Talk:Identity of indiscernibles/Archive 1

Confusion

This article handles the identity of indiscernibles and the indiscernibility of identicals together. The two are separate doctrines deserving separate articles. The indiscernibility of identicals, i.e., Leibniz's law, is indeed one of the two great metaphysical principles of Leibniz. The identity of indiscernibles is not one of the two great metaphysical principles of Leibniz, though Leibniz also accepted it (he thought it followed from the Principle of Sufficient Reason; he was probably wrong about that).

Moreover, it is crucial in the article to distinguish between the almost trivial version of identity of indiscernibles and the non-trivial. The almost trivial version is that if x and y have the same properties, they are identical, and this is how it is stated in the article. This version is easily shown to be true if one is liberal about what properties there are. Let P be the property of being identical with x. If x and y have the same properties, then because x has P, so does y. But then y is identical with x, since P is the property of being identical with x. To avoid such trivialization, the identity of indiscernibles needs to be restricted to purely qualitatively properties, i.e., ones that do not involve the existence of particular rigidly designated things, places, times, etc. It's hard to make this precise, but making it precise is necessary for stating the identity of indiscernibles.

I don't have the time for these revisions right now, but someone should do them. 141.161.84.89 20:23, 30 April 2007 (UTC)

Mention of duck

I'm going to delete this text:

So "if it looks like a duck, walks like a duck, and quacks like a duck, then it is a duck".

Why? Because the text is about classification, not about identity. This may be the case: If someone walks like a duck and quacks like a duck then that person is to be classified as a duck.

Controversial applications

what kind of logic is this? the first 3 statements are about bill's world the conclusion is not!

we would be correct in concluding "bill believes 49/7 and the square root of 49 are two different things. And that is really how the world is!

Leibniz was a genius. We have gone from an age of enlightenment to an age of darkness. We now live in a world of Wikipedia half-wits RWS

I agree with you there, however you are raising a philosophical reply, some people do believe what is in the article disputes Leibniz's law. Make a new section and call it replies if you want. --Aceizace 20:54, 19 February 2006 (UTC)

I think you are mistaken and the argument does not raise a valid philosophical reply. Note what you think problem with the argument is : “"bill believes 49/7 and the square root of 49 are two different things” and therefore “And that is really how the world is!” This is EXACTLY the point the criticism is trying to make. The critique says that if we accept “identity of indiscernible” (Leibniz’s law) we will be led into absurd proposition that what Bill thinks makes the world that way. And since this is absurd(“what kind of fucking logic is this?” being your quote) the Leibniz’s law is wrong.

The correct response to this attack on Leibniz's law is to claim that what a person thinks about the object is not the property of an object.--Hq3473 04:12, 20 February 2006 (UTC)

I just stumbled across this page and I also found the argument found in this section to be, uh, weak. I'll try to express it a little more mathematically.

The claim in step 6 is that "$49 \over 7$ is not identical to ${\sqrt {49}}$ is absurd".

This is not absurd. They *aren't* identical. One has a 7 and a horizontal line, the other has a line with a bunch of corners. Just looking at them you can see that they are different.

To be more precise, $49 \over 7$" and ${\sqrt {49}}$ mathematical expressions, and they are *different* expressions.

In some contexts, these expressions reduce to the same integer, but in others they don't. For example, if the default base is hexadecimal instead of decimal, these expressions yield different numbers, neither of which is an integer. In other contexts, not all operators are defined, which is what is going on with poor Bill. Once you introduce some more complicated operations, Gödel's incompleteness theorems shows that even if you know how to preform all operations and have a well defined context, there exists two expressions that are equal, but that you can not prove that they are equal. (Also see the halting problem.)

It is important to distinguish between "identical" and "equivalent (under some context)".

So, on a very simplistic visual level, you can see that $49 \over 7$ is not identical to ${\sqrt {49}}$, and on a much higher mathematical level you can understand that, indeed, determining if two expressions are the same can be a very hard problem. It is only fairly basic formulas that people automatically do the reductions and mentally classify them as "the same" and then make the incorrect leap to thinking they are "identical". Wrs1864 05:21, 18 November 2006 (UTC)

I changed this to a different example that does not involve evaluating math expressions, but presreves the basic problem of imperfect knowledge.--Hq3473 20:22, 18 November 2006 (UTC)

Thanks, I like your example *much* better. I think it is proably a good idea to leave the dispute tag for a little while to make sure that others agree, but as far as I'm concerned, my objections have been satisfied. Wrs1864 03:56, 19 November 2006 (UTC)

I there is no further objections i will remove the tag in a couple of days.--Hq3473 20:51, 19 November 2006 (UTC)

I am removing the tag--Hq3473 21:23, 22 November 2006 (UTC)

Good thing too, since the word problem for groups shows that there are situations where such identities are undecidable (uncomputable), and are thus, in a sense, unknowable. That is, the expressions may be equal, but it might not be knowable that they are equal. 67.198.37.16 (talk) 20:14, 22 July 2016 (UTC)

Leibniz?

I find it strange that Descartes lived and wrote Meditations before Leibniz was around, yet even the article itself says that Descartes used this reasoning. Might someone who knows more be able to include an explanation on why it is attributed to Leibniz? --Aceizace 20:54, 19 February 2006 (UTC)

The principle existed LONG before Descartes, probably can be attributed to Plato. His theory of Forms had a similar concept. The law got called Leibniz law, for his formulation not for content. Therefore it is not weird that Descartes uses the principle before Leibniz formulation.--Hq3473 15:34, 23 February 2006 (UTC)

From Subjective to Objective

This principle of the identity of indiscernibles makes the claim that a subjective judgment is to be taken as correctly describing the objective world. It claims that what appears to one person has true being for everyone. Perception is reality. However, that is precisely the problem that is to be solved by almost all philosophy. Kant's whole philosphy was written in order to determinine the correctness of assuming that subjective opinions are objective. Einstein's Relativity is also about the subjective observer and his experience of objects. Berkeley, Schopenhauer, Descartes, and many others have dealt with subjectivity and its relation to objectivity. For Leibniz to proclaim the identity of indiscernibles was, itself, an attempt to assert that his own subjective observations should be considered as being truly descriptive of the objective world of experience.Lestrade 01:43, 3 June 2006 (UTC)Lestrade

Feel free to edit the article acordingly. And do not forget to site your sources!--Hq3473 18:15, 7 June 2006 (UTC)

No, it does not make that claim. It does not say "seem to have all the same properties", it says "have all the same properties". It does not presuppose being able to observe all those properties; indeed, with our knowledge of modern physics (Heisenberg's uncertainty principle) we know that one can't observe all properties at the same time. But that doesn't affect the correctness of Leibniz's claim, which is a definitional claim not a claim about human observations. greenrd 01:52, 15 February 2007 (UTC)

Sure, i agree, in my understanding identity of indiscernibles is a metaphysical principal rather then epistemological one, but if someone find sites of famous philosophers thinking otherwise, the article should reflect it. --Hq3473 03:46, 15 February 2007 (UTC)

According to greenrd, Leibniz is making a dogmatic, ontological,objective assertion about the way that the world is constituted, rather than a hypothetical, subjective statement about his own perspective of the world. Then, greenrd brings in the well–known distinction between psychology and logic. This was often used by Russell to relegate his opponents to the class of subjective, introspective psychologists, while he triumphantly stood on the firm ground of universally objective logic.Lestrade 15:48, 27 September 2007 (UTC)Lestrade

Epistemological Version

The articles gives the above rather than an ontological principle:

  The identity of indiscernibles is an ontological principle that states that if there is no way of
  telling two entities apart then they are one and the same entity. That is, entities x and y are
  identical if and only if any predicate possessed by x is also possessed by y and vice versa.

Yours truly,--Ludvikus 03:53, 14 December 2006 (UTC)

Ontological principle

I've modified/corrected the opening sentence from the above, to the following:

    The identity of indiscernibles is an ontological principle; i.e., that if (two
    or more) object(s), or entity/ies have all thier/its property/ies in
    common then they (it) are identical (are one and the same entity). That is, entities x and
    y are identical if and only if any predicate possessed by x is also possessed by y and
    vice versa.

Yours truly,--Ludvikus 04:05, 14 December 2006 (UTC)

So you're the one I should kick in the balls for making it needlessly illegible. Great. I'll change it back to English now. --76.224.107.34 20:36, 10 June 2007 (UTC)

Criticism Counterexample

The proposed criticism is: "Opponents of this counterexample would claim that a contradiction can be found between proposition (2) and (3) (i.e. Lois Lane cannot have opposite thoughts about the same object, regardless of the name)." To me this objection seems like begging the question. Lane think that the person can and can't fly at the same time because she does not know that it is the same person. So she DOES have opposite thoughts, and denying it begs the question: I.E. it is arguing for "Identity of indiscernibles" like this: "I know that Identity of indiscernibles is true, and therefore your counterexample(no matter what it is) cannot work". Thus i propose deleting this weak objection. --Hq3473 23:24, 1 March 2007 (UTC)

Quine's Variation

The most well spoken version of the identification of indiscernibles I have encountered is found in Quine's "Identity, Ostension, and Hypostasis," as follows: "Objects indistinguishable from one another within the terms of a given discourse should be construed as identical for that discourse." This gets us away from descriptions about properties and the like, which of course invite the confusion of supposing that the creation of two objects with identical sets of properties might disprove the proposition (Liebniz would argue that, for this to be the case, you would have to find a way to have two identical objects occupying the same spatio-temporal location as well, which makes a refutation of this kind rather hard to manage, unless you can imagine two individual objects occupying the same space), or suggesting that a single object, seen, say, from two different perspectives, would also disprove the proposition. Of course, the Quinean version is not ontological in the sense of defining specificity to real objects in the physical universe. It is a deliberately broad definition, intended to deal with another set of representational philosophical problems that are only partly related to what Liebniz was interested in demonstrating. Nevertheless, it would seem to me a worthy candidate for admission in this article, for some plucky chap willing to add it in.

Feel free to write about Quine's version, its advantages and shortcomings, etc. Make sure to source to Quine.--Hq3473 03:46, 19 April 2007 (UTC)

Descartes' argument

I don't think that Descartes' argument should be described as an application of the identity of indiscernables. Note that the conclusion, that the body and the mind are different, states that two things are not identical. If anything, this would be an instance of principle 1, the indiscernibility of identicals. Zarquon 03:48, 19 April 2007 (UTC)

The definition in the first paragraph says: "The identity of indiscernibles is an ontological principle: that if and only if. Not that the "if and only if" part makes the Identity of indiscernibles work both ways. --Hq3473 04:59, 19 April 2007 (UTC)

"Controversial Applications" not true

Entities x and y are identical if and only if any predicate possessed by x is also possessed by y and vice versa. Clark Kent is Superman's secret identity; that is, they're the same person (identical) but people don't know this fact. Lois Lane thinks that Clark Kent cannot fly. Lois Lane thinks that Superman can fly. Therefore Superman has a property that Clark Kent does not have, namely that Lois Lane thinks that he can fly. Therefore, Superman is not identical to Clark Kent. Since in proposition 6 we come to a contradiction with proposition 2, we conclude that at least one of the premises is wrong. Either: Leibniz's law is wrong; or else A person's knowledge about x is not a predicate of x, thus undermining Descartes' argument.

The conclusion "Since in proposition 6 we come to a contradiction with proposition 2, we conclude that at least one of the premises is wrong." has been obtained ridiculously. To show that this is an invalid argument, firstly we consider the statement "Therefore Superman has a property that Clark Kent does not have, namely that Lois Lane thinks that he can fly.". Simply put a property of an object must be inherent to itself and not based on some observers view. It is also possible that we cannot confirm that an object has a certain property or not, in which case be contradictory by saying that an electron is a wave and not a particle or vice versa, then when observed we "think" it is a wave or particle, thus appearing contradictory based on the identity of indiscernibles. In that case we cannot say whether the electron is identical to itself and cannot make any conclusions.

Nicholaslyz 10:30, 9 July 2007 (UTC)

Note that you say that " Simply put a property of an object must be inherent to itself and not based on some observers view", this is the point of the "proof" precisely. Descartes tried to rely on a human belief about an object as a property of an object, this exact line of reasoning the "proof" aims to debunk.--Hq3473 20:24, 9 July 2007 (UTC)

Contradictory and incorrect(?) definitions; Proposed article split

The lead defines identity of indiscernables as being: two objects are equal if and only if they have all properties in common. However, further down, identity of indiscernables is distinguished from indiscernability of identicals: the two halves of the if-and-only-if. But it can't be half of itself...

Moreover, many authors use Leibniz's Law to mean only indiscernability of identicals, and the first comment on this very talk page says that identity of indiscernables is not one of Leibniz's great metaphysical principles, although he accepted it.

I think it would make sense to split this page into two separate articles: identity of indiscernables and indiscernability of identicals. I mean, Black's objection is directed at the identity of indiscernables, and the Superman confusion relates to the indiscernability of identicals. Vaccillation between covering the two principles makes for a confusing article.

Let me ask the question: Is there any evidence that any reliable source apart from Wikipedia has treated these two principles together - or that the value of doing so outweighs any confusion created?—greenrd 01:45, 27 October 2007 (UTC)

Oppose split. The introduction can be altered from "if and only if" to the correct statement. I can see where the confusion lies in the Superman example. As for splitting it, I think it would be better to just rename the article to something more appropriate, and have a distinct separation within the article itself. Necessary and sufficient accomplishes this. — metaprimer (talk) 13:18, 27 October 2007 (UTC)

First, thanks for removing the self-contradiction. I think there is an important difference between this article and necessary and sufficient - this article could probably benefit from more expansion (e.g. where they have been applied to try and prove various statements, other controversies about them, etc.); and if it is likely to become a long article containing two sub-articles without much overlap between them (and with potential for confusion!) it makes sense to split it up into two articles.

My point about the Superman section - which I didn't make very clearly, I admit - was that both Descartes' argument and the Superman "paradox" are applications of the indiscernibility of identicals (contrary to what the section currently says).

Also, what would be a good new name, if we kept this article as one article? "Identity of indiscernibles and indiscernibility of identicals" is too long and awkward, in my opinion.—greenrd 13:42, 27 October 2007 (UTC)

Response to Black's critique

Is there a reference for this "response", or is it original research?--Hq3473 22:50, 28 October 2007 (UTC)

It's original research. I'm aware of WP:NOR, but I added it in the hope that no-one would object, in the spirit of WP:IAR. Feel free to remove it.—greenrd 08:19, 29 October 2007 (UTC)

I will remove, because it addition to being OR it does not seem a particularly strong response to Max Black. Sure the universe can be looped, but this just goes to show that Identity of indiscernibles will lead to weird counter-intuitive results. --Hq3473 13:26, 29 October 2007 (UTC)

Black is rather obviously wrong in that he first defines a universe model that contains two distinct objects (say, two parts containing "identical spheres", because that is what reflection symmetry suggests) only to then claim the spheres in both objects are one and the same. To then go on to "refute" that by constructing yet another bilaterally symmetrical universe wherein you place two objects, and also that you have no way to spatially tell them apart when you've just defined them as being spatially distinct, doesn't really help people see the point. I seem to recall Hacking exposed that rather more elegantly and elaborately than the article now suggests. JeR (talk) 19:54, 31 March 2010 (UTC)

Hahah I hadn't thought of that critique. Do you think you could clarify Hacking's argument on the main article?--Heyitspeter (talk) 20:05, 31 March 2010 (UTC)

Secret identity.

The example in the article concludes;

Leibniz's law is wrong; or else

A person's knowledge about x is not a predicate of x, thus undermining Descartes' argument.

However it seems to me that it might just as well be the claim that superman is equal to clark kent that is wrong. Ie. the claim that they are the same person is weaker than the claim that they are equal.

An example that does not involve other peoples believes would be the Supreme Governor of the Church of England and the Paramount Chief of Fiji. The first having the right to formally appoint high-ranking members of the church of England. Taemyr (talk) 17:56, 6 April 2008 (UTC)

I don't like this. Supreme Governor of the Church of England and the Paramount Chief of Fiji COULD be different in, for example, an unlikely event that Fiji succeeds. Clark Kent and Superman or any other real or fictional person with a secret identity are the same people no matter what.--Hq3473 (talk) 16:53, 12 August 2008 (UTC)

Try Supreme governor of England in the year 2000 and the Paramount Chief of Fiji in the same year. Or for an even clearer example, two pointers that point to the same variable. The label is different from the thing. Unless this counter argument is sourced I will remove it as OR. Taemyr (talk) 05:32, 11 September 2008 (UTC)

Secret identity is a common counterexample to desecrates argument. I do not remember which particular article i was quoting at the time. But i believe this article is a sufficient source(although this one take batman as an example): " Alter Egos and Their Names, David Pitt, The Journal of Philosophy, Vol. 98, No. 10 (Oct., 2001), pp. 531-552, page 550", you can find the artcile in full at [1].--Hq3473 (talk) 13:41, 11 September 2008 (UTC)

It seems that this source argues my point though;

Moreover, I do not share Saul’s puzzlement about these cases; for it seems to me that the most straightforward explanation of the substitution failures – namely, that ‘Superman’ and ‘Clark Kent’, ‘Bruce Wayne’ and ‘Batman’ are not coreferential – is correct.

— David Pitt, Alter Egos and Their Names

Although presumably Saul takes an other view. Taemyr (talk) 00:28, 12 September 2008 (UTC)

The point is that the article lists the "alter-ego" as an example of a counter argument to a certain use of Identity of indiscernibles. The article also criticizes this counter-example. So i feel this example should stay, as opposed to making up a different example. As for the criticism, feel free to add it and reference the same source.--Hq3473 (talk) 13:33, 12 September 2008 (UTC)

The Principle "states that two or more objects...are identical..."?!

Surely the principle doesn't state, as the article now says it does, that "two or more objects or entities are identical if...." If it really does state that, then it's clearly absurd; for how can two objects be identical? Isokrates (talk) 20:56, 19 April 2008 (UTC)

It is usual in formal arguments to interpret "two objects" as "two objects that might be instances of the same object." When you require them to be two seperate object this usually needs to be stated. So a relation ≤, is antisymmetric if for any two objects a≤b and b≤a implies a=b. Compare to ... for any two objects a≤b and b≤a is a contradiction. Taemyr (talk) 19:36, 20 April 2008 (UTC)

• Your example is not as helpful as it appears. Every two objects are instances of a<b. And every one object is an instance of a=a. No object(s) is/are instances of both. --Ludvikus (talk) 21:22, 20 April 2008 (UTC)
• I'm saying that you really cannot instantiate the relation you give above - although you do conform by it to standard practice (it's as if 2 contradictions are wiping each other out). --Ludvikus (talk) 21:30, 20 April 2008 (UTC)
• What a strange relation: "something is greater than or equal to something else"! --Ludvikus (talk) 21:32, 20 April 2008 (UTC)
  • Isn't that a tautology? --Ludvikus (talk) 21:37, 20 April 2008 (UTC)

I should perhaps not have used the symbol ≤. Remember that we are defining our relation. So don't pressupose arithmetic "less than or equal", arithmetic "less than or equal" is simply an instance of an antisymetric relation. And no, every two objects need not be instances of where a and b is different. You usually has to specify it explicitly when you want to say that you are reasoning about pairs of un-equal elements. Taemyr (talk) 07:28, 21 April 2008 (UTC)

Two objects are one if such and such is the case.

I just want to simplify that ordinary language version of the alleged apparent self-contradiction. --Ludvikus (talk) 21:16, 20 April 2008 (UTC)

How about Hepsherus and Phosporus both being Venus? Hesperus#"Hesperus is Phosphorus".--Hq3473 (talk) 01:08, 21 April 2008 (UTC)

"Two objects are one", I think this is the heart of the reason why the above poster sees a contradiction. His view is that two objects are never one. So getting around the percieved contradiction while retaining formal correctness would require something like "Two differing objects never share every property." Taemyr (talk) 07:33, 21 April 2008 (UTC)

I personally like the definition from Quine given earlier on this talk page. "Objects indistinguishable from one another within the terms of a given discourse should be construed as identical for that discourse." Because this definition includes which properties are of intererest or not. Taemyr (talk) 07:39, 21 April 2008 (UTC)

Kripke

About this, added and removed twice now;

Saul Kripke, in Naming and Necessity, argued that Superman and Clark Kent are rigid designators that both refer to the same person. Whatever Lois Lane believes about Superman, she necessarily believes about Clark Kent, though she is not aware of that fact. In other words, she does believe that Clark Kent can fly, because when she is forming beliefs about Superman, she is in fact referring to Clark Kent. They are the same person. Under this construction, Lois Lane has a pair of conflicting beliefs, but there is no property that Superman has that Clark Kent lacks, and vice versa

This is largely irrelevant. If Lois Lane is capable of holding conflicting beliefs about the properties of Clark Kent due to her beliefs about Superman. Then Descartes is capable of holding conflicting beliefs about the entity that is his body and the entity that is Descartes. Taemyr (talk) 21:43, 20 December 2008 (UTC)

Pitt, in the article that the argument is sourced on does however raise a valid objection. Since it questions whether the fact that Clark Kent is the same person as Superman is sufficient for Clark Kent and Superman to be corefferential. Using it to source this would be OR though, since the argument is not treated directly by Pitt.Taemyr (talk) 21:47, 20 December 2008 (UTC)

I agree perhaps it is not the best reference. But the argument i quite common in modern philosophy. For example: [2]. I also think this might mentioned here: Steinhart, E. (2002) Indiscernible persons. Metaphilosophy 33 (3), 300 - 320. Will find better references later. --Hq3473 (talk) 19:45, 29 January 2009 (UTC)

Have any sources drawn comparisons with the Masked man fallacy? Taemyr (talk) 13:26, 30 January 2009 (UTC)

Suggestions for Rewrite

There is a profound evolution of thought surrounding the principle of Identity of Indiscernibles spanning more than 2500 years in the West, and successive formulations range from trivial, tautologically true constructions to the metaphysically-laden statement invented by Leibniz. As an axiomatic law of thought, any given instance of this principle can be understood and analyzed only in context, within the given metaphysical framework for that instance. That is, no meaningful discussion of this principle can take place outside of the historical philosophical traditions in which the various instances of this principle have been conceived.

This article fails in this regard, and by implication more or less equates Leibniz' Rule with its own statement of the principle of Identity of Indiscernibles, which is quite different. By declaring merely that "a form" of this rule also was presented by Leibniz, while failing to identify any difference in Leibniz' statement, it is likely readers will wrongly conclude that any nuance adopted by Leibniz is of little import. The actual statement of Leibniz' Rule is as follows:

For any individuals, x and y, if for any intrinsic, non relational property f, x has f if and only if y has f, then x is identical with y.

Consequently for Leibniz, if x and y are distinct they must differ in terms of some intrinsic, non relational property. If the editors had included such detail then the article would not have invited to no avail such sophomoric (at times puerile) banter and facile epistemological refutation.

A useful article on the Identity of Indiscernibles should enumerate and order its most important formulations, and for each provide some metaphysical context for its motivation and limits of application. Thus, in the section on Leibniz' Rule, a minimal outline of his metaphysics, giving special attention to his ontology (real entities, well-founded phenomena, actual existents, i.e. monads), as well as to his meaning of relational and non relational properties, is essential to understanding his formulation of the principle. For instance, Leibniz sought to avoid commitment to space as an independent (ontological) entity, relying instead upon the notion of relational properties between material objects. As such, this whole discussion of Black's thought experiment, utterly divorced as it is from any pertinent context, is absurd. This is because the axioms of Black's imaginary "universe", at least as far as these have been presented in this article, are incomplete, and his assertion is undecidable, as we have not sufficient ground for comparing the intrinsic, non relational properties, whatever these are, of the two hypothetical spheres. I suspect that a first hand reading (not a wiki) of Black would reveal a far deeper and nuanced position than that presented by the editors thus far. Someone should check this. Next, properties such as "x believes N about y" are extrinsic and relational and thus cannot be used in the formulation of so-called thought experiments designed to refute Leibniz' Rule, which precludes such arguments out of hand. Black's impact on other formulations of the principle could be examined.

The article begins as follows:

The identity of indiscernibles is an ontological principle which states that two or more objects or entities are identical (are one and the same entity), if they have all their properties in common.

This statement thus tells us that whenever two entities share all properties in common then they are the same entity, and from this we can derive the contrapositive assertion that if the entities are not identical then they must differ with respect to some property; however, the statement does not say what conclusion can be drawn from the converse, that is, when the objects differ with respect to some property. What then? Since the structure of Descartes' reasoning as it applies here conforms to the unstated converse of the principle given in the article (e.g. If some property is not shared between two objects then they are not identical), the later statement from the article, "one famous application of the indiscernibility of identicals was by René Descartes in his Meditations on First Philosophy," is not supported. Again, if anything meaningful is to obtain from that allusion then one would need to precisely articulate the particular law of thought Descartes was relying upon and then directly compare this with the appropriate formulation and its embedding metaphysical context by now included in the enumeration of formulations of the principle. Nor do we know what constitutes "entities" or "properties" within the sparse construct given in this article. Beyond this, we are not given any epistemological context, which then opens the floodgates to all the tired controversies between rationalists, empiricists, foundationalists, pragmatists, ... ad infinitum. In sum, the present form of this article is poorly conceived, and this whole tangent involving modal logic and intensional contexts is misplaced.

Finally, the opening discussion of First Order logical representations, which may or may not apply to any given formulation of the principle, lacks motivation, is somewhat misleading, and should come later, probably under a heading such as 'applications' or 'mathematical representations' or the like. Including this discussion at the very beginning without qualification suggests that any given formulation of the Identity of Indiscernibles principle, including Leibniz' Rule, is essentially an axiom or theorem of First Order Logic, which is not the case. Moreover, the discussion of tautological identity without sufficient exposition suggests that Leibniz' Rule permits this, which it does not. I suggest a complete rewrite of this article. I should mention that I am not calling for original research here but rather what one at minimum would expect from an accessible, peer-reviewed exposition: namely, an informed discussion of the principle, its history, metaphysical context, applications, and current status as a viable precept.

G.W. Leibniz, 'On the Principle of Indiscernibles', in Leibniz: Philosophical Writings, ed. and tr. G.H.R. Parkinson and M. Morris (London 1973). C.D. Broad, Leibniz: An Introduction (Cambridge 1975). Oxford Companion to Philosophy, ed. Honderich, Ted, (Oxford 1995).

--Devala1 (talk) 23:01, 24 May 2010 (UTC)

Be bold! As long as you cite your sources it should be received positively.--Heyitspeter (talk) 23:46, 24 May 2010 (UTC)

Rm Kant: why

I took out:

In his Critique of Pure Reason, Immanuel Kant argues that it is necessary to distinguish between the thing in itself and its appearance. Even if two objects have completely the same properties, if they are at two different places at the same time, they are numerically different (see: identity)

which had the tag "A citation linking this argument explicitly to identity of indiscernables is required here.", though I'd taken it out before reading that. Firstly, "it is necessary to distinguish between the thing in itself and its appearance" is irrelevant, and must presumably have been a confusion on the part of someone. Secondly, "...they are numerically different" is wrong, in that the use of "numerically" is wrong / meaningless. Third, it isn't at all clear that Kant makes the second argument "Even if two objects have completely the same properties, if they are at two different places at the same time", and I hope he doesn't, because clearly location is a property. And fourth, that point is made in the following section William M. Connolley (talk) 11:17, 17 August 2011 (UTC)

Kant claimed that Leibniz mistakenly considered two perceived objects as though they were mere concepts. Two similar concepts can be thought to be the same or indiscernible. However, if the two similar perceived objects are correctly considered to be phenomena, then they would be known as being in different places in space. In this way, they would not be considered to be the same or indiscernible.

In Critique of Pure Reason, A 264, Kant wrote: "Leibniz took phenomena to be things by themselves, intelligibilia, that is, objects of the pure understanding … and from that point of view his principle of their indiscernibility (principium identitas indiscernibilium) could not be contested. As, however, they are objects of sensibility, and the use of the understanding with regard to them is not pure, but only empirical, their plurality and numerical diversity are indicated by space itself, as the condition of external phenomena. For one part of space, though it may be perfectly similar and equal to another, is still outside it …."

Lestrade (talk) 12:00, 17 August 2011 (UTC)Lestrade

Rm Quantum

I removed this:

Unfortunately this is arguing from the existence of macroscopic objects with possibly hidden variables which are in fact different. And indeed, this just proves the impossibility of there being only 2 distinct objects in a universe. Indeed in quantum mechanics, the relation is between an object (say the object's electric field) and the superposition of all possible identical objects relative to it. So e.g. one could argue that an isolated charged particle has a total field through the spherical surface on its center which is invariant (independent of radius of the sphere). And this is the content of the integral equation version of the divergence of the electric field (Maxwell's 1st law discovered before him).

Whoever wrote that appears to not understand what hidden variables are, what quantum mechanics is, or Maxwell's electrodynamics. Seems to be some original research. 67.198.37.16 (talk) 20:04, 22 July 2016 (UTC)

Revising the Descartes section

I removed this:

A response may be that the argument in the Meditations on First Philosophy isn't that Descartes cannot doubt the existence of his mind, but rather that it is beyond doubt, such that no being with understanding could doubt it. This much stronger claim doesn't resort to relational properties, but rather presents monadic properties, as the foundation for the use of Leibniz's law. One could expound an infinite list of relational properties that may appear to undermine Leibniz's law (e.g., Lois Lane loves Clark Kent, but not Superman. etc.) but nonetheless any approach focused on monadic properties will always produce accurate results in support of Descartes' claim.

The argument is incorrect, as doubt is not a monadic property; not surprisingly, it is also not the argument made by Carriero. Ben Standeven (talk) 15:35, 27 January 2017 (UTC)

References

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The indiscernibility of identicals

For any x and y, if x is identical to y, then x and y have all the same properties.

$\forall x\,\forall y\,[x=y\rightarrow \forall P(Px\leftrightarrow Py)]$

How about the property "has proper name ‹some name›"? For instance, in Peano arithmetic the number "two" can have "name" either $SS0$ or $S0+S0$. The equality $SS0=S0+S0$ is valid, but the property "has proper name $SS0$" is valid only for the first closed term. I have no questions about first-order schema of the indiscernibility of identicals: The property "has proper name $SS0$" cannot be expressed by a formula of Peano arithmetic. But the second-order axiom (cited above) embraces all properties, including inexpressible by formulas.

Eugepros (talk) 07:08, 8 April 2018 (UTC)

Symmetric Universe

This looks like an individual's own rather eclectic research and reasoning, involving what seems to be a fair bit of digression. I do not understand all the concepts mentioned in this section, but reading the "symmetric universe" section is hugely less comprehensible than the rest of the article, and at the very least needs to be explained more slowly and in much greater depth. The first paragraph is ok, and I can just about follow the second one, but the discussion of how there "must" be a "quantum asymmetry" in Black's universe, the curvature of space-time and the origins of calculus leave me behind after countless readings. Indeed, in a discussion of pure logic, the involvement of so many physical experiments seems to me inexplicable.

Additionally, what seem to be bold claims are made, with a very low footnote density. The footnotes available don't all have page numbers, and one is just a google drive link. Finally, I have never seen the "--" punctuation anywhere on wikipedia before, which does not give me great confidence in the writer's familiarity with wikipedia standards and practice.

Under the "Be Bold" policy wikipedia recommends, I will delete the "symmetric universe" section in a week or so, if nobody has any other thoughts. I will leave in the first paragraph, and perhaps point the reader to the issues involved in the later discussion, if I can figure out what they are. If an article offers poorly-evidenced text that seems like gibberish in comparison with the rest of the article, it surely should not be part of any encyclopedia.

If someone can explain to me what the "symmetric universe" section is about, clearly and with sources, I suggest they do so then replace the original text with their explanation. It is possible that I am just being very thick, feel free to tell me so. — Preceding unsigned comment added by 79.66.61.34 (talk) 16:10, 22 August 2018 (UTC)

Link to the Yoneda lemma ?

As a mathematician, reading this article, I can't help but notice that this principle is similar in philosophy to the Yoneda lemma in category theory, which can be seen as a formalization of this principle. That lemma states (as a corollary) that two objects who can't be distinguished based on how they interact with their surroundings (i.e. X,X' satisfy Hom(X,Y)~Hom(X',Y) for all Y, naturally) have to be isomorphic (i.e. X~X', which is the mathematical way of saying "they are the same object with another name"). Maybe speaking about the link could be interesting ? A "serious" reference might be needed. 2A01:E0A:2F0:4C0:3D15:3C47:C8AE:903E (talk) 10:40, 6 March 2020 (UTC)

Critique

Max Black has argued against the identity of indiscernibles by counterexample. Notice that to show that 2. is false, it is sufficient that one provide a model in which there are two distinct (non-identical) things that have all the same properties. He claimed that in the symmetric universe where only two symmetrical spheres exist, the two spheres are two distinct objects, even though they have all the properties in common.

---

I know that Max Black is correct because I am in possession of a wonderful counterexample from pure mathematics--in other words, I have an elegant simple model--which proves, conclusively and persuasively, that there is at least one pair of numerically distinct objects which--nevertheless--have all their properties in common. And as soon as I have my proof published, or submitted, to a scholarly peer-reviewed philosophical journal, I look forward of the opportunity of publishing it here in this excellent Wikipedia article. Ludvikus 03:50, 2 September 2006 (UTC)

---

    I've transcribed here the above from the Article page - before reversion. I have written the
    comment before having become an experienced Wikipedian, understanding and following WP policy.
    Nevertheless, my observation remains true. But like Fermat? - No space to ellaborate?
    Yours truly,--Ludvikus 03:22, 14 December 2006 (UTC)

---

I would say that Mr. Black's critique doesn't hold water, as the two spheres he describes obviously occupy different locations in space. As location in space counts as a property, then the two spheres do not have the same properties. Anyone disagree? -Tim —Preceding unsigned comment added by 218.219.191.130 (talk) 00:07, 10 September 2007 (UTC)

Yeah, there is no such thing as "space", the only way to define space is in relation to other objects. So in the world with only 2 objects the only space for a spehere is defined by "distance to the other sphere" but the ther sphere will have the same prperty, so we still cannot distinguish them. See Theory of relativity.--Hq3473 02:09, 10 September 2007 (UTC)

Ok, it took me a bit of thinking to figure out what seemed wrong about your response, and here it is: First, Black says that the only two things that exist in this hypothetical universe are the two spheres. This, however, cannot be technically accurate, as the properties that we are using to describe those spheres must also exist. So what properties exist? Obviously, numerical, spatial, and physical ones exist, as the spheres exist in space and have size, shape and numerosity. Of course, their size and shape are the same. However, logical properties must also exist. And the critical flaw in Black's example is that with the very act of saying that two spheres exist, he imbues them with the logical property of not being the same object. Sphere A is sphere A. Sphere B is sphere B. Sphere A is not sphere B, and vice-versa. What made me realize this was your response to me in which you wrote "distance to the OTHER sphere." In order for "other" to have any meaning, there would have to be some property that differed between the spheres that allowed us to tell them apart - and that property was the logical one that they have been defined as two separate objects from the start. Any objections to that? - Tim —Preceding unsigned comment added by 125.201.152.222 (talk) 11:48, 14 September 2007 (UTC)

No by saying there are two sphere Black does NOT give you the power to differentiate spheres. Sure if a spectator were to appear in the Black's world he would immediately identify spheres as 1 and 2. But there is no spectator. Think about it this way. Say you pick a sphere and call it Spehere 1 and the other one Spehere 2. Then you leave the world, and then come back again. WOuld you be able to tell which one is Sphere 1 and which one is spehere 2? No you would not. Because Max's world has no way to differentiate the spheres. --Hq3473 13:28, 20 September 2007 (UTC)

I'm sorry, I think I worded my comment above somewhat badly. What I meant when I wrote "there would have to be some property that differed between the spheres that allowed us to tell them apart" was not that we would be able to tell which sphere was A and which was B (after having labeled them and then re-entered Max's world). You are right; we would not be able to tell.

Let me put my argument in other words: If more than one object exists in a universe, then those objects will always be identifiable as different by means of logical properties. This is why: We know from Max's definition of his world that the Sphere A and Sphere B are separate objects. If so, then Sphere A logically *must* have the property of being "not equal to Sphere B." Likewise, Sphere B must have the property of being "not equal to Sphere A." Without these properties, we would be literally unable to conceive of Spheres A and B as being two separate objects; we would have to conclude that "Sphere A" and "Sphere B" were simply two different names for the exact same thing. In case you aren't convinced, take the example of an object lacking, say, a certain mathematical property. Let us say that this object has no numerosity. It is not a single object, nor is it many; the idea of numerosity simply does not apply. Can you imagine it? I can't. I can imagine one object and I can imagine more than one, but no matter how I try, I cannot conceive of an object without numerosity. (Nor can I talk about it! Notice how I had to use singular pronouns and verb conjugations to describe the object.) Sphere A and Sphere B are in the same boat, with reference to logical properties. We cannot conceive of their being separate objects unless each has the property of being not equal to the other.

Logical properties are so taken-for-granted that they are easy to forget. Think of a person debating whether the Law of Noncontradiction is true, not realizing that they are assuming it's true in order to have the debate. Max Black must have forgotten about logical properties, or not thoroughly understood them, when he made his argument against the law of indiscernibles.

One last thing I could say, although this argument shouldn't be necessary given the above, is that Sphere A and Sphere B *do* have different properties as per their location, despite what Hq3473 wrote before. Consider that Sphere A has the property of being 0 distance from Sphere A, while Sphere B has the property of being some non-zero distance from Sphere A. There's something else they don't have in common. -Tim

You seem to have begged the question at "Sphere A logically *must* have the property of being "not equal to Sphere B." Such thing does not follow from "Sphere A and Sphere B are separate objects." This is the whole argument that Max is trying to make -- A and B are separate objects yet Sphere A does NOT has a property of being not equal to B, in fact it IS equal to B. This is the whole point --to show that by using identity of indiscernibles we get two separate object which are nevertheless equal, and to straight up assume otherwise amounts to saying "A and B are not equal because they are not equal." In the end Max's attack works, because either you have to accept that Spheres are "separate but equal" or you eviscerate the Identity of indiscernibles by saying that all distinct objects have the property of being different from other objects and thus are different from other objects. How would such law be useful?--Hq3473 15:23, 27 September 2007 (UTC).

I agree with the grandparent, the argument doesn't hold water. Max Black must construct a space in which to embed to objects, even if this is the topological/geometric/set/etc. construct of "just two spheres." However, if we were to refer to just the simple constructivist approach of S={A,B}, where A,B are elements of the set "Sphere" then we can ask what properties they have in common (up to the Leibniz equality). However, Max Black is implicitly adding more properties: symmetry and an embedding in "the universe". Now, the description in Wikipedia is too weak to make any meaningful conjecture, but knowing the sort of reasoning logicians/philosopher's use, he's probably thinking of a closed, bounded, infinite symmetric space like unit cell of P2; see Crystallography. In that case, while P2 does not discern handedness, orientation, etc., it still provides an infinite number of exactly equivalent metric embeddings. In any of these embeddings we can determine the vector offset (for free, with no additional assumptions), uniquely between the two pairs. If you assume that no meaningful embedding occurs, then you must add in the assumption to the set construction that A does not equal B, and thus, Max Black (and the parent) are begging the question. —Preceding unsigned comment added by 128.194.143.200 (talk) 16:54, 4 March 2008 (UTC)

The last time i checked there is no such thing as "absolute space". Space is only meaning-full if the reference frame is well determined (Introduction_to_special_relativity#Reference_frames_and_Lorentz_transformations:_relativity_revisited). In the world described by Black there is no well defined "space" untill you fix a frame to any one of the spheres. In any case you are welcome to present critique of Max Black's work if you find an appropriate authoritative source. --Hq3473 (talk) 17:10, 12 August 2008 (UTC)

hahaha okay I suppose I shouldn't open up this debate again, but I'm going to try and lay out the argument more clearly: this universe is the set of two items, {a,b}. Each item has the property of being a sphere of a specific size. So the universe is described as, {Sa,Sb}. These are all the properties that the objects have when taken individually, so we can't differentiate them there. Now let's think spatial. If a has the property of being one meter away from the other sphere, then b has the property of being one meter away from the other sphere, and vice versa ([Oa→Ob]&[Ob→Oa]). So giving one sphere the property of being spatially separated from the other doesn't mean that the two spheres actually have different properties. In fact, giving the property to one sphere implies that the other sphere has the exact same property. And if we say, a has the property of not being identical with b, then this implies that b has the property of not being identical with a ([a≠b]→[b≠a]), and vice versa ([b≠a]→[a≠b]). If we take these cases as exhaustive (and if space isn't absolute, which it isn't, they are exhaustive), then we have a case in which the antecedent of leibnez' second law is true Pa<-->Pb but the consequent is false a≠b, making this a counter-example to the "law." Fair enough?--Heyitspeter (talk) 05:59, 1 April 2009 (UTC)

But now of course I've thought of a counterexample to that formulation haha. We can just say, a has the property of being one meter from b. a having this property does not imply b having this property, so there we are. There is a property that one sphere has that the other doesn't implicitly have, so Black's example is finished... Has anyone read his article? That would probably help. I have no idea how he would deal with this.--Heyitspeter (talk) 06:03, 1 April 2009 (UTC)

You can't think spatially. The spheres are not spatially separated, if they where then they are discernible as separate. Consider the property P(a,b) which is true iff a and b are at different places. Then P(Sa,Sb) is true, and P(Sa,Sa) is false. So Sb would have a property that Sa lacks. Taemyr (talk) 23:20, 1 April 2009 (UTC)

Nah wouldn't work that way because P(Sa,Sb)←→P(Sb,Sa) would be true, along with ~P(Sa,Sa)←→~P(Sb,Sb) [here, because the two are logical truths], so in either case the left half of Leibniz' conditional is true but the right is false, creating a counterexample, affirming Black's hypothesis even when the sphere's are spatially separated. But still, one has the property of not being a, and the other doesn't have this property. This is the case even when the sphere's are not in different places...--Heyitspeter (talk) 06:02, 2 April 2009 (UTC)

Mmm sorry I see what you mean. P(Sa,Sb) does not imply P(Sb,Sb), so having the property of being separate from Sb is not shared. But still, we have (a≠b) not implying (b≠b), no matter where the two spheres are... --Heyitspeter (talk) 06:06, 2 April 2009 (UTC)

Yes but Identity of indiscernible defines ≠. So it's a property that you can't use to define indiscernability because if you do you create a circular argument. Taemyr (talk) 06:29, 2 April 2009 (UTC)

All I'm saying is: Suppose identity is a property. Then there is a property that one sphere has that the other does not, so we do not have a case in which the antecedent of Leibniz' theorem is true but the consequent false. This isn't begging the question. Identity of Indiscernible doesn't define identity, it's just a rule by which we can know that identity obtains. --Heyitspeter (talk) 08:27, 2 April 2009 (UTC)

My take on this is that The indiscernibility of identicals and The identity of indiscernibles together define identity. But even if you just look at it as a property that indiscernible elements have then you should not include the identity property. The identity of indiscernibles states; "For any x and y, if x and y have all the same properties, then x is identical to y." If you include the identity as a property then this can be strengthened to "For any x and y, if x and y are identical, then x is identical to y.", which is a rather trivial observation. Identity of indiscernibles is only interesting when it can lead you to conclude that x and y is identical, which means that you can't have a rule that requires you to know if they are identical in order to apply it. Taemyr (talk) 03:06, 3 April 2009 (UTC)

Mmm so Black is saying that all properties short of identity can be shared by two objects, but that those objects can nevertheless be non-identical. Leibniz' second law being wrong. Got it. Thanks!--Heyitspeter (talk) 08:55, 3 April 2009 (UTC)

Still supposing identity is a property there is no logically possible counter-example. This is a really weird situation.--Heyitspeter (talk) 08:37, 5 April 2009 (UTC)

This is my first wiki comment, so I do hope I don't break any important conventions. I have only two points to make. The first point concerns the discussion about symmetric universes. I have been puzzling long over this problem (about 30 years). Can an entity that in no way interacts with the universe be said to exist? If not, then one cannot talk of any other universe. To my thinking 'universe' means 'everything that is'. It would then be nonsense to talk about the properties of another universe. One might say that the only things that can have the same properties are all those things that do not exist; but this makes no sense either.

Problem with Superman example

---

My second comment is about the Clarke Kent and Superman example. If I am not mistaken, one of them wears a tight costume and the other doesn't. So there is actually a difference between them in that sense. If there was absolutely no difference there could be no reason why any person would thinks the other person is able to fly and not able fly at the same time. What is more, as far as I am aware, no one can have two distinct and contradictory thoughts at the same time, so the woman that thinks one thing is not actually the same as the woman that thinks another thing - as she is now in the future and has therefore changed. All of this becomes very complicated. — Preceding unsigned comment added by 81.164.118.56 (talk) 02:02, 18 December 2020 (UTC)

Łukaszyk–Karmowski metric

"calling the Łukaszyk–Karmowski function a metric although it isn't positive definite is a matter of naming, not a critique"

This distance function is not “a metric”, as it does not satisfy the 1st metric axiom (albeit satisfying the remaining two). So perhaps Łukaszyk–Karmowski metric should be moved to Łukaszyk–Karmowski distance (cf. ) and appropriately rephrased. Ł-K distance is positive definite for Dirac delta distributions. The only point here is that there exists a distance function (Ł-K distance) that does not follow the identity of indiscernibles ontological principle/1st metric axiom. That can be considered as a critique of this principle.Guswen (talk) 11:23, 21 January 2021 (UTC)

References

Other stuff

"Considerations about how not to define a metric in mathematics are not relevant enough to merit their own section."

We're not considering how to define (or how not to define) a metric in mathematics. We're talking about Identity of indiscernibles as a general principle. — Preceding unsigned comment added by Guswen (talk • contribs) 00:56, 23 January 2021 (UTC)

The earlier section "1st axiom of a metric" mentioned that the first axiom of a metric is called "identity of indiscernibles". It gave a critique of this axiom by giving an example of distance function without it: the Łukaszyk–Karmowski metric. Both the definition of a metric and the Łukaszyk–Karmowski metric belong to mathematics. The principle of identity of indiscernibles, as discussed here, is an ontological principle. Neither Metric (mathematics) nor Łukaszyk–Karmowski metric mention ontology. If you are certain that how to define a metric has direct relevance for this article from an ontological perspective then it would be good to have a reliable source stating this. Phlsph7 (talk) 03:39, 23 January 2021 (UTC)

Łukaszyk–Karmowski distance can be considered as a distance between quantum particles. Quantum Physics is related to the Identity of Indiscernibles (For a reliable source see [3]): "quantum objects are indistinguishable in a much stronger sense in that it is not just that two or more electrons possess the same intrinsic properties but that – on the standard understanding – no measurement whatsoever could in principle determine which one is which. (cf. Identical particles) (...) This has immediate implications for the Principle of the Identity of Indiscernibles which, expressed crudely, insists that two things which are indiscernible, must be, in fact, identical." and "the Principle of Identity of Indiscernibles (ultimately applies) only to monads (things that came into existence), which (are) the fundamental entities of his (Leibniz) ontology", not to perceivable objects or entities.

Again, it's not the point "how to define (or how not to define) a metric" that has a direct relevance for this article from an ontological perspective, but the facts that (1) the applicability of the principle in the quantum domain is controversial (see [4]), (2) Łukaszyk–Karmowski distance does not follow the Identity of indiscernibles axiom, and (3) Łukaszyk–Karmowski distance is a distance between quantum particles. Guswen (talk) 09:58, 24 January 2021 (UTC)

Maybe given all kinds of assumptions concerning quantum physics and Leibniz's philosophy of monads a connection can be drawn between the Principle of the Identity of Indiscernibles and the Łukaszyk–Karmowski distance. But these assumptions are not part of the Identity of Indiscernibles or the Łukaszyk–Karmowski distance. This may be an interesting topic for a research paper, but not for a wikipedia article, see WP:ORIGINAL. Phlsph7 (talk) 04:15, 25 January 2021 (UTC)

Adding a section about whether the Principle of the Identity of Indiscernibles can be applied to quantum mechanics might be a valuable addition to this article. There should be reliable sources available for this. And if the problems for applying it to quantum mechanics have something to do with the Łukaszyk–Karmowski distance then mentioning it may be justified, if there are sources for this. But the section would be primarily about the application to quantum mechanics. I think this course of action would avoid the objections raised so far. Phlsph7 (talk) 04:36, 25 January 2021 (UTC)

I agree. I will add a section about whether the Principle of the Identity of Indiscernibles can be applied to quantum mechanics, based on the sources concerning experimentally confirmed lack of observer-independent facts, but avoiding mentioning Ł-K metric (there are no reliable sources to support this, so far). Guswen (talk) 09:05, 25 January 2021 (UTC)

That sounds like a good idea. The following free sources might be helpful: [5][6][7] Phlsph7 (talk) 11:08, 25 January 2021 (UTC)

archiving the talk page

This page is getting very long, it might be a good idea to archive it. I would use User:ClueBot_III unless there are objections. Phlsph7 (talk) 04:59, 24 January 2021 (UTC)

I have an objection. I believe that the discussion on the relevance of the Ugly duckling theorem and the Identity of indiscernibles principle is ongoing (it began just two days ago). Perhaps other Wikipedians would like to contribute to this discussion as well before archiving the page. Guswen (talk) 11:25, 24 January 2021 (UTC)

Sorry for not explaining it properly, ClueBot automatically archives old discussions. I was going to set it to discussions that have been inactive for more than 1 year. Phlsph7 (talk) 11:37, 24 January 2021 (UTC)

I added the template. It's taken from User:ClueBot_III#Example:_Numbered_archives_(with_archive_box), I changed the age value to 1 year. It might take a few days before something happens. Phlsph7 (talk) 03:07, 26 January 2021 (UTC)

Ugly duckling theorem

"on the contrary: the proof of the ugly-duckling theorem *uses* the identity of indiscernibles"

Indeed, the proof of the ugly-duckling theorem uses the identity of indiscernibles principle to arrive at its contradiction. The proof assumes a set of 2^n objects, each having properties different than the other (no two objects in this set have all their properties in common). Each property of an object is a considered to be a Boolean-valued predicate, and thus the set form a Boolean {0, 1}^n address space, wherein each address (object) is indeed (logically) separate from the others. But the Ugly duckling theorem proves that any two addresses (objects) in this set are equally similar, as they share the same number of compound predicates, all the logical functions that can be formed from the properties of these objects, with connectives of negation, conjunction and disjunction. (*) Therefore any two addresses (objects) in this set are two things under one name. On the contrary to the statement that “to suppose two things indiscernible is to suppose the same thing under two names”.

The Ugly duckling theorem proves that no discernibility (understood as distinguishability, recognizability, identifiability, distinctability, classifiability, etc.) is possible without some sort of bias.Guswen (talk) 12:20, 21 January 2021 (UTC)

I agree to your above presentation of the proof (except that the set should have n objects, not 2^n, to arrive at 2^n properties), up to, but excluding the text starting with "Therefore" (I marked it with "(*)").

I studied Watanabe's theorem while working on Four Cubes. He meant 2^n objects. Each object can be thought of as a vertex of {n}-cube (a regular n-cube but with each vertex joining all the other vertices; a complete graph) which clearly has 2^n vertices in n dimensions. It is clear that no vertex is special w/r/t the others and (in particular, if you draw this graph as a circular graph) it is clear that any two are equally similar to each other.Guswen (talk) 20:09, 22 January 2021 (UTC)

I think the theorem does not contradict the identity of indiscernibles principle. The questions "are there two different objects that share all properties?" and "are there groups of objects such that two objects from one group share more properties than two objects from different groups?" are quite different. I think the theorem answers "no" to the second question, and its proof assumes that the first question is answered "yes".

These questions are indeed different.

The UDT answer to "are there two different objects (two different vertices of {n}-cube) that share all properties (have the same binary address)?" is "no" ((2) 01 is different than (3) 11 if n=2; square with 2 diagonals in Gray code, etc.). In Watanabe terms "property"="starting predicate", "object"="atomic predicate" (a vertex).

The UDT answer to "are there groups of objects such that two objects from one group share more properties than two objects from different groups?" is "yes" (Assume n=2, 1st group are objects (2) 01 and (3) 11, 2nd group are objects (1) 00 and (3) 11. Clearly (2) 01 and (3) 11 share 1 property, while (1) 00 and (3) 11 share 0 properties; Hamming distance d_HM([01],[11])=1, d_HM([00],[11])=2.

But that's not the point. The UDT [[8]] says that "Any two objects, in so far as they are distinguishable are equally similar.". A square with 2 diagonals is isomorphic to a tetrahedron. Any pair of distinct vertices is equally similar to any other pair of distinct vertices. Vertices (1) and (3) share one edge, two triangles and the whole tetrahedron. Similarly vertices (2) and (3), etc. I think that Watanabe has not considered this in terms of dimensions. Guswen (talk) 14:44, 23 January 2021 (UTC)

Using the example from the picture in Ugly duckling theorem, each two ducklings can be distinguished by some property (that is owned by one, but not by the other). Nevertheless (more precisely: for this reason, among others), and contrary to common prejudice, there is no nontrivial "natural" grouping of ducklings; in particular the two white ducklings are not more similar than a white and the black duckling. - Jochen Burghardt (talk) 08:42, 22 January 2021 (UTC)

By "two same/different objects" you mean objects located in distinct regions of spacetime. But these objects are just perceived by you *now* to be in these distinct regions of spacetime. "Concepts of physical space, time, velocity, particles, position, momentum, etc., used to model perceived nature and express these models and observations in classical terms should be used with extreme caution, as they introduce axioms of their own" [9] Guswen (talk) 20:27, 22 January 2021 (UTC)

As I understand it, the Ugly duckling theorem asserts that there are no differences in degree of similarity (given certain assumptions about the nature of properties and the number of objects): for any a, b, c, the degree of similarity between a & b is the same as the degree of similarity between b & c. But that doesn't mean that they are indiscernible, i.e. exactly similar. To assert that an apple is just as similar to a mango as it is to an orange (say: degree=0.5 in both cases) doesn't assert that apples and mangoes are indiscernible (degree=1). Phlsph7 (talk) 09:16, 22 January 2021 (UTC)

But apples and mangos are equally similar as plums and lawnmowers. This theorem simply says that one cannot discern a plum from a lawnmower. This stands in blatant contradiction with experience of an adult person but certainly not with experience of, say, fifth month human fetus, who's never seen a plum or a lawnmower. We simply learn to discern in order to survive and evolve, while biological evolution is possible only in 4D.

This inevitable conclusion of the UDT must have puzzled Watanabe who proposed that one has to ponderate (give weights to) the predicates so that one can say that in order for two objects to be similar to each other they have to share more important (weighty) predicates. He even called this proposition a corollary of his own theorem. It's clearly not a corollary but a statement about the biological evolution.

If discernibility is mathematically (logically) impossible by the UDT, as such, why even discuss the Identity of indiscernibles? Guswen (talk) 15:16, 23 January 2021 (UTC)

Maybe our disagreement turns on how to define indiscernibility. Two things are indiscernible if they have all their properties in common, see here. So two things are discernible if they don't have all their properties in common. The apple is red and the mango is yellow, so they are discernible. In the duckling example here, B is white and C is non-white, so they are discernible. The theorem doesn't say that "one cannot discern a plum from a lawnmower" given this definition of discernibility. Phlsph7 (talk) 16:20, 23 January 2021 (UTC)

Your own source states that "Recent work on the interpretation of quantum mechanics suggests that the applicability of the principle in the quantum domain is controversial". Well, the whole nature is the quantum domain that, as it is commonly assumed, approximates to classical domain in macro scale, etc.

The apple has never been on Saturn, and neither has mango. The apple is a fruit and so is mango. The number of meaningless questions that one may ask not to be able to tell an apple from a mango is infinite. Guswen (talk) 17:19, 23 January 2021 (UTC)

If my interpretation is correct then the ugly duckling theorem is not directly relevant to the principle of identity of indiscernibles. So it can't contradict it, as Jochen Burghardt stated. Phlsph7 (talk) 09:31, 22 January 2021 (UTC)

I agree with Phlsph7. Identity of indiscernibles is a necessary assumption for the theorem: if two indiscernible objects would exist, they had all properties in common, while two discernible objects can have at most all but one properties in common - they must disagree on at least one property. - Jochen Burghardt (talk) 12:11, 22 January 2021 (UTC)

You're still talking about existence. Space as the boundless three-dimensional extent in which objects and events have relative position and direction. Or time as a continued progress of existence. Such intuitive but naive classical preconceptions are hampering the scientific progress.

Not to mention that Ugly Duckling is a mathematical (logical) theorem. But although it has nothing to do with physics, it invalidates such naive thinking about space and time.

Exotic R4 (again, a mathematical property) explains why we live in 4D. Guswen (talk) 20:27, 22 January 2021 (UTC)

There are no observer-independent facts (Experimentally confirmed in [10]). Consider this in the context of observer-independent existence in space.Guswen (talk) 20:39, 22 January 2021 (UTC)

As far as I can tell, you (Guswen) haven't addressed the issues pointed out by Jochen Burghardt or me. Neither we nor the Ugly duckling theorem talks about problems of spacetime, perception, etc. We both gave arguments that the theorem is not directly relevant to the identity of indiscernibles and that it doesn't contradict it. If you feel that we are wrong and that the reference to the Ugly duckling theorem should remain in this article then please point out why our arguments are wrong, ideally without introducing new concepts not mentioned by us or the theorem. Phlsph7 (talk) 03:55, 23 January 2021 (UTC)

Gentlemen, I believe that I have now addressed the issues that you pointed, providing arguments for the relevance of the UDT and the IOI. The UDT invalidates the concept of discernibility (classifiability) of objects (in space), therefore rendering discussion about IOI redundant. At least for this reason it can be regarded as a critique of IOI principle.

Your arguments are based on the concept of existence (cf. Jochen Burghardt "if two indiscernible objects would exist..."), while mine are based on the concept of "perception". I find the latter concept more persuasive: "Cogito, ergo sum", not the other way around. Guswen (talk) 16:10, 23 January 2021 (UTC)

@Guswen: Just a reply to your recent edit.

Your said "apples and mangos are equally similar as plums and lawnmowers", and I agree that this is a consequence of the UDT. Then you continue "one cannot discern a plum from a lawnmower", and this is where I disagree. In my view, the UDT says that a plum and a lawnmower *can* be discerned, and so can an apple and a mango, and the former pair shows the same degree of dissimilarity than the latter.

A plum and a lawnmower *can* be discerned if and only if one assigns weights to the questions that one uses to discern. This is Watanabe's corollary to the UDT. For example the question "Can they both hear well?" is not very informative, while the question "Are they both edible?" is important and facilitates the discerning process. Guswen (talk) 16:49, 23 January 2021 (UTC)

By analogy, to say "any two points of this regular tetrahedron have the same distance" does not imply "any two points of it are equal"; the former is true, the latter is false for almost all tetrahedra. - Jochen Burghardt (talk) 16:06, 23 January 2021 (UTC)

Indeed, the former is true. But the latter is false not for regular tetrahedrons but for an observer who is somehow oriented in space in which he observes this tetrahedron. This spatial orientation enables him to address (assign labels) to particular vertices of this tetrahedron, so from his perspective the vertices are not equal. Guswen (talk) 16:49, 23 January 2021 (UTC)

It seems to me that the objections to the arguments made by Jochen Burghardt and me haven't been successful so far. Burghardt's regular-tetrahedron-example is good analogy for the distinction between equal distance and zero distance. Conflating these two notions for the similarity measure might lead one astray concerning the relevance of the ugly duckling theorem for the identity of indiscernibles.

Jochen Burghardt's example does not mention any "zero distance".

Any two vertices of a regular-tetrahedron are joined by the same length edge. As long as these vertices are unnamed/unaddressed in space by an observer, they are all equal. After they can be named/addressed in space by an observer, they differ. Guswen (talk) 09:19, 24 January 2021 (UTC)

I suggest that we remove the entry concerning the Ugly duckling theorem in the "See also"-section. The reason cited for including the ugly duckling theorem here was the claim that it contradicts the identity of indiscernibles. Since this is a contested issue, we would need reliable sources to state this explicitly for it to be included. But so far no such references have been presented. Phlsph7 (talk) 03:49, 24 January 2021 (UTC)

The reason cited for including the Ugly duckling theorem at least in the "See also"-section was not just that it contradicts the identity of indiscernibles. It contradicts discernibility as such (no discernibility is possible without some sort of bias). Therefore it also invalidates any additional ontological principles based on discernibility or indiscernibility, such as the identity of indiscernibles. Guswen (talk) 09:19, 24 January 2021 (UTC)

Just to explain my analogy: "point" (in the tetrahedron example) corresponds to "object" (in the UDT world), "distance" corresponds to "degree of similarity", "zero distance" between (or, what is the same in Eulidean geometry, equality of) two points corresponds to "indiscernibility" of objects. Orientation in space is irrelevant; if you like, assume distance to be measured by meterstick, not as distance on the observer's retina; or boil down the example to two dimensions (equilateral triangle, observed from its mid point).

If you're argument is based on existence (which is experimentally invalid by [11]) then you're right. But in the UTD world "zero distance" corresponds to distinguishability ("in so far as they are distinguishable"), while "degree of similarity" corresponds to "indiscernibility" ("are equally similar")[[12]]. Any two objects are distinguishable and yet indiscernible. It's the same kind of fact as 2+2=4; think about the consequences of 4 (known from 1969, at least), not about how to invalidate it. Guswen (talk) 20:11, 24 January 2021 (UTC)

I don't think the paper you cited mentions indiscernibility. Your definition of indiscernibility is very different from the one used in the principle of identity of indiscernibles. Phlsph7 (talk) 04:52, 25 January 2021 (UTC)

The example shows that "equal distance / equal degree of similarity" is a concept different from "zero distance / indiscernibility". The UDT proves the former, but not the latter. For this reason (and for the lack of sources), I agree to Phlsph7 that the UDT should be removed from the "See also"-section. - Jochen Burghardt (talk) 10:08, 24 January 2021 (UTC)

Again. In the UDT world "zero distance" = "distinguishability" (a distinguishable object, seen by a given observer); "degree of similarity" (between two objects, seen by a given observer) = "an edge between these objects". If there is an edge these objects are indiscernible even though they are distinguishable. And these edges are the same for any two objects in a set that we consider. One learns to discern. We should be able to find a lot of credible sources, since 1969. I've quoted some. Guswen (talk) 20:43, 24 January 2021 (UTC)

I'll go ahead and remove entry in the "See also"-section. We can review this decision once reliable sources concerning the alleged contradiction have been presented. Phlsph7 (talk) 10:35, 24 January 2021 (UTC)

Please do not go ahead unless someone else than you opt for "going ahead". No consensus has been reached. And this is not a "Critique" section but just "See also" section. Guswen (talk) 19:43, 24 January 2021 (UTC)

Even if you're right to the bone and the UDT, as you claim, does not invalidate the IOI, maybe someone gets any inspiration from their distinguishability/discernibility. Again, this is just the "See also" section: Ugly duckling theorem - no discernibility is possible without some sort of bias. Guswen (talk) 21:08, 24 January 2021 (UTC)

The material in this article should be relevant to the topic and correct. Both of these points have been contested concerning the discussed addition. So far no reliable sources have been presented to dispel these concerns despite repeated requests to do so. Jochen Burghardt and I are in agreement that the material should be removed, see WP:NOTUNANIMITY. If you (Guswen) have reliable sources to present now then we can discuss them. If not then the contested material should be removed until sources are presented. Phlsph7 (talk) 03:58, 25 January 2021 (UTC)

I think as a minimum we would need a reliable source for the relation between the UDT and indiscernibility (in the sense discussed here). Phlsph7 (talk) 06:42, 25 January 2021 (UTC)

Well, the relation between the UDT ("No discernibility is possible without a bias") and indiscernibility, seems self-evident to me. But, it's true that there's no reliable source to quote. So please remove the contested material. Guswen (talk) 09:10, 25 January 2021 (UTC)

References

IndiscIDs  !→ Refl

Hi there Jochen, sorry about my last edit deleting that entry from the Proof Box! When I saw it was still there after my previous edit, I assumed I had overlooked it by mistake, as often happens. I didn't actually mean to delete it twice  : ]

Although I agree with you that indiscernibility of identicals does not imply reflexivity, I'm not quite following the proof summary you put in that section. The axiom of reflexivity has no predicate or relation variables, and references no predicates or relations aside from equality. This means that _all_ predicates and relations satisfy that axiom: equality does so explicitly, while everything else does so vacuously. The things that must actually satisfy it are the domain objects -- i.e., numbers -- that can stand in for x. So long as every number equals itself, relations of any arity can be defined in any manner consistent with the other axioms of logic. Even when the _domain_ is completely empty, reflexivity holds vacuously by virtue of the universal quantifier on x.

Since the other proofs in the box are both premised on reflexivity, it might make more sense if the entry that establishes reflexivity (i.e., IdIndscs → Refl) were placed up front. The titles of the other proofs ("IndscIDs ∧ Refl → Foo") would then make it clear that indiscernibility of identicals doesn't imply Refl... because, if it did, the "∧ Refl" would be superfluous. What do you think? Hobeewahn (talk) 00:26, 8 January 2022 (UTC)

@Hobeewahn: In my opinion, the proof box, or even the whole section, or maybe even the whole article, is about properties of the equality relation, and in particular about the question which properties can be used to define it. We are striving for a definition like "The equality relation is the only relation R that satisfies p(R) and q(R) and ...", where p(.), q(.) etc. are properties of relations, like reflexivity, symmetry, IndscId, IdIndsc, etc. The first 3 proofs in the box show that each relation R satisfying both IndscId and IdIndsc is an equivalence, i.e. also satisfies Refl, Symm, and Trans.

That is, the symbol "=" in the proof box is actually a variable. Maybe, it should better be called "R", and mentioned explicitly as parameter to the properties, like "∀R. IndscId(R) ∧ Refl(R) → Symm(R)". This is what is actually proven in the box. If "=" would be used as a constant referring to "the" equality relation, we wouldn't need to prove e.g. its symmetry, since it is well known.

In words, e.g. the first claim reads "Every relation R that is reflexive and satisfies indiscernibility of identicals is necessarily also symmetric", and the last one reads "Not every relation R that satisfies indiscernibility of identicals is necessarily also reflexive". To prove the last one, it is sufficient to give a counter-example, and this is what I did. — So much for my view on the proofs, intended as a reply to your 2nd paragraph (I inserted newlines in your source) above.

Moving the proof of IdIndsc → Refl to the top is a good idea. Besides your arguments above, it would also group all (non-)consequences of IndscId together. I still find the name "IndscId" hard to distinguish from "IdIndsc" at first glance; but when they are grouped together it is slightly easier. (As a side remark: maybe, renaming them to "IndscID" and "IDIndsc" is a good idea, as the two upper-case letters are better to recognize? You originally used a capital "D" in "IndscIDs", but I changed it to lower case, maybe this was wrong.) - Jochen Burghardt (talk) 09:03, 8 January 2022 (UTC)

@Jochen: Thank you for this detailed explanation! At last I understand the thrust of your proof, and it makes perfect sense. To make it totally explicit what the empty relation is standing in for, I would just preface it with "When used as '='," ...

I agree that "ID" would look better than "Id" in both of those identifiers, so I will change it as per your suggestion. (There is something of a religious war going on at my workplace about whether acronyms should be uppercased within CamelCase identifiers-- but that is a long story ;) ) And thanks for fixing the oversized arrows left by my earlier edit -- they look way better now.

Hobeewahn (talk) 21:48, 9 January 2022 (UTC)

btw: I initially made a flurry of edits to this page, not knowing how to switch between editing and previewing before publishing the final result. Now that i know how to do that, i frequently forget to comment my edits, because the comment text box disappears when you scroll down to preview them. The "Publish" button meanwhile remains visible, and lets you publish sans comment without raising any warning. Sorry for the inconvenience, and thanks for bearing with me while i get the hang of this... Hobeewahn (talk) 22:55, 9 January 2022 (UTC)