It's as if Johnson Refuted Berkeley by Kicking at an Abstract Idea!

Bob Murphy "refutes" my post noting that the Cosmos writers blundered big-time in saying Newton invented the calculus in the Principia:

"So, I will give Tyson (and his writers) the benefit of the doubt on this one. From further investigations, it seems that Newton used the idea of a limit of shrinking geometric shapes, which one could plausibly say is, or is not, calculus."

I am flabbergasted. First of all, the claim on the table was, again, that Newton had invented the calculus in the Principia. Of course, he had invented it years earlier, and far from inventing it there, he didn't use it there. We can tell because there is no calculus in the Principia: what there are is the geometric ideas ("limit of shrinking geometric shapes") that were used to solve problems of derivation and integration before Newton and Leibniz invented the calculus. Of course, these ideas are the steps that led up to the calculus, and so they are quite "calculus-like." (In fact, in the more general sense of the term "calculus," "a method of calculation," they certainly are a calculus.) But these are not the ideas that Newton and Leibniz are given credit for inventing. And these geometrical ideas Newton used were not invented by him. The very fellow Bob cites to refute me writes, "On the other hand, some propositions of the Principia are framed in a geometric language which appears to be very easily translatable into calculus concepts."

"Translatable": now why would you have to translate the ideas in the Principia into calculus, I wonder? Because they are not expressed in the calculus. Of course, they are expressed in a calculus: but this is quibbling over words, as this was not what Newton invented about which it is said "Newton invented calculus."

It is as though we had the following sequence:

Tyson: Newton wrote the Principia in beautiful English.

Callahan: Oh boy: no, he wrote it in Latin.

Murphy: Ah, ah, let's give Tyson the benefit of the doubt: I found this paper that says Newton's ideas are "easily translatable" into English!

UPDATE: To clarify, Newton used techniques in the Principia, some of which may be called part of calculus, but which were not invented by him. What he (and Leibniz) had invented in the 1660s was not used in the Principia: "Newton (co-)invented calculus in the late 1660s, and he wrote Principia in the late 1680s. It would be natural to expect that Newton used the calculus in Principia. But it seems that he didn’t. Instead, Newton wrote Principia in the style of Euclid’s Elements, that is, using Classical Greek geometry."

Those techniques used are a calculus. So someone might say, "Well, Newton did use calculus in the Principia." The key point here is that he did not use the calculus that he had invented. So for Tyson to say he "invented" calculus in the Principia is absurd: what calculus he had invented was not used in the Principia. What calculus he used in the Principia he had not invented.

UPDATE II: Here is the most convincing case that Newton did not employ his new discoveries from the 1660s in the Principia:

"By the help of the new Analysis [i.e. algebraic calculus] Mr. Newton found out most of the Propositions in his Principia Philosophiae: but because the Ancients for making things certain admitted nothing into Geometry before it was demonstrated synthetically, he demonstrated the Propositions synthetically, that the System of the Heavens might be founded upon good Geometry. And this makes it now difficult for unskilful men to see the Analysis by which those Propositions were found out."

I think this is pretty decisive, because it is Newton himself explaining why he did not employ the new calculus in that work.

The timeline is roughly as follows:

In the 1660s, Newton is moving towards the development of analytical calculus. Newton seems to arrive at the fundamental theorem during this time.

In the 1670s, Leibniz begins working on calculus. He and Newton are aware of each others work at this time.

In 1684, Leibniz publishes his first display of his new techniques.

In 1687, Newton publishes the Principia. As he notes in the quote above, he did not employ his new analytical calculus in writing the book, but instead relied on a geometrical calculus.

In 1694, Newton finally begins publishing his work in analytical calculus.


  1. Gene, you need to calm down.

    The chief argument by which you and the other blogger demonstrated how stupid Tyson (and/or his writers) was, was to say that calculus wasn't even in the Principia.

    The other guy said:

    Tyson informs us in his authoritative manner that Principia also contains Newton’s invention of the calculus. Given the amount of printer’s ink that had been used up in the academic discussion as to why Newton wrote the Principia in Euclidian geometry and not calculus this is an unforgivable gaff. I repeat for those who have not been paying attention there is no calculus in Newton’s Principia.

    Then in your post you wrote:

    For me, the most stunning howler was that Neil deGrasse Tyson says that the Principia was where Newton introduced calculus. Of course, it has no doubt been the topic of many a PhD thesis to ask "Given Newton had already invented calculus, why didn't he use it at all in the Principia?

    So both you and that other blogger were placing the entire weight of your claim to Tyson's idiocy on the (alleged) fact that calculus wasn't in the Principia at all. Not only that this was a fact, but that this was a well-known fact, such that it was the subject of dissertations.

    That's why I said in the comments to your post that this was inconceivable.

    But it turns out that it is not such an open-and-shut case after all.

    Look, it is possible that Newton invented calculus in his private diaries or whatever, and never published it before the Prinicipia. That's what I thought you and the other blogger were claiming.

    For sure, when you and he wanted to show just how stupid Tyson was, you didn't point to something published before the Principia that had as much "pre-calc" reasoning as it did.

    No, you both stressed that it was well-known that there was no calculus in the Principia, and that indeed this was a well-known mystery.

    So, yeah, I think it's relevant that it was NOT such a mystery, since apparently many historians of science would say it's a nuanced issue.

    If you want to say, "OK fine, but still, 0% of the historians of science would say calculus was INTRODUCED TO THE WORLD in Principia, so Tyson is still wrong!!" OK fine. But again, that's not the case you and the other blogger made. You were putting it all on the fact that calculus wasn't even in the Principia, so when I learned that this was not necessarily correct, I objected.

  2. Just one more clarification, please, for neophytes: When you say he co-invented calculus in the late 1660s, was this published somewhere, either by Newton or the other arguable inventor?

    This is the part that is, I now realize, tripping me up. E.g. if Newton and some other guy invented calculus in their private journals etc., but then the first time the world saw something looking like that (but not the fully monty) was in Principia, that would be one thing. I could see why Tyson's claim is still wrong, but it might be an understandable mistake.

    But if Newton and/or the other guy(s) published more "calculus-ish" stuff for the world to read before Newton published his toned-down calculus-ish stuff in Principia, then yes Tyson's statement is inconceivably stupid.

  3. One more thing and I'll drop it: In case you're wondering why I keep persisting on this, it's not out of (mere) stubbornness. The reason is that *I* would have told you, a month ago, that Newton unleashed calculus on the world in Principia.

    Now obviously, I wouldn't have said such a statement even in a personal YouTube video, without doing some research and making sure I knew what the hell I was talking about. So hopefully I would've realized my claim was wrong, or at best, extremely misleading, and I would've caught myself before embarrassing myself.

    But this is why I'm trying to figure out just how big a blunder Tyson's statement was, because this is not some random thing that he and his staff invented, I am pretty sure somebody "taught" me this fact when I was younger. So I'm trying to get to the bottom of just how bad it is.

  4. I googled. Gene is right. Aside from that, if Newton had introduced calculus in the Principia he would have had to explain it. And then there would be no argument at all about whether he even mentions it, would there?
    So Tyson certainly blundered, but so did any of his defenders who didn't do a simple check.