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. — Terrapin Station
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. — SophistiCat
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. — SophistiCat
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. — fdrake
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. — fdrake
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. — SophistiCat
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." — Kornelius
Please let me know if there is any step that isn't clear! — Kornelius
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. — Metaphysician Undercover
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. — Metaphysician Undercover
-WikiIn 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.
There's a lot going on in the question. — fdrake
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. — fdrake
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: — Kornelius
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. — Kornelius
the law of identity states that each thing is identical with itself
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. — SophistiCat
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. — SophistiCat
The law of identity states that a thing is the same as itself. — Metaphysician Undercover
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. — Metaphysician Undercover
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. — Metaphysician Undercover
You must apply inductive logic to "each thing is identical to itself, to derive "all things are identical to themselves". — Metaphysician Undercover
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. — Kornelius
Do you not recognize the difference between "a thing", and "all things"? — Metaphysician Undercover
Proposition: (1) is not logically valid, where (1) refers to the proposition (∃x)(x=x) (∃x)(x=x)(\exists x) (x=x) — Kornelius
The Law of identity is held as a law that is logically true. — Kornelius
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. — fdrake
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. — fdrake
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. — fdrake
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. — SophistiCat
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."
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.”
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! — SophistiCat
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. — Metaphysician Undercover
Can you make clear exactly what that last clause means? — tim wood
And to be sure, the law of identity is proved above. That is, it is a valid conclusion in logic. — tim wood
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. — tim wood
So what exactly are you claiming is, and what exactly are you claiming isn't, and what is your argument? — tim wood
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. — Metaphysician Undercover
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. — Metaphysician Undercover
How is this not a case of equivocation on and confusion over the meaning of the word "same"? — tim wood
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? — tim wood
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? — tim wood
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?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. — Metaphysician Undercover
I take it this your ontological principal. But in what sense is it just an assumption - and not an induction?The law states that a thing is the same as itself. — Metaphysician Undercover
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. — Metaphysician Undercover
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. .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. — Metaphysician Undercover
So to say anything about the object itself, is to simply make an assumption about it. — Metaphysician Undercover
Under which and guided by which,in this Aristotelian manner — Metaphysician Undercover
we can proceed toward understanding what this real existence consists of, what validates this assumption. — Metaphysician Undercover
To start, would you not say that an assumption is a species of proposition? — tim wood
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? — tim wood
I take it this your ontological principal. But in what sense is it just an assumption - and not an induction? — tim wood
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.) — tim wood
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. . — tim wood
Even on your approach, no. On your approach, you don't have access to an object, so comments about an assumption about an object is an assumption on and about an assumption. You've left yourself no back door to the object. — tim wood
Reading the rest of your post, I see we "assume" the subject into real existence, real objective reality, — tim wood
Sure, in your Aristotelian sense. — tim wood
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? — tim wood
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. — tim wood
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. — tim wood
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