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1st case Consider two objects made from the same factory without any difference (an ideal scenario). Can we say that the two objects are the same? I would say no because one may be produced earlier and also after being produced they don't have the same position. Can we say that they are identical?

2nd case Consider 5 apples on the table and 5 apples on a chair. We can say that 5 apples = 5 apples but the apples are not the same. I mean the apples on the table could be green and these on the chair red etc. So can we say that the term "same" applies only to mathematical objects?

What is the difference between same, identical and equal?

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  • The issues you cite are resolved by distinguishing numerical and qualitative identity, but one can equally well use "sameness" instead of "identity". "Equality" usually refers to identification that explicitly disregards all but selected aspects. – Conifold May 21 '20 at 1:03
  • I think the question here is can the terms identical and equal be misunderstood or confused by someone else namely a listener. People usually speak in some acceptable context to be understood. Culture as well as location make a difference on contexts. Here the CONCEPTS of identity and EQUIVALENCY seem to be expressed. Apple's will always be apples but the details about some apples may make a distinction as on a chair or on a table. Equal does not mean equivalent always. Equal can mean IDENTICAL as in a mirror image of the same attributes with no exceptions. A=A expresses the same attributes. – Logikal May 21 '20 at 1:42
  • Equivalent expresses that the same meaning of a language can be derived in more than one way. 3+3=6 but so does 7-1. These express the same answer or idea but use different symbols or words. Sentences can Express the same thing but be distinct sentences: you are fired and you are terminated express distinct sentences & express the same idea in a context. Identity expresses there is no distinction whatsoever between 2 objects. Any slight distinction means the 2 object are not identical. Essentially object a & b are the same object (perhaps with different names) such as Mark Twain & Sam Clemens. – Logikal May 21 '20 at 1:49
  • We can say the MODEL or TEMPLATE used to create a laptop is the same but each laptop from the same manufacturer would have a distinct serial number for instance. They are not identical but they are equivalent if they are the same model. The word SAME could be either identical or equivalent depending on the context. If basketball player 1 & 2 are on the SAME team does that mean IDENTICAL TEAM or a team equivalent? Most English speaking people would say identical. You & I have the same laptop does not Express we use the exact identical laptop but they are equivalent in make model & attributes – Logikal May 21 '20 at 2:00
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    The mathematician Barry Mazur wrote an insightful essay called, "When is one thing equal to some other thing?" people.math.harvard.edu/~mazur/preprints/when_is_one.pdf – user4894 May 21 '20 at 22:14
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A real world factory doesn’t produce identical objects. There are always differences. They are similar. Even a million needles from the same factory won’t contain two identical needles.

When we go down to the nuclear level, atoms are identical. Molecules are identical.

Sameness is strange: Me yesterday, me today, and me tomorrow are the same, but we are different. Not much, but different.

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Same

'... two objects made from the same factory' cannot be one and the same object. Whatever their sameness in the sense that each object has all and only qualities exactly similar to the other's, they are numerically distinct.

Identical

'Identical' is an ambiguous term. It can connnote exact similarity as when each object has all and only qualities exactly similar to another's - or it can mean that X and Y as identical are one and the same object as in Frege's example of the Morning Star and the Evening Star's being identical because they are one and the same object differently referred to, i.e. the planet Venus.

Sameness

This is not confined in its application to mathematical objects. 'The same ...' is incomplete. The same what ? Two automobiles can be of the same model, where there is nothing mathematical about a model (in this sense). Three people can belong to the same family - another non-mathematical object. Mathematics enters necessarily at one point only: countability is implicit in the concept of sameness. Am I the same person that I was yesterday? Yes, I am one and the same person at two different times. If I am in New York and move to LA, have I been in two locations in one and the same country? Yes, of course, the USA.

Equality

'Equality' can mean 'equality in numerical value' as in your apples example but when I say that two models of automobile are of equal utility for your purposes - one's as good, as functional, as the other it's hard to see what strict metric I could use to justify my claim. I just mean, one's as reliable as the other, one's as easy to get serviced or repaired as the other, they equally enhance your image, &c.

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When we use the word 'same' precisely (in the case of objects), there should be no change in the thing.

E.g:

1. If 'A' and 'B' are brothers, their mothers are usually one and the same person.

2. In the case of objects: The first object made in the factory today and the one I put in this box are the same.

If the word 'identical' means 'similar in every detail or exactly alike', there is no such object. But we can use it if they are almost similar or almost alike. E.g: identical twins. So it is mostly used in Zoology, Logic. Mathematics etc. Even if the pins the same machine made not have exactly the same size, we can say they are identical.

In the case of objects, the word 'equal' is used for comparing mathematically.

E.g:

1. 5 - 2 and 1 + 2 are equal.

2. These tables have equal length. (... or the same length; but not identical length).

3. The number of apples on the table and the chair are equal (... or the numbers are the same).

Since there is no change in the number, we usually don't say they are identical. That means, we can't use the word 'identical' if the object mentioned is one and the same thing.

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On a fundamental level there are two notions here, which I'll call sameness and equivalence. The word equivalence can refer to other, coarser, relations in other contexts, but I think using it here will clear things up.

Sameness is the quality of being the same thing. If I have one apple and I have the same apple, then I only have one apple total. Two things that are the same ... are really one thing. Thirty things that are the same are also one thing.

The word same is frequently used with this meaning in English, as in your first sentence.

Consider two objects made from the same factory without any difference (an ideal scenario).

In this sentence there are not two factories that are identical, there is only one.

The next notion, which I'm calling equivalence, is the condition of two things being identical but not necessarily being the same thing.

In your first example, the two objects coming out of the factory are equivalent. They're different objects, but they're totally indistinguishable and share all properties.

There are two fundamental problems here:

  • Consistently distinguishing sameness and equivalence in language is hard.
  • When we're talking about objects, sometimes we're talking about objects "situated in the universe" and sometimes we're talking about objects "divorced from the universe and considered on their own".

Trying to talk about sameness and equivalence in natural language is hard. English has a few words that can refer to either notion depending on the context: same, identical, equal and a few that can't refer to sameness like equivalent. But there are a lot of implicit rules about which expressions can co-refer and which can't. Trying to distinguish the two ideas consistently is also hard.

The second problem is how we talk about objects. If we consider "objects situated in the universe", then the distinction between sameness and equivalence is crisp. I can have two completely identical apples that aren't the same ... and the fact that they are in the universe differently is what's different between them.

If I take my equivalent apples, though, and sever their connection to the universe and consider them completely on their own, are they now the same? Is there now one apple or two? I think the answer to that question is mostly irrelevant, since we can't ask coherent questions about objects severed from the universe that refer to things outside the objects themselves.

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  • What about fermions – CriglCragl Feb 2 at 0:44
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Terms like 'same', 'equal', 'identical', etc. are reference comparisons. Assume we have two (language) terms Sigma and Gamma that (ostensibly) refer to real-world objects, and we want to know something about how the referred-to objects relate to each other. We then have four distinct cases:

  • Sigma and Gamma are different terms for distinct objects or classes of objects: e.g. (and obviously), 'cat' refers to a cat and 'dog' refers to a dog.
  • Sigma and Gamma are different terms for a singular, unique object: e.g., 'Morning Star' and 'Evening Star' are two names for the planet Venus
  • Sigma and Gamma are different terms for a class of objects that have similar functions or characteristics: e.g., 'car' and 'automobile' can be used interchangeably to refer to a range of different objects
  • Sigma is a singular term that refers differentially to distinct objects: e.g., 'president', which might refer to any one of the various presidents of the US, Brazil, Namibia, a corporation, a book club, etc.

With that in mind, we can say this:

Identity: As a rule in both logic and common language 'identity' is reserved for case #2. Sigma is identical to Gamma if they point at the same singular, unique object.

Equality: Equality deals with classes of objects. Reference terms are considered to be equal if:

  1. They point to distinct objects that fall within the same class of objects, as defined for the discussion
  2. They point to distinct classes of objects that fall within the same super-class, as defined for the discussion

It's difficult to apply the concept of 'identity' to a class of objects, since classes of objects invariably have fuzzy boundaries. Equality fills that gap, by allowing us to say that two otherwise distinct things may be treated as (pseudo)identical — interchangeable — within a given context or for a particular purpose.

Sameness: This is a loose, colloquial term meant to imply equivalence without erasing distinctiveness. If we say that Ann and Bob got the same test score we don't mean their tests are identical; we mean the tests are equal on one dimension. If we sat that SUVs and luxury cars are not in the same class, we mean that the objects they point to are the same on some dimensions (the ones that define them as 'cars'), but not the same on other dimension (the ones that define them as SUVs or luxury cars).

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Sameness is informal and merely colloquial. Identity is continuous. Equality is discrete.

So, setting "sameness" aside, Identity may be thought of as self-sameness, unity, singularity. As a singularity it does not really fit into mathematical systems of discrete units unified by, say, an equal sign. It is indivisible. In Leibniz's formulation, what reveals absolutely no "discernible" differences is identical. A person is self-identical insofar as they are essentially indivisible and consist of a unity of consciousness through experiences, or, in Kant's phrase a "transcendental unity of apperception."

Equality contains a difference, as indicated in the space separating the two parallel and otherwise "identical" lines in the (=) symbol standardized by the 16th century mathematician Robert Recorde. He adopted it saying it represented the most identical that two "discernible" things could be. Even in 1 = 1 we can "discern," for example, a left-side 1 from a right-side 1. The "equation" is thus paradoxical, both abstracting from and retaining identity, just as a geometric "point" is both "there" and "not there," a "place" of nonexistent precision.

Combining the "identity" of "identity and difference" as Hegel calls it, is an ancient philosophical conundrum dating back to the "one and many" of Parmenides. Many paradoxes ensue, including the paradoxes of liberal governance in which self-identical, indivisible "persons" are "equal" under the law.

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Math has it right in this case. There are no true definitions of these terms, only conventions. The meanings of these things differ by context. Things are the same, identical, or equal according to the 'equivalence relation' appropriate to the context in which one is speaking.

Famously Leibniz defines identity as having exactly the same values for all the same properties, and having no other properties. But that is in the context of propositional logic, where things have properties. It is the equivalence relation for that domain.

It does not match Kant's definition for identity, which allows for continuity across time, because age is a property, and two instances of you from different times have different ages. But the context is different. Kant is talking about beings, and so there is a different equivalence relation.

Your sets of parts are no different. How you declare their identities depends on the intended use. If an older part has more oxidation, and is of lower quality in some important way, the two are not identical. But if the point is just interchangeability in normal situations, they probably are identical in the relevant sense.

The infamous grade-school question is whether 1 = 0.999999... This is very important once you realize that 1/3 is 0.333.... and three times that is 0.9999... Clearly, formally, as strings of digits, these are not equal, but the equivalence relation between real numbers saves you. Two real numbers are equal if they differ by zero. The difference between these is less than 1/10^n for any n, so it can only be zero. So our two favorite definitions of the reals, as infinite sequences of digits, and as the points on the number line are not the same unless you create an explicit equivalence relation.

This seems like a special case, like the difference between integers, for instance does not require such a hack. But we forget that there are no integers in our universe. There are numbers of cars, and of fingers, and of... And we make an equivalence relation by finding pairings between the set we want to count, and some other set we have already counted. 'Bijection' is the equivalence relation between discrete numbers (of all sizes, even infinities.)

There is no proper, best, or only definition of any of these terms. They are all different words for being alike in specific ways, and those ways are relative to the intended purpose. So trying to discern them in any abstract sense is a misunderstanding.

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