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(I don't know how to ask this other than by laying out my worldview. Needless to say, this is here to be dissected and disemboweled. I realize the broad sweeps will irk people. I hope that by presenting this question as a weak hypothesis, fuzzy conceptions about equivalence, information and order will be challenged, in turn answering the question in the title.)

This has bugged me for more than 20 years. To me, equivalence is the fundamental currency of logic, information and all order.

Some branches of logic don't deal in absolutes, but they have equivalence as their limit or minima (as in AI local minima).

A single bit is an equivalence determination. Bit-data establishes property equivalence, when it already exists.

I've read in one statistics or philosophy book or-other that equivalence and equivalence relations underpin all of set theory, which I understand underpins most if not all mathematics.

In my view, equality is the most fundamental "unit" in nature. The most basic causally relevant state.

If equality is a causally relevant state, "order" is a higher-level (based on usage) state of equivalence. Whether order initiated by natural selection, in spontaneous chemical processes or even driven by gravity, it is the state of property equivalence. The Moon is an orderly arrangement of matter held together by gravity. It is matter with equivalent properties (position and momentum).

Subatomic, atomic and molecular structures represent property equivalence that can be disrupted by heat.

So property equivalence describes the building blocks of matter as with particles, atoms and molecules. Those property equivalences lead to higher levels of order and ultimately to information.

I believe information is too broadly defined. In this treatment, it is closer to the end of the chain.

As an example of overuse of the word information: Yes quantum information exists, but only if we perform calculations with it. It is an overstatement to say that all spontaneous quantum processes are informational.

This last point is to weaken information's grip it has on virtually everthing. Being loosely and nearly universally applied, it disrupts our understanding of more fundamental processes. It is much more revealing to consider how organismal information flow has equivalence determinations at its foundation.

For example, actual neural networks are probabilistic. It is easier to visualize a hierarchy of equivalence (math-like) steps rather than "information" flowing through brain circuits.

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  • Well, equality is an important concept in mathematics and philosophy, which means that you can see it everywhere. But it's not the only concept like that. Greater than/less than relations, real numbers, functions, sets, commutative operators, etc. - pick any of those concepts and you can see it everywhere too. You're likely tempted to say, "but I can see how each of those depends on equivalence!" - but they depend on each other, too, and equivalence depends on them. Equivalence relations are often derived from level sets of a function. Equiv is only one of many important interdependent ideas
    – causative
    Commented Aug 22, 2023 at 4:18
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    Equality stated as the law of identity in ancient logic preserved by modern FOL has always been the fundamental unit/principle in application to nature and various sciences apparently along with LNC. Also equality is not strictly equal to equivalence formally, yet the art of how to pick out equivalence or even a gigantic tower of infinite equivalences required for your intended purposes functionally equals your own judgement call or inspired from some non-empirical divine oracle ultimately... Commented Aug 22, 2023 at 5:21
  • "Some branches of logic don't deal in absolutes, but they have equivalence as their limit or minima (as in AI local minima)." What is an "AI local minima"? Having said that, mathematical theories deal with structures i.e. class of objects that share some specific mathematical properties. From this point of view, what matters is "equivalence" i.e. sameness of behavior wrt relevant properties, irrelevant of the facts that "objects" are really identical or "looking the same" from the spectacles of the theory. Commented Aug 22, 2023 at 8:14
  • Basically, we have different meanings of "equality": one is the [mathematical one](Equality (mathematics)) that is axiomatized by predicate logic. This is strictly related to the mathematical concept of Equivalence relation. Commented Aug 22, 2023 at 8:30
  • The other meaning is that of Identity that usually is considered from a wider philosophical approach. Maybe relevant: Identity in Physics: A Historical, Philosophical, and Formal Analysis. Commented Aug 22, 2023 at 8:31

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I always worry when I see a word written in quotation marks, as with your "unit", that its author isn't quite sure what they are trying to say. Equality is fundamental to mathematics, physics and the mathematical sciences. When we make models of reality in physics, we do so with equations, which are no more than statements of equalities. The notion of equality, therefore, is an essential component of our understanding of virtually everything. However, I think you are perhaps going too far, and in a wrong direction, when you try to characterise equality as the fundamental currency or unit. Too far, in the sense that there are other concepts which one might consider as being fundamental, such as quantity, time, mass and so on. In the wrong direction, in the sense that 'currency' and 'unit' are both inadequate or misleading metaphors for the role played by equality.

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  • Thanks for the interesting reply. Yes I agree about the quotation marks. What class to put something so fundamental into? "Unit" implies a part of a greater whole. I should `have stuck with "state." The existence of equivalent states in the astronomic or quantum worlds doesn't necessitate measurement or depend on us.
    – Nick Mirro
    Commented Aug 23, 2023 at 23:58
  • "other fundamental concepts...apply to quantity, time, mass and so on." Of these three examples, quantity seems the least fundamental. Quantity is a property on which one necessarily draws equivalence relations. That relationship seems to go in that one direction. Quantity requires equivalence, by definition.
    – Nick Mirro
    Commented Aug 24, 2023 at 2:38
  • Mass and time are both quantifiable and so also subject to equivalence relations. So... I agree that 'currency' and 'unit' were bad choices. Trying to envision a world in which humans are not drawing equivalences, does matter (mass) rely on equivalence for its existence? I'd say yes, at least because of symmetry. Another... entanglement is shared spacial proximity.
    – Nick Mirro
    Commented Aug 24, 2023 at 2:38
  • I disagree, in principle a Empty Universe V, would still admit the principle V = V, without having any sense of quantity, time, mass, etc. So, the most "fundamental" relationship is the relationship to onself, x = x. things like quantity, time, mass- all require other objects for measurement purposes, and so are less fundamental. Commented Aug 31, 2023 at 18:17
  • I love "an Empty Universe V." The deepest possible abstraction. I don't disagree, but also don't see how self-identity might be causal. I think causation starts with relations, and the equivalence relations are the causal ones. What I love about this conversation is, we are converging on mandatory usage definitions.
    – Nick Mirro
    Commented Sep 5, 2023 at 4:51
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This is an excellent and profound question. Let's go first with the trivial issues.

Because equality is not "a fundamental unit / principle in nature". Equality is subjective, which means two OBJECTs can be equal for SUBJECT-1, but they can be different for SUBJECT-2. Who is correct? None, or both, because equality is not on the object, but on the subject. It is not in nature, but in reason.

Look your own statement: The Moon is an orderly arrangement of matter held together by gravity. It is matter with equivalent properties (position and momentum).

To start, it must be clear that we speak about a system that has parts (a system is a set of interrelated parts), parts, that, you argue, have "equivalent properties (position and momentum)", whether if the parts are chunks of matter, sand particles or atoms.

In any case, it would appear that all parts are equal. But strictly, from a physical standpoint, there are no two physical objects that are equal. Two apples will always have differences. Two grains of sand might like identical, but on a closer look, they are different. In final terms, strictly, no thing is equal to another.

In your example, one atom on the moon could have

momentum = 7 x 10e-16 p kg m²/s

and the next one could have

momentum = 7.000000001 x 10e-16 p kg m²/s

due to imperceptible but persistent waves of vibrations on the moon caused by falling meteorites.

Of course, here you will say "that's stupid, both numbers just round to the same". Yes, but to what extent? When you choose a rounding factor, you are being subjective (you can say 16 digits, and I will say 160, so objects are equal for SUBJECT-1 and are different for SUBJECT-2). In order to be objective, exact values, WITH ALL DIGITS should be compared.

You can also say that "we can't measure that". But that doesn't mean that the objects are equal, not even thinking about being "fundamentally equal". It just means that we are not able to measure their differences, which necessarily exist.

This simple example is more elegant:

1 apple + 1 apple = 2 apple

According to your perspective, this would mean that the two apples are equal.

But did you know that this only works in your head? Empirically, this is impossible, since there are no two apples that are identical, either in weight, radius, size, etc. We take for granted that apples addition is trivial, but it's not. Reason makes a lot of processes to allow you to translate a physical phenomena, which we even don't know if it really exists, into a pair of abstract objects, identical only in the mind.

That's why equality is not "a fundamental unit / principle in nature", but a subjective judgement.

Anyway, you have an excellent point about equality, but you seem having missed a key aspect about it. This is the deep and interesting side of the question.

Even if equality is not a fundamental feature of nature, IT IS a fundamental feature of reason and Logic (ergo, Mathematics!). If you know Kant, I would state in the context of the Transcendental Doctrine of Elements that the rational capability of detecting equality is possibly the most elementary transcendental potential reason features. Kant missed that. Even Aristotle formulated his principles of reason taking equality for granted. I am currently working on a book about metaphysics, where I propose an equivalent idea, but from the opposite standpoint: difference.

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  • Let me ask you a question that came up in my reasoning and that you honed right in on. It is simply, shouldn't we specify our criteria for equivalence? Property equivalence suggests different things share one property. For the property, it is a self-referential relation. Equivalence in math is more broad in that it allows sets, such as ranges of attributes that fall within a specification. If you define a set, its members are equal with regard to the definition. All matter comprising a moon are equal in that they have like positions and momentums. This is still an equivalence relation.
    – Nick Mirro
    Commented Aug 24, 2023 at 2:49
  • Clearly there are different types of equivalence. I believe this only underscores my basic premise. One can reference collections of objects and apply a wide array of similarities. Two carbon atoms are extremely similar but not equal. Sets are membership based . A member is either in or out. So equivalence describes membership, not all individual member properties.
    – Nick Mirro
    Commented Aug 24, 2023 at 2:56
  • We need to distinguish entities and properties. Yes properties are abstractions. It is not the abstractions that "cause." I agree abstractions are descriptions of physical things. Regardless of our subjectivity, it is the common (equivalent) properties that propel matter the moon is comprised of around our planet as a single entity.
    – Nick Mirro
    Commented Aug 24, 2023 at 3:08
  • @NickMirro The question is if equality is a fundamental principle in nature. My answer is no, equality is a subjective judgement, so, obviously, there are multiple types of equivalence, depending on the selected judgement. "shouldn't we specify our criteria for equivalence?" We do, but that doesn't make equality fundamental in nature. We're just agreeing about the judgement, not about what is fundamental in nature.
    – RodolfoAP
    Commented Aug 24, 2023 at 3:17
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    Thank you for your thoughts. They leave me with quite an interesting perspective to consider. Commented Sep 1, 2023 at 12:31

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