## Law Of Identity And Mathematics Of Change

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• 33
He's just saying that if you use the variable to refer to something, then that thing exists as something, whether it's just an idea or description or whatever it is.

But the variable need not refer in certain models, since certain models may be empty. But it is true nonetheless since it is a quantified statement. Consider my previous post. But to be clear, a variable does not refer to a particular object. It is open for a semantic assignment (i.e. open to be assigned an object, though not in quantified statements, since the variable does not occur free). It has been a while for me, but I think all this is correct...

Now as it concerns "referring to ideas, descriptions, etc.", I am not too sure I follow and I am not sure how this would impact the discussion and the law of identity (which applies to objects, not descriptions, etc.).
• 3.6k
I think you have it a little backwards. We should think of time in relation to physical "clocks," such as heartbeats, diurnal cycles, pendulums or electromagnetic oscillations - because how else can we think of it? That this can be expressed in the form of the chain rule when modeling processes using differentiable functions is just a consequence. The backwards reasoning from a mathematical model to reality is inherently perilous, because mathematics can model all sorts of unphysical and counterfactual things.

You know, the bolded bit amused me, because avoiding the reification of a time beyond or above the unfolding of processes was precisely what I wanted to do. The idea that a clock is simultaneously a measurement of and a definer of time is a bit weird (@Banno @Luke @Fooloso4 @StreetlightX for Wittgenstein thread stuff :) ). I think it's better to think of periodic phenomena as operationalisations of a time concept which is larger than them; ways to index events to regularly repeating patterns.

Another thing the chain rule there reveals, though obviously in some poetic sense, is that there are multiple 'times' and their rates of unfolding differ. Everest slowly increasing in height is effectively a zero rate from the buzz of city life, but from the perspective of stellar accretion Everest's process of increasing height is like driving past the speed limit.

Yes, except that when you ask what "rate" is, time creeps back in. I don't think you can completely eliminate time from consideration, reduce it to something else. You can put it in relation to something else, such as a clock (heartbeats, etc.), but that relationship is not reductive: it goes both ways. Clocks are just as dependent on time as time is on clocks.

Thought experiment here - suppose that the universe is a process of unfolding itself, how can there be a time separate from the rates of its constitutive processes? What I'm trying to get at is that we should think of time as internal to the unfolding of related processes, rather than as an indifferent substrate unfolding occurs over. Think of time as equivalent to the plurality of linked rates, rather than a physical process operative over all of them. Just like 'the kidney' is not an organ of the body, but kidneys are.
• 1.2k
The idea that a clock is simultaneously a measurement of and a definer of time is a bit weird (@Banno Luke @Fooloso4 @StreetlightX for Wittgenstein thread stuff :) ). I think it's better to think of periodic phenomena as operationalisations of a time concept which is larger than them; ways to index events to regularly repeating patterns.

Yes, exactly, clocks (periodic processes) don't define time in the way definitions usually work, i.e. by completely reducing one concept to one or more other concepts; instead they operationalize time.

Thought experiment here - suppose that the universe is a process of unfolding itself, how can there be a time separate from the rates of its constitutive processes? What I'm trying to get at is that we should think of time as internal to the unfolding of related processes, rather than as an indifferent substrate unfolding occurs over. Think of time as equivalent to the plurality of linked rates, rather than a physical process operative over all of them.

I agree with you here: it wouldn't do to think of time as just mathematical time of scientific models, or as an abstract metaphysical entity that exists independently of the world of physical processes. Just as there is no movement without there being moving things, there is no time without there being processes, unfoldings, etc.

And yet... how can there be processes, what could unfolding possibly mean, what are we to make of rates - without referring to the concept of time? I still insist that, although all these physical concepts in the first part of the sentence - let's refer to them as clocks for brevity - serve to operationalize time, they do not define time away; they are not more primary in our understanding than time itself is. And while we cannot understand time without referring to clocks, neither can we understand clocks without referring to time.
• 3.6k
And yet... how can there be processes, what could unfolding possibly mean, what are we to make of rates - without referring to the concept of time? I still insist that, although all these physical concepts in the first part of the sentence - let's refer to them as clocks for brevity - serve to operationalize time, they do not define time away; they are not more primary in our understanding than time itself is. And while we cannot understand time without referring to clocks, neither can we understand clocks without referring to time.

There's a lot going on in the question.

(1) There's an epistemological issue - which Kantian/phenomenological considerations fit into - how are clocks (operationalisations of time) interpretively pre-structured by the categories of the understanding or by experiential temporality.

(2) There's a cultural issue - what are the origins of the unified concept of time, what kind of understandings do people have to learn to grok time?

(3) Then there's an ontological (well, also ontic) issue about unfolding/becoming being dependent upon time for it to unfold.

The interesting issue here is (3), but we need to talk about what not to do given (1) and (2).

I would like to posit that insofar as (1) experiential temporality, or the transcendental structure of time are related to the issue, we shouldn't index ontical unfolding - natural time/temporality - as a development of experiential temporality. Only our understanding of ontic time is facilitated by experiential temporality. Experiential temporality allows the issue to be raised in the first place, but is otherwise irrelevant to providing a good exegesis of the interdependence of time and unfolding. The first eye opening was one event in the natural flows that subtend our existence.

In regards to (2), just like we can't say that a mathematical entity must have a corresponding entity in a nature for a theory of nature which has that mathematical entity in it for that theory to be correct (example: infinite plane waves, sum over 'all histories' approach in quantum mechanics), we should not be so sure that cultural artefacts and norms of interpretation vouchsafe the necessary existence of a referent of words. Nature informs our vocabulary through our understanding, but light's frequencies are nevertheless not arranged in a colour wheel.

I'm quite suspicious, therefore, that something like time would have a unique ontic correlate - for there to be a pattern of nature which is time - just because the unfolding of processes requires a time concept to think. To me, it appears that something like the type-token distinction is at work here; the word time is a sortal we learn that synthesises the operationalisations that we are first exposed to, the mathematical abstractions of periodic processes, numbers inscribed on clockfaces, the rhythm of our hearts and so on. Our understanding is densely populated with things and strategies of thought that are not in concordance with the unfolding of nature, and do not help us to reveal its structure.

From this I think we should resist saying that the progression of the physical entity of a clock depends upon a concept we have derived from the clock; as if the clock would not tick without the operationalisation of time that it embodies in our understanding. Or if it would not tick without experiential temporality stretching along with it.

This speaks to learning what time is by learning the role it plays in (our interpretations of) life, rather than the role it plays in nature itself. I think it suggests we should reject the ontic relevance of time as a unified concept, just like we can reject the idea of mathematical entities necessarily having an existence in nature (if someone kicks over a rock and discovers $\aleph_{\omega}$ I would be incredibly surprised). As the chain rule thing shows, it doesn't matter whether we have $t$ or (smooth, bijective) $f(t)$ in our physical theories, as it just requires scaling the laws (imagine if seconds were instead 2 seconds, divide the time term in a law by a half or multiply by 2 depending on the context, sorted).

I think it's important to think ontic time immanently, and processes being 'clocks' for each other might provide a vantage point from which to do this.
• 7.1k
I apologize: I should not have assumed you were familiar with this; that is completely on me. I am employing standard first-order logic notation. The statement (∀x)(x=x) (∀x)(x=x)(\forall x)(x=x) says "for all x, x is identical to x."

This is the problem then. That is not the law of identity. The law of identity does not allow that there is more than one X. When you say "for all X...", you have already allowed the possibility of more than one X, thus breaking the law.

Please let me know if there is any step that isn't clear!

What is not clear is how you get from the law of identity, as commonly stated, to your formulation of it. And I'm sorry to be the one to inform you of this, but your example fails because it utilizes a formulation of the law of identity which is already itself in violation of the conventional law of identity.
• 33
This is the problem then. That is not the law of identity. The law of identity does not allow that there is more than one X. When you say "for all X...", you have already allowed the possibility of more than one X, thus breaking the law.

?

I am not sure where you are getting this and why you think it is true. Could you clarify? In no suitable formulation of the law of identity would it be valid only in a model with exactly one and only one object. How would you even formulate this? I take it something like this:

$(\exists x) (x=x) \wedge \neg(\exists y)\neg(x=y)$

But this is no law of logic, and certainly not a law of identity. It is fairly simple to provide a model for which the statement is false. Therefore, it is not a law of logic. Logical laws are true in every model, not just some models.

What is not clear is how you get from the law of identity, as commonly stated, to your formulation of it. And I'm sorry to be the one to inform you of this, but your example fails because it utilizes a formulation of the law of identity which is already itself in violation of the conventional law of identity.

I am employing the very familiar and standard notion from logic.

In logic, the law of identity states that each thing is identical with itself. It is the first of the three laws of thought, along with the law of noncontradiction, and the law of excluded middle. However, no system of logic is built on just these laws, and none of these laws provide inference rules, such as modus ponens or DeMorgan's Laws.

In its formal representation, the law of identity is written "a = a" or "For all x: x = x", where a or x refer to a term rather than a proposition, and thus the law of identity is not used in propositional logic. It is that which is expressed by the equals sign "=", the notion of identity or equality. It can also be written less formally as A is A. One statement of such a principle is "Rose is a rose is a rose is a rose."

In logical discourse, violations of the law of identity result in the informal logical fallacy known as equivocation.[1] That is to say, we cannot use the same term in the same discourse while having it signify different senses or meanings and introducing ambiguity into the discourse – even though the different meanings are conventionally prescribed to that term. The law of identity also allows for substitution, and is a tautology.
-Wiki

If your position requires you to reject a basic principle of formal logic, I would reconsider it carefully, especially since you admitted that you are not familiar with logic. We have an obligation not to be confident in our pronouncements if we are not entirely sure about all that goes into asserting them. Moreover, we should be open to reconsidering our position. So far in our discussion, everything is pointing to the conclusion that you (and not I) are confused about the law of identity. The formulation I have given for the law of identity is not mine, it is the one learned by everyone in the first course on logic in philosophy or mathematics.

I believe I have provided relatively clear explanations for what are elementary concepts in logic (i.e. the law of identity). If anything is unclear, please let me know.
• 1.2k
There's a lot going on in the question.

Yes, and thank you for a comprehensive response.

From this I think we should resist saying that the progression of the physical entity of a clock depends upon a concept we have derived from the clock; as if the clock would not tick without the operationalisation of time that it embodies in our understanding. Or if it would not tick without experiential temporality stretching along with it.

Oh but I don't think that we derive the concept of time from the clock. From the moment of the first eye opening we already have some intuitive understanding of time. Observing clocks helps us to further contextualize, structure, and quantify that understanding, and more careful observation and reflection leads to more sophisticated understanding of the structure and measure of time in terms of mathematical models and measuring devices.

So when you ask yourself, "What is time?" you can point to periodic processes or to theoretical models, but then if you ask, "What validates those explanations?" you still have to go back to the phenomenology (including, of course, the phenomenology of clocks), because what else would we go back to? That doesn't mean, of course, that we have to hang on to every prejudice and intuition, but our explanations have to be true to something, or else they just hang free, like abstract mathematical entities.

What does a clock show? What does it mean to say that this iteration is prior to that? If we reject mathematical models as inadequate for exhaustively answering empirical questions, I am afraid that an answer can only be provided by gesturing, tautologically, towards some sort of unfolding. Tautologically because, of course, our notion of unfolding is already informed by the notion of periodic processes.
• 5.9k
As someone already mentioned, the law of identity is not a metaphysical claim of absence of change because, as you rightly said, change is inescapable. Even then we may be able to accommodate the law of identity because the form of matter may change e.g. wood can be made into a chair, window, table, etc. but the substance, wood, doesn't change does it?

Anyway the law of identity is simply a constraint put on logical entities e.g. proper nouns and propositions so that valid inferences can be made. For instance, if the proposition P = It is raining, then as we construct an argument around P the gist of P must remain unchanging throughout. Don't you think?
• 7.1k
I am not sure where you are getting this and why you think it is true. Could you clarify? In no suitable formulation of the law of identity would it be valid only in a model with exactly one and only one object. How would you even formulate this? I take it something like this:

The law of identity states that a thing is the same as itself.

But this is no law of logic, and certainly not a law of identity. It is fairly simple to provide a model for which the statement is false. Therefore, it is not a law of logic. Logical laws are true in every model, not just some models.

That is our point of disagreement. My claim is that the law of identity is not a law of logic, it's a metaphysical assumption. You think it's a law of logic. Because of this disagreement, I do no think we will ever find an expression of the law of identity which we both agree with.

This is from your wiki quote:
the law of identity states that each thing is identical with itself

So we seem to agree at this point. My question to you is how do you proceed from the proposition "each thing...", to your formulation "for all x...."? Notice that the former refers to particular, individual things, and the latter refers to a group of things. Your formulation appears to have a veiled inductive conclusion, inherent within. You must apply inductive logic to "each thing is identical to itself, to derive "all things are identical to themselves". Therefore your formulation is one which has been polluted by inductive logic.
• 3.6k
What does a clock show? What does it mean to say that this iteration is prior to that? If we reject mathematical models as inadequate for exhaustively answering empirical questions, I am afraid that an answer can only be provided by gesturing, tautologically, towards some sort of unfolding. Tautologically because, of course, our notion of unfolding is already informed by the notion of periodic processes.

I don't want to reject mathematical models, far from being a mere philosophical point; if I thought that I would have to change job! Specifically, I think mathematical models really do allow us to find things out about nature. What I was trying to highlight was that the use of time in mathematical models doesn't really tell us much about it, as any smooth bijective function of time could be used to parametrise them.

All that really says is that the time parameter in mathematical models is often rather arbitrary, and when thinking about what ontological commitments to form based on mathematical models, we should be very careful with attributing existence to something which may be chosen so freely.

In terms of the Lotka Volterra example earlier, the relevant dynamic the equations seek is the reciprocal dependence of predator numbers on prey numbers. Predator numbers and prey numbers are something it makes sense to have a commitment about, and the rate of change of one with respect to the other is the target of the model. It's what the equations try to capture.

Time in the model, in that regard, is a useful independent parameter that you can evaluate both populations at. I'm not saying we should do away with it.

So when you ask yourself, "What is time?" you can point to periodic processes or to theoretical models, but then if you ask, "What validates those explanations?" you still have to go back to the phenomenology (including, of course, the phenomenology of clocks), because what else would we go back to? That doesn't mean, of course, that we have to hang on to every prejudice and intuition, but our explanations have to be true to something, or else they just hang free, like abstract mathematical entities.

What I have in mind is a few procedures for giving an account of the unified concept of time.

(A) One takes the plurality of rates, synthesises that through some phenomenological considerations, and outputs a concept of time which is necessary in our understanding.
(B) One takes the time variable, synthesises that through some phenomenological considerations, and outputs a concept of time which is necessary in our understanding.

(C) One takes the plurality of rates, synthesises that through our capacities of understanding more generally, and outputs a time concept which is tied speculatively to time in nature.
(D) One takes a time variable, synthesises that through our capacities of understanding more generally, and outputs a concept of time which is tied speculatively to time in nature.

You can see that (A,B) and (C,D) are grouped structurally, I don't really care which approach is taken within (A,B) or (C,D), they denote the development of a phenomenological understanding of time indexed to humans and a use of whatever that time concept is to understand time in nature.

What I'm trying to point out here is that we should not take answers from the (A,B) group of questions as answers to the (C,D) group of questions. Even if one has, like in Kant, linked the unity of the time concept to the sensory manifold and the transcendental unity of apperception, one still has the independent branch of questions about time in nature; like what Riemann and Einstein and even Bergson aimed at; that cannot be given answers in this way. (C,D) questions are possible to address, and are of philosophical merit. They just require a different workflow to address than the 'link to a priori structure of experience' machine, as there is time in nature irrelevant of experiential temporality.

The problems posed by (C,D) do influence how we should think of experiential temporality - perhaps it is not 'primary' in all senses, humans evolved in the presence of a time which is not our own, and in that regard the 'merely ontic' notion of time targeted in (C,D) is primary. But here what really matters is that they're different question groups with different methodologies to attack. (C,D) weaponise experiential temporality to 'carve nature at its joints'.

My love of the chain rule example is that it suggests one way to exploit the arbitrarity of the time variable to 'internalise' it to other concepts; of differentials of unfolding. While time and unfolding are probably interdependent, time is often seen as unitary whereas unfolding is a plurality of links which we know have affective power in nature. It invites an immanent thought of time, whereas the times thought in (A,B) and the hypostatised 'indifferent substrate' of time are both marred by their transcendental character.

Edit-imprecise summary: time is something empirically real, not just something transcendentally ideal. The empirically real component requires different methodology to attack than the usual Kantian/phenomenological interpretive machines, and is still of philosophical interest.
• 33
The law of identity states that a thing is the same as itself.

All things are identical to themselves. Which is exactly the formulation I discussed and exactly the principle that implies nothing with respect to the number of existant objects.

That is our point of disagreement. My claim is that the law of identity is not a law of logic, it's a metaphysical assumption. You think it's a law of logic. Because of this disagreement, I do no think we will ever find an expression of the law of identity which we both agree with.

The law of identity is a law of logic. What was stated: an object is identical to itself, IS a law of logic. Period. This is why I think you may not be expressing what you really want to express clearly. I think you may have a different identity claim in mind that may, in fact, be a metaphysical claim. The one you expressed, however, is not.

My question to you is how do you proceed from the proposition "each thing...", to your formulation "for all x...."? Notice that the former refers to particular, individual things, and the latter refers to a group of things.

They are equivalent. "For all x" and "each x" is logically equivalent. "Each" is a universal quantifier expression. I could easily have said that $(\forall x)(x=x)$ says: each x is self-identical OR
for each x, x is identical to x.

You must apply inductive logic to "each thing is identical to itself, to derive "all things are identical to themselves".

Not at all. You are misconstruing the statement "each thing" for "each thing I observe now". That is NOT the law of identity. That is, "each thing observed up to this point has been identical to itself" is NOT at all what "each object is identical to itself" states.

For this reason, no induction is required AT ALL to get the law of identity. As stated, the law of identity is an axiom of logic. It is a logical principle through and through because, as I showed, it is true in every model.

That being said, if you want to discuss the empirical claim regarding the objects we've observed, then by all means. However, we do not come to the conclusion that an object is identical to itself via observation anyway.

Otherwise, I am not sure what you are getting at. I think, perhaps, you are trying to say that claims about persistent identity are metaphysical. Here I am in complete agreement with you. Identity over time is metaphysics.
• 7.1k
All things are identical to themselves. Which is exactly the formulation I discussed and exactly the principle that implies nothing with respect to the number of existant objects.

Do you not recognize the difference between "a thing", and "all things"?

The law of identity states that a thing is the same as itself. What justifies you formulation, that all things are the same as themselves, other than induction? The law of identity is not an inductive principle.
• 33
Getting Clear on how to Formulate the Law of Identity, and why we need to be logically precise

In logic, it is important to note that arbitrary reference is equivalent to referring to all objects in a domain of discourse.

This is why it is crucial to be precise when employing the indefinite article. When one uses an indefinite article like "a", one must precisely articular whether one is using one of two quantificational statements: $\exists$ (for some, there exists a, etc.) or $\forall$ (for all, every, etc.)

The reason this is important is that some may want to argue that the law of identity does not employ the unversal quantification. If I say "a thing is identical to itself", do I mean to say:

1) $(\exists x) (x=x)$

OR

2) $(\forall x)(x=x)$

MU argues that the identity principle is just the statement (1), and he further states that someone who says it is (2) is confused:

Do you not recognize the difference between "a thing", and "all things"?

It is very important to note that everyone agrees that (2) is the formulation for the law of identity. It is easy to show why. So here we go:

Proposition: (1) is not logically valid, where (1) refers to the proposition $(\exists x) (x=x)$

PROOF: By definition, a proposition is logically valid if and only if for any model $\mathcal{M}$, (1) is satisfied (is made true) in that model, i.e., $\vDash_{M} (\exists x) (x=x)$.

Consider a model $\mathcal{M}$ with domain $\varnothing$. Since the domain is empty, we have that $\vDash_{M} \neg(\exists x) (x=x)$ from which it follows that $\nvDash_{M} (\exists x) (x=x)$. Thus, (1) is not valid in $\mathcal{M}$, and thus not a logical truth. $\square$

INFORMAL ARGUMENT: Since the proof may use some technical devices we may not be familiar with, here is the essential idea. In order to show that (1) is not a logical truth, we just have to show that we can imagine a possible situation in which (1) is false. Indeed, in a universe where no objects exist, (1) is false. Therefore, (1) cannot be a logical truth.

What is interesting to notice as well, is that (2) does not imply (1). If this were the case, then any model in which (2) is true, (1) is also true. However, in the model I just showed you (see proof above) (2) is true (see one of my previous posts in this discussion if you want the technical proof) but (1) is false. Therefore, (2) does not imply (1).

The reason (2) does not imply (1) is because (1) implies the existence of at least one objects. (2) does not imply the existence of an object. Here are some logical equivalences that might help you see this:

\begin{align*}& (\exists x)(x=x) \Leftrightarrow \neg(\forall x)\neg(x=x)\\& (\forall x)(x=x) \Leftrightarrow \neg(\exists x)\neg(x=x)\end{align*}

The Law of identity is held as a law that is logically true. Indeed, (2) is logically true, i.e., it is true in all models.

The proper way to state the law of identity is:

All objects are self-identical (whether many, one or no objects exist)

Or, alternatively, we can say:

There exists no object that is not identical to itself (whether many, one or no objects exist).

Both of these statements are logically equivalent and, most importantly, do not imply that an object exists.
• 7.1k
Proposition: (1) is not logically valid, where (1) refers to the proposition (∃x)(x=x) (∃x)(x=x)(\exists x) (x=x)

That's the point, the law of identity is a metaphysical assumption, so of course it's not logically valid.

The Law of identity is held as a law that is logically true.

No, the law of identity is not held as a law which is logically valid. It is held to be true, and perhaps even sound, depending on how you define "sound", but it is not held to be logically valid. The three fundamental laws, identity, non-contradiction, and excluded middle, are all held to be true, but not one of them, on its own, is logically valid.
• 1.4k

Suppose you went back in time and encountered your 2-year old self. Two distinct individuals standing side be side, with clear physical differences cannot be considered the identical person. You do not even share the same set of memories, you only share a 2-year subset (and your memories of that shared subset are fuzzier). Your DNA isn't even identical - our DNA gradually changes a little over time.

Even without time travel, to maintain identity over time, there must be something that endures. What is that?
• 1.2k
don't want to reject mathematical models, far from being a mere philosophical point; if I thought that I would have to change job! Specifically, I think mathematical models really do allow us to find things out about nature. What I was trying to highlight was that the use of time in mathematical models doesn't really tell us much about it, as any smooth bijective function of time could be used to parametrise them.

Well, there is this position, to which I am somewhat sympathetic, that the abstract (mathematical) entities that we find to be indispensable in explaining (modeling) the world thereby exist. Of course, as you point out, time may not even be all that indispensable, or even if some time was necessary, there is no one definite form of it that we are forced to adopt. But then the latter problem is basically what Einstein's relativity tackles, where time is quite substantive, even if it is very much a reference- and coordinate-dependent entity.

My love of the chain rule example is that it suggests one way to exploit the arbitrarity of the time variable to 'internalise' it to other concepts; of differentials of unfolding. While time and unfolding are probably interdependent, time is often seen as unitary whereas unfolding is a plurality of links which we know have affective power in nature. It invites an immanent thought of time, whereas the times thought in (A,B) and the hypostatised 'indifferent substrate' of time are both marred by their transcendental character.

You don't even need a smooth function in order to convey this idea: really, what it comes down to is variable substitution: expressing one quantity in terms of another. This works even for ragged and discontinuous relationships. However, to return to my reservations about this thought as a justification for what is, I think, a physical and/or metaphysical thesis, the same abstract manipulation can be applied in ways that are less physically meaningful and certainly don't warrant a parallel conclusion. For example, in the famous predator-prey example, instead of looking at populations of wolves and hares, we could look at the population of wolves and the amount of manure excreted by hares, which of course is closely related to the population of hares. Does this mean that we can therefor dispense with hares in this system? Well, we could for the sake of modeling the population of wolves (or the amount of shit, if that is what we are interested in), but surely our ability to do so doesn't indicate that hares lack substance!

(By the way, for me the Lotka-Volterra problem was one of the more memorable experiences from learning mathematics. It becomes even more dynamically interesting in 3D, if you add another variable into the system, such as grass.)

Edit-imprecise summary: time is something empirically real, not just something transcendentally ideal. The empirically real component requires different methodology to attack than the usual Kantian/phenomenological interpretive machines, and is still of philosophical interest.

Thanks for this, I know I haven't addressed much of what you've said - but that's because I would like to think more about it.
• 3.6k
Well, there is this position, to which I am somewhat sympathetic, that the abstract (mathematical) entities that we find to be indispensable in explaining (modeling) the world thereby exist. Of course, as you point out, time may not even be all that indispensable, or even if some time was necessary, there is no one definite form of it that we are forced to adopt. But then the latter problem is basically what Einstein's relativity tackles, where time is quite substantive, even if it is very much a reference- and coordinate-dependent entity.

I don't think relativity tackles the problem, really. To be sure, it makes time immanent, which is a good step. It makes space, motion, time and mass have reciprocal relationships and intricate interdependence. But it makes it immanent by fleshing out couplings between time and space and motion and mass in an abstract 4 dimensional vector space of which time is an independent direction of variation. You can still do the same trick with a smooth bijection to get another 'time' and make, say, the time direction a function of the oscillation between hyperfine states of a hydrogen atom (as we do to operationalise it now), or through any other physical process of unfolding.

I would like to have my cake and eat it too, and say that time is relational in a deeper sense, but that it still makes sense to think of it as an independent direction of variation in the relativity sense. Less Wrong has an interesting thought experiment on the matter:

But what would it mean for 10 million "years" to pass, if motion everywhere had been suspended?

Does it make sense to say that the global rate of motion could slow down, or speed up, over the whole universe at once—so that all the particles arrive at the same final configuration, in twice as much time, or half as much time? You couldn't measure it with any clock, because the ticking of the clock would slow down too.

Do not say, "I could not detect it; therefore, who knows, it might happen every day."

Say rather, "I could not detect it, nor could anyone detect it even in principle, nor would any physical relation be affected except this one thing called 'the global rate of motion'. Therefore, I wonder what the phrase 'global rate of motion' really means."

This 'global rate of motion' being seen as pregnant in the above understand of general relativity is just what I would like to problematise. I think this is consistent with general relativity, as to think of the universe as having a 'global time coordinate' or 'global rate of change' forgets that time is one of the directions of variation of the universe; it's already baked in.

When we imagine the universe unfolding over time, we fix our frame of reference to the mind's eye independent of it all, and this is good as we have freedom of choice to define how we measure one process with another - and what processes we use for such measurement - but it hides that such an independent direction of variation must still be pregnant in the processes which make up the universe rather than exterior to them all.

That we could externalise time in a manner 'exterior to them all' is more about our imagination than about the ontic status of time as transcendent/immanent with respect to the universe's processes, time is already something interior; so it must have something to do with the plurality of processes which unfold.

One clue that time is relational capacity of systems would be that in the absence of a suitable relationship of coupling or correlation, no unfolding would be observable, and to my knowledge this is just what we see:

One clue comes from theoretical insights arrived at by Don Page and William Wootters in the 1980s. Page, now at the University of Alberta, and Wootters, now at Williams, discovered that an entangled system that is globally static can contain a subsystem that appears to evolve from the point of view of an observer within it. Called a “history state,” the system consists of a subsystem entangled with what you might call a clock. The state of the subsystem differs depending on whether the clock is in a state where its hour hand points to one, two, three and so on. “But the whole state of system-plus-clock doesn’t change in time,” Swingle explained. “There is no time. It’s just the state — it doesn’t ever change.” In other words, time doesn’t exist globally, but an effective notion of time emerges for the subsystem.

A team of Italian researchers experimentally demonstrated this phenomenon in 2013. In summarizing their work, the group wrote: “We show how a static, entangled state of two photons can be seen as evolving by an observer that uses one of the two photons as a clock to gauge the time-evolution of the other photon. However, an external observer can show that the global entangled state does not evolve.”

Other theoretical work has led to similar conclusions. Geometric patterns, such as the amplituhedron, that describe the outcomes of particle interactions also suggest that reality emerges from something timeless and purely mathematical. It’s still unclear, however, just how the amplituhedron and holography relate to each other.

The bottom line, in Swingle’s words, is that “somehow, you can emerge time from timeless degrees of freedom using entanglement.”

So the relational character of time is something that comes out of general relativity conceptually and quantum experiments demonstrably. I would like to say something like this is poetically suggested by basic calculus too:

$\frac{df}{df}=1=\frac{dg}{dg}$

the evolution of the function is indiscernible when you measure that evolution through its own unfolding.
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You don't even need a smooth function in order to convey this idea: really, what it comes down to is variable substitution: expressing one quantity in terms of another. This works even for ragged and discontinuous relationships. However, to return to my reservations about this thought as a justification for what is, I think, a physical and/or metaphysical thesis, the same abstract manipulation can be applied in ways that are less physically meaningful and certainly don't warrant a parallel conclusion. For example, in the famous predator-prey example, instead of looking at populations of wolves and hares, we could look at the population of wolves and the amount of manure excreted by hares, which of course is closely related to the population of hares. Does this mean that we can therefor dispense with hares in this system? Well, we could for the sake of modeling the population of wolves (or the amount of shit, if that is what we are interested in), but surely our ability to do so doesn't indicate that hares lack substance!

You actually have to be very careful with how you transform variables to preserve their meaning. You could surject the real line onto {0,1} and lose so much that the new scale is no longer a clock, it's an indicator of a discrete property. In order to preserve trends, for example, the variable transformation should be sufficiently smooth for the problem tackled and definitely monotonic. The smoothness varies, if one requires to estimate the second derivative of a function from a curve you should only transform using functions which have at least a differentiable second derivative.

The take home message here is that the ability to use any smooth bijection of time equivalently to time is actually rather odd in these terms; most variable substitutions which preserve the interpretable relationship between the variables and the model definitely don't have this property.
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The three fundamental laws, identity, non-contradiction, and excluded middle, are all held to be true, but not one of them, on its own, is logically valid.

Can you make clear exactly what that last clause means?
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Can you make clear exactly what that last clause means?

Isn't it obvious? Not one of the three fundamental laws of logic is a valid logical conclusion. For example, suppose there are rules which must be followed in order to produce a valid logical conclusion. It is impossible that the rules themselves are valid logical conclusions, because they are necessarily prior to any logical conclusion.
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If you're suggesting that the three laws are axioms, then I suppose, but that seems to me to be an incidental point. And to be sure, the law of identity is proved above. That is, it is a valid conclusion in logic.

But I kept my question short and unadorned. You must have some reason for your claim, what exactly is your reason?

As to the law of non-contradiction, it's not difficult to show that if both p and not-p, then you can prove anything. It follows, then, as a conclusion that you cannot have both p and not-p.

That leaves excluded middle. Again a reductio ad absurdum leads to the law of the excluded middle as a valid logical conclusion.

So what exactly are you claiming is, and what exactly are you claiming isn't, and what is your argument?
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And to be sure, the law of identity is proved above. That is, it is a valid conclusion in logic.

The problem is that the formulation of the law of identity, which Kornelius used in the proof that the law of identity is a valid logical conclusion, is not a true representation of the law of identity. The law states that a thing is the same as itself. Konelius' formulation stated "for all things". So in stating that all things have something in common, they are the same in this sense, Kornelius has already violated the law of identity which states that "sameness" can only refer to the relationship between a thing and itself.

As to the law of non-contradiction, it's not difficult to show that if both p and not-p, then you can prove anything. It follows, then, as a conclusion that you cannot have both p and not-p.

That you might be able to prove anything without such a law does not prove that the law is a valid logical conclusion. It only points to the usefulness of the law as a tool for understanding.

So what exactly are you claiming is, and what exactly are you claiming isn't, and what is your argument?

The point being argued was the nature of the law of identity. I said it is an ontological principle, and Kornelius argued that it is a logical principle. My point was that despite the fact that the principle may be adapted and used by logic, it is grounded by, and justified by ontology. Therefore it is an ontological principle, not a logical principle, in the same way that ontological principles which are used in science, are not scientific principles despite the fact that they are used by science.
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The law states that a thing is the same as itself. Konelius' formulation stated "for all things". So in stating that all things have something in common, they are the same in this sense, Kornelius has already violated the law of identity which states that "sameness" can only refer to the relationship between a thing and itself.

How is this not a case of equivocation on and confusion over the meaning of the word "same"?

My point was that despite the fact that the principle may be adapted and used by logic, it is grounded by, and justified by ontology.

But not just adapted and used, but proved within. Not merely borrowed, but thereby made a member of the family. Without (yet) addressing your claim of its being an ontological principle, why cannot it on these grounds just mentioned be a logical principle?

Looking on line, I find these:
"In many respects, the Ontological Principle is simply another way of formulating the Ontic Principle. If there is no difference that does not make a difference, then it follows that “to be” is to both differ and produce difference."

"In other words, the Ontological Principle is an affirmation that banishes any sort of overdetermining cause that affects all other beings without itself, in turn, being affected."

and others of varying length and opacity. Without taxing you to comment on these, what is meant by saying the law of identity is an ontological principle? I might be confusing "Principle" with "principle," here.)

Anyway, what is an ontological principle?
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How is this not a case of equivocation on and confusion over the meaning of the word "same"?

If Kornelius changes the meaning of "same" from how it is properly expressed in the law of identity, to prove that the law of identity is logically valid, then this proof is based on an equivocation and therefore invalid due to that fallacy.

But not just adapted and used, but proved within. Not merely borrowed, but thereby made a member of the family. Without (yet) addressing your claim of its being an ontological principle, why cannot it on these grounds just mentioned be a logical principle?

No acceptable proof has been demonstrated yet. As I explained, the proof provided is based on an adapted version of the law of identity. And, as I've argued this adapted version actually violates the law of identity as expressed in its proper form.

Without taxing you to comment on these, what is meant by saying the law of identity is an ontological principle? I might be confusing "Principle" with "principle," here.)

Anyway, what is an ontological principle?

An ontological principle is a statement, or proposition which claims something about the nature of being. The point I was making is that it is an assumption, rather than something proven by logic.

I believe that to understand why an ontological principle is a fundamental assumption rather than an inductive conclusion requires an analysis of the difference between subject and predicate. Once the subject is distinguished from the predicate as that which is described in the act of predication, then we can proceed toward understanding the distinction between the subject and the object (this might be described in Kantian terms of phenomenon/noumenon). An inductive conclusion is based on predication, and therefore makes a statement concerning a commonality in predication. The sameness which leads to the inductive conclusion is found in the predicate. So the sameness which is referred to with inductive conclusions is a sameness which is produced by predication.

Now we must validate the sameness of the subject, and this is the fact that we call distinct objects by the same name, because they are the same type of object. But this type of sameness can only be validated by predication as well, they are the same type of object, because the same thing can be predicated about them. This leaves us in a vicious circle whereby the object itself is inaccessible, and nothing can be validly said about the object, as Kant described with the concept of noumena.

So to say anything about the object itself, is to simply make an assumption about it. The first assumption that we make is that it is an object, a being, a thing, and therefore it has an identity as such. This is the law of identity, it's based in the assumption that there are real existing things, and that they have within themselves, their own identity, independent of the identity which we give them, which is as a subject. Once we have given the real object real existence, through this assumption, in this Aristotelian manner, we can proceed toward understanding what this real existence consists of, what validates this assumption. This is first and foremost, temporal extension, which the concept of "matter" accounts for.
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An ontological principle is a statement, or proposition which claims something about the nature of being. The point I was making is that it is an assumption, rather than something proven by logic.
I'm going to try to find a bottom in your post. To start, would you not say that an assumption is a species of proposition?

Or perhaps I'm confused: "which claims something about being." What claim can there be about being that is not actually a claim about something else? That is, being, being the supremum genus, has no species and no accidents. How can you predicate anything of being?

The law states that a thing is the same as itself.
I take it this your ontological principal. But in what sense is it just an assumption - and not an induction?

I believe that to understand why an ontological principle is a fundamental assumption rather than an inductive conclusion requires an analysis of the difference between subject and predicate. Once the subject is distinguished from the predicate as that which is described in the act of predication, then we can proceed toward understanding the distinction between the subject and the object (this might be described in Kantian terms of phenomenon/noumenon). An inductive conclusion is based on predication, and therefore makes a statement concerning a commonality in predication. The sameness which leads to the inductive conclusion is found in the predicate. So the sameness which is referred to with inductive conclusions is a sameness which is produced by predication.

If you're suggesting - arguing - that predication attributes to a subject, and neither subject nor attribution "touch" the object, then the ultimate predication, being, is also similarly ungrounded. If you deny induction and call it all assumption, then you rule out reason-as-process. For what indeed can you reason about but predication? (The reasoning itself - if you allow for such - being mainly governed by logic.)

Now we must validate the sameness of the subject, and this is the fact that we call distinct objects by the same name, because they are the same type of object.
Nope. You just ruled this out. More accurately, on your approach, is that we recognize samenesses in the predications. Which is exactly what you say just above. .

So to say anything about the object itself, is to simply make an assumption about it.

Reading the rest of your post, I see we "assume" the subject into real existence, real objective reality,
in this Aristotelian manner
Under which and guided by which,
we can proceed toward understanding what this real existence consists of, what validates this assumption.

But your argument against "For all x: x+x" apparently - near as I can tell - is based on a non-understanding of what it says. You above state "that a thing is the same as itself." You call that a law. Is this true of only some things and not others? Or is it instead true of every thing? If it is true of every thing, then it is true for all things. And you can complete this. So how, exactly, do you disqualify your ontological law of identity from being a law of logic?

Now, under Aristotelian logic, the assumption is that every category has at least one member. So that on the square of opposition, A implies I. That is, given all, you extract some, at least one - it is all at least existentially qualified. Kornelius, however, informs us that these days existential qualification means at least one, whereas universal qualification does not mean at least one. It means all without affirming that there are any. Which is interesting. I take him as correct in what he says.

In sum, it appears your argument has about it a dog-in-the-manger quality. You claim a "law" as your own (in ontology), which is very clearly a closed circle of argument, and at the same time deny it outside that circle. But the grounds for that denial are as confined as the denial itself. And it seems pretty clear that whatever you claim for, is based in what you claim. Tough circle to get out of, not to be escaped by mere assertion.

i can barely handle long posts. If you reply to this, perhaps consider just setting out succinctly your argument against the law of identity being a law of logic. I will grant you have done this in Aristotelian terms - a different argument. But now do it in terms of logic.
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To start, would you not say that an assumption is a species of proposition?

No I don't think that this is the case, because a proposition is a type of statement, and one can hold an assumption without stating it. But I don't think this distinction is relevant anyway.

Or perhaps I'm confused: "which claims something about being." What claim can there be about being that is not actually a claim about something else? That is, being, being the supremum genus, has no species and no accidents. How can you predicate anything of being?

Yes, this is the difficult thing. We do make claims about such general things, universals. What does it mean to be a human being, to be an animal, to be alive, etc.. Notice that I phrased it as "what does it mean", There are many such examples, what does "colour" mean, what does "number" mean. When we make a statement which claims something about these ideas, we are generally trying to clarify the meaning of the term. Do you agree that this type of expression, clarifying the meaning of terms, is distinct from predication? These claims which we have, hold, or make, about the meaning of the terms, are what I call assumptions.

So if someone makes a claim about "being" this is an expression of what that person believes is the meaning of the term. Maybe it could be called defining the term. If you look, you'll notice that such definitions are generally assumptions. For example, let me take something very simple, like numbers, and start with the numeral "one". That word refers to a unity, an individual. Next we have "two". Two refers to one individual together with another, making an artificial unity of "two". Notice that I distinguished the unity which is referred to by "one", from the unity which is referred to by "two", by calling the latter "artificial" (whether or not this term is adequate is not the point). It is necessary to do this because the use of "unity" which refers to one is distinct from the use of "unity" which refers to two. These are two completely distinct types of unity. "Two" implies that the unity referred to as two, is already intrinsically divided into two, whereas "one" implies divisibility (of infinite possibility), with no such division having been made already. So the unity referred to by "two" is a false unity because it is of necessity already divided. In the use of "two", we must recognize a sort of contradiction, a unity, one thing referred to with "two", which already has a defined division into two equal parts, so it is not really a unity. Whereas "one" represents a unity without any such division. Therefore the "unity" of one is distinct from the "unity" of two, three, four, etc.., and we cannot say that "two" refers to a unit in the same way that we say "one" refers to a unit without equivocation. These are some of my "assumptions" concerning numbers.

I take it this your ontological principal. But in what sense is it just an assumption - and not an induction?

This is a good question as well, and I'll tell you what I assume is the answer to it. The problem is that we do not have access to see, touch, or in any way sense the vast majority of things in existence. Therefore we do not have the capacity to make proper inductive conclusions concerning "all things". (Incidentally this is the biggest problem with what I consider the best arguments for God, formulations of the cosmological argument. They start from principles which appear to be inductive principles, but are really not drawn from sound induction, and so are dismissed by atheists as faulty assumptions). This is why ontological principles are better characterized as assumptions rather than inductive conclusions. If we start allowing that these are proper inductive conclusions, it sets a bad example.

Instead, ontological assumptions are drawn from examining all sorts of evidence, and drawing conclusions from who knows what sort of logic, mixed in with different intentions and pragmatic concerns. So it's better to call them assumptions than inductive conclusions.

If you're suggesting - arguing - that predication attributes to a subject, and neither subject nor attribution "touch" the object, then the ultimate predication, being, is also similarly ungrounded. If you deny induction and call it all assumption, then you rule out reason-as-process. For what indeed can you reason about but predication? (The reasoning itself - if you allow for such - being mainly governed by logic.)

I don't agree with this. It may be the case that predication is required for deductive reasoning, but there are other forms of reasoning as well. Induction for example, though it often involves predication, does not require it. But, as mentioned it is difficult to draw a line between good induction and faulty induction. We can apply induction, for example, to different activities, deciding whether certain activities are successful for achieving desired ends. The process of trial and error allows us to focus in on the successful activity, and when it is found that a certain activity consistently produces the desired result, we might produce an inductive conclusion concerning cause and effect. The process of determining the correct activity is not a matter of predication, though it is a matter of reasoning.

Nope. You just ruled this out. More accurately, on your approach, is that we recognize samenesses in the predications. Which is exactly what you say just above. .

You must have misunderstood what I said. The "sameness" recognized through predication is a false sameness. It is the "sameness" which is found within inductive reasoning (which is really similarity), and is not the "sameness" expressed by the law of identity. That's the problem, Kornelius switched the "sameness" of the law of identity (often called numerical identity), for the "sameness" of inductive reasoning (often called qualitative identity, which is really a similarity), so that the formulation of the law of identity expressed by Kornelius was based in an equivocation of the word "same".

I don't see the basis for this claim, I think it's drawn from a misunderstanding of what I said.

Reading the rest of your post, I see we "assume" the subject into real existence, real objective reality,

No, it's the object we assume into existence. The subject has real presence to us, within our minds, but the object is what is assumed. That's why there is such a thing as radical skepticism concerning the sensible world.

We're discussing the law of identity, and this was expressed by Aristotle, and the proper expression of it is maintained as the Aristotelian expression even today. So if we are to understand "the law of identity" we need to understand the Aristotelian principles behind that law. But if your intent is to replace that law with something else, then we ought not call it "the law of identity", because of the risk of creating ambiguity and equivocation.

You above state "that a thing is the same as itself." You call that a law. Is this true of only some things and not others? Or is it instead true of every thing? If it is true of every thing, then it is true for all things. And you can complete this. So how, exactly, do you disqualify your ontological law of identity from being a law of logic?

Let me explain the difference. We can define "thing" as "that which is the same as itself", or we can look at different individual things and make the inductive conclusion that all things are the same as themselves. The latter, as explained above, is a faulty inductive conclusion because it is very likely that the vast majority of individual things are hidden from our senses. So, the law of identity, which defines what a "thing" is, is not supported by inductive logic, it's more of a stipulation. Therefore it is not a logical principle, i.e. it is not a logical conclusion. I will not deny that it is supported by some sort of reasons, and some sort of "necessity", but it is more of a necessity in the sense of "needed for" the purpose of understanding, and not in the sense of a necessary conclusion, which requires some sort of understanding as a prerequisite for logical process.

Now, under Aristotelian logic, the assumption is that every category has at least one member. So that on the square of opposition, A implies I. That is, given all, you extract some, at least one - it is all at least existentially qualified. Kornelius, however, informs us that these days existential qualification means at least one, whereas universal qualification does not mean at least one. It means all without affirming that there are any. Which is interesting. I take him as correct in what he says.

In sum, it appears your argument has about it a dog-in-the-manger quality. You claim a "law" as your own (in ontology), which is very clearly a closed circle of argument, and at the same time deny it outside that circle. But the grounds for that denial are as confined as the denial itself. And it seems pretty clear that whatever you claim for, is based in what you claim. Tough circle to get out of, not to be escaped by mere assertion.

Again, I do not understand the relevance of this.

can barely handle long posts. If you reply to this, perhaps consider just setting out succinctly your argument against the law of identity being a law of logic. I will grant you have done this in Aristotelian terms - a different argument. But now do it in terms of logic.

How would this be possible? To discuss the law of identity in terms of logic would be to reformulate it into logical terms, which would destroy its essence, as Kornelius did.
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