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.
I am Ilya Shambat, and I have always been Ilya Shambat. However I am different now in many respects than I was when I was a toddler, and different also in many respects than I was five years back. A=A in some ways but not in others. A more complete understanding therefore is this:
Something is itself in addition to the changes that it has undergone over time.
Mathematically, this can be seen by taking the A=A equation and replacing it with A1=A0+D. A0 is the initial state; A1 is the later state; and D is the change that has taken place between the initial and the later state.
D - the total change - is a multiple of the time that the change has occurred and the rate of change: D=t*r, where t is time and r is rate of change. The rate of change does not have to be constant, and it does not have to be positive. Change occurs, in all sorts of directions, all the time. And just as, in physics, "work" can be positive and negative, there is also positive change and negative change.
The faster the rate of change, or the greater the time that the change has happened, the greater the change. A big change that takes a short period of time, or small changes accumulated over greater time, both result in a large change.
When there is no time - when t is 0 - then D is also 0, and A is again equal to A. Same is the case when the rate of change is 0. The exception to that rule is if t=0 and r is infinite, or if r=0 and t is infinite; in which case t*r, and thus D, can be anything at all. If either term is infinite and the other term is non-zero, then infinite change is realized.
Change takes time; it also takes speed of change. For any non-infinite time, zero rate of change will produce zero change; and for any non-infinite rate of change, zero time will produce zero change as well.
As Newtonian physics is a subset of larger physics when taken over small speeds and distances, so the law of identity is a subset of change mathematics where either the time or the rate of change is zero and the other term is not infinity. A=A when no change has happened; A1=A0+D when change has.
What A1=A0+D means in reality is that something is itself as it was at the original state, plus or minus the changes that have taken place since that time. This should come as no surprise; but many people do not realize the D factor - the factor of change that takes place in all living beings and in many non-living ones. Based on this miscalcuation people tend to treat others the way they'd known them years previously and not realize the change that they may have undergone during that time. This kind of attitude prevents growth and improvement in people and pigeon-holes them in places that are no longer appropriate. Thus, people may treat contemporary Germans as if they were Nazis, or treat contemporary Jews as if they were Caiaphas, when vast changes have happened in Germans since Second World War and in Jews since 1st century AD. A man may treat his wife based on how she may have been 20 years prior and not realize that she no longer has the same attitudes as she did back then. A person may treat an ex-classmate, 30 years down the road, as though he were still what he was when he was 7. And further on down the line.
The rational response to this misuse of the law of identity is: Living things change. Over time, and with any rate of change, A<>A. A1=A0+D.
The failure to compute change results in all sorts of destructive outcomes. Things are treated as if they were what they'd been in the past without realizing that a lot in them has altered. The attitude of failing to acknowledge change prevents positive change in people from occurring. But it also keeps people from being able to exercise creative intelligence and implement positive changes or keep up with the changes that take place in the world.
The other part of this equation is that A is still A for as long as A exists as itself. Whatever changes I undergo as a person, I am still identifiable as myself. When I am no longer identifiable as myself, I cease to exist. In this case, A1=0; D=(-A0).
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.
With the law of identity remaining in place, it is possible to look at another, less obvious, feature: And that is as follows. As change becomes embedded into the fabric of things, so is the time that the change has taken to transpire. Time, through this mechanism, becomes part of things as they are and is encoded in the reality of things. One obvious example is the year rings that we see in the trees; but the same dynamic can be found in all sorts of less obvious situations. An 80-year-old person carries the mark of time, which a toddler does not.
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.