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Berkeley on the Metaphysics and Logic of Newton’s Calculus in the Principia

Scott Harkema – harkema.2@osu.edu

Lecturer – The Ohio State University

1. The Received View

Many commentators on Berkeley’s critique of the calculus (most notably Douglas Jesseph) hold the following tenets:

Berkeley offers two distinct critiques of Newton’s calculus: the metaphysical and the logical critiques.
Berkeley’s metaphysical critique is weak.
Berkeley’s logical critique is very strong.

Metaphysical	Logical
Focuses on mathematical ontology Problematic entity: infinitesimals Argument: One cannot clearly conceive of infinitesimals. If something can’t be clearly conceived, it can’t be part of a geometrical demonstration. Therefore, infinitesimals cannot be part of a geometrical demonstration.	Focuses on formal consistency Argument: Newton’s method requires treating increments as positive quantities. Newton’s method requires treating increments as zeros. Therefore, Newton’s method contains a contradiction. Therefore, Newton’s method does not meet the standard of a geometrical demonstration.

2. My View

3. Newton’s Calculus in the Principia

Taken as an independent argument, Berkeley’s logical critique fails as a critique of Newton’s method.

Newton and other Newtonians were aware of the problem of formal consistency of Newton’s method.
Newton rationalizes the alleged inconsistency with a theory of limits, called the method of “prime and ultimate ratios” (Principia I.i.1 – I.i.11)
So, Newton himself anticipated the logical critique, and offered a response to it.

Berkeley’s critique engages with Newton’s rationalization of the apparently inconsistent method of the calculus, and his critique of it is irreducibly metaphysical in nature.
So, we should read the logical and metaphysical critiques as complimentary, rather than as independent critiques.

This opens up room for a reevaluation of the strength of Berkeley’s metaphysical arguments against Newton.
This more accurately represents the structure of the text of Berkeley’s Analyst.
This avoids attributing a reading of Newton to Berkeley that misrepresents/misunderstands how Newton thought of the calculus.

Goal of this Paper:

To show that one particular case – Berkeley’s response to Newton’s remarks on the calculus in the Principia– supports the picture sketched above.

½a

½b

Newton’s Proof of the Product Rule:

Avoiding Infinitesimals:

Although Newton freely used infinitesimals in his early mathematical works, in the Principia he claims that “since the hypothesis of indivisibles is problematical and this method is therefore accounted less geometrical, I have preferred to make the proofs of what follows depend on the ultimate sums and ratios of vanishing quantities and the first sums and ratios of nascent quantities.”

Newton clearly thinks his method in the Principia is geometrical, and for this reason banishes infinitesimals
Newton offers a theory of “prime [first] and ultimate ratios” which provides a method for conceiving of instantaneous rates-of-change that does not rely on infinitesimals.

4. Berkeley’s Response

5. Upshots

Standard Method (a la Leibniz)

Take the product p of two changing quantities, A and B, so that p=AB. The derivative of p (dp) can be found as follows:

Introduce infinitesimal increments a and b, so that dp = (A+a)(B+b) – AB
Distribute: dp = (AB + Ab +Ba + ab) – AB
Simplify: dp = Ab + Ba + ab
Disregard ab, since it is the product of two infinitesimals and thus negligible with respect to the rest.
Therefore, dp = Ab + Ba

Newton’s Method (Principia 648)

Take the product p of two changing quantities, A and B, so that p=AB. The “moment” or “fluxion” of p (fp) can be found as follows:

Introduce increments a and b, so that

dp = (A+½a)(B+½b) – (A-½a)(B-½b)

Distribute: dp = (AB + ½Ab + ½Ba + ¼ab) – (AB – ½Ab – ½Ba + ¼ab)
Simplify: fp = Ab + Ba

Notice that Newton’s Method cleverly avoids Step (4) of the standard method – a step that Berkeley typically uses to motivate the Logical Critique.

Berkeley’s remarks on the product rule demonstration show that he clearly recognizes that the standard logical critique does not apply. Newton’s method does not require attributing inconsistent properties to the increments used in the proof.

Berkeley criticizes the product rule demonstration in two ways:

He claims it is “illegitimate and indirect” (not clear that this is necessarily metaphysical)
He claims it relies on a mathematical ontology that is “not easy to conceive” (this is obviously metaphysical)

So, in this particular case, we can not say with tradition that Berkeley’s logical critique is effective and the metaphysical critique is not. To engage with Berkeley’s criticism here, we must engage with his concerns about Newton’s problematic mathematical ontology.

On Infinitesimals:

“The Points or mere Limits of nascent Lines are undoubtedly equal, as having no more Magnitude one than another, a Limit as such being no Quantity. If by a Momentum you mean more than the very initial Limit, it must be either a finite Quantity or an Infinitesimal. But all finite Quantities are expressly excluded from the Notion of a Momentum. Therefore the Momentum must be an Infinitesimal. And indeed, though much Artifice hath been employ'd to escape or avoid the admission of Quantities infinitely small, yet it seems ineffectual. For ought I see, you can admit no Quantity as a Medium between a finite Quantity and nothing, without admitting Infinitesimals. An Increment generated in a finite Particle of Time, is it self a finite Particle; and cannot therefore be a Momentum. You must therefore take an Infinitesimal Part of Time wherein to generate your Momentum.” (The Analyst 11, emphasis mine)

Berkeley argues that Newton’s moments [i.e. derivatives] must be infinitesimals.
Berkeley’s argument shows an awareness of Newton’s attempt to rationalize the demonstrations of theorems of the calculus with his theory of limits – the method of “prime and ultimate ratios.”
The argument is metaphysical in nature, and also links his criticism of the calculus in the Principia to his metaphysical critique of infinitesimals.

On The Proof of the Product Rule:

“But it is plain that the direct and true Method to obtain the Moment or Increment of the Rectangle AB, is to take the Sides as increased by their whole Increments, and so multiply them together, A+a by B+b, the Product whereof AB + aB + bA + ab is the augmented Rectangle; whence if we subduct AB, the remainder aB + bA + ab will be the true Increment of the Rectangle, exceeding that which was obtained by the former illegitimate and indirect Method

by the Quantity ab. And this holds universally be the Quantities a and b what they will, big or little, Finite or Infinitesimal, Increments, Moments, or Velocities. Nor will it avail to say that ab is a Quantity exceedingly small: Since we are told that in rebus mathematicis errores quam minimi non sunt contemnendi [errors, tho’ ever so small, are not to be neglected in Mathematicks].” (The Analyst 9)

“It is said, the Magnitude of Moments is not considered: And yet these same Moments are supposed to be divided into Parts. This is not easy to conceive, no more than it is why we should take Quantities less than A and B in order to obtain the Increment of AB, of which proceeding it must be owned the final Cause or Motive is very obvious; but it is not so obvious or easy to explain a just and legitimate Reason for it, or shew it to be Geometrical.” (The Analyst 11)

Berkeley claims Newton’s method is “illegitimate and indirect”
Berkeley uses the language of conceivability to criticize the division of the increments, linking his criticism to his metaphysical critique.

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