But the difference is that now I have made the natural relativity of the question of identity explicit. — apokrisis

So the idea is that the context of x is not-x, and defining the identity of x as not-not-x recognizes this, rather than making it a contextless tautology? "x is x" does not apply to the contextual, but "x is not-not-x" does apply as an apophatic alternative?

While I am at it, do you agree or disagree with my other "first cut" definitions of "contextual" that parallel what Peirce wrote about "vague" and "general"? — aletheist

But contextuality leaves it open whether the further possibility is 1ns or 2ns. It could a future condtional (the coming battle with the Persian fleet) or it could be some event already fixed by a determination (what will I discover when I finally check my ticket for the lottery drawn last week?). — apokrisis

Did something get accidentally deleted from your post? I do not see how your response here addresses my question.

Of course then along came relativity to demonstrate all this classical definiteness was relativistically contextual and quantumly indeterminate. That is why Peirce gets credit for foreseeing the physical revolutions about to come. — apokrisis

Indeed, "frame of reference" has been in my mind throughout our discussion of contextuality.

So the idea is that the context of x is not-x, and defining the identity of x as not-not-x recognizes this, rather than making it a contextless tautology? "x is x" does not apply to the contextual, but "x is not-not-x" does apply as an apophatic alternative? — aletheist

Your understanding seems not incorrect.

Mind you, if you claim that everything actually does matter to you, excuse me if I think that is patent bullshit. Does it make any difference to you if I wear a red or blue shirt tomorrow? Do you need that to be another determinate fact ... or do you believe in free will in contradiction to your what you just posted? — apokrisis

If we are talking about identity, and the overriding purpose is, that we want to know the truth about the matter, then of course every difference matters. That's no bullshit, it's reality. If we allow that some differences do not matter, then we allow that two distinct things can have the same identity. Since giving two distinct things the same identity is a mistake, then in relation to identity, there is no such thing as a difference which does not matter.

You claim that the "identity of indiscernibles" cannot be upheld, but this is only supported by the claim that some differences do not matter. If we allow into our principle of identity, the notion of "some differences do not matter", then we have a compromised law of identity. Failure to hold fast to strong logical principles allows vagueness to creep into the logic. Such vagueness hinders our ability to determine the truth. Therefore, if our purpose is to determine the truth, we must uphold the principle of identity to the strongest of our capacities, and assume that every difference matters.

Your "Peircean flip" is this act of compromise. It takes identity from the particulars of the individual, and loses it into the vagueness of the general. By claiming some differences don't matter, you claim that we can disregard accidentals to focus on what is essential, so what is identified is a generality. But since the principal purpose of identification is to identify the particular, distinguishing it from other similar things, you negate the capacity to fulfill this fundamental purpose of identity, with that process, the flip.

Yeah, so you will be with those who feel that nature frustrates you with its fundamental quantum indeterminism and general relativity. You want existence to be exact and totallly knowable, even if that has already been discovered to be a kind of mania.

Yep. The problem with Spinoza is that he was right about there having to be a "One", but wrong in conceiving of that basic materiality as a singular substance rather than as the vagueness of unbounded action. So it is material cause ... in its most insubstantial form. So action utterly lacking in form or purpose. An everythingness that is a singular being only because we call its fundamental disunity or lack of direction a single property or characteristic.

Vagueness is the canonical many. And when the question is asked of how many manys there are, the answer that comes back is "I am only counting the one". — apokrisis

I think Spinoza did not conceive of substance as "basic materiality", but rather as infinite activity, but in any case he certainly did not use the language of "vagueness". He did write somewhere in the Ethics that God is the efficient cause of all things; which amounts to saying that substance is the efficient cause of every modification of its attributes. Spinoza says that substance/ God has "infinite attributes" (which I think should be taken in both the quantitative and the qualitative senses) of which we can know only two: thought and extension. I guess this means that substance must also be considered the material cause of all things, but Spinoza does not allow of any final cause. Substance has no purposes, beyond the purposes of its modes.

As to the question of the oneness of substance, Spinoza wrote (quoted from Hegel or Spinoza by Pierre Macherey: page 103) :

"Nothing can be called one or single unless some other thing has first been conceived, in relation to it, as having the same definition (so to speak) as the first. But since the existence of God is his essence itself, and since we can form no universal idea of his essence, it is certain that he who calls God one or single shows either that he does not have a true idea of him, or that he speaks of him improperly."

Yeah, so you will be with those who feel that nature frustrates you with its fundamental quantum indeterminism and general relativity. You want existence to be exact and totallly knowable, even if that has already been discovered to be a kind of mania. — apokrisis

Wow, that's your counterargument? You think I am going to be frustrated by not be able to figure out specific things which I believe are actually knowable in principle? Meanwhile, you deny the fundamental principles of logic, which might be used to figure these things out, and you satisfy yourself with the claim that such things can't be known. Relax, you're absolutely right, of course they can't be known, when you deny yourself the capacity to know them. Ha, ha, ha, humour me some more, it relieves me of my frustration. So, go ahead then, pour yourself a nice drink, and congratulate yourself, you seem to have convinced yourself that you already know all that it is possible for you to know.

God has infinite attributes. To only be one would be a contradiction with God's very nature. It would be like saying: "God only has thought" or "God only has a bookcase." God's nature is always to be more than one, to never have "only a" and have everything all at once. In this respect, God is discrete. Not a "One" of a distinct and separate state, but a defined whole of many without end or beginning.

Being of infinite attributes, God cannot be accounted for by giving the singular. One gets caught neither here nor there if they try. Does God end at the computer? At the book case? In thought? In the death of the sun? No. God is infinite. God cannot be said to begin or end at any point. It's anything but vague.

The vague account comes out of trying to treat God as a singular "One." It takes the obvious truth that God has many singular attributes and try to account for God through them. In the presence of a single attribute, mode or semiotic expression, someone claims to have discovered God, after all it belongs to God. Only they can't say what God is because the singular attribute, mode or semiotic expression is clearly not enough to be God, so they claim God must be vagueness that belongs to or underpins the singular attribute, mode or semiotic expression.

In truth, they have not discovered God at all. They have confused what belongs to God (singular attribute, mode, a semiotic expression) for God itself, taken the singular that points towards God and suggested it amounts to comprehending God. God gets morphed into this "fabric of vagueness" which is prior in causality rather than realised as the discrete infinite that's true regardless of time.

Thanks, but none of that really clarifies the issue.

There looks a good article on exactly this topic - Firstness, Evolution and the Absolute in Peirce’s Spinoza by Shannon Dea - http://files.bloodedbythought.org/texts/On%20Peirce/Dea-44.4-Peirces%20Spinoza.pdf

I'll give it a read, but put the link here in case you want to consider it too.

If you want to be taken seriously, talk sense.

OK, thanks for the clarification, Willow. :)

Thanks for linking that article, apo. It looks very interesting.

Some excerpts....

Peirce observed that, among those metaphysics that recognize all three categories, “there are other philosophies which seem to do full jus- tice to Categories Second and Third and to minimize the first, and among these perhaps Spinoza and Kant are to be included” (PPM 172). However, by the next lecture, Peirce had changed his mind. He listed as proponents of the ontology that recognizes only Secondness and Thirdness “Cartesianism of all kinds, Leibnizianism, Spinozism, and the metaphysics of the Physicists of today” (PPM 190), but listed Kantian- ism and especially Aristotelianism (to which Peirce this time paid particular attention) as among the metaphysical systems that accept the reality of all three categories (PPM 190).

So the Peirce initially thought Spinoza didn't get firstness, then later wanted to change his mind...

Thus, Peirce not only identified metaphysical systems that embrace all three categories as fundamentally Aristotelian; he also linked Aristotle’s metaphysics (and, by extension, those metaphysics that embrace Firsts, Seconds and Thirds) with evolutionism.16 During the same period, he made the difference between real Aristotelianism and the “imaginary” Aristotelianism of the scholastic period to rest in the for- mer’s evolutionism and the latter’s rejection of same. Finally, in a text from the same period, he praised Spinoza’s “slightly modified” Aris totelianism, maintaining that Spinozism shows no trace of influence by the scholastics.

So the true Aristotle got it, the scholastics didn't. And Spinoza was lucky in being uninfluenced...

[Peirce...M]y chief avocation in the last ten years has been to develop my cosmology. This theory is that the evolution of the world is hyperbolic, that is, proceeds from one state of things in the infinite past, to a different state of things in the infinite future. The state of things in the infinite past is chaos, tohu bohu, the nothingness of which consists in the total absence of regularity. The state of things in the infinite future is death, the nothingness of which consists in the complete triumph of law and absence of all spontaneity. (CP 8.317)

So here, from logical considerations, Peirce describes the trajectory from a hot Big Bang to a cold Heat Death about 50 years before science confirmed it....

Elliptic cosmologies accept the reality only of percepts and reject both the origins and the telos of those percepts as fictions.24 Peirce in more than one text identified this position with Epicureanism,25 although we might think of Humean and statistical mechanical cosmologies as likewise exemplars of this type.

The second possible cosmology also accepts the reality of percepts but sees these as emerging not randomly but from a real origin. This position is, however, analogous to a parabolic curve in that its origin and terminus are coincident. Parabolic cosmologies hold that the universe’s telos just is its origin—that the universe will end as it began. For parabolic thinkers, there is no genuine Firstness, only Secondness and Thirdness. Peirce labelled this position pessimistic.26 However, those infused with Nietzschean amor fati would call it optimistic. It is a position with considerable Stoic affinities,27 and one, it is worth observing, that most would identify with Spinoza.

The final cosmology that Peirce laid out is his own. This is the view of those who regard Absolute Firstness and Absolute Secondness as both real and as really divergent from one another. In geometrical terms, the curve described by two points infinitely distant from one another is hyperbolic. On Peirce’s account, if you hold “that the whole universe is approaching in the infinitely distant future a state having a general character different from that toward which we look back in the infinitely distant past, you make the absolute to consist in two distinct real points and are an evolutionist” (CP 1.362).

And here Peirce employs non-Euclidean geometry to model the various metaphysical cum cosmological alternatives in explicit fashion. He really was running rings around absolutely everyone....

In an 1891 article for The Monist, entitled “The Architecture of Theories,” in a section on the nature of space, Peirce inferred from the revolution in geometry an anti-deterministic revolution in metaphysics. “It is evident,” he wrote, “. . . that we can have no reason to think that every phenomenon in all its minutest details is precisely determined by law. That there is an arbitrary element in the universe we see—namely, its variety. This variety must be attributed to spontaneity in some form” (CP 6.30).

In your face MU. :)

But then the paper struggles to identify any grounds by which the Spinozean absolute is Peircean firstness in another guise. It shows Peirce thought Spinoza wasn't fooled by his own "Euclidean" concreteness. But the reasons for being so charitable are not then adduced in any convincing fashion.

Dea says this...

Spinoza’s commitment to conatus underwrites his criticism of Cartesian mechanics. On Spinoza’s account, Descartes was mistaken to regard matter as inert. For Spinoza, matter, like mind, is active; it is in its very essence dynamic. The important role that Spinoza accords to dunamis in his physics no doubt influenced Peirce’s linking of Spinoza with “historical Aristotelianism.” And, since Peirce cites Aristotle’s own principle of dunamis in support of his
attribution to him of evolutionism, so the traces of Aristotelian dunamis in Spinoza’s principle of conatus almost certainly played a role in Peirce’s association of Spinozism with hyperbolic cosmologies.

Then offers a sketch....

To say that Spinoza was a possibilist is not to deny that he was a necessitarian. He was a necessitarian in the sense that he recognized necessity as real. However, he was also a possibilist, who regarded possibility as real and as extending beyond actuality—just as Peirce did. The details of Spinoza’s possibilism go well beyond the scope of this essay, and will have to wait for another time. However, here is a sketch of how the story goes.

For Spinoza as for Peirce, being is at bottom indeterminate; individual things are not substances. Indeed—and here we glimpse another aspect of Spinoza’s pragmati(ci)sm—they are only individuals to the extent that they have effects. For Spinoza, however, for a thing to have a determinate effect is for other possible effects to be closed off to that thing. Thus, to be an individual thing, on Spinoza’s view, is not to perdure (like a substance) but to have limitations....To be a substance, for Spinoza, is to be utterly unlimited—to be pure possibility.

Yep. But now Spinoza in his own words - where it gets less convincing....

All of the passages that are usually adduced in support of the necessitarian, mechanistic-deterministic account of Spinoza confirm this. CM 1,iii: “The Possible and the Contingent are not affections of things [rerum].” E1P33: “Things [res] could not have been produced by God in any other way or in any other order than is the case.” E1P33S1: “I have shown here more clearly than the midday sun that in things [rebus] there is absolutely nothing by virtue of which they can be said to be ‘contingent’. . . . a thing [res] is termed ‘contingent’ for no other reason than the
deficiency of our knowledge.” These passages all explicitly make reference to things [res]. Things are not possible but fully determined. In Spinoza’s idiom, this is not a grand metaphysical claim; it simply follows analytically from the definition of “thing.”

So we seem to end up with the claim that Spinoza defined substance in untraditional fashion - not the formed secondness of hylomorphism or even Peirceanism - but as the pure potentiality of a vagueness.

Yet I don't think that adds up. At best, Spinoza might have dimly realised the need for pure potential (he was Aristotelean), but still made the mistake of thinking Firstness was still some kind of "material stuff", hence already in the hands of thirdness or the habits of form/purpose.

And this is not surprising given a theistic goal where the stuff of existence must be an expression of some mind's meaning - even immanently.

So summary is that Peirce certainly said by the end that Spinoza seemed a fellow triadicist. But so far no evidence to show that Spinoza really got it at a deep and explicity level. The similarity stems mainly from being a developmental process philosopher trying to make an immanent conception of the divine work out. So any logical properties of the metaphyics are consequences of that general orientation - that goal! - not well worked through arguments that carried the day in their own right.

If you want to be taken seriously, talk sense. — apokrisis

Oh, that's a gas, coming from the one who's reply to my post was: nature's going to frustrate you.
If you're at all serious, then address the points of my post, and quit making a joke of yourself by saying that I'm the one who's not being serious.

In case you've forgotten, I'm waiting to hear justification for overruling the "identity of indiscernibles" with the claim that some differences don't matter. Obviously, if overlooking these differences allows you to deny the "identity of indiscernibles", then they do matter.

What's the point of repeating what you can't understand?

The very thing of a purpose defines its own epistemic boundaries - the point at which differences don't make any difference. And if you can't follow that argument, then that's your problem.

For Spinoza possibility is necessary. It never ends or ceases. At any time all possibilities are possible, even when there is a necessary truth. Unlike a lot accounts of possibility, a possibility doesn’t cease when an event is determined. When the die rolls six, is it still possible any other number might have been rolled. Even though I choose to make this post, it is still possible that I could have chosen not to. And so and so on.

In Spinoza’s philosophy, the actual world (causality, determinism) is not opposed to possible worlds(possibilities, truths of what might occur). The one “inevitable” outcome (i.e. the future that will exist) is true alongside the possibility of every outcome. Possibility is always true for Spinoza. At any point, the world may be just about anything. Even though the sun rose this morning, and this is a necessary truth (the “inevitable,” the one future that occurs), every possibility where the sun doesn’t rise (e.g. it “pops” out of existence, the Earth’s orbit or rotation stops, etc.,etc.) is also true-- the didn't occur, but it's still true they are possible. "Potential" never ceases for Spinoza. With every state that occurs, event that is caused to thing that exists, there is the potential to be otherwise.

Realising the necessity of potential, Spinoza also points out potential cannot be "firstness." Why? Well, because it never begins nor ends. There is no time where a cut between a "firstness" of potential and the "secondary" or "tertiary" of actuality can be made. Potential is just as true for any time. It cannot be that which ends to form actual and discrete states of the world. The world of today must have just as much potential as any quantum foam of the distant past.

Spinoza understood the need for pure potential more than anyone else. He realised it must be beyond "firstness" (or "secondary" or "tertiary" ), finally realising potential's poisonous grip on metaphysics, where it thought to be something a force (i.e. a final cause) must "add" to the world for anything to make sense.

When we seek the truth, differences never cease to matter. — Metaphysician Undercover

What matters to someone is always a function of that person's purposes. Surely you can agree that some differences matter to you more than others; and since everyone has finite resources (including time), we have to prioritize which differences - as well as which similarities - are significant enough within a given context to warrant our focused attention.

A difference, by its very nature, as a difference, is a difference, and therefore it must be treated as a difference. If one adopts the perspective that a difference may be so minute, or irrelevant, that it doesn't matter, and therefore doesn't qualify as a difference, then that person allows contradiction within one's own principles (a difference which is not a difference), and the result will be nothing other than confusion. — Metaphysician Undercover

Read more carefully - in the comment that you referenced, did not say anything about a difference not being a difference; he was talking about a difference not making a difference. Do you see the difference (pun intended)? There are times when a difference really is so minute, or irrelevant, that it does not matter in that context; i.e., it is not worth taking into account, given one's purposes. It still is a difference, but it does not make a difference. I care not one whit about the color of paint that is going to be applied to a steel beam when I am analyzing it by means of a mathematical model to determine whether it can carry the forces that the building code says it must be able to withstand as part of an actual structure. The architect has a different purpose, and therefore might have a different stance - although typically he or she just wants to hide everything above the ceiling anyway.

If we allow that some differences do not matter, then we allow that two distinct things can have the same identity. Since giving two distinct things the same identity is a mistake, then in relation to identity, there is no such thing as a difference which does not matter. — Metaphysician Undercover

Why would this always be a mistake? Standardization and mass production are all about minimizing unimportant differences, such that we can treat different things as effectively identical. When I select a particular section for that beam, I am counting on the fact that it is irrelevant which mine produced the iron ore, which cars and washing machines provided the scrap metal, which mill melted all of that together to make the steel, which service center stored it after rolling, which fabricator assembled it, or which erector installed it. None of those differences make a difference in the finished product, as long as it meets certain minimum specifications - i.e., there are no differences that would make a difference - and that is a good thing!

Failure to hold fast to strong logical principles allows vagueness to creep into the logic. Such vagueness hinders our ability to determine the truth. Therefore, if our purpose is to determine the truth, we must uphold the principle of identity to the strongest of our capacities, and assume that every difference matters. — Metaphysician Undercover

But what if it turns out that vagueness is a fundamental and ineliminable aspect of reality? What if the truth is that vagueness constitutes an actual limitation on our ability to determine the truth? In that case, your dogmatic insistence on assuming that every difference matters hinders your ability to determine the truth about vagueness.

But since the principal purpose of identification is to identify the particular, distinguishing it from other similar things, you negate the capacity to fulfill this fundamental purpose of identity, with that process, the flip. — Metaphysician Undercover

The very act of distinguishing one thing from other things already involves neglecting differences that do not make a difference. Why do we pick out this chair or that table or this book or that door as individual objects, rather than always and only referencing them at a molecular, atomic, or even quantum level? Because the difference between one particle and those adjacent to it within the object is irrelevant to our purpose in picking out that object as a single object. You do this all the time, but it comes so naturally that you do not realize it. No one is capable of paying attention to every single difference among phenomena, because there are far too many of them to do so - even just within your field of vision during the passing of one second.

For Spinoza possibility is necessary. It never ends or ceases. — TheWillowOfDarkness

If that is truly Spinonza's view then at least we can cross him off the list. ;)

Spinoza understood the need for pure potential more than anyone else. He realised it must be beyond "firstness" (or "secondary" or "tertiary" ), finally realising potential's poisonous grip on metaphysics, where it thought to be something a force (i.e. a final cause) must "add" to the world for anything to make sense. — TheWillowOfDarkness

It would be nice if you could support your claims with references or quotes for once. This sounds wildly made-up.

God has infinite attributes. To only be one would be a contradiction with God's very nature. — TheWillowOfDarkness

And yet the traditional/classical conception of God is that He is absolutely simple; His attributes are not discrete in the way that you seem to be suggesting.

God is infinite. God cannot be said to begin or end at any point. It's anything but vague. — TheWillowOfDarkness

With all due respect, that seems rather ... vague to me.

Realising the necessity of potential, Spinoza also points out potential cannot be "firstness." Why? Well, because it never begins nor ends. — TheWillowOfDarkness

This seems like a case where Peirce's attempt to use generic terminology for his categories may have been misleading. They are not called 1ns, 2ns, and 3ns because they always and only come about in that order; on the contrary, my interpretation of his cosmology is that in the hierarchy of being, 3ns is primordial relative to the other two. In any case, 1ns/possibility does not "end" where 2ns/actuality "begins," they are both - along with 3ns/necessity - indispensable and irreducible ingredients of ongoing existence.

The very act of distinguishing one thing from other things already involves neglecting differences that do not make a difference. Why do we pick out this chair or that table or this book or that door as individual objects, rather than always and only referencing them at a molecular, atomic, or even quantum level? Because the difference between one particle and those adjacent to it within the object is irrelevant to our purpose in picking out that object as a single object. You do this all the time, but it comes so naturally that you do not realize it. No one is capable of paying attention to every single difference among phenomena, because there are far too many of them to do so - even just within your field of vision during the passing of one second. — aletheist

For the purpose of understanding the nature of nature, we need precision otherwise we miss the boat.

It is alright to say that a book, for practical purposes, has the same identity before and later. But it is more precise to say that the book has changed and continuously changes so that it is never is the same in duration. After all, we are trying to understand nature and not simply make arguments.

For the purpose of understanding the nature of nature, we need precision otherwise we miss the boat. — Rich

I am surprised that you would say this, considering that we started the thread with your comments to the effect that discrete mathematics cannot properly represent the continuity of nature. Precision is a matter of measurement, and measurement is a matter of discrete mathematics; but the continuous is indeterminate.

It is alright to say that a book, for practical purposes, has the same identity before and later. But it is more precise to say that the book has changed and continuously changes so that it is never is the same in duration. — Rich

I made this point earlier; the contextuality of actuality entails that it is not necessarily true that this object from one point of view, or at one place and time, is identical to this object from another point of view, or at another place and time. Perhaps we can agree that it is more precise (in your sense) to recognize the imprecision (in my sense) of reality.

The very thing of a purpose defines its own epistemic boundaries - the point at which differences don't make any difference. And if you can't follow that argument, then that's your problem. — apokrisis

The purpose, as you yourself, described in that passage, is identity. If you are prepared to say, that two things with the exact same identity, are not in fact the exact same thing, (according to the identity of indiscernibles), because of some differences which do not matter, then you only defeat the purpose of identity, which is to distinguish one thing from another.

Read more carefully - in the comment that you referenced, ↪apokrisis did not say anything about a difference not being a difference; he was talking about a difference not making a difference. — aletheist

Correct, and whether or not a difference matters, depends on the context. And the context in which this was stated was in relation to a point where the "identity of indiscernibles", is supposed to no longer be relevant.

So the laws of thought presume the brute existence of the indiscernible difference that secures the principle of identity. And Peirceanism flips this to say indiscernability kicks in at the point where some 3ns ceases to have a reason to care, and so 1ns is left undisturbed. — apokrisis

But in this situation there is no such thing as a difference which doesn't make a difference. Consider identity A and identity B. If these two identities are the same, then according to the principle of identity of indiscernibles, they are one and the same thing. If you deny that principle of identity, and say A and B are really not the same thing, because of some difference between them which does not matter, and is therefore not part of the identity, (the identity being one and the same), then how is it true to say that this difference does not matter? It is only by claiming that there is a difference between them, which does not matter, that you can say they are two distinct things, rather than necessarily one and the same thing, as stipulated by the "identity of indiscernibles". So it is false that this difference does not matter, because it is the only difference which makes them two distinct things.

Therefore, apokrisis' claim, from Peirce, is that two distinct things can have the very same identity, if we allow that there are differences which do not matter. But of course these differences really do matter, because these are the differences whereby we distinguish the two things as distinct. And it is simple contradiction to say that these differences do not matter.

Why would this always be a mistake? Standardization and mass production are all about minimizing unimportant differences, such that we can treat different things as effectively identical. When I select a particular section for that beam, I am counting on the fact that it is irrelevant which mine produced the iron ore, which cars and washing machines provided the scrap metal, which mill melted all of that together to make the steel, which service center stored it after rolling, which fabricator assembled it, or which erector installed it. None of those differences make a difference in the finished product, as long as it meets certain minimum specifications - i.e., there are no differences that would make a difference - and that is a good thing! — aletheist

The point is, that with respect to the principle of identity these minimal differences are the differences which really are important. If you do not respect this fact, then you allow that all mass produced items are in fact, the very same entity, because you are insisting that they are identical. My car is the same object as your car, because they are mass produced and identical. Your desire is to claim that the factors which differentiate them (the differences of the particular) do not actually differentiate them, and identify them as distinct, as those differences are unimportant. So you will claim that they have the very same identity, yet you will also claim that they are two distinct objects. They are distinct objects not by being different though, because those differences don't matter, they have the same identity. What would justify the claim that they are different then? Or is it the case that my car and your car might really be the very same object?

The purpose of the law of identity is so that we can distinguish one object from another, and come to know that object as the thing it is. To claim that we can overlook some minor differences such that numerous objects may have the same identity only defeats this purpose. We simply deny ourselves the capacity to tell these objects apart.

But what if it turns out that vagueness is a fundamental and ineliminable aspect of reality? What if the truth is that vagueness constitutes an actual limitation on our ability to determine the truth? In that case, your dogmatic insistence on assuming that every difference matters hinders your ability to determine the truth about vagueness. — aletheist

You should consider that perspective as rather nonsensical. Even if vagueness is real and fundamental, we will not know this until it is proven. And we cannot prove its reality without identifying it.

The very act of distinguishing one thing from other things already involves neglecting differences that do not make a difference. Why do we pick out this chair or that table or this book or that door as individual objects, rather than always and only referencing them at a molecular, atomic, or even quantum level? Because the difference between one particle and those adjacent to it within the object is irrelevant to our purpose in picking out that object as a single object. You do this all the time, but it comes so naturally that you do not realize it. No one is capable of paying attention to every single difference among phenomena, because there are far too many of them to do so - even just within your field of vision during the passing of one second. — aletheist

Yes of course, distinguishing one thing from another usually involves neglecting differences which do not matter. But here, we are in the context of the principle of the identity itself, the idenity of indiscernibles. So we are, according to that defined context, dealing with things which appear to be the same. We can conclude, as apokrisis implies, that we cannot tell them apart, because they have the very same identity, yet they are not really the very same thing, due to some differences which do not really matter. Or, we can uphold the principle of the identity of indiscernibles, and conclude that if they are distinct entities, then there must be real differences, which matter, by which we can tell them apart.

I am surprised that you would say this, considering that we started the thread with your comments to the effect that discrete mathematics cannot properly represent the continuity of nature. Precision is a matter of measurement, and measurement is a matter of discrete mathematics; but the continuous is indeterminate. — aletheist

But measurements are imprecise and cannot ever be precise which is exactly the point of this thread. It is impossible to stop anything (continuity and continues flow are embedded in nature)in order to retrieve precise measurements. Measurements are always approximations and it is why measurements specifically and mathematics in general (because of its discrete nature) are very poor tools for understanding nature. Straying from this understanding ultimately will always render a poor understanding of nature, the worse one of which by far is turning humans into number machines.

I made this point earlier; the contextuality of actuality entails that it is not necessarily true that this object from one point of view, or at one place and time, is identical to this object from another point of view, or at another place and time. Perhaps we can agree that it is more precise (in your sense) to recognize the imprecision (in my sense) of reality. — aletheist

Agreed. In any discussion of nature one should recognize continuous flow and change, which brings us back to the OP. Discrete and nature are like oil and order - so don't try to mix them up.

Yes, back to the op. Peirce employs this notion, of a difference which doesn't matter, to support the proposition that a continuity is divisible. If we can divide a continuity, at 2 for example, such that we have <2 and >2, then there cannot be any real difference between <2 and >2 or else that difference would indicate that there was no continuity here in the first place. Peirce proposes that we can assume a difference which does not matter, such that <2 and >2 may be identified as different, but because this difference doesn't matter, <2 and >2 can be said to be the same, so that there is no real difference between them, and there is continuity through 2.

Peirce appears to face the problems which are associated with the idea widely accepted in mathematics, that a continuity is divisible. But instead of following through to where his investigations lead, and coming to the proper conclusion, that a true continuity is truly indivisible, he compromises and proposes this principle of a difference which does not matter. That is a mistake. The proper decision would have been to accept the metaphysical principle that continuity is indivisible, because to divide it proves it to be discontinuous, regardless of what mathematicians want to do.

I don't see a problem. Nor does one appear when we make a second cut at >2. We now have three pieces: <2, 2, >2.

Nor is a problem introduced when considering continuity. My simple understanding is that a line is continuous if it is differentiable. Well, the limit of <2 as it approaches 2 is 2. It does not seem problematic. — Banno

The problem with this representation, is that 2 here is not part of the continuity, it is a point of difference, differentiating one part from the other.. There is a supposed continuity which goes right through 2. If we divide that continuity at 2, then 2 is not part of the continuity, but a point of difference. A continuity cannot have a point of difference because this would make it discontinuous.

Agreed. Peirce, as does many others, attempts to apply mathematics (discrete symbolism) to a continuity and predictability arrives at a statement that does not describe nature.

If you are prepared to say, that two things with the exact same identity, are not in fact the exact same thing, (according to the identity of indiscernibles), because of some differences which do not matter, then you only defeat the purpose of identity, which is to distinguish one thing from another. — Metaphysician Undercover

This sentence makes no sense to me. Differences that do not matter enable us to treat two things that are not identical as if they were identical, for a particular purpose; this is the opposite of claiming that two identical things are not, in fact, the same thing. If our purpose is to distinguish two things, then obviously more differences will matter.

It is only by claiming that there is a difference between them, which does not matter, that you can say they are two distinct things, rather than necessarily one and the same thing, as stipulated by the "identity of indiscernibles". — Metaphysician Undercover

Again, this is backwards. The point is not to claim that there is a difference that does not matter in order to distinguish two things that are really identical, it is to treat two things as identical because the real differences between them do not matter within the context of a particular purpose.

My car is the same object as your car, because they are mass produced and identical. Your desire is to claim that the factors which differentiate them (the differences of the particular) do not actually differentiate them, and identify them as distinct, as those differences are unimportant. — Metaphysician Undercover

Again, it depends entirely on our purposes, which depend entirely on the context. For the most part, the difference that makes a difference in this example is that one car is yours and the other is mine - a human convention, not something intrinsic to the objects themselves. If they were sitting side-by-side on a dealer's lot - same year, make, model, trim, colors, options, condition, mileage, price, etc. - then there would be no differences that make a difference, until you (arbitrarily) choose which one to buy.

The purpose of the law of identity is so that we can distinguish one object from another, and come to know that object as the thing it is. To claim that we can overlook some minor differences such that numerous objects may have the same identity only defeats this purpose. — Metaphysician Undercover

It defeats that particular purpose, but it can be useful for other purposes. By acknowledging that the law of identity has a particular purpose, rather than being an absolute and intrinsic feature of the universe regardless of the context, you are effectively agreeing with the point that we have been discussing.

Measurements are always approximations and it is why measurements specifically and mathematics in general (because of its discrete nature) are very poor tools for understanding nature. — Rich

I continue to be skeptical of your claim that all mathematics is inevitably discrete. In the last several decades, category theory - as a more general alternative to (discrete) set theory for the foundations of mathematics - has facilitated developments like synthetic differential geometry or smooth infinitesimal analysis, which show great promise for more faithfully representing the continuous as continuous.

Peirce employs this notion, of a difference which doesn't matter, to support the proposition that a continuity is divisible. — Metaphysician Undercover

He does? Where? Please cite his writings to support this claim. Did Aristotle also employ this notion, since he likewise held that a continuum is (infinitely) divisible, though actually undivided?

If we can divide a continuity, at 2 for example, such that we have <2 and >2, then there cannot be any real difference between <2 and >2 or else that difference would indicate that there was no continuity here in the first place. — Metaphysician Undercover

No one is disputing that actually dividing a continuum introduces a discontinuity. However, that discontinuity is not there until we break the continuity by that very act of division.

Peirce proposes that we can assume a difference which does not matter, such that <2 and >2 may be identified as different, but because this difference doesn't matter, <2 and >2 can be said to be the same, so that there is no real difference between them, and there is continuity through 2. — Metaphysician Undercover

Again, citations please. As far as I can tell, you have no clue about what Peirce had to say regarding these matters.

A continuity cannot have a point of difference because this would make it discontinuous. — Metaphysician Undercover

Indeed, but what you still refuse to acknowledge is that a continuum does not contain any points at all.

No one is disputing that actually dividing a continuum introduces a discontinuity. However, that discontinuity is not there until we break the continuity by that very act of division. — aletheist

Could you give an example of how you "actually" divide a continuum, and "introduce a discontinuity"?

Marking a point on a line, or breaking it into two lines that have points at their separated ends.