So I asked the question: what is an example of the types of things that don't have a beginning? Because I can't really see why an argument based on this is 'disingenuous'. — Wayfarer

The uncaused cause is God. That would be Craig's point. If you find it sensible I guess we'll agree to disagree.

hey fishfry, what kinds of things exactly don't 'begin to exist'? I think if you look at any object in the known universe, then all of them 'began to exist' at some point in time, didn't they? Even atoms began to exist, we are told. So, without any hand-waving, what kinds of things, generally speaking, don't begin to exist? — Wayfarer

I'm stating William Lane Craig's argument in order to characterize it as disingenuous and silly. Was that unclear in my post?

"Premise one: "Whatever begins to exist has a cause."

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

Premise-1: Everything in the world has a cause. — Brian A

Disclaimer, I haven't read the rest of the thread and I don't know where it's gotten to by now.

I did happen to look at William Lane Craig's cosmological argument a while back. His first premise is:

* Everything that begins to exist must have a cause.

Eventually his argument gets around to saying that there must be something that never BEGAN to exist ... it has always existed. That must be God.

Now this is a very disingenuous argument. The phrase "begins to exist" sounds a little odd and most people just dismiss it. But it is actually a sneaky rhetorical maneuver. Because there are now two classes of things in the universe: things that "began" to exist, and therefore have causes; and things that exist but never actually "began" to exist. That thing would be an uncaused cause, which we call God.

That in effect is Craig's argument, and frankly it's silly. The conclusion is baked into the premise. If you right away distinguish between things that "began" to exist and things that didn't; it's easy to wave your hands and obfuscate around for a while and then end up proving the existence of something that exists but did not "begin" to exist.

Your paraphrase misses all of this and it's therefore not Craig's argument.

As far as whether there can be a chain of things infinite to the left, I like the simple mathematical example of the negative integers: ..., -3, -2, -1. This anti-sequence or backwards sequence has an order type called *ω, pronounced "star omega." That's because ω is the order type of the natural numbers; and *ω is the reverse of ω.

*ω provides us with a perfectly sensible mathematical model of a system in which each thing has a predecessor and there is no firstmost element. I'm not saying that's how the universe works. I"m just saying that Craig can't say it doesn't. If people would contemplate the negative integers, they would gain insight and familiarity with infinite regress. It's really no more complicated or strange than the fact that if you have a negative number you can keep on subtracting 1 as many times as you like, and you'll never get to the beginning. Because there is no beginning.

Craig's argument is just sophistry. I'm astonished that anyone takes his argument seriously.

On the other hand your own argument or paraphrase is not Craig's argument; and I haven't looked at yours in detail. So I can't tell if you mean to give your own version of the argument or are inaccurately paraphrasing Craig.

Either way, infinite regress ain't no thang. Just subtract 1. Or take one step left on the number line. And here's another mathematical model. Start with 1 and keep dividing by 2. You get 1/2, 1/4, 1/8, 1/16, ... Once again, there is no first element. Although you can take the limit and say that 0 is God. If that makes anyone happy.

I think we also want a way to talk about fiction (hypothesis, supposition, etc) "in world." So there are two answers, say, to "Does Santa fly in a helicopter?" One is "No, because he doesn't exist," but another is "No, it's a sleigh pulled by flying reindeer." Both have their use. — Srap Tasmaner

I think it might be helpful to distinguish the issue of whether existence is a predicate, with the issue of how philosophers handle fiction. That latter in itself is a huge topic.

If I say, "Ahab is the cabin boy of the Pequod," that statement is false. Yet Ahab does not exist, he's a fictional character. And someone could write fan fiction in which Ahab is the cabin boy, or the whale, or Gregory Peck. ("He tasks me. He heaps me!")

Now analyzing the truth values of statements about fictional characters is a deep business, but it's not really the same question as whether existence is a predicate.

After all, "Ahab is captain of the Pequod" is a shorthand for: "In Melville's novel Moby Dick, Ahab is captain of the Pequod." That statement is true, and there is no ambiguity or confusion. So perhaps one way out of the fiction dilemma is to fully qualify all statements. "Kirk is captain of the Enterprise in the original Star Trek tv show."

But all of this discussion of the truth values of statements about fictional entities, are red herrings (IMO) in the discussion of whether existence is a predicate.

How about if you wanted to state that a set contains at least one member? — Joseph

∃x∈S (x ∈ S)

That would be the formal way, since x ∈ S is a predicate.

The only sensible way to write it is ∃x∈S. — Joseph

Technically that is a convenient abuse of notation. Everyone writes it that way, everyone knows what it means. But it's not actually correct if one is being precise.

That is a case of a standalone quantifier. — Joseph

No. Besides, it has a completely different form than your earlier example of writing ∃x without any predicate at all. You are giving two completely different examples. ∃x by itself is is meaningless. ∃x∈S is technically wrong but everyone understands that it's a shorthand for ∃x(x∈S).

Next, how would you state that a set is empty in predicate logic? — Joseph

¬∃x(x∈S).

∀x∈S[¬∃x] — Joseph

I do not believe the existential (or universal) quantifiers may stand alone. Rather there must be a unary predicate P so that you can write ∃xP(x).

If you have a reference illustrating a use of a standalone quantifier like ∃x I would appreciate a link or specific reference.

An expression such as ¬∃x is meaningless, as is ∀x∈S[¬∃x]. That's just not a well-formed formula in predicate logic.

If the general form of the existential quantifier is ∃xP(x) then in your example you are saying that ¬∃x may be a predicate. So you are assuming the thing you are trying to prove. In fact ¬∃x is not a predicate.

Well, I too think my grasp of Russell's paradox isn't up to mark to continue the discussion into anything fruitful. — TheMadFool

Ok. But now that I've explained it to you, your understanding should be excellent :-)

As for time being involved in logic, I think I'm correct. — TheMadFool

It's true that there are logics where time is modeled, for example temporal logic. But ordinary sentential logic doesn't involve time. Can't add any more, it's all out there on Wiki.

I'm saying if we simply switch from ''pause mode'' (sentential logic) to ''play mode'', as we do when watching movies, the contradiction disappears because time changes. — TheMadFool

Yes I think I see your point. If we think of it as a process, as in the execution of a computer program, then we are just flipping states back and forth. Is that what you mean?

The problem is that we're no longer doing standard logic and set theory. We're doing something else.

The Wikipedia quote you gave seems misleading. "At the same time" doesn't really have to do with time. Perhaps the Wiki article misspoke itself. You can't take an inaccurate sentence in Wikipedia and extrapolate an entirely different meaning of sentential logic than the rest of the world uses.

I still want to clarify. Are you confused about standard logic or just objecting to it? Do you understand that you say one thing and the entire rest of the world says another?

That's ok if that's what you're doing. But if you think your understanding is the standard one, you should study some logic so at least you can understand what you're objecting to.

Also when you quote Wiki, can you please link the article? I'd like to see the context of that quote since it is profoundly misleading. There is no time involved in sentential logic. It's really important for you to understand that.

ps -- I should mention for clarity that when it comes to the Liar paradox, I have no opinion and virtually no interest in the topic. I take no position with your reformulation of the problem in terms of time and causality.

But on Russell's paradox, you are factually wrong on both the history and the math. Russell's paradox and set theory take place in the context of logic in which time and causality are not relevant. Russell's paradox is a historical event in the early history of set theory and it means what it means and not something else.

— TheMadFool

X being inside AND outside is a contradiction. But the rule doesn't say that. It simply alternates the two states on a timeline, at different points of time. — TheMadFool

Logical implication is not causation. They're two completely different things.

Have you ever seen the truth table for implication?

I am wondering, do you think your ideas are how logic works? Or do you understand that you are making up your own logic that's different than standard logic?

There is no time or causation in sentential logic. If P and Q are propositions, we say P => Q is true in case either P is false or Q is true. That's all it means. There need be no causality or connection between them. If 2 + 2 = 5 then I am the Pope. That's a true implication, because the antecedent is false.

This is not a difference of opinion between us. The way I described it is how sentential logic is done.

If you're using some other system of logic like MF logic, please make that clear.

If you are under the impression that standard sentential logic is as you say it is, you need to understand that you are wrong about that. There is no time and no causality in sentential logic or set theory.

IF you assume that ''set contains itself'' and ''the set doesn't contain itself'' imply each other THEN we have a contradiction. — TheMadFool

A contradiction in sentential (aka propositional) logic, a contradiction is the statement "P ^ not-P" for some proposition P.

There is no requirement that they "imply each other." This is something you are making up.

I don't mean to say that you are wrong. You are entitled to make up your own rules of logic, or to use the word "contradiction" in a different way than logicians do.

But when you do so, you are abandoning the generally agreed-on meaning of contradiction.

Do you understand this? Again, I'm very open-minded. Tell me the rules of your system of logic and we'll play. But we're not doing standard logic then. Because in standard logic, a contradiction is when you can prove P and also not-P.

But contradictions are impossible — TheMadFool

On the contrary we use contradictions all the time. You could hardly do math without the famous technique of proof by contradiction, or reductio ad absurdum. The idea is to assume something and show that the assumption leads to a contradiction. We then conclude that the thing we assumed could NOT have been true.

Everyone's seen the classic proof (which I won't reproduce here) that the square root of 2 is irrational. We start by assuming sqrt(2) is rational, and we derive a contradiction. We conclude that sqrt(2) can not be rational.

In the case of Russell, we assume that we may form a set by taking all the objects that satisfy some predicate. For example if G(x) stands for the statement "x is a giraffe" then we may form the set {x : G(x)}. This is read as "The set of all x such that x is a giraffe." In other words this is the set of all giraffes.

It's intuitively tempting to want to say that we can always form a set from a predicate. But in fact we can NOT do this; because if we let P(x) mean "x is not an element of x" then we get a contradiction.

This shows that we may NOT in fact be so permissive with predicates.

The solution in formal set theory is the axiom schema of specification, which says that if X is a set and P is a predicate, we may form the set {x in X : P(x)}. In other words if we start with a set we can always cut it down by a predicate. But we can not just have a predicate by itself. That way leads to contradiction.

Russell himself proposed type theory, which remained somewhat of a backwater for most of the twentieth century, but is now getting renewed interest from the field of computer science.

When you say "contradictions are impossible" and then conclude that they "don't imply each other," that is not standard logic, that's Mad Fool logic, or MF logic if you will.

In standard logic, the contradiction shows that we can not use unrestricted comprehension, as it's called, to form sets.

Therefore, as for the liar paradox, there's a time gap between ''it contains itself'' and ''it doesn't contain itself''. — TheMadFool

No. The assumption that we can form such a set leads to a contradiction, showing that unrestricted comprehension can not be allowed. Please read the Wiki article on Russell's paradox that I linked earlier.

That's not Russell's paradox.

I think I understand now. Small differences in initial states have vastly different outcomes. For example, the temperature may differ by 0.000007 degrees but this tiny difference can mean the difference between fair weather and storms. The butterfly is simply a metaphor for this small difference in a variable. — TheMadFool

Yes exactly.

Our intuition is that a tiny change in the input conditions will smooth out over time and make no difference to the future evolution of the system. But it turns out that our intuition is wrong. Even a tiny change in the initial conditions can lead to a hugely different outcome in the future.

The butterfly story is misleading in that respect. It makes people think that the flapping wings "cause" a storm. That is not true. What is true is that the universe in which the butterfly flaps its wings, and the universe in which the butterfly does not flap its wings, may be profoundly different after a number of iterations of deterministic rules.

Isn't that the gist of your post? — TheMadFool

If you quote you should quote literally. If you are paraphrasing someone, you should not make it look like you're quoting them. A stylistic nitpick, to be sure. Something professional journalists do ... when you put something in quotes it should be the exact words the person said. Not the reporter's paraphrase, which is bound to include the reporter's own worldview and spin.

In this case I also feel that you misconstrued my words. Your paraphrase changed my meaning.

You attributed to me the claim that "Small changes making big differences."

But this is absolutely false. There is no "making" involved. There is no causal relation. It's just that nearby points in the plane have qualitatively different behavior under iteration. Likewise similar states of the universe may nevertheless evolve in profoundly different ways.

Nothing is "making" anything change; and if this is what you took from what I wrote, then I did not express myself with sufficient clarity.

Nearby points may have huge differences in their evolution.

That is what I said, that is all I said. That's what chaos is about, that's what the Mandelbrot set is about and that's what this butterfly story is about. The butterfly story is only a simplification for the public. It's not to be taken as meaning that anything causes anything. It represents a more subtle truth that isn't always understood by thinking the butterfly causes anything.

I did my best to explain this in my post but perhaps I can do better if you tell me why you think anything "makes" anything here. Nearby points may have radically different fates under iteration. That's all.

↪fishfry Small changes making big differences.

So, can I change the fate of the universe by blinking my eye? — TheMadFool

Why are you quoting words I didn't say in my post?

I think this would be better

"it is impossible to calculate the long term effect on the weather of a butterfly flapping its wings". — Jake Tarragon

A more precise statement can be made. It is that under a deterministic iterated system, points (or states) that start out very close together may end up with radically different fates. One point might remain bounded, while a nearby point spirals off to infinity.

With respect to the butterfly:

If you have two universes that are exactly the same; but in one the butterfly flaps its wings, and in the other it doesn't; then when you let your deterministic world run according to its rules and equations, those two universes may have vastly different fates. One may be coherent and stable and the other may disolve into randomness. One may support life and the other not.

It's not about causation. It's about the behavior of nearby points under iterated deterministic systems.

It's counterintuitive. We think that if we have some deterministic system and two points (or states) start out close together, their behavior can't be too different under iteration of the rules. But this turns out to be false. Nearby points can have vastly different futures.

We think that "a tiny difference will dissipate or smooth out over time." But the opposite is true. The tiniest difference in the initial conditions can lead to dramatically different futures.

The classic example is the Mandelbrot set. We've all seen those beautiful pictures. Here's a nice specimen. Hopefully this will post the link and not the image, which is fairly large and worth clicking on.

http://upload.wikimedia.org/wikipedia/commons/a/a4/Mandel_zoom_11_satellite_double_spiral.jpg

Without going into detail, the Mandelbrot set (the part traditionally rendered in black) is the set of points that remain bounded under a given iteration. All the other points eventually wander off to infinity. [That just means they are unbounded, there's nothing mystical or actually infinite going on here, it's just an expression meaning unbounded].

The part of the plane that isn't black, the complement of the Mandelbrot set, can be colored like this: You color each point according to how many iterations (within some range) it takes to get a distance of 2 from the origin. And that assignment of colors results in the wild Mandelbrot pictures we see, which have all this crazy fractal detail at every possible zoom level.

Wikipedia has an excellent writeup. https://en.wikipedia.org/wiki/Mandelbrot_set

And the point is that you have a perfectly deterministic and actually very simple rule that lets you trace out the evolution of a point under continued iteration of the rule. But points that are very close together can end up with wildly different fates. Some remain forever within a circle of radius 2. And others wander off to infinity. [Once an iteration goes outside the circle of radius 2, it will eventually go to infinity].

That's what the butterfly story is illustrating. The universe in which the butterfly flaps its wings; and the universe in which the butterfly doesn't flap its wings; may have wildly and qualitatively different evolutions.

Yet if you step back you can see much deeper patterns in the apparent randomness. That's chaos.

As a historical note, all of this was anticipated and visualized in the mind of Henri Poincaré. He discovered chaos in the 1880's, a century before anyone dreamed of graphical software that could render images of these kinds of sets.

It seems you don't understand either category theory (at a philosophical level) or semiosis and are just seeking to nitpick with contradictory sounding quotes. — apokrisis

Your erudition seems to have overtaken your common sense and your manners. You are incapable of explaining, only insulting. I'm using individual quotes as an alternative to copy/pasting pages and chapters of Zalamea. His entire thesis is that category theory has resurrected Peircean synthesis. Now you are right, I'm just trying to learn what this means. But your unwillingness to explain anything of your jargon-filled posts says something about you.

Is it time for me to say fuck you to you again? I've had enough. Fuck you.

Hot damn! ;) — apokrisis

In the Zalamea paper on Peirce's continuum, Zalamea says on page 8:

"As we shall later see, this synthetical view of the continuum will be fully recovered
by the mathematical theory of categories, in the last decades of the XXth century."

"This" above refers to Peirce's concept of the continuum. So Zalamea's understanding of category theory seems radically different from yours.

If it is a theorem that has been proved, then it follows, does it not? What am I missing? — aletheist

Your full quote earlier was:

For any value of N whatsoever, 2^N > N. Therefore, a power set always has more members than the set from which it is derived. This is true even if N is infinite; the power set of the natural numbers must have more members than the (infinite) set of the natural numbers. — aletheist

There is only one way to read this.

* For any value of N whatsoever, 2^N > N.

* Therefore, a power set always has more members than the set from which it is derived. This is true even if N is infinite.

That is flat out false. You said "therefore ..." and that's wrong. Surely you see that. The correct statement is: "And in fact we can prove that this holds for infinite sets as well." In fact the very definition of 2^N needs to be changed to make this work. Rather than talking about natural number exponentiation, we now change the meaning to redefine 2^N as the collection of subsets from a set of cardinality N to the set 2.

There's no "therefore" in this. You need to make a new definition of the notation and then prove a theorem. You started with the expression 2^N meaning natural number exponentiation, then changed definitions in midstream to redefine 2^N as the powerset of N.

It's a bad argument. Suppose I like to juggle, and the local government forbids juggling. I say, "I have a right to juggle, especially in the privacy of my home." And the answer comes back, "Juggling is not innate to humans. There is no juggling gene. Therefore we may legitimately forbid juggling."

That's the reason the "innateness" argument is a terrible argument to oppose laws forbidding homosexuality. A far better principle is "consenting adults." I have the right to juggle in my own home since I'm not bothering anyone else and I'm an adult capable of making that choice.

Therefore, a power set always has more members than the set from which it is derived. This is true even if N is infinite; the power set of the natural numbers must have more members than the (infinite) set of the natural numbers. — aletheist

That does not follow. It must be proved. That's Cantor's theorem. Well worth looking at since it's a beautiful little proof that gives us an endless hierarchy of transfinite cardinalities.

the irreducible triadicity of a sign relation. — apokrisis

Well when you put it THAT way it's totally CLEAR. LOL.

I see that Peirce has some jargon associated with him. From Googling around I think being triadic is what a mathematician would call ternary, a relation that inputs three objects and outputs T or F. Like equality is binary, it inputs (5,3) and outputs F, inputs (2,2) and outputs T. A ternary relation takes three inputs. Does that have anything to do with this?

If you can briefly explain some of these technical terms it would help.

I wanted to mention that at one point Zalamea basically says that Peirce is doing category theory, or category theory is Peircean. Now I still don't know what Peirce is about, but this is a fantastic connection. I've had some exposure to category theory. Not much but enough to know that modern math is done very differently than anything you see as an undergrad math major. Equations are replaced by arrow diagrams. It's a very different point of view. I've always understood category theory to be loosely related to structuralism. We no longer care what things are made of, we care about their relationships to other things; and about very general patterns in those relationships that tie together previously unrelated areas of math.

Why have I never heard of Peirce before? I've been in lots of online discussions about the philosophy of math, and I've heard the name but never knew he anticipated the math of the future in some deep way.

ps ... I'm randomly reading sections in Zalamea and I come to this: "The triadic Peircean phenomenology ..."

To read that phrase used by an author who speaks so knowledgeably about Grothendieck ... this is breathtaking. Why isn't Zalamea famous? He doesn't even have a Wiki page. This guy has moved the philosophy of math forward fifty years.

I still don't know what triadic Peircean phenomenology is. Can this be explained simply?

Just struggling a bit with how 'sign relations' come into the picture outside of biology..... — Wayfarer

Biology, that's interesting. I thought sign relations were some kind of postmodern talk I don't know anything about other than that Searle thinks Derrida is full of sh*t. That much I know.

How could a set have a cardinality if it's not possible to enumerate that entire set? — Metaphysician Undercover

This is actually a great mathematical question. If you assume the Axiom of Choice (AC), then all sets have a well-defined cardinality that is the smallest ordinal that bijects to the set. If you take AC as false, then there exist sets that can't be well ordered, hence sets that don't have well-defined cardinalities. Or if they do, they're not comparable to the standard Alephs. I'm a little fuzzy on that point.

You have good insight into this. For a set to have a sensible cardinality, it needs to be able to be enumerated.

This is so only because in principle the set of, say, even numbers, can be placed in one to one correspondence with the set of both even and odd numbers. — John

I don't know what "in principle" means. If we are in standard set theory, we can biject the naturals to the evens by mapping each natural n to the natural 2n. If we are not in standard set theory, then you'd have to say what the rules are for that system.

It doesn't make any sense to say that "in principle" you can do something that's legal in set theory. If set theory says you can do it, you can do it. If you are using some other framework for talking about numbers and sets, you have to say what that is.

To make this clear, I interpret the phrase "in principle" as indicating a lack of clarity in specifying what domain we are in. If we're in set theory we can biject, and if we're not, what are the rules? It's like sitting down to play chess and they won't tell you how the pieces move. Tell me what your rules are for defining functions between sets, and I'll tell you whether there's a bijection between the naturals and the evens.

If you read up on Peirce's synecheism - as his model of continuity - it gets clearer. The "continuity" then is of the systematic "constraints plus freedoms" kind that I employ. — apokrisis

I started reading the Zalamea paper, Googled around, and found a pdf of his awesome book Synthetic Philosophy of Contemporary Mathematics. I'm enthralled. A mathematician who actually understands the conceptual revolution that happened in math in the second half of the twentieth century; and writes brilliantly and clearly about philosophy. And he's written a lot about Peirce as well. So at the moment I'm coming to Peirce by way of Zalamea's exposition of the mathematical philosophy of Grothendieck. This is sublime. I truly thank you for this reference.

Different point ... when you talk about points coming in and out of existence, that reminds me of the intuitionists. Which I regard as a somewhat mystical strain of thought. For them, real numbers come into existence when they are thought of or needed or used. In the modern incarnation, a number comes into existence when it's computed. Intuitionism is coming back into style.

Now here is my question.

* It's my understanding that the intuitionistic real line has fewer points than the standard real line. After all only countably many reals are computable, for the reason that there are only countably many Turing machines. Where the standard real line has noncomputable numbers, the intuitionistic line has holes.

* It's also my understanding that the hyperreal line (this is the only nonstandard model of the reals I know anything about) has MORE numbers than the standard reals. After all the hyperreals have all the real numbers, plus a cloud of infinitesimals around each one. (I've heard these are Leibniz's monads, but I don't know anything about Leibniz, being more of a Newton fan. Maybe that explains a lot :-))

* So my question is: Is Peirce a restrictionist, squeezing the noncomputables out of the standard reals and only creating reals when they pop into his intuition; or is he an expansionist, blowing wispy clouds of infinitesimals onto the real line?

I will take your word for it, but my understanding is that the precise nature of the relationship between Peirce's continuum and nonstandard analysis is still not fully settled. — aletheist

You can take my word for it that .999... = 1 is a theorem of nonstandard analysis. But actually you don't need to take my word, I provided a proof above. The nonstandard reals are a model of the first-order theory of the reals. .999... = 1 is a theorem of that theory, hence it's true in any model of that theory. That's Gödel's completeness theorem, as opposed to his more famous incompleteness theorem. If you have a syntactic proof of a statement from some axioms, that statement is true in every model of those axioms,. The converse holds as well. If you don't have a proof, then there are models where it's true and models where it's false.

Please ask for more detail if this proof isn't clear (and you care one way or the other).

I'm afraid I don't know anything about Peirce's continuum but I will certainly try to educate myself in the near future. But remember, any field containing infinitesimals must be incomplete. Does Peirce know his continuum has holes in it? It's logically necessary.

On an unrelated meta-note, it would not be good to allow this discussion to degenerate into a .999...-fest. That's a tremendous distraction.

We typically treat the two values as equal, but arguably there is an infinitesimal (non-zero) difference between them. As you might have guessed, the Peircean continuum is non-Archimedean. — aletheist

As it happens, .999... = 1 is a theorem even in nonstandard analysis. This is easily shown. The hyperreals are a model of the first-order theory of the reals; and .999... = 1 is a theorem of that theory. You see we can reason logically about infinitesimals and we just debunked the claim that the hyperreals (a particular flavor of non-Archimedean field) invalidate .999... = 1.

Another problem with infinitesimals as a model for the continuum is that any field containing infinitesimals must necessarily be topologically incomplete. There are Cauchy sequences that don't converge. Any real line containing infinitesimals is shot full of holes. This is fairly easy to prove.

Last year I was in one of those endless and moronic .999... debates (I have a firm personal policy NOT to get involved in those, but on that one occasion I broke my own rule and needless to say ended up regretting it). As part of that debate I got tired of always hearing about the hyperreals so I went and learned a little about them. I can tell you for a fact that they will not help anyone's argument that .999... is anything other than 1, if you give those symbols their standard meaning. Of course if you change the interpretation of the symbols you can have .999... = 47 if you like. That's another point that's generally missed. If you interpret the symbols using their standard technical meaning, then .999... = 1. There's just no question about it. The only question is how you're allowed to manipulate the symbols.

I'm not saying that the .999... deniers don't have a philosophical point or two. I'm just saying that although .999... deniers often mention the hyperreals, the hyperreals don't help their argument. .999... = 1 is just as true in the hyperreals as it is in the reals.

I am actually much less familiar with other non-Archimedean systems such as the surreals, but the formalization of the surreals is fairly murky. I've never seen anyone claim that .999... is one thing or another in the surreals. Maybe I'll see if I can look that up.

You are very sensitive. I apologise if you have feelings that are easily hurt. But is this my problem or your problem? — apokrisis

My problem! LOL. I haven't read anything after this morning. No harm no foul. For what it's worth, I've often wondered about the relation between the mathematical formalisms that allow you to add up infinitely many 0-dimensional points to get a line of nonzero length. We can integrate dx from 0 to 1 and the answer is 1, we can even teach that to high school students. But it's really the most mysterious equation in the world.

I don't happen to know much about what philosophers have said about this, other than some very nodding acquaintance with intuitionism which I find murky in the extreme. I did try to study free choice sequences once and gave up.

So I'm ignorant but not apathetic. I don't know but I do care. And yes I'm way too sensitive for my own good.

what I am looking for from you is a genuine counter-argument, — apokrisis

What? I'm basically in agreement with you. Maybe you can tell me what you think my thesis is. You might be misunderstanding me. I'm totally baffled by this remark. That's why I was so startled by your saying I don't care. Nothing I've written has supported that conclusion unless I'm expressing myself very poorly.

Is accusing people of not caring part of the academic give and take?

a restatement of a particular institutional view that is widely held for the pragmatic reasons I've previously stated. — apokrisis

Just baffled. When a mathematical point has needed to be made, I've made it. I have never and I repeat never said that I think mathematics has the slightest thing to say about reality. I'm often a formalist. I don't think set theory necessarily applies to the real world. I said that earlier, maybe in the other thread. Why do you think I'm maintaining otherwise?

So this is why the Toms and Fishfrys are so content with what they learn in class. — apokrisis

Your characterization of me is quite unfair. One, to lump me in with Tom, whose erroneious and confused mathematical misunderstandings I've refuted and corrected numerous times already; and two, to claim that I "feel utterly justified not to care."

I'll forego the opportunity to start a pissing match here and you earned a lot of good will with me for pointing me to the Zalamea paper. But really, was this necessary? Maybe I should just say fuck you or something. Would that serve any purpose? I've spent years online, you think I don't know how to return a gratuitous insult? What were you thinking here?

Perhaps, but it seems to me that we have then already conceded that the real numbers do not and cannot constitute a true continuum. They are now just labels that we have assigned to particular locations along the line, not parts of the line itself. — aletheist

I agree with that. Calling real numbers locations seems to avoid a lot of the philosophical issues about the nature of points.

I've started reading the Zalamea paper and I'm spend some time with it. I looked him up, he's a mathematician who's familiar with modern set theory and other foundations (not just the Cantorian set theory of the 1880's but actual contemporary practice) and also the philosophical ideas of the continuum. It's good stuff.

So you defined a point as a howling inconsistency - the very thing that can't exist? The zero dimensionality that somehow still occupies a place within a continuity of dimensionality? — apokrisis

Good point. No pun intended. The answer is in the concept of "occupies a place." If we view the real numbers as specifying locations on a line, and we stop talking about points, perhaps things are less muddled. I agree with you that nobody knows how a line of dimension 1 can be made up of points of dimension 0. But math has formalisms to work around this problem. Would you at least agree that if math hasn't answered this objection, it's been highly successful in devising formalisms that finesse or bypass the problem?

If you can divide the point on one of its sides ... — apokrisis

A point is that which has no part. Euclid was right about that. You can't subdivide a point and a point has no sides. It's sophistry to claim otherwise. If we can all agree on anything, it's that a point has no sides.

I'll read your link. I actually do agree that there is something unsatisfactory about the set theoretical view of the continuum. But at least it's clear and sensible. "Good sense about trivialities is better than nonsense about things that matter." Something I read once on a sign outside a math professor's office. Perhaps there's a sort of Heisenberg uncertainty between truth and precision. What we can say truthfully is imprecise; and what we can say precisely isn't true.

ps ... Glanced at the link. Large cardinals, category theory, sheaf theory. Now this looks like an interesting read. Thanks much.

Is there any way that mathematics could evolve going forward that would enable it to deal with continuity more successfully? — aletheist

Can you explain (so that a philosophical simpleton like me could understand it) how mathematics has failed to successfully deal with continuity? Modern topology and real analysis have been wildly successful in dealing with continuity, at least in the practical sense of physics and engineering. Just ask any freshman who's had to slog through epsilons and deltas :-)

Of course one might argue that math hasn't solved the philosophical problems, but math isn't philosophy. It's like asking why physics hasn't solved the problem of tooth decay. That's the job of dentistry, not physics. Can't blame math for not solving the problems of philosophy. Although in fact it has been incredibly successful in doing so. We DO have a satisfactory theory of continuity in math.

I read the Peirce article you linked and my impression was that math has clarified all these confusions. When you divide a line at a point, the point stays with one segment and not the other. As someone trained in math, it's hard for me to understand how this answer isn't satisfactory.

It's a very important result in mathematics. The continuum has the cardinality of the power set or the natural numbers. It's a much bigger infinity. — tom

What do you mean by "bigger?" Bearing in mind that there are countable models of the reals? https://en.wikipedia.org/wiki/L%C3%B6wenheim%E2%80%93Skolem_theorem

As is the unfortunate custom in this thread, a word with a highly technical meaning in mathematics is being conflated with the same word in its everyday usage. In math one set is "bigger" than another if the smaller set can be injected but not surjected to the larger. But that does not actually correspond to the everyday notion of "bigger," which the Lowenheim-Skolem result shows.

Along these lines ...

One of these infinities is bigger than the other, much bigger. In fact the measure of the natural numbers on the continuum is zero. — tom

What do you make of the Cantor set, an uncountable set of measure zero?

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

Bijection does not preserve measure. You can see that by simply multiplying each element of the unit interval by 2. Now you have a bijection between sets of measure 1 and 2, respectively.

Therefore we can conclude that the set of natural numbers is not countable.
— Metaphysician Undercover

And yet set theory explicitly says otherwise. — aletheist

You are each using different definitions. This is the fallacy of ambiguity. Surely we need not argue about this any more. I have humbly offered the word foozlable as standing in for the set theoretic definition, because it carries no semantic baggage from any common meaning.

It's no different than a doctor examining you and finding your condition "unremarkable." That might be an insult in daily life but it's the best news you can get in a doctor's office. Surely you understand this. It's a medical term of art.

That said, the question of whether the natural numbers can be "counted" in any meaningful sense of the word -- stipulating that technical conditions in formal set theory are not necessarily meaningful -- is a good one.

I don't think you can count past 200. You'd get bored. You can't count to a zillion. You just can't. It couldn't be done in the age of the universe. If counting is an activity that takes place in time, then a finite universe doesn't give you enough time to count any more than some finite number. There are 10^80 hydrogen atoms in the universe. That's a very small natural number. You can't count it.

What can it mean to count to 10^80? Mathematically you can count a set if you can order-biject it to a natural number, or in a more general context to some ordinal. Sure, you can count 10^80 that way. But that's just set theory. [Note that counting via numbers involves finding an order-preserving bijection, not just a bijection].

If you want to say that it means something other than formal set theory to count to 10^80, I'd like you to tell me what it is.

Myself I tend to be a formalist. I like set theory but I don't think it actually means anything. I don't think there are sets in the real world. It's the famous singleton problem. There is no such thing as an apple and "the set containing an apple." There's madness down that road.

I'm fascinated to read a little on Wiki about Bergson. "Bergson is known for his influential arguments that processes of immediate experience and intuition are more significant than abstract rationalism and science for understanding reality." I actually agree with that point of view. I'm opposed to scientism. I've got a lot of reading to do. I'm afraid I can't pick up those many volumes that have been suggested, but I will definitely Google around.

one should never take too much notice of a mathematician making extrapolations of a metaphysical nature. They are bound to be misguided just because they hold in their hands a very impreessive hammer and so are looking about for some new annoying nail to bang flat. — apokrisis

I'll certainly take that to heart :-)

Can you (or anyone) supply some of relevant Bergson and Pierce links that would shed light on the relation between the mathematical real numbers and the philosophical idea of the continuum?

So I would still like to see the alleged proof that the real numbers form a true continuum as Peirce defined it, which (as I understand it) is similar but not identical to the intuitionistic continuum. — aletheist

Oh I'm in no position to do that. I'm not familiar with the philosophical thinking on the continuum. You did note that I questioned whether philosophers accept the real numbers as the correct model of the continuum. I didn't claim that the real numbers form a true continuum, someone else did. I'm aware that there are philosophical objections but I don't know much about them.

No, you claimed the reals can be disordered — tom

Please locate that quote of mine, I can't find it and don't remember saying it. I probably said reordered and definitely well-ordered, but not disordered.

I may have misspoken myself to say that a well-order would be discrete. I don't believe this is true.

However you can certainly put a discrete topology on the reals, even in their usual order. Just define the discrete metric d(x,y) = 1 if x ≠ y and d(x,x) = 0. This metric induces the discrete topology on any set. In the discrete topology every point is an isolated point.

If you think you can disorder the reals, then pleas indicate the number following this one, and suggest between which two numbers you might place it: — tom

We can well-order the reals. https://en.wikipedia.org/wiki/Well-order#Reals

I mention this because it's a counterexample to the intuition that a set can be "counted" if its members can be lined up so that there's one after another. The real numbers can be well-ordered. That means that they can be ordered such that every nonempty subset has a least element. So there's a first real, a second real, and so forth. Now when you run through the natural numbers, you take a limit ordinal and keep on going. Cantor worked all this out.

As I say, the only relevance of this example is to attack the argument that only the natural numbers may be well-ordered. In fact any set may be well-ordered. And even if you don't believe that because you reject Choice, we can still find at least one uncountable set that can be well-ordered without Choice. So the idea of "lining things up one after another" is far stranger than it first appears, and is not a reliable intuition for what can be "counted," whatever that might mean. I don't think it means anything at all, which is why there are now several pages of confusion derailing the original topic of this thread.

Yeah sure, that's the name you gave instead of the name "countable". But I'm not sure that I would agree with the assumption that there is a substantial difference between a foozlable infinity, and an unfoozlable infinity. We can call them countable and uncountable infinities if that's easier. — Metaphysician Undercover

I don't know if it's a "substantial" difference. It's certainly a difference. The rationals are foozlable and the reals aren't. Even in countable models of the real numbers, and yes such things exist, the reals are not foozlable. So yes it's a pretty important difference in math.

But if you insist that "countable" is to be used with its everyday meaning, then we should be careful not to confuse this with foozlability, which is a technical condition used by specialists in set theory.

Countable means capable of being counted. If it cannot be counted, as is the case with something infinite, or endless, it is not capable of being counted. Therefore the infinite is not countable. — Metaphysician Undercover

You do agree they're foozlable, right? I just want to make sure I'm understanding you.

I think the issue here has been metaphysical — apokrisis

I really miss your bug-eyed avatar from the old forum :-)