I’m familiar with the limit explanation. There are still plenty of cases (esp. in physics) where something like a dx is treated as exactly equal to zero in one place, and also cancels itself out elsewhere in the same equation, without causing any inaccuracy. You say that zero is just the limit, but it’s at the limit that Calculus provides a mathematically precise description of, for instance, the area under a curve. Of course this doesn’t mean that Calculus is not rigorous; only that in its typical symbolic expression, though you can usually manipulate things algebraically, when dealing with the magnitudes of the dx-type terms you have to pay attention to what you’re actually saying. Algebra was invented so that you wouldn’t have to pay attention to what you’re saying, only follow the rules. But in actual fact you still have to look at the meaning of the equation from time to time. You can’t be totally indifferent to the content of “x.”
In other words, the limit rule by which you reconcile the Calculus may be internal to the Calculus, but it is external to the Algebra by which the Calculus is usually expressed.
The case is the same with the logic paradoxes, such as the Russell one; it’s only by looking at what you’re really saying that you can dismiss a logical problem as missing the point.
@Roko,
I’m familiar with the limit explanation. There are still plenty of cases (esp. in physics) where something like a dx is treated as exactly equal to zero in one place, and also cancels itself out elsewhere in the same equation, without causing any inaccuracy. You say that zero is just the limit, but it’s at the limit that Calculus provides a mathematically precise description of, for instance, the area under a curve. Of course this doesn’t mean that Calculus is not rigorous; only that in its typical symbolic expression, though you can usually manipulate things algebraically, when dealing with the magnitudes of the dx-type terms you have to pay attention to what you’re actually saying. Algebra was invented so that you wouldn’t have to pay attention to what you’re saying, only follow the rules. But in actual fact you still have to look at the meaning of the equation from time to time. You can’t be totally indifferent to the content of “x.”
In other words, the limit rule by which you reconcile the Calculus may be internal to the Calculus, but it is external to the Algebra by which the Calculus is usually expressed.
The case is the same with the logic paradoxes, such as the Russell one; it’s only by looking at what you’re really saying that you can dismiss a logical problem as missing the point.