I wouldn't bet against true AI; that is, genuine, conscious thinking machines — counterpunch

I'm happy to take the other side of that bet.

or something so similar, it's impossible to tell the difference. — counterpunch

That of course is the idea of the Turing test, a purely behavioral standard. But that test does not distinguish between a conscious person and a philosophical zombie, or a clever chatbot.

Ultimately, a rationalist has to suppose that the brain is a machine - a biological machine — counterpunch

A biological machine. You seem to agree that there is something special about biology, about life. No computer is alive. You seem to be offering talking points that support my view and not yours.

In any event, a rationalist must admit that there are many things about the world we don't yet know. We may discover in the future principles of physics that go beyond the principles of computation. There's no evidence the world is computable. That's an assumption on your part. As I say, the simulation argument is going to be laughable a hundred years from now, just as the phlogiston theory of heat is today.

That is already one level above the binary substrate. — counterpunch

Skyscrapers are more complex than mud huts, what of it? Clever organizations of programs are still programs, and are not conscious.

I suspect someone will simulate a brain, — counterpunch

Ah but that's exactly the point. Such a simulation would not necessarily be conscious, any more than a perfect simulation of gravity will attract nearby bowling balls. We can simulate particle physics, we simulate the weather, we simulate football games. In no case is the simulation the real thing. If you perfectly simulated a human brain down to the neuron, you would find that externally, if you stimulate its optic nerve, a particular region of the cortex associated with vision lights up. But it would not have the subjective experience of seeing.

That's another problem with the simulation argument. It's a simulation, not the real thing. You can program a computer to simulate a black hole, but it doesn't suck in nearby matter. Likewise you might eventually be able to simulate a brain, but it would not be conscious.

Simulations of things are not the things themselves. You don't get wet when you run a simulation of a hurricane. I hope I don't have to belabor this point, but perhaps I do since you fell into exactly that confusion. I'll grant that you might be able to simulate a brain, but you still have not told me how you plan to implement a mind.

What did I miss? Who got banned?

I could understand, perhaps, creating a simulation of a brain - and wonder if a sufficiently detailed simulation of a brain might think. — counterpunch

Bostrom's argument makes an analogy between the primitive video games of the 1980s, and the realistic ones of today. But that's silly. Nobody thinks Ms. PacMan has a subjective experience of pleasure eating white dots, or terror at being gobbled by monsters. On the contrary, in terms of implementing self-awareness, we have made ZERO progress in all this time. Bostrom's argument collapses immediately. We have no idea if it's possible to implement consciousness, nor do we have any idea how to do it even if it could be done. So his "zillions of ancestor simulations" idea fails. Its core premise is false. His argument is valid but not sound.

What's the answer to the universe? Oh it's the great programmer in the sky. I'm baffled anyone takes this nonsense seriously. Just as the Romans, who had great waterworks, thought that the mind was a flow; and post-Newtonians thought the universe was a great clockwork machine; we, who have mastered computer technology, think the world must be a computer. The idea suffers from presentism, with no awareness of how silly it will look in a century.

If you're in hell it's much more likely that the creator of your troubles is you, and not some mythical programmer in the sky.

are we in a simulation or not — Echarmion

Simulation of what?

I'm trying to figure out MU's position a little better. — javra

Good luck.

Thereby making basic numbers (e.g., 2), as well as their basic relations (e.g., 2 + 2 = 4), non-arbitrary. — javra

I haven't followed the posts in this thread for a while but just so you know, @Meta doesn't think 2 + 2 and 4 refer to the same mathematical object. You may be assuming too much if you think he agrees with the rest of the world about this.

What we have on our hands (nowhere) is a place which is no place. Paradox! — Todd Martin

Everybody knows
This is nowhere

Some questions just remind me of songs.

https://www.youtube.com/watch?v=ZwQyX_osSLY

Yes, I always read the example of integer numbers, but this is an ordered series, the universe is not. And even if it's not granted, I guess the probabilities of it to occur are really high, like 99.999...%. — Philosophuser

Order makes no difference whatsoever to the argument. But if you insist, if the set of regions is countably infinite, it's ordered by picking any bijection with the natural numbers. Or if it has the cardinality of the reals, you can give it the usual order on the reals. If it has any cardinality whatsoever, you can biject it with an ordinal, assuming the axiom of choice, and give it that order. If the axiom of choice is false and the set of regions is not well-orderable, it's a very weird set. You have to pick one.

Or perhaps -- and this is my belief -- when physicists use the word infinity, they mean something other than mathematical infinity entirely, and haven't thought through the issues of basic set theory. I see this all the time.

The mainl point is that order is irrelevant in the argument. In fact my argument assumed a countably infinite set of regions and I didn't consider any other cases. It's hard to imagine an uncountable set of finite-sized regions, that's a very weird space. The real line can be divided into countably many finite-length intervals, delimited by the integers for example. But uncountably many? That's a weird topological space known as the long line. Like I say, physicists never think about the implications of their ideas about infinity.

Yes, I scratched my head at the seeming contradiction between the uncountability of the irrationals and the definition of the Dedekin cut, but of course by definition each cut either has a lower set with no upper bound or a greater set with no lower, et voila: uncountability. I guess I'd never really thought about it, thanks. — Kenosha Kid

I'm reminded of another example that illustrates the weirdness of the way the rationals sit in the reals.

Consider the rationals in the unit interval. Between any two irrationals there is a rational, infinitely many of them in fact. Suppose we were to put an interval, no matter how small as long as it's nonzero, around each rational. The intervals don't need to be the same size, as long as they're all nonzero. Then it's perfectly obvious that the intervals must overlap, and that their union must cover the entire unit interval.

Perfectly obvious, and false. Here's the counterexample. Since the reals are countably infinite, let $r_1, r_2, r_3, \dots$ be an enumeration of the rationals in the unit interval.

Let $\epsilon \gt 0$ be any tiny positive real number. Put an interval of length $\frac{\epsilon}{2}$ around $r_1$. Put an interval of length $\frac{\epsilon}{4}$ around $r_2$. Put an interval of length $\frac{\epsilon}{8}$ around $r_3$. In general, put an interval of length $\frac{\epsilon}{2^n}$ around $r_n$.

What is the sum of the lengths of all the intervals? Since there might be overlaps, the total length is at most $\displaystyle \sum_{n=1}^\infty \frac{\epsilon}{2^n} = \epsilon \sum_{n=1}^\infty \frac{1}{2^n} = \epsilon \cdot 1 = \epsilon$.

But $\epsilon$ is tiny, it's as small as we want! So we can cover the rationals in the unit interval with a collection of nontrivial intervals whose total length is as close to zero as we want! This is the definition of measure zero. And that means that the remaining elements of the unit interval, namely the irrationals, have length, or measure, 1. That's what it means to say that the probability of randomly picking a rational from the unit interval is 0; and the probability of picking in irrational is 1.

I love this example. The math is perfectly clear. I can not for the life of me visualize how the intervals could behave this way. But they do.

I think you'd have to order them the other way. They arise in group theory as the additive inverse of the natural numbers. — Garth

I'm sorry, what? You don't think the negative integers are ordered as I said? Which is smaller, -4 or -3? Did you miss that day in 8th grade? Even as an (ordered) additive group, the integers are ordered the negative integers followed by 0, followed by the positive integers in their usual order. I hope this is not new material to you. I'm afraid I could not find the sense of your statement. Are you familiar with the real number line and the usual order on the integers? What's the smallest negative integer?

God, as a proposed entity, is always defined as always existing. God is never defined as having a moment of birth. So I don't see where the conclusion is baked in other than perhaps in assumptions about what and which entities have beginnings -- other than God. — Garth

My reading of section 1, Form of the Argument, seems to agree with my interpretation. But I am no expert on this, I think Craig is a charlatan. In any event the OP doesn't want to discuss Craig and I have nothing else to contribute. Except to say that personally I believe in infinite regress, since I believe in the negative integers: $\dots, -4, -3, -2, -1$. In this case each "event" has a cause, namely its immediate predecessor, yet there is no first cause.

This is totally unrelated to the teleological argument. Please read the thread accordingly. — Varese

There are many variations of the cosmological argument. If you want discussion on a specific version, you should post a link to, and/or summarize, the specific version of the argument in which you are interested. Nobody can read your mind.

everything has to have a cause to exist, — Varese

That's not what William Lane Craig says. He says that everything that BEGINS to exist has a cause. He is thereby sneakily baking in his conclusion to his premise. You see, God never BEGAN to exist, God ALWAYS existed, and therefore God requires no cause. It's pure sophistry.

It would be helpful, though, if you want to initiate a discussion, to link or present the version of the cosmo argument you wish to discuss. Else confusion will be generated, since we won't all be talking about the same thing.

I guess you don't remember the key points (from my perspective) of or previous discussions. — Metaphysician Undercover

My mind has blocked them out as traumatic experiences.

What I objected to was calling things like what is represented by 4, as "objects". I made this objection based on the law of identity, similar to the op here. You insisted it's not an "object" in that sense of the word, it's a "mathematical object". And I insisted that it ought not be called an object of any sort. So you proceeded with an unacceptable interpretation of the law of identity in an attempt to validate your claim. What I believe, is that "mathematical object" is an incoherent concept. — Metaphysician Undercover

What's incoherent is you objecting to 4 = 4 as an instance of the law of identity; and claiming that 2 + 2 and 4 refer to different mathematical objects, when in fact I showed you a formal proof that they represent exactly the same mathematical object. A proof that you refuse to this day to even acknowledge, let alone refute or discuss. That's to your shame. You pretend to be an honest conversant but you are not.

This depends on what = represents. — Metaphysician Undercover

Bill Clinton on his slimiest day never reached such heights of bullshit. Pardon my French.

Does it represent "is the same as", or does it represent "is equal to"? — Metaphysician Undercover

There's no difference. And you have failed to articulate even a plausible argument for a difference.

From our last discussion, you did not seem to respect a difference between the meaning of these two phrases. — Metaphysician Undercover

You got that right.

If you're still of the same mind, then there is no point in proceeding until we work out this little problem. — Metaphysician Undercover

A point on which we agree. There's no point in proceeding. You're trolling me as you have been for two years now.

This is why I say context of the symbol is important. When the law of identity is represented as a=a, = symbolizes "is the same as". But when we write 2+2=4, = symbolizes "is equal to". If we assume that a symbol always represents the very same thing in every instance of usage, we are sure to equivocate. Clearly, "is the same as" does not mean the same thing as "is equal to". — Metaphysician Undercover

Bullpucky to da max.

No, of course not, that's clearly a false representation of what math is. That would be like saying that 2+2=4 could be considered to be valid regardless of what the symbols mean. That's nonsense, it's what the symbols mean which gives validity to math. — Metaphysician Undercover

So you're right and Hilbert and Euclid are wrong. Ok. Forgive me if I choose to disagree.

Yes, that is exactly the case it can even mean something different in the same sentence. When some says "I reserved a table for 4 at 4", each instance of 4 means something different to me. — Metaphysician Undercover

LOL. That's a good example and I take your point. Natural language is often ambiguous. In a given mathematical context, a given symbol holds the exact same meaning throughout. Dinner isn't a mathematical context. But yeah that's a good example of the problem with ambiguity of natural language. And thanks for the chuckle.

And, as I explained to you already, when someone says 2+2=4, each instance of 2 must refer to something different or else there would not be four, only two distinct instances of the very same two, and this would not make four. — Metaphysician Undercover

That's utter nonsense. Interestingly though, I came across this identical line of reasoning this morning. I don't know what the source or full context of this page is, but you either have a kindred spirit, or you wrote this. Author is objecting to the equation 2 = 1 + 1. He says:

The existence of two requires two copes of one. Then there are two objects which are separable and yet absolutely indistinguishable.

But I insist that this appeal to eternal abstract beings which exist in copies is illicit. If you claim the existence of two copies of one, what differentiates them? There would have to be a feature to distinguish them; yet arithmetic insists there is no such feature. As eternal abstract beings they have to be identical in every respect. Then how can they be plural? I conclude that there can’t be identical copies of the pure number one.

Hard to believe there are two people who assert this nonsense, not just you alone. Unless as I say you are the author. But I concede that IF you are not the author, then you are not unique in your confusion regarding mathematical objects.

http://www.henryflynt.org/meta_tech/that1=2.html

By the way there is a standard formalism for obtaining multiple copies of the same object, you just Cartesian-product them with a distinct integer. So if you need two copies of the real line $\mathbb R$, you just take them as $\mathbb R \times \{1\}$ and $\mathbb R \times \{2\}$. It's not that mathematicians haven't thought about this problem. It's that they have, and they have easily handled it. As usual you confuse mathematical ignorance with philosophical insight.

In any event, you avoided (as you always do when presented with a point you can't defend) my question. If set theorists are not only wrong but morally bad, is Euclid equally so? You stand by your claim that set theorists are morally bad? Those are your words. Defend or retract please.

Clearly, "is the same as" does not mean the same thing as "is equal to".
— Metaphysician Undercover

I say it does. I think you're splitting hairs for the sake of argument. — Wayfarer

Thanks man, sometimes @Meta makes me question my own sanity.

I guess I'd never really thought about it, thanks. — Kenosha Kid

You're welcome. Honestly it confuses me too, it's just one of those things that you "get used to," as von Neumann said. He famously said that in math you don't understand things, you just get used to them.

50%, surely? Otherwise you couldn't have a unique Dedekind cut for each real. — Kenosha Kid

Doesn't work that way. There are only countably many rationals and uncountably many irrationals. The rationals have measure zero in the unit interval and the irrationals have measure one. It's perfectly true that between any two reals there's a rational and vice versa, and this leads people into the common error you've made.

See for example

https://math.stackexchange.com/questions/124458/theres-a-real-between-any-two-rationals-a-rational-between-any-two-reals-but

https://math.stackexchange.com/questions/18969/why-are-the-reals-uncountable/18973

https://math.stackexchange.com/questions/1173495/are-there-many-fewer-rational-numbers-than-reals/1173502

If the universe is infinite in space, there are a lot of "yous", birthing continually, — Philosophuser

I regularly need to debunk this entirely fallacious claim, which is ubiquitous on the Internet and everywhere else.

Consider the infinite sequence 1, 2, 3, 4, 5, 6, ... It's infinite, but 2 never recurs, neither does anything else. Everything happens only once. For all you know, the world is just like that. Each of us really is a special snowflake after all.

Ok, you tighten your argument. In each finite region of space there are only finitely many atoms hence only finitely many possible configurations of atoms. If there are infinitely many such regions, what happens?

Consider that there are only two states, 0 and 1. Consider the sequence 0,0,1,0,0,0,0,0,0,0,0... Clearly there are infinitely many 0's. But 1 never reappears again. SOME state must recur infinitely often, but not every state necessarily does.

So even if the universe is infinite in extent; and we divide up space into infinitely many finite-volume regions, each containing only finitely many atoms that can take only finitely many configurations; it's true that SOME configuration must recur infinitely many times; but not the configuration representing YOU. Some sea slug gets replicated, but not me or you. Make sense?

Then again, there may well be laws of nature we don't yet know, that place constraints on which configurations can occur. So again, some sea slug gets duplicated but not you, because there is an as yet unknown law of nature that precludes humans from being replicated more than once, or left-handers, or primates, or whatever. Once atoms organize into chains of organic molecules, a biochemist will tell you that some combinations just can't occur at all.

Finally there is a probabilistic argument. You might say that statistically it's almost certain that every configuration would be duplicated infinitely many times. And you are correct. In the scenario of infinitely many regions, each of finite volume and containing only finitely many atoms; and in the absence of natural constraints as to what configurations may occur, so that we assume every configuration is equally likely; it is true with probability 1 that every possible configuration happens infinitely many times. That's the infinite monkey theorem.

However, that's not good enough. In infinitary probability theory, probability 1 events may fail; and probability 0 events might happen. The probability that you randomly select a real number from the unit interval and it turns out to be rational, is zero. Yet there are infinitely many rational numbers. You might pick one. I just threw a dart at the unit interval and it hit the point 2/3. A measure-zero miracle!

So the absolute best you can say is that IF all configurations of matter are equally likely (highly dubious IMO), the it is almost certain that every configuration will occur infinitely often; but it still might not. "Almost certain" is a technical term meaning that the probability is 1. But in infinitary probability, the event may still not occur. It's almost certain that a random real number is irrational; but rationals exist, lots of them. They just have what's called measure zero on the real line.

I really hope a few people read this and become disabused of this notion that in an infinite sample space everything must happen infinitely often. It's not true.

https://en.wikipedia.org/wiki/Infinite_monkey_theorem

https://en.wikipedia.org/wiki/Almost_everywhere

tl;dr: Some sea slug gets replicated infinitely often. But possibly you don't. Even in MWI, you might be the last copy of you alive. So make the best of it. Don't count on quantum immortality, which is based on bad math and arguably bad physics as well.

Well, I wouldn't go so far as to call it a violent agreement. The point of the op I believe, is that it's incorrect to call mathematical objects "objects" at all, because they do not fulfill the requirement of identity. — Metaphysician Undercover

I confess I didn't understand the OP at all. They seemed to be saying that there is only one mathematical object, all of math. I couldn't parse that. I didn't join the thread till you were talking about something I could at least understand.

But now you say, "it's incorrect to call mathematical objects "objects" at all, because they do not fulfill the requirement of identity." When a while back you disagreed that 2 + 2 and 4 represent the same mathematical object (regarding which you are totally wrong but nevermind), that was one thing. But now you seem to be saying that 4 = 4 is not valid to you because mathematical objects don't fulfill the law of identity. Am I understanding you correctly? Do you agree that 4 = 4 and that both sides represent the same mathematical object? Or are you saying that since there aren't any mathematical objects, 4 = 4 does not represent anything at all?

And so, if we start talking about them as if they are objects, and believe that they have identities as objects, and treat them that way, when they do not, there is bound to be problems which arise. — Metaphysician Undercover

4 = 4 is true by the law of identity, yes or no? I can't believe I'm even having this conversation. You've never convinced me that your mathematical nihilism isn't an elaborate troll.

Where we agree is that they are "abstract", but the problem is in where we go from here. — Metaphysician Undercover

Probably nowhere, as I knew this would. I imagine you knew it too.

Here is where the difference between us appears to arise. You are saying that there are "abstract things represented by the symbols". That's Platonism plain and simple, the "abstract things" are nothing other than Platonic Ideas, or Forms. See, you even allow that there are relations between these things. — Metaphysician Undercover

On my Platonist days I say that. But if you object, I am perfectly willing to do exactly the same math, but regarding it as a purely formal game of symbol manipulation, no different in principle than chess. Then you have no philosophical objections, any more than you would to formal games like chess or Go or Parcheesi. 4 + 4 and 2 + 2 = 4 are legal moves in my game. It doesn't matter to me. Do you at least accept that math can be regarded as a formal game without regard to meaning? It's actually often helpful to think of it that way even if you secretly believe otherwise. One can be pragmatic regarding one's philosophy.

But from my perspective, a symbol has meaning, — Metaphysician Undercover

Well then you are the Platonist. What is the meaning of the way the knight moves in chess? Clearly there is no real world referent. Nor is there any real world referent for many of the constructs of higher math. I'm willing to stipulate, for purposes of this discussion, that there are no real world referents for any of the constructs of math. So what? As long as the rules are consistent and the game is fun, we can all play.

and meaning is itself a relation between a mind and the symbol. So I see that you've jumped to the conclusion that this relation between a symbol and a mind, is itself a thing, — Metaphysician Undercover

No, you are the one saying that. I'm saying that if you don't believe 4 represents an abstract mathematical object, then it's perfectly ok to regard it as a meaningless symbol subject to the laws of arithmetic, which can be mindlessly encoded in a computer program like your calculator. When you punch '4' into your pocket calculator, the circuitry doesn't know what 4 means, but it perfectly manipulates 4 according to the rules with which it's programmed.

and you then proceed to talk about relations between these supposed things which are really just relations, and not things at all, in the first place. — Metaphysician Undercover

Ok fine, it's all a meaningless formal game. It makes no difference to me. But if you don't think 4 is a thing, you are most definitely a mathematical nihilist. When you go to the store and buy a dozen eggs, do you make these same points at the checkout stand?

If you can follow what I said above, then I'll explain why there's a real problem here. — Metaphysician Undercover

Truly I stopped following you back when you claimed that 2 + 2 and 4 don't represent the same mathematical object, when in fact they do. And when I gave you a purely formal syntactic proof that they represent the same thing, and you refused to even engage with my argument. You didn't say, "I reject the Peano axioms," or "I have it on good authority that Giuseppe Peano cheated at cribbage and is therefore not to be trusted," or "You made a mistake on line 3," or anything like that. You simply ignored the argument entirely, despite my asking you several times to respond. You have yet to demonstrate that you're having a serious conversation with me.

The relation between a symbol and a mind, which is how I characterized the abstract above, as meaning, is context dependent. When you characterize this relation, the abstract, as a thing, you characterize it as static, unchangeable. — Metaphysician Undercover

I have no trouble with time-dependent assignment of meaning. But, are you claiming that 4 means one thing to you today and other thing tomorrow? What ever are you talking about?

This is what allows you to say that it is the same as manipulating symbols devoid of meaning, the symbol must always represent the exact same thing. — Metaphysician Undercover

Well yes, that's your nihilism speaking again. If I say 4 = 4 and you assert that the symbol '4' may have different meanings on each side of the equation, you are the crazy one. I don't mean that in pejorative sense, it's an accurate description.

Of course a symbol like 'x' may mean one thing in one context and a different thing in another, but I truly hope you are not thinking that this is a very deep or significant point. Within a given context, a symbol only has one particular meaning; otherwise we can't do math, we can't do science, we can't even get on the bus. "Oh, today the #4 bus goes to Liverpool. You must want yesterday's #4 bus that went to Bristol." Come on, how can you expect me to take you seriously when you assert such nonsense?

But here is where this thing represented, the abstract, fails the law of identity, the meaning, which is the relation between the mind and the symbol, is context dependent and does not always remain the exact same. — Metaphysician Undercover

In the string 4 = 4, does the symbol '4' refer to the same thing on each side of the equation? Is this or is this not an instance of the law of identity?

I think we've discussed this enough already, for you to know that I denounce all set theory as ontologically unsound, fundamentally. — Metaphysician Undercover

Well ok you're a mathematical nihilist and you don't deny it. But I doubt you actually live like that. You couldn't pay your bills or read the paper, if the meaning of the symbols keeps changing for you.

It doesn't mean a whole lot though, only that I think it's bad, like if I saw a bunch of greedy people behaving in a way I thought was morally bad, I might try to convince them that what they were doing is bad. — Metaphysician Undercover

Set theorists are morally bad people? Who need to be shown the error of their ways? Wow you are far gone my friend. Are you a type theorist? A category theorist? Or do you feel that those people are morally bad too? Not just wrong, but morally bad. I'm genuinely curious about this. Is it just set theory? Or do you feel this way about all historical attempts at mathematical systemization and formalization, from Euclid on down to the present?

However, if it served them well, and made their lives easy, I'd have a hard time convincing them. — Metaphysician Undercover

You haven't convinced me that you're serious about anything you write. At least when you converse with me.

Ok to make this short, can you please just respond to these two questions:

* Is 4 = 4 an instance of the law of identity; or does the symbol '4' have a different meaning on each side?

* Is Euclid morally bad by virtue of attempting mathematical synthesis?

What a sad joke. — hypericin

Nice having an intelligent chat with you.

I think we've been through this before. — Metaphysician Undercover

Good point, I knew better than to start. Don't know what I'm thinking. This can't end well.

You insisted on an unreasonable separation between "objects" and "mathematical objects", such that mathematical objects are not a type of object. — Metaphysician Undercover

Well, mathematical objects are abstract objects. But I agree that numbers aren't like rocks. That doesn't mean that numbers don't exist. It only means that numbers are abstract. And, per structuralism and Benacerraf's famous essay, What Numbers Cannot Be, numbers are not any particular thing. They're not actually sets, even though they are typically represented as sets. Numbers are the abstract things represented by sets. I suspect you and I might be in violent agreement on this point, but I'm not sure.

We could start with the axiom of extensionality. Any axiom which treats numbers as elements of a set, treats the numbers as objects. — Metaphysician Undercover

Only sets can be elements of sets in pure set theory. So we can represent numbers by sets and say that the $\in$ relation holds between a pair of sets; but you are reading more into that than is intended. In high school set theory we would say that the students are members of the school, and we'd call the school a set and the students elements of that set. And I doubt that you'd disagree. But i formal set theory we are not reifying, I think that's the word, the things that are elements of sets. We're just saying that the membership relation holds between the abstract things represented by the symbols. You can, if you like, view the entire enterprise as an exercise in formal symbol manipulation that could be carried out by computer, entirely devoid of meaning. It would not make any difference to the math.

The issue though, is that set theory treats them as "individual things", therefore Platonism is implied. Set theory relies on Platonism because it cannot proceed unless what 2, 3, 4, refer to are objects, which can be members of a set. — Metaphysician Undercover

Maybe you'd like category theory better. In categorical set theory there are no elements at all, only relationships among sets. But in category theory they call things "objects" and that might make you unhappy.

We could go down a rabbit hole here but just tell me this. Do you believe that if E is the set of even positive integers, then E = {2, 3, 4, 6, ...}. Do you agree with that statement? Or do you deny the entire enterprise? I'm trying to put a metric on your mathematical nihilism. Is it naughty to put numbers into sets? Why? Some numbers are prime, some are even, some are solutions to various equations. Why can't we collect them into sets? Either conceptually or, if you don't like that, formally?

The right political class has therefore become peddlers of cheap conspiracy theories to a base of insane people. — hypericin

And the left, the home of Russiagate and Hunter Biden's laptop as Russian disinformation, are the modern rationalists? I'll take the other side of that proposition.

The problem with the phrase "conspiracy theory" is this. Someone asserts that the earth is flat and that the world is run by lizard people. They get called conspiracy theorists. Then later, the government wishes to lie the country into war. They assert falsely that Saddam has WMDs or that the North Vietnamese attacked a US naval vessel at the Gulf of Tonkin; then they label political dissent as conspiracy theories, to smear legitimate dissent by association with nutballery. That's the move you're making here. You define conspiracy theory as any idea you don't like, or any idea that conflicts with the official status quo that you happen to favor.

Many mathematical axioms such as those of set theory rely on the assumption of mathematical objects — Metaphysician Undercover

Can you name one such? Structuralism is in these days. It doesn't matter if you call sets "beer mugs" as Hilbert pointed out. It's the properties and relations that matter, not the nature of individual things.

whether there was something which we refer to with "3", and something which we refer to with "prime", which existed prior to the existence of these words, and that is a difficult metaphysical question without a straight forward answer. — Metaphysician Undercover

Hey, something on which we agree!

Platonic objects lack clear identity conditions — Pneumenon

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. It's perfectly simple to define a similarity class of triangles or a congruence class of triangles or a particular triangle. I don't follow your examples.

A classic Platonic object in math is the unit circle. It's a circle of radius one centered at the origin. Now you're right, coordinate systems are arbitrary so in fact we could locate the unit circle anywhere on the plane. In that sense "the" unit circle is arbitrary. But once you fix a rectangular coordinate system, you have a unique unit circle and you can then define the trigonometric functions, Fourier series, topological groups, and all the rest of the interesting mathematical concepts that generalize or abstract from the unit circle.

Math isn't concerned at all with particular objects; only with abstract forms. It doesn't matter what numbers "are," only how they behave and how they relate to other numbers. This is mathematical structuralism. 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.

To those throwing rocks at mathematical Platonism (as I do when I'm taking the other side of this debate), was 3 prime before there were intelligent life forms in the universe? If that's too easy (I don't think it is), were there infinitely many primes? Or at least no largest prime?

I’m well aware of the technical distinction but they tend to blur in the popular imagination. — Wayfarer

I just wanted to note that for general info, especially since the linked article and the article that it referenced both made the same error.

BTW there’s a really good intro to Everett’s notion — Wayfarer

Sean Carroll, a rockstar physicist who makes terrific expository Youtube videos, is a big proponent of many worlds. I like to think of him as a great physicist but a not-so-great philosopher. In any event, the Schrödinger equation is physics. Interpretations of the Schrödinger equation are metaphysics, a point lost on many physicists. Remember that Newton did not make this error. He had his theory of gravity, which accurately predicted how massive bodies behave, up to the limits of observation at the time. But when pressed to say what gravity actually was, he famously said, "I frame no hypotheses." Meaning that science describes, it does not explain. Contemporary scientists still have a lot to learn from old Isaac.

The Multiverse Idea is Rotting Culture — Wayfarer

A point of clarification. The multiverse and the many worlds interpretation are two entirely separate ideas. The multiverse refers to the fact that parts of the early universe after the big bang moved away from each other very fast, and are no longer accessible to each other given the limitations of light speed. There's one big universe emanating from the big bang, and many separate islands within it, each unreachable from the others.

Many worlds is an interpretation of quantum physics that says that rather than an observation collapsing the wave function to one particular outcome, all possible outcomes occur, each in its own world that branches off at that moment.

Multiverse and many worlds are completely separate ideas. I'm not even sure that anyone has explored their interactions.

ps -- I read the article. The author, Sam Kriss -- whose work I know and enjoy -- conflates and confuses the multiverse and many worlds. He seems to be talking about many worlds in the article but sometimes calls it the multiverse.

Not only that. The New Scientist article Kriss links makes exactly the same mistake, talking about many worlds but calling it the multiverse. I believe there's quite a bit of confusion about this floating around.

Is it ethical to purchase factory farmed animal products and fund this? — Down The Rabbit Hole

No worse than typing on a computer whose components are made in exploitive third-world factories. You might recall that a while back, Foxconn employees were committing suicide. They solved the problem by installing suicide nets on the roof.

The global supply chain is something you don't want to look too closely at.

Whoops! Misread this as "Quantum Immorality". Now there's a metaphysical topic worth pursuing! :nerd: — jgill

Good one! You can misbehave all you like because in some other world you were Godly.

'Quantum immortality'). — Philosophuser

I would not believe a word anyone writes about the subject unless they put their money where their mouth is and kill themselves. Since no sane person would do that, the idea stands refuted. Even the people who claim to believe it, don't. Maybe that nutball in Nashville was testing out the theory, eh what? They say he believed in the lizard people. Compared to QI/MWI, that's practically sane.

QI is just wrong. The only you is the one in this world. The other ones may be identical copies, but they are not you. It's like forking a process in a Unix system. You get a copy identical in every way to the original; but it's a distinct process that has its own future after the moment of the fork. Even in MWI there is only one of you, the one in this world. The other one has the identical state as you (exact same configuration of atoms, identical life experiences, memories, thoughts) up to the moment where its future diverges from yours. To put it in theological terms, each "copy" has its own soul, if that metaphor helps to make my point more clear.

Two configurations of atoms may be identical, but they are not the same person. They're two distinct individuals that happen to have the same configuration up to some moment of time. Relates to the transporter problem. You die at one end and a copy with your memories and experiences is created at the other; but they are not the same individual. You can see that just by imagining a nondestructive scan at the sending end, resulting in two copies. Two souls identical in experience and mentality up to the moment of cloning, but with distinct diverging histories thereafter.

I got plenty of nothing
And nothing's plenty for me
I got no car - got no mule
I got no misery
Folks with plenty of plenty
They've got a lock on the door
Afraid somebody's gonna rob 'em
While there out (a) making more - what for
I got no lock on the door - that's no way to be
They can steal the rug from the floor - that's OK with me
'Cause the things that I prize - like the stars in the skies - are all free
I got plenty of nothing
And nothing's plenty for me
I got my gal - got my song
(I) Got heaven the whole day long
- Got my gal - got my love - got my song

https://www.youtube.com/watch?v=bx1YfQF0WNQ

"What if, after you died, a man removed a helmet from your head and asked 'So how was the simulation, how many units can we put you down for?'" — Outlander

How is this substantially different from waking up from a dream? Are you a man dreaming you're a butterfly, or a butterfly dreaming you're a man? How do you know you're not dreaming right now?

You are a married, middle-aged man with a nice house, two cars, a wife, and two children — Outlander

And you may find yourself living in a shotgun shack
And you may find yourself in another part of the world
And you may find yourself behind the wheel of a large automobile
And you may find yourself in a beautiful house, with a beautiful wife
And you may ask yourself, "Well... how did I get here?"

https://genius.com/Talking-heads-once-in-a-lifetime-lyrics

Gotta go, it's feeding time in my vat.

What do physicists mean when they talk about nothing? : when physicists talk about nothing, they mean empty space (vacuum). — Gnomon

Krauss is full of bullpucky. He presupposes the laws of physics and the quantum soup, neither of which are nothing. He's another one of these dumb smart guys. Much taken with himself. "Look how clever I am," is his message to the world.

What is the entitlement of the poor to relief for rent and food and other basics? — tim wood

A better question, given the nature of the 2008 and 2020 bailouts, is what is the entitlement of the rich to a claim on the public trough? As the Occupy marchers chanted, "Banks got bailed out and we got sold out." Truer words were never spoken. In 2020, for every crumb tossed to the working class, the billionaires got the whole cake. The numbers bear that out. 2008 and 2020 were both massive upward transfers of wealth. Local stores are shut down while Walmart and Amazon are prospering like never before. Makes you wonder, doesn't it? Why are we talking about those "greedy poor" whose livelihoods were just destroyed by the shutdowns? You might have seen the recent news story about the woman who put up a tent conforming to all safe outside dining practices who had her restaurant shut down anyway. While directly next door, a film crew was allowed to put up an identical tent. Makes one suspect that the lockdowns are less about public safety and more about crushing the working class and rewarding the politically connected.

What passes for capitalism today is a far cry from the invisible hand of Adam Smith. It's one hand greasing the palms of politicians and the other hand strangling the little people. No wonder the kids are into socialism.

No, it is right. Yes, set theory is consistent if and only if there is a model of set theory. But if set theory is consistent then set theory itself doesn't prove that it has a model;. That's Godel's 2nd Incompleteness Theorem. — GrandMinnow

Oh I see, you're right. My parser got confused.

Okay. In preparation for this response I read the introductory material in Hamkins's paper, Pointwise Definable Models in Set Theory. This is his formal paper on the material he discussed informally in the Mathoverflow link I posted earlier. Hamkins is a particularly lucid and engaging writer and this material is accessible even to nonspecialists.

Hamkins shows that (assuming set theory is consistent) there is a model of set theory -- uncountably many models, in fact -- in which everything is definable. Every element, every set, every collection of sets, etc. In particular, every real number in this model is definable. Of course if set theory is not consistent it has no models at all, so it's necessary to assume consistency to get the discussion off the ground.

Now such a model must necessarily be countable, because there are after all only countably many possible definitions or predicates. That's easy to see, there are only countably many finite strings over a countable alphabet.

But, in such a model, the real numbers are still uncountable! That's because Cantor's theorem is a theorem of ZF, and is therefore true in any model of ZF. Cantor's theorem says that the reals, being bijectively equivalent to the powerset of the natural numbers, are uncountable.

So what's going on? The trick is the famous Lowenheim-Skolem theorem, which says that if a collection of axioms has an infinite model, it has models of all cardinalities. In particular there are countable models of set theory. Yet in those models, the real numbers are still uncountable. How does that work?

What does it mean for a set to be uncountable? It means there is no bijection between the set and the natural numbers.

Conceptually here is what's going on. Suppose that we have some countable set that is a model of set theory. Remember that a bijection is just a special kind of function; and that a function is just a particular kind of set (namely a set of ordered pairs in which each first element only appears once). So what we can do is go into the model, and remove all the bijections between the natural numbers and our set of interest.

If we are careful to remove bijections in such a way that the remaining elements still form a model of set theory, we'll have a countable universe that contains a set that is uncountable. In other words there are no bijections between the set and the natural numbers, because we've carefully removed them. But the universe is still a countable set. Of course the "carefulness" involved in throwing out the bijections while still satisfying the axioms of set theory can be made rigorous.

As seen from the outside, our uncountable set is actually countable. But as seen from inside the model, it's uncountable, because we've removed the bijections. This shows that the notion of countability is a "relative" property. It depends on whether you're looking at it from inside or from outside a given model.

See https://en.wikipedia.org/wiki/Absoluteness

and in particular, https://en.wikipedia.org/wiki/Absoluteness#Failure_of_absoluteness_for_countability

So in a model where everything is definable, that model is must necessarily be countable as seen from the outside. But it does contain the real numbers (which exist in any model of set theory); and by Cantor's theorem, the real numbers are uncountable (as seen from within the model) yet all real numbers are definable.

I mention all this to provide context and hopefully some clarity to my following comments.

That thread you linked to includes an argument that uses 'least undefinable ordinal' to throw shade on the "naive" notion of definability. — GrandMinnow

By "throwing shade" you mean "flat out falsifying," in the same way that Russell's paradox "throws shade" on Frege's notion of unrestricted set formation, right? In other words Hamkins is not just casting vague innuendo. He's providing a technical argument that falsifies a commonly held belief. Just to be clear about this.

But one would not claim that 'definabie' itself is a predicate in the theory. — GrandMinnow

Correct, that one of Hamkins's points.

I only mentioned a certain syntactical fact - regarding formulas of a certain form. I wouldn't say in set theory itself, about set theory, that there exists a definable something or other. To even speak of that "something' is to speak of an object that exists per a set theoretic model, — GrandMinnow

I'm afraid I couldn't quite glean the meaning of this.

but indeed, as we well know, set theory (if it is consistent, which I take as a "background" assumption) does not prove the existence of a model of set theory. (Though, I'm not expert enough to defend against possible other complications in the matter.) — GrandMinnow

No, that's not right. Set theory is consistent if and only if there's a model. That's Gödel's completeness theorem. If it's consistent there's a model, and if there's a model it's consistent.

I changed this post greatly:
So to make the argument work that there are only countably many definable real numbers, maybe something like this in a set theoretic meta-theory for set theory: — GrandMinnow

Remember, there are only countably many real numbers as seen from outside a Hamkins model; but within the model there are uncountably many reals (by Cantor's theorem) and they're all definable. Tricky stuff.

Let 'Rx' be the set theory formula 'x is a real number'. Let M be any model of set theory such that any subset S of the universe of M satisfies 's is countable' if and only if S is countable. Let D (the set of definable reals) be the subset of the universe of M by D = {d | there exists a formula F of set theory such that (E!x(Fx & Rx) is a theorem of set theory & d satisfies Fx and d satisfies Rx). Then D is countable. — GrandMinnow

There is a model of set theory in which everything is definable. In particular, each real number is definable. In such a model the real numbers are countable as seen from outside the model; but they are uncountable as seen from inside the model. You are using the word countable without regard for the fact that countability is a relative property. I think your argument is ambiguous because of that. In any Hamkins model, the reals are uncountable (inside the model) yet they're all definable. And that's because they're "really" countable as seen from outside the model.

tl;dr: There is a model of set theory in which all real numbers are definable. Such a model is actually countable, as seen from the outside; nevertheless within the model, Cantor's theorem is true and the reals are uncountable.

I think he made a typo and actually meant 'between the natural numbers and the computable numbers'. — GrandMinnow

Yes sorry typo.

So to make the countability argument work, maybe something like this in a set theoretic meta-theory for set theory: — GrandMinnow

Duty calls for other stuff this evening, will get back to you later tonight or tomorrow to continue this interesting convo. But who on earth ever suggested that the set of noncomputable reals was anything other than uncountable? Surely not me and not the OP as far as I could tell. The claims (mine anyway) are:

* The set of computable reals is countable.
* The set of computable reals is NOT computably countable; and
* The question of whether the set of computable reals is definable is Hamkins-murky and way beyond my pay grade. Though if I understand Hamkins correctly, there are models of the reals in which everything is definable. That's why it's murky.

The negative statement was deliberately chosen. — GrandMinnow

I only meant to add clarity. It's a small point. The OP has disappeared and I was hoping he'd return long enough to comment on my observation that the proof of the existence of a noncomputable real is itself computable. That is, proofs are computations, hence a computational intelligence would be able to prove the existence of noncomputable phenomena, given the standard axioms and inference rules of math.

the merely ostensive (and not specified by actual mathematical description) list you gave is either actually not definable or it's finite. — GrandMinnow

I believe your assertion that there is no definable list of noncomputable reals may well be true but I'm not sure that the proof would be at all elementary. Do you happen to have such a proof? There's surely no computable list, since that would solve the Halting problem. But why not a definable list? I don't know if what you said is true, but I do know that it's not trivial either way. Definability is very murky as Hamkins points out. See his brilliant response in this thread.

https://mathoverflow.net/questions/44102/is-the-analysis-as-taught-in-universities-in-fact-the-analysis-of-definable-numb

it is not the case that there does not exist a definable uncomputable real number. — GrandMinnow

That's a lot of negations, but for clarity, there does exist a definable yet noncomputable real, namely Chaitin's Omega. I haven't mentioned it because I didn't want to further complicate the conversation.

https://en.wikipedia.org/wiki/Chaitin%27s_constant

Suppose the following is the complete list of computable irrational numbers between e and pi
— TheMadFool
Right from the start of your argument, the merely ostensive (and not specified by actual mathematical description) list you gave is either actually not definable or it's finite. — GrandMinnow

No that's not right. There are a countably infinity of Turing machines hence a countable infinity of computable numbers, hence a bijection between the natural numbers and the noncomputable numbers.

As I indicated earlier, there is no computable bijection; but there is a bijection.

Definability is a much more subtle concept and is best left out of the discussion. For one thing, first-order definability is not itself formally first-order definable. There are models in which everything is definable. Joel David Hamkins has written on this topic. It's not relevant here and would take us far beyond the topic.

I found this on wikipedia:

While the set of real numbers is uncountable, the set of computable numbers is classically countable and thus almost all real numbers are not computable
— wikipedia — TheMadFool

I have a point that might be of interest to you.

Wiki is correct that if you take the standard rules of math, you can logically deduce the existence of noncomputable numbers. You've quoted an outline of the proof. Cantor's diagonal argument shows that the reals are uncountable; but we can show that there are only countably many Turing machines. So there must be real numbers whose digits can not be cranked out by a Turing machine. QED.

But what is a logical deduction? You start from some axioms, which are strings of symbols. You accept some rules of inference. You mechanically apply the rules of inference to your axioms, and conclude the result. The entire process of deduction could be carried out by a computer. You input the axioms as strings of symbols; you program in the rules of inference; and the computer can determine whether or not that's a valid proof.

So that's the thing. The proof that noncomputable real numbers exist, is itself a computation. In general, proofs are computations. Symbol manipulation according to formal rules.

In other words: A computational intelligence would eventually prove to itself that noncomputable real numbers exist given the rules of standard math.

What do you think about that?

It's much simpler to show that there are uncomputable numbers in [e,pi] — f64

You gave an existence proof without naming any specific noncomputable number. And in order to do so you needed a cardinality or a measure theoretic argument, neither of which are physically meaningful.

The point is that a number whose existence is shown only through an existence proof has a lesser claim on mathematical existence than one one built by construction.

Of course his post is finite so it's not likely that he's specified any particular noncomputable real. But the larger point is that a number that encodes an infinite amount of information has a lesser claim to mathematical existence than one that encodes only a finite amount of information.

And either way, mathematical existence is not physical existence, A computer could put in our minds the idea of a flying horse, Captain Ahab, Captain Kirk, and noncomputable numbers. But since those things don't exist in the physical world, they are not evidence that the world is not a computer.

Of course numbers in general are abstract and even fractions like 2/3 are not instantiated in the world. You can't measure 2/3 of anything, unless you're going to refer to a quark of 2/3 spin or charge. In which case I'll just use 3/4 as an example of a number whose mathematical existence is on solid ground but whose physical existence is doubtful.

Please remember that all physical measurement is approximate. Even the positive integers are murky. I can show you three oranges or three planets but I can't show you the number three. Numbers have only abstract existence; so the mathematical existence of any type of esoteric number can never tell us anything about the physical world.

ps -- Just to anticipate @TheMadFool's objection: Just as a computer could put in our minds the idea that the world is or isn't a computation; why couldn't it put in our minds the idea that noncomputable numbers might or might not exist? I truly don't follow your argument that just because the great computer in the sky puts some contradictory idea in our head, that this is evidence of the nonexistence of the great computer. After all, God created atheists!

In other words there exists a non-computable irrational number between e and pi, existing in the same sense as e or pi. — TheMadFool

Fine, name one. All you have is an existence proof; and an existence proof is a weaker class of metaphysical existence than a constructive proof like showing that 2/3 or pi exists.

Display v [a number, any number] — TheMadFool

You mean an infinite decimal representation of a number? I'm afraid I didn't follow your algorithm at all. Perhaps you can give an example or explain it more clearly.

Note: After the first display operation for v, subsequent v's are attached to the previous v. So if the first v = 2, the second v = 23, the third v = 2345 or 2325, the fourth v = 23451267 or v = 23251246, ad infinitum.

And that's as far as I managed to get...comments?! — TheMadFool

As your final number is the output of an algorithm, it's surely not noncomputable. Though I'm not sure I really understand the details of your idea. Would like to see a more clear exposition.

But if you are generating a number from an algorithm, you haven't generated a noncomputable.


Let me leave you with an interesting example.

Suppose someone claims they have the following algorithmic procedure to generate a noncomputable number.

* Enumerate the computable numbers. We can do this because they are countably infinite.

* Form the antidiagonal according to a deterministic rule. Replace each digit n with n+1 (mod 10). That is, 0 is replaced by 1, 1 is replace by 2, ..., and 9 is replaced by 0.

Since we have enumerated all the computable numbers, and the antidiagonal is not on the list,
we have seemingly devised a perfectly deterministic procedure that has generated a noncomputable number!

What is the flaw? It's subtle. There is an enumeration of the computable numbers; but there is no computable enumeration of the computable numbers! That is, by a cardinality argument there is a bijection from the positive integers to the computable numbers. But that bijection can not itself be computable! Why not? Because to form such a list we have to look at every Turing machine and generate each digit of the corresponsing computable number by successively inputting 0, 1, 2, 3, ... and seeing what digits it outputs. But how do we know which TMs will halt and which will loop or go on forever without outputting a digit? We can't, because the Halting problem is unsolvable. Turing worked this out in 1936

There is no computable function that enumerates all and only those TMs that halt. So there is no computable enumeration of the computable numbers. And our "deterministic" generation of a noncomputable number doesn't work.


Finally: If you want to prove that we are not computations, all you have to do is figure out how to solve the Halting problem. We already know that no computation can solve it. If a human can, then we are not computations. The problem is that nobody's ever figured out how to solve the Halting problem.

This idea is intimately related to Gödel's first incompleteness theorem. No mechanical procedure can determine all mathematical truth. Penrose thinks this shows that we are not computations. Nobody buys his argument; but everyone agrees that Penrose's bad ideas are better than most people's good ones.

If say x, an non-computable irrational number, exists, I mean, limiting myself to the current domain of discourse, that it has the same ontolological status as, say, the number 2 or the square root of 2 or pi or e. — TheMadFool

The latter are all computable and encode only a finite amount of information. In fact that's exactly why you can name and identify specific ones.

Can you name or identify any specific noncomputable number? If not, then you're wrong that they have the same ontological status as computable numbers. In this regard I find agreement with the constructivists. A number that requires an infinite amount of information to specify has a weaker ontological status than one that only requires a finite amount of information. Even you agree with this point. If you claim noncomputatlble numbers exist, name one.

Of course noncomputable numbers have mathematical existence in that we can prove (given the standard rules of math) that they exist; but that's only an existence proof that gives no clue of how to find one. That is exactly the constructivists' complaint.