There has been debate since ancient Greece and India over when or if something is a pure unity. If it has pure unity does it have parts at all? If something can be potentially divided by us or an angel or God, isn't it divided in itself yet held together by some kind of cohesion? Geometry seems to say that plurality of parts is equal in footing as the object's unity. So math and common sense say anything physical has a 3 fourths of itself in reality. Physics though, quite interestingly, computes the total information in something as proportional to its surface area, not its volume. But anyone who says they have the final answer to these questions has probably lied to himself. Zeno and chariots can be torn apart by the heavens

I'm missing your point also. What's your gripe about the innocuous Riemann sphere? :chin: — jgill

To make infinite numbers into a circle is to make a vicious circle. It is to say that the beginning is the same as the end. — Metaphysician Undercover

How can one argue about this? It is so silly. :lol:

So the circles vicious as opposed to innocuous?

:cool:

So the circles vicious as opposed to innocuous? — Manuel

That's the world that MU lives in. :roll:

Again, this is the difference between fiction and fact. — Metaphysician Undercover

But we're not talking "fact," if by that you mean the real world. The subject was set theory, which is an artificial formal theory. Set theory is not any part of any physical theory. I pointed out to you that in set theory, everything is a set, including the elements of sets. You responded by saying you hadn't realized that. I thought we were therefore making progress: You acknowledged learning something you hadn't known before. And now you want to revert back to "fact," as if set theory has an ontological burden. It does not.

We can imagine infinite regress, and imagine time extending forever backward, but it isn't consistent with the empirical evidence. — Metaphysician Undercover

But I never claimed it did. I offered the mathematical example of the integers. Are you a disbeliever in sufficiently small negative numbers? Do you believe in -47? -48? -4545434543? Where does your belief stop? Of course this is not a physical example, it's a mathematical example; in fact, an example that illustrates the difference between physics and math.

That's the problem with infinite regress, it's logically possible, — Metaphysician Undercover

Ok! Then we are in agreement. Since I have made absolutely no other claims. So just to satisfy my curiosity, do you believe in the negative integers? They believe in you.

but proven through inductive (empirical) principles (Aristotle's cosmological argument for example) to be impossible. — Metaphysician Undercover

Discussion for another time, but I have made no claims about the world. Why do you argue as if I did?

I beg to differ. Didn't we go through this already in the Gabriel's horn thread. — Metaphysician Undercover

That was a lengthy thread from a while ago. Can you remind me of the specifics? It's not possible that "we went through this" about the Riemann sphere. Stereographic projection is a commonplace idea among every mapmaker since antiquity who's wrestled with the dilemma of representing a spherical earth on a flat map. Can you remind me of what on earth you might be talking about? You place a sphere above a plane. From the north pole of the sphere, you draw a straight line through a point on the sphere and extend the line to a point on the plane. You thereby have a mapping from the sphere to the plane. In cartography it's a basic technique. In complex variables theory, it's a way of visualizing the complex numbers as a sphere. There is no mysticism or "vicious circle" or any such nonsense as you claim.

It seems like you haven't learned much about the way that I view these issues. — Metaphysician Undercover

If you would say what you're talking about, I can respond. The Gabriel's horn thread was lengthy and long past. Tell me what you're talking about.

In any event, I've learned far too much about how you view things.

You write very well, but your thinking hasn't obtained to that level. Another example of the difference between form and content. — Metaphysician Undercover

Your ignorance is only matched by your ill manners. Going forward, if you can't be civil, put a sock in it.

Are you denying the contradiction in what you wrote? — Metaphysician Undercover

I repeatedly said that the only thing they have in common is being elements of the given set. So why are you acting like I haven't said that every single time?

If they are members of the same set, then there is a meaningful similarity between them. — Metaphysician Undercover

Only in a sophistic sense. I already pointed out to you that if "meaningful similarity" or "property" or "predicate" is interpreted as referring to an idea expressible in a finite-length string of symbols, there are more subsets of the natural numbers than there are properties. Therefore most sets are entirely random. Their elements have nothing at all in common except for being gathered into the given set.

Being members of the same set constitutes a meaningful similarity. — Metaphysician Undercover

Ok fine, on that definition. I'll agree. But it's a pretty trivial point. Especially for you to be going on about it.

You said "the elements of a set need not be 'the same' in any meaningful way. — Metaphysician Undercover

Other than being in the same set. You deliberately quote me out of context to make a point. Disingenuous much?

The only thing they have in common is that they're elements of a given set." Can't you see the contradiction? — Metaphysician Undercover

No. What I said is perfectly accurate.

If they are said to be members of the same set, then they are the same in some meaningful way. — Metaphysician Undercover

Only that they are members of the same set. So what? You are being childish to go on like this.

It is contradictory to say that they are members of the same set, and also say that they are not the same in any meaningful way. — Metaphysician Undercover

Yeah yeah.

Another example of this same sort of contradiction is when people refer to a difference which doesn't make a difference. If you apprehend it as a difference, and speak about it as a difference, then clearly it has made a difference to you. Likewise, if you see two things as elements of the same set, then clearly you have apprehended that they are the same in some meaningful way. To apprehend them as members of the same set, yet deny that they are the same in a meaningful way, is nothing but self-deception. Your supposed set is not a set at all. You are just saying that there is such a set, when there really is no such set. You are just naming elements and saying "those are elements of the same set" when there is no such set, just some named elements. Without defining, or at least naming the set, which they are members of, there is no such set. And, naming the set which they are elements of is a designation of meaningful sameness. — Metaphysician Undercover

Why are you going on like this? Let me remind you of the conversation. You expressed realization that in set theory, everything's a set. Then you claimed that leads to infinite regress. I pointed out that one, there's nothing logically wrong with infinite regress. I gave the negative integers as an example.

Then I pointed out that in set theory, we adopt the axiom of foundation to explicitly rule out infinite regress. You totally ignored both those points to go off on this trivial and pointless tangent.

Here is a feature of imaginary things which you ought to learn to recognize. I discussed it briefly with Luke in the other thread. An imaginary thing (and I think you'll agree with me that sets are imaginary things, or "pure abstraction" in your terms) requires a representation, or symbol , to be acknowledged. And, for an imaginary thing, to exist requires being acknowledged. However, the symbol, or representation, is not the imaginary thing. The imaginary thing is something other than the symbols which represent it. So the imaginary thing necessarily has two distinct aspects, the representation, and the thing itself, the former is called form, the latter, content. And this is necessary of all imaginary things. — Metaphysician Undercover

Well, for sake of discussion, it's not clear to me that every imaginary thing has a referent. Sets, for example. The empty set is an imaginary formal thing, but I don't know that it has a referent. Certainly not in the physical world.

The important point is that you cannot claim to remove one of these, from the imaginary thing, because both are necessary. So a purely formal system, or pure content of thought, are both impossibilities. And when you say "these things are elements of the same set", you have in a sense named that set, as the set which these things are elements of, thereby creating a meaningful similarity between them. The point being that a meaningful similarity is something which might be created, solely by the mind and that is how the imagination works in the process of creating fictions. But when something is a creation, it must be treated as a creation. — Metaphysician Undercover

Focus. Focus. You said that the fact that in set theory everything is a set, leads to infinite regress. I pointed out that the negative integers are an example of an unproblematic negative regress; and that the axiom of foundation rules out infinite regresses of set membership.

You have avoided both those points to go off on trivialities and irrelevancies. And personal insults. What's the point?

Again, incoherency fishfry. Can't you see that? There is necessarily a reason why you place them in the same set, and this 'reason why' is something other than actually being in the same set. — Metaphysician Undercover

Oh no, not at all. The powerset of the natural numbers is uncountable. There are more sets than reasons. Most sets have no reason at all.

You've gone from saying that the elements have something in common, namely being in the given set -- which I agree with -- to now saying that there's some OTHER reason in addition to that. You're simply wrong about that. The powerset of the natural numbers exists, that's an axiom of set theory. Every set has a powerset, the set consisting of all the set's subsets. And the powerset is far larger than the set itself. There aren't enough "reasons" or predicates or explanations to cover them all, by a countability argument.

You are not acknowledging that "being gathered into a set" requires a cause, — Metaphysician Undercover

You're thinking of the south and the Civil war. A side in a war needs a cause. A set needs no cause. Show me in the axioms for set theory where it says that. This is just something you made up. Again, you're trying to reify sets; but sets are only imaginary formal entities whose behavior is entirely determined by the axioms.

and that cause is something other than being in the same set. — Metaphysician Undercover

You're just making that up. And changing the subject.

I challenged you on your claim that the idea that sets contain only other sets leads to infinite regress. I pointed out that the axiom of foundation precludes infinite regress of set membership. You changed the subject.

So the relation that the things have to one another by being in the same set is not the same as the relation they have to one another by being caused to be in the same set. — Metaphysician Undercover

You can say the knight flies over the moon, but that's not in the rules of chess. There are no "causes" in the axioms of set theory. So you're just making this up and then typing in crap, and wasting my time trying to get you to focus on the actual conversation we were having, which for a brief moment got substantive before you reverted to just making things up.

And things which are in the same set necessarily have relations to each other which are other than being in the same set, because they have relations through the cause, which caused them to be in the same set. — Metaphysician Undercover

There are not enough predicates to cover all the sets that there are. Most sets have no reason or cause at all; they're pure randomness.

It appears like you didn't read what I said. — Metaphysician Undercover

I could say the same about you. But I have read what you've said. What you've said is wrong; and your repeating it doesn't make it any less wrong.

That a word is not defined does not mean that it has no meaning. As I said, it may derive meaning from its use. If the word is used, then it has meaning. So if "set" derives it's meaning from the axioms, then there is meaning which inheres within, according to its use in the axioms. — Metaphysician Undercover

Ok. Fine. But there are no "causes" in the axioms.

What we do not agree on is what "inherent order" means. — Metaphysician Undercover

Don't start that crap again. I can't help it if you reject modern math. I can't do anything about that.

i really do not see how you get from the premise, that "set" is not defined, but gets its meaning from its use, to the conclusion that a set might have no inherent order. In order for the word "set" to exist, it must have been used. Therefore it is impossible for "set" not to have meaning, and we might say that there is meaning (order, if order is analogous to meaning, as you seem to think), which inheres within. Wouldn't you agree with this, concerning the use of any word? If the word has been used, there is meaning which inheres within, as given by that use. And, for a word to have any existence it must have been used. — Metaphysician Undercover

You've worn me out. I'm losing interest.

It appears like you misunderstood. I didn't say every set is a number, to the contrary. I said that if we proceed under the precepts of set theory, every number is a set. — Metaphysician Undercover

Well as Bill Clinton said, that depends on what the meaning of "is" is. If you mean that a number literally is a set, no, that's not true, as Benacerraf so insightfully pointed out. If you mean that in set theory a number is represented by a set, then that's true. Important for you to make that distinction.

Therefore we cannot say that "number" is undefined because "set" is now a defining feature of "number", just like when we say every human beings is an animal, "animal" becomes a defining feature of "human being". — Metaphysician Undercover

In set theory, a number is defined as a particular type of set. Just because set is an undefined term doesn't mean that we can't use it to define other things. Just as point is an undefined term in Euclidean geometry, but a line is made of points. Right? Right.

Didn't it strike you that I was in a very agreeable mood that day? — Metaphysician Undercover

Yes, that didn't last long. But you were more than agreeable the other day. You actually achieved some insight. You realized that a set has no definition, and that its meaning is derived from the axioms. You realized that the members of sets are also sets.

Now I'm back to my old self, pointing out your contradiction in saying that things could be in the same set without having any meaningful relation to each other, other than being in the same set. You just do not seem to understand that things don't just magically get into the same set. There is a reason why they are in the same set. — Metaphysician Undercover

You're just wrong about that. Provably wrong, since there aren't enough reasons to cover uncountably many sets.

Maybe at some point we'll discuss the supposed empty set. How do you suppose that nothing could get into a set? — Metaphysician Undercover

By the axiom of pairing, which has as a consequence the fact that if $X$ is a set, so is $\{X\}$. Everything's given by the axioms.

I have no idea what you mean by "nothing." That's not in the axioms. The empty set is not nothing. It's the empty set. A particular thing.

Actually I do not agree with general relativity, so I would ban that first. — Metaphysician Undercover

Charming. You don't believe in abstract math, you don't believe in physics.

You keep saying things like this, the Pythagorean theorem is not true, now Euclidian geometry in general is not true. I suppose pi is not true for you either? Until you provide some evidence or at least an argument, these are just baseless assertions. — Metaphysician Undercover

You probably shouldn't bring up pi. You said the other day that pi is not a particular real number. That's a statement so monumentally ignorant that I either have to ignore it or stop responding to you altogether. So far I'm just trying to ignore it. Why you'd bring it up again, I don't know. You're just reminding me what a monumental waste of time this is.

On what basis do you say they are a unity then? [/url}

The axiom of powersets.
— Metaphysician Undercover

You have a random group of natural numbers. Saying that they are a unity does not make them a unity. — Metaphysician Undercover

Every subset of the natural numbers is a set.

So saying that they are a "set" does not make them a unity. This is where you need a definition of "set" which would make a set a unity. — Metaphysician Undercover

I'm afraid "unity" is not mentioned in the axioms. You keep making things up. You are unable to focus on what's in the axioms. It's like someone trying to teach you chess and you say, "Well the knight must wear armor and save damsels," or "The knight must be "a man who served his sovereign or lord as a mounted soldier in armor." No no no no no. The knight in chess is exactly what the rules say the knight is. You don't get the concept of formal rules, fine. I doubt you're like this in real life, and you're quite tedious to regress to this infantile obfuscatory state here. I thought we'd moved a little past that, but apparently not.

Then you have no basis to your claim that a set is a unity. — Metaphysician Undercover

But I never said a set is a unity. I don't know what a unity is. It's not mentioned in the axioms.

And you cannot treat a set as a unified whole. If a set is supposed to be a unified whole, then you cannot claim that "set" is not defined. — Metaphysician Undercover

I agree that objection has been raised against set theory. It's not a point I'm interested in debating. Thoralf Skolem pointed out that the concept of set is far less coherent than people imagine. Many mathematicians and philosophers have made the point. For purposes of discussion, I'll even concede the point. But it's irrelevant. It doesn't diminish or change set theory, which is a particular formal system that need not have any referent or even be entirely sensible. It's just a list of formal symbols and the game is to derive their logical consequences.

If you don't want to play chess, that's fine. But for you to stand on a soapbox in the middle of town and rant and rail about chess, that's another thing entirely. You don't like set theory, you get no argument from me. I like set theory but I don't think others need to. But your vociferous objections to the reality of set theory are a waste of time. I don't make any claims it's real. It's just a formal system that some people find interesting, and that gained 20th century mindshare as the foundation of math. In fact set theory is all the more interesting lately, "now that it's been relieved of its ontological burden," as one set theorist put it.

ps -- @Meta let me sum this up. A couple of weeks ago I noticed that you are taking Frege's side in the great Frege-Hilbert controversy; namely, that you claim axioms must mean something or refer to something. Hilbert says no, that the theorems must be true of beer mugs and tables.

Since you feel that way, it's not something I can talk you out of. There is no right or wrong position. In real life Frege refused to "get" modern math and Hilbert stopped returning his letters. Likewise you don't want to get modern math. That's your right. But there is no point in your repeating these same talking points. The axioms of set theory are what they are. There are no "causes" or "reasons" nor "inherent order." I can't argue these points with you anymore.

We made a bit of progress when you started to at least acknowledge the reality of modern set theory. But if you don't want to build on that, I can't argue you out of your position and I wouldn't if I could.

pps --

I do reject fractions, — Metaphysician Undercover

LOL.

That's the world that MU lives in. — jgill

:100:

I did mention that ordinal less-than is membership:

df: k is ord-less-than j <-> k e j — TonesInDeepFreeze

Df. If x and y are ordinals, then x precedes y (x is less than y) iff x is an element of y. — TonesInDeepFreeze

Another poster made it appear as if I hold that words (such as 'least') or symbols don't have explicit definitions, and that ambiguity results. My point though is that, other than primitives, symbols do have definitions, with very strict rules as to what constitutes a definitions, and those rules prevent ambiguity. And I gave an explicit definition of 'least'.

Hi.

So it seems to me a number is a "unity" and a set is not a noun but more like a verb. It's our action of containing a unity or many unities or unities and containers (verbs). I've been considering the "set of all sets that do not contain themselves" vs the "set of all sets the do contain themselves". This leads to what I see as Hilbert's position (contra Frege) of our rational power of humans to think of thinking of thinking of thinking and on to infinity. The set\verb would take precedence over the unity\number we place before our eyes as an object.

So it seems to me a number is a "unity" — Gregory

I always get into trouble with these philosophically loaded terms. Any number can be broken up into parts. 2 = 1 + 1, 1 = 1/2 + 1/2. So nothing in math is "indivisible." If anything at all is, it would be a pure Euclidean point, or a single real number representing a location on the number line. But what of it? Making a mystery or a big deal out of the idea that something is a "unity" doesn't speak to me; and it's one of the points where I do get in trouble in these philosophical discussions. A number is a number. It might be a real number, representing a signed distance on the number line. Or it could be a complex number, representing a rotation and stretching operator in the plane. Or a quaternion, used by game programmers as a nice formalism for rotating things in 3-space. I know the formalism of how numbers are represented in set theory; and I know that numbers aren't "really" sets; rather, they're abstract things that are pointed to by their various representations. What that means, I don't worry about too much. I've read a bit of the literature, I was reading up on structuralism the other day when that came up in one of the discussions on this site.

But I don't know what a unity is, or whether a number is one. I looked it up on Google, and it says that, "Unity, or oneness, is generally regarded as the attribute of a thing whereby it is undivided in itself and yet divided from others."

Well ok. But other than 0 and an individual point in space, I don't know what it means for any mathematical entity to be "undivided in itself." Actually only a point on a line has no parts. A point in space, say 3-space, is given by three spacial coordinates (x,y,z). And that's three things! I can take a point's projection onto the x, y, and z axes, to find that its "components" are x, y, and z, respectively. So even a point in space has components. I don't know what philosophers make of that.

All it all I can't agree or disagree that a number is a unity. I don't even know what that means.

and a set is not a noun but more like a verb. It's our action of containing a unity or many unities or unities and containers (verbs). — Gregory

I see what you mean. You have an apple and an orange, and forming a set {apple, orange} is an act of gathering. It's quite mysterious and not entirely coherent, a point @Metaphysician Undercover has made and that I somewhat agree with. I don't know what it means to form a set out of individual objects. When pressed I can fall back on the formalism of the axiom of pairing, one consequence of which is the fact that if I have a mathematical object $X$, I'm allowed to form $\{X\}$, "the set containing" $X$. I totally understand and accept that mathematically. Metaphysically, I don't know for sure that it's even a coherent concept.

That in fact is one of @Metaphysician Undercover's frequent points. What he doesn't understand is that I totally agree with him. Or at least I do for sake of discussion. It's not a hill I need to die on. I make no claim that set theory is coherent or sensible. Only that it's a formal system of rules that some people find interesting. I don't reify set theory or put it on a pedestal or make any claims about it. Like the novel Moby Dick. It's not a true story, but it's worthwhile nevertheless. It's based on a true incident, but only very loosely. If someone wants to tell me that set theory is incoherent, I don't object to that point of view. It doesn't matter. It's interesting on its own terms; and massively useful in formalizing most of modern math. What more can you ask of a formalism?

Which is to say that if set-collection is regarded by you as a verb, I do see your point. The act of gathering individuals into a mathematical set is a great act of abstraction that leads to many counterintuitive results. It's a powerful concept, even if not entirely coherent.

I've been considering the "set of all sets that do not contain themselves" vs the "set of all sets the do contain themselves". — Gregory

Russell's paradox just shows that we can't form the set of all sets; and in fact that we can't form sets out of arbitrary predicates. Now that's very profound. Originally it was thought that if P is a predicate, then the collection of all the things that satisfy P form a set. That turns out to lead to a contradiction. Rather, a set is nothing more or less than exactly what the axioms of set theory say they are. Which for some philosophers is not a very satisfactory state of affairs. What I do know is that "high school sets," which are collections of similar or related objects, are nothing like actual mathematical sets. Actual mathematical sets are far stranger than that.

This leads to what I see as Hilbert's position (contra Frege) of our rational power of humans to think of thinking of thinking of thinking and on to infinity. The set\verb would take precedence over the unity\number we place before our eyes as an object. — Gregory

I was trained in modern mathematical abstraction and have a hard time understanding Frege's point of view. It upsets some people (Frega, @Meta) that mathematical axioms don't necessarily "mean" anything or "refer" to anything. As Hilbert said, "“One must be able to say at all times
— instead of points, straight lines, and planes — tables, chairs, and beer mugs.” Whether he truly believed that, or was only retreating behind the formalist view because a realist mathematical stance is untenable, I don't know. Hilbert's formalist dreams were blown up by Gödel. There's a realm of mathematical truth that exists outside of anything we can capture with axioms.

It upsets some people (Frega, Meta) that mathematical axioms don't necessarily "mean" anything or "refer" to anything. — fishfry

And it should do, for classical set theory and real analysis are misleading and unrepresentative nonsense, unless cut down to the computationally meaningful content. Students who are taught those subjects aren't normally given the proviso that every result appealing to the axiom of choice is nonsensical, question-begging and of use only to pure mathematicians and historians.

And it should do, for classical set theory and real analysis are misleading and unrepresentative nonsense, unless cut down to the computationally meaningful content. — sime

I'm perfectly happy to stipulate so for purposes of discussion. After all, there are no infinite sets in physics, at least at the present time. So, what of it? The knight doesn't "really" move that way. Everybody knows that knights rescue damsels in distress, a decidedly sexist notion in our modern viewpoint. Therefore chess is misleading and unrepresentative nonsense. Nevertheless, millions of people enjoy playing the game. And millions more enjoy NOT playing the game. What I don't understand is standing on a soapbox railing against the game. If math is nonsense, do something else. Nobody's forcing you to do math, unless you're in school. And then your complaints are not really about math itself, but rather about math pedagogy. And I agree with you on that. When I'm in charge, a lot of state math curriculum boards are going straight to Gitmo.

Students who are taught those subjects aren't normally given the proviso that every result appealing to the axiom of choice is nonsensical, question-begging and of use only to pure mathematicians and historians. — sime

May well be so. I still think the way the knight moves is nonsensical too. What of it? You don't find me down at the park yelling at the chess players. Why is this a concern to you?

I might point out, though, that assuming the negation of the axiom of choice has consequences every bit as counterintuitive as assuming choice. Without choice you have a vector space that has no basis. An infinite set that changes cardinality if you remove a single element. An infinite set that's Dedekind-finite. You lose the Hahn-Banach theorem, of vital interest in functional analysis, which is the mathematical framework for quantum mechanics. The axiom of choice is even involved in political science via the Arrow impossibility theorem.

Besides, if you have a nation made up of states, can't you always choose a legislature? A legislature is a representative from each state. If there were infinitely many states, couldn't each state still choose a representative? The US Senate is formed by two applications of the axiom of choice. The House of Representatives is a choice set on the 435 Congressional districts. The axiom of choice is perfectly true intuitively. If you deny the axiom of choice, you are asserting that there's a political entity subdivided into states such that it's impossible to form a legislature. How would you justify that? It's patently false. If nothing else, each state could choose a representative by lot.

Thanks for sharing your wisdom on these types of threads

But we're not talking "fact," if by that you mean the real world. The subject was set theory, which is an artificial formal theory. Set theory is not any part of any physical theory. I pointed out to you that in set theory, everything is a set, including the elements of sets. You responded by saying you hadn't realized that. I thought we were therefore making progress: You acknowledged learning something you hadn't known before. And now you want to revert back to "fact," as if set theory has an ontological burden. It does not. — fishfry

I'm not reverting back. Just because I understand better what I didn't understand as well before, doesn't mean that I am now bound to accept the principles which I now better understand.

Focus. You said that the fact that in set theory everything is a set, leads to infinite regress. I pointed out that the negative integers are an example of an unproblematic negative regress; and that the axiom of foundation rules out infinite regresses of set membership. — fishfry

I suggest you look into the concept of infinite regress. The negative numbers are not an example of infinite regress.

Yes, that didn't last long. But you were more than agreeable the other day. You actually achieved some insight. You realized that a set has no definition, and that its meaning is derived from the axioms. You realized that the members of sets are also sets. — fishfry

No, you said "set" has no definition, as a general term, and I went along with that. But I spent a long time explaining to you how a set must have some sort of definition to exist as a set. You seem to be ignoring what I wrote. Since you haven't seriously addressed the points I made, and you claim not to be interested, I won't continue.

So "2" cannot refer to two distinct but same things? — Luke

Of course not, that's contradictory. According to the terms of the law of identity, two distinct things are not the same thing, so "two distinct but same things" is contradictory if we adhere to the definition of "same" provided by the law of identity.

You cannot have 2 apples or 2 iPhones, etc? — Luke

Those are similar but different things, therefore not the same.

The categories we use are either discovered or man-made. If they are discovered, then how can we be "wrong in an earlier judgement" about them; why are there borderline cases in classification; and why does nothing guarantee their perpetuity as categories? — Luke

I still don't see your point, or the relevance.

An infinite regress in the real world would simply be past time encompassing the negative numbers as they move by the laws of physics. You might find such a past unsatisfactory without perfect unity undergirding it but that comes from your particular spirituality

This is Stanford Encyclopedia of Philosophy on infinite regress. "An infinite regress is a series of appropriately related elements with a first member but no last member, where each element leads to or generates the next in some sense."

Notice that there is a starting point, and this is why infinite regress is a logical problem, there is generally an assumption which requires something else for justification, and this requires something else etc.. Numbers in themselves, do not constitute an infinite regress because a number itself does not require a next number for justification. We may justify with the prior number, and finally the concept of "one", "unity", which is grounded in something other than number. So infinite regress in numbers is axiom dependent.

Peano’s axioms for arithmetic, e.g., yield an infinite regress. We are told that zero is a natural number, that every natural number has a natural number as a successor, that zero is not the successor of any natural number, and that if x and y are natural numbers with the same successor, then x = y. This yields an infinite regress. Zero has a successor. It cannot be zero, since zero is not any natural number’s successor, so it must be a new natural number: one. One must have a successor. It cannot be zero, as before, nor can it be one itself, since then zero and one would have the same successor and hence be identical, and we have already said they must be distinct. So there must be a new natural number that is the successor of one: two. Two must have a successor: three. And so on … And this infinite regress entails that there are infinitely many things of a certain kind: natural numbers. But few have found this worrying. After all, there is no independent reason to think that the domain of natural numbers is finite—quite the opposite. — Stanford Encyclopedia of Philosophy

Notice the statement that "few have found this worrying". This is because, as fishfry demonstrates, "pure mathematicians" are wont to create axioms with total disregard for such logical problems which are entailed by those axioms. In other words, there are many issues which philosophers see as logical problems, but mathematicians ignore as irrelevant to mathematics. As pure mathematicians proceed in this way, the logical problems accumulate. This has created the divide between mathematics and philosophy which fishfry and I touched on in the other thread, in reference to the Hilbert-Frege disagreement.

Quick review: supposing and setting aside that some philosophers worry about some mathematics, still though, what problems does that mathematics cause for mathematicians?

I'm not reverting back. Just because I understand better what I didn't understand as well before, doesn't mean that I am now bound to accept the principles which I now better understand. — Metaphysician Undercover

You seemed to be reverting back to the Frege-Hilbert paradigm, which is a pointless discussion because there is no right or wrong, just a different worldview. I can't talk you out of yours nor would I if I could.

I suggest you look into the concept of infinite regress. The negative numbers are not an example of infinite regress. — Metaphysician Undercover

In fact they are. They are often used philosophically as a model of infinite regress of causation. If you say cause -47 causes -46 which causes -45 etc., you have a model of causality in which (1) every effect has a direct cause, yet (2) there is no first cause. That's infinite regress.

ps -- I looked at the SEP article. That is utterly bizarre. An infinite regress goes backward without a beginning. Going forward without end like the Peano axioms is not an infinite regress. I refer to all the standard cosmological arguments, for example William Lane Craig's Kalam cosmological argument, where he argues against infinite regress going backwards. I have never seen infinite regress defined incorrectly as going forward as in this SEP article. The author made a mistake.

You might check the Wiki article on infinite regress, which is itself a little vague but at least correct. The SEP piece confuses induction with infinite regress. That's false. Induction always has a base case. Infinite regress fails to have a base case, that's what makes it an infinite regress.

No, you said "set" has no definition, as a general term, and I went along with that. But I spent a long time explaining to you how a set must have some sort of definition to exist as a set. — Metaphysician Undercover

You are equivocating two senses of "definition." The word set has no definition in set theory. You can consult the axioms of Zermelo-Fraenkel set theory to verify this.

On the other hand, some sets do have definitions, or more accurately, specifications. For example the set of prime numbers, the set of even numbers, the set of counterexamples to Fermat's last theorem. That latter by the way is the empty set. See the axiom schema of specification to understand how SOME sets may indeed have a specification.

But MOST sets can't possibly have specifications, because there are more sets than specifications, a point I've made several times and that you prefer not to engage with. There are uncountably many sets and only countably many specifications. There simply aren't enough specifications to specify all the sets that there are. Most sets are simply collections of elements unrelated by any articulable property other than being collected into that set.

And a set can't ONLY be given by a specification, because then I'll give you the specification $x \notin x$ and get the Russell paradox.

It's a bit like numbers. There is no general definition of number; but there are specific definitions of the real numbers, the complex numbers, the p-adic numbers, the quaternions, and so forth.

You seem to be ignoring what I wrote. — Metaphysician Undercover

I carefully read everything you write. And either refute it or place it in context or give my own point of view. And then you come back with the same misunderstandings and ignore my refutations.

Since you haven't seriously addressed the points I made, — Metaphysician Undercover

I have addressed all of your points and either refuted them or placed them in context.

and you claim not to be interested, — Metaphysician Undercover

I'm not interested in recapitulating the Frege-Hilbert dispute, since it's a matter of worldview and I could not change your mind because there is no right or wrong about the matter. And I'm in good company, because in the end Hilbert simply stopped responding to Frege's letters. If you reject abstract axiomatic systems, nobody can talk you out of that viewpoint.

I won't continue. — Metaphysician Undercover

Thanks for the chat, then. All the best.

Thanks for sharing your wisdom on these types of threads — Gregory

You're welcome. Very much appreciated.

Logical regress is a logical problem. An infinite past is not a logical problem because it doesn't have to have a completion like a thought does. Lastly, logical regress does not apply in mathematics. It's purely about logic and how our thoughts must rest in noncontradiction

I looked at the SEP article. That is utterly bizarre. An infinite regress goes backward without a beginning. — fishfry

I agree. It's nonsense. Regress means going backward. I am more than familiar with these notions, as I investigate dynamical processes going forward as well as those going backward.

"Lastly, logical regress does not apply in mathematics" - OK, thank goodness.

I looked at the SEP article. That is utterly bizarre. An infinite regress goes backward without a beginning. Going forward without end like the Peano axioms is not an infinite regress. — fishfry

I agree. It's nonsense. Regress means going backward. I am more than familiar with these notions, as I investigate dynamical processes going forward as well as those going backward. — jgill

Another example of the division between mathematics and philosophy. But the Wikipedia entry is consistent with the SEP.. You two just seem to twist around the concept, to portray infinite regress as a process that has an end, but without a start, when in reality the infinite regress is a logical process with a start, without an end.

Perhaps it is the idea of "forward" and "backward" which is confusing you. There is no forward and backward in logic, only one direction of procedure because to go backward may result in affirming the consequent which is illogical.

But MOST sets can't possibly have specifications, because there are more sets than specifications, a point I've made several times and that you prefer not to engage with. There are uncountably many sets and only countably many specifications. There simply aren't enough specifications to specify all the sets that there are. Most sets are simply collections of elements unrelated by any articulable property other than being collected into that set. — fishfry

This is what I've argued is incoherent, the assumption of an unspecified set, and you've done nothing to justify your claim that such a thing is coherent. I will not ask you to show me an unspecified set, because that would require that you specify it, making such a thing impossible for you. So I'll ask you in another way.

We agree that a set is an imaginary thing. But I think that to imagine something requires it do be specified in some way. That's the point I made with the distinction between the symbol, and the imaginary thing represented or 'specified' by the symbol. The symbol, or in the most basic form, an image, is a necessary requirement for an imaginary thing. Even within one's own mind, there is an image or symbol which is required as a representation of any imaginary thing. The thing imagined is known to be something other than the symbol which represents it. So, how do you propose that an imaginary thing (like a set), can exist without having a symbol which represents it, thereby specifying it in some way? Even to say "there are sets which are unspecified" is to specify them as the sets which are unspecified. Then what would support the designation of unspecified "sets" in plural? if all such sets are specified as "the unspecified", what distinguishes one from another as distinct sets? Haven't you actually just designated one set as "the unspecified sets"?

I won't continue — Metaphysician Undercover

Ah, the good old daze. That didn't last long.

Another example of the division between mathematics and philosophy. But the Wikipedia entry is consistent with the SEP.. You two just seem to twist around the concept, to portray infinite regress as a process that has an end, but without a start, when in reality the infinite regress is a logical process with a start, without an end. — Metaphysician Undercover

Perhaps it is the idea of "forward" and "backward" which is confusing you. There is no forward and backward in logic, only one direction of procedure because to go backward may result in affirming the consequent which is illogical. — Metaphysician Undercover

It's the distinction between two linear orders:

... < a4 < a3 < a2 < a1 < a0.

There's no first element but there is a last element. The earth (a1) rests on the back of turtle a1, which rests on the back of turtle a2, and so forth. It's turtles all the way down.

The SEP article reverses this:

a0 < a1 < a2 < a3 < ...

The earth (a0) has a turtle on it, a1; which has a turtle on its back, a2, and so forth. It's turtles all the way UP!

Now I recognize that in some sense these structures are "the same," in the sense that they just a mirror image of each other. Technically we would say that there is an order anti-isomorphism between them.

But the semantics are completely different. In the first model, there is no uncaused cause. William Lane Craig would argue (sophistically, but whatever) that this is impossible; that there MUST be a first cause, which is not only God, but is the Christian God. That's the argument.

So my contention is that the SEP article flips the direction of what an infinite regress is. And that's not necessarily a mathematical view. It's the philosophical view too. Turtles all the way down versus turtles all the way up.

This is what I've argued is incoherent, the assumption of an unspecified set, and you've done nothing to justify your claim that such a thing is coherent. I will not ask you to show me an unspecified set, because that would require that you specify it, making such a thing impossible for you. So I'll ask you in another way. — Metaphysician Undercover

I will gladly show you an unspecified set, one of the classic cases. It's called the Vitali set. Consider a binary relation $\sim$ defined on pairs of real numbers by:

$x \sim y$ if $x - y \in \mathbb Q$

That is, two real numbers $x$ and $y$ are related by $\sim$ if their difference is rational. For example $\pi + \frac{1}{2} \sim \pi$. That's because their difference, 1/2, is rational.

You can verify that the relation $\sim$ is reflexive (every real number is related to itself, since the difference with itself is zero, which is rational); symmetric: if $x \sim y$ then $y \sim x$; since the difference in one direction is just the negative of the difference in the other, so they're either both rational or neither are; and transitive, meaning that if $x \sim y$ and $y \sim z$ then $x \sim z$. You should see if you can convince yourself that this is true.

A binary relation that is reflexive, symmetric, and transitive is called an equivalence relation. There is a basic theorem about equivalence relations, which is that they partition a given set into a collection of pairwise disjoint sets whose union is the original set.

So $\sim$ partitions the real numbers into a collection of pairwise disjoint subsets, called equivalence classes, such that every real number is in exactly one subset. By the axiom of choice there exists a set, generally called $V$ in honor of Giuseppe Vitali, who discovered it, such that $V$ contains exactly one member, or representative, of each equivalence class.

You can tell me NOTHING about the elements of $V$. Given a particular real number like 1/2 or pi, you can't tell me whether that number is in $V$ or not. The ONLY thing you know for sure is that if 1/2 is in $V$, then no other rational number can be in $V$. Other than that, you know nothing about the elements of $V$, nor do those elements have anything at all in common, other than their membership in $V$.

Is this example important? Yes, it's part of the foundation of modern probability theory.

$V$ is the classic example of a nonconstructive set. I don't expect you to regard this as particularly intuitive. It's an example typically shown to first-year grad students in math. It takes a while to get your mind around it. But "whether you like it or not," as Gavin Newsom said about gay marriage, $V$ is a perfectly legitimate set in ZFC, and actually turns out to be of theoretical importance.

You have now seen the classic example of a nonconstructive set.

We agree that a set is an imaginary thing. But I think that to imagine something requires it do be specified in some way. — Metaphysician Undercover

You're wrong. I just demonstrated a specific example, one that is not only famous in theoretical mathematics, but that is also important in every field that depends on infinitary probability theory such as statistics, actuarial science, and data science.

I know you have an intuition. Your intuition is wrong. One of the things studying math does, is refine your intuitions.

That's the point I made with the distinction between the symbol, and the imaginary thing represented or 'specified' by the symbol. The symbol, or in the most basic form, an image, is a necessary requirement for an imaginary thing. Even within one's own mind, there is an image or symbol which is required as a representation of any imaginary thing. The thing imagined is known to be something other than the symbol which represents it. So, how do you propose that an imaginary thing (like a set), can exist without having a symbol which represents it, thereby specifying it in some way? Even to say "there are sets which are unspecified" is to specify them as the sets which are unspecified. Then what would support the designation of unspecified "sets" in plural? if all such sets are specified as "the unspecified", what distinguishes one from another as distinct sets? Haven't you actually just designated one set as "the unspecified sets"? — Metaphysician Undercover

What distinguishes one set from another as distinct sets? Their elements, as expressed by the axiom of extensionality, as I've explained to you at least a dozen times in the past year.

It's like being out in a field picking daisies. You pick this daisy, you pick that daisy. When you're done, you have a basket full of daisies. Must they have some particular property in common for you to have picked them? No, you picked them randomly. The only thing they have in common is that you picked them. For no reason at all. It's just like this week's winning lottery numbers. They have nothing in common other than that they were picked randomly. Once you start thinking about it that way, you'll find many such examples in daily life of perfectly random sets. A bunch of people check into a hotel.What do they have in common that distinguishes them from all other human beings? Nothing at all, except that they all checked into the hotel.

If you know nothing else about mathematical sets, know this: A set is entirely characterized by its elements.

As Judge Reinhold said to Sean Penn in Fast Times at Ridgemont High: Learn it. Know it. Live it.

Footnote 1 of the SEP article says: "Talk of ‘first’ and ‘last’ members here is just a matter of convention. We could just as well have said that an infinite regress is a series of appropriately related elements with a last member but no first member, where each element relies upon or is generated from the previous in some sense. What direction we see the regress going in does not signify anything important."

Footnote 1 of the SEP article says: "Talk of ‘first’ and ‘last’ members here is just a matter of convention. We could just as well have said that an infinite regress is a series of appropriately related elements with a last member but no first member, where each element relies upon or is generated from the previous in some sense. What direction we see the regress going in does not signify anything important." — TonesInDeepFreeze

Awfully good catch, thank you. Especially since the footnotes don't appear on the article page and must be clicked on to see them at all. I commend your attention to detail in clicking on the footnotes.

I am of two minds on this. On the one hand yes, the footnote is correct and there is fundamentally no difference mathematically. The two interpretations are order anti-isomorphic, just mirror images of each other.

Still, to me the semantics are profoundly different. I guess I have to accept that in the end the difference is not important. Still @Metaphysician Undercover must also agree that when he says that @jgill and I have infinite regress wrong, he's incorrect about that too. If both interpretations are the same, everyone's right.

Thanks for clicking on that footnote!

ps -- I just can't agree with SEP, period. The article is wrong and I'm right. My basis for this belief is the concept of well-foundedness, which is essential to set theory and is encoded as the axiom of foundation.

In set theory it is legal to have infinitely upward membership chains $x_0 \in x_1 \in x_2 \in \dots$

In fact this is not only legal, it's standard, as exemplified by the finite von Neuman ordinals $0 \in 1 \in 2 \in \dots$.

Whereas it is expressly forbidden to have infinitely downward membership chains $\dots \in x_2 \in x_1 \in x_0$.

Even though the two conditions are mirror-images of each other, set theorists strongly distinguish between the two; considering one situation normal and the other illegal. I rest my case, and will probably drop the author a note with my two cents.

So ∼∼ partitions the real numbers into a collection of pairwise disjoint subsets, called equivalence classes, such that every real number is in exactly one subset. By the axiom of choice there exists a set, generally called VV in honor of Giuseppe Vitali, who discovered it, such that VV contains exactly one member, or representative, of each equivalence class. — fishfry

You are specifying "the real numbers". How is this not a specification?

You're wrong. I just demonstrated a specific example, one that is not only famous in theoretical mathematics, but that is also important in every field that depends on infinitary probability theory such as statistics, actuarial science, and data science.

I know you have an intuition. Your intuition is wrong. One of the things studying math does, is refine your intuitions. — fishfry

Actually, you're wrong, your set is clearly a specified set.

You can tell me NOTHING about the elements of VV. Given a particular real number like 1/2 or pi, you can't tell me whether that number is in VV or not. The ONLY thing you know for sure is that if 1/2 is in VV, then no other rational number can be in VV. Other than that, you know nothing about the elements of VV, nor do those elements have anything at all in common, other than their membership in VV. — fishfry

This is not true, you have already said something else about the set, the elements are real numbers.

Still Metaphysician Undercover must also agree that when he says that @jgill and I have infinite regress wrong, he's incorrect about that too. If both interpretations are the same, everyone's right. — fishfry

I'll agree with Tones, the two ways are just different ways of looking at the same thing. That's why I said the Wikipedia article is consistent with the SEP. I do believe there are metaphysical consequences though, which result from the different ways, or perhaps they are not consequences, but the metaphysical cause of the difference in ways. The principal consequence, or cause (whichever it may be), is the way that we view the ontological status of contingency.

You are specifying "the real numbers". How is this not a specification? — Metaphysician Undercover

The real numbers include some numbers that are in $V$ and many that aren't. In what way does that specify $V$? That's like saying I can specify the people registered at a hotel this weekend as the human race. Of course everyone at the hotel is human, but humanity includes many people who are not registered at the hotel.

Actually, you're wrong, your set is clearly a specified set. — Metaphysician Undercover

How so? I gave an existence proof. In no way did I tell you how to determine which real numbers are in it and which aren't. Can you explain your thought process?

This is not true, you have already said something else about the set, the elements are real numbers. — Metaphysician Undercover

And the people at the hotel are humans. As are all the people not at the hotel. If that's all you mean by specification, that all I have to do is name some arbitrary superset of the set in question, then every set has a specification. If that's what you meant, I'll grant you your point. But it doesn't seem too helpful. It doesn't tell me how to distinguish members of a set from non members.

It's like an exclusive club that only allows in certain people. You run the club. You hire a doorman. He asks, "How can I tell who's a member or not?" And you say, "Oh just let everybody in." What kind of specification is that?

I'll agree with Tones, the two ways are just different ways of looking at the same thing. That's why I said the Wikipedia article is consistent with the SEP. I do believe there are metaphysical consequences though, which result from the different ways, or perhaps they are not consequences, but the metaphysical cause of the difference in ways. The principal consequence, or cause (whichever it may be), is the way that we view the ontological status of contingency. — Metaphysician Undercover

I agree that there is a point of view by which it makes no difference whether you define infinite regress as going forward or backward. And a sense in which it makes a huge difference, such as well-foundedness. It can be argued either way.

Perhaps it is the idea of "forward" and "backward" which is confusing you. There is no forward and backward in logic, only one direction of procedure because to go backward may result in affirming the consequent which is illogical. — Metaphysician Undercover

Here's a counterpart of this idea in mathematics. (1) is called left composition or outer composition, and (2) is called right composition or inner composition. (1) and (2) are very different ideas. But to compound difficulties in language or notation, (2) is usually numerically evaluated using backward recursion, which is very efficient:

(1) ${{G}_{n}}(z)={{g}_{n}}\circ {{g}_{n-1}}\circ \cdots \circ {{g}_{1}}(z),\text{ }G(z)=\underset{n\to \infty }{\mathop{\lim }}\,{{G}_{n}}(z)$

(2) ${{F}_{n}}(z)={{f}_{1}}\circ {{f}_{2}}\circ \cdots \circ {{f}_{n-1}}\circ {{f}_{n}}(z),\text{ }F(z)=\underset{n\to \infty }{\mathop{\lim }}\,{{F}_{n}}(z)$

I still don't see your point, or the relevance. — Metaphysician Undercover

The point is that basing your mathematical "principles" on empiricism or reality demonstrably leads to absurdity, including your rejection of fractions, negative numbers, imaginary numbers, infinity, circles, probabilities, possible set orderings, and potentially all mathematics. Instead of coming to realise that this indicates a serious problem with your principles and position, you continue in your delusion that you possess a superior understanding of mathematics.

I still don't see your point, or the relevance.
— Metaphysician Undercover

The point is that basing your mathematical "principles" on empiricism or reality demonstrably leads to absurdity, including your rejection of fractions, negative numbers, imaginary numbers, infinity, circles, probabilities, possible set orderings, and potentially all mathematics. Instead of coming to realise that this indicates a serious problem with your principles and position, you continue in your delusion that you possess a superior understanding of mathematics. — Luke

:100: :100: :100: :100: :100: :100: :100: :100: :100: :100: :100:

Take heart, MU. You may think yourself alone and ridiculed, but you have only to find the Yellow brick Road and follow it to Emerald City where your ideas will find acceptance. :nerd: