". But there are plenty of other identity conditions that pick out the same object: "That unique office occupied by { x ∈ Q : x < 4 } in the reals" picks it out as well. At this point, the numbers themselves are identity conditions for offices!

Maybe I don't need to reduce math to one object after all...

I hope I didn't wander off too far. And I hope that this is, at least, interesting.

Not at all. Just on the one hand, feeling totally inadequate to discuss the philosophical side of this ... Quine and Spinoza for example. — fishfry

Look, I'm not educated. At all. I have neither a degree nor a high school diploma. I have what Americans call a "GED", and that's it. My job is to sit at home in my underwear and write code. If I tried to show up at an academic conference, they'd kick me out just based on smell. I can make computers go bleep bloop, though!

Although I am wearing this shirt right now:



If that counts for anything.

As modern math shows us, concepts like identical and same and "can't tell them apart" depend on the context. They are not absolute. — fishfry

Let me make sure I understand you. You're saying this:

"There's this thing called identity. Math tells us that identity is contextual. So we should also say that the identity of people is contextual, since math tells us about identity. For example, Clark Kent and Superman are contextual identities."

Is that right?

Easily falsified in-universe. Lois Lane has no idea. It's the glasses, apparently. A little willing suspension of disbelief on that point. How can she possibly not see that they're the same guy? But she can't tell. — fishfry

I don't understand. "Lois is unaware of the fact that Clark Kent is Superman. Therefore, Clark Kent is not Superman". That's not what you're saying, is it? If I put on clown makeup and then kill someone, I'm still guilty of a crime, even after I wash off the makeup.

Reading over your other points, I have a hunch about your position. You have used the term "representation" a few times. And you seem to think that, if A and B both represent C, then A and B are identical insofar as they represent C. You seem to identify how something is represented with what it is.

Two things.

1. Do Clark Kent and Superman represent one thing? If so, what is that one thing?
2. Would horses have four legs if nobody counted them? If so, which four would it be? The natural number 4? The rational number 4? The real number 4?

You know, you might have missed the mark in tagging me for your resurrection of this thread. I made an offhand remark about Hilbert's famous beer mug quote, and I may have mentioned the turn towards structuralism of latter 20th century math. But I am not proclaiming myself the Lord High Defender of these ideas, nor am I particularly knowledgeable about these matters, nor do I even have any particularly strong opinions about them. You don't like structuralism, that's ok by me. You are a Platonist, so was Gödel, and so are most working mathematicians.

...

Why me? All I did was quote-check the beer mug. I might as well have worn Hilbert's famous hat.

...

I think you are misconstruing the hell out of mathematical structuralism; flailing at a straw man; and expecting me to engage or to be an enthusiastic advocate of something or other.

...

Ok. Fine. Whatever. Tell me why you think I am the right target for this conversation. I can't hold up my end. I'm defenseless. — fishfry

I just wanted to bother you 'cause I thought you'd be fun to talk to. I figured you'd just ignore me if you weren't into it. I didn't think you'd start screaming "OH GOD WHY DID YOU CHOOSE ME I WANT NO PART IN THIS MADNESS" like I jumped out of the bushes and throttled you or something.

It's okay. You can just ignore me if you want. But I think you'll have fun.

You are using "numerically distinct" in some kind of crazy way. 5 and 6 are numerically distinct. Past that I have no idea what you mean.

Okay, so some philosophers use the terms "qualitative identity" and "quantitative identity" (or numerical identity). Qualitative identity is where you can't tell them apart. Quantitative is where they're the exact same thing like Superman and Clark Kent. "Quantitative" is kinda misleading, there, so I'll drop these terms.

Instead, I'll use "indiscernible" and "identical". Two things are indiscernible if you cannot tell them apart. They're identical if they are one and the same thing. So two "identical" twins can be indiscernible. But they're not identical, strictly speaking, cause they're two different people. Do I still sound like raving lunatic?

You've used that phrase [identity condition] a few times, and I should admit that I have no idea what it means.

What is an identity condition? Can you explain what you mean? Can you give me an example or two, say with regard to some familiar mathematical examples like the set of natural numbers, or the cyclic group of order 4, or the Riemann sphere, or any other example you care to name.

I just have no idea what you mean by an "identity condition," and how this idea stands in opposition to the idea of structuralism.

An identity condition is a statement that only applies to one thing. The statement "a prime number between 2 and 5" is an identity condition, because it applies only to one mathematical object, which is 3.

That sounds very simple, right? But it gets complicated. See below.

(Apropos of nothing, have you ever read Augustin Rayo's The Construction of Logical Space? You might enjoy it.)

Do you mean perhaps that the two representations of the cyclic group of order 4 I gave earlier should be properly regarded as two separate things? But of course they are. They also happen to have the same group-theoretic structure, so they represent the same group. It's all about the context in which we use the word "same."

I think we agree that the set {0, 1, 2, 3} is not identical to {1, i, -1, -i}. And we agree that both of those are instances of the cyclic group with order 4, if they have the right operations defined on them.

Now, here are some questions:

1. Is "{0, 1, 2, 3} under addition mod 4" identical to the cyclic group with order 4?
2. Is "{0, 1, 2, 3} under addition mod 4" an instance of said group?

Nobody is arguing with this. Clark Kent and Superman are the same and they are different. It all depends on the context.

They're only "different" in a figurative sense. I think Clark Kent is identical to Superman at all times. They're both one dude. "Clark Kent" shares a referent with "Superman".

I don't think that identity is dependent on context like that. There's nothing you can do to make something stop being identical to itself.

Have I ever done so?

Nah, man. It was just a tangential aside. It's okay. I'm not trying to throttle you. I'm just talking to you. I'm just writing this on my laptop. I'm just under your floorboards. Ominously cracking my long, gnarled fingers. And salivating.

So are you arguing against Platonism? Or constructivism? — fishfry

I want to start with this: I'm defending Platonism, bro. The objection based on identity conditions really bothers me, so I want to say find some way to get around it and still be a Platonist.

I think it's analogous to Quine's observation about modality:

Take, for instance, the possible fat man in the doorway; and again, the possible bald man in the doorway. Are they the same possible man, or two possible men? How de we decide? How many possible men there are in that doorway? Are there more possible thin ones than fat ones? How many of them are alike? — WVO Quine

I think this is a deep problem. Deep problems manifest in multiple places – that's how you know they're deep. The fact that something similar happens with modality as with math ought to tip us off that this is not a surface-level concern.

So how does it manifest for math? Like this:

Antiplatonist: "We cannot say that mathematical objects exist in a mind-independent sense. This is because there are no clear identity conditions for them."
Platonist: "Identity is structural. Two mathematical objects are numerically identical iff they are structurally identical."
Antiplatonist: "Here are two objects. They are numerically distinct and structurally identical. Therefore, numerical identity is not structural identity. Therefore, you have not answered my objection."

My solution to this dilemma, per the first post, is simple: no two mathematical objects are numerically distinct. There's only one, Math. Any two apparently distinct ones are just logically valid observations of the same object. A duck and a rabbit, if you like Wittgenstein. Attributes, if you like Spinoza. Emanations, if you like Plotinus.

(I like Spinoza. )

This is the famous two spheres argument against the identity of indiscernibles. — fishfry

I agree, with a caveat.

Black's argument is this: "Two things can be indiscernible, in every single respect, and still numerically distinct". He's saying that two things can be indiscernible in every possible way and still be two things.

My argument is a lot more constrained. I'm saying, "Two things can be indiscernible, in terms of mathematical structure, and still be numerically distinct". And this is proved, IMO, by the graph example.

I don't think anyone is saying there can't be two of something in math. There's only one set of real numbers, defined axiomatically as the unique (up to isomorphism) Dedekind-complete ordered field.

Yet we have no trouble taking two "copies" of the real numbers, placing them at right angles to each other, and calling it the Cartesian plane. Or n copies to make Cartesian n-space.

For that matter we say that a line is determined by two points. But one point is exactly like any other, except for their location, yet nobody thinks there's only one point. There are lots of points. — fishfry

I mean, you're not wrong. Everybody agrees that there can be two of something in math. As you note, points differ by their location. Location is structural. Therefore, points are structurally distinct. And I agree with you on that. I'm not saying, "Ha! Here are two different points. Suck it, structuralists!"

What I am saying is, "Here are two objects. They are numerically distinct, but structurally identical. Therefore, structural identity is not numerical identity".

If it is possible to individuate two objects without appeal to structure, then we can find two such objects that are structurally identical. Which sinks structuralism. So the burden of the structuralist is to do math in such a way that you must appeal to structure in order to individuate two objects.

You did this for points, I think. If points are individuated by location, and location is structural, then numerical identity for points is structural identity. It works.

But this doesn't apply to the graph-theoretic example. You can simply say, "There exists a graph with two vertices and no edges". If any graphs exist, this one does. And now we've individuated two vertices without appeal to structure. The structuralist must show that one vertex has a structural property that the other doesn't. By definition, this cannot be done.

You can't make structure itself a primitive notion, by the way. That defeats the point of structuralism. The whole point was to abstract away from particulars and deny intrinsic properties. If you make structure primitive, then you've basically made it an intrinsic property.

One last point: when I say, "Therefore, the two vertices are numerically distinct", I'm speaking ex hypothesi. In fact, I do not think the vertices are numerically distinct. I don't think any two mathematical objects are numerically distinct. There's only one. Any talk of distinction is just a way of talking about how that one object relates to itself.

I'm not sure how to cash that out, exactly. Perhaps I should say: math relates to itself in every logically valid way, for every coherent system of logic. Therefore, every coherent logical system expresses math.

As David Hilbert said, "One must be able to say at all times--instead of points, straight lines, and planes--tables, chairs, and beer mugs."

That quote answers your concern that we can't identify particular things. We don't care about identifying particular things. Rather, we only care about structural and logical relationships among particular things; and those relations are independent of the things themselves. — fishfry

Four years later, I had a whim to come back here. I just wanted to explain why this is wrong.

Take graph theory. I show you a graph with two vertices and no edges. By hypothesis, the two vertices are two different things. Those two vertices, however, are structurally indiscernible. Which makes them the same vertex, according to structuralism. Contradiction.

Therefore, either mathematical objects are not identified by their structure, or the stated graph can't be defined. The latter is false. So mathematical objects are not identified by their structure.

That's the fundamental problem with structuralism. You cannot escape the need for identity conditions by focusing solely on relations rather than particulars, because relations are particulars. "I don't have to count objects, because I go by kinds" – and if I ask how many kinds there are...?

It seems to me that the only way to justify self-defense is to either (1) abandon stipulation #1 or (2) reject #3. — Bob Ross

Maybe you could weasel out like this: "If someone is doing something bad, and I stop them, then I'm not really harming them. It's better for someone to be killed in self-defense than to successfully become a burglar."

Some weird virtue-ethical argument like that.

You don't defeat a charge of circularity by going in a circle again.

I agree with you, but a die-hard materialist would consider this circular reasoning.

I don't remember ever seeing idea being thought of as something that exists mind-independent — Lionino

Have you... heard of Plato?

What definition? Above I reply to your question. — jkop

The definition you gave. Go back and re-read. I'm not recapping what we just said. Re-read.

we recognize these words that we type by some (but not all) features that they possess — jkop

And now you're introducing "features", which are another abstraction introduced to explain the first. This is exactly what I said you'd do.

Any actual set that exemplifies the description. — jkop

Circular definition.

Again, it's not gonna work.

Descriptions are actual material objects. — jkop

I get what you're trying to do – reduce the abstract to the concrete. It ain't gonna work. Any reduction that successfully reduces abstracta fails to explain them, and any reduction that sufficiently explains them fails to reduce them.

Case in point: can a description be spoken more than once? If we both use the same description, which set of physical events is the description? You'll find that your response to that question either introduces some new abstraction to get rid of ideas, or fails to explain multiple realizability of abstractions.

Try reading what I wrote.

I prefer masturbating over semantics so I can score a point and ignore what's actually under discussion.

I don't really care what the current jargon is. ¯\_(ツ)_/¯

Yep, it does create a cycle of violence. But the cycle continues precisely because both parties are left with no choice. You can't step in and say, "Break the cycle by allowing the other guy to hit you and get away with it!" That just ain't gonna work.

First vs third world = difference in level of technology

Israel is a first-world country in a third-world area.

Ideally, the situation would be for Israel to have advanced weapons and use them to hold off attacks from their technologically backward neighbors. That's what the Iron Dome is meant to do. "Launch all the rockets you want, and we'll just sit here and shoot them down." Hamas says, "Alright, then. Rockets don't work? We'll just charge in and slaughter as many of you as we can."

What is Israel supposed to do?

Yes, people will die. Children will be orphaned. Children will die. Many people will be maimed, crippled, impoverished, and immiserated. Homes and shops will burn. It's going to be horrible, and I think the Israelis know this. But still: what are they supposed to do? Just sit there and take it? Thus encouraging a second strike? No. They have to hit back, and it has to hurt.

What makes this difficult is the fact that Israel is overwhelmingly more powerful than Hamas. This makes the situation difficult for anyone with a conscience. It makes the situation especially difficult for Western leftists, who see everything through a prism of oppressed/oppressor logic. If the first question you ask is always, "Which person is the cop and which person is George Floyd?" then your moral prism will be skewed.

I think that P(psyop) > P(aliens). Between those two flavors of crazy, I pick the less crazy one.

Oh, they're doing the alien thing again. What am I being distracted from?

Ukraine must be losing.

Okay, so when I contemplate an object, I don't need identity conditions for it because my intellectual faculty perceives that object in its naked Being.

But what if somebody else comes up to me and says, "I, too, have contemplated mathematical objects, and I apprehend that there is no form of specific polygons, only a form of The Polygon, and all shapes, like squares and triangles, participate in it."

Then another person says, "I have also contemplated mathematical objects, and the opposite is true: there are separate forms for scalene and right triangles, and I have perceived them both."

Without positing some kind of identity conditions for abstracta, how do I even begin arguing with those two?

The question is, how do you know which form you are contemplating?

I don't know what that means. Your example was triangles. Triangles are identified up to similarity by their angles; and up to congruence by the lengths of their sides; and identified uniquely by their congruence class and position and orientation in space. — fishfry

Is there a distinct Platonic form of the isosceles triangle, or just of triangles in general? Take the exact shape that my shoe has at some instant (since the particles in it move around). Is there a Platonic form of that shape?

I understand what you mean by mathematical structuralism. But relations between objects are only identifiable if you have identity conditions for the objects between which the relations obtain.

If it helps: I'm primarily interested in answering the identity-condition objection to Platonism. The dialectic goes kinda like this, where P is a Platonist and AP is an anti-Platonist:

P: The triangle is a mathematical object that exists.
AP: All of the triangles, or just one?
P: All of them.
AP: Okay. So how do we tell the difference between two triangles? Do all acute triangles answer to the form of the acute triangle, or are there multiple acute triangles?

And so on. The two big objections to Platonism that arise from conversations like this are that Platonic objects lack clear identity conditions and that the ontology is profligate, a crowded slum, what Quine called Plato's Beard. Reducing every object to Math should answer both objections.

But what happens when one modifies the logic? — Marchesk

This is a good question. I would say that modifications to the logic introduce new subdomains. Provided that translation functions can be constructed between those domains, there shouldn't be any problem with all of this being and expression of Math.

EDIT: or perhaps, instead of translation functions, we can say that any logical space that gives us valid derivations from Math is constructed via identity statements.

Whence the boundary of the white triangle? In the perception or in the judgement? — Banno

Dualism of scheme and content. Davidson would tut at you.

Maybe I can say something.

Mysticism is harder to discuss than other subjects because, by definition, it's about stuff you can't talk about. It would be very Wittgensteiny of me to say that that settles it, but that would be premature. The word "ineffable" is a logical contradiction, because it's a word for things there aren't words for, but we can still use it.

I would say that talk about mysticism must be rooted in mystical practice, and ultimately circle back to it.

Yes, I care. I think that the OP's observation about non-combativeness make don't apply to mystical discussion in particular. Non-combativeness should be the rule, not the exception. This is coming from someone who spent years arguing with people on the internet, on this forum and its precursor. I can have discussions, and even debates, and the debates can even get heated, but I don't do "internet arguments" anymore.

Anyway, I think that talking about mysticism is like talking about sex; why talk when you can do?

If you want to be a panpsychist, the best way to do so is to attack emergentism as hard as you can. If you can say that emergentism isn't true, and that consciousness is real, then you can say that consciousness is fundamental.

Replace "such" by "most."

The part that vexes me: is that a bad thing? I'm no Humean, but other thinkers (and feelers) cast their shadow...

Personally I think it would help to normalize part-time work. If a software developer can do their job in 4 hours each day, why the hell should they be at the office for 8?

That's like saying medicine's therapeutic project failed because there are still sick people. — Banno

I think this is a tad uncharitable. Wittgenstein wants to cure a specific illness. The vaccination campaign against smallpox succeeded and there's no more smallpox.

Perhaps we could say that Wittgenstein's project is ongoing, in that it has to dissolve problems as they arise, and problems will always keep arising. But only a minority of professional philosophers are on board with this project, and it looks as if it will stay that way. I think you're right about the reason why.

I think that, assuming we agree on what Wittgenstein's therapeutic project is, it's helpful to ask what motivates it. Does it just see professional philosophy as a waste of time for a lot of really smart people? Is the point just to make the intelligentsia more productive? That seems wrong. It ought to be something more significant than that.

Well, the cure is one that must be self-administered. Many are loathe to be cured. They're like anti-vaxxers in that respect. — Ciceronianus the White

Quite so. Wittgenstein is completely correct that there are no philosophical problems.

But if everybody is just gonna ignore him, then it doesn't help to say that. And in that respect, his therapeutic project has failed.

I don't read him like that. I never thought he was commanding people what they're allowed to do. — jacksonsprat22

Certainly not commanding, but if he was able to do what he said he wanted to do, then you would no longer want to do philosophy after reading him.

He had some important points to make, but his therapeutic project failed; people still do the kind of philosophy that Wittgensteinian therapy was supposed to "cure."

We can argue back and forth all day about whether or not he's correct in his view of traditional philosophy. But the proof, I think, is in the pudding. 69 years later and the philosophers continue to philosophize. If philosophy is an illness, it appears to be terminal, for all of Ludwig's well-intentioned mental oncology.

Okay, let's try an example: the successor axiom in Peano arithmetic says that if a is a number, then so is its successor. And the induction axiom says that if s contains 0, and also every successor of every one of its elements, then s contains all the numbers.

So does Witty's constructivism make the induction axiom nonsense, or does it mean we have to construct the induction axiom from an intension and the number 0? The successor axiom, presumably, is or contains an intension.

SO a Turing Machine could be set up to calculate 1+1, and would halt - hence 1+1 has an extension; but if set up to find root 2, it would not, and hence root 2 has no extension... or something like that. — Banno

So the existence of potential infinites is secured by our ability to grasp a rule, and that rule becomes an intension in the sense used in the SEP article. If the rule allows to construct a finite extension, then we can get extensions from it, too.

So the extension of the set of integers is always finite, although it can be continued arbitrarily. And now I'm being assaulted by that giddiness of logical legerdemain that Witty talks about...

Maybe this is swinging too hard, but: the motivation for this eludes me. Abstracta are spooky, but so are ineffable rules grasped without interpretation. Why does Wittgenstein like this spook more than the Platonic spook?

In other words, how is it that finite signs, as expressed by finite beings, have a sense of infinity. This has more to do with Wittgenstein's later philosophy, i.e., what it means to master a technique or practice — Sam26

There is a means of grasping a rule that is not an interpretation, but is exhibited in following and going against the rule in actual cases, says (paraphrased) Witty.

Did he have anything to say about the Halting Problem? I have a sudden, strong hunch that it's related to this. Maybe I'm just seeing things, but grasping that a Turing machine goes on forever without doing any calculations seems to be a case of grasping a rule in Wittgenstein's sense.

Does anyone else find his absence from the forums a bit odd? His star status was once far greater than any other philosopher, with the possible exception of Russell. Yet he never gets a mention now. — Banno

Absent from the forums, yes, but still read by young people in crisis. When I was in high school, I had a friend who was obsessed with Sartre. So he still plays that role.

Of course you did. Reread your OP. — SophistiCat

Well, no. "I believe the probability is 50/50." This statement is not a probability. You'd have to be pretty dense to confuse those two.

Anyway, either you won't or can't figure this out, so I'm done. Have a nice day.

Maybe I'm just dense, but: the reals are uncountably infinite, which means, if we're Wittgensteinian constructivists, that the set of real numbers doesn't exist, since it can't be paraphrased as shorthand for something that can be done with an intension and a finite set of extensions.

But you said that those two things are unrelated. How come? How do you save the set of reals if it can't be constructed recursively?