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I keep having a thought, which is probably silly. It would be good if you could put it to rest, or point me to an area of discourse where this is discussed.

Take two things in the abstract, which differ in some ways that are known. If we remove those things which make them different, do we have the same thing?

If we do the same in the physical world with two objects that are different and remove those things which make them different, do they become the same thing?

As an example, if we take two different photons and remove the things which make them different, do they become the same photon?

Could this explain what is happening with Quantum Mechanical weirdness?

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    Maybe is better to avoid using "thing" to denote the object and its attributes (properties ?). Having said that, in order e.g. to "conflate" two different photons, you have to remove also their spatio-temporal locations ? – Mauro ALLEGRANZA Oct 24 '16 at 12:24
  • That is what I was thinking, the object is the photon, it's properties which differentiate it from another photon include its spatio-temporal location. (edit for clarity) – Marcus Oct 24 '16 at 13:02
  • But if removing spatio-temporal coordinates from two photon, the two "conflate" into one single entity, then two different spatio-temporal "instances" of the same photon are not belonging to the "same" object ... – Mauro ALLEGRANZA Oct 24 '16 at 13:04
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    Are you familiar with Leibniz's Law plato.stanford.edu/entries/identity-indiscernible? – Dave Oct 24 '16 at 13:17
  • I was imagining that the conflation would happen through overlap of the spatio-temporal uncertainty. So the two photons have a location which is uncertain and overlapping, so that the photons are partially the same object. I know this is incredibly wishy-washy, I'm embarrassed for myself. But at the same time the thought won't go away. – Marcus Oct 24 '16 at 13:20
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David Deutsch has tried to explain quantum mechanics along the lines you have considered in his book "The Beginning of Infinity" chapter 11.

A physical system can exist in multiple instances that are identical in every measurable respect - Deutsch describes them as fungible. A quantum system changes over time by rules that dictate how many of the fungible instances of a system change their values. The rule doesn't dictate which instances out of the fungible set change their values since there is no way to distinguish them before the change. Deutsch uses this idea to explain features of quantum interference and other features of quantum mechanics.

For a blog post explaining how some of these ideas correspond to the formalism of quantum mechanics, see

https://conjecturesandrefutations.com/2015/04/04/fungibility-in-quantum-mechanics/

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Equality is a highly polysemic word and might have to be defined on a case by case basis, according to your needs. A key question is whether you consider that two objects that do not share the same location and time can be equal. In other words you need to define an answer to the question: "can this distinct apple replace this distinct apple?" Does it matter to me, at the grocery store, whether I take this one or this one (because I just want any apple) or do these two apples have distinct properties that are relevant to me?

It's a matter of the definition that you chose to apply to your particular model, which depends on your purpose. Assuming two things equal when you can, allows you to greatly simplify your model (principle of economy). This allows you in particular to count items, de-emphasizing the fact that they are distinct.

For example, in Newtonian mechanics, two objects that are not in the same coordinates at time t, may be (roughly) equal in mass, energy, etc. are clearly not the same. You assume that F and mass x acceleration are "equal" because that is very useful.

It is relatively clear to me that relativity requires to think about "sameness" with more attention, since time is no longer absolute; what may happen in Quantum physics is beyond my current knowledge (in the latter case, I suspect that physicists try not to wander too far off the formulas, for fear of getting lost in philosophical mazes).

Sometimes the location (or some other secondary property) matters, sometimes it doesn't. In particular, we have the notion (useful in finance) of fungibility. Fungibility, as the word suggests (from Latin fungi "to serve [in place of]"), depends on the function of the object you are looking at. A one-dollar bill is absolutely "equal" to another one-dollar bill, provided it belongs to an issue after such and such date, and it is not damaged, etc. Or you don't want them to be "equal" because you are interested in their serial number. Indeed, the whole idea of finance is to disregard location as much as possible: whether your one-dollar bill is at bank A or bank B or in your pocket, it is still "equal" to another one-dollar bill. This is also how you get "clearing" between receivables and debts.

In any case, sameness (which intuitively, for a physical object requires sharing same location in space and time) is only one of the possible definitions of equality.

If you want to get a sense of what the various meanings could be (and assuming you are knowledgeable about programming or willing to become so) you could take a basic Lisp course. It is highly symbolic language where the creators had to bang their head about on the concept of equality, resulting in an impressive list of predicate functions with different nuances. Or perhaps you have already done so.

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I believe that the concept of equality is universal and does not depend on any prior assumptions. Two things that have identical properties, even if they occupy the same location, can be different, but it may be impossible to know this.

Often equality is conflated with equivalence, which is a property that is reflexive, transitive, and symmetric. But we have options in defining an equivalence relation. It is artificial and arbitrary. Equality is an equivalence relation, but so is the relation where all objects are equivalent. Equivalence relations behave in many ways like equality, and they are tailored to be useful in a certain situation. For example, in modular arithmetic we say that two numbers are equivalent (or congruent) modulo n if they differ by a multiple of n. There is special notation to indicate that 5 is congruent to 8 modulo 3, and it is not correct to say that 5=8.

Equivalence relations can always be defined in terms of equality. Since 5 is congruent to 8 modulo 3, we may write [5]=[8], where [k] denotes the equivalence class of k modulo 3. Two objects are equivalent if and only if their equivalence classes are equal. For convenience in more advanced mathematical exposition this distinction is dropped notationally and we may indeed write 5=8, as long as the intended meaning is understood.

I can't define equality in a way that isn't circular, but I do not think it is something that we can pick and choose. Any observer will detect things that are equal as being the same, and some things that are not equal may be perceived as equal by an observer even though they are not.

  • How do you account for the fact that the natural number 1 is not the same set as the integer 1, which is not the same set as the rational number 1, which is not the same set as the real number 1, which is not the same set as the complex number 1? What theory of equality lets you say that 1 is always the same as 1 even though it's represented by many different sets? You might have a look at this. abel.math.harvard.edu/~mazur/preprints/when_is_one.pdf – user4894 Oct 29 '16 at 19:30
  • @iser4894 The different incarnations of 1 are not equal, it's an abuse to say they are. However the abuse is so useful that it's not regarded as so. – Matt Samuel Oct 29 '16 at 19:35
  • @user4894 In terms of a "theory of equality" declaring them to be equal, there is for example a unique injective homomorphism of rings from the integers to the real numbers, and we say that the real numbers in the image of the homomorphism are the same as the corresponding integers in the domain. They are not the same but the homomorphism identifies them. This is very common in mathematics. While my perspective on this may be a bit extreme, certainly most mathematicians will agree that the integers do not form a subset of the rationals, nor the reals, etc. but they can be naturally embedded. – Matt Samuel Oct 29 '16 at 20:17
  • That is precisely my point. They are the same even though they are different sets. If you claim otherwise you put the map (set theory) before the territory (the actual nature of 1). The set theoretical 1 (natural, rational, etc.) is a model. It's not the actual number. That's why 1 is always 1 even though it has many different set theoretic representations. You cannot substitute pedantry for truth. The business with the injections is a formalism but it doesn't actually tell us what 1 is. – user4894 Oct 29 '16 at 22:11
  • most mathematicians will agree that the integers do not form a subset of the rationals --- No you are wrong about this. That's the point. Everyone agrees that 1 is 1 is 1; and that when pressed, we agree that in the set-theoretic formalism they are different sets with natural injections smoothing out the philosophical problems of set theory. – user4894 Oct 29 '16 at 22:13
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See when you say photons, i believe the current assumption is that they exist in multiple states simultaneously until they are measured. So in that sense potentially they are the same but have equal potential to be different. Also photons can be entangled and not really sure the physical effects on composition of quantum entanglement. To answer your question in short no thing of equal composition are not identical because different forces affect it at a given point in time that is generally not identical but does have the potential to be. So possibly no one really has a way of know with precision. Also depends on your perception of time just a thought.

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In W & R's Principia Mathematica, x = y is defined as (φ).φ!(x) ⇒ φ!(y), where (φ) means all φ, i.e, y satisfies all the predicates which x satisfies, or y possesses all the properties which x possesses. When Wittgenstein and Ramsay criticised this definition, Russell acknowledged the weight of their criticism but later stood by his original definition. The following discussion speculates the insights behind W&R's definition and their defence against Wittgenstein and Ramsey's attack. This is totally a speculation, I'm not sure whether W&R really thought this way.

  1. The notion of thing is a mental construct. Where one thing ends and another thing begins depends entirely on the mind. E.g, when we say a soccer team has 11 members, we implicitly assume everyone knows the qualifying property of a member so that no one counts limbs as members.

  2. All that we can know about a thing is that it is the causal origin of a bundle of qualities, each of which can be known by acquaintance. There is reason to believe that a thing also has qualities which can never been known by us.

    2.1. When we speak of things, we are actually speaking of bundles of qualities.

    2.2. When we compare things, we are comparing two bundles that already have many common qualities - i.e., we are comparing two members of the same class which has a defining quality. For example, when we compare two soccer players, we are comparing two bundles each of which posses the common quality which we call "is a human."

  3. When we ask "if two things are identical," what we really mean is whether two qualities we are currently acquainted with belong to the same bundle. For example, you saw a stranger who walked on Lake Tahoe, and months later you saw a person walked on Lake Michegan, and you wonder if the strangers you saw in separate locations were the self-same person - in other words, whether the quality of "walked on lake Tahoe" and the quality of "walked on Lake Michigan" belong to one and the same bundle of qualities. Now you know these two persons apparent have one unusual common property, i.e. "can walk on water"; the next thing you do is to recall more details about these two encounters and see if there are more qualities in common - the more qualities they have in common, the more likely these two persons apparent are self-same.

    3.1 When we say x and y are the same thing, we mean x and y denote the same thing. When we speak of things, we either use proper names or descriptions.

    3.1.1 To say two things that they are identical is false but not nonsense. 
      Two things implies they are different; identical implies there is only 
      one thing under consideration; "two thing are identical" is the 
      same as asserting "x≠y and x=y" which is false but not nonsense.
      (see ✳3.24 Law of contradiction)
    
    3.1.2 To say "one thing is identical with itself" is true and   
       significant and requires proof, thus is not to say nothing.
       Since x=x has the greatest degree of self-evidence, the fact  
       that from PM's primitive propositions one can deduce x=x gives  
       reason for believing PM's premises and the definition of identity.
       See (PM ✳2.08 and ✳13.15).
    
  4. Let A and B denote classes; if each member from the class which A denotes belongs to the class which B denotes and vice versa, we say A and B denote the same class.

  5. Let A and B denote bundles of qualities; if each quality from the bundle which A denotes belongs to the bundle which B denotes and vice versa, we say A and B denote the self-same bundle.

    5.1 Let x denote a thing and φ denote a predicate, we say x possesses the quality of φ when φ(x).

    5.2 φ's are the qualities which we call universals; some universals can be acquainted by us if they are sensible.

    5.2.1 Although x's are believed to be the causal origins of φ's, φ's are the beginning of our knowledge of the world (x's), and all we can say about an x are its φ's.

    5.3 When we say a bundle of qualities, we mean qualities joined
    together by logical AND, i.e. φ1.φ2.φ3...φn...

    5.3.1 A predicate φ defines a class α each of whose member x satisfies φ(x): α = {x | φ(x) }.

    5.3.2 All the classes defined by predicates which x possesses constitute a higher class k: {α | x ∈ α and α = {x|φ!(x)} } where φ!(x) is a predicate of x.

    5.3.2.1 it is necessary to ensure that α is defined by a predicate of x because all the functions that can take x as an argument form an illegitimate totality (See PM Introduction. Chapter II, V)

    5.3.2.2 x is the only member of the class p'k, where p'k = {x|(α).α∈k⇒x∈α}, i.e. p'k is the product of its members. (see PM ✳40.01)

     5.3.2.2.1 It is possible that, in the sensible world,   
                 y possesses all the sensible qualities of x and is 
                 different from x because the qualities which y has 
                 and x doesn't is not sensible. In this case the
                 differences between x and y is indiscernible.
    
     5.3.2.2.2 But it is logically impossible that x and y are 
                 different and still have all the qualities in common
                 because if x≠y then y does not have the property of   
                 "≠y" which x has. Here Russell pointed out that 
                 Wittgenstein implicitly assumed that ≠ is more
                 fundamental: in PM, ≠ is defined on the basis of =; 
                 thus as soon as ='s definition is rejected, ≠ lost 
                 its meaning; Wittgenstein spoke of two while 
                 rejecting the definition of =, thus W implicitly 
                 assumed diversity is indefinable (or more fundamental).
    

    5.3.2.3 x is an instance of each of the qualities x possesses; k is the abstraction of all the qualities x possesses.

  6. Let x and y denote things; if the thing denoted by y possesses every quality possessed by the thing denoted by x, we say x and y denote the self-same thing. I.e., IF (φ).φ!(x) ⇒ φ!(y) THEN x = y, where (φ) means for all φ.

    6.1 It seems necessary to use ⇔ (iff) in the place of ⇒, but this is not the case because if (φ).φ(x) ⇒ φ(y), then ~φ(x) ⇒ ~ φ(y), where (φ) means for all φ.

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