Newton publishes his work on calculus immediately

Most accounts concede that Newton started his work several years before Liebniz (Newton no later than 1666; Liebniz no earlier than 1673), and had already developed the theory of tangents by the time Liebniz got going. Suppose, then, Newton published almost immediately--say, 1670? Does this accelerate the development of mathematics, physics, and engineering incrementally or by an order of magnitude or more?
 
I don't know if it accelerates the development of mathematics, physics, etc at all. Newton's formulation was inferior to that of Leibniz, so today we exclusively use Leibniz's version of the calculus (or actually its descendants).

It's possible that if Newton publishes in 1670, Leibniz never publishes (on that topic) and we are stuck with the Newton version, slowing down later developments.
 
I don't know if it accelerates the development of mathematics, physics, etc at all. Newton's formulation was inferior to that of Leibniz, so today we exclusively use Leibniz's version of the calculus (or actually its descendants).
Well...the math is the same either way, what differs is the notation. Additionally, it's not true that we exclusively use Leibniz notation; in physics, Newtonian dot notation is often used to indicate derivatives with respect to time in mechanics (I distinctly remember being shocked when I first saw this in an upper-level mechanics course).

In any case, calculus saw a very considerable amount of further development after either Leibniz or Newton, for example in the development of analysis in the 19th century, and there have been a number of further notations developed that are neither from Leibniz nor Newton. Without the priority controversy, more people might be working on or with the calculus instead of arguing about it, and they might develop new notations that are more convenient than anything Newton or Leibniz dreamed up.
 
Most accounts concede that Newton started his work several years before Liebniz (Newton no later than 1666; Liebniz no earlier than 1673), and had already developed the theory of tangents by the time Liebniz got going. Suppose, then, Newton published almost immediately--say, 1670? Does this accelerate the development of mathematics, physics, and engineering incrementally or by an order of magnitude or more?

Somehow I doubt that this difference of a few years, when the "natural sciences" was still the pursuit of gentlemen of leisure, is going to have much of a difference in the grand scheme of things.

It's possible that if Newton publishes in 1670, Leibniz never publishes (on that topic) and we are stuck with the Newton version, slowing down later developments.

3 years isn't all that long for a document to be transmitted from England to Germany; it seems very reasonable that Leibniz would not have heard of Newton's work by the time he was publishing. Even if he had, the minor differences in their works might constitute enough of a distinction to merit publishing anyway. Honestly, the ability of Leibniz's notation to specify which variable the differential was being taken with respect to was huge (in contrast to Newton's assumption that it was t).

Well...the math is the same either way, what differs is the notation. Additionally, it's not true that we exclusively use Leibniz notation; in physics, Newtonian dot notation is often used to indicate derivatives with respect to time in mechanics (I distinctly remember being shocked when I first saw this in an upper-level mechanics course).

Newton's notation is used by physicists for brevity and ease; serious work is done using Leibniz-inspired notation.

In any case, calculus saw a very considerable amount of further development after either Leibniz or Newton, for example in the development of analysis in the 19th century, and there have been a number of further notations developed that are neither from Leibniz nor Newton. Without the priority controversy, more people might be working on or with the calculus instead of arguing about it, and they might develop new notations that are more convenient than anything Newton or Leibniz dreamed up.

I'm not entirely certain how much time people spent "arguing about notation instead of working on it".

We also, for example, use Lagrange's f'(x) notation pretty often, and Euler's notation is used sometimes in, e.g., differential geometry. Some of the more abstract and abtruse ideas of "differential" also have their own specialized symbols (or, for a not very abstract example, gradients and ∇)
 
Newton's notation is used by physicists for brevity and ease; serious work is done using Leibniz-inspired notation.
So...physics isn't "serious work"?

Speaking for myself, my physics degree was considerably more "serious" in any sense, including the mathematical, than my math degree.

I'm not entirely certain how much time people spent "arguing about notation instead of working on it".
Um, quite a lot? It's only the most famous and vicious priority controversy of all time...
 
So...physics isn't "serious work"?

Speaking for myself, my physics degree was considerably more "serious" in any sense, including the mathematical, than my math degree.

Certainly physics is serious work (though if your physics degree was more serious in the mathematical sense than your math degree, that's a failing on the part of your university's math department).

But every physicist I've ever known has used Leibnitz notation for their serious work. Newton dot notation is used for quickly writing things out; actually working through things is typically done in the notation that allows you to indicate which variable you're differentiation or integrating with respect to; this is particularly critical in physics, because most real physical problems involve 3-4 dimensions. It's all well and good to write out the Heat Equation as \dot{y} = \alpha \nabla^2 y for brevity, but you can't even begin to actually approach it in that form - and no physicist I know would try to.

EDIT: for absolute clarity, I mean that while Newton's notation is used sometimes by physicists even in the modern day, serious work, including by physicists, is done using Leibnitz's notation.

Um, quite a lot? It's only the most famous and vicious priority controversy of all time...

"Of all time". Yes, no other controversy has ever inspired as much viciousness or fame as that of notation between 2 17th century mathematicians.

Anyway, my point wasn't "I don't know that people spent a lot of time bitching about it"; rather, I don't know that they spent a lot of time arguing instead of working. It seems unlikely to me that Bernoulli was spending less time working because he was spending so much time writing pamphlets about why Netwon's notation was stupid.
 
Certainly physics is serious work (though if your physics degree was more serious in the mathematical sense than your math degree, that's a failing on the part of your university's math department).
I think I slipped through the cracks, honestly, but nevertheless that was the case. I only really had one serious math class, a two-semester PDE course; otherwise everything was pretty easy, even the mandatory (one-semester) analysis course. Of course I wasn't doing proofs in the physics classes (that was strictly a graduate thing), but the overall difficulty and complexity of the problems was rather higher.

But every physicist I've ever known has used Leibnitz notation for their serious work. Newton dot notation is used for quickly writing things out; actually working through things is typically done in the notation that allows you to indicate which variable you're differentiation or integrating with respect to; this is particularly critical in physics, because most real physical problems involve 3-4 dimensions. It's all well and good to write out the Heat Equation as \dot{y} = \alpha \nabla^2 y for brevity, but you can't even begin to actually approach it in that form - and no physicist I know would try to.
I don't know, what are you defining as "serious work"? What is "serious work" here?

In any case, the dot, as I have seen it used in physics, is always and invariably used to refer to differentiation with respect to time, which means it's actually pretty clear what variable is being manipulated when it's in use.

"Of all time". Yes, no other controversy has ever inspired as much viciousness or fame as that of notation between 2 17th century mathematicians.
Maybe you should try reading what I wrote? I didn't say it was the most vicious controversy of all time flat out, I said it was the most vicious priority controversy of all time. You know, academic fight. It shouldn't be surprising that the most vicious examples of such arguments date back to the beginning of the scientific endeavor, when the social norms of the field were developing.

Anyway, my point wasn't "I don't know that people spent a lot of time bitching about it"; rather, I don't know that they spent a lot of time arguing instead of working. It seems unlikely to me that Bernoulli was spending less time working because he was spending so much time writing pamphlets about why Netwon's notation was stupid.
Any amount of time he was spending writing pamphlets was time that he could have been using to do research. More seriously, the whole affair greatly damaged communications between the mathematicians of Britain and the rest of Europe for a time, and avoiding it altogether would probably lead to collaborations forming that were aborted IOTL. The overall effect would be, as I said, more people working on the calculus rather than arguing about it.
 
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