The law of identity is one of the most basic laws in mathematics. The law of identity states that a thing is itself: A=A. While this is true absolutely of things that don't change, the living things (and many non-living things) are constantly changing; and, as impacting on the living things - as well as many non-living things - that change, there needs to be a supplement to this law. — Ilya B Shambat
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change ...
... a system in which a function describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a lake.
Change takes non-zero time, and a non-zero rate of change, for all non-infinite situations. The faster the change, and the greater the time over which it takes place, the greater the change that transpires; the greater the difference between the object at the initial state and the object at a later state. — Ilya B Shambat
because despite the fact that the thing is changing it still remains the thing that it is, i.e.the same as itself.
I believe you are importing metaphysical claims into the law of identity. The law itself is completely neutral with respect to whether or not an object is the same (or different) after undergoing certain change. — Kornelius
We could take a radical metaphysical position and insist that objects can only be self-identical for any given time slice t tt. But here too, the law of identity would apply at any given time slice. The law is completely neutral here. — Kornelius
If perhaps you are pointing out that this is somewhat of a bogus or artificial abstraction, I quite agree. After all nobody knows whether time itself is accurately modeled by the standard model of the mathematical real numbers. That's a philosophical assumption made by science. It bumps into quantum theory. There are good reasons to doubt the mental model of static states as a function of time, and the standard real numbers as the official model of time. That viewpoint has been pragmatically successful for a few hundred years, but as to its ultimate truth, that's unknown. — fishfry
I think the law of identity is itself a metaphysical claim. So it's not a matter of me importing metaphysical claims into the law of identity, it already is a metaphysical claim. — Metaphysician Undercover
The law of identity is a law of logic; it is not an ontological principle. Perhaps you mean Leibniz's law of indiscernibles? — Kornelius
The law of identity is most certainly a principle of logic, not of metaphysics. — Kornelius
Yep. Or basically what we talk is about a bijection. Or set theory.The law of identity, by the way, is not a law of mathematics. It's more primitive than that, it's a law of logic. Mathematics inherits the law of identity from logic; math doesn't posit or explicitly assume it.
The law of identity operates at a much "lower level" than that of modeling changing systems like weather or biology. — fishfry
No.The law of identity is therefore a subset of reality — Ilya B Shambat
No, I mean Aristotle's law of identity, which is an ontological principle. It states that a thing is the same as itself. It is ontological because it assumes the existence of the thing. Without the existence of the thing the principle makes no sense. So if any logicians make use of this principle, they are making use of an ontological principle. — Metaphysician Undercover
It may be the case that logicians make use of the principle, but to classify the principle itself, we need to see what validates it, and that is an ontological assumption about the existence of a thing. — Metaphysician Undercover
But we know now, because of mathematical advances in logic, that this principle does not assume the existence of anything. The statement (∀x)(x=x) (∀x)(x=x)(\forall x) (x=x) is made true by any model that assumes no objects: it would be vacuously satisfied, and therefore true. — Kornelius
It is simply incorrect to say that the statement that every object is identical with itself implies (or presumes) that an object exists. It does not. — Kornelius
I am sorry to be blunt, but this is simply incorrect. As I said: every model validates it, no matter whether no objects, some objects or infinitely many (countable or uncountable) objects exist. — Kornelius
The law of identity is therefore a subset of reality; something that happens when either the time or the speed of change is zero, and the other term is not infinite. In a larger picture, things both change and remain the same. This is something of course that many people understand intuitively; but it takes reasoning and mathematics to understand it at a rational level.
The law of identity operates at a much "lower level" than that of modeling changing systems like weather or biology.
— fishfry
Yep. Or basically what we talk is about a bijection. Or set theory. — ssu
One can imagine measuring the time it takes a kettle to boil by the heartbeat of the person watching it, — fdrake
the clock measuring both factors out. In that regard time's an instrumental variable for any bijective continuously differentiable function of it. — fdrake
Ah but no. The continuity of the real numbers are the mathematical model of time. But we don't know for sure if time itself is continuous. That was my point. I don't necessarily take differential equations for reality. It's the map/territory thing. — fishfry
I am Ilya Shambat, and I have always been Ilya Shambat. — Ilya B Shambat
So, since it's arbitrary for the math, you can think of time relationally; as the pairing of systems creating an index; rather than as the index by which systems evolve. — fdrake
Edit: or if you want it put (overstated) metaphysically, instead of conceiving as becoming as being changing over time, you can consider time as being's rates of becoming. — fdrake
Kornelius os correct as far as logic based on 'set theory' irrespective of whether an 'object' or ' member of a set' can be said to 'exist in the world'. Indeed 'existence' is a whole other ball game transcendent of the one we usually call 'formal logic' — fresco
I can't see that the law of identity makes any ontological claim at all other than that 'objects' might have static fixed identity rather than dynamic continued functionality. But that is the essence of the OP and the basis of the pseudo-problem of the Ship of Theseus. If that is what you are driving at then I agree. — fresco
I'm not familiar with your use of symbols, but there is an object assumed, or else there is nothing identified. The object need not be a physical object, are you familiar with mathematical objects? If your statement identifies a mathematical object, then this is an ontological statement, it gives reality to that mathematical object, as an identified object. Perhaps your symbol is the object itself, I don't know what your symbol symbolizes. And a model with no objects makes no sense to me, because the model is itself an object. — Metaphysician Undercover
You cannot claim that a specified object is identical to itself, and also say that there is no such object, without launching yourself into nonsense. — Metaphysician Undercover
You can say that, but your claim is wrong. Try to demonstrate it, why don't you? Show me a model with no objects which validates the law of identity. — Metaphysician Undercover
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