Necessary assumption - independently.

[Harman]

If an author conclusion is:1) Leibniz and Newton both independently discovered Calculus,
I understand that a necessary assumption would be that some third source did not help either Leibniz or Newton in anyway because that would not be independent.

What if however, the conclusion was that:2) Leibniz and Newton discovered calculus independently from each other? Would the assumption made above be necessary for this conclusion as well? I thought it isn't but am not 100% sure

If a third source helped Leibniz, wouldn't that not change the fact that Leibniz still discovered it independently from Newton?

Thanks.

[Nikki Siclunov]

Hi Harman,

Let's take a look at both conclusions carefully:

Leibniz and Newton both independently discovered Calculus

This sentence is improperly constructed; it should say, "Leibniz and Newton each independently discovered Calculus." You are correct in that this conclusion relies on the assumption that some third party did not help either of them make their discovery: each made the discovery independently of any third source, and also independently of each other. You are also assuming that Newton didn't help Leibniz, and vice versa.

This brings me to your second example:

Leibniz and Newton discovered calculus independently from each other

Essentially, you aren't saying that each of them made the discovery entirely alone, only that they didn't rely on each other's work. The assumption in this case is not the same as in your first example. Even if someone else helped Newton, for instance, the conclusion still stands - as long as that someone else wasn't Leibniz. Here, the assumption is that neither Leibniz nor Newton relied on the other's work in making his discovery.

By the way, your second example also contains an error (it should read, "of each other").

Hope this helps! Let me know.

Thanks!