Multiplication, then, is something more than just an efficient method of addition? What is multiplication doing that addition cannot do? Or is the question(s) I'm asking too simple to answer. For example, am I assuming something I should not? — tim wood

I have not as yet acquired sufficient technical understanding to answer this question. In fact it's a point on which I'm stuck myself. It's a point of logic involving induction. At one point I read the Wiki page and a couple of articles about Presburger arithmetic and thought I had a vague understanding of what was going on; but if I did, it's certainly not stored in my brain cells right now.

The best place to start is the Wiki page and see if it sheds insight. Also I remember that in my earlier research, I discovered that the Wiki page on the Peano axioms actually hints at the difficulty with defining induction, but again these are memories of a few months ago.

All I really know is that when you make a recursive definition of multiplication in PA, in the manner I'd always assumed you can do, you are actually adding some kind of secret sauce without realizing it. You can see how little of this I understand.

I said there are scientific theories that don’t rule it out. That doesn’t mean they say it is definite so. — Pfhorrest

I'm going to stop posting in this thread till I read the Susskind paper. But again, if by "it" you mean a physical theory that incorporates infinity, then my earlier remarks would have to apply. Namely, that I suspect that the physicists are using the word infinity in a manner inconsistent with how mathematicians think of infinity. But in philosophical discussions, the physics infinity acquires the trappings of respectability of mathematical infinity over the past 140 years. This I believe is a logic trap. We must remember that when physicists talk about infinity, they generally have no idea what they're talking about mathematically. Just like with everything else they do. Physicists use math, they don't do math; and one should not rely on physicists for the logically correct foundations of things.

Ah! So is Meta's contention simply that Presburger arithmetic is the whole of Mathematics? — Banno

Regarding @Metaphysician Undercover thought process. From Apocalypse Now:

Capt. Benjamin Willard: They told me that you had gone totally insane, and that your methods were unsound.

Colonel Kurtz: Are my methods unsound?

Capt. Benjamin Willard: I don't see any method at all, sir.

I did say "all mathematics." My bad. I meant all arithmetic. And mine is really more a question than a claim arising out the the earlier discussion. — tim wood

I know a technical context in which that's not true.

There's a theory weaker than Peano arithmetic called Presburger arithmetic that allows for only addition. It's strictly weaker than PA; and in fact has the remarkable property that it's logically complete in the sense of Gödel. Every statement in Presberger arithmetic can either be proven or disproven.

When you add in multiplication, you get PA and that is logically incomplete.

It's commonly believed that in PA you recursively define multiplication based on addition. But it turns out that at a technical level (which I haven't yet grokked) the particular use of recursion is not strictly within the allowable rules in Presberger arithmetic and so is more folklore than truth. In fact the theory of addition is strictly weaker than the theory of addition and multiplication.

Meta's still playing with rocks while the rest of us have pointy sticks. — InPitzotl

He sure gets into a lot of people's heads.

It makes as much sense as an infinite but bounded universe, which is hardly a foreign idea in cosmology. — Kenosha Kid

Rather than my spouting off more, I'm going to read the Susskind paper because he's someone I respect AND he's not Penrose so his ideas would be more mainstream. I'm not really up on what the speculative cosmologists do these days.

I will say that mathematically I'm troubled by the casual use of infinity in speculative physics. My understanding is that when physicists say infinity, they generally do not mean what mathematicians regard as infinity. If someone thinks there are infinitely many "instants" in time, are there countably many or uncountably many? Is the continuum hypothesis true or false about these infinitely many instants? Or if time is infinite in one direction or the other, what exactly does that mean if there aren't infinitely many instants. Are there infinitely many Planck times, or what exatly?

I don't think physicists think clearly about infinity otherwise they'd ask themselves these questions. In mathematics to get the theory of infinity off the ground we must assume a powerful axiom call the axiom of infinity; and there is no evidence that the axiom of infinity is true about the physical world.

So I do have a lot of misgivings whenever I hear physicists talking about infinity. And when I take the trouble to dig deeper into the details, I generally find that they're not using the word the same way mathematicians do.

But these are just impressions, and as I say I don't know much about speculative cosmology.

Do you think that in Bostrom's simulated universes, it's TMs all the way down? — A Raybould

I discussed this at length. You chose not to engage with my questions, my points, or my arguments. You failed to demonstrate basic understanding of the technical terms you're throwing around. You repeatedly failed to define your terms "process" and "simulation" even after my repeated requests to do so.

This is no longer productive.

I rather suspect that's true, unfortunately. — A Raybould

I haven't seen your handle much before. People who know me on this board know that I'm perfectly capable of getting into the mud. I'm sorely tempted at this moment but will resist the urge. I say to you again:

A TM is not a physical device. It's an abstract mathematical construction. A computation, by definition, is anything that a TM can do. This isn't me saying this, it's Turing followed by 80 years worth of computer science saying that.

If you think there's something that counts as a computation, that

a) Can not be implemented by a TM; and

b) Is consistent with known physics;

then by all means tell me what that is to you.

What part of 'a computation is what a Turing machine does, not what it is' do you not understand? At least until we sort that out, I am not going to read any more of this jumble. — A Raybould

Best you don't, since I couldn't have been more clear.

A TM is not a physical device. It's an abstract mathematical construction. A computation, by definition, is anything that a TM can do. This isn't me saying this, it's Turing followed by 80 years worth of computer science saying that.

If you think there's something that counts as a computation, that

a) Can not be implemented by a TM; and

b) Is consistent with known physics;

then by all means tell me what that is to you.

You seemed to think that it would be absurd to even think that it could be infinitely old, and I don't see why that or any other "actual infinity" would be absurd. — Pfhorrest

I don't think it's absurd at all.

I do think that you haven't backed up your claim that there's a scientific theory to that effect, other than Penrose's. If my knowledge of the literature is insufficient, @Kenosha Kid has pointed me to a paper of Susskind which I'm looking forward to reading, since I know Susskind is a serious physicist.

I'm still confused by where you're coming from. Sometimes you claim an infinite past is a plausible or at least published scientific theory; but recently you said that you aren't talking about science, only philosophy.

For my part I never imagine I'm wise enough to have an opinion on when the world started or how. I'm only discussing the state of the art of science. If there's more speculation about an infinite past than I've been formerly aware of, then I'll stand corrected on this point. But please be clear. I'm not arguing against an infinite past, an infinite future, or an infinite number of turtles all the way down.

I'm arguing against the claim that an infinite past is a serious scientific theory. I'll know more after I read the Susskind paper.

You can check Guth's 2007 review where he also discusses inflation fields that are past eternal but bounded. — Kenosha Kid

If they're bounded then I already acknowledged that example as in 1/2, 1/4, 1/8, ... in which there's no "first step" but is nonetheless bounded below. This surely isn't what the other poster, I think it was @Pfhorrest, meant by infinite past.

"Past eternal but not bounded?" Sorry that doesn't make a lot of sense.

I'll read Susskind's paper and I appreciate the reference to a physicist I take seriously.

I'm willing to stipulate that Susskind and others (Penrose for sure) have theories positing and endless sequence of universes before the big bang.

Even in Bostrom's simulation argument, neither brains nor minds are TMs: in that argument, — A Raybould

If the word simulation means something other than computation, you need to state clearly what that is; and it has to be consistent either with known physics; or else stated as speculative physics.

I'll agree that Bostrom and other philosophers (Searle included) don't appear to know enough computability theory to realize that when they say simulation they mean computation; and that when they say computation they must mean a TM or a practical implementation of a TM. If not, then what?

When we simulate gravity or the weather or a first person shoot-'em-up video game or Wolfram's cellular automata or any other simulation, it's always a computer simulation. What other kind is there?

And when we say computation, the word has a specific scientific meaning laid out by Turing in 1936 and still the reigning and undefeated champion.

Now for the record there are theories of:

* Supercomputation; in which infinitely many instructions or operations can be carried out in finite time; and

* Hypercompuation; in which we start with a TM and adjoin one or more oracles to solve previously uncomputable problems.

Both supercomputation and hypercomputation are studied by theorists; but neither are consistent with known physical theory. The burden is on you to be clear on what you mean by simulation and computation if you don't mean a TM.

I (or, rather, what I perceive as myself) is a process (a computation being performed), and what I perceive as being the rest of you is just data in that process. — A Raybould

But what do you mean by computation? Turing defined what a computation is. If you mean to use Turing's definition, then you have no disagreement with me. And if you mean something else, then you need to clearly state what that something else is; since the definition of computation has not changed since Turing's definition.

To confuse a process (in either the computational sense here, or more generally) with the medium performing the process is like saying "a flight to Miami is an airplane." — A Raybould

I have done no such thing. I don't know why you'd think I did. A computation is not defined by the medium in which it's implemented; and in fact a computation is independent of its mode of execution. I genuinely question why you think I said otherwise.

If you agree that you and I are "processes," a term you haven't defined but which has a well-known meaning in computer science with which I'm highly familiar, then a process is a computation. You can execute Euclid's algorithm with pencil and paper or on a supercomputer, it makes no difference. It's the same computation.

A computation is distinct from the entity doing the computation (even if the latter is a simulation - i.e. is itself a computation - they are different computations (and even when a computation is a simulation of itself, they proceed at different rates in unending recursion.)) — A Raybould

You're arguing with yourself here. I have never said anything to the contrary. A computation is independent of the means of its execution. What does that have to do with anything we're talking about?

I recognize that this loqution is fairly common - for example, we find Searle writing "The question is, 'Is the brain a digital computer?' — A Raybould

Searle also, in his famous Chinese room argument, doesn't talk about computations in the technical sense; but his argument can be perfectly well adapted. Searle's symbol lookups can be done by a TM.

And again, so what? You claim the word simulation doesn't mean computation; and that computation isn't a TM. That's two claims at odds with reality and known physics and computer science. The burden is on you to provide clarity. You're going on about a topic I never mentioned and a claim I never made.

And for the purposes of this discussion I am taking that question as equivalent to 'Are brain processes computational?" - but, as this quote clearly shows, this is just a manner of speaking, — A Raybould

But a computation is a very specific technical thing. If I start going on about quarks and I say something that shows that I'm ignorant of physics and I excuse myself by saying, "Oh that was just a manner of speaking," you would label me a bullshitter.

If you mean to use the word computation, you have to either accept its standard technical definition; or clearly say you mean something else, and then say exactly what that something else is.

and IMHO it is best avoided, as it tends to lead to confusion (as demonstrated in this thread) — A Raybould

I'm not confused. My thinking and knowledge are perfectly clear. A computation is defined as in computer science. And if you mean that we are a "simulation" in some sense OTHER than a computation, you have to say what you mean by that, and you have to make sure that your new definition is compatible with known physics.

and can prime the mind to overlook certain issues in the underlying question (for example, if you assume that the brain is a TM, it is unlikely that you will see what Chalmers is trying to say about p-zombies.) — A Raybould

I understand exactly what Chalmers is saying about p-zombies now that I re-acquainted myself with the topic as a result of this thread.

But you're going off in directions.

What do you mean by simulation, if not a computer simulation? And what do you mean by a computation, if not a TM?

To me, Searle's first version of his question is little more than what we now call click-bait. — A Raybould

Whatever. I'm not Searle and he got himself into some #MeToo trouble and is no longer teaching. Why don't you try talking to me instead of throwing rocks at Searle?

Some inflationary models do still posit a singularity at the beginning of time, in which case the universe had a beginning just as in ordinary non-inflationary big bang models. But there is also a model of eternal inflation, where there wasn't necessarily any start of time, just a local stop of inflation, which is the "big bang" for all intents and purposes as we usually mean it, in such a model. It seems to be in that context of eternal inflation specifically that the term "big bang" is used that way, rather than to mean a singularity.

You can start here for info on eternal inflation:

https://en.wikipedia.org/wiki/Eternal_inflation — Pfhorrest

This is the third time you've linked the same article which directly contradicts your claim.

Science aside, I don't get the philosophical objection to the possibility of an "actual infinity". As far as we can tell, the universe is consistent with the possibility of it being infinite in spatial extent: it is, at the very least, so big that our current measurements can't distinguish between how big it is and it being infinitely big. Can you elaborate on what would be wrong with supposing that it might be infinitely big (or that it might be infinitely old, etc)? — Pfhorrest

I have no philosophical objection. The claim made by you was that science posits an infinite past, which is false. If we're not doing science, you can say anything you like since it's unprovable and unfalsifiable. I'm challenging you on your claim that there's a scientific theory positing an infinite past. I challenged you to provide a link. You linked a Wiki article that directly contradicts your claim. When I noted that, you linked the same Wiki article again. And now you say you're NOT talking about science but rather philosophy,

I think the confusion is between the inflation of the universe, which is described above as being eternal into the future but not the past, i.e. the universe had a start but will have no end, and the eternal inflation field which may have caused the start of this universe, which may be eternal into the past, and may or may not be eternal into the future depending on whether it fully and universally collapsed into the vacuum of our universe. Multiverse theory says it did not do so, and continued making new universes before and after ours through quantum superposition and/or local collapse. — Kenosha Kid

You'll have to provide a link for the absurd (and false) claim that there is a reputable theory of physics positing an infinite past.

Sorry, but I have no idea what you're talking about fishfry. The stuff you claim here makes no sense to me at all. When did I say I was just kidding? — Metaphysician Undercover

In the very post I was replying to.

You know, ZF is only one part of mathematics. If axioms of ZF contradict other mathematical axioms, then there is contradiction within mathematics. In philosophy we're very accustomed to this situation, as philosophy is filled with contradictions, and we're trained to spot them. So we might reject one philosophy based on the principles of another, or reject a part of one philosophy, and so on. There is no reason for an all or nothing attitude. Likewise, one might reject ZF, or parts of it, based on other mathematical principles. — Metaphysician Undercover

Fine. Find a statement P such that there's a mathematical proof of both P and its negation. That's the only way you can demonstrate that mathematics is inconsistent. I'm still waiting.

You can start here for info on eternal inflation:

https://en.wikipedia.org/wiki/Eternal_inflation — Pfhorrest

Yes thank you. I haven't read it all yet so perhaps this point is explained. In the little bit that I read, they said that Guth said that time goes infinitely forward but not necessarily backward. And I thought that was inconsistent with what you are saying. But perhaps there are infinitely backward models too. In which case they must be metaphysical speculation rather than actual physics, because there is no theory of actual infinity in the physical world, universe or multiverse. You might as well invoke turtles all the way down as an "infinity of time into the past" in this context. There's no scientific meaning to the phrase.

ps -- My argument rules out forward infinity too as being a physical theory, rather than metaphysical speculation more suitable for the pub than the seminar. Physics has no actual infinity, period. The modern theory of infinite sets does not refer to the physical world. I believe I've even made this very point to @Metaphysician Undercover once or twice. Mathematics does not necessarily refer to the real world; and in this instance -- pending some development yet to be made -- it does not.

So just to be clear, I don't object to going backward or forward; but positing an actual infinity of time in either direction is simply non-physical. It is a metaphysical assumption. It is by definition outside the realm of science.

I don't think that I said I believe in the rationals. I was arguing using principles consistent with the rationals, so you inferred that I believe in the rationals. But arguing using principles which are consistent with one theory doesn't necessarily mean that the person believes in that theory. So I don't see your point here, I think you just misunderstood. — Metaphysician Undercover

I agree that just because you argue from certain premises doesn't mean you agree with them. But you are being disingenuous here. I could easily go back to our older discussions and show you where you accepted the rationals in order to deny $\sqrt 2$. I don't take this as a serious remark. Your prior posts don't support your claim that "I was only kidding about the rationals." You are retconning your posts and I'm not buying it.

What I argue against is inconsistency in the rules. And, if someone asked me to play chess, and I noticed inconsistencies in the rules, I would point them out. — MetaphysicianUndercover

But that is fantastic! If you have discovered a specific inconsistency in the ZF axioms, you would be famous. Gödel showed that set theory can never prove its own consistency. To make progress we must either assume the consistency of ZF; or else, equivalently, posit the existence of a model of ZF. This by the way is what some readers may have heard of in passing as "large cardinals." For example there's a thing called an inaccessible cardinal. It can be defined by its properties, but it can't be shown to exist within ZF. If we assume that one exists, it would be a model for the axioms of ZF; showing that ZF is consistent.

So this is the state of the art on what's known about the consistency of ZF.

If you have found an inconsistency, you will be famous. I'd be glad to help you express it mathematically and we can both be famous.

The problem is that so far you have not demonstrated an inconsistency in ZF. You've only made a sequence of increasingly bizarre and nihilistic assertions about mathematics, none of which are remotely true as concerning that discipline.

To show ZF inconsistent, here is what you must do: Produce a proposition $P$, a well-formed formula of the first order predicate calculus plus the axioms of ZF; such that there is a proof within ZF of both $P$ and of $\neg P$.

You made a bold claim. That's what you need to do to back it up. I'll be glad to help with the translation of your idea to math; if you actually have an idea.

I wonder what claim you think it being asserted by .999... = 1. — MetaphysicianUndercover

I don't think any claim is being asserted beyond the fact that the equation is derivable line by line from the axioms of set theory and predicate calculus. You're the one who thinks it "means" something. I have no idea what you are even thinking. The equation refers to nothing in the real world and I never claimed that it does. You're punching at a strawman.

As I've demonstrated, we can still object to a specific set of mathematical rules, using a different set of mathematical rules to make that objection. — MetaphysicianUndercover

Of course. You could use a different model of the real numbers such as the hyperreals. But .999... = 1 is a theorem in the hyperreals as well. You could try intuitionist math. .999... = 1 is most probably a theorem of intuitionist math but I confess ignorance on this point. I can never make sense of the intuitionists and it's not for lack of trying.

So if you want to work in some alternative framework I'm perfectly open to it. There are in fact a number of interesting variants of chess, too. Like the 3D chess they play on the Enterprise.

This is due to inconsistency in the rules of mathematics. Look at how many different systems of "numbers" there are. — MetaphysicianUndercover

There's no general definition of "number" in mathematics.

We do have exact definitions for natural numbers and integers, rationals, reals, complex numbers, quaternions, octonions, p-adic numbers, transfinite numbers, hypereal numbers, and probalby a lot more I don't even know about. But ironically, and confusing to many amateur philosophers, there is no general definition of number. A number is whatever mathematicians call a number. The history of math is an endles progressions of new things that at first we regard with suspicion, and then become accusotomed to calling numbers.

You know @Meta, you seen to deny any understanding of math as a social activity of humans. But that's exactly what it is. Perhaps there's a Platonic math out there and perhaps not; but either way, mathematics included the history of people who do mathematics, going back to the first cavedweller who put a mark in the ground when he killed a mastodon.

in any event there are dictionary definitions of number, but there is no general mathematical definition of number. Particular kinds of numbers, yes. Number in general, no.

I don't agree with this analogy at all. We apply mathematics toward understanding the world, and working with physical materials in the world. This is completely different from the game of chess. — MetaphysicianUndercover

You're confusing pure math with applied math. And it's true that chess doesn't apply to the world; but I could pick a better analogy. Take sailing. Recreational sailors are playing a game that has no actual consequences outside of the game. But their game arises out of the accumulated knowledge of thousands of years of sailing, most of which was done for trade and exploration. So that's a formal game, if you like, with connections to the real world.

But really, you are saying that there is no math other than applied math. You miss a lot from that perspective. And a lot of abstract pure math becomes very practical hundreds or even thousands of years later. So you can't really make the distinction you are making. Euclid studied the factorization of integers into primes; but it wasn't till the 1980s that someone had the idea of applying prime factorization to the security of digital communications. Today number theory underlies the security of the Internet. If you'd been in charge back then you'd have told Euclid to stop fooling around and build a wheel or something, and humanity wouldn't have learned any number theory and would not today be able to secure the Internet.

You're not only a mathematical nihilist. You're a mathematical Philistine. "One who has no appreciation for the arts." You deny the art of mathematics. You know nothing of mathematics.

If the principles of mathematics were not to some degree "true of the world", they would not be useful in the world. There is no such requirement in the game of chess. So it's completely acceptable to criticize the principles of mathematics when they are not "true of the world", because mathematics is used for purposes which require them to be true of the world. But the game of chess is not used in this way. So if I were to criticize the rules of the game of chess, it would be if I thought they were deficient for serving their purpose. — MetaphysicianUndercover

You're arguing that BECAUSE math is sometimes useful, it may ONLY exist if it is useful. What's your evidence for that proposition?

That's like saying that abstract art is ok as long as it's useless; but the moment anyone uses a painting to cover a hole in their wall, only practical art is permitted. You know you are speaking nonsense.

Physicists and others find math useful. That doesn't place any limits on what math can be or what mathematicians can do.

Euclid wasn't trying to solve the problem of Internet security 2200 years ago; but that's where his mathematical thinking led. You simply never know when a piece of math will eventually be indispensable, as they say, in the world.

Nobody claims that math = physics. That hasn't been true since Riemann and others developed non-Euclidean geometry in the 1840s. Surely you must know a little about this.

This is a nonsensical analogy. The rules of mathematics are used for a completely different purpose than the rules for chess. And the rules of math, to whatever degree they are not true of the world, lose there effectiveness at serving their purpose. The rules of chess are not used in that way. — MetaphysicianUndercover

You're just confusing pure and applied math. And missing the lessons of history that what is abstract nonsense in one era may well and often does become the fundamental engineering technology of a future time.

You know when Hamilton discovered quaternions, nobody had any use for them at all. Today they're used by video game developers to do rotations in 3-space. Did you know that? Are you pretending to be ignorant of all of this? That when you run the world nobody will do any math that isn't useful today?

Man you are a nihilist true.

In contemporary inflationary cosmology, the universe isn't "everything that happened after the big bang". The big bang is just an important early event in the history of the universe as we know it (which may be only a small part of the total universe, which may be infinitely old, if eternal inflation is correct). — Pfhorrest

Have you a specific reference to that meaning of the universe? I know that Lawrence Krauss says that the universe "came from nothing," meaning in a technical sense that it spontaneously big banged out of the fluctuating quantum fields. He was certainly using universe in the sense I did: whatever happened after the big bang.

Other than Penrose's cyclic conformal cosmology, there is an endless succession of universes But even then, a universe is what happens between successive big bangs.

I'd like to see a reference to the definition of the universe as something that includes the big bang plus something else, other than Penrose's theory. This would be new to me.

Secondly, do you have a reference for any physicist claiming (while doing physics, not metaphysical speculation) that the universe is infinite in duration? Did it have a beginning or is there an infinite regress? Perhaps it goes back as 1/2, 1/4, 1/8, etc., so that although there's an infinite regress yet there is also a greatest lower bound.

Since I don't believe there are any physics grants going to postdocs to investigate these questions, I must -- at the risk of appearing overly blunt -- call bullpucky on each of your two claims that (1) the word universe commonly or standardly or even occasionally means something other than whatever happened after the big bang or the most recent big bang; and (2) that anyone seriously claims the universe is infinite either in the past or the future.

If you have references I'd be glad for the education.

ps -- The Wiki page on eternal inflation contradicts your claim about an infinite past.

"Alan Guth's 2007 paper, "Eternal inflation and its implications",[3] states that under reasonable assumptions "Although inflation is generically eternal into the future, it is not eternal into the past."" My emphasis.

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

If everything is relative, than everything is crooked and there is no truth about what a person is, what he has done, and what he deserves. The world would therefore be entirely abstract and meaningless if there was no objective truth. Is this enough to prove relativism wrong? — Gregory

You're not following the news lately? There's no truth. There's just the mob, backed by spineless and complicit politicians. This won't end well.

That's interesting. So is dark matter somehow outside of the universe? If the universe is everything that happened after the big bang? Was the dark matter always there? This raises a lot of new questions.

Quantum computers are based on such an idea. In normal probability theory you take 1 and 0 to be certainty and impossibility, and a probability is a real number between them. In quantum theory, a probabilty amplitude is a complex number (that is, of the form a + bi where $i^2 = - 1$); and the square of its length is the probability that some event will be observed. They use this to implement qubits, which are bits that are neither 1 nor 0 until they are, or some such popularized explanation.

Different thread, different argument. What makes you think that I believe in any sort of mathematics? What I believe is that it's about time for a good dose of healthy skepticism to be directed at mathematical axioms. — Metaphysician Undercover

Point1: Ok a fair answer but still a deflection. The question is why you earlier believed in the rationals, but now do not believe in 1/9. Since 1/9 is a rational number, being the ratio of two integers, 1/9 is rational.

If I've caught you in a little inconsistency, or your thinking has changed, or if I'm misunderstanding this point, I'd like to understand. Specifically with respect to 1/9 and its rationality.

Point 2: Now your deflection is interesting. You changed the subject to claim that I have no reason to believe you believe in mathematics. But it's perfectly obvious that you do. I don't believe in tennis. It doesn't interest me. I don't hang out on tennis forums and tell players that their game is nonsense and their rules are unsound. I simply don't watch tennis matches and don't click on tennis-related news. The last tennis match I paid attention to was Bobby Riggs versus Billie Jean King. So when I see you passionately arguing your point -- whatever it may be, since your mathematical nihilism is hard to fathom -- I assume you must care a lot about mathematics.

Point 3: Do you regard the rules of chess as needing a "good dose of skepticism?" Why or why not? Perhaps you are putting more ontological certainly into math than math itself claims. I personally don't think that .999... = 1 is "true" in any meaningful sense. In the real world the notation isn't defined at all, since there are no infinite series because as far as we know, the axiom of infinity is false.

So YOU are the one setting up strawman claims on behalf of math, that math itself doesn't claim.

How can you complain about the rules of a formal game? How could one be "skeptical" about the rules of baseball? What does that even mean?

Sure, why would I deny this? It's been shown to me in so many different ways. — Metaphysician Undercover

But that's great. Then you and I are in absolute agreement. The proposition here is that ".999... = 1 is a logical consequence of the ZF axioms." You are agreeing with this. From a formalist perspective, it is no different than saying that a particular chess position can be legally reached.

But if you agree with this, then you and I have no disagreement. Because I make no other claims!

I wonder what claim you think it being asserted by .999... = 1. It's a statement in the formal game of modern math. You can no more object to it than you can object to the rules of chess.

So tell me: What extra secret sauce are you imbuing the symbolism with? Why do you think there's some "true" meaning out there in Platonic land? Do you think there's a real way that the knight moves and the rules of chess have got it wrong?

Can't you see that if you agree that the formalism is valid, then we're in agreement. I myself make no claim of the soundness of mathematics; only its validity. You're arguing against a strawman of your own invention.

But if you have good reason to believe that the consequence is a falsity, then it's just evidence of the faults of those axioms. — Metaphysician Undercover

No. Math isn't true or false any more than chess is true or false. If you criticize math for having rules that are not technically true of the world, you must make the exact same criticism of chess. Do you?

Do you agree, that if the the formal consequence of the axioms is to produce a falsity (whether or not you believe the present example is a falsity), then there is likely fault in the axioms? — Metaphysician Undercover

Suppose for sake of argument I say yes. The axioms of math are faulty by virtue of not being true of the world. Will you then grant me that the rules of chess are likewise faulty by virtue of not being true of the world?

...and here is the gift the crackpot gives to the world. Occasionally. — Banno

Exorcisms cheap. Or, go home and read my book, How to Keep One Step Ahead of Your Mind and you'll feel better in the morning. Or, best, three ounces or so of a decent single-malt Scotch. — tim wood

I'm a grizzled veteran of .999... threads. I was dismayed to find one had sprouted up here, but I'm powerless to resist. I've sworn off them many times without success.

Still trying to nail jello to the wall? — tim wood

@Metaphysician Undercover's in my head for sure. But I think it's a good clarifying question. Whether he accepts the formalism on its own terms even if not in any ultimate sense.

You don't seem to understand. "One" does not represent a quantity which can be divided. — Metaphysician Undercover

Curious about your 1/9 concerns. A while back you told me you believe in rationals but not sqrt(2). But now you don't seem to believe in rationals. What's up?

Secondly, can you give me a yes or no response to this question? Do you agree, either by personal understanding or by taking my word for it, that regardless of whether .999... = 1 is "true" in any metaphysical sense, it is still the case that it's a formal consequence of the axioms of ZF set theory?

A question mark? — A Seagull

And the Mysterians!

https://en.wikipedia.org/wiki/%3F_and_the_Mysterians

I’m interested in the role of fashion in regards to racism. — Pinprick

I'm reporting this to the paragraph police.

My point was that ".999..." has a different meaning from "1". InPitzotl insisted that it is two names referring to the same thing. Clearly it is not, because .999... is derived from 1/9 in the op, and 1 has a simple meaning without any such baggage. — Metaphysician Undercover

I could prove from first principles that .999... and 1 refer to the same real number. You choose not to engage with the argument. Nowhere to go with that.

I didn't claim this is a problem, that was Pitzotl''s misinterpretation. I said that if the same thing has two distinct names, there is a reason for that. — Metaphysician Undercover

Would you agree that the fact that a thing has more than one nam is no argument against the two expressions or representations designating the same thing?

I thought you were making that argument earlier but now you don't seem so sure.

In any event, I offer you this. .999... = 1 is a theorem of ZF set theory; for exactly the same reason that the knight moves the way it does in chess. There is no "truth" to the situation; rather there are only the rules of a formal game. If you made different rules you could defined .999... to be 47 and you could make the knight move differently. it would be fine.

The acceptance that .999... = 1, and of the consequences of the ZF axioms in general, is based on utility. When we accept ZF we can build up most of known mathematics and provide rigor to what the physicists do (usually a century or two after they've already done it). That's a pragmatic argument for accepting the axioms.

If you want to say that .999... = 1 offends your sensibilities, you are free to do that. As long as you are willing to grant the proposition that .999... = 1 is a theorem of ZF. That is a matter of objective fact that could be verified by a computer program. That is, there's a finite sequence of verifiable steps from the axioms to the conclusion.

That's really all it means; and even if you think that somehow "deep down" the equality is false; you must still admit that the statement follows logically from the axioms of ZF. So that you'd have to conclude either that ZF is inconsistent (which as far as we know, it may be) or that it's simply the wrong set of axioms for mathematics. In which case you're free to propose different ones.

If you want to read Chalmers' own words, he has written a book and a series of papers on the issue. As you did not bother to read my original link, I will not take the time to look up these references; you can find them yourself easily enough if you want to (and they may well be found in that linked article). I will warn you that you will find the papers easier to follow if you start by first reading the reference I gave you. — A Raybould

You're right, will do as the inclination strikes.

That is a different question than the one you asked, and I replied to, earlier. The answer to this one is that a TM is always distinguishable from a human, because neither a human, nor just its brain, nor any other part of it, is a TM. A human mind can implement a TM, to a degree, by simulation (thinking through the steps and remembering the state), but this is beside the point here.[ — A Raybould

Oh my. That's not true. But first for the record let me say that I agree with you. A TM could perhaps convincingly emulate but never implement an actual human mind. I don't believe the mind is a TM.

But many smart people disagree. You have all the deep thinkers who believe the entire universe is a "simulation," by which they mean a program running on some kind of big computer in the sky. (Why do these hip speculations always sound so much like ancient religion?) We have many people these days talking about how AI will achieve consciousness and that specifically, the human mind IS a TM. I happen to believe they're all wrong, but many hold that opinion these days. Truth is nobody knows for sure.

I've read many arguments saying that minds (and even the entire universe) are TMs. Computations. I don't agree, but I can't pretend all these learned opinions aren't out there. Bostrom and all these other likeminded characters. By the way I think Bostrom was originally trolling and is probably surprised that so many people are taking his idea seriously.

If you had actually intended to ask "...indistinguishable from a human when interrogated over a teletype" (or by texting), that would be missing the point that p-zombies are supposed to be physically indistinguishable from humans (see the first paragraph in their Wikipedia entry), even when examined in the most thorough and intrusive way possible. This is a key element in Chalmers' argument against metaphysical physicalism. — A Raybould

I'm perfectly willing to stipulate that a p-zombie is physically indistinguishable. I made the assumption, which might be wrong, that their impetus or mechanism of action is driven by a computation. That is, they're totally human-like androids run by a computer program.

If you are saying the idea is that they're totally lifelike and they have behavior but the behavior is not programmed ... then I must say I don't understand how such a thing could exist, even in a thought experiment. Maybe I should go read Chalmers.

As a p-zombie is physically identical to a human (or a human brain, if we agree that no other organ is relevant), then it is made of cells that work in a very non-Turing, non-digital way. — A Raybould

You and I would both like to believe that. But neither of us has evidence that the mind is not a TM, nor do we have hours in the day to fight off the legion of contemporary intellectuals arguing that it is.

Roger Penrose is famous for arguing that the mind is not a computation, and that it has something to do with Gödel's incompleteness theorem being solved in the microtubules. Nobody takes the idea seriously except as a point of interest. Sir Roger's bad ideas are better than most people's good ones.

But you can't be unaware that many smart people argue that the mind is a TM.

We don't know of anything that works in a "non-Turing, non digital way." There are mathematical models of hypercomputation or supercomputation in which one assumes capabilities beyond TMs. But there's no physics to go along with it. Nobody's ever seen hypercomputation in the physical world and the burden would be on you (and I) to demonstrate such.

Chalmers believes he can show that there is a possible world identical to ours other than it being inhabited by p-zombies rather than humans, and therefore that the metaphysical doctrine of physicalism - that everything must necessarily be a manifestation of something physical - is false. — A Raybould

I recall this argument. It's wrong. If our minds are a logical or necessary consequence of our physical configuration, and a p-zombie is identical to a human, then the p-zombie must be self-aware.

Otherwise there is some "secret sauce" that implements consciousness; something that goes beyond the physical. Some argument for physicalism. You just refuted it. Maybe I"m misunderstanding the argument. But if the mind is physical and a p-zombie is physically identical, then a p-zombie has a mind. If a p-zombie is physically identical yet has no mind, then mind is NOT physical. Isn't that right?

Notice that there is no mention of AI or Turing machines here. — A Raybould

Only, in my opinion, because not every philosopher understands the theory of computation.

What animates the p-zombie? If it's a mindless machine, it must have either a program, or it must have some noncomputable secret sauce. If the latter, the discovery of such a mechanism would be the greatest scientific revolution of all time. If Chalmers is ignoring this, I can't speak for him nor am I qualified to comment on his work.

P-zombies only enter the AI debate through additional speculation: If p-zombies are possible, then it is also possible that any machine (Turing or otherwise), no matter how much it might seem to be emulating a human, is at most emulating a p-zombie. — A Raybould

As I mentioned earlier, it's entirely possible that my next door neighbor is only emulating a human. We can never have first-hand knowledge of anyone else's subjective mental states.

I still want to understand what is claimed to animate a p-zombie. Is it a computation? Or something extra-computational? And if it's the latter, physics knows of no such mechanism.

As the concept of p-zombies is carefully constructed so as to be beyond scientific examination,
p/quote]

Ah. Perhaps that explains my unease with the concept. My understanding is that p-zombies are logically incoherent. They are identical to human enough to emulate all human behavior, but they don't implement a subjective mind. In which case, mind must be extra-computable. Penrose's idea. I tend to agree that the mind is not computable. But how do p-zombies relate?
— A Raybould

such a claim may be impossible to disprove, but it is as vulerable to Occam's razor as is any hypothesis invoking magic or the supernatural. — A Raybould

I think you've motivated me to at least go see what Chalmers has to say on the subject. Maybe I'll learn something. I can see that there must be more to the p-zombie argument than I'm aware of.

ps -- I just started reading and came to this: "It seems that if zombies really are possible, then physicalism is false and some kind of dualism is true."

https://plato.stanford.edu/entries/zombies/

This tells me that my thinking is on the right track. If we are physical and p-zombies are physically identical then p-zombies are self-aware. If they aren't, then humans must have some quality or secret sauce that is non-physical.

I would raise an intermediate possibility. The mechanism of mind might be physical but not computable. So we have three levels, not two as posited by the p-zombie theorists:

* Mind is entirely physical.

* Mind is entirely physical but not necessarily computable, in the sense of Turing 1936. It might be some sort of hypercomputation as is studied by theorists.

* Mind might be non-physical. In which case we are in the realm of spirits and very little can be said.


pps --

AHA!!

" Proponents of the argument, such as philosopher David Chalmers, argue that since a zombie is defined as physiologically indistinguishable from human beings, even its logical possibility would be a sound refutation of physicalism, because it would establish the existence of conscious experience as a further fact."

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

This is exactly what I'm understanding. And I agree that I probably made things too complicated by inserting computability in there. The p-zombie argument is actually much simpler.

Well that counts as research for me these days. A couple of paragraphs of SEP and a quote-mine from Wiki. Such is the state of contemporary epistemology. If it's on Twitter it's true.

With you top of the pile. — Professor Death

"Better to reign in Hell than serve in Heaven" -- Milton, the stapler guy in Office Space.

Seems you have a short memory. What we previously determined is that I do not believe that 2+2 is the same thing as 4. Remember? You argued that 2+2 is identical to 4, ignoring the difference between equivalent and identical. — Metaphysician Undercover

I remember all too well, which is why I'm not joining in with the rest of the gang arguing with you. Nice job trolling them all though. Your objection isn't to the theory of convergent infinite series of real numbers. Your objection starts with 2 + 2 = 4. You seem to agree.

I surely didn't ignore the difference between equivalent and identical. On the contrary I wrote many posts carefully explaining the difference; and showing through actual mathematical proof, as well as other methods, that 2 + 2 and 4 designate the same mathematical object. One object, two names.

You do agree that a single object or thing may have more than one name. Earlier you claimed this as some kind of problem. Your position seems absurd on its face.

It's not nonsense to say that what you need to do both is eyes, hands, a knife, and the ability to coordinate what you see with your eyes and the knife in your hand. You have to be much better at that to be a surgeon than to be a kid stabbing a doll, but the kinds of things you need are the same, differing in quality, not kind. — Pfhorrest

Ok. I'm willing to agree that a couple of kids dribbling a basketball on the playground differs from an NBA game only in quality and not in kind. I don't actually believe that, but it's at least arguable.

But what professional philosophers do and what people on an online forum like this do is, I'd maintain, different in kind and not just in quality. I guess we can agree to disagree because it's ultimately a matter of opinion. All I can say is that I've read some papers on philosophy and I've read a lot of posts on this forum and I can't imagine changing my opinion on this.

Suppose a patient attends a psychiatry assessment and says "Doc I'm insane." The doctor asks why they believe such a thing. They continue to explain quite reasonably several reasons why they are insane. The doctor agrees that these signs and behaviours indicate insanity. Except now (s)he is face with a clinical dilemma.
If he agrees with the patients beliefs about the concept of insanity is the patient insane? Surely only a sane person could describe insanity and I couldn't imagine a truly insane person ever admitting they arent perfectly fine. — Benj96

Surely you've read Catch-22. If you're crazy, you can get out of flying more bombing missions. But if you don't want to fly any more bombing missions that shows you're sane!

If that's the case, then why have two distinct representations for one and the same thing? — Metaphysician Undercover

I'd point out that 2 + 2 = 4, but we've previously determined that you don't even believe that.

They compute limits which are not the same as sums. A limit is what a finite sum converges to. — EnPassant

The sum of an infinite series is defined as the limit of the sequence of partial sums of the series.

I don't understand your objection. You are entirely right that there is no such thing as an infinite sum until we define it. And how do we define it? We first define the limit of a sequence, then we define the sum of an infinite series as the limit of sequence of the partial sums. To me this seems rather clever. We have something that at first makes no sense, then we MAKE it make sense with a clever definition. What exactly are you objecting to?

Earlier on someone wrote a very convincing 'proof':
x = 0.999...
10x = 9.999...
10x = 9 + 0.999...
10x = 9 + x
9x = 9
x = 1

All well and good. But what does x = 0.999... mean? In terms of infinite sums let the sum be S.
The last two lines give:
9S = 9
S = 1. — EnPassant

This standard "proof" is of course bullpucky. It's true, but not actually a proof at this level. Why? Well, as you yourself have pointed out, the field axioms for the real numbers say that if $x$ and $y$ are real numbers, then so is $x + y$. By induction we may show that any finite sum is defined. Infinite sums are not defined at all.

To define infinite sums, we do the following:

* We accept the axiom of infinity in ZF set theory, which says that there is an infinite set that models the Peano axioms. We call that set $\mathbb N$. The axiom of infinity is a very powerful assumption that allows us to get higher mathematics off the ground. However, the axiom of infinity is manifestly false in the physical world. It's precisely at this point that mathematics diverges from physics. It doesn't matter how useful math is for physics. One must realize that no mathematical truth can have ontological significance in the physical world. The fact that .999... = 1 is in the end no more meaningful than asking why the knight moves as it does in chess. It's just a consequence of the rules of a formal game.

* Having modeled $\mathbb N$ within set theory, we use equivalence relations to build up the sets $\mathbb Z$ and $\mathbb Q$ of integers and rationals, respectively.

* Using Dedekind cuts (or any of a number of other constructions) we define the real numbers $\mathbb R$ and show that they are a complete, totally ordered, Archimedean field. Such a field is categorigal, meaning that any other such field is isomorphic to the one we've constructed. Complete in this context means that the reals have the least upper bound property, which says in effect that there are no "holes" in the real numbers. This is the property that characterizes the reals and distinguishes them from the other famous densely ordered set, the rationals.

* Having rigorously defined the real numbers and shown that they are essentially unique, we then define the limit of a sequence of real numbers via the usual epsilon definition.

* Having defined the limit of a sequence, we then define (as you have pointed out) the sum of an infinite series as the limit of the sequence of partial sums.

* Having done all that, we then prove a theorem that says that if we have a convergent sequence, we can multiply each of its terms by a constant, and the resulting sequence converges to that constant times the sum of the original sequence.

That last theorem is the ONLY WAY to justify $10 \times .999... = 9.999...$

So using that proof requires mathematical principles and reasoning far more sophisticated than the mere fact that .999... = 1. This "proof" is at best a heuristic for beginners. I wouldn't object to it if it were presented this way. But every time someone writes $10 \times .999... = 9.999...$, they are implicitly invoking the theorem on term-by-term multiplication of a convergent infinite series by a constant; and they are leading students into confusion.

As you have noted, addition in the real numbers is only defined for finite sums. To define infinite sums requires a whole lot of technical machinery based ultimately on the axiom of infinity. We must in fact make a powerful conceptual leap, one that contradicts everything we know about the real world, to get a satisfactory theory of the real numbers.

Same remarks for the 1 = 3 x 1/3 = 3 x .333... = .999... proof. A heuristic for beginners, but hopelessly bogus as an actual mathematical proof. Not because it's wrong, but rather because it is secretly invoking mathematical principles that are deeper and more sophisticated than the fact claiming to be proved.

I understand the mandatory discerning nature of this forum — Outlander

I see little evidence of such.

A particularly stupid and offensive comment. — unenlightened

The competition's stiff around here.